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Feedback in living systems

https://www.youtube.com/watch?v=eHsYuPEYXgE

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WEBVTT Kind: captions Language: en 00:00:00.520 --> 00:00:02.300 - [Instructor] So last weekend, my family and I 00:00:02.300 --> 00:00:04.410 went out hiking in the desert. 00:00:04.410 --> 00:00:06.070 And as you can tell from these pictures 00:00:06.070 --> 00:00:09.410 I snapped along the way, it was a gorgeous hike. 00:00:09.410 --> 00:00:13.340 We made our way to this lake around a small canyon range 00:00:13.340 --> 00:00:15.430 and up and down this mountain trail. 00:00:15.430 --> 00:00:17.460 Now, all of this was really great, 00:00:17.460 --> 00:00:19.360 but there was just one problem. 00:00:19.360 --> 00:00:21.500 It got super hot. 00:00:21.500 --> 00:00:24.340 And because we were exercising out in the hot sun, 00:00:24.340 --> 00:00:26.470 we started sweating buckets. 00:00:26.470 --> 00:00:29.580 And all I wanted to do after a while was to find some water 00:00:29.580 --> 00:00:31.910 and shade as soon as possible. 00:00:31.910 --> 00:00:34.320 So finally, after we sat down to have 00:00:34.320 --> 00:00:35.940 a nice picnic in the shade 00:00:35.940 --> 00:00:39.160 and our sweat provided some evaporative cooling, 00:00:39.160 --> 00:00:42.590 our bodies were able to cool down without overheating. 00:00:42.590 --> 00:00:45.140 So you might be wondering why did our bodies 00:00:45.140 --> 00:00:47.350 in our behavior respond that way? 00:00:47.350 --> 00:00:49.730 Why did we sweat and want to find shade? 00:00:49.730 --> 00:00:53.670 Well, the answer is our bodies were protecting us from harm. 00:00:53.670 --> 00:00:55.810 The human body isn't able to function 00:00:55.810 --> 00:00:57.460 at too high of a temperature. 00:00:57.460 --> 00:01:00.580 So our bodies helped cool us down through a combination 00:01:00.580 --> 00:01:04.220 of physiological and behavioral responses. 00:01:04.220 --> 00:01:07.087 Physiological responses being the internal, chemical, 00:01:07.087 --> 00:01:09.270 and physical changes that our bodies 00:01:09.270 --> 00:01:11.160 carry out unconsciously, 00:01:11.160 --> 00:01:13.530 and behavioral responses being the actions 00:01:13.530 --> 00:01:17.490 we carry out consciously in response to what our body needs. 00:01:17.490 --> 00:01:19.980 So in this case, the physiological response 00:01:19.980 --> 00:01:22.130 would be sweating, which our body does 00:01:22.130 --> 00:01:23.860 in order to cool itself down. 00:01:23.860 --> 00:01:27.570 And in addition to sweating, other physiological responses 00:01:27.570 --> 00:01:31.400 were also happening such as, our blood vessels were dilating 00:01:31.400 --> 00:01:33.500 and we were getting thirsty. 00:01:33.500 --> 00:01:35.940 And the behavioral responses were our attempts 00:01:35.940 --> 00:01:39.823 to find shade and get out of the sun and to drink water. 00:01:40.780 --> 00:01:45.080 This tendency of an organism to maintain internal conditions 00:01:45.080 --> 00:01:48.350 within an acceptable range despite changes 00:01:48.350 --> 00:01:52.840 in its external environment is called homeostasis. 00:01:52.840 --> 00:01:54.700 And I'll write down our definition. 00:01:54.700 --> 00:01:59.700 So it's the tendency to maintain internal conditions 00:02:00.060 --> 00:02:04.320 despite changes in external conditions. 00:02:04.320 --> 00:02:08.360 So homeostasis is incredibly important because without it, 00:02:08.360 --> 00:02:12.170 we could have overheated and been in real danger. 00:02:12.170 --> 00:02:15.300 So in other words, homeostasis is necessary 00:02:15.300 --> 00:02:17.870 in order for organisms to survive. 00:02:17.870 --> 00:02:20.550 Now, you might also be wondering how living things 00:02:20.550 --> 00:02:24.600 generally maintain this homeostatic condition of theirs. 00:02:24.600 --> 00:02:28.500 And this usually involves negative feedback loops. 00:02:28.500 --> 00:02:30.160 So let me draw this diagram for us. 00:02:30.160 --> 00:02:33.070 We have our stimulus, we have a detection 00:02:33.070 --> 00:02:34.953 and then a response. 00:02:37.400 --> 00:02:42.330 So in negative feedback, a stimulus or a detectable change 00:02:42.330 --> 00:02:45.950 in internal conditions triggers the body to carry out 00:02:45.950 --> 00:02:50.190 a response that will counteract or oppose this change. 00:02:50.190 --> 00:02:53.990 So it'll bring conditions back within an ideal range. 00:02:53.990 --> 00:02:56.100 And this is what is represented right here 00:02:56.100 --> 00:02:58.873 by this blocking symbol in the diagram. 00:02:59.890 --> 00:03:01.990 So going back to my family's hiking trip, 00:03:01.990 --> 00:03:04.590 we can say that the stimulus was the increase 00:03:04.590 --> 00:03:06.920 in our body temperatures as a result 00:03:06.920 --> 00:03:09.110 of hiking in the hot desert. 00:03:09.110 --> 00:03:11.980 Our bodies detected that our internal temperature 00:03:11.980 --> 00:03:15.220 was moving outside of the acceptable range, 00:03:15.220 --> 00:03:18.410 which typically falls between 97.7 00:03:18.410 --> 00:03:23.410 to 99.5 degrees Fahrenheit or 36.5 to 37.5 degrees Celsius. 00:03:26.850 --> 00:03:28.130 And the cool thing is that, 00:03:28.130 --> 00:03:30.570 once our bodies detected the stimulus, 00:03:30.570 --> 00:03:33.230 they produced a response to counteract 00:03:33.230 --> 00:03:35.800 this change through negative feedback. 00:03:35.800 --> 00:03:38.970 We were hot, so we wanted to become less hot. 00:03:38.970 --> 00:03:41.120 And in this case, the negative feedback loop 00:03:41.120 --> 00:03:44.310 caused responses like sweating that helped 00:03:44.310 --> 00:03:45.890 cool our body temperatures down 00:03:45.890 --> 00:03:49.580 to the acceptable or the ideal range. 00:03:49.580 --> 00:03:51.280 It's also worth noting that our bodies 00:03:51.280 --> 00:03:54.870 can elicit negative feedback mechanisms in response 00:03:54.870 --> 00:03:57.940 to our body temperature dropping too low. 00:03:57.940 --> 00:04:01.300 So if our body temperature drops below the ideal range 00:04:01.300 --> 00:04:03.800 or our body temperature decreases, 00:04:03.800 --> 00:04:06.590 then the body counteracts this change through responses 00:04:06.590 --> 00:04:09.850 like shivering and blood vessel constriction, 00:04:09.850 --> 00:04:12.520 all with a goal of helping to keep us warm. 00:04:12.520 --> 00:04:15.320 So negative feedback mechanisms help cool us down 00:04:15.320 --> 00:04:19.760 when we get too hot or they warm us up when we get too cold. 00:04:19.760 --> 00:04:23.330 So they help to keep our body temperatures just right. 00:04:23.330 --> 00:04:25.890 And this process of maintaining body temperature, 00:04:25.890 --> 00:04:28.660 otherwise known as thermoregulation, 00:04:28.660 --> 00:04:30.230 let me write that out for us, 00:04:30.230 --> 00:04:33.340 it can be seen in all different kinds of organisms. 00:04:33.340 --> 00:04:36.120 You might've seen dogs pant when they're hot 00:04:36.120 --> 00:04:39.030 or spotted lizards sunbathing to stay warm. 00:04:39.030 --> 00:04:41.260 And these are all homeostatic responses 00:04:41.260 --> 00:04:43.960 that help keep the organism's body temperature 00:04:43.960 --> 00:04:47.190 within the acceptable range that we talked about. 00:04:47.190 --> 00:04:50.170 Another really awesome example of a negative feedback loop 00:04:50.170 --> 00:04:53.940 is osmoregulation, specifically in salmon. 00:04:53.940 --> 00:04:55.810 And here's a picture. 00:04:55.810 --> 00:04:59.270 Now, salmon spend part of their lives in freshwater streams 00:04:59.270 --> 00:05:02.540 and the other part of their lives in salt-water oceans. 00:05:02.540 --> 00:05:05.800 So in fresh water, the salt concentration of the water 00:05:05.800 --> 00:05:08.630 is lower than the salt concentration you would find 00:05:08.630 --> 00:05:11.500 in the fish's internal body fluid. 00:05:11.500 --> 00:05:14.760 While in saltwater, the salt concentration of the water 00:05:14.760 --> 00:05:18.930 is higher than this fish's internal salt concentration. 00:05:18.930 --> 00:05:20.830 So this means that in fresh water, 00:05:20.830 --> 00:05:23.140 the fish will tend to absorb water 00:05:23.140 --> 00:05:25.290 and lose salts through their skin. 00:05:25.290 --> 00:05:28.340 Well, the opposite is true in saltwater. 00:05:28.340 --> 00:05:31.110 Any large change in a fish's internal salt 00:05:31.110 --> 00:05:33.440 or water levels could be fatal. 00:05:33.440 --> 00:05:35.810 So how exactly can salmon tolerate 00:05:35.810 --> 00:05:39.280 these extremely different environmental conditions? 00:05:39.280 --> 00:05:42.960 Well, they also use negative feedback mechanisms. 00:05:42.960 --> 00:05:45.560 So salmon have a negative feedback system, 00:05:45.560 --> 00:05:49.100 which detects changes in internal salt concentrations 00:05:49.100 --> 00:05:52.380 and causes a response that involves either taking up 00:05:52.380 --> 00:05:54.940 or excreting salt through the gills 00:05:54.940 --> 00:05:57.650 or having more or less dilute urine 00:05:57.650 --> 00:06:01.600 in order to reestablish ideal internal salt concentrations. 00:06:01.600 --> 00:06:05.840 And this is otherwise known as osmoregulation. 00:06:05.840 --> 00:06:08.320 So again, we have a feedback loop that acts 00:06:08.320 --> 00:06:11.110 to oppose a stimulus, which in this case 00:06:11.110 --> 00:06:14.750 is the change in internal salt concentrations. 00:06:14.750 --> 00:06:17.620 So now we know that homeostatic mechanisms 00:06:17.620 --> 00:06:20.250 usually involve negative feedback loops, 00:06:20.250 --> 00:06:22.720 but what about positive feedback loops? 00:06:22.720 --> 00:06:25.890 Well, many organisms actually use positive feedback loops 00:06:25.890 --> 00:06:28.490 to bring processes to completion. 00:06:28.490 --> 00:06:31.590 So while negative feedback loops dampen stimuli 00:06:31.590 --> 00:06:35.730 or oppose stimuli, positive feedback loops do the opposite. 00:06:35.730 --> 00:06:38.450 They amplify stimuli. 00:06:38.450 --> 00:06:40.700 And as you can tell from this diagram, 00:06:40.700 --> 00:06:43.380 instead of having a blocking symbol here, 00:06:43.380 --> 00:06:45.210 we have an arrow to indicate 00:06:45.210 --> 00:06:48.000 the amplification of the stimulus. 00:06:48.000 --> 00:06:51.090 So in humans for instance, a positive feedback loop 00:06:51.090 --> 00:06:53.060 is used for childbirth. 00:06:53.060 --> 00:06:54.940 So as you can see from this diagram, 00:06:54.940 --> 00:06:57.740 the stimulus in childbirth comes from the baby's head, 00:06:57.740 --> 00:07:00.610 which presses against the cervix here. 00:07:00.610 --> 00:07:03.340 And this stimulates neurons in the cervix, 00:07:03.340 --> 00:07:06.070 which send a signal for the brain to release 00:07:06.070 --> 00:07:09.200 a special kind of hormone called oxytocin. 00:07:09.200 --> 00:07:11.920 Now, oxytocin is responsible for causing 00:07:11.920 --> 00:07:15.151 the uterus to contract, which as you might've guessed 00:07:15.151 --> 00:07:17.670 causes more pressure on the cervix 00:07:17.670 --> 00:07:19.660 which sends more neural signals, 00:07:19.660 --> 00:07:22.050 which releases more oxytocin. 00:07:22.050 --> 00:07:24.750 And this loop continues on and on 00:07:24.750 --> 00:07:28.070 all the way until the baby is born. 00:07:28.070 --> 00:07:31.050 So when the baby is born, because the baby's head 00:07:31.050 --> 00:07:33.400 isn't pressing up against the cervix 00:07:33.400 --> 00:07:35.380 and the pelvic floor anymore, 00:07:35.380 --> 00:07:37.200 the neuron stops sending the signal 00:07:37.200 --> 00:07:38.720 and the brain stops triggering 00:07:38.720 --> 00:07:41.190 the release of so much oxytocin. 00:07:41.190 --> 00:07:44.593 So that's how the loop will eventually come to an end. 00:07:45.950 --> 00:07:48.150 So to recap on what we've talked about. 00:07:48.150 --> 00:07:50.320 Today, we learned that organisms maintain 00:07:50.320 --> 00:07:53.660 their internal conditions through homeostasis. 00:07:53.660 --> 00:07:55.060 And this is usually accomplished 00:07:55.060 --> 00:07:56.820 through negative feedback loops, 00:07:56.820 --> 00:08:00.420 which dampen or oppose stimuli as we talked about 00:08:00.420 --> 00:08:04.000 with thermoregulation and osmoregulation. 00:08:04.000 --> 00:08:06.730 But on the other hand as we saw in childbirth, 00:08:06.730 --> 00:08:10.200 positive feedback loops work to amplify stimuli 00:08:10.200 --> 00:08:13.103 in order to bring processes to completion.

Inertial Mass vs. Gravitational Mass

https://www.youtube.com/watch?v=Ws3yB3QsKY4

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https://www.youtube.com/api/timedtext?v=Ws3yB3QsKY4&ei=3FWUZeP5ArK2vdIP2ua60A0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=D40D3EE7F5E277DE16AEFF253FDDB0FCF9EA5155.138C2A91079077BCD442CF4B5546AABFE687C8A1&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.180 --> 00:00:01.520 - [Instructor] Knowing the mass of an object 00:00:01.520 --> 00:00:05.110 actually tells you two independent things about that object. 00:00:05.110 --> 00:00:08.610 For instance, if you knew that this truck had a large mass 00:00:08.610 --> 00:00:11.000 you'd know that it has a large amount of inertia, 00:00:11.000 --> 00:00:11.890 that is to say, 00:00:11.890 --> 00:00:14.310 it'd be very reluctant to being accelerated. 00:00:14.310 --> 00:00:16.120 It'd be difficult to speed up. 00:00:16.120 --> 00:00:17.570 And once you got it up to speed, 00:00:17.570 --> 00:00:19.160 it'd be very difficult to stop. 00:00:19.160 --> 00:00:20.880 It would take a large amount of force 00:00:20.880 --> 00:00:23.910 that's 'cause it has a large amount of inertial mass. 00:00:23.910 --> 00:00:27.240 And this idea of inertial mass is best exemplified 00:00:27.240 --> 00:00:29.180 with Newton's second law. 00:00:29.180 --> 00:00:34.170 So acceleration equals the net force divided by M. 00:00:34.170 --> 00:00:36.830 This M right here, down here in the denominator, 00:00:36.830 --> 00:00:39.670 this is the inertial mass, 00:00:39.670 --> 00:00:41.920 because it's telling you how reluctant that thing is 00:00:41.920 --> 00:00:43.070 to being accelerated more. 00:00:43.070 --> 00:00:46.340 Inertial mass would give you less acceleration, 00:00:46.340 --> 00:00:48.110 but mass also tells you something else. 00:00:48.110 --> 00:00:51.080 It tells you how much that object is gonna interact 00:00:51.080 --> 00:00:52.020 via gravity. 00:00:52.020 --> 00:00:55.280 So if this truck has a large mass that also tells you 00:00:55.280 --> 00:00:58.430 its force of gravity is going to be very large. 00:00:58.430 --> 00:01:02.880 So the force of gravity FG on this truck is equal to MG 00:01:02.880 --> 00:01:06.320 that this M right here is not inertial mass. 00:01:06.320 --> 00:01:09.350 This M here, is telling you how much this truck interacts 00:01:09.350 --> 00:01:11.430 via gravity with other objects. 00:01:11.430 --> 00:01:14.120 And that means this is the gravitational mass. 00:01:14.120 --> 00:01:16.750 Now in our universe for a given object, 00:01:16.750 --> 00:01:19.860 these two values inertial mass and gravitational mass 00:01:19.860 --> 00:01:20.810 are gonna be the same. 00:01:20.810 --> 00:01:24.410 So this trucks, inertial mass measured in kilograms 00:01:24.410 --> 00:01:27.930 is gonna be the exact same value as this trucks, 00:01:27.930 --> 00:01:30.920 gravitational mass measured in kilograms, 00:01:30.920 --> 00:01:32.300 but it didn't have to be that way. 00:01:32.300 --> 00:01:35.100 I mean, these two ideas are conceptually different. 00:01:35.100 --> 00:01:37.530 One, the inertial mass tells you how much inertia 00:01:37.530 --> 00:01:39.850 or reluctance to acceleration something has, 00:01:39.850 --> 00:01:42.140 but the gravitational mass tells you how much that object 00:01:42.140 --> 00:01:43.580 interacts via gravity. 00:01:43.580 --> 00:01:45.410 So you could imagine a universe 00:01:45.410 --> 00:01:47.900 or maybe there's no force gravity, 00:01:47.900 --> 00:01:49.620 but objects still have a reluctance 00:01:49.620 --> 00:01:52.240 to being accelerated by other forces. 00:01:52.240 --> 00:01:54.280 Or maybe you can imagine a universe 00:01:54.280 --> 00:01:56.200 where there is a force of gravity, 00:01:56.200 --> 00:01:57.510 but the number that tells you 00:01:57.510 --> 00:01:59.970 how much something interacts via gravity, 00:01:59.970 --> 00:02:02.150 could have been different from the number that tells you 00:02:02.150 --> 00:02:04.890 how reluctant that object is to being accelerated. 00:02:04.890 --> 00:02:07.920 But for our universe, these two numbers are the same. 00:02:07.920 --> 00:02:09.420 I mean, scientists to this day 00:02:09.420 --> 00:02:11.750 are still doing very delicate experiments 00:02:11.750 --> 00:02:14.400 to try to decern any small differences 00:02:14.400 --> 00:02:15.770 between these two. 00:02:15.770 --> 00:02:19.110 But as far I can tell, to the best experiments up to date, 00:02:19.110 --> 00:02:21.020 these two numbers are exactly the same 00:02:21.020 --> 00:02:23.380 even though they're conceptually different. 00:02:23.380 --> 00:02:24.900 So this is good to keep in mind. 00:02:24.900 --> 00:02:26.290 If you're gonna do an experiment, 00:02:26.290 --> 00:02:28.390 you're gonna be measuring either inertial mass, 00:02:28.390 --> 00:02:31.390 or gravitational mass typically, how would you know 00:02:31.390 --> 00:02:34.010 in a given experiment if you measured one or the other? 00:02:34.010 --> 00:02:35.920 Well, I mean, if you just use a simple experiment, 00:02:35.920 --> 00:02:39.700 like take a spring scale, measure the force you're exerting 00:02:39.700 --> 00:02:42.600 on a cart and then measure the acceleration of that cart 00:02:42.600 --> 00:02:46.310 using meter sticks and stopwatches or a motion sensor. 00:02:46.310 --> 00:02:48.810 And if you just plug this into Newton's second law, 00:02:48.810 --> 00:02:52.120 so if you know that acceleration from a motion detector, 00:02:52.120 --> 00:02:53.640 stopwatches and rulers, 00:02:53.640 --> 00:02:55.870 and you measure the force with the spring scale 00:02:55.870 --> 00:02:58.760 and you solve for this M well, 00:02:58.760 --> 00:03:01.650 this is the denominator of Newton's second law. 00:03:01.650 --> 00:03:04.540 That means you just solve for inertial mass, 00:03:04.540 --> 00:03:08.030 'cause you solved in a formula that contained inertial mass. 00:03:08.030 --> 00:03:09.940 How would you experimentally determine 00:03:09.940 --> 00:03:12.020 the gravitational mass of this cart? 00:03:12.020 --> 00:03:13.300 Well, it's even easier. 00:03:13.300 --> 00:03:15.320 All you have to do take a scale, you know, 00:03:15.320 --> 00:03:17.930 just a digital scale, take your cart, 00:03:17.930 --> 00:03:19.900 put your cart on the digital scale 00:03:19.900 --> 00:03:21.820 and just measure how much the scale reads 00:03:21.820 --> 00:03:24.020 because you know that the force of gravity 00:03:24.020 --> 00:03:25.870 is gonna be measured by the scale. 00:03:25.870 --> 00:03:28.030 That's the number you get out of the scales 00:03:28.030 --> 00:03:30.500 telling you how much weight this object has. 00:03:30.500 --> 00:03:32.350 So the scale would just read this 00:03:32.350 --> 00:03:33.560 and if you know what planet you're on, 00:03:33.560 --> 00:03:34.550 you know what G you've got. 00:03:34.550 --> 00:03:37.960 So if you know, G is 9.8 and you solve for this M, 00:03:37.960 --> 00:03:40.300 well, look at you solve for the gravitational mass, 00:03:40.300 --> 00:03:42.690 how much this thing interacts via gravity. 00:03:42.690 --> 00:03:44.230 So whenever you put something on a scale, 00:03:44.230 --> 00:03:46.370 weigh it like that and get M, 00:03:46.370 --> 00:03:47.810 you're getting gravitational mass. 00:03:47.810 --> 00:03:49.690 If you do the other way with Newton's second law, 00:03:49.690 --> 00:03:51.550 you're getting inertial mass. 00:03:51.550 --> 00:03:53.710 People get this mixed up, but it's pretty easy. 00:03:53.710 --> 00:03:56.510 If you ever use a formula that involves little G 00:03:56.510 --> 00:03:59.620 or like big G, gravitational constant big G, 00:03:59.620 --> 00:04:02.440 that means you've solved M in that formula 00:04:02.440 --> 00:04:03.920 for gravitational mass. 00:04:03.920 --> 00:04:06.960 If there isn't a G, then you're solving for inertial mass. 00:04:06.960 --> 00:04:08.490 So for instance, (mumbles) 00:04:08.490 --> 00:04:10.930 you do some experiment where you try to very delicately 00:04:10.930 --> 00:04:13.940 measure the force of gravity between two spheres. 00:04:13.940 --> 00:04:14.773 This would be hard. 00:04:14.773 --> 00:04:15.840 You probably wouldn't set it up like this. 00:04:15.840 --> 00:04:17.200 You'd have to be more sophisticated, 00:04:17.200 --> 00:04:19.890 but let's say you could just measure the force of gravity. 00:04:19.890 --> 00:04:22.220 These two spheres exert on each other. 00:04:22.220 --> 00:04:24.850 The formula for that would be big G, M one 00:04:24.850 --> 00:04:27.330 times M of the other divided by the distance 00:04:27.330 --> 00:04:28.720 between them squared. 00:04:28.720 --> 00:04:30.280 You'd have to know one of the masses, 00:04:30.280 --> 00:04:32.370 but the spring scale could give you the force. 00:04:32.370 --> 00:04:35.090 You can measure the distance between them with a ruler 00:04:35.090 --> 00:04:37.720 big G you know, it's a constant of the universe. 00:04:37.720 --> 00:04:41.490 If you knew one of the other masses and solved for this one, 00:04:41.490 --> 00:04:43.297 you'd be getting the gravitational mass 00:04:43.297 --> 00:04:46.120 you could use the formula that's got big G. 00:04:46.120 --> 00:04:49.140 Any formula with big G or with little G 00:04:49.140 --> 00:04:52.060 like force of gravity is MG. 00:04:52.060 --> 00:04:55.010 These are all formulas that tell you how much the object M 00:04:55.010 --> 00:04:57.380 is gonna interact via gravity. 00:04:57.380 --> 00:05:01.300 Or you could even imagine gravitational field is big GM 00:05:02.630 --> 00:05:03.910 over R squared. 00:05:03.910 --> 00:05:05.440 All of these M's here, 00:05:05.440 --> 00:05:08.560 this M here, that M there, that M there, 00:05:08.560 --> 00:05:11.170 and this M here all gravitational mass, 00:05:11.170 --> 00:05:13.370 'cause there's either big G or little G 00:05:13.370 --> 00:05:16.240 involved in that fundamental equation. 00:05:16.240 --> 00:05:18.340 If there's a fundamental equation that doesn't have big G 00:05:18.340 --> 00:05:19.270 or little G, 00:05:19.270 --> 00:05:21.530 you're not talking about how something interacts by gravity, 00:05:21.530 --> 00:05:23.090 you're talking about it's inertia, 00:05:23.090 --> 00:05:24.310 and that would be a natural mass. 00:05:24.310 --> 00:05:26.350 So for instance, if you did some other experiment 00:05:26.350 --> 00:05:28.190 maybe you slam two carts together 00:05:28.190 --> 00:05:31.240 and use conservation of momentum to solve for M 00:05:31.240 --> 00:05:33.540 well, momentum is MV. 00:05:33.540 --> 00:05:36.250 This formula has nothing to do with little G or big G, 00:05:36.250 --> 00:05:38.060 no gravitational constants here. 00:05:38.060 --> 00:05:40.340 So if you use this collision experiment 00:05:40.340 --> 00:05:42.150 and solve for the mass of one of the carts, 00:05:42.150 --> 00:05:45.190 you've solved for the inertial mass of the cart. 00:05:45.190 --> 00:05:47.500 Similarly, if you use kinetic energy, 00:05:47.500 --> 00:05:50.630 this formula has nothing fundamentally to do with gravity. 00:05:50.630 --> 00:05:53.150 One half MV squared, there's no big G or little G 00:05:53.150 --> 00:05:55.700 this M here would be inertial mass. 00:05:55.700 --> 00:05:58.190 If you did the period of a mass on a spring, 00:05:58.190 --> 00:06:01.400 is two PI root M over K. 00:06:01.400 --> 00:06:03.810 There's no little G or big G to be found in here. 00:06:03.810 --> 00:06:06.330 That means this is also inertial mass. 00:06:06.330 --> 00:06:07.980 So unless there's a little G or big G 00:06:07.980 --> 00:06:10.760 in your fundamental equation here, your basic equation, 00:06:10.760 --> 00:06:12.540 that mass is gonna be inertial mass 00:06:12.540 --> 00:06:13.960 if there is a little G or big G 00:06:13.960 --> 00:06:15.920 you're talking about gravitational mass. 00:06:15.920 --> 00:06:16.900 Now, if you're clever, 00:06:16.900 --> 00:06:19.700 you could do a single experiment with two phases 00:06:19.700 --> 00:06:22.820 and get both masses at once for instance, 00:06:22.820 --> 00:06:25.130 let's say you got a spring of known spring constant 00:06:25.130 --> 00:06:26.470 and you hung a block on it 00:06:26.470 --> 00:06:28.930 and you lowered it gently until it hangs 00:06:28.930 --> 00:06:31.140 at a certain distance, unless you measured. 00:06:31.140 --> 00:06:33.100 How much did this thing stretch? 00:06:33.100 --> 00:06:34.600 Well, if you measure that with a ruler, 00:06:34.600 --> 00:06:38.740 then you know it this position, the spring force KX 00:06:38.740 --> 00:06:42.600 had better be equal to the gravitational force, MG. 00:06:42.600 --> 00:06:45.030 And so KX would just equal MG 00:06:45.030 --> 00:06:46.730 if the spring constance known 00:06:46.730 --> 00:06:48.520 and you measured X with a ruler, 00:06:48.520 --> 00:06:49.590 and you know what planning you're on 00:06:49.590 --> 00:06:51.090 'cause G is 9.8 on earth. 00:06:51.090 --> 00:06:53.470 If you solve for this and look at G is right here, 00:06:53.470 --> 00:06:55.530 you multiplied by the G and this formulate 00:06:55.530 --> 00:06:57.300 came from a gravitational formula, 00:06:57.300 --> 00:07:00.130 you would have sold for gravitational mass. 00:07:00.130 --> 00:07:02.260 And now you know the gravitational mass of the object, 00:07:02.260 --> 00:07:04.080 how could you get the inertial mass? 00:07:04.080 --> 00:07:06.380 Well, let's say you just pull down a little extra. 00:07:06.380 --> 00:07:08.390 You pull this down a little extra, you let go. 00:07:08.390 --> 00:07:10.830 And then it's gonna oscillate at a certain period. 00:07:10.830 --> 00:07:12.990 Let's say you measure that period with a stopwatch. 00:07:12.990 --> 00:07:16.090 You measure how long it takes to go through one full cycle. 00:07:16.090 --> 00:07:19.690 That's got to equal two PI, root M over K. 00:07:19.690 --> 00:07:21.670 Now, there's no little G or big G here. 00:07:21.670 --> 00:07:23.000 This has nothing to do with gravity. 00:07:23.000 --> 00:07:24.890 So if you measure this period with a stopwatch 00:07:24.890 --> 00:07:26.320 and you know the spring constant 00:07:26.320 --> 00:07:28.160 and you solve for this M, 00:07:28.160 --> 00:07:32.030 well, now you've solved for the inertial mass of that block. 00:07:32.030 --> 00:07:33.010 And now you know both. 00:07:33.010 --> 00:07:35.300 One stage got us the gravitational mass, 00:07:35.300 --> 00:07:37.780 'cause it came from MG the M did. 00:07:37.780 --> 00:07:39.480 The second stage, got us the inertial mass 00:07:39.480 --> 00:07:42.020 'cause it comes from two PI, root M over K, 00:07:42.020 --> 00:07:43.970 and this formula has nothing to do with gravity. 00:07:43.970 --> 00:07:46.720 So this would be a way you could find both masses at once. 00:07:46.720 --> 00:07:49.670 So recapping, inertial mass and gravitational mass 00:07:49.670 --> 00:07:53.190 are identical numbers, but different conceptually. 00:07:53.190 --> 00:07:55.870 One, tells you how reluctant an object is 00:07:55.870 --> 00:07:57.200 to being accelerated. 00:07:57.200 --> 00:07:59.630 And the other tells you how much the object will interact 00:07:59.630 --> 00:08:00.700 via gravity. 00:08:00.700 --> 00:08:03.160 And if the mass shows up in a basic formula 00:08:03.160 --> 00:08:05.540 that involves little G or big G 00:08:05.540 --> 00:08:07.890 that's gonna be the gravitational mass. 00:08:07.890 --> 00:08:10.773 Otherwise it's gonna be the inertial mass.

Work-Energy Principle Example

https://www.youtube.com/watch?v=BYkxKrbETb4

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en

WEBVTT Kind: captions Language: en 00:00:00.180 --> 00:00:01.310 - [Instructor] So, the work-energy principle 00:00:01.310 --> 00:00:03.700 states that the net work done on an object 00:00:03.700 --> 00:00:06.800 is gonna equal the changing kinetic energy of that object. 00:00:06.800 --> 00:00:08.680 And this works for systems as well. 00:00:08.680 --> 00:00:10.949 So, the net work done on a system of objects 00:00:10.949 --> 00:00:14.250 is gonna equal the change in the total kinetic energy 00:00:14.250 --> 00:00:16.030 of the objects in that system. 00:00:16.030 --> 00:00:18.480 Now, that sounds really complicated and technical, 00:00:18.480 --> 00:00:19.313 but I like to think 00:00:19.313 --> 00:00:21.420 about the work-energy principle's a shortcut. 00:00:21.420 --> 00:00:23.500 This is a really nice shortcut that lets me 00:00:23.500 --> 00:00:26.500 determine the change in kinetic energy without having to do 00:00:26.500 --> 00:00:29.720 a bunch of complicated conservation of energy equations 00:00:29.720 --> 00:00:31.530 or kinematic formulas. 00:00:31.530 --> 00:00:33.040 The catch is that I need to know 00:00:33.040 --> 00:00:35.410 how to figure out what the net work is. 00:00:35.410 --> 00:00:36.960 So, how do you figure out net work? 00:00:36.960 --> 00:00:40.010 Well, the formula for work done is F d cosine theta. 00:00:40.010 --> 00:00:42.020 Since this formula, work-energy principle, 00:00:42.020 --> 00:00:43.400 relies on net work, 00:00:43.400 --> 00:00:45.840 this has to be magnitude of the net force 00:00:45.840 --> 00:00:48.000 times magnitude of the distance traveled 00:00:48.000 --> 00:00:49.470 times cosine of theta. 00:00:49.470 --> 00:00:53.500 Remember this theta has to be angle between, not any angle, 00:00:53.500 --> 00:00:55.810 angle between the net force direction 00:00:55.810 --> 00:00:57.286 and the direction of motion. 00:00:57.286 --> 00:00:59.190 And so, let's try this out. 00:00:59.190 --> 00:01:00.330 How do you use this thing? 00:01:00.330 --> 00:01:01.860 Let's kick the tires. 00:01:01.860 --> 00:01:03.050 Let's say there's a satellite. 00:01:03.050 --> 00:01:04.320 It's moving to the right 00:01:04.320 --> 00:01:06.690 and there's a net force on this satellite. 00:01:06.690 --> 00:01:08.940 Now, this net force could go in any direction. 00:01:08.940 --> 00:01:11.909 If the net force has a component in the direction of motion, 00:01:11.909 --> 00:01:14.560 then, the net work is gonna be positive. 00:01:14.560 --> 00:01:17.570 And if so, anything here from like negative 90, 00:01:17.570 --> 00:01:20.120 well, like it's negative 89.9, 00:01:20.120 --> 00:01:21.670 because 90 would be perpendicular for many, 00:01:21.670 --> 00:01:25.640 for like 89.9 negative to positive 89.9, 00:01:25.640 --> 00:01:28.440 you've got a component in the direction of motion. 00:01:28.440 --> 00:01:30.857 That means you're gonna be doing positive net work. 00:01:30.857 --> 00:01:32.660 And that means the change of kinetic energy 00:01:32.660 --> 00:01:35.180 will be positive because it just equals that number. 00:01:35.180 --> 00:01:37.290 That means kinetic energy increases. 00:01:37.290 --> 00:01:38.300 You're gonna be speeding up 00:01:38.300 --> 00:01:40.420 And that kind of, it makes sense intuitively. 00:01:40.420 --> 00:01:42.570 If your force is in the direction of motion, 00:01:42.570 --> 00:01:43.780 you're speeding up. 00:01:43.780 --> 00:01:44.760 What about the other case? 00:01:44.760 --> 00:01:45.900 What if your net force points 00:01:45.900 --> 00:01:47.430 in the opposite direction of motion? 00:01:47.430 --> 00:01:49.540 Well, now, the net work is gonna be negative. 00:01:49.540 --> 00:01:52.490 You'll have a negative change in kinetic energy. 00:01:52.490 --> 00:01:54.500 In other words, you're gonna slow down 00:01:54.500 --> 00:01:56.900 and if the net force points perpendicular, 00:01:56.900 --> 00:01:58.490 well, then, you're not doing any work 00:01:58.490 --> 00:02:01.430 because cosine of 90 is gonna be zero. 00:02:01.430 --> 00:02:03.040 No net work would be done. 00:02:03.040 --> 00:02:04.920 There's gonna be no change in kinetic energy. 00:02:04.920 --> 00:02:05.890 That doesn't mean you stop. 00:02:05.890 --> 00:02:08.860 It just means you're not going to speed up or slow down. 00:02:08.860 --> 00:02:09.693 This does something. 00:02:09.693 --> 00:02:10.650 You might be like, "Don't you do something?" 00:02:10.650 --> 00:02:11.810 Yeah, you're gonna drift upward. 00:02:11.810 --> 00:02:13.740 You're gonna start changing your direction, 00:02:13.740 --> 00:02:16.090 but this is not gonna be doing any work on you 00:02:16.950 --> 00:02:18.890 at that moment. 00:02:18.890 --> 00:02:20.280 And so, just to be clear, I mean, 00:02:20.280 --> 00:02:21.920 let's just try a complicated one here. 00:02:21.920 --> 00:02:24.280 Let's say this force goes in some direction. 00:02:24.280 --> 00:02:25.790 Let's say your velocity even goes down. 00:02:25.790 --> 00:02:28.400 So, maybe your satellite's heading downward 00:02:28.400 --> 00:02:30.810 and your force is gonna go in any direction. 00:02:30.810 --> 00:02:32.580 Well, if it goes this way, 00:02:32.580 --> 00:02:35.370 exactly backwards, it's gonna be 180. 00:02:35.370 --> 00:02:37.040 You're gonna be doing negative work. 00:02:37.040 --> 00:02:39.250 You're gonna be slowing down, decreasing kinetic energy. 00:02:39.250 --> 00:02:41.130 And you're not gonna change direction. 00:02:41.130 --> 00:02:44.120 If you're like this, you have a component opposite. 00:02:44.120 --> 00:02:47.271 So, you're gonna be slowing down and changing direction. 00:02:47.271 --> 00:02:49.360 This will just be changing direction. 00:02:49.360 --> 00:02:51.676 You're not speeding up or slowing down at that moment. 00:02:51.676 --> 00:02:54.890 This will speed you up and change your direction. 00:02:54.890 --> 00:02:57.320 And finally, this will just be speeding you up 00:02:57.320 --> 00:02:59.430 and you will not be changing your direction. 00:02:59.430 --> 00:03:01.670 So, the work-energy principle's convenient 00:03:01.670 --> 00:03:04.120 to just get a conceptual or qualitative idea 00:03:04.120 --> 00:03:05.050 of what's going on. 00:03:05.050 --> 00:03:07.510 And it can obviously also give you an idea 00:03:07.510 --> 00:03:09.380 of how to calculate things. 00:03:09.380 --> 00:03:11.970 So, let's try one where you actually have to get a number. 00:03:11.970 --> 00:03:14.000 So, let's say there's a hot air balloon 00:03:14.000 --> 00:03:16.350 and it's a 300 kilogram hot air balloon. 00:03:16.350 --> 00:03:17.230 Drifting to the left, 00:03:17.230 --> 00:03:19.970 it had an initial speed of seven meters per second, 00:03:19.970 --> 00:03:22.170 and it's traveling a total of 50 meters 00:03:22.170 --> 00:03:24.150 to the left during this journey. 00:03:24.150 --> 00:03:25.930 Now, there's gonna be forces on this hot air balloon. 00:03:25.930 --> 00:03:29.010 Obviously, there's gonna be gravity and some buoyant force, 00:03:29.010 --> 00:03:31.250 but because these are perpendicular 00:03:31.250 --> 00:03:32.510 to the direction of motion, 00:03:32.510 --> 00:03:33.420 they do no work. 00:03:33.420 --> 00:03:35.650 And so, when we're gonna use this work-energy principle, 00:03:35.650 --> 00:03:36.690 they're not even gonna factor in. 00:03:36.690 --> 00:03:38.120 We don't even have to know these 00:03:38.120 --> 00:03:40.160 since they were perpendicular and did no work. 00:03:40.160 --> 00:03:42.727 You only consider the forces in the direction of motion. 00:03:42.727 --> 00:03:44.730 So, let's say there was a wind gust 00:03:44.730 --> 00:03:47.330 helping you to the left here of 200 newtons, 00:03:47.330 --> 00:03:50.150 but this is a big, bulky balloon, not that aerodynamic. 00:03:50.150 --> 00:03:51.890 And so, there was a drag force 00:03:51.890 --> 00:03:55.100 from air resistance of 104 newtons to the right. 00:03:55.100 --> 00:03:57.530 And what we wanna know is we wanna determine 00:03:57.530 --> 00:03:59.150 the final speed of the hot air balloon 00:03:59.150 --> 00:04:02.170 after it travels 50 meters directly to the left 00:04:02.170 --> 00:04:03.560 with the forces shown. 00:04:03.560 --> 00:04:05.490 Now, there's lots of different ways to do this. 00:04:05.490 --> 00:04:06.820 You know, Newton's laws, 00:04:06.820 --> 00:04:10.330 you can do a kinematic formula, there's all kinds of stuff, 00:04:10.330 --> 00:04:12.170 even momentum, technically, impulse, 00:04:12.170 --> 00:04:14.399 but the easiest, I'm pretty sure the easiest way to do this 00:04:14.399 --> 00:04:17.310 is just gonna be the work-energy principle, 00:04:17.310 --> 00:04:19.900 which states that the net work done 00:04:19.900 --> 00:04:22.690 is gonna equal the change in kinetic energy. 00:04:22.690 --> 00:04:23.620 So, let's go ahead and do it. 00:04:23.620 --> 00:04:25.430 So, we know that net work is equal 00:04:25.430 --> 00:04:27.360 to the magnitude of the net force 00:04:27.360 --> 00:04:29.580 times the magnitude of the distance traveled 00:04:29.580 --> 00:04:31.490 times cosine of the angle between them. 00:04:31.490 --> 00:04:33.050 What's change in kinetic energy mean? 00:04:33.050 --> 00:04:35.970 Well, change in anything is final minus initial. 00:04:35.970 --> 00:04:39.530 So, since kinetic energy is 1/2 m v squared, 00:04:39.530 --> 00:04:42.530 this is just gonna be 1/2 m v final squared, 00:04:42.530 --> 00:04:46.710 minus 1/2 m v initial squared. 00:04:46.710 --> 00:04:48.470 So, that's gonna be the change in kinetic energy, 00:04:48.470 --> 00:04:49.760 final minus initial. 00:04:49.760 --> 00:04:50.910 All right, let's plug in numbers here. 00:04:50.910 --> 00:04:52.660 So, net force, how do we get that? 00:04:52.660 --> 00:04:53.580 Vertical pieces don't matter. 00:04:53.580 --> 00:04:54.860 We're just looking horizontally here. 00:04:54.860 --> 00:04:56.340 Those were the only ones that are gonna affect it. 00:04:56.340 --> 00:04:58.040 These vertical ones just cancel. 00:04:58.040 --> 00:05:01.180 So, we have 200 to the left, 104 to the right. 00:05:01.180 --> 00:05:02.320 So, we're gonna have to subtract those 00:05:02.320 --> 00:05:04.270 to get 96 to the left. 00:05:04.270 --> 00:05:05.370 And we just want magnitude. 00:05:05.370 --> 00:05:09.280 So, I'm gonna get 96 newtons to the left, 00:05:09.280 --> 00:05:11.770 not negative or anything, I'm just taking magnitude, 00:05:11.770 --> 00:05:13.020 times the distance traveled. 00:05:13.020 --> 00:05:14.207 We know that's 50. 00:05:14.207 --> 00:05:18.720 So, we get times 50 meters, cosine of the angle. 00:05:18.720 --> 00:05:19.553 Now, this is careful. 00:05:19.553 --> 00:05:20.860 You might be like, "Oh 180 here." 00:05:20.860 --> 00:05:23.330 But no, the net force points to the left. 00:05:23.330 --> 00:05:25.060 This 200 is winning here. 00:05:25.060 --> 00:05:27.570 So, leftward net force and the leftward direction, 00:05:27.570 --> 00:05:29.468 the angle between those two is zero, 00:05:29.468 --> 00:05:31.820 cosine of zero is just one. 00:05:31.820 --> 00:05:33.010 We're maxed out here. 00:05:33.010 --> 00:05:36.200 So, net force points in the direction of motion. 00:05:36.200 --> 00:05:38.797 So, equals, let's plug in the rest of this, 00:05:38.797 --> 00:05:42.410 1/2, the mass is 300 kilograms, 00:05:42.410 --> 00:05:44.040 times v final squared, 00:05:44.040 --> 00:05:45.570 that's what we wanna determine, 00:05:45.570 --> 00:05:49.883 minus 1/2, 300 kilograms, 00:05:51.630 --> 00:05:54.890 times the initial speed is seven meters per second. 00:05:54.890 --> 00:05:58.359 So, we got seven meters per squared. 00:05:58.359 --> 00:06:01.467 Well, 96 times 50 is gonna come out to 4,800. 00:06:02.590 --> 00:06:04.960 That means that's the net work done. 00:06:04.960 --> 00:06:06.370 Notice that's joules. 00:06:06.370 --> 00:06:08.210 That's how much energy we've added. 00:06:08.210 --> 00:06:09.840 That's the change in kinetic energy. 00:06:09.840 --> 00:06:11.790 So, we know the net work is change in kinetic energy. 00:06:11.790 --> 00:06:14.864 We've added 4,800 joules of kinetic energy. 00:06:14.864 --> 00:06:19.864 That's gonna have to equal, half of 300 is 150 kilograms, 00:06:21.160 --> 00:06:23.630 times v final squared, 00:06:23.630 --> 00:06:28.630 minus, if you take a half of 300 times seven squared, 00:06:28.700 --> 00:06:31.710 you're gonna get 7,350. 00:06:31.710 --> 00:06:32.890 So, this is how much energy 00:06:32.890 --> 00:06:35.410 the hot air balloon started with initially. 00:06:35.410 --> 00:06:36.650 So, after we moved that to the left, 00:06:36.650 --> 00:06:38.070 we add those together. 00:06:38.070 --> 00:06:41.760 I'm gonna get 12,150 joules 00:06:41.760 --> 00:06:45.050 is how much kinetic energy the balloon ends with. 00:06:45.050 --> 00:06:48.740 And that's gotta equal 150 kilograms 00:06:48.740 --> 00:06:50.800 times v final squared. 00:06:50.800 --> 00:06:55.800 I could divide 12,150 by 150 and you get exactly 81 00:06:56.743 --> 00:06:59.530 and that's gonna equal v final squared. 00:06:59.530 --> 00:07:01.050 And if you take a square root of that, 00:07:01.050 --> 00:07:02.440 you get exactly nine. 00:07:02.440 --> 00:07:04.664 So, the final velocity of this hot air balloon 00:07:04.664 --> 00:07:06.970 is nine meters per second. 00:07:06.970 --> 00:07:07.894 It sped up. 00:07:07.894 --> 00:07:09.630 That's not surprising. 00:07:09.630 --> 00:07:11.850 This net force was directed leftward 00:07:11.850 --> 00:07:13.890 and the object was moving leftward. 00:07:13.890 --> 00:07:14.930 So, we're doing positive work. 00:07:14.930 --> 00:07:16.640 We're increasing the kinetic energy. 00:07:16.640 --> 00:07:17.928 We started with seven meters per second. 00:07:17.928 --> 00:07:20.210 We ended up with nine meters per second. 00:07:20.210 --> 00:07:21.043 And this is an example 00:07:21.043 --> 00:07:23.180 of how you use the work-energy principle. 00:07:23.180 --> 00:07:24.160 So, to recap, 00:07:24.160 --> 00:07:25.470 the work-energy principle states, 00:07:25.470 --> 00:07:28.110 the net work is equal to the change in kinetic energy. 00:07:28.110 --> 00:07:29.520 This can help you just conceptually 00:07:29.520 --> 00:07:31.350 or qualitatively determine 00:07:31.350 --> 00:07:33.200 whether something is gonna speed up, slow down, 00:07:33.200 --> 00:07:34.960 or change direction or both. 00:07:34.960 --> 00:07:36.010 And then, quantitatively, 00:07:36.010 --> 00:07:37.930 you can use this to specifically solve 00:07:37.930 --> 00:07:40.140 for the change in kinetic energy, 00:07:40.140 --> 00:07:42.830 as well as the final or initial speed 00:07:42.830 --> 00:07:44.053 something might've had.

Changes in Momentum Worked Examples

https://www.youtube.com/watch?v=5_bdL_g-X9I

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en

WEBVTT Kind: captions Language: en 00:00:00.250 --> 00:00:01.430 - [Instructor] So, here's a pink ball 00:00:01.430 --> 00:00:03.610 rolling toward a green cube 00:00:03.610 --> 00:00:05.878 that's sitting at rest on a frictionless surface. 00:00:05.878 --> 00:00:09.920 When the pink ball hits and slams into the green cube, 00:00:09.920 --> 00:00:12.490 it's gonna exert a force to the right on the green cube 00:00:12.490 --> 00:00:14.270 and the green cube's gonna speed up. 00:00:14.270 --> 00:00:16.250 But because of Newton's third law, 00:00:16.250 --> 00:00:17.970 whatever force the pink ball 00:00:17.970 --> 00:00:19.860 exerts on the green cube to the right 00:00:19.860 --> 00:00:22.090 has to be equal and opposite to the force 00:00:22.090 --> 00:00:25.600 the green cube exerts backward on the pink ball. 00:00:25.600 --> 00:00:27.860 So, the green cube's gonna gain some momentum, 00:00:27.860 --> 00:00:29.800 but the pink ball is gonna lose some momentum. 00:00:29.800 --> 00:00:31.520 And here's the cool thing. 00:00:31.520 --> 00:00:34.390 Whatever momentum is gained by this green cube 00:00:34.390 --> 00:00:36.310 has to be the same amount of momentum 00:00:36.310 --> 00:00:38.410 that's lost by the pink ball. 00:00:38.410 --> 00:00:40.510 So, if the green cube gained 00:00:40.510 --> 00:00:42.560 three units of momentum to the right, 00:00:42.560 --> 00:00:44.210 and then, the pink ball has to lose 00:00:44.210 --> 00:00:45.870 three units of momentum to the right. 00:00:45.870 --> 00:00:46.960 And this is true, 00:00:46.960 --> 00:00:49.540 whether these are the same mass or different masses, 00:00:49.540 --> 00:00:50.760 or whether the green cube 00:00:50.760 --> 00:00:52.890 was moving left to start or right to start. 00:00:52.890 --> 00:00:53.840 However they collide, 00:00:53.840 --> 00:00:55.340 if two objects collide, 00:00:55.340 --> 00:00:57.670 the change in momentum between the two objects 00:00:57.670 --> 00:00:59.760 has to be equal and opposite. 00:00:59.760 --> 00:01:01.300 And this is why we love momentum. 00:01:01.300 --> 00:01:02.750 Whatever momentum is gained by one 00:01:02.750 --> 00:01:04.230 has to be lost by the other. 00:01:04.230 --> 00:01:05.840 And so, how is that possible? 00:01:05.840 --> 00:01:07.000 You might be like, "How can that work?" 00:01:07.000 --> 00:01:08.250 Well, let's go up here. 00:01:08.250 --> 00:01:09.620 Think about these forces. 00:01:09.620 --> 00:01:12.150 The forces on each object are equal and opposite. 00:01:12.150 --> 00:01:13.300 Here's the cool part though. 00:01:13.300 --> 00:01:17.150 The time they're in contact also has to be equal 00:01:17.150 --> 00:01:19.786 because as soon as one of them loses contact with the other, 00:01:19.786 --> 00:01:22.580 the other loses contact with the one. 00:01:22.580 --> 00:01:26.050 And in physics, force times time is called the impulse. 00:01:26.050 --> 00:01:28.260 Now, this is the change in momentum. 00:01:28.260 --> 00:01:29.870 This will equal the change in momentum. 00:01:29.870 --> 00:01:31.100 So, if you wanna know how much momentum 00:01:31.100 --> 00:01:33.508 was gained by this green cube, 00:01:33.508 --> 00:01:36.490 multiply the force exerted on it by the time, 00:01:36.490 --> 00:01:38.630 you'll get the change in momentum of the green cube. 00:01:38.630 --> 00:01:39.780 That's true for this pink ball. 00:01:39.780 --> 00:01:41.710 And look, the force is backwards. 00:01:41.710 --> 00:01:44.368 It's just the negative of this other impulse. 00:01:44.368 --> 00:01:46.900 So, this is why the pink ball, 00:01:46.900 --> 00:01:49.170 no matter what it's mass or how fast it's going, 00:01:49.170 --> 00:01:50.244 same with green cube, 00:01:50.244 --> 00:01:54.300 these changes in momentum must be equal and opposite 00:01:54.300 --> 00:01:56.980 simply due to Newton's third law. 00:01:56.980 --> 00:01:58.520 So, to be clear, what I'm saying is this. 00:01:58.520 --> 00:02:00.490 If you are to graph the momentum, 00:02:00.490 --> 00:02:03.340 this pink ball would have started off with some momentum. 00:02:03.340 --> 00:02:05.600 And that would have been constant before the collision 00:02:05.600 --> 00:02:07.520 if there's no friction or resistance. 00:02:07.520 --> 00:02:08.840 But then, it lost some momentum. 00:02:08.840 --> 00:02:11.520 So, this goes down and maybe end up right here. 00:02:11.520 --> 00:02:13.160 And then, afterward, it'll maintain 00:02:13.160 --> 00:02:14.910 a constant amount of momentum. 00:02:14.910 --> 00:02:17.220 The green box, the green cube here, 00:02:17.220 --> 00:02:19.188 started with zero momentum, it was at rest. 00:02:19.188 --> 00:02:21.980 During this collision, it gained some momentum. 00:02:21.980 --> 00:02:24.110 So, it's gonna jump up maybe to here. 00:02:24.110 --> 00:02:27.430 What I'm saying is that, and afterward, it stays constant, 00:02:27.430 --> 00:02:29.759 what I'm saying is that these two jumps are the same. 00:02:29.759 --> 00:02:31.780 You know, if this thing lost, 00:02:31.780 --> 00:02:34.900 let's say it lost four units of momentum, 00:02:34.900 --> 00:02:37.140 well then, this green cube has to gain 00:02:37.140 --> 00:02:38.357 four units of momentum. 00:02:38.357 --> 00:02:40.010 And you might be like, "Okay, I don't care." 00:02:40.010 --> 00:02:41.190 But here's why you should care. 00:02:41.190 --> 00:02:44.260 If you are to graph the total momentum, 00:02:44.260 --> 00:02:47.010 what this means is that the total momentum, 00:02:47.010 --> 00:02:48.930 which was just the pink ball initially, 00:02:48.930 --> 00:02:51.150 is gonna remain constant the whole time. 00:02:51.150 --> 00:02:55.450 It's as if the momentum never realized a collision occurred. 00:02:55.450 --> 00:02:58.660 This total momentum just remains constant. 00:02:58.660 --> 00:03:01.400 And this is why conservation and momentum is a thing. 00:03:01.400 --> 00:03:02.750 It is why it's really useful. 00:03:02.750 --> 00:03:05.600 The total initial momentum in a system, 00:03:05.600 --> 00:03:09.000 even if a collision occurs between objects in that system, 00:03:09.000 --> 00:03:11.460 the total initial momentum must be equal 00:03:11.460 --> 00:03:13.690 to the total final momentum. 00:03:13.690 --> 00:03:16.330 As long as the forces are only internal, 00:03:16.330 --> 00:03:18.277 that is to say between objects in the system, 00:03:18.277 --> 00:03:20.280 and there's no external forces, 00:03:20.280 --> 00:03:21.820 then, this will always be true. 00:03:21.820 --> 00:03:24.160 So, this is a super powerful tool 00:03:24.160 --> 00:03:26.110 we can use to problem-solve. 00:03:26.110 --> 00:03:28.070 This saves us a lot of time and trouble. 00:03:28.070 --> 00:03:29.240 If a collision is occurring, 00:03:29.240 --> 00:03:32.060 this is one of our best methods to solve for things 00:03:32.060 --> 00:03:33.825 is conservation of momentum. 00:03:33.825 --> 00:03:35.170 Now, keep in mind. 00:03:35.170 --> 00:03:37.350 Even though the total momentum in a system 00:03:37.350 --> 00:03:38.461 has to stay the same, 00:03:38.461 --> 00:03:40.730 the momentum of an individual object 00:03:40.730 --> 00:03:42.220 does not have to stay the same. 00:03:42.220 --> 00:03:44.679 These objects can exchange momentum, but again, 00:03:44.679 --> 00:03:47.010 the reason this is gonna be conserved 00:03:47.010 --> 00:03:48.500 is that they do so equally. 00:03:48.500 --> 00:03:51.540 If one gains five, the other loses five and so on, 00:03:51.540 --> 00:03:53.550 and the total amount stays the same. 00:03:53.550 --> 00:03:55.230 So, let me show you how this works real quick. 00:03:55.230 --> 00:03:57.200 So, to give you an idea numerically here. 00:03:57.200 --> 00:03:59.450 So, let's say the pink ball was two kilograms 00:03:59.450 --> 00:04:01.565 and it was going five meters per second start. 00:04:01.565 --> 00:04:05.330 And the green cube had a mass, m, and afterward, 00:04:05.330 --> 00:04:08.110 let's say the pink ball's going four meters per seconds. 00:04:08.110 --> 00:04:09.040 So, it's slowed down. 00:04:09.040 --> 00:04:10.650 And the green cubes speeds up. 00:04:10.650 --> 00:04:12.620 Let's say, it's going eight meters per second. 00:04:12.620 --> 00:04:14.640 You might think, "Wait, this isn't conserved. 00:04:14.640 --> 00:04:18.310 The pink ball only lost one, five to four, 00:04:18.310 --> 00:04:20.130 but the green cube gained eight." 00:04:20.130 --> 00:04:22.040 But, remember, we're not conserving velocity. 00:04:22.040 --> 00:04:24.130 We're conserving momentum. 00:04:24.130 --> 00:04:26.690 So, momentum is m times v, 00:04:26.690 --> 00:04:28.740 momentum is mv, 00:04:28.740 --> 00:04:30.060 and it's a vector, you gotta be careful. 00:04:30.060 --> 00:04:31.430 It has a direction. 00:04:31.430 --> 00:04:33.620 So, I'm not saying the amount of velocity is conserved 00:04:33.620 --> 00:04:34.453 or anything like that. 00:04:34.453 --> 00:04:35.520 I'm saying momentum is conserved. 00:04:35.520 --> 00:04:38.270 So, what I'm saying is that p initial total 00:04:38.270 --> 00:04:41.040 has to equal p final total. 00:04:41.040 --> 00:04:43.560 So, only the pink ball had momentum initially. 00:04:43.560 --> 00:04:46.370 So, it had two kilograms 00:04:46.370 --> 00:04:49.920 times five meters per second of momentum. 00:04:49.920 --> 00:04:52.520 To start with, the green cube had none. 00:04:52.520 --> 00:04:53.978 So, this has gotta equal. 00:04:53.978 --> 00:04:57.390 Afterward, the pink ball has two kilograms 00:04:57.390 --> 00:05:00.170 times four meters per second. 00:05:00.170 --> 00:05:02.320 And the green cube does have momentum afterward. 00:05:02.320 --> 00:05:03.153 Now, we gotta add it up. 00:05:03.153 --> 00:05:04.130 There's gotta be the total momentum. 00:05:04.130 --> 00:05:06.800 So, plus mass of the cube 00:05:06.800 --> 00:05:09.570 times its eight meters per second. 00:05:09.570 --> 00:05:10.640 Well, so this isn't too hard. 00:05:10.640 --> 00:05:11.473 The math here is easy. 00:05:11.473 --> 00:05:13.080 So, this is gonna be 10 units of momentum, 00:05:13.080 --> 00:05:15.410 10 kilogram meters per second 00:05:15.410 --> 00:05:17.160 is what the ball started with. 00:05:17.160 --> 00:05:18.020 And then, the ball is gonna end 00:05:18.020 --> 00:05:21.220 with eight kilogram meters per second. 00:05:21.220 --> 00:05:22.053 So, we can see right here, 00:05:22.053 --> 00:05:24.330 the ball lost two units of momentum. 00:05:24.330 --> 00:05:26.500 That means the green cube better gain 00:05:26.500 --> 00:05:27.333 two units of momentum. 00:05:27.333 --> 00:05:29.980 So, it's gonna be plus m times eight. 00:05:29.980 --> 00:05:30.813 And so, this is what I mean 00:05:30.813 --> 00:05:32.200 when this can help you solve problems, 00:05:32.200 --> 00:05:33.360 we can just solve for the mass now. 00:05:33.360 --> 00:05:35.530 Now, we can know what the mass of the cube had to be. 00:05:35.530 --> 00:05:38.600 So, 10 minus eight, it's gonna be two units. 00:05:38.600 --> 00:05:41.976 It's how much momentum this green cube has to gain. 00:05:41.976 --> 00:05:44.530 And if we divide two by eight, 00:05:44.530 --> 00:05:46.740 we get that the mass of the green cube 00:05:46.740 --> 00:05:50.610 had to be .25 kilograms. 00:05:50.610 --> 00:05:55.436 So, indeed, if we took the .25 kilograms times the eight 00:05:55.436 --> 00:05:58.130 that this cube ended with, .25 times eight 00:05:58.130 --> 00:06:00.140 really does give us two, positive two, 00:06:00.140 --> 00:06:03.270 it gained two units of momentum. 00:06:03.270 --> 00:06:05.660 This is why momentum is conserved. 00:06:05.660 --> 00:06:07.434 Whatever a gain in momentum one thing gets, 00:06:07.434 --> 00:06:09.709 there's a corresponding loss in the other. 00:06:09.709 --> 00:06:12.190 And so, this is gonna be equal. 00:06:12.190 --> 00:06:15.770 Now, how could you ever make it so that this was not equal? 00:06:15.770 --> 00:06:17.580 Well, this will always be equal 00:06:17.580 --> 00:06:19.140 if the only force is being exerted 00:06:19.140 --> 00:06:20.480 are internal to your system. 00:06:20.480 --> 00:06:23.320 The only way you make this nonequal 00:06:23.320 --> 00:06:25.420 is to have external forces. 00:06:25.420 --> 00:06:26.530 What would that look like? 00:06:26.530 --> 00:06:30.300 Well, imagine the ball and the cube were on a ramp now. 00:06:30.300 --> 00:06:32.220 So, now that this thing's inclined, 00:06:32.220 --> 00:06:35.370 gravity is gonna be exerting an external force. 00:06:35.370 --> 00:06:37.150 Let's say it sit like 30 degrees. 00:06:37.150 --> 00:06:40.222 And if we graph the total momentum of the ball and cube, 00:06:40.222 --> 00:06:42.740 it's not gonna be a straight line anymore. 00:06:42.740 --> 00:06:44.770 It is not gonna look like this. 00:06:44.770 --> 00:06:45.780 That's what it looked like before 00:06:45.780 --> 00:06:47.470 when there were no external forces. 00:06:47.470 --> 00:06:49.540 This time, it's gonna be angled up. 00:06:49.540 --> 00:06:51.430 Let's just call down the ramp positive. 00:06:51.430 --> 00:06:53.270 This time, it's gonna look like this. 00:06:53.270 --> 00:06:55.330 Gravity's pulling down the ramp. 00:06:55.330 --> 00:06:57.100 Let's call that the positive direction. 00:06:57.100 --> 00:06:58.860 If we're measuring the momentum 00:06:58.860 --> 00:07:00.993 in that parallel to the ramp direction, 00:07:00.993 --> 00:07:03.690 we're gonna see an increase in momentum that direction, 00:07:03.690 --> 00:07:04.523 because that's the way 00:07:04.523 --> 00:07:07.856 that this external force of gravity points. 00:07:07.856 --> 00:07:09.810 So, how much will this increase? 00:07:09.810 --> 00:07:11.840 How could you figure out what the change 00:07:11.840 --> 00:07:14.120 in the total momentum is gonna be? 00:07:14.120 --> 00:07:16.040 One way to do it is to find the impulse. 00:07:16.040 --> 00:07:19.770 So, remember, force times time would give you the impulse. 00:07:19.770 --> 00:07:21.550 Let's just say we were looking at this 00:07:21.550 --> 00:07:23.730 for about one second of the time 00:07:23.730 --> 00:07:25.130 that these were on the ramp. 00:07:25.130 --> 00:07:26.820 And let's say they have the values they did before. 00:07:26.820 --> 00:07:30.620 So, two kilograms and .25 kilograms. 00:07:30.620 --> 00:07:34.790 That would mean the force down the ramp is mg sine theta. 00:07:34.790 --> 00:07:38.060 So, the force along this parallel direction is m, 00:07:38.060 --> 00:07:42.440 which the total mass of our system is 2.25 kilograms. 00:07:42.440 --> 00:07:47.380 So, m times g, 9.8, times sine of 30. 00:07:47.380 --> 00:07:48.940 So, you need to take this sine of 30 00:07:48.940 --> 00:07:50.640 if you wanna know the change in momentum 00:07:50.640 --> 00:07:52.090 along this ramp direction. 00:07:52.090 --> 00:07:57.030 So, the force of gravity parallel is mg sine theta. 00:07:57.030 --> 00:07:58.220 And then, that's the force. 00:07:58.220 --> 00:07:59.670 So, that's the force parallel to the ramp. 00:07:59.670 --> 00:08:01.970 That would give us the change in momentum parallel. 00:08:01.970 --> 00:08:04.803 We just have to multiply by the time, which is one second. 00:08:04.803 --> 00:08:09.140 And that gives us about 11 units of momentum. 00:08:09.140 --> 00:08:12.690 So, this system would gain 11 units of momentum. 00:08:12.690 --> 00:08:13.673 That doesn't tell you who's gonna get it. 00:08:13.673 --> 00:08:15.520 It's not like the two gets all of it, 00:08:15.520 --> 00:08:17.260 or the .25 gets all of it. 00:08:17.260 --> 00:08:18.470 But the total system, 00:08:18.470 --> 00:08:19.860 if you watched for a second, 00:08:19.860 --> 00:08:21.689 while it's on this ramp, frictionless ramp, 00:08:21.689 --> 00:08:23.448 it would increase its momentum. 00:08:23.448 --> 00:08:25.397 It'd be changing its total momentum 00:08:25.397 --> 00:08:28.780 because there was an external force. 00:08:28.780 --> 00:08:32.090 So, recapping, if there are no external forces on a system, 00:08:32.090 --> 00:08:33.534 then, the total momentum initial 00:08:33.534 --> 00:08:37.110 will equal the total momentum final. 00:08:37.110 --> 00:08:39.520 So, using p equals mv, 00:08:39.520 --> 00:08:41.380 you can add up these contributions 00:08:41.380 --> 00:08:43.070 and set them equal to problem-solve. 00:08:43.070 --> 00:08:46.020 And in a case where there is an external force, 00:08:46.020 --> 00:08:48.040 you can find how much that system 00:08:48.040 --> 00:08:50.030 will gain or lose in momentum 00:08:50.030 --> 00:08:52.100 by taking that external force 00:08:52.100 --> 00:08:54.550 times the time that force was applied. 00:08:54.550 --> 00:08:57.140 That would give you the total change in momentum 00:08:57.140 --> 00:08:59.153 of that system over that time.

Torque Basics

https://www.youtube.com/watch?v=TQQXpFhACSU

vtt

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en

WEBVTT Kind: captions Language: en 00:00:00.250 --> 00:00:01.540 - [Instructor] Imagine you've got a door here 00:00:01.540 --> 00:00:03.310 with a blue door knob. 00:00:03.310 --> 00:00:05.370 Any one of these 10 newton forces 00:00:05.370 --> 00:00:09.170 will cause the door to rotate around the hinge or the axis, 00:00:09.170 --> 00:00:11.320 or sometimes this is called the pivot point. 00:00:11.320 --> 00:00:13.930 Any one of these forces will cause the door to rotate. 00:00:13.930 --> 00:00:16.090 My question is, if you could insert one of these 00:00:16.090 --> 00:00:17.640 in one of these locations, 00:00:17.640 --> 00:00:19.650 which one of these forces, if any, 00:00:19.650 --> 00:00:23.310 would cause the most angular acceleration of this door? 00:00:23.310 --> 00:00:26.070 And you might think, "Oh, well, 10 newtons is 10 newtons. 00:00:26.070 --> 00:00:28.750 They'll all cause the same amount," but that's not true. 00:00:28.750 --> 00:00:30.080 It turns out we put door knobs 00:00:30.080 --> 00:00:32.210 at the end of doors for a reason. 00:00:32.210 --> 00:00:34.280 This red 10 newtons at the outside edge 00:00:34.280 --> 00:00:36.840 will cause the most angular acceleration. 00:00:36.840 --> 00:00:39.570 It'll cause this door to speed up most rapidly. 00:00:39.570 --> 00:00:40.630 And this used to bother me. 00:00:40.630 --> 00:00:43.230 I was like, "How come this is getting an advantage?" 00:00:43.230 --> 00:00:45.010 I think the best way to think about it is this. 00:00:45.010 --> 00:00:46.930 Even though these forces are all going through 00:00:46.930 --> 00:00:49.597 the same angle, so they've gone through 20 degrees, 00:00:49.597 --> 00:00:51.870 and now they've all gone through 30 degrees, 00:00:51.870 --> 00:00:55.760 now they've all gone through 45, 60 and 90. 00:00:55.760 --> 00:00:57.120 Even though these forces 00:00:57.120 --> 00:00:58.710 have all gone through the same angle, 00:00:58.710 --> 00:01:01.320 they have not gone through the same distance. 00:01:01.320 --> 00:01:02.153 Some of these forces 00:01:02.153 --> 00:01:03.860 have been exerted through a larger distance. 00:01:03.860 --> 00:01:04.760 So just look at it. 00:01:04.760 --> 00:01:06.830 If you imagine rotating this thing, 00:01:06.830 --> 00:01:09.770 that red force, this outside pink force here 00:01:09.770 --> 00:01:12.230 goes through a much larger distance 00:01:12.230 --> 00:01:14.410 than that inner yellow force. 00:01:14.410 --> 00:01:17.190 This force has gone through very little distance whatsoever. 00:01:17.190 --> 00:01:18.840 And you might think, well, why does that matter? 00:01:18.840 --> 00:01:22.260 Well, it matters because if you remember work done. 00:01:22.260 --> 00:01:24.670 Work done is proportional to the amount of force, 00:01:24.670 --> 00:01:25.780 but these are all 10 newtons 00:01:25.780 --> 00:01:27.370 so that doesn't really matter here. 00:01:27.370 --> 00:01:29.560 And it's also proportional to the amount of distance 00:01:29.560 --> 00:01:31.280 through which that force is applied. 00:01:31.280 --> 00:01:33.360 And because this outside force 00:01:33.360 --> 00:01:35.590 has gone through so much more distance 00:01:35.590 --> 00:01:37.440 than these inner forces, 00:01:37.440 --> 00:01:40.240 it's done more work over the same angle. 00:01:40.240 --> 00:01:42.180 And if you do more work, 00:01:42.180 --> 00:01:44.930 you input more kinetic energy into the door, 00:01:44.930 --> 00:01:48.560 it's gonna be moving faster for the same angle 00:01:48.560 --> 00:01:51.610 compared to what's caused by these other forces here. 00:01:51.610 --> 00:01:53.550 And this is why in angular mechanics 00:01:53.550 --> 00:01:55.370 you can't just think about forces, 00:01:55.370 --> 00:01:58.160 you have to think about something called torque. 00:01:58.160 --> 00:02:00.880 The symbol for torque is this fancy T. 00:02:00.880 --> 00:02:02.820 It's the Greek letter tau. 00:02:02.820 --> 00:02:05.210 And the amount of torque caused by a force, 00:02:05.210 --> 00:02:07.380 so you need a force to cause a torque, 00:02:07.380 --> 00:02:08.740 but it's more than just force. 00:02:08.740 --> 00:02:10.620 You have to multiply r, 00:02:10.620 --> 00:02:13.190 the distance from the access to the force 00:02:13.190 --> 00:02:15.170 by the amount of force 00:02:15.170 --> 00:02:17.790 in order to find how much torque is being exerted 00:02:17.790 --> 00:02:19.350 by a given force. 00:02:19.350 --> 00:02:21.220 The more torque that's exerted, 00:02:21.220 --> 00:02:23.900 the more angular acceleration you'd get, 00:02:23.900 --> 00:02:25.960 the faster you'd get something to speed up. 00:02:25.960 --> 00:02:26.793 Now you might wonder like, 00:02:26.793 --> 00:02:29.210 "Okay, I get that more force gives me more torque, 00:02:29.210 --> 00:02:31.690 how come this is just r and not like r squared? 00:02:31.690 --> 00:02:34.100 It seems kind of random, maybe its like square root of r." 00:02:34.100 --> 00:02:37.240 Well, if you remember arc lengths from back in the day, 00:02:37.240 --> 00:02:39.550 arc length is r times theta. 00:02:39.550 --> 00:02:42.650 So if I'm twice as far away from an axis, 00:02:42.650 --> 00:02:44.400 I get twice the arc length. 00:02:44.400 --> 00:02:47.170 If I get twice the arc length, I get twice the work done. 00:02:47.170 --> 00:02:48.180 You get twice the work done, 00:02:48.180 --> 00:02:50.350 you get twice the input kinetic energy 00:02:50.350 --> 00:02:52.470 and it turns out twice the kinetic energy 00:02:52.470 --> 00:02:54.940 will give you twice the angular acceleration. 00:02:54.940 --> 00:02:57.680 This is why everything is just proportional to r 00:02:57.680 --> 00:03:00.510 in terms of torque, it's not like r squared here. 00:03:00.510 --> 00:03:03.830 So for example, let's just say the distance from the axis, 00:03:03.830 --> 00:03:04.663 because that's what matters, 00:03:04.663 --> 00:03:07.080 to this 10 newtons here was one meter. 00:03:07.080 --> 00:03:10.210 And from the access to the purple force was two meters. 00:03:10.210 --> 00:03:14.180 And from the access to this doorknob force was three meters. 00:03:14.180 --> 00:03:16.070 What this torque formula means is that even though 00:03:16.070 --> 00:03:17.330 these are all 10 newtons, 00:03:17.330 --> 00:03:19.170 they'd all be exerting different amounts of torque. 00:03:19.170 --> 00:03:21.680 I'd have to take the one meter times 10 newtons 00:03:21.680 --> 00:03:24.160 would give me 10 newton meters. 00:03:24.160 --> 00:03:27.710 So the unit for torque is meters times newtons, 00:03:27.710 --> 00:03:29.970 but we usually write it as newton meters. 00:03:29.970 --> 00:03:31.240 If you buy a torque wrench, 00:03:31.240 --> 00:03:34.170 you could set it in newton meters or in foot pounds, 00:03:34.170 --> 00:03:36.470 if you're doing the US system. 00:03:36.470 --> 00:03:38.420 And then this purple force, even though it's 10 newtons, 00:03:38.420 --> 00:03:41.360 you'd have to take two meters times 10 newtons. 00:03:41.360 --> 00:03:45.290 This would exert a torque of 20 newton meters 00:03:45.290 --> 00:03:47.280 and the doorknob force wins the battle 00:03:47.280 --> 00:03:51.100 because it would have three times 10 would exert a torque 00:03:51.100 --> 00:03:53.140 of 30 newton meters. 00:03:53.140 --> 00:03:56.070 So the same size force can exert a different amount 00:03:56.070 --> 00:03:59.070 of torque, depending on how far away it is from the axis. 00:03:59.070 --> 00:04:01.020 So one area you have to be careful of this torque 00:04:01.020 --> 00:04:02.480 technically is a vector. 00:04:02.480 --> 00:04:03.313 It has a direction, 00:04:03.313 --> 00:04:04.710 it could be positive or negative. 00:04:04.710 --> 00:04:07.690 If you're doing full-blown engineering 3D physics, 00:04:07.690 --> 00:04:10.450 technically these torques would point out of the screen here 00:04:10.450 --> 00:04:13.350 out of the page, but for intro algebra-based physics, 00:04:13.350 --> 00:04:15.700 and for most problems, you can usually get away with 00:04:15.700 --> 00:04:19.730 just considering counterclockwise or clockwise 00:04:19.730 --> 00:04:21.810 as being the direction of the torque. 00:04:21.810 --> 00:04:25.410 That is to say these forces were making this object rotate 00:04:25.410 --> 00:04:27.840 in the counterclockwise direction, 00:04:27.840 --> 00:04:29.630 so they were all at the same sign. 00:04:29.630 --> 00:04:32.590 The convention is to call counterclockwise positive. 00:04:32.590 --> 00:04:34.230 So we'd call these all positive. 00:04:34.230 --> 00:04:37.440 If there were any forces that tried to rotate the system 00:04:37.440 --> 00:04:40.140 clockwise, you'd call those torques negative. 00:04:40.140 --> 00:04:42.600 You can do it either way, as long as you're consistent. 00:04:42.600 --> 00:04:44.740 Most books pick this as the convention though, 00:04:44.740 --> 00:04:45.890 so you should be aware of that. 00:04:45.890 --> 00:04:47.560 And then the last little bit to be careful about, 00:04:47.560 --> 00:04:50.180 I'm drawing all these forces nice and perpendicular 00:04:50.180 --> 00:04:53.960 to the r, and if that's the case, you just do r times F. 00:04:53.960 --> 00:04:56.200 If your force has different components, 00:04:56.200 --> 00:04:58.080 you need to make sure that the only component 00:04:58.080 --> 00:05:00.360 you plug in here is the perpendicular piece. 00:05:00.360 --> 00:05:02.360 So if this had some weird angle here, 00:05:02.360 --> 00:05:05.420 you'd only want the piece that was directly into 00:05:05.420 --> 00:05:10.060 this perpendicular lever arm here at a perpendicular angle. 00:05:10.060 --> 00:05:11.480 We'll talk about that more later. 00:05:11.480 --> 00:05:13.980 For now, let's just try some problems to kick the tires 00:05:13.980 --> 00:05:15.560 and get used to this formula. 00:05:15.560 --> 00:05:16.850 So imagine this example here 00:05:16.850 --> 00:05:18.900 where you've got the fancy door, you know, 00:05:18.900 --> 00:05:20.520 with a fancy hotel or restaurant 00:05:20.520 --> 00:05:22.930 that's a rotating circle and you can go in 00:05:22.930 --> 00:05:23.840 from either direction. 00:05:23.840 --> 00:05:25.470 This would be a bird's-eye view. 00:05:25.470 --> 00:05:27.550 Now, imagine you go into the hotel, 00:05:27.550 --> 00:05:28.383 you're pushing over here, 00:05:28.383 --> 00:05:29.670 you took physics, you know what to do. 00:05:29.670 --> 00:05:31.410 So you exert this 20 newtons over here. 00:05:31.410 --> 00:05:33.310 Let's say someone else comes in from the other edge. 00:05:33.310 --> 00:05:35.680 It's all awkward and they're trying to go in the other way 00:05:35.680 --> 00:05:36.740 and it's a stalemate. 00:05:36.740 --> 00:05:39.920 You're both pushing with forces, but nothing's happening. 00:05:39.920 --> 00:05:41.810 And that doesn't mean the two forces are equal. 00:05:41.810 --> 00:05:44.650 If you're in a stalemate here in terms of angular motion, 00:05:44.650 --> 00:05:47.660 that means your torques are equal and opposite. 00:05:47.660 --> 00:05:49.890 They're opposed, they have the same magnitude, 00:05:49.890 --> 00:05:51.630 but they'll have opposite directions of torque. 00:05:51.630 --> 00:05:53.180 So if you're locked in a stalemate here, 00:05:53.180 --> 00:05:55.950 that means the torque that you exert with your 20 newtons 00:05:55.950 --> 00:05:59.650 has to be equal to the torque from the other person. 00:05:59.650 --> 00:06:01.370 So let's try to figure out how much force 00:06:01.370 --> 00:06:02.860 would this person have to exert? 00:06:02.860 --> 00:06:04.600 It's not gonna be 20 newtons. 00:06:04.600 --> 00:06:06.480 They're pushing closer to the axis here 00:06:06.480 --> 00:06:08.110 so they're gonna have to push with more force. 00:06:08.110 --> 00:06:09.230 How much more force? 00:06:09.230 --> 00:06:11.810 Well, we can use the formula for torque to find it. 00:06:11.810 --> 00:06:13.060 The torques have to be equal. 00:06:13.060 --> 00:06:15.240 If there's no rotation here, you're balanced out. 00:06:15.240 --> 00:06:16.630 If your force is 20 newtons, 00:06:16.630 --> 00:06:19.910 you're exerting a force three meters away from the axis. 00:06:19.910 --> 00:06:22.170 That's your r, would be three meters 00:06:22.170 --> 00:06:25.770 times 20 newtons means you're exerting 20 times three, 00:06:25.770 --> 00:06:27.970 so 60 newton meters of torque. 00:06:27.970 --> 00:06:29.940 That means the other person has to be exerting 00:06:29.940 --> 00:06:33.000 60 newton meters of torque, but there r isn't two. 00:06:33.000 --> 00:06:35.140 Be careful here, you always have to measure from the axis, 00:06:35.140 --> 00:06:36.560 the point where you're rotating about. 00:06:36.560 --> 00:06:38.640 That'd be one meter. 00:06:38.640 --> 00:06:40.210 This door's all symmetric here. 00:06:40.210 --> 00:06:42.590 So it'd be one meter times F. 00:06:42.590 --> 00:06:45.410 And if you take this 60 newton meters 00:06:45.410 --> 00:06:47.260 and you divide by one meter, 00:06:47.260 --> 00:06:48.450 you're gonna get that this force here 00:06:48.450 --> 00:06:50.040 is gonna have to be 60 newtons. 00:06:50.040 --> 00:06:52.040 So this person is gonna have to exert more force. 00:06:52.040 --> 00:06:55.280 In fact, they pushed three times closer to the axis, 00:06:55.280 --> 00:06:57.010 so they're gonna have to exert three times 00:06:57.010 --> 00:06:58.310 the force that you do. 00:06:58.310 --> 00:07:02.030 You have a three times advantage here in holding this door 00:07:02.030 --> 00:07:03.770 compared to the other person. 00:07:03.770 --> 00:07:05.070 All right, let's try one more 00:07:05.070 --> 00:07:06.590 just to make sure we understand it. 00:07:06.590 --> 00:07:08.190 Let's say it's now rush hour, you know, 00:07:08.190 --> 00:07:10.410 bird's eye view here, same circular door. 00:07:10.410 --> 00:07:12.240 Three people are trying to go through at once. 00:07:12.240 --> 00:07:13.660 It's gonna be a mad house. 00:07:13.660 --> 00:07:16.020 This time I want to know, it's not gonna be a stalemate. 00:07:16.020 --> 00:07:17.780 This door is gonna rotate in some direction. 00:07:17.780 --> 00:07:20.130 I want to know what the net torque is. 00:07:20.130 --> 00:07:21.660 So just like you can find net force, 00:07:21.660 --> 00:07:24.120 you can find the net torque, but you gotta be careful. 00:07:24.120 --> 00:07:25.290 These might have different signs, 00:07:25.290 --> 00:07:27.320 you gotta add or subtract accordingly. 00:07:27.320 --> 00:07:28.370 So start over here. 00:07:28.370 --> 00:07:31.020 How much torque would be from this 10 newtons? 00:07:31.020 --> 00:07:34.600 Well, it's exerted three meters away from the axis, 00:07:34.600 --> 00:07:35.990 so it's r is three meters. 00:07:35.990 --> 00:07:38.750 So the torque from that force would be three meters 00:07:38.750 --> 00:07:42.010 times 10 newtons, and since this is directed 00:07:42.010 --> 00:07:45.270 counterclockwise, I'm just gonna call that positive 00:07:45.270 --> 00:07:47.110 and I'll have to be consistent with that choice. 00:07:47.110 --> 00:07:48.700 So now let's consider this eight newtons. 00:07:48.700 --> 00:07:50.960 You might think it would have an oppositely directed 00:07:50.960 --> 00:07:53.120 sign of torque from this 10 newtons 00:07:53.120 --> 00:07:54.880 'cause the eight is down, the 10 is up, 00:07:54.880 --> 00:07:57.440 but it's also trying to rotate this door 00:07:57.440 --> 00:07:59.270 in the counterclockwise direction. 00:07:59.270 --> 00:08:00.780 So in terms of forces, 00:08:00.780 --> 00:08:03.560 this 10 newton and eight newton are oppositely directed, 00:08:03.560 --> 00:08:06.220 but in terms of torques, they are the same direction. 00:08:06.220 --> 00:08:08.890 They're both causing rotation counterclockwise. 00:08:08.890 --> 00:08:11.770 So if I called this torque from the 10 newtons positive, 00:08:11.770 --> 00:08:14.620 I've gotta call the torque from this eight newtons positive 00:08:14.620 --> 00:08:16.780 'cause it's trying to exert a torque in the same direction. 00:08:16.780 --> 00:08:20.080 So I'd have one meter is the r for the eight 00:08:20.080 --> 00:08:22.780 times eight newtons would be the torque 00:08:22.780 --> 00:08:24.250 from the eight newtons. 00:08:24.250 --> 00:08:25.500 And then I have one more force here. 00:08:25.500 --> 00:08:28.960 This five newton is trying to rotate clockwise. 00:08:28.960 --> 00:08:31.130 Since I called counterclockwise positive, 00:08:31.130 --> 00:08:32.760 I'm gonna have to make this a negative torque, 00:08:32.760 --> 00:08:36.470 so minus three meters, the r from the access 00:08:36.470 --> 00:08:39.180 to this five newtons is three meters 00:08:39.180 --> 00:08:41.250 multiplied by five newtons. 00:08:41.250 --> 00:08:44.360 And if you take 30 plus eight minus 15, 00:08:44.360 --> 00:08:45.230 you're gonna get a total 00:08:45.230 --> 00:08:48.760 of positive 23 newton meters of torque, 00:08:48.760 --> 00:08:49.930 so this is not a stalemate. 00:08:49.930 --> 00:08:52.920 There will be an amount of angular acceleration 00:08:52.920 --> 00:08:55.120 caused by this net torque. 00:08:55.120 --> 00:08:58.410 So to recap, just like net forces 00:08:58.410 --> 00:09:01.010 can cause regular acceleration, 00:09:01.010 --> 00:09:04.140 net torques can cause angular acceleration. 00:09:04.140 --> 00:09:05.400 If there is no net torque, 00:09:05.400 --> 00:09:07.950 that means there is no angular acceleration. 00:09:07.950 --> 00:09:10.820 The way you find the torque from a given force 00:09:10.820 --> 00:09:13.880 is you take r, the distance from the axis 00:09:13.880 --> 00:09:16.360 to where that force is applied and you multiply 00:09:16.360 --> 00:09:19.970 by the amount of force, as long as it's that amount of force 00:09:19.970 --> 00:09:23.170 that runs perpendicular to this lever arm 00:09:23.170 --> 00:09:25.240 or this r direction. 00:09:25.240 --> 00:09:27.240 Be careful that torque is a vector. 00:09:27.240 --> 00:09:30.840 We typically count counterclockwise as positive 00:09:30.840 --> 00:09:34.610 and clockwise as negative, but if you're consistent, 00:09:34.610 --> 00:09:36.050 you can call whichever one of these 00:09:36.050 --> 00:09:37.450 you want to be positive 00:09:37.450 --> 00:09:39.600 as long as you call the other one negative.

Period of a Pendulum

https://www.youtube.com/watch?v=mzatvUid9Pw

vtt

https://www.youtube.com/api/timedtext?v=mzatvUid9Pw&ei=3FWUZYiKD6CsmLAP75iC0AQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=201A549503A22AB95F92B45F0727E4170B235EFE.EEF60365923DB329951F27E772A36B98C3E1C8DF&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.150 --> 00:00:01.310 - [Instructor] So, a simple pendulum 00:00:01.310 --> 00:00:03.540 is just a mass hanging from a string. 00:00:03.540 --> 00:00:05.210 And if you are to pull this mass, 00:00:05.210 --> 00:00:07.004 sometimes it's called a pendulum bob, 00:00:07.004 --> 00:00:09.380 if you are to pull it back and then let go, 00:00:09.380 --> 00:00:11.860 gravity would act as a restoring force 00:00:11.860 --> 00:00:15.310 and this mass would swing back and forth over and over. 00:00:15.310 --> 00:00:17.220 And because this simple pendulum 00:00:17.220 --> 00:00:19.920 is a simple harmonic oscillator, it's motion. 00:00:19.920 --> 00:00:21.552 Its angle as a function of time 00:00:21.552 --> 00:00:25.320 would be accurately described by a sine or a cosine graph. 00:00:25.320 --> 00:00:26.153 So, in other words, 00:00:26.153 --> 00:00:30.120 if you are to pull this back initially by 15 degrees, 00:00:30.120 --> 00:00:32.600 you might get a graph that looks something like this. 00:00:32.600 --> 00:00:34.840 Now, every simple harmonic oscillator 00:00:34.840 --> 00:00:38.010 has a characteristic period of motion. 00:00:38.010 --> 00:00:40.670 Now, the period of motion is the time it takes 00:00:40.670 --> 00:00:42.800 to complete one full cycle. 00:00:42.800 --> 00:00:45.210 So, the time it takes to swing from here 00:00:45.210 --> 00:00:46.043 all the way to there, 00:00:46.043 --> 00:00:47.330 all the way back to here, 00:00:47.330 --> 00:00:49.660 would be one full period. 00:00:49.660 --> 00:00:52.570 And in this graph, I've written it as .5 seconds, 00:00:52.570 --> 00:00:54.690 but the period of every simple pendulum 00:00:54.690 --> 00:00:56.290 is not gonna be .5 seconds. 00:00:56.290 --> 00:00:57.930 The period of a simple pendulum 00:00:57.930 --> 00:01:00.687 depends on the characteristics of that pendulum 00:01:00.687 --> 00:01:02.520 and the environment that it's in. 00:01:02.520 --> 00:01:05.760 So, to derive this formula for the period of a pendulum, 00:01:05.760 --> 00:01:07.300 you would need calculus. 00:01:07.300 --> 00:01:08.420 So, I'm just gonna write it down 00:01:08.420 --> 00:01:10.390 and give you a quick tour and compare it 00:01:10.390 --> 00:01:12.560 with the period formula for a mass on a spring, 00:01:12.560 --> 00:01:13.600 to the period of a pendulum 00:01:13.600 --> 00:01:18.600 is gonna be equal to two pi times the square root 00:01:18.740 --> 00:01:21.870 of the ratio of the length of the pendulum, L, 00:01:21.870 --> 00:01:25.070 so the length of that string here, the length L, 00:01:25.070 --> 00:01:28.145 divided by g, the gravitational acceleration 00:01:28.145 --> 00:01:31.530 of the planet that the pendulum is being used on. 00:01:31.530 --> 00:01:32.510 Now, if you look at this 00:01:32.510 --> 00:01:33.830 and you've been paying attention in physics, 00:01:33.830 --> 00:01:35.800 you might be like, "Wait, that looks really similar 00:01:35.800 --> 00:01:38.620 to the formula for the period of a mass on a spring." 00:01:38.620 --> 00:01:40.199 So, if you are to take a mass on a spring, 00:01:40.199 --> 00:01:42.130 displace it 15 centimeters, 00:01:42.130 --> 00:01:44.098 you'd get a similar graph 00:01:44.098 --> 00:01:47.880 and this would also have a characteristic period of motion. 00:01:47.880 --> 00:01:51.020 And if you wrote down the period for a mass on a spring, 00:01:51.020 --> 00:01:53.590 it looks like this, its period for a mass on a spring 00:01:53.590 --> 00:01:56.470 is also two pi, so that's identical. 00:01:56.470 --> 00:01:59.120 And then, it's also the square root of a ratio, 00:01:59.120 --> 00:02:02.294 but instead of L over g for the mass on a spring, 00:02:02.294 --> 00:02:06.340 it's m, the mass of the block connected to the spring, 00:02:06.340 --> 00:02:10.690 divided by k, the spring constant of the spring. 00:02:10.690 --> 00:02:13.021 So, one obvious similarity between these two formulas 00:02:13.021 --> 00:02:14.630 is just their format. 00:02:14.630 --> 00:02:18.110 They're both a two pi times a square root of a ratio, 00:02:18.110 --> 00:02:20.810 but another important similarity between these 00:02:20.810 --> 00:02:22.140 that might not be evident 00:02:22.140 --> 00:02:23.840 is that neither of these formulas 00:02:23.840 --> 00:02:26.090 depend on the amplitude of the motion. 00:02:26.090 --> 00:02:28.090 So, the period of a pendulum 00:02:28.090 --> 00:02:30.360 does not depend on the amplitude of the motion 00:02:30.360 --> 00:02:32.000 and the period of a mass on a spring 00:02:32.000 --> 00:02:35.020 also does not depend on the amplitude of the motion. 00:02:35.020 --> 00:02:35.853 What I mean by that 00:02:35.853 --> 00:02:37.850 is if you are to pull this pendulum back, 00:02:37.850 --> 00:02:38.920 instead of 15 degrees, 00:02:38.920 --> 00:02:41.370 pull it back 20 degrees and let go, 00:02:41.370 --> 00:02:43.100 it would have farther to swing. 00:02:43.100 --> 00:02:45.350 So, its motion might look something like this, 00:02:45.350 --> 00:02:47.830 and we get down farther and it would get back up, 00:02:47.830 --> 00:02:50.120 but it would take the exact same amount of time. 00:02:50.120 --> 00:02:51.790 The period would not change 00:02:51.790 --> 00:02:54.320 if you pull this amplitude back farther. 00:02:54.320 --> 00:02:56.490 Same goes with this mass on a spring. 00:02:56.490 --> 00:02:58.300 Instead of pulling it 15 centimeters, 00:02:58.300 --> 00:03:00.630 let's say you pulled it 20 centimeters, again, 00:03:00.630 --> 00:03:02.060 this would start up higher. 00:03:02.060 --> 00:03:03.960 It would get down lower. 00:03:03.960 --> 00:03:06.900 But the time it takes to complete one full cycle 00:03:06.900 --> 00:03:09.730 would not vary as you vary this amplitude. 00:03:09.730 --> 00:03:10.690 And that might seem weird. 00:03:10.690 --> 00:03:12.410 You're like, "Wait, don't these objects 00:03:12.410 --> 00:03:14.210 have farther to go now 00:03:14.210 --> 00:03:15.990 that you've pulled it back to a larger amplitude?" 00:03:15.990 --> 00:03:18.780 That's true, they'll both have farther to travel, 00:03:18.780 --> 00:03:20.550 but they'll be going faster now. 00:03:20.550 --> 00:03:24.632 And faster motion over a bigger distance is gonna offset. 00:03:24.632 --> 00:03:27.499 And the amplitude does not affect the period 00:03:27.499 --> 00:03:30.760 of the motion of a pendulum or of a mass on a spring. 00:03:30.760 --> 00:03:33.220 As for the differences, well, 00:03:33.220 --> 00:03:35.370 the denominator here for the mass on a spring 00:03:35.370 --> 00:03:38.690 depends on k, and that's the spring constant k, 00:03:38.690 --> 00:03:39.710 and that should make sense. 00:03:39.710 --> 00:03:43.150 More spring constant means you get more restoring force, 00:03:43.150 --> 00:03:45.400 the spring force is the restoring force here. 00:03:45.400 --> 00:03:47.170 More restoring force means this mass 00:03:47.170 --> 00:03:48.800 is gonna be moving faster. 00:03:48.800 --> 00:03:51.280 That means it's gonna take less time 00:03:51.280 --> 00:03:52.940 to go through a full period. 00:03:52.940 --> 00:03:56.000 So, dividing by a bigger number, bigger spring k, 00:03:56.000 --> 00:03:57.520 gives you less period. 00:03:57.520 --> 00:03:59.660 That's also true here but it's not k. 00:03:59.660 --> 00:04:02.270 The force that's the restoring force for a pendulum 00:04:02.270 --> 00:04:04.980 isn't a spring, it's the force of gravity. 00:04:04.980 --> 00:04:07.220 So, mg depends on the g. 00:04:07.220 --> 00:04:10.810 So, a bigger g would give you a bigger restoring force 00:04:10.810 --> 00:04:12.310 over here for the pendulum. 00:04:12.310 --> 00:04:14.700 That means the pendulum could be moving faster. 00:04:14.700 --> 00:04:17.400 Moving faster means it's gonna take less time 00:04:17.400 --> 00:04:18.920 to complete a full period. 00:04:18.920 --> 00:04:21.250 So, dividing by a bigger g, if you took this pendulum 00:04:21.250 --> 00:04:23.570 to the surface of Jupiter or something, 00:04:23.570 --> 00:04:25.070 where the g is bigger, 00:04:25.070 --> 00:04:27.480 it would swing back and forth faster. 00:04:27.480 --> 00:04:29.560 It would take less time to complete it 00:04:29.560 --> 00:04:31.420 because that restoring force is bigger. 00:04:31.420 --> 00:04:33.890 So, even though these denominators are different letters, 00:04:33.890 --> 00:04:35.760 they're arriving from the same source. 00:04:35.760 --> 00:04:38.880 They're both arriving because the restoring force 00:04:38.880 --> 00:04:41.520 is larger when you increase these denominators, 00:04:41.520 --> 00:04:43.460 which increases the speed of the object. 00:04:43.460 --> 00:04:45.320 Now, maybe the biggest difference here 00:04:45.320 --> 00:04:47.610 is that the numerator here 00:04:47.610 --> 00:04:49.620 for the mass on a spring depends on mass, 00:04:49.620 --> 00:04:51.730 but nowhere is mass to be found 00:04:51.730 --> 00:04:53.010 in this period of a pendulum. 00:04:53.010 --> 00:04:56.020 The period of the pendulum does not depend on the mass. 00:04:56.020 --> 00:04:57.480 Now, this is kind of interesting. 00:04:57.480 --> 00:05:00.520 So, you know, really big, heavy person gets on a swing, 00:05:00.520 --> 00:05:01.730 swings back and forth, 00:05:01.730 --> 00:05:04.240 very light child gets on the same swing, 00:05:04.240 --> 00:05:06.240 they will take the same amount of time 00:05:06.240 --> 00:05:07.730 to complete a full cycle. 00:05:07.730 --> 00:05:09.800 Their mass does not factor in here 00:05:09.800 --> 00:05:10.790 to the period of pendulum, 00:05:10.790 --> 00:05:13.840 but it does for the mass on a spring, why is that? 00:05:13.840 --> 00:05:16.440 Well, bigger mass on a spring 00:05:16.440 --> 00:05:18.770 gives you more inertia in the system. 00:05:18.770 --> 00:05:20.430 If you have more inertia in the system, 00:05:20.430 --> 00:05:22.130 it's more sluggish to movement. 00:05:22.130 --> 00:05:23.510 It's gonna go slower. 00:05:23.510 --> 00:05:25.460 That means it's gonna take more time 00:05:25.460 --> 00:05:27.210 to complete a full cycle. 00:05:27.210 --> 00:05:28.490 Now, you might think, "Wait a minute. 00:05:28.490 --> 00:05:29.660 don't that hold true up here? 00:05:29.660 --> 00:05:33.470 Look, if we have a bigger mass pendulum bob, 00:05:33.470 --> 00:05:35.982 that should increase the rotational inertia. 00:05:35.982 --> 00:05:37.970 And so, that should make this take longer. 00:05:37.970 --> 00:05:39.560 It should be more sluggish to movement. 00:05:39.560 --> 00:05:41.680 It should take longer to go through a full period." 00:05:41.680 --> 00:05:45.370 But look at the restoring force is also, for a pendulum, 00:05:45.370 --> 00:05:46.970 proportional to mass. 00:05:46.970 --> 00:05:48.510 So, if you increase the mass of this pendulum, 00:05:48.510 --> 00:05:49.810 you do get more inertia, 00:05:49.810 --> 00:05:51.370 but you're getting more restoring force 00:05:51.370 --> 00:05:53.120 because the restoring force is gravity. 00:05:53.120 --> 00:05:54.084 Those completely offset 00:05:54.084 --> 00:05:57.930 mass does not end up showing up into this pendulum formula, 00:05:57.930 --> 00:05:59.730 even though it does down here. 00:05:59.730 --> 00:06:02.140 So, the spring force is not proportional to mass. 00:06:02.140 --> 00:06:03.510 Spring force is kx. 00:06:03.510 --> 00:06:06.487 Increasing the mass of this block on the end 00:06:06.487 --> 00:06:09.140 does not increase the spring restoring force. 00:06:09.140 --> 00:06:11.020 So, this mass stays in the numerator here, 00:06:11.020 --> 00:06:13.640 but it does not affect the period of a pendulum. 00:06:13.640 --> 00:06:15.590 So, why does this L show up then? 00:06:15.590 --> 00:06:18.210 Why is it length of the pendulum in the numerator? 00:06:18.210 --> 00:06:20.880 Well, the rotational inertia does get increased 00:06:20.880 --> 00:06:23.280 when you increase the length of the pendulum. 00:06:23.280 --> 00:06:24.210 But increasing that length 00:06:24.210 --> 00:06:26.550 that does not increase the force of gravity. 00:06:26.550 --> 00:06:27.940 If you wanna get technical, 00:06:27.940 --> 00:06:31.270 rotational inertia is proportional to length squared, 00:06:31.270 --> 00:06:33.840 but the torque would only be proportional to length. 00:06:33.840 --> 00:06:36.040 That's why only one L shows up here. 00:06:36.040 --> 00:06:39.920 Long story short, if you increase the length of a pendulum, 00:06:39.920 --> 00:06:42.150 you're gonna increase the inertia of that pendulum. 00:06:42.150 --> 00:06:43.380 That's gonna make it take longer 00:06:43.380 --> 00:06:44.730 to go through a full cycle. 00:06:44.730 --> 00:06:45.900 This is while I go into the park 00:06:45.900 --> 00:06:47.650 and finding the long swings. 00:06:47.650 --> 00:06:50.410 The longer the swing, the longer it actually takes, 00:06:50.410 --> 00:06:52.670 the more time it actually takes to swing back and forth. 00:06:52.670 --> 00:06:55.480 I think that's more fun than the little short swings 00:06:55.480 --> 00:06:57.380 that go back and forth really quick. 00:06:57.380 --> 00:06:58.670 So, let's try a sample problem 00:06:58.670 --> 00:07:01.160 to see how this period formula works. 00:07:01.160 --> 00:07:02.660 Let's say you went to the park. 00:07:02.660 --> 00:07:03.900 You're 60 kilograms. 00:07:03.900 --> 00:07:04.990 You're swinging on a swing 00:07:04.990 --> 00:07:07.880 and your friend pulls you back 20 degrees 00:07:07.880 --> 00:07:10.370 on a swing that's one meter in length. 00:07:10.370 --> 00:07:12.370 Let's find the period of the motion. 00:07:12.370 --> 00:07:13.680 So, in other words, the time it takes 00:07:13.680 --> 00:07:15.150 to go all the way to here 00:07:15.150 --> 00:07:16.720 and then all the way back to there. 00:07:16.720 --> 00:07:19.710 We use the period formula for a pendulum. 00:07:19.710 --> 00:07:23.600 It's two pi, root L over g. 00:07:23.600 --> 00:07:27.670 And so, we would do two pi times the square root, 00:07:27.670 --> 00:07:29.860 the length here is the length of the string here. 00:07:29.860 --> 00:07:31.800 So, one meter, technically, 00:07:31.800 --> 00:07:34.040 it'd be to wherever your center of mass is, 00:07:34.040 --> 00:07:35.820 but we're gonna assume our center mass 00:07:35.820 --> 00:07:37.250 is right here at the end, 00:07:37.250 --> 00:07:40.260 divided by g, well, we're on earth, I'm assuming, 00:07:40.260 --> 00:07:43.420 so, 9.8 meters per second squared. 00:07:43.420 --> 00:07:44.359 If you solve all of that, 00:07:44.359 --> 00:07:48.070 you get a period of about two seconds exactly. 00:07:48.070 --> 00:07:51.490 So, this swing would have a period of about two seconds. 00:07:51.490 --> 00:07:54.480 Now, notice we did not use this 20 degrees. 00:07:54.480 --> 00:07:55.670 That's the amplitude. 00:07:55.670 --> 00:07:58.020 This period does not depend on the amplitude. 00:07:58.020 --> 00:08:01.200 We also did not use the fact that I was 60 kilograms. 00:08:01.200 --> 00:08:03.410 The period of a pendulum also does not depend 00:08:03.410 --> 00:08:06.250 on the mass of the bob at the end of the pendulum. 00:08:06.250 --> 00:08:08.640 We only use the length and if you're on earth, 00:08:08.640 --> 00:08:11.260 the denominator here is always gonna be 9.8. 00:08:11.260 --> 00:08:14.310 Now, one thing I should be clear about is that a pendulum 00:08:14.310 --> 00:08:17.100 is technically not a perfect simple harmonic oscillator. 00:08:17.100 --> 00:08:19.760 It's only approximately a simple harmonic oscillator. 00:08:19.760 --> 00:08:22.780 So, this formula here is only approximately correct, 00:08:22.780 --> 00:08:25.400 but for small angles, it's almost perfect. 00:08:25.400 --> 00:08:26.560 So, at 20 degrees, 00:08:26.560 --> 00:08:28.610 this formula is only gonna be off. 00:08:28.610 --> 00:08:31.066 The error's only going to be about 1%, 00:08:31.066 --> 00:08:32.230 which isn't bad. 00:08:32.230 --> 00:08:34.420 If you get this back to like 70 degrees, 00:08:34.420 --> 00:08:37.920 even then, the error's only around 10%. 00:08:37.920 --> 00:08:41.190 So, this is a really, really good approximation 00:08:41.190 --> 00:08:44.710 if you're in small angles, like around 20 to 30 degrees. 00:08:44.710 --> 00:08:46.340 The bigger the angle gets though, 00:08:46.340 --> 00:08:47.960 the worst the approximation gets. 00:08:47.960 --> 00:08:51.310 For small angles, most physicists just treat a pendulum 00:08:51.310 --> 00:08:53.656 as if it's a perfect simple harmonic oscillator. 00:08:53.656 --> 00:08:55.560 As you get to those higher angles though, 00:08:55.560 --> 00:08:56.700 you gotta be careful. 00:08:56.700 --> 00:08:59.470 This can start deviating more significantly 00:08:59.470 --> 00:09:00.870 from the actual value. 00:09:00.870 --> 00:09:02.860 So, to recap, the period of a pendulum 00:09:02.860 --> 00:09:04.930 depends on the length of the pendulum 00:09:04.930 --> 00:09:07.650 and the surface gravity of the planet that you're on. 00:09:07.650 --> 00:09:10.110 It does not depend on the amplitude 00:09:10.110 --> 00:09:12.040 or the mass of the pendulum. 00:09:12.040 --> 00:09:13.940 And the form it takes is very similar 00:09:13.940 --> 00:09:16.070 to the period of a mass on a spring 00:09:16.070 --> 00:09:17.890 where the numerator increases 00:09:17.890 --> 00:09:19.530 due to increased inertia 00:09:19.530 --> 00:09:21.680 and the denominator increases 00:09:21.680 --> 00:09:24.253 due to increased restoring force.

Dihybrid cross and the Law of Independent Assortment

https://www.youtube.com/watch?v=ZkyX3TmmMag

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https://www.youtube.com/api/timedtext?v=ZkyX3TmmMag&ei=3FWUZcG6BcSkvdIPuJW6uAs&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=58F3E661750ACB523C38CDDACC06C84974860328.5C17A64722736B620D5DE8EA4833EA5689C52D7C&key=yt8&lang=en&name=Default&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:00.300 --> 00:00:02.070 - [Instructor] In this video, we're going to build 00:00:02.070 --> 00:00:03.977 on our understanding of Mendelian genetics 00:00:03.977 --> 00:00:05.390 and Punnett squares 00:00:05.390 --> 00:00:08.340 by starting to think about two different genes. 00:00:08.340 --> 00:00:09.820 So we're going back to the pea plant, 00:00:09.820 --> 00:00:12.070 and we're gonna think about the gene for pea color 00:00:12.070 --> 00:00:15.090 and the gene for pea shape. 00:00:15.090 --> 00:00:18.430 So let's say that in the parental generation, 00:00:18.430 --> 00:00:21.640 you have one parent that is homozygous dominant 00:00:21.640 --> 00:00:24.210 for both of these genes. 00:00:24.210 --> 00:00:28.500 So their genotype is capital Y, capital Y, 00:00:28.500 --> 00:00:32.260 and capital R, capital R. 00:00:32.260 --> 00:00:35.640 So their phenotype for sure is going to be yellow round, 00:00:35.640 --> 00:00:36.840 and we also see the genotype. 00:00:36.840 --> 00:00:39.680 We see which alleles it has. 00:00:39.680 --> 00:00:41.270 Now, let's say that that is crossed 00:00:41.270 --> 00:00:44.100 with the homozygous recessive parent. 00:00:44.100 --> 00:00:47.710 So in this case, it's going to be green-colored peas. 00:00:47.710 --> 00:00:49.950 It's counterintuitive to write green with a y, 00:00:49.950 --> 00:00:52.350 a lowercase y in the color yellow, 00:00:52.350 --> 00:00:54.730 but the lowercase y represents green. 00:00:54.730 --> 00:00:57.930 And also these are wrinkled green peas. 00:00:57.930 --> 00:01:01.410 So I will write that with the lowercase r here. 00:01:01.410 --> 00:01:03.370 And so the phenotype here is going 00:01:03.370 --> 00:01:06.900 to be green and wrinkled for sure. 00:01:06.900 --> 00:01:09.320 Now, what's going to happen when they cross? 00:01:09.320 --> 00:01:12.110 What does the F1 generation look like? 00:01:12.110 --> 00:01:15.460 Well, we know from Mendel's law of segregation 00:01:15.460 --> 00:01:19.180 for each of these genes, that when a gamete is created, 00:01:19.180 --> 00:01:23.290 it randomly gets one copy of each of these genes. 00:01:23.290 --> 00:01:24.123 So for this first one, 00:01:24.123 --> 00:01:27.150 it's going to randomly get one of these capital Ys. 00:01:27.150 --> 00:01:29.430 So it's going to get a capital Y for sure 00:01:29.430 --> 00:01:30.620 from this first parent. 00:01:30.620 --> 00:01:34.460 And it's also going to randomly get one of these capital Rs. 00:01:34.460 --> 00:01:36.860 So it's going to get a capital R for sure 00:01:36.860 --> 00:01:38.370 from that first parent. 00:01:38.370 --> 00:01:40.370 And then by the same logic, 00:01:40.370 --> 00:01:43.340 it's going to randomly get one of these two lowercase y's. 00:01:43.340 --> 00:01:46.390 So it's going to get a lowercase y for sure, 00:01:46.390 --> 00:01:47.900 and it's going to randomly get one 00:01:47.900 --> 00:01:49.650 of these two lowercase r's, 00:01:49.650 --> 00:01:52.470 so it's going to get a lowercase r for sure. 00:01:52.470 --> 00:01:57.060 So this is the genotype for all of the F1 generation. 00:01:57.060 --> 00:01:58.893 This is often known as a dihybrid. 00:01:59.976 --> 00:02:03.120 It is heterozygous in both genes. 00:02:03.120 --> 00:02:04.830 Now, what's the phenotype here? 00:02:04.830 --> 00:02:06.450 Well, we know yellow is dominant, 00:02:06.450 --> 00:02:08.050 and we know round is dominant. 00:02:08.050 --> 00:02:10.210 So if we looked at these plants right over here, 00:02:10.210 --> 00:02:12.430 their peas would still be yellow round, 00:02:12.430 --> 00:02:15.380 just like this homozygous parent over here. 00:02:15.380 --> 00:02:17.080 Now, what's interesting is when you do 00:02:17.080 --> 00:02:19.760 what's known as a dihybrid cross 00:02:19.760 --> 00:02:22.890 when you cross one of this F1 generation 00:02:22.890 --> 00:02:25.130 with itself or with each other. 00:02:25.130 --> 00:02:26.020 And to do that, 00:02:26.020 --> 00:02:30.440 I'm gonna create a four by four Punnett square here. 00:02:30.440 --> 00:02:34.210 And so one parent here is going to be hybrid in 00:02:34.210 --> 00:02:37.540 or heterozygous in the color gene 00:02:37.540 --> 00:02:40.470 and also heterozygous in the shape gene. 00:02:40.470 --> 00:02:42.900 And that's going to be true of the other parent as well. 00:02:42.900 --> 00:02:46.380 Heterozygous or hybrid in the color gene 00:02:46.380 --> 00:02:49.890 and also heterozygous in the shape gene. 00:02:49.890 --> 00:02:52.160 And so that's why this is called a dihybrid cross. 00:02:52.160 --> 00:02:53.300 You're crossing things 00:02:53.300 --> 00:02:56.330 that are hybrid in two different genes. 00:02:56.330 --> 00:02:59.300 Now, we've already talked about the law of segregation. 00:02:59.300 --> 00:03:03.320 The gamete is randomly going to get one copy of each gene. 00:03:03.320 --> 00:03:07.190 Now, Mendel also has the law of independent assortment, 00:03:07.190 --> 00:03:08.580 which tells us the alleles 00:03:08.580 --> 00:03:12.100 of different genes segregate independently. 00:03:12.100 --> 00:03:14.020 So for this parent here, 00:03:14.020 --> 00:03:17.230 whether it contributes a capital Y or a lowercase y 00:03:17.230 --> 00:03:19.910 is independent of whether it contribute a capital R 00:03:19.910 --> 00:03:21.330 or a lowercase r. 00:03:21.330 --> 00:03:22.920 Now, there is a little bit of a asterisk, 00:03:22.920 --> 00:03:24.490 a little caveat on there. 00:03:24.490 --> 00:03:27.020 We now know that genes sit on chromosomes. 00:03:27.020 --> 00:03:28.810 One chromosome will have many genes on it. 00:03:28.810 --> 00:03:31.630 And this law of independent assortment only applies 00:03:31.630 --> 00:03:34.690 to genes that are actually sitting on different chromosomes. 00:03:34.690 --> 00:03:36.710 If they sit on the same chromosome, 00:03:36.710 --> 00:03:40.230 they generally are not going to assort independently. 00:03:40.230 --> 00:03:43.040 But let's just assume the law of independent assortment 00:03:43.040 --> 00:03:45.380 'cause this is true for most genes. 00:03:45.380 --> 00:03:49.920 So this first parent can contribute a capital Y 00:03:49.920 --> 00:03:51.530 out of this first gene 00:03:51.530 --> 00:03:54.900 and a capital R out of the second gene, 00:03:54.900 --> 00:03:57.380 or they could contribute the lowercase copy 00:03:57.380 --> 00:03:58.600 of the first gene 00:03:58.600 --> 00:04:02.780 and the capital R copy of the second gene, 00:04:02.780 --> 00:04:06.000 this capital R, the round allele of the second gene. 00:04:06.000 --> 00:04:08.210 And we could go through every combination here. 00:04:08.210 --> 00:04:12.580 It could also contribute the yellow allele 00:04:12.580 --> 00:04:14.363 and the wrinkled allele. 00:04:15.490 --> 00:04:20.290 Or it could contribute the green allele 00:04:20.290 --> 00:04:22.543 and the wrinkled allele as well. 00:04:23.540 --> 00:04:26.000 And the same would be true for this other dihybrid, 00:04:26.000 --> 00:04:28.130 this other parent right over here. 00:04:28.130 --> 00:04:29.510 So let me just write that down. 00:04:29.510 --> 00:04:33.440 They could contribute capital Y in two of the scenarios. 00:04:33.440 --> 00:04:35.270 They could contribute a lowercase y 00:04:35.270 --> 00:04:37.590 or the green allele in two of the scenarios. 00:04:37.590 --> 00:04:39.920 And they could contribute a capital R 00:04:39.920 --> 00:04:42.360 in two of the scenarios, a round allele, 00:04:42.360 --> 00:04:44.700 or a lowercase r in two of the scenarios, 00:04:44.700 --> 00:04:46.100 a wrinkled allele. 00:04:46.100 --> 00:04:48.050 So you have all of the different combinations 00:04:48.050 --> 00:04:49.690 that each of them can contribute. 00:04:49.690 --> 00:04:51.730 Once again, whether you get the yellow 00:04:51.730 --> 00:04:53.110 or the green is independent 00:04:53.110 --> 00:04:55.260 of whether you get the round or the wrinkled. 00:04:55.260 --> 00:04:58.083 So these are all equally probably right over here. 00:04:59.100 --> 00:05:01.270 When the two gametes from these two parents merge, 00:05:01.270 --> 00:05:03.000 we can then look at what the genotype 00:05:03.000 --> 00:05:04.350 of the offspring is going to be, 00:05:04.350 --> 00:05:06.900 really the genotype of the F2 generation 00:05:06.900 --> 00:05:10.470 'cause we're crossing two members of the F1 generation. 00:05:10.470 --> 00:05:12.210 So I encourage you to pause this video 00:05:12.210 --> 00:05:13.720 and fill in this grid. 00:05:13.720 --> 00:05:15.820 See if you can figure the different genotypes 00:05:15.820 --> 00:05:16.803 that will result. 00:05:18.140 --> 00:05:20.270 All right, now let's do this together. 00:05:20.270 --> 00:05:21.710 So this scenario right over here, 00:05:21.710 --> 00:05:25.060 you're getting a capital Y from both parents, 00:05:25.060 --> 00:05:29.810 and you're getting a capital R from both parents. 00:05:29.810 --> 00:05:30.730 This scenario over here, 00:05:30.730 --> 00:05:33.330 you're getting a capital Y from this parent, 00:05:33.330 --> 00:05:35.840 lowercase y from that parent, 00:05:35.840 --> 00:05:38.663 and then you're getting a capital R from both. 00:05:39.650 --> 00:05:43.940 This scenario over here, capital Y from both parents. 00:05:43.940 --> 00:05:45.500 Capital Y, capital Y. 00:05:45.500 --> 00:05:49.940 And you're getting a capital R from this parent 00:05:49.940 --> 00:05:52.480 and a lowercase r from that parent. 00:05:52.480 --> 00:05:54.460 And then this scenario over here, 00:05:54.460 --> 00:05:57.550 you're getting gonna capital Y allele from this parent 00:05:57.550 --> 00:05:59.640 and lowercase y from that parent, 00:05:59.640 --> 00:06:03.130 and you're getting a capital R from this parent 00:06:03.130 --> 00:06:05.630 and a lowercase r from that parent. 00:06:05.630 --> 00:06:07.810 And now I'm just going to speed up the video 00:06:07.810 --> 00:06:10.923 and just fill in the rest of these using the same logic. 00:06:14.440 --> 00:06:17.080 All right, now that we've filled out this Punnett square, 00:06:17.080 --> 00:06:19.470 let's think about the different phenotypes. 00:06:19.470 --> 00:06:20.530 How many of these plants are going 00:06:20.530 --> 00:06:22.560 to produce yellow round peas? 00:06:22.560 --> 00:06:24.310 Pause the video and think about it. 00:06:25.320 --> 00:06:27.310 Well, it's yellow and round. 00:06:27.310 --> 00:06:31.260 It has to have at least one capital Y and one capital R. 00:06:31.260 --> 00:06:33.620 So that one's going to be yellow and round. 00:06:33.620 --> 00:06:35.560 This is going to be yellow and round. 00:06:35.560 --> 00:06:37.970 This is going to be yellow and round. 00:06:37.970 --> 00:06:40.270 That's yellow and round as well. 00:06:40.270 --> 00:06:42.500 This is yellow and round. 00:06:42.500 --> 00:06:44.500 That's yellow and round. 00:06:44.500 --> 00:06:46.610 This is yellow and round. 00:06:46.610 --> 00:06:49.560 And that one is yellow and round. 00:06:49.560 --> 00:06:53.040 And then last but not least, I think this is the last one 00:06:53.040 --> 00:06:55.430 that is both yellow and round. 00:06:55.430 --> 00:06:57.090 And actually let me make a little color code here. 00:06:57.090 --> 00:07:00.240 Yellow plus round. 00:07:00.240 --> 00:07:01.850 And here we're talking about the phenotype. 00:07:01.850 --> 00:07:03.840 You can see we have different genotypes here, 00:07:03.840 --> 00:07:06.730 but because both yellow and round are dominant, 00:07:06.730 --> 00:07:09.170 as long as you have at least one Y and one R, 00:07:09.170 --> 00:07:10.340 you're going to have a yellow 00:07:10.340 --> 00:07:12.330 plus round phenotype over here. 00:07:12.330 --> 00:07:14.300 So you have one, two, three, four, 00:07:14.300 --> 00:07:16.540 five, six, seven, eight, nine. 00:07:16.540 --> 00:07:19.580 And I will say there's nine of these over here. 00:07:19.580 --> 00:07:24.580 Now, how many of these are going to be yellow plus wrinkled? 00:07:25.490 --> 00:07:28.820 Pause the video and think about that, that phenotype. 00:07:28.820 --> 00:07:30.000 So yellow and wrinkled, 00:07:30.000 --> 00:07:32.300 you're going to have to have a capital Y 00:07:32.300 --> 00:07:35.610 and two lowercase r's in order to be wrinkled. 00:07:35.610 --> 00:07:39.310 So you have at least one capital Y and two lowercase r's, 00:07:39.310 --> 00:07:42.860 and least one capital Y and two lowercase r's. 00:07:42.860 --> 00:07:47.170 Let's see, at least one capital Y and two lowercase r's. 00:07:47.170 --> 00:07:50.390 It looks like you have exactly three of them, 00:07:50.390 --> 00:07:51.550 that phenotype. 00:07:51.550 --> 00:07:54.390 And then what about the other way around? 00:07:54.390 --> 00:07:58.410 What if we are looking for, I'll do it in this green color, 00:07:58.410 --> 00:08:03.410 green plus round? 00:08:03.840 --> 00:08:06.180 How many of them exhibit that phenotype? 00:08:06.180 --> 00:08:07.450 Well, to be green and round, 00:08:07.450 --> 00:08:09.590 you have to have two lowercase y's, 00:08:09.590 --> 00:08:12.370 and you have to have at least one capital R. 00:08:12.370 --> 00:08:15.950 So this would be green and round. 00:08:15.950 --> 00:08:18.090 This would be green and round. 00:08:18.090 --> 00:08:20.270 And then this would be green and round as well. 00:08:20.270 --> 00:08:21.800 So you have three of those. 00:08:21.800 --> 00:08:23.300 And then how many of them are going 00:08:23.300 --> 00:08:25.430 to be both green and wrinkled? 00:08:25.430 --> 00:08:28.290 Well, I think you see that one scenario over here 00:08:28.290 --> 00:08:30.020 that is both green and wrinkled, 00:08:30.020 --> 00:08:33.450 having that homozygous recessive phenotype. 00:08:33.450 --> 00:08:35.290 And so if you were to do this many times, 00:08:35.290 --> 00:08:37.870 you'd expect the ratios between these various phenotypes 00:08:37.870 --> 00:08:41.400 to be nine to three to three to one. 00:08:41.400 --> 00:08:43.000 And when Mendel and many other people 00:08:43.000 --> 00:08:45.180 since Mendel have done these types of experiments, 00:08:45.180 --> 00:08:47.060 they have seen that statistically, 00:08:47.060 --> 00:08:49.870 this is what you see in that F2 generation. 00:08:49.870 --> 00:08:51.820 Now, you're unlikely to get exactly a nine 00:08:51.820 --> 00:08:53.590 to three to three to one ratio. 00:08:53.590 --> 00:08:54.950 It's all probabilistic. 00:08:54.950 --> 00:08:58.650 Every one of these 16 scenarios are equally likely, 00:08:58.650 --> 00:09:00.530 so you would expect this nine to three 00:09:00.530 --> 00:09:01.950 to three to one ratio, 00:09:01.950 --> 00:09:04.870 but you're not always going to get that exact ratio. 00:09:04.870 --> 00:09:06.970 You'll probably get something close to it.

Energy at the microscopic scale

https://www.youtube.com/watch?v=VeueBB5Abt4

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https://www.youtube.com/api/timedtext?v=VeueBB5Abt4&ei=3FWUZeSaDLWvvdIPzsKD-Aw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=1C2CE84C6708A498AA6DA31CAE0C6CA6B1E8CA4E.407CDB4CF072C219B270648D8A2B351FB83A9F8D&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.400 --> 00:00:01.560 - [Instructor] Welcome. 00:00:01.560 --> 00:00:04.300 Today, we're going to take a look at forms of energy, 00:00:04.300 --> 00:00:07.270 such as kinetic, electrical, thermal, 00:00:07.270 --> 00:00:09.370 gravitational, potential energy. 00:00:09.370 --> 00:00:11.300 It turns out when you start thinking 00:00:11.300 --> 00:00:15.230 about energy on smaller scales, or the microscopic level, 00:00:15.230 --> 00:00:17.930 all of these forms of energy are basically two things: 00:00:17.930 --> 00:00:21.380 one, kinetic energy, particles moving around, 00:00:21.380 --> 00:00:25.170 and two, potential energy, energy stored by a field, 00:00:25.170 --> 00:00:27.713 such as electric, magnetic or gravitational. 00:00:28.800 --> 00:00:30.980 Let's start with a deceivingly simple example 00:00:30.980 --> 00:00:33.693 and explore properties of energy as we zoom in. 00:00:36.830 --> 00:00:38.563 I'm drawing a glass of water. 00:00:39.450 --> 00:00:40.900 This water has a temperature. 00:00:41.830 --> 00:00:44.310 I can warm up the water by adding energy 00:00:44.310 --> 00:00:45.813 from an electric stove, 00:00:46.710 --> 00:00:49.850 or cool down the water by putting it in the fridge, 00:00:49.850 --> 00:00:52.920 thus removing energy in the liquid water. 00:00:52.920 --> 00:00:55.210 From this example, we can see that temperature 00:00:55.210 --> 00:00:56.830 is related to energy. 00:00:56.830 --> 00:01:00.083 But how is it related to kinetic or potential energy? 00:01:01.010 --> 00:01:02.620 Hmm. 00:01:02.620 --> 00:01:05.110 This big picture, the macroscopic scale, 00:01:05.110 --> 00:01:07.500 allows us to look at energy in terms of temperature. 00:01:07.500 --> 00:01:09.830 But let's go to the microscopic or small-scale 00:01:09.830 --> 00:01:12.293 to get a better picture of the physics at play. 00:01:14.330 --> 00:01:16.130 As we look inside this liquid, 00:01:16.130 --> 00:01:19.103 we see lots of water molecules moving around. 00:01:20.070 --> 00:01:21.740 Zoom, zoom, zoom. 00:01:21.740 --> 00:01:23.690 Well, they don't actually make a sound. 00:01:25.950 --> 00:01:28.650 The average speed at which these molecules move 00:01:28.650 --> 00:01:31.160 is related to their kinetic energy and its temperature. 00:01:31.160 --> 00:01:32.543 This is thermal motion. 00:01:34.220 --> 00:01:38.700 If we zoom in again, we can look into the strong chemical 00:01:38.700 --> 00:01:40.710 bonds within molecules. 00:01:40.710 --> 00:01:44.130 Here, the individual atoms, in this case for water, 00:01:44.130 --> 00:01:46.420 is hydrogen and oxygen. 00:01:46.420 --> 00:01:49.160 They can vibrate back and forth and rotate, 00:01:49.160 --> 00:01:50.883 so they also have kinetic energy. 00:01:51.740 --> 00:01:52.900 Let's look at another example 00:01:52.900 --> 00:01:55.773 between macro and microscale energy interactions. 00:01:56.710 --> 00:01:59.660 When you burn something, think of a fire. 00:01:59.660 --> 00:02:01.400 A chemical reaction takes place, 00:02:01.400 --> 00:02:03.310 and it releases a lot of energy. 00:02:03.310 --> 00:02:04.960 How does it do that? 00:02:04.960 --> 00:02:08.220 Let's look at the microscopic scale to find out. 00:02:08.220 --> 00:02:10.520 In this example, I'm gonna burn methane gas, 00:02:10.520 --> 00:02:12.530 and the chemical reaction that takes place 00:02:12.530 --> 00:02:17.380 is methane, CH4, and oxygen, O2. 00:02:17.380 --> 00:02:20.910 They rearrange to create water, H2O, 00:02:20.910 --> 00:02:24.103 and carbon dioxide, CO2, plus energy. 00:02:25.750 --> 00:02:26.860 Before the reaction, 00:02:26.860 --> 00:02:28.720 there is a greater chemical potential energy 00:02:28.720 --> 00:02:29.730 than afterwards. 00:02:29.730 --> 00:02:30.563 But don't worry. 00:02:30.563 --> 00:02:32.970 Energy is still conserved because that potential energy 00:02:32.970 --> 00:02:35.383 is converted to kinetic energy and radiation. 00:02:37.330 --> 00:02:40.600 What is the source of chemical potential energy? 00:02:40.600 --> 00:02:43.040 At this level, we can think of individual bonds 00:02:43.040 --> 00:02:44.920 between atoms storing energy 00:02:44.920 --> 00:02:47.670 so that energy can be absorbed or released 00:02:47.670 --> 00:02:49.803 as bonds are broken and reformed. 00:02:50.750 --> 00:02:54.280 But where does the energy and chemical bonds come from? 00:02:54.280 --> 00:02:55.793 We need to zoom in again. 00:02:57.260 --> 00:02:58.410 In a single atom, 00:02:58.410 --> 00:03:02.530 there's a nucleus that contains protons and neutrons 00:03:02.530 --> 00:03:05.200 and overall has a positive charge. 00:03:05.200 --> 00:03:07.113 This creates an electromagnetic field. 00:03:07.950 --> 00:03:10.600 The interaction of other charged particles, 00:03:10.600 --> 00:03:12.430 like negatively charged electrons 00:03:12.430 --> 00:03:15.130 relatively far away from the nucleus, 00:03:15.130 --> 00:03:18.140 with this field provide potential energy. 00:03:18.140 --> 00:03:19.950 You can think of this electric potential energy 00:03:19.950 --> 00:03:22.480 as the same kind of concept as a potential energy 00:03:22.480 --> 00:03:24.583 of a mass and a gravitational field. 00:03:25.700 --> 00:03:28.550 Zooming back out, each molecule has its own 00:03:28.550 --> 00:03:30.640 particular configuration of charged particles 00:03:30.640 --> 00:03:33.700 within the electromagnetic fields, right? 00:03:33.700 --> 00:03:36.603 This means it has an associated potential energy. 00:03:37.570 --> 00:03:39.960 As we've seen, this chemical potential energy 00:03:39.960 --> 00:03:42.513 is the result of energy stored in fields. 00:03:43.790 --> 00:03:46.210 Okay, so we've covered electrical, chemical, 00:03:46.210 --> 00:03:47.250 thermal types of energy, 00:03:47.250 --> 00:03:49.300 but there's other forms out there, right? 00:03:50.220 --> 00:03:52.600 For instance, what about sound waves? 00:03:52.600 --> 00:03:56.023 Here's a speaker and an ear. 00:03:57.890 --> 00:04:00.070 The energy in sound waves is transferred 00:04:00.070 --> 00:04:01.060 through the vibrations, 00:04:01.060 --> 00:04:03.840 a back and forth motion of molecules in the air. 00:04:03.840 --> 00:04:07.370 Another example could be elastic potential energy 00:04:07.370 --> 00:04:09.470 or the energy stored in a spring. 00:04:09.470 --> 00:04:11.010 At the microscopic level, 00:04:11.010 --> 00:04:12.750 as you stretch the spring, 00:04:12.750 --> 00:04:14.490 see the hand stretching the spring? 00:04:14.490 --> 00:04:17.140 The atoms are being pulled out of their equilibrium position 00:04:17.140 --> 00:04:20.330 within a solid and thus gain potential energy 00:04:20.330 --> 00:04:21.610 from the electromagnetic force 00:04:21.610 --> 00:04:23.670 that holds the solid together. 00:04:23.670 --> 00:04:24.810 How neat is this? 00:04:24.810 --> 00:04:27.170 We can describe all these energies 00:04:27.170 --> 00:04:29.363 as just kinetic or potential. 00:04:30.370 --> 00:04:32.470 So there's one more microscopic form of energy 00:04:32.470 --> 00:04:33.730 that we need to talk about, 00:04:33.730 --> 00:04:36.350 and it might seem a little complicated at first. 00:04:36.350 --> 00:04:38.010 Let's go back to the combustion example 00:04:38.010 --> 00:04:39.780 we talked about earlier. 00:04:39.780 --> 00:04:43.860 As I said before, this process releases radiant energy. 00:04:43.860 --> 00:04:45.210 We can see burning objects. 00:04:45.210 --> 00:04:46.430 They glow brightly. 00:04:46.430 --> 00:04:48.690 We can also put our hand near the burning object. 00:04:48.690 --> 00:04:49.620 Don't touch it! 00:04:49.620 --> 00:04:51.690 And feel the radiant heat. 00:04:51.690 --> 00:04:54.653 This radiation that's emitted carries energy with it. 00:04:55.650 --> 00:04:58.630 So how do we explain the radiant energy? 00:04:58.630 --> 00:05:00.590 Does it fit it into one of these two categories, 00:05:00.590 --> 00:05:02.460 either kinetic or potential energy 00:05:02.460 --> 00:05:03.810 that we've been discussing? 00:05:04.690 --> 00:05:08.500 Well, it turns out it sort of fits into both groups. 00:05:08.500 --> 00:05:12.500 Whoa, electromagnetic radiation, such as light, 00:05:12.500 --> 00:05:14.830 can be modeled in a couple of different ways, 00:05:14.830 --> 00:05:17.410 which we'll go into more detail in another video. 00:05:17.410 --> 00:05:19.320 But one way to model the light 00:05:19.320 --> 00:05:22.423 is as a wave of electric and magnetic fields. 00:05:23.830 --> 00:05:26.240 Another way to think about light 00:05:26.240 --> 00:05:28.800 is being made up of particles called photons. 00:05:28.800 --> 00:05:31.410 In this instance, the particles are carrying the energy. 00:05:31.410 --> 00:05:32.850 So with both of these models, 00:05:32.850 --> 00:05:34.480 radiant energy can be explained 00:05:34.480 --> 00:05:36.550 by the same microscopic interactions 00:05:36.550 --> 00:05:39.110 that cause the other forms of energy. 00:05:39.110 --> 00:05:41.960 In conclusion, we can see energy at the macroscopic scale, 00:05:41.960 --> 00:05:44.240 like temperature or light being emitted. 00:05:44.240 --> 00:05:46.500 However, we must look at the microscale 00:05:46.500 --> 00:05:49.010 to observe the different forms of energy 00:05:49.010 --> 00:05:51.550 that we experience are really just the result 00:05:51.550 --> 00:05:54.750 of kinetic and potential energy of particles. 00:05:54.750 --> 00:05:55.883 How cool!

Energy and fields

https://www.youtube.com/watch?v=rMNJfR168_U

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https://www.youtube.com/api/timedtext?v=rMNJfR168_U&ei=3FWUZbStDYmpp-oP-7iHkAs&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=7EAFEA599A0FFA0F55AE8BFFCECC87B714FB7CED.A6E336A330632F9BAC43E8B6D5616183FEC482B2&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.410 --> 00:00:02.850 - [Instructor] In previous videos we have already defined 00:00:02.850 --> 00:00:04.920 or provided a definition for energy 00:00:04.920 --> 00:00:07.780 as the capacity to do work. 00:00:07.780 --> 00:00:11.870 We have also talked about the notion of a field. 00:00:11.870 --> 00:00:14.460 We have talked about things like an electric field 00:00:14.460 --> 00:00:16.710 or a gravitational field. 00:00:16.710 --> 00:00:19.080 And these are really mental constructs 00:00:19.080 --> 00:00:23.700 that we have produced to explain force at a distance. 00:00:23.700 --> 00:00:26.910 For example, if I have a planet here 00:00:26.910 --> 00:00:30.830 and then I have some other object here that has some mass, 00:00:30.830 --> 00:00:35.830 we know that these are going to exert forces on each other, 00:00:36.030 --> 00:00:38.830 and actually equal and opposite forces on each other. 00:00:38.830 --> 00:00:41.680 And scientists said, well, they're not touching each other. 00:00:41.680 --> 00:00:44.130 How are they exerting forces on each other? 00:00:44.130 --> 00:00:46.660 And so they introduced this notion of a field 00:00:46.660 --> 00:00:49.480 that each of these objects produce, 00:00:49.480 --> 00:00:52.230 a gravitational field of sorts. 00:00:52.230 --> 00:00:53.980 Now Einstein came later and said, 00:00:53.980 --> 00:00:56.530 well, actually they're warping space, time, 00:00:56.530 --> 00:00:57.363 et cetera, et cetera. 00:00:57.363 --> 00:00:59.760 But a field is one way to think 00:00:59.760 --> 00:01:02.860 about how they're able to induce a force 00:01:02.860 --> 00:01:04.670 so to speak in each other. 00:01:04.670 --> 00:01:08.060 Similarly, if you have two electric charges, 00:01:08.060 --> 00:01:11.650 let's say you have two negative point charges like that. 00:01:11.650 --> 00:01:14.370 We know that they push away on each other, 00:01:14.370 --> 00:01:16.850 that like charges repel. 00:01:16.850 --> 00:01:18.110 Well, they're not touching each other. 00:01:18.110 --> 00:01:20.757 How do they know to have a force being applied 00:01:20.757 --> 00:01:23.160 to them in opposite directions? 00:01:23.160 --> 00:01:25.140 So once again, there's this idea 00:01:25.140 --> 00:01:27.460 that each of these produces a field, 00:01:27.460 --> 00:01:30.830 the other one is in the other electric charges field. 00:01:30.830 --> 00:01:35.090 And then that field somehow applies that force 00:01:35.090 --> 00:01:37.720 or makes that force happen to the other thing. 00:01:37.720 --> 00:01:39.840 Notice, the field is a useful concept 00:01:39.840 --> 00:01:41.490 to predict what will happen 00:01:41.490 --> 00:01:43.327 and to quantify how it could happen. 00:01:43.327 --> 00:01:45.600 But it really is just something 00:01:45.600 --> 00:01:48.794 in our minds to make sense of the universe. 00:01:48.794 --> 00:01:50.180 So with that out of the way, 00:01:50.180 --> 00:01:52.410 let's look at this water wheel right over here. 00:01:52.410 --> 00:01:55.290 You can see that the water comes down from here 00:01:55.290 --> 00:01:56.610 and then it falls. 00:01:56.610 --> 00:01:58.460 And as it falls, it pushes, 00:01:58.460 --> 00:02:00.640 it fills up these things right over here 00:02:00.640 --> 00:02:01.840 which then pushes it down. 00:02:01.840 --> 00:02:03.800 And then the whole wheel turns. 00:02:03.800 --> 00:02:07.470 And then that wheel could do work. 00:02:07.470 --> 00:02:09.180 Actually could do useful work. 00:02:09.180 --> 00:02:12.086 In a physics context, not all work is necessarily useful. 00:02:12.086 --> 00:02:14.780 But this could actually do useful work. 00:02:14.780 --> 00:02:18.660 So what I wanna think about is two different drops of water. 00:02:18.660 --> 00:02:19.870 I have a drop of water here, 00:02:19.870 --> 00:02:21.445 maybe the same drop of water. 00:02:21.445 --> 00:02:26.310 When it's up here versus once it has gone all the way down 00:02:26.310 --> 00:02:28.030 and has been dumped into what I'm assuming 00:02:28.030 --> 00:02:31.150 is a stream down here. 00:02:31.150 --> 00:02:35.380 Now, which one has a higher capacity to do work? 00:02:35.380 --> 00:02:37.280 Pause this video and think about that. 00:02:39.490 --> 00:02:42.880 Well, I just told you that when the water drop is up here, 00:02:42.880 --> 00:02:45.650 it has the capacity as it falls 00:02:45.650 --> 00:02:48.080 because of the gravitational field, 00:02:48.080 --> 00:02:49.850 which is pulling down on it. 00:02:49.850 --> 00:02:52.360 And by the way, if the gravitational field 00:02:52.360 --> 00:02:54.020 is pulling down on the water drop, 00:02:54.020 --> 00:02:57.328 that water drop is also pulling up on earth. 00:02:57.328 --> 00:02:59.780 But this gravitational field of earth 00:02:59.780 --> 00:03:02.000 is pulling down on that water drop. 00:03:02.000 --> 00:03:03.310 And because of that, 00:03:03.310 --> 00:03:05.570 if the water drop is not supported 00:03:05.570 --> 00:03:07.947 it can actually do work in this example 00:03:07.947 --> 00:03:12.700 on its way to being in this position right over here. 00:03:12.700 --> 00:03:14.050 Now this position right over here, 00:03:14.050 --> 00:03:16.640 in theory, it could maybe still do work. 00:03:16.640 --> 00:03:19.050 Maybe there's a cliff right over here 00:03:19.050 --> 00:03:20.640 and it can continue to pour down. 00:03:20.640 --> 00:03:22.890 But the water drop up here clearly has the capacity 00:03:22.890 --> 00:03:25.330 to do more work because it has the potential work 00:03:25.330 --> 00:03:28.210 that it can do from going from here to here. 00:03:28.210 --> 00:03:29.930 And then obviously it could then continue 00:03:29.930 --> 00:03:33.270 to do any work that this position would allow it to have. 00:03:33.270 --> 00:03:35.887 So we would say that this water drop 00:03:35.887 --> 00:03:38.289 by virtue of its position, 00:03:38.289 --> 00:03:42.180 has a higher capacity to do work and has more energy. 00:03:42.180 --> 00:03:43.840 And what is a form of that energy? 00:03:43.840 --> 00:03:47.100 Well, in this case, it's gravitational potential energy. 00:03:47.100 --> 00:03:49.930 It's energy that is stored. 00:03:49.930 --> 00:03:51.100 And I put that in quotes 00:03:51.100 --> 00:03:52.380 because it's not like you're going to be able 00:03:52.380 --> 00:03:54.940 to open that water drop and all of a sudden see energy, 00:03:54.940 --> 00:03:58.500 but it's energy that's stored by virtue of its position. 00:03:58.500 --> 00:03:59.840 Another way to think about it 00:03:59.840 --> 00:04:01.890 is instead of imagining that the energy 00:04:01.890 --> 00:04:03.340 is stored in the water drop, 00:04:03.340 --> 00:04:05.520 and it is really happening in our minds, 00:04:05.520 --> 00:04:08.770 is to say that that energy is stored in the field. 00:04:08.770 --> 00:04:11.810 In this case, this gravitational field. 00:04:11.810 --> 00:04:15.070 Now the gravitational field is pulling on this water drop. 00:04:15.070 --> 00:04:16.900 So the direction of motion 00:04:16.900 --> 00:04:19.950 would actually reduce the energy in the field. 00:04:19.950 --> 00:04:23.550 So if we just let things happen, earth's gravitational field 00:04:23.550 --> 00:04:25.810 is going to pull on this water drop. 00:04:25.810 --> 00:04:28.020 And actually that water drop has a gravitational field 00:04:28.020 --> 00:04:29.480 that's going to pull up on earth, 00:04:29.480 --> 00:04:32.420 but as that water drop gets pulled down, 00:04:32.420 --> 00:04:34.580 the total amount of energy stored 00:04:34.580 --> 00:04:36.640 in the field is going to go down. 00:04:36.640 --> 00:04:37.790 Now what happened to that energy? 00:04:37.790 --> 00:04:40.280 That energy gets transferred out of the field 00:04:40.280 --> 00:04:42.470 into kinetic energy of this wheel, 00:04:42.470 --> 00:04:44.970 which could then be transferred into other things.

What is energy?

https://www.youtube.com/watch?v=zKb3QRyIDnM

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https://www.youtube.com/api/timedtext?v=zKb3QRyIDnM&ei=3FWUZeGQDPe1vdIPot2CsA0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=8B2C683DA7166F5147C682D4FE3DF45C52E632EB.16A03906BA077B0ECB888F25E80DBE41FB8709B6&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.120 --> 00:00:02.160 - [Instructor] Energy is a word we hear all the time 00:00:02.160 --> 00:00:03.560 in seemingly different contexts, 00:00:03.560 --> 00:00:04.900 almost every single day. 00:00:04.900 --> 00:00:07.170 We hear about renewable energy on the news, 00:00:07.170 --> 00:00:08.630 and particularly in the winter, 00:00:08.630 --> 00:00:10.400 we hear people talking about their energy bills, 00:00:10.400 --> 00:00:11.500 because they're worried 00:00:11.500 --> 00:00:13.830 about how much it's going to cost to heat their homes. 00:00:13.830 --> 00:00:14.950 So this brings about the question 00:00:14.950 --> 00:00:18.530 of what is energy that we can talk about it so often 00:00:18.530 --> 00:00:20.890 and in seemingly such different ways? 00:00:20.890 --> 00:00:23.590 So in physics, we actually have a specific definition 00:00:23.590 --> 00:00:25.040 of what energy is, 00:00:25.040 --> 00:00:26.470 and you'll see it's really not that different 00:00:26.470 --> 00:00:29.060 from how we talk about energy day to day. 00:00:29.060 --> 00:00:33.130 Energy in physics is defined as the ability to do work. 00:00:33.130 --> 00:00:35.800 We can't talk about energy without talking about work, 00:00:35.800 --> 00:00:37.890 so we should probably define that right now, 00:00:37.890 --> 00:00:39.550 because work is another one of these words 00:00:39.550 --> 00:00:41.730 that we use an awful lot, but once again, 00:00:41.730 --> 00:00:44.193 physics has a specific definition for it. 00:00:45.130 --> 00:00:47.010 In physics, work is performed 00:00:47.010 --> 00:00:50.120 when you apply force over a distance. 00:00:50.120 --> 00:00:52.170 We can actually write this as an equation, 00:00:52.170 --> 00:00:56.140 W equals F times d, where W is work, F is force, 00:00:56.140 --> 00:00:58.323 and d is distance or displacement. 00:00:59.260 --> 00:01:01.130 If you've ever moved a box, a suitcase, 00:01:01.130 --> 00:01:03.780 or really any object in your room across the floor 00:01:03.780 --> 00:01:06.590 just to get it out of your way, you performed work. 00:01:06.590 --> 00:01:09.170 You had to apply force to that box to move it 00:01:09.170 --> 00:01:11.490 whatever distance it took to get out of your way. 00:01:11.490 --> 00:01:13.030 If it was a short distance, 00:01:13.030 --> 00:01:14.380 you can see from the equation, 00:01:14.380 --> 00:01:16.330 that that's going to be less work 00:01:16.330 --> 00:01:18.650 than if you have to move it across your entire room, 00:01:18.650 --> 00:01:21.070 down the hall, into another room. 00:01:21.070 --> 00:01:23.240 And in order to perform this work to move the box 00:01:23.240 --> 00:01:25.910 out of your way, you had to have energy available. 00:01:25.910 --> 00:01:27.710 That energy is enabling you to do the work, 00:01:27.710 --> 00:01:29.570 because you're going to transfer the energy 00:01:29.570 --> 00:01:32.930 from yourself to that box in order to move it. 00:01:32.930 --> 00:01:35.670 So, another way that we can think about work and energy 00:01:35.670 --> 00:01:38.240 is that the change in energy of the system, 00:01:38.240 --> 00:01:39.700 in this case, you and the box, 00:01:39.700 --> 00:01:42.250 is actually equal to the work done. 00:01:42.250 --> 00:01:43.990 When we define energy this way, 00:01:43.990 --> 00:01:46.850 it allows us to do a lot of interesting things. 00:01:46.850 --> 00:01:49.800 We've set up a way to measure and calculate energy, 00:01:49.800 --> 00:01:52.890 so it's actually a quantifiable property. 00:01:52.890 --> 00:01:54.870 Let's go back to the example of a box, 00:01:54.870 --> 00:01:57.480 and let's say that instead of just trying to move a box 00:01:57.480 --> 00:01:58.940 out of your way in your room, 00:01:58.940 --> 00:02:01.010 you're actually going to pack up everything in your room, 00:02:01.010 --> 00:02:02.120 because you're going to move 00:02:02.120 --> 00:02:03.660 to a completely different house. 00:02:03.660 --> 00:02:06.000 And now you have 10 boxes to move. 00:02:06.000 --> 00:02:09.320 We can actually calculate the energy required 00:02:09.320 --> 00:02:12.000 to move all of those boxes. 00:02:12.000 --> 00:02:14.500 You might be saying to yourself, "Wait a minute. 00:02:14.500 --> 00:02:16.230 I'm now moving 10 boxes. 00:02:16.230 --> 00:02:18.030 That actually sounds kind of tiring. 00:02:18.900 --> 00:02:20.150 And if I'm getting tired, 00:02:20.150 --> 00:02:22.550 does that mean I'm actually losing energy?" 00:02:22.550 --> 00:02:25.560 It turns out that energy is coming from somewhere, 00:02:25.560 --> 00:02:27.750 and in this case, it's going to come from food. 00:02:27.750 --> 00:02:29.440 So as you're moving these boxes, 00:02:29.440 --> 00:02:31.310 you may find yourself getting hungry, 00:02:31.310 --> 00:02:33.090 so you should probably grab a snack, 00:02:33.090 --> 00:02:35.620 something like, I don't know, an apple, 00:02:35.620 --> 00:02:37.920 let's pretend that's what I've drawn there, 00:02:37.920 --> 00:02:39.730 so that you can get more energy 00:02:39.730 --> 00:02:41.913 to move the rest of those 10 boxes. 00:02:43.070 --> 00:02:46.530 And you might be thinking now, "Wait a minute. 00:02:46.530 --> 00:02:50.290 This energy coming from food to me seems different 00:02:50.290 --> 00:02:52.970 than energy between me and moving a box." 00:02:52.970 --> 00:02:56.060 And that's because you can actually see the box moving, 00:02:56.060 --> 00:02:58.410 which brings us to our next point, 00:02:58.410 --> 00:03:00.130 energy comes in various forms 00:03:00.130 --> 00:03:02.330 and they don't all look the same. 00:03:02.330 --> 00:03:04.740 We have equations to quantify the energy 00:03:04.740 --> 00:03:05.810 of these various forms, 00:03:05.810 --> 00:03:08.300 and we'll talk about those in another video, 00:03:08.300 --> 00:03:10.610 but the key here is that energy 00:03:10.610 --> 00:03:12.450 can transfer between objects 00:03:12.450 --> 00:03:14.750 and it can also convert between different forms, 00:03:14.750 --> 00:03:18.240 such as when you eat and get energy from that apple, 00:03:18.240 --> 00:03:21.510 and then you use that energy to move a box. 00:03:21.510 --> 00:03:25.530 So to summarize, energy is the ability to do work. 00:03:25.530 --> 00:03:28.810 Work is done when you apply force across a distance, 00:03:28.810 --> 00:03:31.090 and we can write that as an equation. 00:03:31.090 --> 00:03:33.010 And because we can calculate the energy 00:03:33.010 --> 00:03:35.040 of a system using equations, 00:03:35.040 --> 00:03:37.883 we now know that energy is a quantifiable property.

Calculating gravitational potential energy

https://www.youtube.com/watch?v=VafUJehX48w

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https://www.youtube.com/api/timedtext?v=VafUJehX48w&ei=3FWUZcXnLIDoxN8PgpamyAs&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=EF4CFC08AAC10E86F83383D048C05682CB1AF526.D3710B0541DD23313F681BD4985577A0C1804166&key=yt8&lang=en&name=Default&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:01.260 --> 00:00:02.400 - [Instructor] In previous videos, 00:00:02.400 --> 00:00:05.640 we have introduced the idea of energy 00:00:05.640 --> 00:00:08.770 as the capacity to do work 00:00:08.770 --> 00:00:11.820 and we have talked about multiple types of energies. 00:00:11.820 --> 00:00:14.830 We've talked about kinetic energy, energy due to motion. 00:00:14.830 --> 00:00:17.330 We've talked about potential energy, 00:00:17.330 --> 00:00:21.150 which is energy by virtue of position. 00:00:21.150 --> 00:00:23.570 And when we're talking about potential energy, 00:00:23.570 --> 00:00:27.130 we're talking about it relative to some other position. 00:00:27.130 --> 00:00:29.150 And in particular, in this video, 00:00:29.150 --> 00:00:31.980 we're going to talk about gravitational potential energy, 00:00:31.980 --> 00:00:34.300 which is potential energy due to position 00:00:34.300 --> 00:00:36.660 in a gravitational field. 00:00:36.660 --> 00:00:41.000 So let's say that this is the surface of the earth. 00:00:41.000 --> 00:00:46.000 Let's say that I have a five kilogram mass right over here, 00:00:47.650 --> 00:00:52.650 and let's say that it is 10 meters above 00:00:53.840 --> 00:00:55.300 the surface of the earth. 00:00:55.300 --> 00:00:58.780 My question to you is how much more potential energy 00:00:58.780 --> 00:01:00.400 does it have in this position 00:01:00.400 --> 00:01:03.210 than when it is in this position, 00:01:03.210 --> 00:01:06.310 when it is sitting on the surface of the earth, 00:01:06.310 --> 00:01:08.430 10 meters lower? 00:01:08.430 --> 00:01:10.630 Pause the video and try to think about that. 00:01:12.610 --> 00:01:15.350 All right, now let's work on this together. 00:01:15.350 --> 00:01:19.300 So our gravitational potential energy is going to be equal 00:01:19.300 --> 00:01:22.950 to our mass times lowercase g, 00:01:22.950 --> 00:01:25.210 which you can view as the constant 00:01:25.210 --> 00:01:28.660 for earth's gravitational field near the surface of earth. 00:01:28.660 --> 00:01:31.060 And the reason why I say near the surface of earth 00:01:31.060 --> 00:01:33.550 is as you get further and further from earth, 00:01:33.550 --> 00:01:35.330 this thing could actually change, 00:01:35.330 --> 00:01:37.030 but near the surface of the earth, 00:01:37.030 --> 00:01:39.460 we assume that it is roughly constant, 00:01:39.460 --> 00:01:43.560 and then you multiply that times your height. 00:01:43.560 --> 00:01:46.070 So calculating this is pretty straightforward 00:01:46.070 --> 00:01:48.270 as long as you know what g is. 00:01:48.270 --> 00:01:53.270 G, we can approximate it as 9.8 meters per second squared. 00:01:55.240 --> 00:01:57.310 So when you multiply all of this out, 00:01:57.310 --> 00:01:59.970 this is going to be equal to your mass, 00:01:59.970 --> 00:02:01.983 which is five kilograms, 00:02:03.240 --> 00:02:06.310 times the gravitational field constant, 00:02:06.310 --> 00:02:09.980 so times 9.8 meters per second squared, 00:02:14.400 --> 00:02:15.560 times your height, 00:02:15.560 --> 00:02:18.010 which in this situation is 10 meters, 00:02:18.010 --> 00:02:20.750 so times 10 meters. 00:02:20.750 --> 00:02:25.653 And so this is going to be equal to five times 9.8 is 49, 00:02:26.530 --> 00:02:30.110 times 10 is 490. 00:02:30.110 --> 00:02:31.453 We have kilograms, 00:02:32.540 --> 00:02:34.810 and then we have meters times meters, 00:02:34.810 --> 00:02:39.280 so times meters squared per second squared. 00:02:39.280 --> 00:02:41.160 And these might seem like strange units, 00:02:41.160 --> 00:02:42.510 but you might recognize this 00:02:42.510 --> 00:02:45.960 as also the units of force times distance, 00:02:45.960 --> 00:02:49.420 which we could also express in terms of joules. 00:02:49.420 --> 00:02:52.340 So this is 490 joules, 00:02:52.340 --> 00:02:57.120 which is our units both for energy and our unit for work. 00:02:57.120 --> 00:03:00.320 Now, let's make sure that this is intuitive sense. 00:03:00.320 --> 00:03:01.680 Well, one way to think about it is 00:03:01.680 --> 00:03:05.990 how much work would it take to go from here to here? 00:03:05.990 --> 00:03:10.380 Well, you're going to be lifting it a distance of 10 meters, 00:03:10.380 --> 00:03:13.180 and as you're lifting it a distance of 10 meters, 00:03:13.180 --> 00:03:15.410 what is the force you're going to have to apply? 00:03:15.410 --> 00:03:16.640 Well, the force you're going to have to apply 00:03:16.640 --> 00:03:18.250 is going to be the weight of the object. 00:03:18.250 --> 00:03:21.430 The weight is its mass times the gravitational field. 00:03:21.430 --> 00:03:24.570 So in order to put it in that position from the ground, 00:03:24.570 --> 00:03:27.320 you're going to have to do its weight times the height, 00:03:27.320 --> 00:03:30.120 or 490 joules of work. 00:03:30.120 --> 00:03:33.930 And so you can do 490 joules of work to get it there 00:03:33.930 --> 00:03:35.300 and then you can think about it 00:03:35.300 --> 00:03:37.830 as that energy being stored this way. 00:03:37.830 --> 00:03:39.460 And now it can then do that work. 00:03:39.460 --> 00:03:40.800 How could it do that work? 00:03:40.800 --> 00:03:42.750 Well, there's a bunch of ways you could do it. 00:03:42.750 --> 00:03:47.750 You could have this attached to maybe a pulley of some kind 00:03:47.850 --> 00:03:51.650 and then if it had another weight right over here, 00:03:51.650 --> 00:03:55.420 and let's just, for simplicity, assume it has the same mass, 00:03:55.420 --> 00:03:59.000 well, if you let this first purple mass go, 00:03:59.000 --> 00:04:00.370 it's going to go down. 00:04:00.370 --> 00:04:02.430 And if you assume that this pulley 00:04:02.430 --> 00:04:04.010 is completely frictionless, 00:04:04.010 --> 00:04:07.130 this mass is going to be lifted by 10 meters. 00:04:07.130 --> 00:04:09.750 And so if you have a five kilogram mass 00:04:09.750 --> 00:04:13.470 that is lifted by 10 meters in Earth's gravitational field, 00:04:13.470 --> 00:04:15.060 near the surface of the earth, 00:04:15.060 --> 00:04:18.270 you would have just done 490 joules of work. 00:04:18.270 --> 00:04:19.890 So hopefully this makes sense 00:04:19.890 --> 00:04:21.490 why you're just really taking the weight 00:04:21.490 --> 00:04:23.040 of the object times its height. 00:04:23.040 --> 00:04:24.700 And hopefully it also makes sense 00:04:24.700 --> 00:04:28.100 that it then has the capacity to do that amount of work. 00:04:28.100 --> 00:04:31.953 And in this case, we said relative to sitting on the ground.

Calculating kinetic energy

https://www.youtube.com/watch?v=3FIFHRrut2s

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https://www.youtube.com/api/timedtext?v=3FIFHRrut2s&ei=3FWUZcL3LN3ixN8P_6C02A0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=5CA164B51A6044FE01594A6FBB1DA6E48D8B0C13.11122D7DA475A85922F46E10C1D8CDB36D6F0865&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.560 --> 00:00:01.410 - [Instructor] In this video, 00:00:01.410 --> 00:00:03.420 we're gonna talk about kinetic energy 00:00:03.420 --> 00:00:06.540 and we're also gonna think about how to calculate it. 00:00:06.540 --> 00:00:09.020 So you can already imagine based on the word kinetic, 00:00:09.020 --> 00:00:10.780 which is referring to motion 00:00:10.780 --> 00:00:13.560 that this is the energy that an object has 00:00:13.560 --> 00:00:15.650 by virtue of its motion. 00:00:15.650 --> 00:00:17.100 And when we talk about energy, 00:00:17.100 --> 00:00:20.570 we're talking about its capacity to do work. 00:00:20.570 --> 00:00:24.390 So just based on that early definition of kinetic energy, 00:00:24.390 --> 00:00:26.300 which of these two running backs 00:00:26.300 --> 00:00:28.920 do you think has more kinetic energy, 00:00:28.920 --> 00:00:33.010 this gentleman on the left whose mass is a 100 kilograms 00:00:33.010 --> 00:00:36.290 and who is traveling at a speed of two meters per second, 00:00:36.290 --> 00:00:37.470 or the gentleman on the right, 00:00:37.470 --> 00:00:39.660 who has a mass of 25 kilograms 00:00:39.660 --> 00:00:43.120 and who's traveling with a speed of four meters per second? 00:00:43.120 --> 00:00:45.020 Pause this video and think about that. 00:00:46.240 --> 00:00:48.550 All right, now let's think about this together. 00:00:48.550 --> 00:00:50.530 So I'm first just gonna give you the formula 00:00:50.530 --> 00:00:53.620 for kinetic energy, but then we are going to derive it. 00:00:53.620 --> 00:00:56.510 So the formula for kinetic energy is that it's equal 00:00:56.510 --> 00:00:59.480 to 1/2 times the mass of the object, 00:00:59.480 --> 00:01:02.250 times the magnitude of its velocity squared, 00:01:02.250 --> 00:01:03.250 or another way to think about it, 00:01:03.250 --> 00:01:05.240 its speed squared. 00:01:05.240 --> 00:01:08.390 And so given this formula, pause the video 00:01:08.390 --> 00:01:10.330 and see if you can calculate the kinetic energy 00:01:10.330 --> 00:01:11.930 for each of these running backs. 00:01:13.100 --> 00:01:15.911 All right, let's calculate the kinetic energy 00:01:15.911 --> 00:01:18.210 for this guy on the left. 00:01:18.210 --> 00:01:22.430 It's gonna be 1/2 times his mass, 00:01:22.430 --> 00:01:27.430 which is 100 kilograms, times the square of the speed, 00:01:28.000 --> 00:01:32.610 so times four meters squared per second squared, 00:01:32.610 --> 00:01:35.010 have to make sure that we square the units as well. 00:01:35.010 --> 00:01:37.570 And this is going to be equal to 1/2 times 100 00:01:37.570 --> 00:01:42.334 is 50 times four is 200 and then the units are kilogram 00:01:42.334 --> 00:01:45.960 meter squared per second squared. 00:01:45.960 --> 00:01:48.150 And you might already recognize that this is the same thing 00:01:48.150 --> 00:01:52.500 as kilogram meter per second squared times meters, 00:01:52.500 --> 00:01:56.450 or these are really the units of force times distance, 00:01:56.450 --> 00:01:58.520 or this is the units of energy 00:01:58.520 --> 00:02:01.700 which we can write as 200 joules. 00:02:01.700 --> 00:02:04.540 Now let's do the same thing for this running back 00:02:04.540 --> 00:02:06.330 that has less mass. 00:02:06.330 --> 00:02:10.170 Kinetic energy here is gonna be 1/2 times the mass, 00:02:10.170 --> 00:02:14.580 25 kilograms times the square of the speed here, 00:02:14.580 --> 00:02:19.220 so that's gonna be 16 meters squared per second squared. 00:02:19.220 --> 00:02:20.930 And then that gets us. 00:02:20.930 --> 00:02:25.120 We're essentially gonna have 1/2 times 16 is eight times 25, 00:02:25.120 --> 00:02:27.600 200, and we get the exact same units 00:02:27.600 --> 00:02:29.960 and so we can go straight to 200 joules. 00:02:29.960 --> 00:02:31.620 So it turns out that they have 00:02:31.620 --> 00:02:33.900 the exact same kinetic energy. 00:02:33.900 --> 00:02:36.610 Even though the gentleman on the right has one fourth 00:02:36.610 --> 00:02:39.270 the mass and only twice the speed, 00:02:39.270 --> 00:02:42.640 we see that we square the speed right over here 00:02:42.640 --> 00:02:44.120 so that makes a huge difference. 00:02:44.120 --> 00:02:46.730 And so their energy due to their motion, 00:02:46.730 --> 00:02:50.360 they have the same capacity to do work. 00:02:50.360 --> 00:02:51.540 Now, some of you are thinking, 00:02:51.540 --> 00:02:53.970 where does this formula come from? 00:02:53.970 --> 00:02:55.930 And one way to think about work and energy 00:02:55.930 --> 00:03:00.310 is that you can use work to transfer energy 00:03:00.310 --> 00:03:02.830 to a system or to an object somehow. 00:03:02.830 --> 00:03:05.710 And then that energy is that object's capacity 00:03:05.710 --> 00:03:07.380 to do work again. 00:03:07.380 --> 00:03:10.710 So let's imagine some object that has a mass m 00:03:10.710 --> 00:03:14.830 and the magnitude of its velocity or its speed is v. 00:03:14.830 --> 00:03:17.920 So what would be the work necessary 00:03:17.920 --> 00:03:20.730 to bring that object that has mass m 00:03:20.730 --> 00:03:25.730 to a speed of v, assuming it's starting at a standstill? 00:03:25.740 --> 00:03:27.840 Well, let's think about it a little bit. 00:03:27.840 --> 00:03:31.360 Work is equal to the magnitude of force 00:03:31.360 --> 00:03:32.720 in a certain direction, 00:03:32.720 --> 00:03:36.860 times the magnitude of the displacement in that direction, 00:03:36.860 --> 00:03:37.810 which we could write like that. 00:03:37.810 --> 00:03:40.290 Sometimes they use s for the magnitude 00:03:40.290 --> 00:03:42.090 of displacement as well. 00:03:42.090 --> 00:03:45.130 And so what is the force the same thing as? 00:03:45.130 --> 00:03:48.050 We know that the force is the same thing 00:03:48.050 --> 00:03:51.970 as mass times the acceleration. 00:03:51.970 --> 00:03:52.980 And we're going to assume 00:03:52.980 --> 00:03:54.500 that we have constant acceleration 00:03:54.500 --> 00:03:57.510 just so that we can simplify our derivation here. 00:03:57.510 --> 00:03:59.960 And then what's the distance that we're gonna travel. 00:03:59.960 --> 00:04:02.530 Well, the distance is gonna be the average magnitude 00:04:02.530 --> 00:04:04.690 of the velocity, or we could say the average speed, 00:04:04.690 --> 00:04:06.420 so I'll write it like this, 00:04:06.420 --> 00:04:11.420 times the time that it takes to accelerate the object 00:04:11.420 --> 00:04:13.640 to a velocity of v. 00:04:13.640 --> 00:04:15.970 Well, how long does it take to accelerate an object 00:04:15.970 --> 00:04:19.130 to a velocity of v if you're accelerating it at a? 00:04:19.130 --> 00:04:22.080 Well, this is just gonna be the velocity divided 00:04:22.080 --> 00:04:23.120 by the acceleration. 00:04:23.120 --> 00:04:23.953 Think about it. 00:04:23.953 --> 00:04:25.820 If you're going, trying to get to a velocity 00:04:25.820 --> 00:04:28.230 of four meters per second, and you're accelerating 00:04:28.230 --> 00:04:30.780 at two meters per second, per second, 00:04:30.780 --> 00:04:34.939 four divided by two is gonna leave you with two seconds. 00:04:34.939 --> 00:04:37.750 And if you're starting at a speed of zero 00:04:37.750 --> 00:04:40.080 and you're going to a magnitude of a velocity 00:04:40.080 --> 00:04:44.180 or a speed of v, and you're assuming constant acceleration, 00:04:44.180 --> 00:04:48.870 your average velocity is just gonna be v over two. 00:04:48.870 --> 00:04:51.710 So this is just v over two. 00:04:51.710 --> 00:04:54.990 And then we get a little bit of a drum roll right over here. 00:04:54.990 --> 00:04:57.890 We see that acceleration cancels with acceleration, 00:04:57.890 --> 00:05:02.890 and we are left with mass times v squared over two, 00:05:03.460 --> 00:05:05.470 mv squared over two, 00:05:05.470 --> 00:05:08.440 which is exactly what we had right over here. 00:05:08.440 --> 00:05:11.540 So the work necessary to accelerate an object 00:05:11.540 --> 00:05:16.540 of mass m from zero speed to a speed of v is exactly this. 00:05:17.490 --> 00:05:19.790 And that's how much energy is then stored 00:05:19.790 --> 00:05:22.730 in that object by virtue of its motion. 00:05:22.730 --> 00:05:24.400 And if you don't have energy loss 00:05:24.400 --> 00:05:27.313 it could in theory, do this much work.

Energy in fields

https://www.youtube.com/watch?v=waw8-hFq1cc

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https://www.youtube.com/api/timedtext?v=waw8-hFq1cc&ei=3FWUZf7XHJXxmLAPw_W_0AI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=3B1EEF227C22794E80FB74E872F605437133A8ED.E4403D141FE7FBB8C4D8C1220ABE0208D62C23DE&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.530 --> 00:00:01.660 - [Presenter] In this video, we're gonna talk 00:00:01.660 --> 00:00:04.290 about how energy is stored in field 00:00:04.290 --> 00:00:07.530 and in particular, how if we change the position of things 00:00:07.530 --> 00:00:11.090 within that field, how it might change the energy. 00:00:11.090 --> 00:00:13.060 So, just as a bit of a refresher, 00:00:13.060 --> 00:00:15.910 let's remind ourselves what energy is. 00:00:15.910 --> 00:00:20.910 It is the capacity to do work. 00:00:21.160 --> 00:00:23.330 And we've also seen that work, 00:00:23.330 --> 00:00:26.320 we can view it as equal to the magnitude of force 00:00:26.320 --> 00:00:30.600 times the displacement in the direction of that force. 00:00:30.600 --> 00:00:34.040 And then we can also remind ourselves what a field is. 00:00:34.040 --> 00:00:35.810 And I'm not just talking about a big lawn 00:00:35.810 --> 00:00:37.970 or something like that, a football field. 00:00:37.970 --> 00:00:40.900 I'm talking about a general idea in physics that's used. 00:00:40.900 --> 00:00:44.040 It's really just a concept that allows us to predict 00:00:44.040 --> 00:00:48.680 and explain how to things that are not touching each other 00:00:48.680 --> 00:00:49.940 are still interacting, 00:00:49.940 --> 00:00:52.640 are still able to exert forces on each other. 00:00:52.640 --> 00:00:54.500 And in other videos, I've also talked about that. 00:00:54.500 --> 00:00:56.670 Really nothing in this universe is touching. 00:00:56.670 --> 00:00:59.850 We just conceptualize that sometimes they are. 00:00:59.850 --> 00:01:01.520 But just as an example of a field, 00:01:01.520 --> 00:01:02.770 we have an electric field here. 00:01:02.770 --> 00:01:04.500 I could have done another type of field, 00:01:04.500 --> 00:01:06.520 we could have done a magnetic field, 00:01:06.520 --> 00:01:08.800 we could have done a gravitational field. 00:01:08.800 --> 00:01:11.160 Although when you study general relativity 00:01:11.160 --> 00:01:12.650 which Einstein gave us, 00:01:12.650 --> 00:01:16.090 we realize that it might not exactly be optimal 00:01:16.090 --> 00:01:17.630 to think about it as a field. 00:01:17.630 --> 00:01:19.250 But in an electric field, right over here, 00:01:19.250 --> 00:01:21.810 we have a positive charge, we have a negative charge. 00:01:21.810 --> 00:01:23.820 We know from experience 00:01:23.820 --> 00:01:26.420 that these two things attract each other 00:01:26.420 --> 00:01:29.610 and the convention is to draw these field lines 00:01:29.610 --> 00:01:33.120 that go from the positive towards the negative. 00:01:33.120 --> 00:01:37.150 Now, if we were to just let go of these two point charges 00:01:37.150 --> 00:01:38.980 over here, what would happen? 00:01:38.980 --> 00:01:42.300 Well, we know that due to the electric field constructed 00:01:42.300 --> 00:01:44.310 or created or that we imagined was created 00:01:44.310 --> 00:01:46.260 by this negative point charge, 00:01:46.260 --> 00:01:49.450 this positive charge would have a force 00:01:49.450 --> 00:01:51.310 acting on it towards the negative charge 00:01:51.310 --> 00:01:53.610 and vice versa due to the electric field 00:01:53.610 --> 00:01:56.370 that is created by the positive charge. 00:01:56.370 --> 00:01:59.330 The negative charge is going to also be attracted 00:01:59.330 --> 00:02:00.170 to the positive charge. 00:02:00.170 --> 00:02:02.840 They're both going to move towards each other. 00:02:02.840 --> 00:02:05.280 And so, when we talk about energy in fields 00:02:05.280 --> 00:02:08.863 or energy stored in fields, in our initial configuration, 00:02:10.582 --> 00:02:12.590 how is there energy in this field? 00:02:12.590 --> 00:02:14.820 How is their capacity to do work? 00:02:14.820 --> 00:02:16.720 Pause this video and think about that. 00:02:17.910 --> 00:02:19.100 Well, think about it this way. 00:02:19.100 --> 00:02:23.480 Imagine if each of these charges were attached 00:02:23.480 --> 00:02:25.280 to some type of a mass, 00:02:25.280 --> 00:02:27.670 let me do this in a color you can actually see. 00:02:27.670 --> 00:02:30.540 So, let's say that this is towing some type of a mass 00:02:30.540 --> 00:02:35.290 and this is towing some type of a mass right over here. 00:02:35.290 --> 00:02:38.590 Well, when you let go and the forces are exerted 00:02:38.590 --> 00:02:40.570 on each of these point charges, 00:02:40.570 --> 00:02:42.530 assuming that the forces are large enough, 00:02:42.530 --> 00:02:45.060 they're going to be able to pull these masses 00:02:45.060 --> 00:02:46.000 towards each other. 00:02:46.000 --> 00:02:48.790 So, there's a potential amount of work that could be done 00:02:48.790 --> 00:02:50.300 and it would essentially keep happening 00:02:50.300 --> 00:02:52.750 until these point charges touch. 00:02:52.750 --> 00:02:55.070 And I used air quotes with my hand 00:02:55.070 --> 00:02:56.310 even though you can't see it, 00:02:56.310 --> 00:02:58.490 or until they can't get any closer to each other 00:02:58.490 --> 00:03:01.920 or some other is keeping them from getting any closer 00:03:01.920 --> 00:03:03.020 to each other. 00:03:03.020 --> 00:03:06.840 So now, let me think about how could I increase the energy 00:03:06.840 --> 00:03:09.210 that is stored in this field. 00:03:09.210 --> 00:03:11.093 Pause the video and think about that. 00:03:12.750 --> 00:03:15.710 Well, what if I were to keep the positive charge 00:03:15.710 --> 00:03:19.350 where it is, but if I were to take the negative charge 00:03:19.350 --> 00:03:22.040 and if I were to move it in a direction opposite 00:03:22.040 --> 00:03:25.940 from the force direction that the field is trying to exert. 00:03:25.940 --> 00:03:27.660 So, instead of the negative charge there, 00:03:27.660 --> 00:03:30.830 what if I moved it all the way out there? 00:03:30.830 --> 00:03:34.080 Once again, let me do that in a color you can actually see. 00:03:34.080 --> 00:03:35.580 Well, if I moved it out here 00:03:35.580 --> 00:03:39.270 and we're still towing some type of a mass, 00:03:39.270 --> 00:03:42.680 you can see that when I moved it against the direction 00:03:42.680 --> 00:03:46.070 that the force of the field is trying to exert, 00:03:46.070 --> 00:03:48.300 that I've increased the energy in the field 00:03:48.300 --> 00:03:50.410 because now I can do more work. 00:03:50.410 --> 00:03:55.410 I can drag this potential mass over a larger distance. 00:03:55.410 --> 00:03:57.530 So, we have a general principle here. 00:03:57.530 --> 00:04:02.120 If we let these charges go in the direction of the forces 00:04:02.120 --> 00:04:04.350 that are being exerted on them due to the field, 00:04:04.350 --> 00:04:07.530 we're going to reduce the energy in the field. 00:04:07.530 --> 00:04:11.430 But then if we are able to move them against the forces 00:04:11.430 --> 00:04:13.190 of the field, and as you can imagine, 00:04:13.190 --> 00:04:14.420 we're going to have to put energy 00:04:14.420 --> 00:04:16.850 into the system to do that. 00:04:16.850 --> 00:04:20.010 But if we do that, then we're storing more energy 00:04:20.010 --> 00:04:22.990 in this field because in that case, 00:04:22.990 --> 00:04:25.300 they're going to be further away from each other 00:04:25.300 --> 00:04:28.410 and so they could drag their little masses 00:04:28.410 --> 00:04:30.553 that they're towing even further.

Electromagnetic radiation emission

https://www.youtube.com/watch?v=WsSMMeYh8jU

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en

WEBVTT Kind: captions Language: en 00:00:00.320 --> 00:00:03.040 - [Norm] Let me ask you a seemingly simple question. 00:00:03.040 --> 00:00:06.060 I have a picture of fire here, and my question is, 00:00:06.060 --> 00:00:08.310 what is fire? 00:00:08.310 --> 00:00:11.190 Well, what would you say if I were to tell you that fire, 00:00:11.190 --> 00:00:13.560 as we see it, these flickering flames, 00:00:13.560 --> 00:00:16.050 it is nothing but hot air? 00:00:16.050 --> 00:00:17.460 And I know what you might be thinking. 00:00:17.460 --> 00:00:18.527 Hot air? 00:00:18.527 --> 00:00:20.140 "Norm, I've seen air that's hot, 00:00:20.140 --> 00:00:21.700 or I experienced air that's hot 00:00:21.700 --> 00:00:24.370 and I don't oftentimes even see the air, 00:00:24.370 --> 00:00:26.460 but here I clearly see something bright, 00:00:26.460 --> 00:00:28.070 something that's emitting light, 00:00:28.070 --> 00:00:31.350 something that's emitting electromagnetic radiation." 00:00:31.350 --> 00:00:33.400 And then what I would say to you, 00:00:33.400 --> 00:00:34.550 if you were thinking that, 00:00:34.550 --> 00:00:38.490 is it actually turns out that anything in our universe 00:00:38.490 --> 00:00:42.480 that has a temperature above absolute zero, zero Kelvin, 00:00:42.480 --> 00:00:43.680 which is pretty much anything 00:00:43.680 --> 00:00:45.970 that you will ever come across in your life, 00:00:45.970 --> 00:00:48.420 emits electromagnetic radiation. 00:00:48.420 --> 00:00:50.740 Objects with temperature aren't the only way 00:00:50.740 --> 00:00:52.700 to create electromagnetic radiation, 00:00:52.700 --> 00:00:55.840 but it is a major way that's happening all around us. 00:00:55.840 --> 00:00:57.960 Even if you were in a pitch-black room, 00:00:57.960 --> 00:01:01.000 you would be emitting electromagnetic radiation. 00:01:01.000 --> 00:01:03.450 A tree outside, even if it was dark outside, 00:01:03.450 --> 00:01:05.810 is emitting electromagnetic radiation. 00:01:05.810 --> 00:01:07.600 You might say, "Wait, but I don't see the tree," 00:01:07.600 --> 00:01:08.900 and that's because your eyes 00:01:08.900 --> 00:01:11.410 can only detect certain frequencies 00:01:11.410 --> 00:01:13.790 of electromagnetic radiation. 00:01:13.790 --> 00:01:16.440 If we look at this diagram right over here, 00:01:16.440 --> 00:01:19.730 we can see how we've categorized many of the frequencies, 00:01:19.730 --> 00:01:21.140 and you can see that up here, 00:01:21.140 --> 00:01:23.060 this is the frequency, this is the wavelength, 00:01:23.060 --> 00:01:25.410 and these are in powers of 10. 00:01:25.410 --> 00:01:28.480 So you can really view this as a logarithmic scale. 00:01:28.480 --> 00:01:29.530 And just over here, 00:01:29.530 --> 00:01:32.300 you can see that our eyes can only visibly see 00:01:32.300 --> 00:01:37.300 a small section of this logarithmic scale of frequencies. 00:01:37.480 --> 00:01:38.760 One of the things I like to wonder, 00:01:38.760 --> 00:01:40.620 if humans didn't have eyes, 00:01:40.620 --> 00:01:42.970 if we weren't able to detect even the small segment 00:01:42.970 --> 00:01:44.590 of the electromagnetic spectrum, 00:01:44.590 --> 00:01:45.720 would we even know 00:01:45.720 --> 00:01:49.300 that something like electromagnetic waves existed? 00:01:49.300 --> 00:01:52.160 But we can see you have gamma rays, x-rays, UV rays, 00:01:52.160 --> 00:01:55.760 infrared rays, microwave, FM, AM radio waves, 00:01:55.760 --> 00:01:57.570 long radio waves. 00:01:57.570 --> 00:01:59.430 In most hot air, 00:01:59.430 --> 00:02:02.570 the frequency isn't high enough for us to see it. 00:02:02.570 --> 00:02:06.610 So most hot air is going to be in the infrared range. 00:02:06.610 --> 00:02:10.380 Only if it gets hot enough will we start to see it, 00:02:10.380 --> 00:02:13.080 and that's what's happening with this fire here. 00:02:13.080 --> 00:02:15.800 And if you look closely at a fire, 00:02:15.800 --> 00:02:18.230 you might actually see that the location 00:02:18.230 --> 00:02:20.800 where the combustion reaction is happening, 00:02:20.800 --> 00:02:22.800 that that actually might be dark. 00:02:22.800 --> 00:02:25.660 And then right above that, you'll see some blue flame, 00:02:25.660 --> 00:02:28.220 and then you'll see, maybe if you look closely, 00:02:28.220 --> 00:02:30.270 some green or yellow flame, 00:02:30.270 --> 00:02:32.740 and then you will see the orange flame, 00:02:32.740 --> 00:02:34.970 and then you will see the red flame. 00:02:34.970 --> 00:02:37.400 And the reason why it might be dark right where 00:02:37.400 --> 00:02:39.180 the combustion reaction is happening 00:02:39.180 --> 00:02:43.210 is that might be very high energy electromagnetic waves. 00:02:43.210 --> 00:02:45.880 That would be in the UV spectrum. 00:02:45.880 --> 00:02:47.930 That would be at a higher frequency than what's visible, 00:02:47.930 --> 00:02:49.250 so to our eyes, it looks dark. 00:02:49.250 --> 00:02:52.190 And then as it cools, it goes through the visible spectrum. 00:02:52.190 --> 00:02:54.870 And then if it cools enough, it goes to infrared. 00:02:54.870 --> 00:02:57.060 But we human beings have built the capability 00:02:57.060 --> 00:03:00.970 to see beyond what our regular eyes can see. 00:03:00.970 --> 00:03:02.510 For example, 00:03:02.510 --> 00:03:05.560 these are what are often known as thermal images, 00:03:05.560 --> 00:03:08.660 but they're really just detecting the infrared range. 00:03:08.660 --> 00:03:10.700 So this is a picture of two dogs. 00:03:10.700 --> 00:03:12.270 It could be pitch-black outside. 00:03:12.270 --> 00:03:13.910 I mean, it could be the middle of the night, 00:03:13.910 --> 00:03:15.760 but because they have temperature, 00:03:15.760 --> 00:03:17.980 they are releasing electromagnetic waves, 00:03:17.980 --> 00:03:18.960 which we can detect. 00:03:18.960 --> 00:03:20.440 And this over here has a scale 00:03:20.440 --> 00:03:21.850 of what the temperature must be. 00:03:21.850 --> 00:03:23.950 So you can see the eyes of the dog 00:03:23.950 --> 00:03:26.090 are the hottest part right over here. 00:03:26.090 --> 00:03:28.990 You can also see thermal imaging of not only a hand, 00:03:28.990 --> 00:03:31.010 but after a hand has touched a wall. 00:03:31.010 --> 00:03:32.830 With our eyes, if you were to touch a wall 00:03:32.830 --> 00:03:34.250 for say 30 seconds, 00:03:34.250 --> 00:03:36.800 it doesn't look like the wall has changed at all, 00:03:36.800 --> 00:03:38.380 but if you were to look at the infrared, 00:03:38.380 --> 00:03:40.160 you would see that you would have heated up parts 00:03:40.160 --> 00:03:41.330 of the wall and you would be able 00:03:41.330 --> 00:03:43.350 to see the shape of the hand. 00:03:43.350 --> 00:03:45.610 And so you can imagine, we human beings, 00:03:45.610 --> 00:03:48.100 because of our ability to detect electromagnetic waves 00:03:48.100 --> 00:03:50.250 and explore electromagnetic waves, 00:03:50.250 --> 00:03:52.360 we've been able to leverage them more and more 00:03:52.360 --> 00:03:53.620 in our everyday lives. 00:03:53.620 --> 00:03:56.430 Thermal imaging itself has a lot of applications. 00:03:56.430 --> 00:03:58.450 Firefighters use it to find people, 00:03:58.450 --> 00:04:01.640 or to find flames in the middle of a lot of smoke. 00:04:01.640 --> 00:04:03.820 We have things like x-rays, 00:04:03.820 --> 00:04:05.920 which are high energy electromagnetic waves, 00:04:05.920 --> 00:04:08.610 which we can use to see through soft tissues. 00:04:08.610 --> 00:04:09.960 So we can see bones. 00:04:09.960 --> 00:04:11.820 This is an old image and it looks like they're using 00:04:11.820 --> 00:04:13.580 the x-rays kind of carelessly. 00:04:13.580 --> 00:04:15.940 You don't wanna be throwing that radiation around, 00:04:15.940 --> 00:04:16.773 but even today. 00:04:16.773 --> 00:04:18.970 I got an x-ray of my teeth just the other day 00:04:18.970 --> 00:04:20.470 when I went to the dentist. 00:04:20.470 --> 00:04:21.960 When you talk on your cell phone, 00:04:21.960 --> 00:04:24.060 the way that your cell phone is able to communicate 00:04:24.060 --> 00:04:26.543 is leveraging electromagnetic waves. 00:04:27.390 --> 00:04:29.180 This is another thing that's mind blowing to me, 00:04:29.180 --> 00:04:30.750 is that my little cell phone 00:04:30.750 --> 00:04:33.020 can actually emit electromagnetic waves 00:04:33.020 --> 00:04:36.180 in the radio part of the spectrum far enough 00:04:36.180 --> 00:04:38.080 to be received by a cell tower that could be 10, 00:04:38.080 --> 00:04:42.750 20, and in certain cases, 30 or 40 miles away. 00:04:42.750 --> 00:04:45.990 Microwave ovens literally released microwaves, 00:04:45.990 --> 00:04:49.240 which are absorbed by our food, which heats up the food. 00:04:49.240 --> 00:04:51.140 And so I'll leave you there. 00:04:51.140 --> 00:04:53.400 The big picture here is that electromagnetic waves 00:04:53.400 --> 00:04:54.730 are all around us. 00:04:54.730 --> 00:04:57.810 It's most obvious to us in the visible spectrum, 00:04:57.810 --> 00:04:59.670 because that's what we can see. 00:04:59.670 --> 00:05:03.570 But there is a large continuum of different frequencies 00:05:03.570 --> 00:05:05.820 that the visible is only a part of. 00:05:05.820 --> 00:05:09.620 And we human beings have leveraged this phenomenon 00:05:09.620 --> 00:05:11.630 in all sorts of interesting ways, 00:05:11.630 --> 00:05:14.160 and I would suspect that we're just at the beginning 00:05:14.160 --> 00:05:15.140 of this exploration. 00:05:15.140 --> 00:05:18.230 Maybe you will come up with a new application 00:05:18.230 --> 00:05:20.163 of electromagnetic waves.

pH and solubility

https://www.youtube.com/watch?v=Z2bDnvDGQHQ

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https://www.youtube.com/api/timedtext?v=Z2bDnvDGQHQ&ei=3FWUZcnjEoG3p-oP4qCB8Ak&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=5D194F1C634A2EFEDBA7EC13522A75304A839532.AE68E506F75106B89BCC2764E30393D25A82E1D9&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.570 --> 00:00:02.270 - [Instructor] Changing the pH of a solution 00:00:02.270 --> 00:00:06.150 can affect the solubility of a slightly soluble salt. 00:00:06.150 --> 00:00:09.610 For example, if we took some solid lead two fluoride, 00:00:09.610 --> 00:00:11.120 which is a white solid, 00:00:11.120 --> 00:00:13.180 and we put it in some distilled water, 00:00:13.180 --> 00:00:15.440 the solid is going to reach an equilibrium 00:00:15.440 --> 00:00:17.700 with the ions in solution. 00:00:17.700 --> 00:00:19.970 Lead two fluoride forms lead two plus ions 00:00:19.970 --> 00:00:24.160 and fluoride anions in a one to two mole ratio. 00:00:24.160 --> 00:00:27.230 So if we have two lead two plus ions in this diagram, 00:00:27.230 --> 00:00:30.160 we need four fluoride anions. 00:00:30.160 --> 00:00:32.270 At equilibrium, the rate of this solution 00:00:32.270 --> 00:00:34.530 is equal to the rate of precipitation, 00:00:34.530 --> 00:00:37.000 and therefore, the amount of solid 00:00:37.000 --> 00:00:41.090 and the concentration of ions in solution remains constant. 00:00:41.090 --> 00:00:45.700 And this forms a saturated solution of lead two fluoride. 00:00:45.700 --> 00:00:47.580 To the system at equilibrium, 00:00:47.580 --> 00:00:49.790 we're gonna add some H plus ions. 00:00:49.790 --> 00:00:51.580 So by increasing the concentration 00:00:51.580 --> 00:00:53.580 of H plus ions in solution, 00:00:53.580 --> 00:00:57.440 we're decreasing the pH of the solution. 00:00:57.440 --> 00:00:59.930 When the H plus ions are added to the solution, 00:00:59.930 --> 00:01:01.990 most of them react with the fluoride anions 00:01:01.990 --> 00:01:03.120 that are present. 00:01:03.120 --> 00:01:06.680 So H plus plus F minus forms HF. 00:01:06.680 --> 00:01:09.760 Comparing the first diagram to the second diagram, 00:01:09.760 --> 00:01:12.360 I just happened to add three H plus ions, 00:01:12.360 --> 00:01:15.130 which will react with three of the fluoride anions 00:01:15.130 --> 00:01:18.320 that are present to produce three HF. 00:01:18.320 --> 00:01:19.950 Notice how the concentration 00:01:19.950 --> 00:01:22.240 of fluoride anions in solution 00:01:22.240 --> 00:01:25.270 has decreased from the first diagram to the second diagram, 00:01:25.270 --> 00:01:28.750 because of the addition of the H plus ions. 00:01:28.750 --> 00:01:30.950 So the system was at equilibrium 00:01:30.950 --> 00:01:35.100 and the concentration of fluoride anions was decreased. 00:01:35.100 --> 00:01:37.210 According to Le Chatelier's Principle, 00:01:37.210 --> 00:01:39.490 the system will shift in the direction 00:01:39.490 --> 00:01:41.460 that decreases the stress. 00:01:41.460 --> 00:01:42.910 So if the stress is decreased 00:01:42.910 --> 00:01:44.880 concentration of fluoride anions, 00:01:44.880 --> 00:01:46.870 the system will shift to the right 00:01:46.870 --> 00:01:49.990 to make more fluoride anions. 00:01:49.990 --> 00:01:51.920 And when the system shifts to the right, 00:01:51.920 --> 00:01:54.100 more lead two fluoride dissolves 00:01:54.100 --> 00:01:56.680 to increase the concentration of Pb two plus 00:01:56.680 --> 00:01:58.710 and fluoride anion. 00:01:58.710 --> 00:02:01.060 We can see that comparing the second diagram 00:02:01.060 --> 00:02:03.220 to the third diagram, 00:02:03.220 --> 00:02:05.850 so the amount of solid has gotten smaller, 00:02:05.850 --> 00:02:08.390 since some of that lead two fluoride dissolved, 00:02:08.390 --> 00:02:11.550 and we've increased the concentration of Pb two plus 00:02:11.550 --> 00:02:14.160 and F minus in solution. 00:02:14.160 --> 00:02:15.730 The solid keeps dissolving 00:02:15.730 --> 00:02:19.520 and the concentration of ions keeps increasing in solution 00:02:19.520 --> 00:02:23.120 until the system reaches equilibrium. 00:02:23.120 --> 00:02:24.400 So for a saturated solution 00:02:24.400 --> 00:02:26.830 of lead two fluoride at equilibrium, 00:02:26.830 --> 00:02:29.830 decreasing the pH or making the solution more acidic 00:02:29.830 --> 00:02:32.780 by increasing the concentration of H plus ions, 00:02:32.780 --> 00:02:36.410 increases the solubility of lead two fluoride, 00:02:36.410 --> 00:02:39.210 which is why we saw more of the solid dissolve 00:02:39.210 --> 00:02:41.540 when the H plus ions were added. 00:02:41.540 --> 00:02:43.280 This effect of decreasing the pH 00:02:43.280 --> 00:02:46.500 and increasing the solubility of a slightly soluble salt 00:02:46.500 --> 00:02:49.350 happens whenever the slightly soluble salt 00:02:49.350 --> 00:02:52.260 contains a basic anion. 00:02:52.260 --> 00:02:55.610 For this example, the basic anion is the fluoride anion, 00:02:55.610 --> 00:02:58.530 which reacts with the added H plus ions. 00:02:58.530 --> 00:03:00.850 And when the basic anion reacts, 00:03:00.850 --> 00:03:04.160 that decreases the concentration of that basic anion, 00:03:04.160 --> 00:03:07.720 which caused the equilibrium to shift to the right. 00:03:07.720 --> 00:03:10.460 And there are many other examples of basic anions, 00:03:10.460 --> 00:03:12.660 two more would be the hydroxide anion 00:03:12.660 --> 00:03:14.870 and the carbonate anion. 00:03:14.870 --> 00:03:17.790 And if a compound contains a basic anion, 00:03:17.790 --> 00:03:21.840 such as the hydroxide anion, hydroxide functions as a base 00:03:21.840 --> 00:03:25.460 and reacts with H plus ions to form H2O. 00:03:25.460 --> 00:03:27.870 So therefore, the solubility of a compound 00:03:27.870 --> 00:03:31.180 containing a hydroxide ion would increase 00:03:31.180 --> 00:03:34.820 as H plus ions are added to the solution. 00:03:34.820 --> 00:03:35.910 It's also important to note 00:03:35.910 --> 00:03:38.010 for this lead two fluoride problem, 00:03:38.010 --> 00:03:42.070 if the pH is decreased at a constant temperature, 00:03:42.070 --> 00:03:46.930 the Ksp value for PbF two remains constant. 00:03:46.930 --> 00:03:50.220 So the molar solubility does increase, 00:03:50.220 --> 00:03:53.530 but the Ksp value remains the same. 00:03:53.530 --> 00:03:55.450 This time, instead of lead two fluoride, 00:03:55.450 --> 00:03:57.790 let's look at lead two chloride. 00:03:57.790 --> 00:04:00.380 Lead two chloride is also a white solid. 00:04:00.380 --> 00:04:02.570 So if we dissolve some in solution, 00:04:02.570 --> 00:04:04.690 eventually, we would reach an equilibrium 00:04:04.690 --> 00:04:07.550 between the solid and the ions in solution. 00:04:07.550 --> 00:04:10.930 So this diagram here shows a saturated solution 00:04:10.930 --> 00:04:14.620 of lead two chloride and the system is at equilibrium. 00:04:14.620 --> 00:04:16.290 And to the system at equilibrium, 00:04:16.290 --> 00:04:21.290 we decrease the pH by adding H plus ions to the solution. 00:04:21.490 --> 00:04:23.570 In this case, the chloride anions 00:04:23.570 --> 00:04:26.870 aren't basic enough to react with the H plus ions. 00:04:26.870 --> 00:04:29.310 Therefore, we do not form HCl 00:04:29.310 --> 00:04:32.440 and the concentration of chloride anions 00:04:32.440 --> 00:04:36.360 remains the same as it was in the original diagram. 00:04:36.360 --> 00:04:39.330 So if the concentration of chloride ions remains the same 00:04:39.330 --> 00:04:41.230 and the concentration of lead two plus ions 00:04:41.230 --> 00:04:44.810 would remain the same, the system is still at equilibrium 00:04:44.810 --> 00:04:47.630 and decrease in the pH had no effect 00:04:47.630 --> 00:04:50.330 on the solubility of the solids. 00:04:50.330 --> 00:04:54.010 So we have the same amount of lead two chloride solid 00:04:54.010 --> 00:04:57.050 on the bottom of the beaker in both diagrams. 00:04:57.050 --> 00:04:59.690 So whenever an anion has an extremely weak base, 00:04:59.690 --> 00:05:01.440 like the chloride anion, 00:05:01.440 --> 00:05:05.530 we say that this is an anion of negligible basicity 00:05:05.530 --> 00:05:07.230 and the solubility of salts 00:05:07.230 --> 00:05:10.010 with anions of negligible basicity 00:05:10.010 --> 00:05:13.353 is unaffected by changes in pH.

The common-ion effect

https://www.youtube.com/watch?v=fI3U9T7LigY

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https://www.youtube.com/api/timedtext?v=fI3U9T7LigY&ei=3FWUZYnbLYq5vdIPxYm3yAI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=931B35B438DA656512B3A7AA036C2042D04B5E46.29B1E8FAE14A372E7CCCA9D1FD3B7B516073159C&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.360 --> 00:00:02.140 - [Instructor] The presence of a common ion 00:00:02.140 --> 00:00:05.040 can affect a solubility equilibrium. 00:00:05.040 --> 00:00:05.873 For example, 00:00:05.873 --> 00:00:09.530 let's say we have a saturated solution of lead II chloride. 00:00:09.530 --> 00:00:11.950 Lead II chloride is a white solid, 00:00:11.950 --> 00:00:14.360 so here's the white solid on the bottom of the beaker. 00:00:14.360 --> 00:00:18.100 And the solid's at equilibrium with the ions in solution. 00:00:18.100 --> 00:00:22.210 So that would be Pb2+ and Cl-. 00:00:22.210 --> 00:00:26.890 Notice how the mole ratio is one-to-two of Pb2+ to 2Cl-. 00:00:26.890 --> 00:00:30.590 So if we have two Pb2+ ions in our diagram, 00:00:30.590 --> 00:00:33.320 there should be twice as many chloride anions. 00:00:33.320 --> 00:00:36.350 At equilibrium, the rate of dissolution is equal to 00:00:36.350 --> 00:00:38.510 the rate of precipitation. 00:00:38.510 --> 00:00:41.250 Therefore, the concentration of ions in solution 00:00:41.250 --> 00:00:43.180 remains constant. 00:00:43.180 --> 00:00:45.220 So our system is at equilibrium. 00:00:45.220 --> 00:00:48.570 And let's add some solid potassium chloride. 00:00:48.570 --> 00:00:50.780 Potassium chloride is a soluble salt. 00:00:50.780 --> 00:00:55.780 So it will dissociate and turn into K+ and Cl- in solution. 00:00:56.320 --> 00:00:58.650 Adding a source of chloride anion means 00:00:58.650 --> 00:01:01.360 the system is no longer at equilibrium. 00:01:01.360 --> 00:01:02.193 So let me write in here, 00:01:02.193 --> 00:01:05.950 not at equilibrium at that moment in time. 00:01:05.950 --> 00:01:07.730 So the system was at equilibrium 00:01:07.730 --> 00:01:10.320 and a stress was added to the system. 00:01:10.320 --> 00:01:13.500 In this case, the stress was increased chloride anion. 00:01:13.500 --> 00:01:17.527 So there's an increase in the concentration of Cl-. 00:01:18.471 --> 00:01:20.410 According to Le Chatelier's principle, 00:01:20.410 --> 00:01:21.980 the system will move in the direction 00:01:21.980 --> 00:01:23.980 that decreases the stress. 00:01:23.980 --> 00:01:25.060 So if the stress is 00:01:25.060 --> 00:01:27.810 increased concentration of chloride anion, 00:01:27.810 --> 00:01:29.610 the system will move to the left 00:01:29.610 --> 00:01:33.000 to get rid of some of that extra chloride anion. 00:01:33.000 --> 00:01:34.720 When the system moves to the left, 00:01:34.720 --> 00:01:39.720 Pb2+ ions will combine with chloride anions to form PbCl2. 00:01:39.747 --> 00:01:42.290 And we can see that down here in the diagram. 00:01:42.290 --> 00:01:43.123 So imagine, 00:01:43.123 --> 00:01:47.480 say this Pb2+ ion combined with these two chloride anions 00:01:47.480 --> 00:01:50.180 to form some more of the white solid. 00:01:50.180 --> 00:01:51.610 Looking at the third diagram, 00:01:51.610 --> 00:01:53.580 the amount of white solid has increased 00:01:53.580 --> 00:01:55.280 from the second diagram, 00:01:55.280 --> 00:01:59.859 and we've lost this Pb2+ ion and these two chloride anions. 00:01:59.859 --> 00:02:02.450 And the amount of our precipitate PbCl2 00:02:02.450 --> 00:02:06.180 will keep forming until equilibrium is reached. 00:02:06.180 --> 00:02:07.700 Let's just say this third diagram, 00:02:07.700 --> 00:02:10.470 it does represent the system at equilibrium. 00:02:10.470 --> 00:02:13.410 So I'll write on here, at equilibrium. 00:02:13.410 --> 00:02:16.750 This is an example of the common ion effect. 00:02:16.750 --> 00:02:20.310 For this problem, the common ion is the chloride anion, 00:02:20.310 --> 00:02:22.640 because there were two sources of it. 00:02:22.640 --> 00:02:25.750 One was from the disillusion of PbCl2. 00:02:25.750 --> 00:02:28.720 If we had dissolved some solid to make a saturated solution, 00:02:28.720 --> 00:02:32.340 the source of these chloride anions would be from PbCl2. 00:02:32.340 --> 00:02:35.470 And the second source is from the added KCl, 00:02:35.470 --> 00:02:38.490 which of course dissolved to form chloride anion. 00:02:38.490 --> 00:02:42.090 So the chloride anion is the common ion. 00:02:42.090 --> 00:02:43.830 And we use Le Chatelier's principle 00:02:43.830 --> 00:02:46.360 to predict the system will move to the left 00:02:46.360 --> 00:02:49.650 to get rid of the extra chloride anion. 00:02:49.650 --> 00:02:50.950 When the system moved to the left, 00:02:50.950 --> 00:02:53.550 we formed more of the solid PbCl2. 00:02:53.550 --> 00:02:55.830 And that's why this amount got bigger over here. 00:02:55.830 --> 00:03:00.040 So if we compare the first diagram with the third diagram, 00:03:00.040 --> 00:03:03.050 the first diagram has more of 00:03:03.050 --> 00:03:05.600 the lead II chloride in solution. 00:03:05.600 --> 00:03:07.830 And the third diagram has less of it. 00:03:07.830 --> 00:03:11.850 Therefore, the addition of the common ion of chloride anion, 00:03:11.850 --> 00:03:16.650 that decreased the solubility of lead II chloride. 00:03:16.650 --> 00:03:18.380 So the common ion effect says that 00:03:18.380 --> 00:03:21.130 the solubility of a slightly soluble salt, 00:03:21.130 --> 00:03:22.610 like lead II chloride, 00:03:22.610 --> 00:03:26.490 is decreased by the presence of a common ion. 00:03:26.490 --> 00:03:28.360 Another way to think about this is using 00:03:28.360 --> 00:03:30.490 the reaction quotient, Q. 00:03:30.490 --> 00:03:34.140 For the diagram on the left, we're at equilibrium. 00:03:34.140 --> 00:03:38.560 Therefore the reaction quotient Qsp is equal to 00:03:38.560 --> 00:03:41.630 the Ksp value for lead II chloride, 00:03:41.630 --> 00:03:44.620 which means the system is at equilibrium. 00:03:44.620 --> 00:03:48.600 Adding chloride anion increases the value for Qsp. 00:03:48.600 --> 00:03:51.790 So now Qsp is greater than Ksp 00:03:51.790 --> 00:03:54.340 and the system is not at equilibrium. 00:03:54.340 --> 00:03:56.700 In order to decrease the value for Q, 00:03:56.700 --> 00:03:59.290 the system needs to move to the left. 00:03:59.290 --> 00:04:01.920 And the system will continue to move to the left 00:04:01.920 --> 00:04:05.890 until Qsp is equal to Ksp again 00:04:05.890 --> 00:04:08.550 and the system is at equilibrium. 00:04:08.550 --> 00:04:13.320 A shift to the left means an increase in the amount of PbCl2 00:04:13.320 --> 00:04:18.030 which therefore decreases the solubility of PbCl2. 00:04:18.030 --> 00:04:20.918 But it doesn't change the value for Ksp. 00:04:20.918 --> 00:04:25.918 Ksp for PbCl2 stays the same at the same temperature. 00:04:26.530 --> 00:04:29.360 Next, let's see how the presence of a common ion affects 00:04:29.360 --> 00:04:32.740 the molar solubility of lead II chloride. 00:04:32.740 --> 00:04:33.573 And to do that, 00:04:33.573 --> 00:04:36.100 let's calculate the molar solubility of lead II chloride 00:04:36.100 --> 00:04:38.110 at 25 degrees Celsius 00:04:38.110 --> 00:04:43.010 in a solution that is 0.10 molar in potassium chloride. 00:04:43.010 --> 00:04:47.464 The Ksp value for lead II chloride at 25 degrees Celsius is 00:04:47.464 --> 00:04:50.490 1.7 times 10 to the negative fifth. 00:04:50.490 --> 00:04:52.500 To help us calculate the molar solubility, 00:04:52.500 --> 00:04:53.920 we're going to use an ICE table, 00:04:53.920 --> 00:04:56.070 where I stands for the initial concentration, 00:04:56.070 --> 00:04:57.880 C is the change in concentration, 00:04:57.880 --> 00:05:00.870 and E is the equilibrium concentration. 00:05:00.870 --> 00:05:01.750 First, let's say that 00:05:01.750 --> 00:05:04.420 none of the lead II chloride has dissolved yet. 00:05:04.420 --> 00:05:05.350 And if that's true, 00:05:05.350 --> 00:05:09.200 the concentration of lead II plus ions would be zero, 00:05:09.200 --> 00:05:11.890 and the concentration of chloride anions 00:05:11.890 --> 00:05:15.600 from the lead II chloride would also be zero. 00:05:15.600 --> 00:05:18.200 However, there's another source of chloride anions, 00:05:18.200 --> 00:05:22.320 and that's because our solution is 0.10 molar in KCl. 00:05:22.320 --> 00:05:25.200 KCl is a soluble salt. 00:05:25.200 --> 00:05:30.190 So KCl associates completely to turn to K+ and Cl-. 00:05:30.190 --> 00:05:34.570 Therefore, if the concentration of KCl is 0.10 molar, 00:05:34.570 --> 00:05:38.830 that's also the concentration of Cl- from the KCl. 00:05:38.830 --> 00:05:43.570 So we can add here plus 0.10 molar. 00:05:43.570 --> 00:05:47.000 And think about that as being from our KCl. 00:05:47.000 --> 00:05:48.560 So there are two sources. 00:05:48.560 --> 00:05:51.340 There's going to be two sources of chloride anions here. 00:05:51.340 --> 00:05:55.450 And so the Cl anion and is our common ion. 00:05:55.450 --> 00:05:58.570 The other source of chloride anion is PbCl2 00:05:58.570 --> 00:05:59.780 when it dissolves. 00:05:59.780 --> 00:06:02.320 So some of the PbCl2 will dissolve, 00:06:02.320 --> 00:06:06.120 we don't know how much, so I like to write -X in here. 00:06:06.120 --> 00:06:07.510 And if some of that dissolves, 00:06:07.510 --> 00:06:12.510 the mole ratio of PbCl2 to Pb2+ is a one-to-one mole ratio. 00:06:12.700 --> 00:06:17.700 So if we're losing X for PbCl2, we're gaining X for Pb2+. 00:06:18.530 --> 00:06:20.100 And looking at our mole ratios, 00:06:20.100 --> 00:06:22.150 if we're gaining X for Pb2+ 00:06:22.150 --> 00:06:24.860 and it's a one-to-two mole ratio, 00:06:24.860 --> 00:06:29.460 we would write in here +2X for the chloride anion. 00:06:29.460 --> 00:06:33.120 So for the equilibrium concentration of Pb2+, 00:06:33.120 --> 00:06:36.400 it would be zero plus X, or just X. 00:06:36.400 --> 00:06:39.590 And for the equilibrium concentration of the chloride anion, 00:06:39.590 --> 00:06:42.830 it would be 0.10 plus 2X. 00:06:42.830 --> 00:06:47.830 So the 0.10 came from the potassium chloride, 00:06:48.000 --> 00:06:52.350 and the 2X came from the dissolution of lead II chloride. 00:06:52.350 --> 00:06:54.410 Next, we need to write a Ksp expression, 00:06:54.410 --> 00:06:57.200 which we can get from the disillusion equation. 00:06:57.200 --> 00:07:02.200 So, Ksp is equal to the concentration of lead II plus ions 00:07:02.850 --> 00:07:05.280 raised to the first power 00:07:05.280 --> 00:07:07.440 times the concentration of chloride anions. 00:07:07.440 --> 00:07:09.550 And since there's a two as a coefficient 00:07:09.550 --> 00:07:10.690 in the balanced equation, 00:07:10.690 --> 00:07:14.020 we need to raise that concentration to the second power. 00:07:14.020 --> 00:07:15.330 Pure solids are left out of 00:07:15.330 --> 00:07:17.080 equilibrium constant expressions. 00:07:17.080 --> 00:07:20.150 So we don't write anything for PbCl2. 00:07:20.150 --> 00:07:22.950 Next, we plug in our equilibrium concentrations. 00:07:22.950 --> 00:07:27.100 So for lead II plus, the equilibrium concentration is X. 00:07:27.100 --> 00:07:29.010 And for the chloride anion, 00:07:29.010 --> 00:07:32.933 the equilibrium concentration is 0.10 plus 2X. 00:07:34.270 --> 00:07:38.860 We also need to plug in the Ksp value for lead II chloride. 00:07:38.860 --> 00:07:41.270 Here we have the Ksp value plugged in. 00:07:41.270 --> 00:07:44.490 X and 0.10 plus 2X. 00:07:44.490 --> 00:07:49.000 And let's think about 0.10 plus 2X for a second here. 00:07:49.000 --> 00:07:50.961 With a very low value for Ksp, 00:07:50.961 --> 00:07:53.440 1.7 times 10 to the negative fifth, 00:07:53.440 --> 00:07:57.450 that means that not very much of the PbCl2 will dissolve. 00:07:57.450 --> 00:08:00.380 And if that's true, X is a pretty small number. 00:08:00.380 --> 00:08:04.090 And if X is a small number, 2X is also pretty small. 00:08:04.090 --> 00:08:05.960 So we're going to make an approximation 00:08:05.960 --> 00:08:09.140 and say that 0.10 plus a pretty small number 00:08:09.140 --> 00:08:12.550 is approximately equal to just 0.10. 00:08:12.550 --> 00:08:15.130 And that's going to make the math easier on us. 00:08:15.130 --> 00:08:18.320 So instead of writing 0.10 plus 2X squared, 00:08:18.320 --> 00:08:20.643 we just have 0.10 squared. 00:08:21.880 --> 00:08:26.450 Solving for X, we find that X is equal to 0.0017, 00:08:27.580 --> 00:08:28.623 which we could just write as 00:08:28.623 --> 00:08:33.623 1.7 times 10 to the negative third molar. 00:08:34.240 --> 00:08:35.530 It's okay to write molar here 00:08:35.530 --> 00:08:38.010 because this X value represents 00:08:38.010 --> 00:08:42.120 the equilibrium concentration of Pb2+. 00:08:42.120 --> 00:08:45.670 And if that's the equilibrium concentration of Pb2+, 00:08:45.670 --> 00:08:48.030 that's also the concentration of lead to chloride 00:08:48.030 --> 00:08:48.920 that dissolved. 00:08:48.920 --> 00:08:51.290 So this number, this concentration, 00:08:51.290 --> 00:08:55.320 is the molar solubility of lead II chloride 00:08:55.320 --> 00:08:57.160 in a solution at 25 degrees 00:08:57.160 --> 00:09:00.810 where the solution is 0.10 molar in KCl. 00:09:00.810 --> 00:09:04.040 Most textbooks leave this -X out of their ICE tables 00:09:04.040 --> 00:09:06.810 because the concentration of a solid doesn't change. 00:09:06.810 --> 00:09:08.250 I like to just leave it in here though 00:09:08.250 --> 00:09:10.550 to remind me that X represents 00:09:10.550 --> 00:09:14.840 the molar solubility of the slightly soluble salt. 00:09:14.840 --> 00:09:16.010 Finally, if we'd calculate 00:09:16.010 --> 00:09:17.990 the molar solubility of lead II chloride 00:09:17.990 --> 00:09:20.620 without the presence of a common ion, 00:09:20.620 --> 00:09:23.310 this 0.10 would have been gone from everything. 00:09:23.310 --> 00:09:25.720 And doing the math that way, we would have found that 00:09:25.720 --> 00:09:29.150 the molar solubility at 25 degrees Celsius 00:09:29.150 --> 00:09:31.820 and using this value for the Ksp, 00:09:31.820 --> 00:09:36.820 the molar solubility comes out to 0.016 molar. 00:09:37.720 --> 00:09:40.300 So comparing these two smaller solubilities, 00:09:40.300 --> 00:09:43.210 0.016 molar versus 0.0017, 00:09:45.150 --> 00:09:48.120 that's approximately a factor of 10. 00:09:48.120 --> 00:09:50.980 Therefore the addition of a common ion 00:09:50.980 --> 00:09:55.980 decreased the solubility by approximately a factor of 10. 00:09:56.980 --> 00:09:59.970 So doing the common ion effect in a quantitative way 00:09:59.970 --> 00:10:01.750 also shows a decrease in 00:10:01.750 --> 00:10:04.280 the solubility of a slightly soluble salt 00:10:05.412 --> 00:10:07.553 because of the presence of a common ion.

Worked example: Predicting whether a precipitate forms by comparing Q and Kₛₚ

https://www.youtube.com/watch?v=vyENVkttVtw

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https://www.youtube.com/api/timedtext?v=vyENVkttVtw&ei=3FWUZcusLK39mLAP25eRiAM&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=B11E3A2E7158ED8FCD2FF9BCCF3F3DB4BF9BAF40.53023259B4220ED7135130989861F1B1177B34C1&key=yt8&lang=en&name=Default&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:00.330 --> 00:00:02.020 - [Instructor] For this problem, our goal is to figure out 00:00:02.020 --> 00:00:04.160 whether or not a precipitate will form 00:00:04.160 --> 00:00:08.810 if we mix 0.20 liters over 4.0 times 10 to the negative 00:00:08.810 --> 00:00:12.140 third Molar solution of lead two nitrate, 00:00:12.140 --> 00:00:16.410 with 0.80 liters of an 8.0 times 10 to the negative third 00:00:16.410 --> 00:00:19.530 Molar solution of sodium sulfate. 00:00:19.530 --> 00:00:21.690 The first step is to figure out the identity 00:00:21.690 --> 00:00:23.840 of the precipitate that might form. 00:00:23.840 --> 00:00:26.630 So we're mixing an aqueous solution of lead two nitrate 00:00:26.630 --> 00:00:30.360 with an aqueous solution of sodium sulfate. 00:00:30.360 --> 00:00:33.228 In the lead two nitrate solution they're lead two plus 00:00:33.228 --> 00:00:37.046 cations and nitrate anions. 00:00:37.046 --> 00:00:39.330 In the sodium sulfate aqueous solution, 00:00:39.330 --> 00:00:43.643 there are sodium cations and sulfate anions. 00:00:44.540 --> 00:00:48.210 So we take the cation from one and the anion from the other. 00:00:48.210 --> 00:00:51.350 So one possible product would be lead sulfate. 00:00:51.350 --> 00:00:55.450 So let's write on here, PbSO4, after we cross over 00:00:55.450 --> 00:01:00.450 our charges and we take the other cation and the other anion 00:01:01.070 --> 00:01:05.080 and so the other product would be sodium nitrate. 00:01:05.080 --> 00:01:07.033 So we write it in here and a NaNO3. 00:01:08.170 --> 00:01:12.930 To balance the equation, we need a two in front of NANO3. 00:01:12.930 --> 00:01:16.962 Since nitrates are soluble, sodium nitrate is an aqueous 00:01:16.962 --> 00:01:21.962 solution and lead sulfate would be our possible precipitate. 00:01:22.120 --> 00:01:24.130 Now that we know are possible precipitate, 00:01:24.130 --> 00:01:26.360 let's go ahead and write a net ionic equation 00:01:26.360 --> 00:01:29.920 showing the formation of that precipitate. 00:01:29.920 --> 00:01:34.040 So lead two plus ions, would come together with sulfate 00:01:34.040 --> 00:01:39.040 anions to form PBSO4. 00:01:39.090 --> 00:01:43.870 So PBSO4 is the possible precipitate. 00:01:43.870 --> 00:01:46.260 Since lead sulfate is our possible precipitate, 00:01:46.260 --> 00:01:50.070 we really only care about the concentration of lead two plus 00:01:50.070 --> 00:01:52.810 ions and sulfate anions in solution. 00:01:52.810 --> 00:01:56.610 We don't need to worry about sodium cations or nitrate 00:01:56.610 --> 00:02:00.180 anions because those are the spectator ions 00:02:00.180 --> 00:02:02.760 in our overall reaction. 00:02:02.760 --> 00:02:05.880 Running the overall equation, and the net ionic equation 00:02:05.880 --> 00:02:08.470 are really optional for a problem like this. 00:02:08.470 --> 00:02:11.350 But we really need to do is identify the precipitate 00:02:11.350 --> 00:02:14.230 and then write out the disillusion equation. 00:02:14.230 --> 00:02:19.230 So PBSO4 would be our possible precipitate. 00:02:19.430 --> 00:02:24.360 And if it dissolves in water, we would form lead two plus 00:02:24.360 --> 00:02:29.360 cations in aqueous solution, and sulfates anions 00:02:29.530 --> 00:02:31.257 in aqueous solutions, right? 00:02:31.257 --> 00:02:33.470 aq over here. 00:02:33.470 --> 00:02:36.180 The reason why it's important to write out the dissolution 00:02:36.180 --> 00:02:40.030 equation is because we can write a KSP expression 00:02:40.030 --> 00:02:40.940 from it. 00:02:40.940 --> 00:02:44.250 So KSP is equal to, it would be the concentration 00:02:44.250 --> 00:02:46.600 of lead two plus raised to the first power 00:02:46.600 --> 00:02:48.100 because we have a coefficient of one 00:02:48.100 --> 00:02:49.710 in the balanced equation, 00:02:49.710 --> 00:02:52.990 times the concentration of sulfate also raised 00:02:52.990 --> 00:02:56.680 to the first power and pure solids are left out 00:02:56.680 --> 00:02:58.680 of equilibrium, constant expressions. 00:02:58.680 --> 00:03:02.830 Therefore we're not going to include lead sulfate. 00:03:02.830 --> 00:03:07.830 For lead two sulfate KSP is equal to 6.3 times 10 00:03:09.380 --> 00:03:14.380 to the negative seven at 25 degrees Celsius. 00:03:15.420 --> 00:03:19.310 The concentrations of lead two plus and sulfate in the KSP 00:03:19.310 --> 00:03:23.060 expression, are equilibrium concentrations. 00:03:23.060 --> 00:03:25.840 For our problem, we're gonna calculate QSP, 00:03:25.840 --> 00:03:27.980 which has the same form as KSP, 00:03:27.980 --> 00:03:29.930 the differences the concentrations can be 00:03:29.930 --> 00:03:31.980 at any moment in time. 00:03:31.980 --> 00:03:34.080 And we're gonna calculate QSP at the moment, 00:03:34.080 --> 00:03:35.960 our two solutions are mixed, 00:03:35.960 --> 00:03:40.200 and then we're going to compare QSP to KSP. 00:03:40.200 --> 00:03:42.640 I've drawn out some diagrams to help us understand 00:03:42.640 --> 00:03:45.720 how QSP compares to KSP and what that means 00:03:45.720 --> 00:03:46.890 for the solution. 00:03:46.890 --> 00:03:49.020 However, these aren't perfect diagrams 00:03:49.020 --> 00:03:51.330 they're just to help get the point across. 00:03:51.330 --> 00:03:55.470 If QSP is less than KSP, the solution is unsaturated, 00:03:55.470 --> 00:03:58.400 which means no precipitate would form. 00:03:58.400 --> 00:03:59.880 For an unsaturated solution, 00:03:59.880 --> 00:04:02.300 you can dissolve more led two sulfate in it. 00:04:02.300 --> 00:04:05.000 So lead two sulfate as a white solid, 00:04:05.000 --> 00:04:08.130 so if we were to put a small amount of lead two sulfate 00:04:08.130 --> 00:04:10.980 in our unsaturated solution, it would dissolve, 00:04:10.980 --> 00:04:15.980 and it would continue to dissolve until QSP is equal to KSP. 00:04:16.450 --> 00:04:19.760 And the system is at equilibrium. 00:04:19.760 --> 00:04:22.760 At equilibrium, the solid is turning into the ions 00:04:22.760 --> 00:04:26.760 at the same rate, the ions are turning back into the solid. 00:04:26.760 --> 00:04:29.780 Since the rate of dissolution is equal to the rate 00:04:29.780 --> 00:04:32.880 of precipitation when the system is at equilibrium, 00:04:32.880 --> 00:04:34.990 the concentrations of lead two plus ions 00:04:34.990 --> 00:04:37.070 and sulfate ions are constant, 00:04:37.070 --> 00:04:40.740 and this represents a saturated solution. 00:04:40.740 --> 00:04:43.300 And since the solution is saturated at equilibrium, 00:04:43.300 --> 00:04:46.060 if we tried to add some more lead two sulfate 00:04:46.060 --> 00:04:47.730 at the same temperature, 00:04:47.730 --> 00:04:49.850 we wouldn't be able to dissolve any more, 00:04:49.850 --> 00:04:52.220 we would just increase the pile of lead two sulfate 00:04:52.220 --> 00:04:54.310 on the bottom of the beaker. 00:04:54.310 --> 00:04:58.180 That concept helps us understand what happens when QSP 00:04:58.180 --> 00:05:00.320 is greater than KSP. 00:05:00.320 --> 00:05:04.730 When QSP is greater than KSP, the solution is oversaturated. 00:05:04.730 --> 00:05:07.490 So it's exceeded the limit of what can dissolve, 00:05:07.490 --> 00:05:10.640 and therefore you can imagine some lead two plus ions 00:05:10.640 --> 00:05:15.290 combining with some sulfate ions to form a precipitate. 00:05:15.290 --> 00:05:18.470 Therefore, when QSP is greater than KSP, 00:05:18.470 --> 00:05:20.600 a precipitate will form. 00:05:20.600 --> 00:05:23.100 The precipitate will continue to form, 00:05:23.100 --> 00:05:25.940 until QSP is equal to KSP, 00:05:25.940 --> 00:05:28.620 and the system reaches equilibrium. 00:05:28.620 --> 00:05:30.840 Next, we need to go back to what we were given 00:05:30.840 --> 00:05:32.240 in our initial problem, 00:05:32.240 --> 00:05:34.750 when we mixed our two solutions together. 00:05:34.750 --> 00:05:37.090 Remember we only cared about the concentration of lead 00:05:37.090 --> 00:05:39.510 two plus ions and sulfate ions. 00:05:39.510 --> 00:05:42.330 So we're gonna calculate the concentration of those two ions 00:05:42.330 --> 00:05:45.950 at the moment in time, when the two solutions are mixed. 00:05:45.950 --> 00:05:48.450 First let's calculate the concentration of lead two 00:05:48.450 --> 00:05:49.790 plus ions. 00:05:49.790 --> 00:05:53.110 The original solution of lead two nitrate 00:05:53.110 --> 00:05:56.280 had a concentration of 4.0 times 10 to the negative 00:05:56.280 --> 00:05:57.610 third Molar. 00:05:57.610 --> 00:06:00.200 So molarity is equal to moles over liters. 00:06:00.200 --> 00:06:02.190 So we can plug in the concentration, 00:06:02.190 --> 00:06:04.550 and we can also plug in the volume of that solution, 00:06:04.550 --> 00:06:08.580 which was 0.20 liters, and solve for X, 00:06:08.580 --> 00:06:13.130 X equal to 8.0 times 10 to the negative fourth moles. 00:06:13.130 --> 00:06:14.930 That's how many moles of lead two nitrate 00:06:14.930 --> 00:06:17.870 there are and that's also how many moles of lead two 00:06:17.870 --> 00:06:19.760 plus ions there are. 00:06:19.760 --> 00:06:22.440 Therefore to find the concentration of lead two plus ions 00:06:22.440 --> 00:06:23.760 after the solutions are mixed, 00:06:23.760 --> 00:06:27.060 we plug in 8.0 times 10 to the negative fourth moles, 00:06:27.060 --> 00:06:30.010 and for the volume we're adding these two solutions 00:06:30.010 --> 00:06:32.430 together, so the total volume of the solution 00:06:32.430 --> 00:06:34.573 is 0.20 plus 0.80. 00:06:35.534 --> 00:06:37.870 So the concentration of lead two plus ions 00:06:37.870 --> 00:06:42.400 will be equal to 8.0 times 10 to the negative fourth Molar. 00:06:42.400 --> 00:06:44.100 We can do the same type of calculation 00:06:44.100 --> 00:06:46.400 to find the concentration of sulfate ions 00:06:46.400 --> 00:06:49.140 after the two solutions have been mixed. 00:06:49.140 --> 00:06:51.560 So we take the concentration of the original solution 00:06:51.560 --> 00:06:54.970 of sodium sulfate and plug that into the molarity equation, 00:06:54.970 --> 00:06:59.110 plug in the volume solve for X and 6.4 times 10 to negative 00:06:59.110 --> 00:07:02.770 third moles is how many moles of sodium sulfate there are. 00:07:02.770 --> 00:07:05.510 That's also how many moles of sulfate ions there are, 00:07:05.510 --> 00:07:08.930 so we plug in that number into the concentration 00:07:08.930 --> 00:07:11.220 for sulfate, and once again, since we're adding 00:07:11.220 --> 00:07:13.470 the two solutions together, we divide that by the total 00:07:13.470 --> 00:07:16.430 volume to get a concentration of sulfate ions 00:07:16.430 --> 00:07:20.220 of 6.4 times 10 to the negative third Molar. 00:07:20.220 --> 00:07:22.850 Now that we know the concentrations of lead two plus ions 00:07:22.850 --> 00:07:26.080 and sulfate ions, after the two solutions have been mixed, 00:07:26.080 --> 00:07:29.930 we can plug those concentrations into our QSP expression 00:07:29.930 --> 00:07:31.990 and solve for QSP. 00:07:31.990 --> 00:07:36.050 So at this moment in time, QSP is equal to 5.1 times 10 00:07:36.050 --> 00:07:37.760 to the negative six. 00:07:37.760 --> 00:07:41.560 At 25 degrees Celsius, the KSP value for lead two sulfate 00:07:41.560 --> 00:07:45.120 is equal to 6.3 times 10 to the negative seventh. 00:07:45.120 --> 00:07:48.160 QSP at this moment in time is 5.1 times 10 00:07:48.160 --> 00:07:49.690 to the negative six. 00:07:49.690 --> 00:07:54.357 Therefore QSP is greater than KSP. 00:07:55.980 --> 00:08:00.210 Since QSP is greater than KSP, we've exceeded the limit 00:08:00.210 --> 00:08:02.930 of what can dissolve and therefore the solution 00:08:02.930 --> 00:08:05.220 is oversaturated. 00:08:05.220 --> 00:08:09.550 So yes, a precipitate will form and the precipitate 00:08:09.550 --> 00:08:12.190 of lead two sulfate will continue to form 00:08:12.190 --> 00:08:15.493 until QSP is equal to KSP.

Worked example: Calculating solubility from Kₛₚ

https://www.youtube.com/watch?v=Ywvw32DaxUk

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https://www.youtube.com/api/timedtext?v=Ywvw32DaxUk&ei=3FWUZffxLKfxvdIP1cyokAo&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=A54865589E4C3C877EECFEBE4CB272C32F4BBB21.D7CC41F3D442C56FEB7314D8BB40B3FE01478C84&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.170 --> 00:00:01.003 - [Instructor] Let's calculate 00:00:01.003 --> 00:00:03.270 the molar solubility of calcium fluoride 00:00:03.270 --> 00:00:05.620 if the Ksp value for calcium fluoride is 00:00:05.620 --> 00:00:10.540 3.9 times 10 to the negative 11th at 25 degrees Celsius. 00:00:10.540 --> 00:00:11.400 The first step is 00:00:11.400 --> 00:00:15.710 to write the dissolution equation for calcium fluoride. 00:00:15.710 --> 00:00:20.170 So, solid calcium fluoride will dissolve in solution 00:00:20.170 --> 00:00:25.170 to form aqueous calcium two plus ions and fluoride anions. 00:00:26.100 --> 00:00:28.960 And to balance that out, we need to make sure 00:00:28.960 --> 00:00:32.840 and include a two in front of the fluoride anions. 00:00:32.840 --> 00:00:34.900 The next step is to set up an ICE table, 00:00:34.900 --> 00:00:38.260 where I stands for initial concentration, 00:00:38.260 --> 00:00:40.930 C stands for the change in concentration, 00:00:40.930 --> 00:00:44.910 and E stands for equilibrium concentration. 00:00:44.910 --> 00:00:47.420 Before any of the solid calcium fluoride dissolves, 00:00:47.420 --> 00:00:50.190 the initial concentrations of calcium two plus ions 00:00:50.190 --> 00:00:52.970 and fluoride anions in solution is zero. 00:00:52.970 --> 00:00:54.830 So we can go ahead and put a zero in here 00:00:54.830 --> 00:00:58.740 for the initial concentration of the ions in solution. 00:00:58.740 --> 00:01:00.680 Some of the calcium fluoride will dissolve, 00:01:00.680 --> 00:01:02.070 and we don't know how much. 00:01:02.070 --> 00:01:06.500 So I like to represent that by writing -X on the ICE table, 00:01:06.500 --> 00:01:09.090 where X is the concentration of calcium fluoride 00:01:09.090 --> 00:01:10.400 that dissolves. 00:01:10.400 --> 00:01:13.350 Looking at the mole ratios, it's a one-to-one mole ratio 00:01:13.350 --> 00:01:16.720 between calcium fluoride and calcium two plus ions. 00:01:16.720 --> 00:01:17.990 So if we're losing X 00:01:17.990 --> 00:01:20.100 for the concentration of calcium fluoride, 00:01:20.100 --> 00:01:21.570 we must be gaining X 00:01:21.570 --> 00:01:24.650 for the concentration of calcium two plus ions. 00:01:24.650 --> 00:01:27.180 And since it's a one-to-two mole ratio 00:01:27.180 --> 00:01:30.300 for calcium two plus ions to fluoride anions, 00:01:30.300 --> 00:01:33.250 if we're gaining +X for calcium two plus, 00:01:33.250 --> 00:01:35.350 we must gain plus +2X for fluoride anions. 00:01:36.890 --> 00:01:40.790 So the equilibrium concentration of calcium two plus ions is 00:01:40.790 --> 00:01:42.930 zero plus X, or just X, 00:01:42.930 --> 00:01:45.870 and the equilibrium concentration of fluoride anions will be 00:01:45.870 --> 00:01:49.070 zero plus 2X, or just 2X. 00:01:49.070 --> 00:01:51.730 The next step is to write the Ksp expression 00:01:51.730 --> 00:01:53.460 from the balanced equation. 00:01:53.460 --> 00:01:55.360 So Ksp is equal to 00:01:55.360 --> 00:01:58.910 the concentration of calcium two plus ions, 00:01:58.910 --> 00:02:00.800 and since there's a coefficient of one 00:02:00.800 --> 00:02:01.883 in the balanced equation, 00:02:01.883 --> 00:02:04.015 that's the concentration of calcium two plus ions 00:02:04.015 --> 00:02:05.865 raised to the first power, 00:02:05.865 --> 00:02:08.906 times the concentration of fluoride anions, 00:02:08.906 --> 00:02:10.615 and since there is a coefficient of two 00:02:10.615 --> 00:02:12.136 in the balanced equation, 00:02:12.136 --> 00:02:14.610 it's the concentration of fluoride anions raised to 00:02:14.610 --> 00:02:16.040 the second power. 00:02:16.040 --> 00:02:17.790 Pure solids are not included 00:02:17.790 --> 00:02:19.760 in equilibrium constant expression. 00:02:19.760 --> 00:02:21.830 So we're going to leave calcium fluoride 00:02:21.830 --> 00:02:24.540 out of the Ksp expression. 00:02:24.540 --> 00:02:27.150 The concentration of ions in our Ksp expression 00:02:27.150 --> 00:02:29.740 are equilibrium concentrations. 00:02:29.740 --> 00:02:31.220 Therefore we can plug in X 00:02:31.220 --> 00:02:34.360 for the equilibrium concentration of calcium two plus 00:02:34.360 --> 00:02:39.000 and 2X for the equilibrium concentration of fluoride anions. 00:02:39.000 --> 00:02:44.000 We can also plug in the Ksp value for calcium fluoride. 00:02:44.490 --> 00:02:45.390 So that would give us 00:02:45.390 --> 00:02:50.390 3.9 times 10 to the negative 11th is equal to 00:02:50.740 --> 00:02:55.330 X times 2X squared. 00:02:55.330 --> 00:02:57.190 Next we need to solve for X. 00:02:57.190 --> 00:03:01.410 So, 3.9 times 10 to the negative 11th is equal to 00:03:01.410 --> 00:03:03.130 X times 2X squared. 00:03:03.130 --> 00:03:04.570 Well, 2X squared is equal to 00:03:04.570 --> 00:03:08.910 4X squared times X is equal to 4X cubed. 00:03:08.910 --> 00:03:13.410 So to solve for X, we need to divide both sides by four 00:03:13.410 --> 00:03:17.980 and then take the cube root of both sides. 00:03:17.980 --> 00:03:21.100 So we'd take the cube root of the left side 00:03:21.100 --> 00:03:25.683 and the cube root of X cubed. 00:03:26.620 --> 00:03:28.562 That gives us X is equal to 00:03:28.562 --> 00:03:33.562 2.1 times 10 to the negative fourth. 00:03:33.980 --> 00:03:36.100 And looking at our ICE table, X represents 00:03:36.100 --> 00:03:39.740 the equilibrium concentration of calcium two plus ions. 00:03:39.740 --> 00:03:43.010 So 2.1 times 10 to the negative fourth molar is 00:03:43.010 --> 00:03:46.700 the equilibrium concentration of calcium two plus ions. 00:03:46.700 --> 00:03:48.110 For the fluoride anions, 00:03:48.110 --> 00:03:50.620 the equilibrium concentration is 2X. 00:03:50.620 --> 00:03:55.620 So two times 2.1 times 10 to the negative fourth is 4.2, 00:03:55.690 --> 00:03:56.832 let me go ahead and write that down here, 00:03:56.832 --> 00:04:00.210 4.2 times 10 to the negative fourth molar 00:04:00.210 --> 00:04:03.860 for the equilibrium concentration of fluoride anions. 00:04:03.860 --> 00:04:04.900 Our goal was to calculate 00:04:04.900 --> 00:04:07.410 the molar solubility of calcium fluoride. 00:04:07.410 --> 00:04:09.300 And molar solubility refers to 00:04:09.300 --> 00:04:12.470 the concentration of our salt that dissolved 00:04:12.470 --> 00:04:16.030 to form a saturated solution at equilibrium. 00:04:16.030 --> 00:04:17.460 So if X refers to 00:04:17.460 --> 00:04:21.430 the concentration of calcium two plus ions at equilibrium, 00:04:21.430 --> 00:04:23.620 looking at our mole ratios, that's also 00:04:23.620 --> 00:04:26.790 the concentration of calcium fluoride that dissolved. 00:04:26.790 --> 00:04:30.020 Therefore, 2.1 times 10 to the negative fourth molar 00:04:30.020 --> 00:04:34.640 is also the molar solubility of calcium fluoride. 00:04:34.640 --> 00:04:37.770 Technically at a constant temperature of 25 degrees, 00:04:37.770 --> 00:04:40.690 the concentration of a solid doesn't change. 00:04:40.690 --> 00:04:44.170 And so you'll see most textbooks not to put in -X 00:04:44.170 --> 00:04:46.580 on the ICE table. I like to just put it in though 00:04:46.580 --> 00:04:49.630 to remind me that X in this case does refer to 00:04:49.630 --> 00:04:51.263 the molar solubility.

Introduction to solubility equilibria

https://www.youtube.com/watch?v=N9a2r01ToZk

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https://www.youtube.com/api/timedtext?v=N9a2r01ToZk&ei=3FWUZeb8GNH2mLAPm6ij6A4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245324&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=2456109890004492CA9A294C4877CE22068ECF97.1C4DE7A8987CF3CDF2D1BB7AD36F521CDCE2C971&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.610 --> 00:00:01.870 - [Instructor] Let's say we have a beaker 00:00:01.870 --> 00:00:04.700 of distilled water at 25 degrees Celsius. 00:00:04.700 --> 00:00:07.750 And to the beaker, we add some barium sulfate. 00:00:07.750 --> 00:00:10.810 Barium sulfate is a white solid. 00:00:10.810 --> 00:00:12.420 A small amount of the barium sulfate 00:00:12.420 --> 00:00:15.870 dissolves in the water and forms Ba2+ ions in solution 00:00:15.870 --> 00:00:17.980 and sulfate ions in solution. 00:00:17.980 --> 00:00:20.350 So let me draw those in on our diagrams. 00:00:20.350 --> 00:00:22.520 So we're gonna form some Ba2+ ions 00:00:22.520 --> 00:00:25.530 and some sulfate anions. 00:00:25.530 --> 00:00:28.620 But most of the barium sulfate remains undissolved 00:00:28.620 --> 00:00:31.180 and so we'll draw that here sitting on the bottom 00:00:31.180 --> 00:00:33.040 of the beaker. 00:00:33.040 --> 00:00:36.650 So barium sulfate can dissolve to form Ba2+ ions 00:00:36.650 --> 00:00:38.820 and sulfate anions in solution. 00:00:38.820 --> 00:00:41.440 And it's possible for the Ba2+ ion 00:00:41.440 --> 00:00:44.850 to combine with the sulfate anion to form a precipitate, 00:00:44.850 --> 00:00:46.460 of barium sulfate. 00:00:46.460 --> 00:00:49.470 When the rate of dissolution is equal to the rate of 00:00:49.470 --> 00:00:53.690 precipitation, the system is at equilibrium. 00:00:53.690 --> 00:00:55.740 These types of equilibria are referred 00:00:55.740 --> 00:00:58.880 to as solubility equilibria. 00:00:58.880 --> 00:01:00.950 And when the system is at equilibrium, 00:01:00.950 --> 00:01:04.060 the concentrations of Ba2+ ions and sulfate anions 00:01:04.060 --> 00:01:05.630 solution are constant. 00:01:05.630 --> 00:01:08.780 And the amount of solid is constant too. 00:01:08.780 --> 00:01:11.170 And this forms a saturated solution. 00:01:11.170 --> 00:01:13.370 The balanced equation shows the dissolution 00:01:13.370 --> 00:01:15.730 of a salt barium sulfate. 00:01:15.730 --> 00:01:17.070 And from the balanced equation, 00:01:17.070 --> 00:01:20.170 we can write an equilibrium constant expression. 00:01:20.170 --> 00:01:23.015 So we would write the equilibrium constant K 00:01:23.015 --> 00:01:27.410 is equal to the concentration of Ba2+ 00:01:27.410 --> 00:01:29.470 and since there's a coefficient of one 00:01:29.470 --> 00:01:30.620 in the balanced equation, 00:01:30.620 --> 00:01:33.200 it'd be the concentration raised to the first power 00:01:33.200 --> 00:01:35.640 times the concentration of sulfate 00:01:35.640 --> 00:01:39.000 also raised to the first power. 00:01:39.000 --> 00:01:42.110 And since pure solids are left out of equilibrium constant 00:01:42.110 --> 00:01:46.510 expressions, we would not include the solid barium sulfate. 00:01:46.510 --> 00:01:48.510 For solubility equilibria, 00:01:48.510 --> 00:01:53.510 we would write Ksp where sp stands for solubility product. 00:01:53.630 --> 00:01:58.372 The solubility product constant Ksp has only one value for a 00:01:58.372 --> 00:02:01.180 given salt at a specific temperature. 00:02:01.180 --> 00:02:04.340 That temperature is usually 25 degrees Celsius. 00:02:04.340 --> 00:02:08.940 And Ksp indicates how much of that salt will dissolve. 00:02:08.940 --> 00:02:11.430 For example, at 25 degrees Celsius, 00:02:11.430 --> 00:02:13.960 the Ksp value for barium sulfate 00:02:13.960 --> 00:02:16.930 is 1.1 times 10 to the negative 10th. 00:02:16.930 --> 00:02:19.670 When the Ksp value is much less than one, 00:02:19.670 --> 00:02:22.120 that indicates the salt is not very soluble. 00:02:22.120 --> 00:02:25.440 So barium sulfate is not a soluble salt. 00:02:25.440 --> 00:02:28.610 If the Ksp value is greater than one, 00:02:28.610 --> 00:02:31.640 like it is for something like sodium chloride, 00:02:31.640 --> 00:02:33.610 that indicates a soluble salt 00:02:33.610 --> 00:02:36.070 that dissolves easily in water. 00:02:36.070 --> 00:02:39.880 The solubility of a substance refers to the amount of solid 00:02:39.880 --> 00:02:42.930 that dissolves to form a saturated solution. 00:02:42.930 --> 00:02:47.930 Usually the units for solubility are in grams per liter. 00:02:48.440 --> 00:02:52.140 Molar solubility refers to the number of moles of the solid 00:02:52.140 --> 00:02:55.460 that dissolve to form one litter of the saturated solution. 00:02:55.460 --> 00:02:59.550 And therefore the units would be moles per one liter 00:02:59.550 --> 00:03:01.133 or you could just write M. 00:03:03.680 --> 00:03:07.030 Ksp values can be used to predict the relative solubilities 00:03:07.030 --> 00:03:11.030 of salts that produce the same number of ions in solution. 00:03:11.030 --> 00:03:13.560 For example, silver chloride, silver bromide, 00:03:13.560 --> 00:03:17.750 and silver iodide all produce two ions in solution. 00:03:17.750 --> 00:03:20.880 Let's look at the dissolution equation for silver chloride 00:03:20.880 --> 00:03:22.470 to see why this is true. 00:03:22.470 --> 00:03:24.720 Solid silver chloride turns into Ag+ and Cl-. 00:03:26.500 --> 00:03:30.370 So that's one Ag+ ion and one Cl- ion for a total 00:03:30.370 --> 00:03:32.590 of two ions in solution. 00:03:32.590 --> 00:03:35.140 And we could write out similar equations for silver bromide 00:03:35.140 --> 00:03:38.920 and silver iodide, so they all produce two ions in solution. 00:03:38.920 --> 00:03:41.500 However, a salt like Lead(II)chloride 00:03:41.500 --> 00:03:43.940 produces three ions in solution. 00:03:43.940 --> 00:03:47.210 So Lead(II)chloride would give one Pb2+ ion 00:03:47.210 --> 00:03:49.680 and two chloride anions in solution. 00:03:49.680 --> 00:03:51.900 One plus two is three ions. 00:03:51.900 --> 00:03:55.890 Since Lead(II)chloride produces three ions in solution, 00:03:55.890 --> 00:03:59.360 we can determine its solubility relative to the other 00:03:59.360 --> 00:04:02.040 three by comparing Ksp values. 00:04:02.040 --> 00:04:04.390 Here are the Ksp values for the three salts 00:04:04.390 --> 00:04:06.250 at 25 degrees Celsius. 00:04:06.250 --> 00:04:09.530 For silver chloride, it's 1.8 times 10 to the negative 10th, 00:04:09.530 --> 00:04:12.950 for silver bromide, it's 5.0 times 10 to the negative 13th. 00:04:12.950 --> 00:04:15.246 And for silver iodide, it's 8.3 times 10 00:04:15.246 --> 00:04:17.270 to the negative 17th. 00:04:17.270 --> 00:04:20.130 When comparing salts that produce the same number of ions, 00:04:20.130 --> 00:04:22.030 the higher the value of Ksp, 00:04:22.030 --> 00:04:25.020 the higher the solubility of the salt. 00:04:25.020 --> 00:04:29.190 And since silver chloride has the highest Ksp value of these 00:04:29.190 --> 00:04:33.520 three, silver chloride is the most soluble salts. 00:04:33.520 --> 00:04:35.340 For some insight into why this is true, 00:04:35.340 --> 00:04:38.200 let's look at the Ksp expression for silver chloride, 00:04:38.200 --> 00:04:40.990 which we can get from the balanced equation. 00:04:40.990 --> 00:04:43.010 The higher the value for Ksp, 00:04:43.010 --> 00:04:46.290 the higher the concentration of these ions at equilibrium, 00:04:46.290 --> 00:04:49.490 which means that more of the solid must have dissolved. 00:04:49.490 --> 00:04:52.540 Therefore, silver chloride has the highest solubility 00:04:52.540 --> 00:04:54.680 out of these three salts. 00:04:54.680 --> 00:04:58.030 Let's say we have some solid calcium fluoride that we add to 00:04:58.030 --> 00:05:01.300 pure water at 25 degrees Celsius. 00:05:01.300 --> 00:05:04.830 Eventually equilibrium is reached and the equilibrium 00:05:04.830 --> 00:05:07.523 concentration of Ca2+ ions is measured 00:05:07.523 --> 00:05:11.290 to be 2.1 times 10 to the negative 4th M. 00:05:11.290 --> 00:05:15.040 Our goal is to calculate the Ksp for calcium fluoride 00:05:15.040 --> 00:05:17.490 at 25 degrees Celsius. 00:05:17.490 --> 00:05:19.240 The first step is to write out the 00:05:19.240 --> 00:05:22.730 dissolution equation for calcium fluoride. 00:05:22.730 --> 00:05:27.180 So we would write CaF2 solid, and we know that calcium forms 00:05:27.180 --> 00:05:32.180 a 2+ cation, so we would write Ca2+ in aqueous solution 00:05:34.040 --> 00:05:37.210 and to balance everything we'd need two fluoride anions. 00:05:37.210 --> 00:05:41.770 So 2F- also in aqueous solution. 00:05:41.770 --> 00:05:43.790 The next step is to use the balanced equation 00:05:43.790 --> 00:05:45.950 to write the Ksp expression. 00:05:45.950 --> 00:05:50.180 So Ksp is equal to, there's a one as a coefficient 00:05:50.180 --> 00:05:53.000 in front of Ca2+ so it'd be the concentration 00:05:53.000 --> 00:05:56.360 of Ca2+ raise to the first power, 00:05:56.360 --> 00:05:59.220 times the concentration of fluoride anion 00:05:59.220 --> 00:06:01.160 and since there's a two as a coefficient, 00:06:01.160 --> 00:06:04.490 this is the concentration of fluoride anion squared. 00:06:04.490 --> 00:06:08.800 For a Ksp expression, these are equilibrium concentrations, 00:06:08.800 --> 00:06:12.260 and we already know the concentration of Ca2+ 00:06:12.260 --> 00:06:16.260 at equilibrium is 2.1 times 10 to the negative 4th. 00:06:16.260 --> 00:06:17.750 So that can be plugged in for the 00:06:17.750 --> 00:06:21.160 equilibrium concentration of Ca2+. 00:06:21.160 --> 00:06:23.760 So here's our expression with 2.1 times 10 00:06:23.760 --> 00:06:25.450 to the negative 4th plugged in, 00:06:25.450 --> 00:06:27.540 and next we need to plug in the equilibrium 00:06:27.540 --> 00:06:30.090 concentration of fluoride anion. 00:06:30.090 --> 00:06:32.370 We're looking at the disillusion equation, 00:06:32.370 --> 00:06:37.370 the mole ratio of Ca2+ to fluoride anion is 1:2. 00:06:37.650 --> 00:06:40.300 Therefore, at equilibrium, there's twice as many 00:06:40.300 --> 00:06:44.440 fluoride ions in solution as there are Ca2+ ions. 00:06:44.440 --> 00:06:47.190 Therefore the equilibrium concentration of the fluoride 00:06:47.190 --> 00:06:51.960 anion would just be twice this concentration for Ca2+. 00:06:51.960 --> 00:06:55.490 So the equilibrium concentration of fluoride anion 00:06:55.490 --> 00:06:59.533 must be 4.2 times 10 to the negative 4th M. 00:07:00.986 --> 00:07:03.360 And when you do the math, you get that Ksp 00:07:03.360 --> 00:07:06.750 for calcium fluoride is equal to 3.7 times 10 00:07:06.750 --> 00:07:10.690 to the negative 11th at 25 degrees Celsius. 00:07:10.690 --> 00:07:13.320 Ksp values can be difficult to measure 00:07:13.320 --> 00:07:16.060 and therefore different sources often give different values 00:07:16.060 --> 00:07:18.990 for Ksp at the same temperature. 00:07:18.990 --> 00:07:22.690 For example, for calcium fluoride at 25 degrees Celsius, 00:07:22.690 --> 00:07:26.110 one source had Ksp equal to 3.5 times 10 00:07:26.110 --> 00:07:27.520 to the negative 11th. 00:07:27.520 --> 00:07:32.090 Another one had 3.9 times 10 to the negative 11th. 00:07:32.090 --> 00:07:35.332 Since we got 3.7 times 10 to the negative 11th, 00:07:35.332 --> 00:07:38.140 this sounds like a pretty good calculation for the numbers 00:07:38.140 --> 00:07:40.390 that we used for our problem. 00:07:40.390 --> 00:07:42.950 Finally, let's think about the molar solubility 00:07:42.950 --> 00:07:44.260 of calcium fluoride. 00:07:44.260 --> 00:07:47.480 So how many moles of our salt dissolve to form 00:07:47.480 --> 00:07:50.460 one litter of our saturated solution? 00:07:50.460 --> 00:07:53.267 Well, the mole ratio of Ca2+ ions 00:07:53.267 --> 00:07:56.349 to calcium fluoride is 1:1. 00:07:56.349 --> 00:08:01.349 Therefore, the concentration of Ca2+ ions in solution 2.1 00:08:01.420 --> 00:08:03.770 times 10 to the negative 4th M, 00:08:03.770 --> 00:08:06.278 that number must also be the molar solubility 00:08:06.278 --> 00:08:08.340 of calcium fluoride. 00:08:08.340 --> 00:08:10.420 Therefore, for this problem, we could say that we use the 00:08:10.420 --> 00:08:13.100 molar solubility of calcium fluoride to calculate 00:08:13.100 --> 00:08:15.673 the Ksp value for calcium fluoride.

Worked example: Using Le Chȃtelier’s principle to predict shifts in equilibrium

https://www.youtube.com/watch?v=YBEoLzMWCdM

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en

WEBVTT Kind: captions Language: en 00:00:00.490 --> 00:00:03.210 - [Instructor] Carbon monoxide will react with hydrogen gas 00:00:03.210 --> 00:00:05.650 to produce methanol. 00:00:05.650 --> 00:00:07.770 Let's say that the reaction is at equilibrium 00:00:07.770 --> 00:00:09.310 and our job is to figure out 00:00:09.310 --> 00:00:11.100 which direction the equilibrium will shift, 00:00:11.100 --> 00:00:13.620 to the left, to the right, or not at all, 00:00:13.620 --> 00:00:17.390 as we try to make changes to the reaction at equilibrium. 00:00:17.390 --> 00:00:19.640 For example, if we add some hydrogen gas 00:00:19.640 --> 00:00:21.650 to our reaction at equilibrium, 00:00:21.650 --> 00:00:25.690 we're increasing the concentration of one of our reactants. 00:00:25.690 --> 00:00:27.360 According to the Le Chatelier's principle, 00:00:27.360 --> 00:00:29.130 the net reaction will move in the direction 00:00:29.130 --> 00:00:32.420 that decreases the stress placed on the system. 00:00:32.420 --> 00:00:34.170 So if the stress is increased amount 00:00:34.170 --> 00:00:35.860 of one of the reactants, 00:00:35.860 --> 00:00:37.900 the equilibrium will shift to the right 00:00:37.900 --> 00:00:40.610 to get rid of some of that reactant. 00:00:40.610 --> 00:00:43.370 In part B, some methanol is removed. 00:00:43.370 --> 00:00:47.790 So if we're decreasing the concentration of our product, 00:00:47.790 --> 00:00:51.150 the equilibrium's gonna shift to make more of our product, 00:00:51.150 --> 00:00:54.840 therefore, the equilibrium will shift to the right. 00:00:54.840 --> 00:00:56.620 Next, the volume is increased 00:00:56.620 --> 00:00:58.780 on the reaction at equilibrium. 00:00:58.780 --> 00:01:03.660 And if we increase the volume, we decrease the pressure, 00:01:03.660 --> 00:01:05.440 therefore, we could consider the stress 00:01:05.440 --> 00:01:07.280 to be decreased pressure. 00:01:07.280 --> 00:01:09.850 Le Chatelier's principle says the net reaction is gonna go 00:01:09.850 --> 00:01:11.640 in the direction that relieves the stress. 00:01:11.640 --> 00:01:13.890 So if the stress is decreased pressure, 00:01:13.890 --> 00:01:18.140 the net reaction is going to shift to increase the pressure. 00:01:18.140 --> 00:01:19.790 And we can figure out which direction that is 00:01:19.790 --> 00:01:21.760 by looking at the balanced equation. 00:01:21.760 --> 00:01:23.000 On the reactant side, 00:01:23.000 --> 00:01:25.340 there's one mole of gas and two moles of gas 00:01:25.340 --> 00:01:27.310 for a total of three moles of gas. 00:01:27.310 --> 00:01:30.040 On the product side, there's only one mole of gas. 00:01:30.040 --> 00:01:32.280 So there's three moles of gas on the left 00:01:32.280 --> 00:01:34.760 and only one mole of gas on the right. 00:01:34.760 --> 00:01:36.940 Since the net reaction is going to try 00:01:36.940 --> 00:01:38.740 to increase the pressure, 00:01:38.740 --> 00:01:41.420 the equilibrium shifts to the left, 00:01:41.420 --> 00:01:44.520 toward the side that's gonna form more moles of gas, 00:01:44.520 --> 00:01:47.300 therefore increasing the pressure. 00:01:47.300 --> 00:01:49.070 Next, we try adding some neon gas 00:01:49.070 --> 00:01:52.020 to our reaction mixture at equilibrium. 00:01:52.020 --> 00:01:53.850 Well, neon gas is an inert gas, 00:01:53.850 --> 00:01:55.010 which means it doesn't react 00:01:55.010 --> 00:01:57.200 with any of our reactants or products. 00:01:57.200 --> 00:01:59.010 And if we look at the expression 00:01:59.010 --> 00:02:01.160 for the reaction quotient Qp, 00:02:01.160 --> 00:02:03.160 neon gas is not included. 00:02:03.160 --> 00:02:05.300 Therefore, adding neon gas is not going 00:02:05.300 --> 00:02:07.590 to change the value for Qp, 00:02:07.590 --> 00:02:10.500 so the reaction remains at equilibrium. 00:02:10.500 --> 00:02:12.470 So the answer is there's no shift 00:02:12.470 --> 00:02:15.280 when an inert gas is added. 00:02:15.280 --> 00:02:17.190 And that might sound a little strange at first 00:02:17.190 --> 00:02:18.980 because adding neon gas means 00:02:18.980 --> 00:02:21.540 that the total pressure would increase, 00:02:21.540 --> 00:02:24.080 the total pressure since we're adding a gas. 00:02:24.080 --> 00:02:26.700 However, the partial pressures stay the same. 00:02:26.700 --> 00:02:29.320 So the partial pressures for methanol 00:02:29.320 --> 00:02:32.290 and carbon monoxide and hydrogen gas actually stay the same 00:02:32.290 --> 00:02:34.760 and therefore Q doesn't change. 00:02:34.760 --> 00:02:38.000 Next, we add a catalyst to our reaction at equilibrium. 00:02:38.000 --> 00:02:39.300 Catalysts speed up reactions 00:02:39.300 --> 00:02:41.420 by lowering deactivation energy. 00:02:41.420 --> 00:02:43.970 However, the catalyst is gonna speed up that the forward 00:02:43.970 --> 00:02:46.310 and the reverse reactions by the same amount 00:02:46.310 --> 00:02:49.720 and therefore the reaction remains at equilibrium. 00:02:49.720 --> 00:02:52.190 So there's no shift when a catalyst is added 00:02:52.190 --> 00:02:54.700 to a reaction at equilibrium. 00:02:54.700 --> 00:02:55.690 And then in part F, 00:02:55.690 --> 00:02:57.580 let's try decreasing the temperature 00:02:57.580 --> 00:02:59.930 on the reaction at equilibrium. 00:02:59.930 --> 00:03:01.740 Well, this reaction is exothermic 00:03:01.740 --> 00:03:03.920 because Delta H is less than 0, 00:03:03.920 --> 00:03:06.680 so we can treat heat as a product. 00:03:06.680 --> 00:03:09.900 So we go ahead and write heat on the product side. 00:03:09.900 --> 00:03:11.880 If we treat heat like a product, 00:03:11.880 --> 00:03:15.210 decreasing the temperature is like decreasing the amount 00:03:15.210 --> 00:03:16.850 of our product, therefore, 00:03:16.850 --> 00:03:19.230 the net reaction will move to the right 00:03:19.230 --> 00:03:21.410 to make more of the product. 00:03:21.410 --> 00:03:23.110 Whether that reaction moves to the right, 00:03:23.110 --> 00:03:25.080 you can think about that being an increase 00:03:25.080 --> 00:03:26.550 in the amount of products 00:03:26.550 --> 00:03:29.740 and therefore a decrease in the amount of reactants. 00:03:29.740 --> 00:03:31.200 And when you increase the products 00:03:31.200 --> 00:03:32.750 and decrease the reactants, 00:03:32.750 --> 00:03:36.130 you increase the value for the equilibrium constant. 00:03:36.130 --> 00:03:40.020 Therefore, lowering the temperature causes an increase 00:03:40.020 --> 00:03:44.390 in the equilibrium constant for an exothermic reaction. 00:03:44.390 --> 00:03:45.740 Note that changing the temperature 00:03:45.740 --> 00:03:48.180 in part F is the only change 00:03:48.180 --> 00:03:50.710 that actually changed the equilibrium constant. 00:03:50.710 --> 00:03:53.250 So in all the other ones, in A through E, 00:03:53.250 --> 00:03:57.233 the equilibrium constant stayed the same value.

Worked example: Calculating the equilibrium total pressure after a change in volume

https://www.youtube.com/watch?v=2q5UPN4tF20

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en

WEBVTT Kind: captions Language: en 00:00:00.220 --> 00:00:02.610 - [Instructor] Phosphorus pentachloride will decompose 00:00:02.610 --> 00:00:06.220 into phosphorus trichloride and chlorine gas. 00:00:06.220 --> 00:00:10.930 Kp for this reaction is equal to .500 at 500 Kelvin. 00:00:10.930 --> 00:00:13.090 Let's say that this reaction is at equilibrium 00:00:13.090 --> 00:00:16.370 and a reaction vessel that has a volume of 2.0 liters 00:00:16.370 --> 00:00:18.450 and the equilibrium partial pressure, 00:00:18.450 --> 00:00:22.100 of PCl5 is equal to .980 atmospheres. 00:00:22.100 --> 00:00:24.670 The equilibrium partial pressure of PCl3 00:00:24.670 --> 00:00:27.310 is equal to .700 atmospheres. 00:00:27.310 --> 00:00:29.840 And the equilibrium partial pressure of chlorine gas 00:00:29.840 --> 00:00:32.930 is equal to .700 atmospheres. 00:00:32.930 --> 00:00:35.090 If we add up those three partial pressures, 00:00:35.090 --> 00:00:38.800 we get the total pressure of the gas mixture at equilibrium, 00:00:38.800 --> 00:00:41.860 which is equal to 2.38 atmospheres. 00:00:41.860 --> 00:00:45.150 And we're gonna call this total pressure, P1. 00:00:45.150 --> 00:00:47.730 If we decrease the volume from two liters 00:00:47.730 --> 00:00:48.950 down to one liter, 00:00:48.950 --> 00:00:52.840 and we keep the temperature constant at 500 Kelvin, 00:00:52.840 --> 00:00:55.360 we decrease the volume by a factor of two, 00:00:55.360 --> 00:00:59.290 which means we increase the pressure by a factor of two. 00:00:59.290 --> 00:01:02.930 So, all of the partial pressures of our gases double, 00:01:02.930 --> 00:01:05.130 and there's a new total pressure, 00:01:05.130 --> 00:01:07.420 which is twice the original total pressure. 00:01:07.420 --> 00:01:11.160 And this new total pressure is equal to 4.76 atmospheres. 00:01:11.160 --> 00:01:14.580 And from now on, we'll call this total pressure, P2. 00:01:14.580 --> 00:01:16.320 So, when the volume was two liters, 00:01:16.320 --> 00:01:18.780 the reaction started out at equilibrium 00:01:18.780 --> 00:01:21.430 and by decreasing the volume to one liter, 00:01:21.430 --> 00:01:22.850 and by doubling the pressure, 00:01:22.850 --> 00:01:25.870 we've introduced a stress to the system. 00:01:25.870 --> 00:01:27.300 And so, at this moment in time, 00:01:27.300 --> 00:01:28.910 when these are the partial pressures, 00:01:28.910 --> 00:01:32.173 the reaction is not at equilibrium. 00:01:33.210 --> 00:01:35.380 Le Chatelier's principle says the net reaction 00:01:35.380 --> 00:01:38.290 will move in the direction that decreases the stress. 00:01:38.290 --> 00:01:41.150 So, if the stress is an increase in the pressure, 00:01:41.150 --> 00:01:43.210 the net reaction will move in the direction 00:01:43.210 --> 00:01:45.500 that decreases the pressure. 00:01:45.500 --> 00:01:46.910 Looking at the balanced equation, 00:01:46.910 --> 00:01:49.320 there's one mole of gas on the reactant side, 00:01:49.320 --> 00:01:52.960 and there are two moles of gas on the product side. 00:01:52.960 --> 00:01:54.154 So, if the net reaction goes 00:01:54.154 --> 00:01:56.560 from the products to the reactants, 00:01:56.560 --> 00:01:58.581 if the net reaction goes to the left, 00:01:58.581 --> 00:02:00.960 the net reaction is going to the side 00:02:00.960 --> 00:02:03.890 with the smaller number of moles of gas, 00:02:03.890 --> 00:02:07.540 which would decrease the pressure and relieve the stress. 00:02:07.540 --> 00:02:09.493 The net reaction keeps moving to the left 00:02:09.493 --> 00:02:13.110 until equilibrium is reestablished. 00:02:13.110 --> 00:02:15.160 And when equilibrium is reestablished, 00:02:15.160 --> 00:02:18.860 there'll be a new total pressure, which we'll call P3. 00:02:18.860 --> 00:02:22.890 So, our goal is to calculate P3, 00:02:22.890 --> 00:02:26.670 so we can compare it to P1 and P2. 00:02:26.670 --> 00:02:29.210 And we're gonna do this in a quantitative way 00:02:29.210 --> 00:02:31.730 and in a more qualitative way. 00:02:31.730 --> 00:02:32.920 Let's use an ICE table 00:02:32.920 --> 00:02:36.590 to help us figure out the final total pressure, P3. 00:02:36.590 --> 00:02:37.423 In an ICE table, 00:02:37.423 --> 00:02:40.210 I stands for the initial partial pressure in this case, 00:02:40.210 --> 00:02:43.700 C is the change and E is the equilibrium partial pressure. 00:02:43.700 --> 00:02:46.360 The initial partial pressure of PCl5, 00:02:46.360 --> 00:02:48.667 after the volume was reduced to one liter, 00:02:48.667 --> 00:02:51.280 was 1.96 atmospheres. 00:02:51.280 --> 00:02:53.890 And the partial pressures of PCl3 and Cl2 00:02:53.890 --> 00:02:57.090 were both 1.40 atmospheres. 00:02:57.090 --> 00:02:58.938 We already used Le Chatelier's principle 00:02:58.938 --> 00:03:03.000 to realize the net reaction's going to go to the left, 00:03:03.000 --> 00:03:04.880 which means we're going to decrease 00:03:04.880 --> 00:03:06.150 in the amount of our products, 00:03:06.150 --> 00:03:09.061 and we're going to increase the amount of the reactants. 00:03:09.061 --> 00:03:12.310 So, if we're gonna increase the amount of PCl5, 00:03:12.310 --> 00:03:13.880 we don't know how much we're gonna increase, 00:03:13.880 --> 00:03:15.080 we're gonna call that x. 00:03:15.080 --> 00:03:16.570 But we know it's going to increase, 00:03:16.570 --> 00:03:20.110 so we write plus x under the change part on the ICE table. 00:03:20.110 --> 00:03:24.450 And since our mole ratio of PCl5 to PCl3 is one to one, 00:03:24.450 --> 00:03:26.930 if we're gaining x for PCl5, 00:03:26.930 --> 00:03:29.860 we must be losing x for PCl3. 00:03:29.860 --> 00:03:31.570 And the same goes for Cl2, 00:03:31.570 --> 00:03:33.170 since there's a coefficient of one. 00:03:33.170 --> 00:03:36.370 So, we write minus x in our ICE table. 00:03:36.370 --> 00:03:39.340 Therefore, the equilibrium partial pressure of PCl5 00:03:39.340 --> 00:03:43.986 would be 1.96 plus x. 00:03:43.986 --> 00:03:46.820 The equilibrium partial pressure of PCl3 00:03:46.820 --> 00:03:50.140 would be 1.40 minus x. 00:03:50.140 --> 00:03:52.500 And the equilibrium partial pressure of Cl2 00:03:52.500 --> 00:03:56.120 would also be 1.40 minus x. 00:03:56.120 --> 00:03:58.381 Next, we can plug in the equilibrium partial pressures 00:03:58.381 --> 00:04:00.869 into our Kp expression. 00:04:00.869 --> 00:04:03.970 So, we can plug in 1.40 minus x 00:04:03.970 --> 00:04:06.793 for the equilibrium partial pressure of PCl3, 00:04:06.793 --> 00:04:11.460 1.40 minus x for the equilibrium partial pressure of Cl2, 00:04:11.460 --> 00:04:14.010 and 1.96 plus x 00:04:14.010 --> 00:04:17.514 for the equilibrium partial pressure of PCl5. 00:04:17.514 --> 00:04:19.930 And we can also plug in the value 00:04:19.930 --> 00:04:22.630 for the equilibrium constant, Kp. 00:04:22.630 --> 00:04:25.360 Here's what our equilibrium constant expression looks like 00:04:25.360 --> 00:04:27.130 with everything plugged in. 00:04:27.130 --> 00:04:29.780 And next, we would need to solve for x, 00:04:29.780 --> 00:04:32.610 which would involve the use of a quadratic equation. 00:04:32.610 --> 00:04:34.100 And when you do all that math, 00:04:34.100 --> 00:04:36.890 you find that x is equal to .330. 00:04:40.140 --> 00:04:42.810 Now that we know x is equal to .33, 00:04:42.810 --> 00:04:45.440 we can solve for the equilibrium partial pressures. 00:04:45.440 --> 00:04:49.760 1.96 plus .33 is equal to 2.29. 00:04:49.760 --> 00:04:52.660 Therefore, the equilibrium partial pressure of PCl5 00:04:52.660 --> 00:04:54.970 is 2.29 atmospheres, 00:04:54.970 --> 00:04:58.003 for PCl3, it's 1.40 minus x, 00:04:58.003 --> 00:05:02.510 1.40 minus 0.33 is equal to 1.07. 00:05:02.510 --> 00:05:05.150 Therefore, the equilibrium partial pressure of PCl3 00:05:05.150 --> 00:05:07.520 is equal to 1.07 atmospheres. 00:05:07.520 --> 00:05:09.990 And it's the same math for Cl2. 00:05:09.990 --> 00:05:13.000 So, the equilibrium partial pressure of Cl2 00:05:13.000 --> 00:05:16.640 is also 1.07 atmospheres. 00:05:16.640 --> 00:05:18.840 So, to find the total pressure, P3, 00:05:18.840 --> 00:05:22.680 we simply need to add up the individual partial pressures. 00:05:22.680 --> 00:05:27.680 So, 2.29 plus 1.07 plus 1.07 is equal to 4.43 atmospheres. 00:05:30.295 --> 00:05:33.878 Therefore, P3 is equal to 4.43 atmospheres. 00:05:38.940 --> 00:05:40.830 Doing the math helps us realize that x 00:05:40.830 --> 00:05:42.295 is not a very large number. 00:05:42.295 --> 00:05:44.730 And the reason why x is not a very large number 00:05:44.730 --> 00:05:49.210 is because the Kp value is equal to .500 for this reaction. 00:05:49.210 --> 00:05:51.270 And when K is close to one, 00:05:51.270 --> 00:05:52.670 there's an appreciable amount 00:05:52.670 --> 00:05:55.410 of both reactants and products at equilibrium. 00:05:55.410 --> 00:05:58.630 And we can see that with our equilibrium partial pressures. 00:05:58.630 --> 00:06:01.200 There's an appreciable amount of both of them. 00:06:01.200 --> 00:06:02.860 And since there has to be a decent amount 00:06:02.860 --> 00:06:05.133 of both reactants and products at equilibrium, 00:06:05.133 --> 00:06:06.900 we're not gonna see huge change 00:06:06.900 --> 00:06:09.580 from these initial partial pressures. 00:06:09.580 --> 00:06:12.860 So, there will definitely be a shift to the left 00:06:12.860 --> 00:06:14.950 to decrease the pressure. 00:06:14.950 --> 00:06:17.236 And when these were the partial pressures, 00:06:17.236 --> 00:06:21.150 if you remember P2 was equal to 4.76, 00:06:21.150 --> 00:06:23.260 so there's gonna be a decrease of pressure. 00:06:23.260 --> 00:06:24.440 So, there's gonna be a decrease, 00:06:24.440 --> 00:06:27.550 so the pressure is going to go down from 4.76, 00:06:27.550 --> 00:06:29.480 but since there's not a large change, 00:06:29.480 --> 00:06:31.660 it's not gonna be a huge change. 00:06:31.660 --> 00:06:36.480 And that's why we saw P3 only dropped to 4.43 atmospheres. 00:06:36.480 --> 00:06:38.240 So, if you go back to the original problem, 00:06:38.240 --> 00:06:40.800 and our goal is to figure out P3 00:06:40.800 --> 00:06:43.320 in relation to P1 and to P2, 00:06:43.320 --> 00:06:45.400 without doing all of that math, 00:06:45.400 --> 00:06:46.560 we could think to ourselves, 00:06:46.560 --> 00:06:49.120 okay, so, we decrease the volume by a factor of two, 00:06:49.120 --> 00:06:51.470 which doubled the total pressure. 00:06:51.470 --> 00:06:53.340 But then, the net reaction moves to the left 00:06:53.340 --> 00:06:54.873 to decrease the pressure. 00:06:54.873 --> 00:06:57.682 Since it's not gonna move much to the left, 00:06:57.682 --> 00:07:00.270 it's not gonna decrease the pressure by a lot. 00:07:00.270 --> 00:07:03.190 Therefore, the final pressure P3 00:07:03.190 --> 00:07:06.095 is going to be a little less than 4.76, 00:07:06.095 --> 00:07:09.953 but greater than 2.38 atmospheres.

Le Chȃtelier’s principle: Changing temperature

https://www.youtube.com/watch?v=j7FOzKIDrg8

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en

WEBVTT Kind: captions Language: en 00:00:00.260 --> 00:00:02.720 - [Instructor] Le Chateliers's Principle says if a stress 00:00:02.720 --> 00:00:05.260 is applied to a reaction at equilibrium, 00:00:05.260 --> 00:00:07.710 the net reaction goes in the direction that relieves 00:00:07.710 --> 00:00:09.060 the stress. 00:00:09.060 --> 00:00:11.580 One possible stress is to change the temperature 00:00:11.580 --> 00:00:13.840 of the reaction at equilibrium. 00:00:13.840 --> 00:00:16.560 As an example, let's look at the hypothetical reaction 00:00:16.560 --> 00:00:20.070 where gas A turns into gas B. 00:00:20.070 --> 00:00:22.980 Delta H for this reaction is less than zero, 00:00:22.980 --> 00:00:27.170 which tells us this is an exothermic reaction. 00:00:27.170 --> 00:00:30.070 And for an exothermic reaction, heat is given off, 00:00:30.070 --> 00:00:31.110 heat is released. 00:00:31.110 --> 00:00:33.520 Therefore we can go ahead and write plus heat 00:00:33.520 --> 00:00:35.620 on the product side. 00:00:35.620 --> 00:00:38.820 Let's say that our hypothetical reaction is at equilibrium, 00:00:38.820 --> 00:00:40.560 and then we change the temperature, 00:00:40.560 --> 00:00:44.350 so we're going to increase the temperature. 00:00:44.350 --> 00:00:45.980 According to Le Chatelier's Principal, 00:00:45.980 --> 00:00:47.750 the net reaction is gonna go in the direction 00:00:47.750 --> 00:00:49.350 that decreases the stress. 00:00:49.350 --> 00:00:52.940 And if we treat heat if we treat like a product 00:00:52.940 --> 00:00:55.140 and we've increased the temperature, 00:00:55.140 --> 00:00:58.260 it says if we've increased the amount of one of our products 00:00:58.260 --> 00:01:01.100 and therefore the net reaction is going to shift to the left 00:01:01.100 --> 00:01:03.670 to decrease a product. 00:01:03.670 --> 00:01:06.720 Let's use particulate diagrams and the reaction quotient Q 00:01:06.720 --> 00:01:09.660 to explain what's going on when we increase the temperature 00:01:09.660 --> 00:01:12.060 on our reaction at equilibrium. 00:01:12.060 --> 00:01:14.360 The first particular diagram shows the reaction 00:01:14.360 --> 00:01:17.680 at equilibrium, and let's prove that by calculating QC 00:01:17.680 --> 00:01:19.360 at this moment in time. 00:01:19.360 --> 00:01:21.820 We can get the expression for the reaction quotient, 00:01:21.820 --> 00:01:24.680 QC by looking at the balanced equation. 00:01:24.680 --> 00:01:26.770 So we have coefficients of one in front of A 00:01:26.770 --> 00:01:29.400 and in front of B, therefore QC is equal 00:01:29.400 --> 00:01:31.440 to the concentration of B to the first power 00:01:31.440 --> 00:01:33.350 divided by the concentration of A, 00:01:33.350 --> 00:01:35.070 also to the first power. 00:01:35.070 --> 00:01:37.400 To find the concentration of B, we know that B 00:01:37.400 --> 00:01:39.230 is represented by blue spheres, 00:01:39.230 --> 00:01:42.150 so there are 1, 2, 3 blue spheres, 00:01:42.150 --> 00:01:45.390 And if each sphere represents 0.1 moles of a substance 00:01:45.390 --> 00:01:49.430 three times 0.1 is equal to 0.3 moles of B 00:01:49.430 --> 00:01:52.644 and at the volumes equal to 1.0 liter, 00:01:52.644 --> 00:01:56.770 0.3 divided by 1.0 liter is 0.3 Molar. 00:01:56.770 --> 00:02:01.770 So the concentration of B is equal to 0.3 Molar. 00:02:02.390 --> 00:02:05.210 There are also three particles of A 00:02:05.210 --> 00:02:07.030 therefore the concentration of A 00:02:07.030 --> 00:02:11.270 is also equal to 0.3 Molar. 00:02:11.270 --> 00:02:13.410 0.3 divided by 0.3 is equal to one. 00:02:13.410 --> 00:02:16.670 So QC at this moment of time is equal to one. 00:02:16.670 --> 00:02:21.100 KC for this reaction is equal to one at 25 degrees Celsius. 00:02:21.100 --> 00:02:24.070 So QC is equal to KC. 00:02:24.070 --> 00:02:28.970 And when QC is equal to KC, the reaction is at equilibrium. 00:02:28.970 --> 00:02:31.570 So for this first particular diagram, 00:02:31.570 --> 00:02:33.353 the reaction's at equilibrium. 00:02:34.200 --> 00:02:36.300 Next, we introduced the stress to the reaction 00:02:36.300 --> 00:02:39.700 at equilibrium and the stress is an increase 00:02:39.700 --> 00:02:41.330 in the temperature. 00:02:41.330 --> 00:02:44.430 In general, for an exothermic reaction, 00:02:44.430 --> 00:02:46.840 increasing the temperature lowers the value 00:02:46.840 --> 00:02:48.930 for the equilibrium constant. 00:02:48.930 --> 00:02:52.440 So for this hypothetical reaction at 25 degrees Celsius, 00:02:52.440 --> 00:02:55.500 KC is equal to one, but since we've increased 00:02:55.500 --> 00:02:59.120 the temperature, the value for the equilibrium constant 00:02:59.120 --> 00:03:00.370 is going to decrease. 00:03:00.370 --> 00:03:04.990 So let's say it goes 2.5, if we increase the temperature 00:03:04.990 --> 00:03:07.863 to 30 degrees Celsius. 00:03:08.770 --> 00:03:13.520 So if we calculate QC for our second particular diagram, 00:03:13.520 --> 00:03:16.270 we still have three blues and three reds, 00:03:16.270 --> 00:03:17.730 and the volume is still the same, 00:03:17.730 --> 00:03:21.740 therefore QC is still equal to one, 00:03:21.740 --> 00:03:24.280 but the difference is KC has now changed, 00:03:24.280 --> 00:03:27.020 so QC is not equal to KC, 00:03:27.020 --> 00:03:30.170 so we are not at equilibrium. 00:03:30.170 --> 00:03:33.830 And in this case, QC is greater than KC 00:03:33.830 --> 00:03:36.583 'cause QC is equal to one and KC is equal to 0.5. 00:03:37.660 --> 00:03:42.660 And when QC is greater than KC, there are too many products 00:03:42.960 --> 00:03:44.890 and not enough reactants. 00:03:44.890 --> 00:03:49.890 And therefore the net reaction goes to the left. 00:03:49.890 --> 00:03:52.030 When the net reaction goes to the left, 00:03:52.030 --> 00:03:55.430 we're going to have be turned into A. 00:03:55.430 --> 00:04:00.120 So we should see one blue sphere turn into one red sphere. 00:04:00.120 --> 00:04:03.190 So if we have three blues and three reds, 00:04:03.190 --> 00:04:08.010 and one blue turns into a red, that gives us only two blues 00:04:08.010 --> 00:04:10.550 and four reds now. 00:04:10.550 --> 00:04:13.810 So when we calculate QC for our third particular diagram, 00:04:13.810 --> 00:04:17.440 the concentration of B would be 0.2 Molar, 00:04:17.440 --> 00:04:21.650 and the concentration of A would be 0.4 Molar. 00:04:21.650 --> 00:04:26.650 So 0.2 divided by 0.4 is equal to 0.5. 00:04:27.010 --> 00:04:30.890 Well, KC is also equal to 0.5. 00:04:30.890 --> 00:04:34.580 Therefore QC is equal to KC, 00:04:34.580 --> 00:04:37.850 and the reaction is at equilibrium. 00:04:37.850 --> 00:04:39.740 And when a reaction is at equilibrium, 00:04:39.740 --> 00:04:43.830 the concentrations of reactants and products are constant. 00:04:43.830 --> 00:04:46.980 Let's go back to our hypothetical reaction at equilibrium, 00:04:46.980 --> 00:04:50.060 but this time we're going to decrease the temperature. 00:04:50.060 --> 00:04:53.130 If we treat heat like a product decrease in the temperature 00:04:53.130 --> 00:04:56.060 is like decreasing the amount of one of the products. 00:04:56.060 --> 00:04:58.460 Therefore the net reaction will go to the right 00:04:58.460 --> 00:05:00.490 to make more product. 00:05:00.490 --> 00:05:02.820 If we approach this problem by thinking about the reaction 00:05:02.820 --> 00:05:06.100 quotient Q for an exothermic reaction, 00:05:06.100 --> 00:05:09.840 a decrease in temperature in general causes an increase 00:05:09.840 --> 00:05:11.930 in the equilibrium constants. 00:05:11.930 --> 00:05:14.570 And if the equilibrium constant increases, 00:05:14.570 --> 00:05:17.130 then Q would be less than K. 00:05:17.130 --> 00:05:19.340 And when Q is less than K, 00:05:19.340 --> 00:05:22.850 the net reaction goes to the right. 00:05:22.850 --> 00:05:25.010 The net reaction would continue to go to the right 00:05:25.010 --> 00:05:26.700 until Q is equal to K, 00:05:26.700 --> 00:05:29.990 and equilibrium has been re-established. 00:05:29.990 --> 00:05:32.240 Next let's look at an endothermic reaction 00:05:32.240 --> 00:05:34.800 where Delta H is greater than zero. 00:05:34.800 --> 00:05:39.140 When six water molecules complex to a Cobalt two plus ion 00:05:39.140 --> 00:05:42.800 the resulting complex ion is pink in color. 00:05:42.800 --> 00:05:46.440 And when for chloride anions complex to a cobalt two plus 00:05:46.440 --> 00:05:50.910 ion, the resulting complex ion is blue in color. 00:05:50.910 --> 00:05:53.830 When the pink ion reacts with four chloride anions, 00:05:53.830 --> 00:05:55.850 the blue ion is formed. 00:05:55.850 --> 00:05:58.050 And since this reaction is endothermic, 00:05:58.050 --> 00:06:01.363 we can put heat on the reactant side. 00:06:02.500 --> 00:06:04.770 We're gonna use these particular diagrams down here 00:06:04.770 --> 00:06:07.210 to help us understand what happens to an endothermic 00:06:07.210 --> 00:06:10.340 reaction at equilibrium when the temperature changes, 00:06:10.340 --> 00:06:12.640 however, these drawings aren't designed to be completely 00:06:12.640 --> 00:06:14.560 accurate for this particular reaction. 00:06:14.560 --> 00:06:18.160 They're just to help us understand what color we would see. 00:06:18.160 --> 00:06:21.410 For example, let's say that this middle particulate diagram 00:06:21.410 --> 00:06:23.870 represents the reaction at equilibrium. 00:06:23.870 --> 00:06:26.710 And if there are decent amounts of both the blue ion 00:06:26.710 --> 00:06:29.070 and the pink ion at equilibrium, 00:06:29.070 --> 00:06:31.280 the resulting equilibrium mixture, 00:06:31.280 --> 00:06:33.170 so this is an aqueous solution 00:06:33.170 --> 00:06:36.343 would appear to be purple or violet. 00:06:38.060 --> 00:06:40.950 If we were to increase the temperature 00:06:40.950 --> 00:06:42.650 for this endothermic reaction, 00:06:42.650 --> 00:06:45.270 we treat heat like a reactant. 00:06:45.270 --> 00:06:47.880 So increasing the temperature is like increasing 00:06:47.880 --> 00:06:49.860 the amount of a reactant. 00:06:49.860 --> 00:06:53.730 And therefore the net reaction will shift to the right 00:06:53.730 --> 00:06:56.620 to get rid of some of that reactant. 00:06:56.620 --> 00:06:58.210 Whether the net reaction goes to the right, 00:06:58.210 --> 00:07:00.800 we're gonna increase in the amount of blue ion, 00:07:00.800 --> 00:07:04.460 and we're going to decrease in the amount of pink ion. 00:07:04.460 --> 00:07:08.240 Therefore looking at this particular diagram on the right 00:07:08.240 --> 00:07:12.720 there are now more blue ions than there are pink ions 00:07:12.720 --> 00:07:16.000 compared to our equilibrium mixture in the middle. 00:07:16.000 --> 00:07:18.480 Therefore for this third particular diagram, 00:07:18.480 --> 00:07:22.420 the resulting aqueous solution is going to look blue. 00:07:22.420 --> 00:07:25.260 If we think about those using Q in general, 00:07:25.260 --> 00:07:28.670 for an endothermic reaction, an increase in temperature 00:07:28.670 --> 00:07:32.490 causes an increase in the equilibrium constant K. 00:07:32.490 --> 00:07:37.490 And if K increases, then the reaction quotient Q 00:07:37.500 --> 00:07:39.410 is less than K. 00:07:39.410 --> 00:07:40.970 And when Q is less than K, 00:07:40.970 --> 00:07:44.180 the net reaction goes to the right. 00:07:44.180 --> 00:07:47.720 Now let's go back to the middle particular diagram. 00:07:47.720 --> 00:07:49.790 And so there are reactions at equilibrium 00:07:49.790 --> 00:07:53.000 and this time we're going to decrease the temperature. 00:07:53.000 --> 00:07:55.090 If we treat heat as a reactant, 00:07:55.090 --> 00:07:56.660 and we decrease the temperature, 00:07:56.660 --> 00:07:59.000 it's like we're decreasing one of our reactants. 00:07:59.000 --> 00:08:01.430 Therefore the net reaction will shift to the left 00:08:01.430 --> 00:08:03.920 to make more of our reactant. 00:08:03.920 --> 00:08:05.840 And when that reaction goes to the left, 00:08:05.840 --> 00:08:08.320 we're gonna decrease in the amount of the blue ion, 00:08:08.320 --> 00:08:12.110 and we're going to increase in the amount of the pink ion. 00:08:12.110 --> 00:08:14.930 So when we compare the middle particulate diagram 00:08:14.930 --> 00:08:17.110 to the one on the left and the one that left, 00:08:17.110 --> 00:08:19.230 there's a lot more of the pink ion, 00:08:19.230 --> 00:08:20.970 then there is of the blue ion. 00:08:20.970 --> 00:08:23.860 Therefore the overall solution, 00:08:23.860 --> 00:08:27.380 the overall aqueous solution is going to appear pink. 00:08:27.380 --> 00:08:29.990 If we think about what's happening using Q 00:08:29.990 --> 00:08:33.020 for an endothermic reaction in general, 00:08:33.020 --> 00:08:34.540 when you decrease the temperature, 00:08:34.540 --> 00:08:37.040 you decrease the equilibrium constant. 00:08:37.040 --> 00:08:40.030 And if the equilibrium constant decreases 00:08:40.030 --> 00:08:42.680 now, Q would be greater than K, 00:08:42.680 --> 00:08:45.410 which means too many products and not enough reactance. 00:08:45.410 --> 00:08:48.670 Therefore the net reaction would go to the left. 00:08:48.670 --> 00:08:51.400 Let's go back to our exothermic reaction. 00:08:51.400 --> 00:08:54.270 At this time let's pretend like we're starting with only A, 00:08:54.270 --> 00:08:56.710 so we start with only A and we have none of B. 00:08:56.710 --> 00:08:59.970 And our goal is to make as much as we possibly can 00:08:59.970 --> 00:09:02.440 and to do it as fast as possible. 00:09:02.440 --> 00:09:05.160 One way to increase the rate of a reaction 00:09:05.160 --> 00:09:07.930 would be to increase the temperature. 00:09:07.930 --> 00:09:10.420 However, for an exothermic reaction, 00:09:10.420 --> 00:09:13.960 increasing the temperature decreases the equilibrium 00:09:13.960 --> 00:09:15.170 constant K. 00:09:15.170 --> 00:09:17.870 And if you decrease the equilibrium constant K, 00:09:17.870 --> 00:09:20.280 you would decrease the amount of B 00:09:20.280 --> 00:09:23.290 that you would have when you reach equilibrium. 00:09:23.290 --> 00:09:25.490 So we can't run our hypothetical reaction 00:09:25.490 --> 00:09:27.770 at too high of a temperature because that would decrease 00:09:27.770 --> 00:09:29.820 the equilibrium constant. 00:09:29.820 --> 00:09:32.250 So instead to speed up the rate of the reaction, 00:09:32.250 --> 00:09:34.633 we could add a catalyst, 00:09:36.110 --> 00:09:39.760 let's look at a graph of concentration of B versus time, 00:09:39.760 --> 00:09:41.970 and we're gonna start with this blue curve here, 00:09:41.970 --> 00:09:44.890 which represents the hypothetical reaction 00:09:44.890 --> 00:09:48.530 without a catalyst, so the uncatalyzed reaction. 00:09:48.530 --> 00:09:51.760 When time is equal to zero, the concentration of B is zero 00:09:51.760 --> 00:09:53.620 because we start with only A. 00:09:53.620 --> 00:09:57.790 And as A turns into B the concentration of B increases 00:09:57.790 --> 00:10:00.810 over time, and eventually the concentration of B 00:10:00.810 --> 00:10:02.360 becomes constant. 00:10:02.360 --> 00:10:04.710 And when the concentration of B becomes constant, 00:10:04.710 --> 00:10:06.710 the reaction reaches equilibrium. 00:10:06.710 --> 00:10:10.740 So this dotted line here represents the concentration of B 00:10:10.740 --> 00:10:12.490 at equilibrium. 00:10:12.490 --> 00:10:14.950 The yellow line represents the reaction 00:10:14.950 --> 00:10:16.760 with a catalyst added. 00:10:16.760 --> 00:10:18.810 So once again, we're starting with only A, 00:10:18.810 --> 00:10:20.190 so when time is equal to zero, 00:10:20.190 --> 00:10:22.410 the concentration of B is equal to zero. 00:10:22.410 --> 00:10:24.850 And as time increases A turns into B, 00:10:24.850 --> 00:10:27.290 so the concentration of B increases 00:10:27.290 --> 00:10:30.860 and eventually the concentration of B becomes constant. 00:10:30.860 --> 00:10:33.390 And the reaction reaches equilibrium. 00:10:33.390 --> 00:10:36.540 Notice that the reaction reached equilibrium much faster 00:10:36.540 --> 00:10:38.410 with the addition of the catalyst 00:10:38.410 --> 00:10:40.900 than it did without the catalyst. 00:10:40.900 --> 00:10:43.730 So the addition of a catalyst allows a reaction 00:10:43.730 --> 00:10:46.340 to reach equilibrium faster. 00:10:46.340 --> 00:10:49.810 However, notice that the final equilibrium concentration 00:10:49.810 --> 00:10:53.220 of B is the same for both the uncatalyzed reaction 00:10:53.220 --> 00:10:55.150 and the catalyzed reaction. 00:10:55.150 --> 00:10:57.730 Therefore, the addition of a catalyst does not change 00:10:57.730 --> 00:11:01.570 the composition of the equilibrium mixture. 00:11:01.570 --> 00:11:03.710 And that's because the catalyst speeds up 00:11:03.710 --> 00:11:07.920 both the forward reaction and the reverse reaction, 00:11:07.920 --> 00:11:10.150 but the rates are still equal. 00:11:10.150 --> 00:11:11.490 And since the rates are equal, 00:11:11.490 --> 00:11:13.350 there's no change in the composition 00:11:13.350 --> 00:11:15.153 of the equilibrium mixture.

Le Chȃtelier’s principle: Changing volume

https://www.youtube.com/watch?v=FmqyKCQo7Tk

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WEBVTT Kind: captions Language: en 00:00:00.955 --> 00:00:02.840 - [Tutor] Le Chatelier's principle says that 00:00:02.840 --> 00:00:06.490 if a stress is applied to a reaction mixture at equilibrium, 00:00:06.490 --> 00:00:08.500 the net reaction goes in the direction 00:00:08.500 --> 00:00:10.730 that relieves the stress. 00:00:10.730 --> 00:00:14.030 One possible stress that we could do is to change the volume 00:00:14.030 --> 00:00:16.590 on our reaction at equilibrium. 00:00:16.590 --> 00:00:20.180 Let's say we have a hypothetical reaction, where solid A, 00:00:20.180 --> 00:00:23.500 which is symbolized by red in our particular diagram, 00:00:23.500 --> 00:00:27.370 turns into solid B, which is symbolized by a blue 00:00:27.370 --> 00:00:29.620 and also C, which is a gas 00:00:29.620 --> 00:00:33.000 and C will be symbolized by white spheres. 00:00:33.000 --> 00:00:35.650 The equilibrium constant Kc for this reaction, 00:00:35.650 --> 00:00:39.750 is equal to 0.4 at 25 degrees Celsius. 00:00:39.750 --> 00:00:40.930 The first particular diagram 00:00:40.930 --> 00:00:43.240 shows the reaction at equilibrium, 00:00:43.240 --> 00:00:46.380 and we can see there's some solid A and some solid B 00:00:46.380 --> 00:00:48.070 at the bottom of the container, 00:00:48.070 --> 00:00:51.930 and there's also some gaseous particles of C. 00:00:51.930 --> 00:00:53.330 Let's introduce a stress, 00:00:53.330 --> 00:00:55.970 to our reaction mixture at equilibrium. 00:00:55.970 --> 00:00:58.870 Let's decrease the volume of the container. 00:00:58.870 --> 00:01:01.970 So looking from the first particular diagram to the second, 00:01:01.970 --> 00:01:03.530 we can see there's been a decrease 00:01:03.530 --> 00:01:05.370 in the volume of the container. 00:01:05.370 --> 00:01:08.790 That's gonna cause an increase in the pressure 00:01:08.790 --> 00:01:11.140 because pressure comes from these gas particles 00:01:11.140 --> 00:01:13.830 slamming into the sides of the container. 00:01:13.830 --> 00:01:15.940 And if we decrease the volume, 00:01:15.940 --> 00:01:18.860 now there's less distance for these particles to travel 00:01:18.860 --> 00:01:21.040 before they slam into the sides of the container, 00:01:21.040 --> 00:01:23.970 which means we increase the collision frequency. 00:01:23.970 --> 00:01:27.130 And therefore the pressure increases. 00:01:27.130 --> 00:01:29.060 According to Le Chatelier's principle, 00:01:29.060 --> 00:01:30.990 the net reaction is gonna go in the direction, 00:01:30.990 --> 00:01:32.750 that relieves the stress. 00:01:32.750 --> 00:01:35.310 So if the stress is increased pressure, 00:01:35.310 --> 00:01:37.690 the net reaction is going to try to move in the direction 00:01:37.690 --> 00:01:40.420 that decreases the pressure. 00:01:40.420 --> 00:01:43.070 Looking at the equation for this hypothetical reaction, 00:01:43.070 --> 00:01:46.110 the two solids aren't really contributing to the pressure. 00:01:46.110 --> 00:01:49.000 So it's only gas C that we need to worry about. 00:01:49.000 --> 00:01:52.770 And there's one mole of gas C on the product side, 00:01:52.770 --> 00:01:56.810 and there are zero moles of gas on the reactant side. 00:01:56.810 --> 00:01:58.530 So if the net reaction went to the right, 00:01:58.530 --> 00:02:01.530 we'd be going from zero moles of gas to one mole of gas. 00:02:01.530 --> 00:02:04.210 So going to the right would increase the moles of gas, 00:02:04.210 --> 00:02:06.450 which would increase the pressure. 00:02:06.450 --> 00:02:08.720 However, that's not what the reaction wants to do. 00:02:08.720 --> 00:02:11.350 The goal of the reaction is to relieve the stress 00:02:11.350 --> 00:02:14.830 and therefore the reaction wants to decrease the pressure. 00:02:14.830 --> 00:02:17.210 So the reactions gonna go to the left, 00:02:17.210 --> 00:02:19.250 to get rid of some of that gas 00:02:19.250 --> 00:02:23.250 and decreasing the amount of gas will decrease the pressure, 00:02:23.250 --> 00:02:26.380 therefore relieving the stress. 00:02:26.380 --> 00:02:28.670 If the net reaction moves to the left, 00:02:28.670 --> 00:02:30.470 we're gonna lose some of our products. 00:02:30.470 --> 00:02:32.220 So we're gonna decrease the amount of C 00:02:32.220 --> 00:02:34.860 and we're gonna decrease in the amount of B 00:02:34.860 --> 00:02:37.270 and we're going to gain some of our reactants. 00:02:37.270 --> 00:02:39.630 So we're gonna increase in the amount of A, 00:02:39.630 --> 00:02:42.810 and we can see all that and the third particulate diagram. 00:02:42.810 --> 00:02:44.530 So going from the second particular diagram 00:02:44.530 --> 00:02:46.150 to the third particular diagram, 00:02:46.150 --> 00:02:47.630 we've decreased the amount of C, 00:02:47.630 --> 00:02:49.370 we've gone from four particles of C 00:02:49.370 --> 00:02:51.110 to only two particles of C. 00:02:51.110 --> 00:02:52.900 We've also decreased the amount of B. 00:02:52.900 --> 00:02:54.790 You can see that this blue, 00:02:54.790 --> 00:02:56.870 this blue solid has gotten smaller, 00:02:56.870 --> 00:03:00.040 and we've increased the amount of A. 00:03:00.040 --> 00:03:02.870 So A, is this, A, is this red sphere here, 00:03:02.870 --> 00:03:04.390 and you can see how it's gotten bigger 00:03:04.390 --> 00:03:07.320 from the second particulate diagram to the third. 00:03:07.320 --> 00:03:09.630 And by going from four particles of C, 00:03:09.630 --> 00:03:11.680 to only two particles of C, 00:03:11.680 --> 00:03:14.150 we've decreased the amount of the gas, 00:03:14.150 --> 00:03:17.000 and therefore we've decreased the pressure. 00:03:17.000 --> 00:03:19.290 To better understand what happens to a reaction mixture 00:03:19.290 --> 00:03:21.590 at equilibrium when a stress is placed on it, 00:03:21.590 --> 00:03:24.140 let's calculate the reaction quotient Q 00:03:24.140 --> 00:03:26.160 for these three particular diagrams, 00:03:26.160 --> 00:03:28.820 for the same hypothetical reaction. 00:03:28.820 --> 00:03:31.550 The expression for the reaction quotient Qc, 00:03:31.550 --> 00:03:35.010 has the same form as the equilibrium constant expression 00:03:35.010 --> 00:03:36.330 for Kc. 00:03:36.330 --> 00:03:37.860 So since solids are left out 00:03:37.860 --> 00:03:40.320 of the equilibrium constant expression, 00:03:40.320 --> 00:03:43.420 we only need to include the concentration of the gas. 00:03:43.420 --> 00:03:45.910 And since there's a coefficient of one in front of C, 00:03:45.910 --> 00:03:48.070 Qc is equal to the concentration of C 00:03:48.070 --> 00:03:50.060 raised to the first power. 00:03:50.060 --> 00:03:51.610 Since there are four particles of C 00:03:51.610 --> 00:03:53.280 in the first particular diagram, 00:03:53.280 --> 00:03:56.430 and if each particle represents 0.1 moles of C, 00:03:56.430 --> 00:03:58.046 four times, 0.1 00:03:58.046 --> 00:03:59.710 is equal to 0.4 00:03:59.710 --> 00:04:00.760 moles of C. 00:04:00.760 --> 00:04:02.650 So to find the concentration of C, 00:04:02.650 --> 00:04:04.100 we take the moles and divide that 00:04:04.100 --> 00:04:07.530 by the volume of the container, which is 1.0 liter. 00:04:07.530 --> 00:04:09.730 So 0.4 divided by 1.0 00:04:09.730 --> 00:04:10.691 is equal to 00:04:10.691 --> 00:04:11.770 0.4 00:04:11.770 --> 00:04:12.603 molar. 00:04:12.603 --> 00:04:14.130 So that's the concentration of C 00:04:14.130 --> 00:04:16.200 in the first particular diagram. 00:04:16.200 --> 00:04:19.170 We plugged that into our expression for Q. 00:04:19.170 --> 00:04:20.240 So Qc 00:04:20.240 --> 00:04:21.073 is equal 00:04:21.073 --> 00:04:21.963 to 00:04:21.963 --> 00:04:23.150 0.4 00:04:23.150 --> 00:04:24.177 and notice that 00:04:24.177 --> 00:04:25.010 Kc 00:04:25.010 --> 00:04:25.843 is also 00:04:25.843 --> 00:04:26.676 equal to 00:04:26.676 --> 00:04:27.509 0.4. 00:04:27.509 --> 00:04:31.100 So at this moment in time, Qc is equal to Kc, 00:04:31.100 --> 00:04:34.940 which tells us the reaction is at equilibrium. 00:04:34.940 --> 00:04:37.170 Next, we think about the stress that was applied 00:04:37.170 --> 00:04:38.900 to the reaction at equilibrium. 00:04:38.900 --> 00:04:41.070 We decreased the volume. 00:04:41.070 --> 00:04:42.410 And if we look at the volumes here, 00:04:42.410 --> 00:04:46.180 we're going from 1.0 liters to 0.5 liters. 00:04:46.180 --> 00:04:49.080 So we're decreasing the volume by a factor of two, 00:04:49.080 --> 00:04:51.520 which would cause an increase in the pressure 00:04:51.520 --> 00:04:53.850 by a factor of two. 00:04:53.850 --> 00:04:57.010 And changing the volume would change the concentration. 00:04:57.010 --> 00:05:00.167 So instead of 0.4, divided by 1.0, 00:05:00.167 --> 00:05:01.243 it'd be 0.4 00:05:01.243 --> 00:05:02.440 divided by 00:05:02.440 --> 00:05:03.600 0.5, 00:05:03.600 --> 00:05:05.084 which is equal to 00:05:05.084 --> 00:05:06.210 0.8 00:05:06.210 --> 00:05:07.043 molar. 00:05:07.043 --> 00:05:09.890 So the concentration has doubled. 00:05:09.890 --> 00:05:13.710 So if we calculate Q for our second particular diagram, 00:05:13.710 --> 00:05:16.970 we plug in the concentration of C, which is 0.8 molar. 00:05:16.970 --> 00:05:17.910 So Qc 00:05:17.910 --> 00:05:19.363 is equal to 00:05:19.363 --> 00:05:20.450 0.8, 00:05:20.450 --> 00:05:21.440 K is equal 00:05:21.440 --> 00:05:22.530 to 0.4. 00:05:22.530 --> 00:05:24.580 So Q is not equal to K, 00:05:24.580 --> 00:05:27.730 therefore the reaction is not at equilibrium 00:05:27.730 --> 00:05:29.357 for our second particular diagram. 00:05:29.357 --> 00:05:34.040 Let me write in here, not at equilibrium. 00:05:34.040 --> 00:05:34.873 In this case, 00:05:34.873 --> 00:05:35.890 Qc 00:05:35.890 --> 00:05:37.690 is greater than Kc, 00:05:37.690 --> 00:05:39.850 which tells us we have too many products 00:05:39.850 --> 00:05:41.340 and not enough reactants. 00:05:41.340 --> 00:05:43.860 Therefore the net reaction goes to the left 00:05:43.860 --> 00:05:45.580 to get rid of some of the products 00:05:45.580 --> 00:05:48.950 and to increase the amount of reactants. 00:05:48.950 --> 00:05:50.890 The net reaction keeps going to the left 00:05:50.890 --> 00:05:53.440 until we reach equilibrium again. 00:05:53.440 --> 00:05:55.330 So if we calculate the concentration of C 00:05:55.330 --> 00:05:57.100 in the third particular diagram, 00:05:57.100 --> 00:05:58.870 here there are only two particles. 00:05:58.870 --> 00:05:59.930 So that'd be 00:05:59.930 --> 00:06:00.763 0.2 00:06:00.763 --> 00:06:01.750 moles 00:06:01.750 --> 00:06:04.780 divided by a volume of 0.5 liters. 00:06:04.780 --> 00:06:05.690 So 0.2 00:06:05.690 --> 00:06:07.150 divided by 0.5 00:06:07.150 --> 00:06:08.660 is equal to 00:06:08.660 --> 00:06:09.493 0.4 00:06:09.493 --> 00:06:10.326 molar. 00:06:10.326 --> 00:06:11.230 So Qc 00:06:11.230 --> 00:06:12.063 is equal 00:06:12.063 --> 00:06:13.100 to 0.4. 00:06:13.100 --> 00:06:17.083 And since Qc is equal to 0.4, Kc is still 0.4. 00:06:18.650 --> 00:06:22.560 So Qc is equal to Kc and we're at equilibrium. 00:06:22.560 --> 00:06:26.180 So equilibrium has been re-established. 00:06:26.180 --> 00:06:27.900 Since the reaction is at equilibrium 00:06:27.900 --> 00:06:29.940 in a third particular diagram, 00:06:29.940 --> 00:06:31.700 the net reaction stops going to the left 00:06:31.700 --> 00:06:35.020 and the concentration of C remains constant. 00:06:35.020 --> 00:06:37.150 Let's apply what we've learned to another reaction, 00:06:37.150 --> 00:06:41.360 the synthesis of ammonia from nitrogen gas and hydrogen gas. 00:06:41.360 --> 00:06:43.940 If we have a mixture of these gases at equilibrium, 00:06:43.940 --> 00:06:45.230 and we introduce a stress, 00:06:45.230 --> 00:06:47.770 the system like a decrease in volume, 00:06:47.770 --> 00:06:49.560 the decrease in the volume of the container 00:06:49.560 --> 00:06:52.150 would cause an increase in the pressure. 00:06:52.150 --> 00:06:54.080 And according to Le Chatelier's principle, 00:06:54.080 --> 00:06:55.940 the net reaction is gonna go in the direction 00:06:55.940 --> 00:06:59.280 that relieves the stress that was placed on the system. 00:06:59.280 --> 00:07:02.120 So if the stress is increased pressure, 00:07:02.120 --> 00:07:03.360 the net reaction says, 00:07:03.360 --> 00:07:06.350 I wanna move in the direction that decreases that pressure. 00:07:06.350 --> 00:07:09.320 So the net reaction moves to the right, 00:07:09.320 --> 00:07:11.900 because there are four moles of gas on the left 00:07:11.900 --> 00:07:13.970 and only two moles of gas on the right. 00:07:13.970 --> 00:07:15.790 And by moving to the right, 00:07:15.790 --> 00:07:18.020 that goes from four moles of gas to two moles of gas, 00:07:18.020 --> 00:07:20.010 which decreases the amount of gas 00:07:20.010 --> 00:07:23.220 and causes a decrease in the pressure. 00:07:23.220 --> 00:07:26.370 If we had a mixture of these gases at equilibrium, 00:07:26.370 --> 00:07:29.530 and we increased the volume and increase in the volume 00:07:29.530 --> 00:07:32.190 would cause a decrease in the pressure. 00:07:32.190 --> 00:07:35.440 So the stress this time, is decreased pressure. 00:07:35.440 --> 00:07:37.780 To relieve the stress, the net reaction, 00:07:37.780 --> 00:07:41.020 wants to move in the direction that increases the pressure. 00:07:41.020 --> 00:07:43.860 Therefore, the net reaction is going to move to the left 00:07:43.860 --> 00:07:45.140 because if it moves to the left, 00:07:45.140 --> 00:07:47.160 we're going from two moles of gas in the right 00:07:47.160 --> 00:07:49.080 to four moles of gas on the left. 00:07:49.080 --> 00:07:52.000 So that's an increase in the moles of gas 00:07:52.000 --> 00:07:53.800 and increase in the amount of gas, 00:07:53.800 --> 00:07:56.730 causes an increase in the pressure. 00:07:56.730 --> 00:07:57.780 Now let's see what happens, 00:07:57.780 --> 00:07:59.584 when we have equal amounts of moles of gas 00:07:59.584 --> 00:08:01.930 on both sides of the equation. 00:08:01.930 --> 00:08:04.630 For example, for the hypothetical reaction, 00:08:04.630 --> 00:08:06.530 where gas A turns into gas B, 00:08:06.530 --> 00:08:09.200 there's one mole of gas on the reactant side, 00:08:09.200 --> 00:08:12.230 and there's one mole of gas on the product side, 00:08:12.230 --> 00:08:15.460 and let's use particular diagrams and reaction quotients 00:08:15.460 --> 00:08:17.080 to understand what's going on here. 00:08:17.080 --> 00:08:19.490 So for our first particular diagram, 00:08:19.490 --> 00:08:21.200 here's the Qc expression. 00:08:21.200 --> 00:08:23.520 It's equal to the concentration of B to the first power, 00:08:23.520 --> 00:08:26.270 divided by the concentration of A to the first power 00:08:26.270 --> 00:08:28.850 and the concentration of B since B is blue, 00:08:28.850 --> 00:08:32.700 there's two blue spheres in this first particular diagram. 00:08:32.700 --> 00:08:36.040 So two times 0.1 moles is 0.2 moles of blue 00:08:36.040 --> 00:08:38.430 divided by a volume of one liter. 00:08:38.430 --> 00:08:43.350 Therefore the concentration of B is equal to 0.2. 00:08:43.350 --> 00:08:46.700 The concentration of A would also be equal to 0.2 molar 00:08:46.700 --> 00:08:50.040 because there's two particles of A in here. 00:08:50.040 --> 00:08:52.360 So 0.2 divided by 0.2 00:08:52.360 --> 00:08:53.560 is equal to 00:08:53.560 --> 00:08:54.730 one. 00:08:54.730 --> 00:08:58.360 And since K is equal to one for this reaction, 00:08:58.360 --> 00:09:01.240 Kc is equal to one at 25 degrees Celsius. 00:09:01.240 --> 00:09:04.950 Qc is equal to Kc at this moment in time. 00:09:04.950 --> 00:09:07.940 Therefore the reaction is at equilibrium 00:09:07.940 --> 00:09:10.220 in this first particular diagram. 00:09:10.220 --> 00:09:13.010 If you introduce a stress to our system at equilibrium, 00:09:13.010 --> 00:09:16.080 let's decrease the volume by a factor of two. 00:09:16.080 --> 00:09:20.050 So we're going from one, one liter to 0.5 liters. 00:09:20.050 --> 00:09:23.190 That's gonna increase the pressure by a factor of two. 00:09:23.190 --> 00:09:26.350 And it's also going to double the concentrations. 00:09:26.350 --> 00:09:28.130 So the concentration of B 00:09:28.130 --> 00:09:29.310 is now 00:09:29.310 --> 00:09:30.143 0.4 00:09:30.143 --> 00:09:30.976 molar 00:09:30.976 --> 00:09:34.620 and the concentration of A is also 0.4 molar. 00:09:34.620 --> 00:09:36.660 So 0.4 divided by 0.4 00:09:36.660 --> 00:09:37.493 is equal 00:09:37.493 --> 00:09:38.326 to 00:09:38.326 --> 00:09:39.159 one. 00:09:39.159 --> 00:09:40.310 So Qc 00:09:40.310 --> 00:09:41.250 is still 00:09:41.250 --> 00:09:42.540 equal to Kc. 00:09:42.540 --> 00:09:46.160 The reaction is still at equilibrium. 00:09:46.160 --> 00:09:49.150 Since changing the volume didn't change the value for Q, 00:09:49.150 --> 00:09:51.320 Qc is still equal to Kc. 00:09:51.320 --> 00:09:53.960 And since there's no change, there's no net reaction 00:09:53.960 --> 00:09:57.080 to the left and there's no net reaction to the right. 00:09:57.080 --> 00:10:00.080 Therefore changing the volume on a reaction at equilibrium, 00:10:00.080 --> 00:10:03.060 when there are equal amounts of moles of gas on both sides, 00:10:03.060 --> 00:10:08.060 has no effect on the composition of the equilibrium mixture. 00:10:09.000 --> 00:10:10.980 Let's go back to the reaction for the synthesis 00:10:10.980 --> 00:10:14.470 of ammonia gas from nitrogen gas and hydrogen gas. 00:10:14.470 --> 00:10:17.050 And let's say this reaction is at equilibrium. 00:10:17.050 --> 00:10:19.170 And we add in some helium gas 00:10:19.170 --> 00:10:22.010 to the reaction mixture at equilibrium. 00:10:22.010 --> 00:10:25.340 Helium is an inert gas, which means it doesn't react 00:10:25.340 --> 00:10:27.770 with any of the gases that we have present. 00:10:27.770 --> 00:10:28.740 It's tempting to say, 00:10:28.740 --> 00:10:32.720 adding this inert gas would increase the total pressure 00:10:32.720 --> 00:10:36.430 and therefore the net reaction would shift to the right 00:10:36.430 --> 00:10:39.100 to get rid of that increased pressure. 00:10:39.100 --> 00:10:42.250 However, notice that helium is not a part of the expression 00:10:42.250 --> 00:10:45.500 for Qc and therefore after you add it, 00:10:45.500 --> 00:10:48.630 you're not actually changing any of these concentrations. 00:10:48.630 --> 00:10:53.630 And so Qc is still equal to Kc after the addition of helium. 00:10:53.890 --> 00:10:55.700 And since Qc is equal to Kc, 00:10:55.700 --> 00:10:57.820 the reaction is still at equilibrium 00:10:57.820 --> 00:11:00.450 and there's no shift either direction. 00:11:00.450 --> 00:11:03.220 Therefore adding an inert gas to a reaction, 00:11:03.220 --> 00:11:05.780 make sure at equilibrium, has no effect 00:11:05.780 --> 00:11:08.753 on the composition of the reaction mixture.

Le Chȃtelier’s principle: Changing concentration

https://www.youtube.com/watch?v=jjmAGbcKEZY

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WEBVTT Kind: captions Language: en 00:00:00.930 --> 00:00:02.710 - [Instructor] Le Chatelier's principle says, 00:00:02.710 --> 00:00:06.420 if a stress is applied to a reaction mixture at equilibrium, 00:00:06.420 --> 00:00:08.770 the net reaction goes in the direction 00:00:08.770 --> 00:00:10.920 that relieves the stress. 00:00:10.920 --> 00:00:13.770 Change in the concentration of a reactant or product 00:00:13.770 --> 00:00:18.010 is one way to place a stress on a reaction at equilibrium. 00:00:18.010 --> 00:00:21.220 For example, let's consider the hypothetical reaction 00:00:21.220 --> 00:00:24.770 where gas A turns into gas B. 00:00:24.770 --> 00:00:27.760 And let's say the reaction is at equilibrium. 00:00:27.760 --> 00:00:29.590 And we suddenly introduce a stress 00:00:29.590 --> 00:00:34.590 such as we increase the concentration of reactant A. 00:00:34.800 --> 00:00:37.083 According to Le Chatelier's principle, 00:00:38.200 --> 00:00:40.180 the net reaction is gonna go in the direction 00:00:40.180 --> 00:00:41.520 that relieves the stress. 00:00:41.520 --> 00:00:44.390 And since we increase the concentration of A, 00:00:44.390 --> 00:00:46.970 the net reaction is gonna go to the right 00:00:46.970 --> 00:00:50.190 to decrease the concentration of A. 00:00:50.190 --> 00:00:51.680 Let's use some particular diagrams 00:00:51.680 --> 00:00:53.060 so we can get into the details 00:00:53.060 --> 00:00:55.440 of how the reaction goes to the right. 00:00:55.440 --> 00:00:58.330 So we're gonna symbolize gas A by red particles 00:00:58.330 --> 00:01:00.970 and gas B by blue particles. 00:01:00.970 --> 00:01:02.980 And for this hypothetical reaction, 00:01:02.980 --> 00:01:05.740 the equilibrium constant is equal to three 00:01:05.740 --> 00:01:09.070 at 25 degrees Celsius. 00:01:09.070 --> 00:01:11.480 Let's start by writing the reaction quotient. 00:01:11.480 --> 00:01:16.480 Qc is equal to, and we get that from our balanced equation. 00:01:16.490 --> 00:01:20.200 That would be the concentration of B to the first power 00:01:20.200 --> 00:01:25.200 divided by the concentration of A, also to the first power. 00:01:28.050 --> 00:01:30.520 Let's calculate the concentrations of B and A 00:01:30.520 --> 00:01:33.010 from our first particular diagram. 00:01:33.010 --> 00:01:35.670 So B is represented by the blue spheres 00:01:35.670 --> 00:01:37.650 and there are three blue spheres. 00:01:37.650 --> 00:01:40.960 If each particle represents 0.1 moles of a substance, 00:01:40.960 --> 00:01:44.310 and the volume of the container is one litter, 00:01:44.310 --> 00:01:46.040 since we have three particles, 00:01:46.040 --> 00:01:49.630 that'd be three times 0.1, which is 0.3 moles 00:01:49.630 --> 00:01:53.260 divided by a volume of one litter is 0.3 molar.. 00:01:53.260 --> 00:01:57.590 So the concentration of B is 0.3 molar. 00:01:57.590 --> 00:02:00.430 For A, we have one particles, that's 0.1 mole 00:02:00.430 --> 00:02:03.530 divided by one litter, which is 0.1 molar. 00:02:03.530 --> 00:02:07.128 So the concentration of A is 0.1 molar. 00:02:07.128 --> 00:02:09.810 And 0.3 divided by 0.1 is equal to three. 00:02:09.810 --> 00:02:14.400 So Qc at this moment in time is equal to three. 00:02:14.400 --> 00:02:16.490 Notice we could have just counted our particles, 00:02:16.490 --> 00:02:20.590 three blues and one red and said three over one. 00:02:20.590 --> 00:02:22.810 That would have been a little bit faster. 00:02:22.810 --> 00:02:27.780 So Qc is equal to three and Kc is also equal to three. 00:02:27.780 --> 00:02:29.351 So I should have written a C in here. 00:02:29.351 --> 00:02:34.160 So when Qc is equal to Kc, the reaction is at equilibrium. 00:02:34.160 --> 00:02:36.730 So in this first particular diagram here 00:02:36.730 --> 00:02:41.730 where Qc is equal to Kc, the reactions are at equilibrium. 00:02:41.890 --> 00:02:43.480 Next, we're gonna introduce a stress 00:02:43.480 --> 00:02:45.500 to our reaction at equilibrium. 00:02:45.500 --> 00:02:48.930 We're going to increase the concentration of A. 00:02:48.930 --> 00:02:51.790 So here, we're gonna add four particles of A 00:02:51.790 --> 00:02:55.810 to the reaction mixture at equilibrium. 00:02:55.810 --> 00:02:58.540 The second particulate diagram shows what the reaction 00:02:58.540 --> 00:03:02.010 looks like right after we add those four red particles. 00:03:02.010 --> 00:03:05.100 So we started with one red particle and we added four. 00:03:05.100 --> 00:03:07.610 So now there's a total of five red particles. 00:03:07.610 --> 00:03:10.910 And we still have the same three blue particles 00:03:10.910 --> 00:03:13.710 that we had in the first particular diagram. 00:03:13.710 --> 00:03:17.200 Let's calculate Qc at this moment in time. 00:03:17.200 --> 00:03:20.300 So just after we introduced the stress. 00:03:20.300 --> 00:03:23.250 Since there are three blue particles and five red particles, 00:03:23.250 --> 00:03:27.633 Qc is equal to three divided by five, which is equal to 0.6. 00:03:29.470 --> 00:03:33.690 Since Qc is equal to 0.6, and Kc is equal to three, 00:03:33.690 --> 00:03:38.241 at this moment in time, Qc is less than Kc. 00:03:38.241 --> 00:03:41.810 So there are too many reactants and not enough products. 00:03:41.810 --> 00:03:46.080 Therefore, the net reaction is going to go to the right 00:03:46.080 --> 00:03:49.100 and we're going to decrease in the amount of A, 00:03:49.100 --> 00:03:52.870 and we're gonna increase in the amount of B. 00:03:52.870 --> 00:03:54.749 The third particular diagram shows what happens 00:03:54.749 --> 00:03:58.120 after the net reaction moves to the right. 00:03:58.120 --> 00:04:00.250 So we said, we're gonna decrease the amount of A 00:04:00.250 --> 00:04:02.880 and increase in the amount of B. 00:04:02.880 --> 00:04:04.910 We're going from three blues 00:04:04.910 --> 00:04:08.957 in the second particular diagram to six blues in the third. 00:04:08.957 --> 00:04:13.957 And we're going from five reds to only two reds. 00:04:14.940 --> 00:04:19.940 Therefore, three reds must have turned into blues 00:04:20.110 --> 00:04:23.000 to get the third particular diagram on the right. 00:04:23.000 --> 00:04:27.540 And if we calculate Qc for our third particular diagram, 00:04:27.540 --> 00:04:30.980 it'd be equal to six divided by two, 00:04:30.980 --> 00:04:33.280 which is equal to three. 00:04:33.280 --> 00:04:37.780 So at this moment in time, Qc is equal to Kc. 00:04:37.780 --> 00:04:39.090 They're both equal to three. 00:04:39.090 --> 00:04:41.827 So equilibrium has been reestablished 00:04:41.827 --> 00:04:44.940 in the third particular diagram. 00:04:44.940 --> 00:04:47.640 It isn't always necessary to calculate Q values 00:04:47.640 --> 00:04:51.730 when doing a Le Chatelier's changing concentration problem. 00:04:51.730 --> 00:04:54.040 However, for this hypothetical reaction, 00:04:54.040 --> 00:04:57.440 it's useful to calculate Q values to understand 00:04:57.440 --> 00:04:59.760 that we're starting at equilibrium 00:04:59.760 --> 00:05:01.520 and then a stress is introduced 00:05:01.520 --> 00:05:04.080 such as changing the concentration of a reaction or product. 00:05:04.080 --> 00:05:08.400 And that means the reaction is no longer at equilibrium. 00:05:08.400 --> 00:05:10.460 Le Chatelier's principle allows us to predict 00:05:10.460 --> 00:05:14.130 which direction the net reaction will go 00:05:14.130 --> 00:05:16.320 or we could also use Q to predict 00:05:16.320 --> 00:05:18.150 the direction of the net reaction. 00:05:18.150 --> 00:05:21.850 The net reaction will continue going in that new direction 00:05:21.850 --> 00:05:23.970 until Q is equal to K again 00:05:23.970 --> 00:05:27.220 and equilibrium has been reestablished, 00:05:27.220 --> 00:05:28.530 let's look at another reaction. 00:05:28.530 --> 00:05:30.570 This is the synthesis of ammonia 00:05:30.570 --> 00:05:34.060 from nitrogen gas and hydrogen gas. 00:05:34.060 --> 00:05:36.800 And let's see the reaction is at equilibrium. 00:05:36.800 --> 00:05:38.810 So let's also look at this on a graph 00:05:38.810 --> 00:05:41.320 of concentration versus time. 00:05:41.320 --> 00:05:44.580 At equilibrium, the concentrations of reactants and products 00:05:44.580 --> 00:05:48.040 are constant, which is why we see these straight lines here 00:05:48.040 --> 00:05:51.983 for the concentration of hydrogen, ammonia, and nitrogen. 00:05:52.830 --> 00:05:56.480 And let's introduce a stress to the system at equilibrium. 00:05:56.480 --> 00:05:58.710 So right now we are at equilibrium 00:05:58.710 --> 00:06:00.980 and all the concentrations are constant. 00:06:00.980 --> 00:06:05.160 And let's increase the concentration of hydrogen. 00:06:05.160 --> 00:06:08.020 So we can see that on our graph. 00:06:08.020 --> 00:06:10.530 So right here, there's a sudden increase 00:06:10.530 --> 00:06:13.670 in the concentration of hydrogen. 00:06:13.670 --> 00:06:16.870 Adding hydrogen means that Q is no longer equal to K 00:06:16.870 --> 00:06:19.870 and therefore the reaction is not at equilibrium. 00:06:19.870 --> 00:06:21.880 So let's go ahead and write over here, 00:06:21.880 --> 00:06:24.173 Now we're not at equilibrium. 00:06:25.050 --> 00:06:27.600 And Le Chatelier's principle allows us to predict 00:06:27.600 --> 00:06:30.560 which direction the net reaction will move. 00:06:30.560 --> 00:06:33.130 So since we added a stress, 00:06:33.130 --> 00:06:36.600 the stress being increased concentration of hydrogen, 00:06:36.600 --> 00:06:38.860 the net reaction is gonna move to the right 00:06:38.860 --> 00:06:42.670 to get rid of some of that hydrogen that was added. 00:06:42.670 --> 00:06:44.460 And when the reaction goes to the right, 00:06:44.460 --> 00:06:46.850 the amount of ammonia will increase. 00:06:46.850 --> 00:06:49.620 And that's what we can see right here on this red line here, 00:06:49.620 --> 00:06:52.170 the amount of ammonia is increasing. 00:06:52.170 --> 00:06:54.340 And the amount of ammonia increases 00:06:54.340 --> 00:06:57.760 because nitrogen and hydrogen are reacting to form ammonia. 00:06:57.760 --> 00:06:59.900 Therefore, the amount of nitrogen and hydrogen 00:06:59.900 --> 00:07:00.851 will decrease. 00:07:00.851 --> 00:07:03.560 Here we can see the amount of hydrogen is decreasing. 00:07:03.560 --> 00:07:05.711 And down here, we can see the amount of nitrogen 00:07:05.711 --> 00:07:07.800 is decreasing. 00:07:07.800 --> 00:07:09.710 The reaction will continue to go to the right 00:07:09.710 --> 00:07:12.060 until equilibrium is reestablished. 00:07:12.060 --> 00:07:14.330 And that happens at the second dotted line here. 00:07:14.330 --> 00:07:16.250 And we know that because we can see 00:07:16.250 --> 00:07:19.830 all of these concentrations are now constant. 00:07:19.830 --> 00:07:23.870 So the reaction has reached equilibrium. 00:07:23.870 --> 00:07:26.720 So far we've only talked about change in the concentration 00:07:26.720 --> 00:07:28.600 of a reactant so for example, 00:07:28.600 --> 00:07:31.632 if we increase the concentration of hydrogen, 00:07:31.632 --> 00:07:34.260 the net reaction goes to the right. 00:07:34.260 --> 00:07:36.360 We could also say shifts to the right. 00:07:36.360 --> 00:07:38.530 So for a reaction at equilibrium, 00:07:38.530 --> 00:07:40.730 if you increase the concentration of reactants, 00:07:40.730 --> 00:07:42.520 such as the concentration of hydrogen 00:07:42.520 --> 00:07:45.750 or the concentration of nitrogen, 00:07:45.750 --> 00:07:49.070 the reaction will shift to the right 00:07:49.070 --> 00:07:52.970 to decrease the amount of one of those reactants. 00:07:52.970 --> 00:07:54.840 And if our reaction is at equilibrium 00:07:54.840 --> 00:07:57.700 and we were to increase the amount of our product, 00:07:57.700 --> 00:07:59.348 increase the amount of ammonia, 00:07:59.348 --> 00:08:00.940 so this time the stress place 00:08:00.940 --> 00:08:03.130 is increased concentration of a product, 00:08:03.130 --> 00:08:05.310 Le Chatelier's principle says the net reaction 00:08:05.310 --> 00:08:08.560 is going to move in the direction that decreases the stress. 00:08:08.560 --> 00:08:11.640 So in this case, the net reaction would go to the left 00:08:11.640 --> 00:08:13.833 to decrease the amount of ammonia. 00:08:14.730 --> 00:08:16.460 And if our reaction is at equilibrium 00:08:16.460 --> 00:08:21.460 and we were to decrease the concentration of our product, 00:08:21.490 --> 00:08:24.520 the net reaction would shift to the right 00:08:24.520 --> 00:08:27.110 to make more of the product. 00:08:27.110 --> 00:08:29.930 Or if we decrease the concentration of one of our reactants, 00:08:29.930 --> 00:08:32.240 let's say of nitrogen, 00:08:32.240 --> 00:08:34.600 in this case the reaction will shift to the left 00:08:34.600 --> 00:08:36.493 to make more of our reactant.

Worked example: Using the reaction quotient to find equilibrium partial pressures

https://www.youtube.com/watch?v=U7U3e0r_8y8

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https://www.youtube.com/api/timedtext?v=U7U3e0r_8y8&ei=3VWUZYHWL-zumLAP_beT6Ac&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245325&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=8A0B4814259B356932423B813BB71225EE03F258.E15C78EFB141418255A9F2DE79BBA200FF049910&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.568 --> 00:00:02.932 - [Tutor] For the reaction of iron two oxide 00:00:02.932 --> 00:00:07.463 plus carbon monoxide goes to solid iron and carbon dioxide. 00:00:07.463 --> 00:00:12.463 The equilibrium constant Kp is equal to 0.26 at 1000 Kelvin. 00:00:12.818 --> 00:00:15.929 Our goal is to find the equilibrium partial pressures 00:00:15.929 --> 00:00:19.531 of our two gasses, carbon monoxide and carbon dioxide. 00:00:19.531 --> 00:00:21.232 If the initial partial pressures 00:00:21.232 --> 00:00:24.210 are 0.80 atmospheres for carbon monoxide 00:00:24.210 --> 00:00:28.140 and 0.40 atmospheres for carbon dioxide, 00:00:28.140 --> 00:00:30.131 we can use the reaction quotient Q, 00:00:30.131 --> 00:00:32.833 to predict which direction that reaction will go 00:00:32.833 --> 00:00:34.819 to reach equilibrium. 00:00:34.819 --> 00:00:36.569 So let's calculate Qp 00:00:37.648 --> 00:00:39.221 and Qp 00:00:39.221 --> 00:00:40.370 is equal to, 00:00:40.370 --> 00:00:42.139 first we think about our products 00:00:42.139 --> 00:00:44.885 and we leave solids out of equilibrium expressions, 00:00:44.885 --> 00:00:47.787 and therefore we also leave it out of our expression for Qp. 00:00:47.787 --> 00:00:49.381 So we're gonna leave out iron. 00:00:49.381 --> 00:00:52.818 We're going to include carbon dioxide since it's a gas. 00:00:52.818 --> 00:00:56.115 So we're gonna write the partial pressure of carbon dioxide, 00:00:56.115 --> 00:00:57.830 and since we have a coefficient of one 00:00:57.830 --> 00:00:59.229 in front of carbon dioxide, 00:00:59.229 --> 00:01:01.685 it's the partial pressure raise to the first power 00:01:01.685 --> 00:01:04.140 divided by, next we look at our reactants, 00:01:04.140 --> 00:01:06.709 and we have a solid, so we're gonna leave that out. 00:01:06.709 --> 00:01:09.231 So we have another gas, carbon monoxide. 00:01:09.231 --> 00:01:12.090 So this will be the partial pressure of carbon monoxide 00:01:12.090 --> 00:01:13.460 raised to the first power, 00:01:13.460 --> 00:01:15.878 since there's also a coefficient of one. 00:01:15.878 --> 00:01:17.749 Next, we plug in our partial pressures 00:01:17.749 --> 00:01:19.218 at this moment in time. 00:01:19.218 --> 00:01:21.593 The partial pressure of carbon dioxide 00:01:21.593 --> 00:01:22.467 is 00:01:22.467 --> 00:01:23.300 0.40, 00:01:24.158 --> 00:01:26.631 and the partial pressure of carbon monoxide 00:01:26.631 --> 00:01:27.464 is 00:01:27.464 --> 00:01:28.297 0.80 00:01:28.297 --> 00:01:29.635 atmospheres. 00:01:29.635 --> 00:01:32.146 So we plug those into our expression for Qp 00:01:32.146 --> 00:01:33.108 and 0.40 00:01:33.108 --> 00:01:34.679 divided by 0.80 00:01:34.679 --> 00:01:36.236 is equal to 00:01:36.236 --> 00:01:37.750 0.50. 00:01:37.750 --> 00:01:39.390 So Qp 00:01:39.390 --> 00:01:43.210 at this moment in time is equal to 0.50. 00:01:43.210 --> 00:01:46.730 Since QP is not equal to Kp at this moment in time, 00:01:46.730 --> 00:01:49.569 the reaction is not at equilibrium. 00:01:49.569 --> 00:01:54.120 So Qp is equal to 0.50 and Kp is equal to 0.26. 00:01:54.120 --> 00:01:54.953 So Qp 00:01:55.850 --> 00:01:57.199 is greater 00:01:57.199 --> 00:01:58.548 than Kp. 00:01:58.548 --> 00:02:00.986 And when Qp is greater than Kp, 00:02:00.986 --> 00:02:04.655 there are too many products and not enough reactants. 00:02:04.655 --> 00:02:09.467 Therefore the net reaction is going to move to the left. 00:02:09.467 --> 00:02:12.373 Next let's fill out our I.C.E table for this reaction. 00:02:12.373 --> 00:02:15.679 I stands for the initial partial pressure in atmospheres. 00:02:15.679 --> 00:02:18.369 And so the initial partial pressure of carbon monoxide 00:02:18.369 --> 00:02:20.276 was 0.80 atmospheres. 00:02:20.276 --> 00:02:22.717 And the initial partial pressure of carbon dioxide 00:02:22.717 --> 00:02:25.068 is 0.40 atmospheres. 00:02:25.068 --> 00:02:26.656 C stands for change. 00:02:26.656 --> 00:02:30.017 And E stands for the equilibrium partial pressure. 00:02:30.017 --> 00:02:32.609 Calculating Qp allowed us to realize 00:02:32.609 --> 00:02:35.306 that the net reaction moves to the left. 00:02:35.306 --> 00:02:37.416 And if the net reaction moves to the left, 00:02:37.416 --> 00:02:39.999 we're going to lose some carbon dioxide 00:02:39.999 --> 00:02:43.569 and we're going to gain some carbon monoxide. 00:02:43.569 --> 00:02:46.506 So first let's think about carbon dioxide. 00:02:46.506 --> 00:02:47.695 We're gonna lose some of it, 00:02:47.695 --> 00:02:50.146 but we don't know how much and therefore 00:02:50.146 --> 00:02:52.169 that's gonna be represented by X. 00:02:52.169 --> 00:02:55.767 So we're gonna write minus X here for carbon dioxide. 00:02:55.767 --> 00:02:57.689 And since the coefficient is a one 00:02:57.689 --> 00:02:59.457 in front of carbon dioxide, 00:02:59.457 --> 00:03:02.177 and it's also one in front of carbon monoxide, 00:03:02.177 --> 00:03:04.536 if we lose X for carbon dioxide, 00:03:04.536 --> 00:03:08.376 we're going to gain X for carbon monoxide. 00:03:08.376 --> 00:03:10.745 Therefore the equilibrium partial pressure 00:03:10.745 --> 00:03:12.619 for carbon monoxide 00:03:12.619 --> 00:03:13.452 would be 00:03:13.452 --> 00:03:14.926 0.80 00:03:14.926 --> 00:03:15.759 plus X. 00:03:16.629 --> 00:03:19.956 And the equilibrium partial pressure for carbon dioxide 00:03:19.956 --> 00:03:20.789 would be 00:03:20.789 --> 00:03:21.622 0.40 00:03:22.646 --> 00:03:23.479 minus X. 00:03:24.417 --> 00:03:27.725 Our next step is to write an equilibrium constant expression 00:03:27.725 --> 00:03:29.235 for this reaction. 00:03:29.235 --> 00:03:34.235 So Kp is equal to the partial pressure of carbon dioxide 00:03:34.543 --> 00:03:38.845 divided by the partial pressure of carbon monoxide. 00:03:38.845 --> 00:03:41.965 So the expressions for Kp and Qp look the same, 00:03:41.965 --> 00:03:43.994 but the difference is for Kp, 00:03:43.994 --> 00:03:46.954 it would be the equilibrium partial pressures only. 00:03:46.954 --> 00:03:48.185 Whereas for Qp, 00:03:48.185 --> 00:03:51.373 it's the partial pressures at any moment in time. 00:03:51.373 --> 00:03:52.434 And since for Kp, 00:03:52.434 --> 00:03:55.535 we're talking about the equilibrium partial pressures, 00:03:55.535 --> 00:03:58.591 we can take those directly from our I.C.E table 00:03:58.591 --> 00:03:59.754 and plug them in. 00:03:59.754 --> 00:04:03.496 So we can plug in the equilibrium partial pressure of CO2 00:04:03.496 --> 00:04:08.496 and the equilibrium partial pressure of carbon monoxide. 00:04:08.833 --> 00:04:12.046 Here we can see our two equilibrium partial pressures 00:04:12.046 --> 00:04:14.622 plugged into our Kp expression 00:04:14.622 --> 00:04:17.114 and also the equilibrium constant Kp 00:04:17.114 --> 00:04:19.634 is equal to 0.26 for this reaction, 00:04:19.634 --> 00:04:21.625 so that's plugged in as well. 00:04:21.625 --> 00:04:23.697 Our next step is to solve for X. 00:04:23.697 --> 00:04:27.736 So we multiply both sides by 0.80 plus X, 00:04:27.736 --> 00:04:31.115 and we get this and then we do some more algebra 00:04:31.115 --> 00:04:35.536 and we get down to 1.26 X is equal to 0.192. 00:04:35.536 --> 00:04:36.369 So 00:04:36.369 --> 00:04:37.202 0.192 00:04:37.202 --> 00:04:38.035 divided by 00:04:38.035 --> 00:04:39.502 1.26 00:04:39.502 --> 00:04:40.335 is equal to 00:04:40.335 --> 00:04:41.470 0.15. 00:04:41.470 --> 00:04:42.303 So X 00:04:42.303 --> 00:04:43.564 is equal to 00:04:43.564 --> 00:04:44.397 0. 00:04:44.397 --> 00:04:45.230 1 00:04:45.230 --> 00:04:46.776 5. 00:04:46.776 --> 00:04:49.189 Now that we know that X is equal to 0.15, 00:04:49.189 --> 00:04:50.879 we can go back to our I.C.E table 00:04:50.879 --> 00:04:54.028 and solve for the equilibrium partial pressures. 00:04:54.028 --> 00:04:55.720 So for carbon monoxide, 00:04:55.720 --> 00:04:59.760 the equilibrium partial pressure is 0.80 plus X. 00:04:59.760 --> 00:05:01.240 So that's equal to 00:05:01.240 --> 00:05:02.348 0.80 00:05:02.348 --> 00:05:03.480 plus 00:05:03.480 --> 00:05:04.313 0. 00:05:04.313 --> 00:05:05.146 1 00:05:05.146 --> 00:05:05.979 5, 00:05:05.979 --> 00:05:07.185 which is equal to 00:05:07.185 --> 00:05:08.018 0. 00:05:08.018 --> 00:05:08.851 9 00:05:08.851 --> 00:05:09.684 5 00:05:09.684 --> 00:05:10.517 atmospheres. 00:05:10.517 --> 00:05:12.325 So that's the equilibrium partial pressure 00:05:12.325 --> 00:05:14.313 of carbon monoxide. 00:05:14.313 --> 00:05:15.733 For carbon dioxide, 00:05:15.733 --> 00:05:19.730 the equilibrium partial pressure is 0.40 minus X. 00:05:19.730 --> 00:05:20.563 So 00:05:20.563 --> 00:05:21.529 0.40 00:05:21.529 --> 00:05:22.809 minus 00:05:22.809 --> 00:05:23.642 0.15 00:05:24.499 --> 00:05:25.796 is equal to 00:05:25.796 --> 00:05:26.629 0. 00:05:26.629 --> 00:05:27.462 2 00:05:27.462 --> 00:05:28.295 5 00:05:28.295 --> 00:05:29.128 atmospheres. 00:05:29.128 --> 00:05:31.135 So that's the equilibrium partial pressure 00:05:31.135 --> 00:05:33.394 for carbon dioxide. 00:05:33.394 --> 00:05:34.227 Finally, 00:05:34.227 --> 00:05:36.223 we can use the reaction quotient Qp 00:05:36.223 --> 00:05:38.183 to make sure that these two answers, 00:05:38.183 --> 00:05:41.393 for equilibrium partial pressures are correct. 00:05:41.393 --> 00:05:44.539 So we can write that Qp is equal to 00:05:44.539 --> 00:05:47.363 the partial pressure of CO2 00:05:47.363 --> 00:05:50.753 divided by the partial pressure of CO. 00:05:50.753 --> 00:05:52.144 And we can plug in those, 00:05:52.144 --> 00:05:54.335 those equilibrium partial pressures. 00:05:54.335 --> 00:05:55.603 So this would be 00:05:55.603 --> 00:05:57.561 0.25 atmospheres, 00:05:57.561 --> 00:06:01.133 was the equilibrium partial pressure of carbon dioxide 00:06:01.133 --> 00:06:04.334 and 0.95 was the equilibrium partial pressure 00:06:04.334 --> 00:06:05.913 of carbon monoxide. 00:06:05.913 --> 00:06:07.913 And 0.25 divided by 0.95 00:06:08.807 --> 00:06:09.987 is equal to 00:06:09.987 --> 00:06:10.820 0. 00:06:10.820 --> 00:06:11.653 2 00:06:11.653 --> 00:06:12.928 6. 00:06:12.928 --> 00:06:13.761 And 00:06:13.761 --> 00:06:14.797 since Kp 00:06:14.797 --> 00:06:16.770 is also equal to 00:06:16.770 --> 00:06:17.603 0.26 00:06:18.487 --> 00:06:19.991 at this moment in time, 00:06:19.991 --> 00:06:20.985 Qp 00:06:20.985 --> 00:06:22.010 is equal 00:06:22.010 --> 00:06:23.075 to Kp 00:06:23.075 --> 00:06:25.963 and the reaction is at equilibrium. 00:06:25.963 --> 00:06:28.076 Therefore we know we have the correct 00:06:28.076 --> 00:06:30.300 equilibrium partial pressures.

Using the reaction quotient

https://www.youtube.com/watch?v=JDUNQMj9Yk0

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https://www.youtube.com/api/timedtext?v=JDUNQMj9Yk0&ei=3VWUZb6OL-H7vdIPr9ixKA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245325&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=2790DDE62B6918ED3C0C88D6B2F46F80AC06D370.545E9B7B2A4493DC7794EE054FFDC746CEF2436F&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.250 --> 00:00:01.240 - [Instructor] The reaction quotient 00:00:01.240 --> 00:00:03.800 is symbolized by the capital letter Q. 00:00:03.800 --> 00:00:07.840 And it tells us whether a reaction is at equilibrium or not. 00:00:07.840 --> 00:00:09.900 If the reaction is not at equilibrium, 00:00:09.900 --> 00:00:11.730 it also allows us to predict 00:00:11.730 --> 00:00:14.160 which direction the net reaction will go 00:00:14.160 --> 00:00:15.900 to reach equilibrium. 00:00:15.900 --> 00:00:17.770 For example, let's look at the hypothetical 00:00:17.770 --> 00:00:21.700 reaction where A gas turns into gas B. 00:00:21.700 --> 00:00:25.200 Gas A will be represented by red circles or red spheres, 00:00:25.200 --> 00:00:28.580 and gas B will be represented by blue spheres. 00:00:28.580 --> 00:00:31.320 The equilibrium constant for this hypothetical reaction 00:00:31.320 --> 00:00:35.280 is equal to 3 at 25 degrees Celsius. 00:00:35.280 --> 00:00:36.880 Let's start by writing out the expression 00:00:36.880 --> 00:00:38.400 for the reaction quotient. 00:00:38.400 --> 00:00:41.450 So we would write out here QC is equal to, 00:00:41.450 --> 00:00:42.840 and this has the same form 00:00:42.840 --> 00:00:45.600 as the equilibrium constant expression. 00:00:45.600 --> 00:00:49.460 So we would put concentration of B to the first power, 00:00:49.460 --> 00:00:54.273 divided by the concentration of A, also to the first power. 00:00:55.640 --> 00:00:58.080 Let's look at our first particulate diagram here, 00:00:58.080 --> 00:00:59.600 and let's think about each particle 00:00:59.600 --> 00:01:02.720 representing 0.1 moles of a substance. 00:01:02.720 --> 00:01:06.960 And the volume of the container is one liter. 00:01:06.960 --> 00:01:10.220 So let's first find the concentration of B 00:01:10.220 --> 00:01:12.980 so we can plug it into our expression for Q. 00:01:12.980 --> 00:01:14.550 B are the blue spheres. 00:01:14.550 --> 00:01:18.370 So we count up one, two, three blue spheres. 00:01:18.370 --> 00:01:21.260 Each sphere or each particle represents 0.1 moles. 00:01:21.260 --> 00:01:24.340 So three times 0.1 is 0.3. 00:01:24.340 --> 00:01:26.390 And then we divide that by the volume 00:01:26.390 --> 00:01:30.380 of one liter to get a concentration of 0.3 moles. 00:01:30.380 --> 00:01:33.370 So we can go ahead and plug in 0.3 mole 00:01:33.370 --> 00:01:35.800 for the concentration of B. 00:01:35.800 --> 00:01:38.660 Next, we divide that by the concentration of A, 00:01:38.660 --> 00:01:41.950 and since there are five red particles, 00:01:41.950 --> 00:01:44.310 and each particle represents 0.1 moles, 00:01:44.310 --> 00:01:47.650 five times 0.1 is 0.5 moles of A 00:01:47.650 --> 00:01:49.490 divided by a volume of one liter, 00:01:49.490 --> 00:01:54.080 it gives a concentration of 0.5 mole for A. 00:01:54.080 --> 00:01:55.610 Notice that we could have just counted 00:01:55.610 --> 00:01:58.820 the number of particles, three blues and five reds, 00:01:58.820 --> 00:02:00.930 and just done three divided by five 00:02:00.930 --> 00:02:05.010 and get the same value for the reaction quotient Q. 00:02:05.010 --> 00:02:08.500 So QC is equal to three fists or 0.6. 00:02:08.500 --> 00:02:11.860 And KC remember was equal to three. 00:02:11.860 --> 00:02:14.360 So Q is not equal to K. 00:02:14.360 --> 00:02:18.180 In this case, QC is less than KC. 00:02:18.180 --> 00:02:21.480 Since Q is not equal to K, at this moment and time, 00:02:21.480 --> 00:02:24.420 the reaction is not at equilibrium. 00:02:24.420 --> 00:02:26.860 In order for this reaction to reach equilibrium, 00:02:26.860 --> 00:02:29.360 Q needs to be equal to K. 00:02:29.360 --> 00:02:32.560 And since Q is a lot smaller than K, 00:02:32.560 --> 00:02:33.440 if you think about it, 00:02:33.440 --> 00:02:36.130 we need to increase the numerator 00:02:36.130 --> 00:02:38.620 and decrease the denominator. 00:02:38.620 --> 00:02:41.650 So that means, we have too many reactants 00:02:41.650 --> 00:02:43.220 and not enough products. 00:02:43.220 --> 00:02:46.970 And so the net reaction, if go back up here 00:02:46.970 --> 00:02:48.480 to the equation here, 00:02:48.480 --> 00:02:51.490 the net reaction is going to move to the right 00:02:51.490 --> 00:02:53.460 to make more products. 00:02:53.460 --> 00:02:55.810 So the net reaction moves to the right 00:02:55.810 --> 00:02:57.920 to make more blue particles 00:02:57.920 --> 00:03:00.650 and therefore the number of red particles would decrease. 00:03:00.650 --> 00:03:02.080 We can see that comparing these 00:03:02.080 --> 00:03:04.020 first two particulate diagrams here. 00:03:04.020 --> 00:03:06.530 So let's compare the first particular diagram 00:03:06.530 --> 00:03:08.670 to the second particular diagram. 00:03:08.670 --> 00:03:10.190 And the first particulate diagram, 00:03:10.190 --> 00:03:12.920 we had three blues and five reds. 00:03:12.920 --> 00:03:15.220 And the second particulate diagram, 00:03:15.220 --> 00:03:19.210 we have five blues and only three reds. 00:03:19.210 --> 00:03:21.460 So that shows the reaction has moved to the right 00:03:21.460 --> 00:03:23.010 to increase the amount of products 00:03:23.010 --> 00:03:25.590 and to decrease the amount of reactants. 00:03:25.590 --> 00:03:28.490 Let's calculate QC at this moment and time 00:03:28.490 --> 00:03:30.310 for our second particular diagram 00:03:30.310 --> 00:03:32.860 to see if we've reached equilibrium yet. 00:03:32.860 --> 00:03:36.230 Well, we have five blue particles and only three reds. 00:03:36.230 --> 00:03:40.890 So QC would be equal to five over three or five thirds. 00:03:40.890 --> 00:03:43.980 Remember that the equilibrium constant KC, 00:03:43.980 --> 00:03:45.690 is equal to three. 00:03:45.690 --> 00:03:49.750 Therefore, QC is still not equal to KC. 00:03:49.750 --> 00:03:53.050 And therefore the reaction is not at equilibrium. 00:03:53.050 --> 00:03:56.300 And Q is actually still less than K. 00:03:56.300 --> 00:03:57.850 Therefore the net reaction 00:03:57.850 --> 00:04:00.290 is going to move to the right again 00:04:00.290 --> 00:04:01.930 to increase the amount of products 00:04:01.930 --> 00:04:04.700 and to decrease the amount of reactants. 00:04:04.700 --> 00:04:06.990 Let's compare our second particulate diagram 00:04:06.990 --> 00:04:09.110 to our third particulate diagram. 00:04:09.110 --> 00:04:10.820 And the second particulate diagram, 00:04:10.820 --> 00:04:12.750 we had five blues and three reds. 00:04:12.750 --> 00:04:14.150 And then the third one here, 00:04:14.150 --> 00:04:18.070 we have one, two, three, four, five, six blues, 00:04:18.070 --> 00:04:19.460 and two reds. 00:04:19.460 --> 00:04:21.890 So we've increased in the amount of blue 00:04:21.890 --> 00:04:24.480 and we've decreased in the amount of red. 00:04:24.480 --> 00:04:27.510 Let's calculate QC for the moment of time 00:04:27.510 --> 00:04:30.450 represented by our third particulate diagram. 00:04:30.450 --> 00:04:33.610 Well, there are six blues and only two reds. 00:04:33.610 --> 00:04:36.080 So QC is equal to six divided by two, 00:04:36.080 --> 00:04:38.283 which is equal to three. 00:04:39.490 --> 00:04:41.840 So QC is equal to three 00:04:41.840 --> 00:04:44.570 and remember K is also equal to three. 00:04:44.570 --> 00:04:49.330 So QC is equal to KC and therefore this reaction 00:04:49.330 --> 00:04:51.520 is now at equilibrium. 00:04:51.520 --> 00:04:52.750 And at equilibrium, 00:04:52.750 --> 00:04:54.520 the reactions turn into the products 00:04:54.520 --> 00:04:57.890 at the same rate the products turn back into the reactants. 00:04:57.890 --> 00:05:00.420 And therefore the concentration of both reactants 00:05:00.420 --> 00:05:03.580 and products remained constant at equilibrium. 00:05:03.580 --> 00:05:05.300 So when Q is less than K, 00:05:05.300 --> 00:05:07.710 the reaction is not at equilibrium. 00:05:07.710 --> 00:05:10.620 There are too many reactants and not enough products. 00:05:10.620 --> 00:05:13.230 Therefore the net reaction goes to the right 00:05:13.230 --> 00:05:15.870 to increase the amount of products. 00:05:15.870 --> 00:05:18.120 The net reaction continues to go to the right 00:05:18.120 --> 00:05:20.790 until Q is equal to K and the reaction 00:05:20.790 --> 00:05:22.600 has reached equilibrium. 00:05:22.600 --> 00:05:25.180 At that point, the concentrations of reactants and products 00:05:25.180 --> 00:05:27.720 stop changing and they remain constant. 00:05:27.720 --> 00:05:31.140 It's also possible for Q to be greater than K. 00:05:31.140 --> 00:05:34.400 And if that's true, the reaction is not at equilibrium, 00:05:34.400 --> 00:05:36.590 but in this case you have too many products 00:05:36.590 --> 00:05:38.450 and not enough reactants. 00:05:38.450 --> 00:05:41.610 Therefore the net reaction goes to the left 00:05:41.610 --> 00:05:43.970 to increase the amount of reactants 00:05:43.970 --> 00:05:46.140 and to decrease the amount of products. 00:05:46.140 --> 00:05:48.540 The net reaction will continue to go to the left 00:05:48.540 --> 00:05:53.170 until Q is equal to K and the reaction reaches equilibrium. 00:05:53.170 --> 00:05:54.420 Let's look at another reaction, 00:05:54.420 --> 00:05:56.800 which is the decomposition of phosgene 00:05:56.800 --> 00:06:00.250 to form carbon monoxide and chlorine gas. 00:06:00.250 --> 00:06:03.300 KC for this reaction is equal to 2.2 times 00:06:03.300 --> 00:06:06.850 10 to the negative 10th at 100 degrees Celsius. 00:06:06.850 --> 00:06:08.820 And let's say we're given concentrations 00:06:08.820 --> 00:06:11.730 of phosgene carbon monoxide and chlorine 00:06:11.730 --> 00:06:12.760 at a moment in time, 00:06:12.760 --> 00:06:16.050 and asked if the reaction is at equilibrium or not. 00:06:16.050 --> 00:06:18.170 And if the reaction's not at equilibrium, 00:06:18.170 --> 00:06:21.440 we need to predict which direction the net reaction will go 00:06:21.440 --> 00:06:23.470 to reach equilibrium. 00:06:23.470 --> 00:06:26.070 Our approach is gonna be to calculate QC 00:06:26.070 --> 00:06:30.410 at that moment in time, and then compare QC to KC. 00:06:30.410 --> 00:06:34.180 So first we need to write our QC expression, 00:06:34.180 --> 00:06:38.720 and this is equal to the concentration of carbon monoxide 00:06:38.720 --> 00:06:40.140 raised to the first power 00:06:40.140 --> 00:06:43.030 times the concentration of chlorine 00:06:43.030 --> 00:06:45.070 raised to the first power 00:06:45.070 --> 00:06:49.070 and that's divided by the concentration of phosgene. 00:06:49.070 --> 00:06:52.283 So the concentration of COCL2. 00:06:53.170 --> 00:06:54.310 So at this moment of time, 00:06:54.310 --> 00:06:57.210 the concentration of carbon monoxide is 3.4 times 00:06:57.210 --> 00:06:58.950 10 to the negative six mole. 00:06:58.950 --> 00:07:01.470 The concentration of chlorine is 6.0 times, 00:07:01.470 --> 00:07:03.150 times 10 to the negative six mole, 00:07:03.150 --> 00:07:04.590 and the concentration of phosgene 00:07:04.590 --> 00:07:08.160 is equal to 2.0 times 10 to the negative third mole. 00:07:08.160 --> 00:07:11.150 When we plug all those into our Q expression and solve, 00:07:11.150 --> 00:07:13.570 we get that QC is equal to 1.0 00:07:13.570 --> 00:07:15.483 times 10 to the negative eighth. 00:07:16.530 --> 00:07:21.460 So in this case, QC is greater than KC 00:07:21.460 --> 00:07:24.790 because QC is equal to 1.0 times 10 to the negative eight 00:07:24.790 --> 00:07:28.610 and KC is equal to 2.2 times 10 to the negative 10th. 00:07:28.610 --> 00:07:30.950 And when QC is greater than KC, 00:07:30.950 --> 00:07:33.870 we have too many products and not enough reactants. 00:07:33.870 --> 00:07:37.270 Therefore the net reaction is going to go to the left 00:07:37.270 --> 00:07:39.150 and there's going to be an increase 00:07:39.150 --> 00:07:41.200 in the amount of phosgene. 00:07:41.200 --> 00:07:43.850 The reaction will continue to go to the left 00:07:43.850 --> 00:07:48.850 until Q is equal to K, and the reaction reaches equilibrium.

Worked example: Calculating equilibrium concentrations from initial concentrations

https://www.youtube.com/watch?v=l81swfmUaek

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en

WEBVTT Kind: captions Language: en 00:00:00.700 --> 00:00:02.400 - [Instructor] For the reaction bromine gas 00:00:02.400 --> 00:00:05.670 plus chlorine gas goes to BrCl, 00:00:05.670 --> 00:00:10.290 Kc is equal to 7.0 at 400 Kelvin. 00:00:10.290 --> 00:00:13.840 If the initial concentration of bromine is 0.60 molar 00:00:13.840 --> 00:00:15.740 and the initial concentration of chlorine 00:00:15.740 --> 00:00:19.140 is also 0.60 molar, our goal is to calculate 00:00:19.140 --> 00:00:24.140 the equilibrium concentrations of Br2, Cl2 and BrCl. 00:00:25.970 --> 00:00:28.220 To help us find the equilibrium concentrations, 00:00:28.220 --> 00:00:30.030 we're gonna use an ICE table, 00:00:30.030 --> 00:00:33.280 where I stands for the initial concentration, 00:00:33.280 --> 00:00:35.740 C stands for the change in concentration 00:00:35.740 --> 00:00:39.750 and E stands for equilibrium concentration. 00:00:39.750 --> 00:00:41.380 For the initial concentrations, 00:00:41.380 --> 00:00:43.423 we have 0.60 molar for bromine, 00:00:43.423 --> 00:00:46.700 0.60 molar for chlorine, and if we assume 00:00:46.700 --> 00:00:48.440 the reaction hasn't started yet, 00:00:48.440 --> 00:00:52.600 then we're gonna put a zero in here for our product, BrCl. 00:00:52.600 --> 00:00:57.040 Next, we think about Br2 reacting with Cl2 to form BrCl. 00:00:58.200 --> 00:01:00.300 Some of the bromine is going to react, 00:01:00.300 --> 00:01:01.930 but we don't know how much, 00:01:01.930 --> 00:01:04.650 so we're gonna call that amount x, 00:01:04.650 --> 00:01:06.760 and we're gonna lose some of that bromine 00:01:06.760 --> 00:01:10.340 when we form our product, so we're gonna write minus x 00:01:10.340 --> 00:01:13.090 under bromine in our ICE table. 00:01:13.090 --> 00:01:14.980 Next, we think about mole ratios. 00:01:14.980 --> 00:01:18.490 In the balanced equation, it's a one to one mole ratio 00:01:18.490 --> 00:01:20.360 of bromine to chlorine. 00:01:20.360 --> 00:01:22.790 Therefore, if we're losing x for bromine, 00:01:22.790 --> 00:01:25.290 we're also going to lose x for chlorine. 00:01:25.290 --> 00:01:29.730 So I can write here minus x under chlorine in the ICE table. 00:01:29.730 --> 00:01:33.640 When Br2 and Cl2 react together, we lose our reactants, 00:01:33.640 --> 00:01:36.630 and that means we're gonna gain some of our products. 00:01:36.630 --> 00:01:39.800 To figure out how much, we need to look at mole ratios. 00:01:39.800 --> 00:01:44.730 So the mole ratio of bromine to BrCl is one to two, 00:01:44.730 --> 00:01:48.270 therefore if we're losing x for Br2, 00:01:48.270 --> 00:01:52.050 we must be gaining two x for BrCl. 00:01:52.050 --> 00:01:55.800 So I can go ahead and write plus two x under BrCl. 00:01:55.800 --> 00:01:58.870 Next, let's think about equilibrium concentrations. 00:01:58.870 --> 00:02:01.450 If the initial concentration of bromine is 0.6 00:02:01.450 --> 00:02:04.780 and we're losing x, the equilibrium concentration 00:02:04.780 --> 00:02:08.870 must be 0.60 minus x. 00:02:08.870 --> 00:02:10.340 And the same thing for chlorine. 00:02:10.340 --> 00:02:14.640 It would be 0.60 minus x. 00:02:14.640 --> 00:02:19.430 For BrCl, we start off with zero, and we gained two x. 00:02:19.430 --> 00:02:22.840 Therefore at equilibrium, the equilibrium concentration 00:02:22.840 --> 00:02:25.730 would be equal to just two x. 00:02:25.730 --> 00:02:27.730 The next step is to use the balanced equation 00:02:27.730 --> 00:02:31.150 to write an equilibrium constant expression. 00:02:31.150 --> 00:02:33.920 So we would write Kc is equal to, 00:02:33.920 --> 00:02:35.690 and then we look at our balanced equation, 00:02:35.690 --> 00:02:38.410 and for our product we have BrCl 00:02:38.410 --> 00:02:39.900 with a two as a coefficient, 00:02:39.900 --> 00:02:44.807 so Kc would be equal to the concentration of BrCl squared, 00:02:46.530 --> 00:02:49.200 and we're gonna divide that by the concentration 00:02:49.200 --> 00:02:51.570 of our reactants, which would be Br2, 00:02:51.570 --> 00:02:55.980 so the concentration of Br2 raised to the first power, 00:02:55.980 --> 00:02:57.550 because the coefficient of one, 00:02:57.550 --> 00:03:00.370 times the concentration of Cl2 00:03:00.370 --> 00:03:03.970 also raised to the first power. 00:03:03.970 --> 00:03:06.960 The concentrations in an equilibrium constant expression 00:03:06.960 --> 00:03:10.560 are equilibrium concentrations, therefore we can plug in 00:03:10.560 --> 00:03:13.720 the equilibrium concentrations from our ICE table. 00:03:13.720 --> 00:03:18.560 So the equilibrium concentration for BrCl was two x, 00:03:18.560 --> 00:03:23.560 the equilibrium concentration for Br2 was 0.60 minus x, 00:03:23.600 --> 00:03:27.830 and the same for chlorine, so we can plug that in as well. 00:03:27.830 --> 00:03:30.130 Here we have our equilibrium concentrations 00:03:30.130 --> 00:03:33.080 plugged into our equilibrium constant expression, 00:03:33.080 --> 00:03:37.830 and also Kc was equal to 7.0 for this reaction at 400 Kelvin 00:03:37.830 --> 00:03:41.220 so 7.0 is plugged in for Kc. 00:03:41.220 --> 00:03:45.460 Our goal is to solve for x, and I've re-written it down here 00:03:45.460 --> 00:03:49.730 because 0.60 minus x times 0.60 minus x 00:03:49.730 --> 00:03:52.970 is equal to 0.60 minus x squared. 00:03:52.970 --> 00:03:56.048 And if you write it this way, it's a little bit easier 00:03:56.048 --> 00:03:57.430 to see that we can solve for x 00:03:57.430 --> 00:04:00.930 by taking the square root of both sides. 00:04:00.930 --> 00:04:04.260 So let's go ahead and take the square root of both sides 00:04:04.260 --> 00:04:06.390 and solve for x. 00:04:06.390 --> 00:04:07.850 Taking the square root of both sides 00:04:07.850 --> 00:04:12.850 gives us 2.65 is equal to two x over 0.60 minus x. 00:04:12.940 --> 00:04:15.190 To solve for x, we would then multiply both sides 00:04:15.190 --> 00:04:18.760 by 0.60 minus x to give us this, 00:04:18.760 --> 00:04:20.520 and then after a little more algebra, 00:04:20.520 --> 00:04:24.317 we get 1.59 is equal to 4.65x. 00:04:24.317 --> 00:04:27.853 So x is equal to 1.59 divided by 4.65, 00:04:28.870 --> 00:04:30.880 which is equal to 0.34. 00:04:33.920 --> 00:04:36.260 Now that we know that x is equal to 0.34, 00:04:36.260 --> 00:04:38.830 we can plug that into our ICE table 00:04:38.830 --> 00:04:42.070 and solve for our equilibrium concentrations. 00:04:42.070 --> 00:04:44.890 So for the equilibrium concentration of Br2, 00:04:44.890 --> 00:04:49.513 it's 0.60 minus x, so that's 0.60 minus 0.34, 00:04:52.400 --> 00:04:56.620 which is equal to 0.26 molar. 00:04:56.620 --> 00:05:00.980 So 0.26 molar is the equilibrium concentration for bromine. 00:05:00.980 --> 00:05:02.970 For chlorine, it would be the same calculation, 00:05:02.970 --> 00:05:07.350 0.60 minus x would be 0.60 minus 0.34, 00:05:07.350 --> 00:05:09.520 so the equilibrium concentration of chlorine 00:05:09.520 --> 00:05:12.330 is also 0.26 molar. 00:05:12.330 --> 00:05:16.990 For BrCl, it's two times x so that's two times 0.34, 00:05:19.030 --> 00:05:23.020 which is equal to 0.68 molar. 00:05:23.020 --> 00:05:27.733 So 0.68 molar is the equilibrium concentration for BrCl.

Worked example: Calculating an equilibrium constant from initial and equilibrium pressures

https://www.youtube.com/watch?v=MKt3X0SGrtw

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en

WEBVTT Kind: captions Language: en 00:00:00.760 --> 00:00:02.370 - [Instructor] Let's say we have a pure sample 00:00:02.370 --> 00:00:04.540 of phosphorus pentachloride, 00:00:04.540 --> 00:00:08.284 and we add the PCl5 to a previously evacuated flask 00:00:08.284 --> 00:00:11.410 at 500 Kelvin. 00:00:11.410 --> 00:00:13.620 And let's say the initial pressure of the PCl5 00:00:13.620 --> 00:00:18.200 is 1.66 atmospheres. 00:00:18.200 --> 00:00:23.159 Some of the PCl5 is going to turn into PCl3 and Cl2. 00:00:23.159 --> 00:00:26.370 Once equilibrium is reached, the total pressure, 00:00:26.370 --> 00:00:30.332 let's say, is measured to be 2.35 atmospheres. 00:00:30.332 --> 00:00:33.930 Our goal is to calculate the equilibrium partial pressures 00:00:33.930 --> 00:00:38.930 of these three substances, so PCl5, PCl3 and Cl2. 00:00:39.250 --> 00:00:41.500 And from those equilibrium partial pressures, 00:00:41.500 --> 00:00:44.140 we can also calculate the Kp value 00:00:44.140 --> 00:00:47.120 for this reaction at 500 Kelvin. 00:00:47.120 --> 00:00:49.430 To help us find the equilibrium partial pressures, 00:00:49.430 --> 00:00:51.650 we're gonna use an ICE table where I stands 00:00:51.650 --> 00:00:54.298 for the initial partial pressure in atmospheres, 00:00:54.298 --> 00:00:56.770 C is the change in partial pressure, 00:00:56.770 --> 00:01:00.620 and E stands for the equilibrium partial pressure. 00:01:00.620 --> 00:01:01.470 We already know we're starting 00:01:01.470 --> 00:01:06.340 with a partial pressure of 1.66 atmospheres for PCl5. 00:01:06.340 --> 00:01:08.960 And if we assume that the reaction hasn't started yet, 00:01:08.960 --> 00:01:11.820 we're starting with zero for our partial pressures 00:01:11.820 --> 00:01:14.473 of PCl3 and Cl2. 00:01:15.570 --> 00:01:18.690 Some of the PCl5 is going to decompose. 00:01:18.690 --> 00:01:20.380 And since we don't know how much, 00:01:20.380 --> 00:01:22.370 we're gonna call that amount x. 00:01:22.370 --> 00:01:24.630 So, we're gonna write minus x here, 00:01:24.630 --> 00:01:27.320 since we're gonna lose some PCl5. 00:01:27.320 --> 00:01:29.350 Next, we need to look at mole ratios. 00:01:29.350 --> 00:01:34.125 So, the mole ratio of PCl5 to PCl3 is one to one. 00:01:34.125 --> 00:01:37.070 So, for losing x for PCl5, 00:01:37.070 --> 00:01:40.755 we must be gaining x for PCl3. 00:01:40.755 --> 00:01:43.040 The same idea with Cl2, 00:01:43.040 --> 00:01:45.460 the coefficient in the balanced equation is a one. 00:01:45.460 --> 00:01:48.280 So, if we're losing x for PCl5, 00:01:48.280 --> 00:01:52.220 that means we're gaining x for Cl2. 00:01:52.220 --> 00:01:55.161 Therefore, the equilibrium partial pressure for PCl5 00:01:55.161 --> 00:01:58.900 would be 1.66 minus x. 00:01:58.900 --> 00:02:01.700 The equilibrium partial pressure for PCl3 00:02:01.700 --> 00:02:04.210 would be zero plus x, which is just x. 00:02:04.210 --> 00:02:06.446 And the equilibrium partial pressure for Cl2 00:02:06.446 --> 00:02:09.780 would be zero plus x, which is also x. 00:02:09.780 --> 00:02:13.050 To figure out what x is, we're going to use Dalton's law. 00:02:13.050 --> 00:02:15.700 And Dalton's law says that the total pressure 00:02:15.700 --> 00:02:18.430 of a mixture of gases is equal to the sum 00:02:18.430 --> 00:02:20.560 of the individual partial pressures 00:02:20.560 --> 00:02:22.850 of the gases in the mixture. 00:02:22.850 --> 00:02:24.360 So, we said that the total pressure 00:02:24.360 --> 00:02:26.856 of all the gases at equilibrium 00:02:26.856 --> 00:02:30.870 is equal to 2.35 atmospheres. 00:02:30.870 --> 00:02:33.600 So, we can plug that into Dalton's law. 00:02:33.600 --> 00:02:36.290 And then, we can take the equilibrium partial pressures 00:02:36.290 --> 00:02:40.800 from our ICE table and plug those into Dalton's law as well. 00:02:40.800 --> 00:02:43.220 So, we're gonna plug in 1.66 minus x 00:02:43.220 --> 00:02:46.154 for the equilibrium partial pressure of PCl5, 00:02:46.154 --> 00:02:49.962 x for the equilibrium partial pressure of PCl3, 00:02:49.962 --> 00:02:53.370 and x for the equilibrium partial pressure of PCl2. 00:02:54.560 --> 00:02:59.110 So, let's plug in 1.66 minus x. 00:02:59.110 --> 00:03:03.340 And then, we have plus x and plus x. 00:03:03.340 --> 00:03:04.450 And let's solve for x. 00:03:04.450 --> 00:03:07.110 Notice how we have a minus x and a plus x. 00:03:07.110 --> 00:03:08.630 So, that cancels out. 00:03:08.630 --> 00:03:13.490 So, we simply subtract 1.66 from 2.35 00:03:13.490 --> 00:03:16.880 and we find that x is equal to .69. 00:03:16.880 --> 00:03:20.220 So, if x is equal to .69, 00:03:20.220 --> 00:03:25.220 the equilibrium partial pressure of Cl2 is .69 atmospheres. 00:03:25.733 --> 00:03:29.061 And the equilibrium partial pressure PCl3 00:03:29.061 --> 00:03:31.061 is also .69 atmospheres. 00:03:32.854 --> 00:03:35.104 And 1.66 minus .69 gives us 00:03:36.162 --> 00:03:39.335 the equilibrium partial pressure of PCl5 00:03:39.335 --> 00:03:42.335 and that's equal to .97 atmospheres. 00:03:44.860 --> 00:03:47.180 Now that we have our equilibrium partial pressures 00:03:47.180 --> 00:03:48.680 for all three gases, 00:03:48.680 --> 00:03:51.930 we can calculate the value for the equilibrium constant 00:03:51.930 --> 00:03:54.470 for this reaction at 500 Kelvin. 00:03:54.470 --> 00:03:57.170 First, we need to write an equilibrium constant expression. 00:03:57.170 --> 00:04:01.630 So, we would write Kp is equal to, and for our products, 00:04:01.630 --> 00:04:05.847 we have PCl3, so this would be the partial pressure of PCl3 00:04:07.446 --> 00:04:10.930 times the partial pressure of our other product, 00:04:10.930 --> 00:04:12.720 which is Cl2, so let's put in there 00:04:12.720 --> 00:04:15.130 the partial pressure of Cl2. 00:04:15.130 --> 00:04:20.130 And all of that is divided by the partial pressure of PCl5. 00:04:21.210 --> 00:04:24.423 So, this would be the partial pressure of PCl5. 00:04:25.890 --> 00:04:28.730 Next, we plug in our equilibrium partial pressures. 00:04:28.730 --> 00:04:33.730 So, the equilibrium partial pressure of PCl3 is .69. 00:04:33.790 --> 00:04:38.220 The equilibrium partial pressure of Cl2 is also .69. 00:04:38.220 --> 00:04:42.664 And the equilibrium partial pressure of PCl5 is .97. 00:04:42.664 --> 00:04:45.090 Once we plug our numbers in and we solve, 00:04:45.090 --> 00:04:50.090 we get that Kp is equal to .49 at 500 Kelvin.

Properties of the equilibrium constant

https://www.youtube.com/watch?v=U39OqCEMjto

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WEBVTT Kind: captions Language: en 00:00:00.400 --> 00:00:02.900 - [Instructor] An equilibrium constant has one value 00:00:02.900 --> 00:00:05.990 for a particular reaction at a certain temperature. 00:00:05.990 --> 00:00:07.640 For example, for this reaction, 00:00:07.640 --> 00:00:11.940 we have oxalic acid turning into two H plus ions 00:00:11.940 --> 00:00:14.320 and the oxalate anion. 00:00:14.320 --> 00:00:17.740 The equilibrium constant, K C, for this reaction 00:00:17.740 --> 00:00:20.490 is equal to 3.8 times 10 to the negative six 00:00:20.490 --> 00:00:22.910 at 25 degrees Celsius. 00:00:22.910 --> 00:00:25.830 However, the value of the equilibrium constant changes 00:00:25.830 --> 00:00:28.300 if we change how we write the balanced equation. 00:00:28.300 --> 00:00:30.250 For example, let's reverse everything. 00:00:30.250 --> 00:00:34.100 So, instead of having oxalic acid on the reactant side, 00:00:34.100 --> 00:00:35.810 let's have it on the product side. 00:00:35.810 --> 00:00:38.970 And instead of having H plus ions in the product side 00:00:38.970 --> 00:00:40.760 and oxalate on the product side, 00:00:40.760 --> 00:00:43.800 let's put them on the reactant side. 00:00:43.800 --> 00:00:46.050 Since we've reversed the reaction, 00:00:46.050 --> 00:00:48.270 there's a new equilibrium constant. 00:00:48.270 --> 00:00:51.570 So we're going to write K C for the reverse reaction, 00:00:51.570 --> 00:00:52.403 is equal to, 00:00:52.403 --> 00:00:54.100 and to figure out the value 00:00:54.100 --> 00:00:56.180 for the new equilibrium constant, 00:00:56.180 --> 00:00:57.600 since we reversed everything, 00:00:57.600 --> 00:01:01.630 we take the inverse of the original equilibrium constant. 00:01:01.630 --> 00:01:03.250 So to find the new K C value, 00:01:03.250 --> 00:01:06.790 we take one over the original equilibrium constant 00:01:06.790 --> 00:01:11.790 of 3.8 times 10 to the negative six. 00:01:13.400 --> 00:01:18.400 And this is equal to 2.6 times 10 to the fifth, 00:01:20.360 --> 00:01:23.490 also at 25 degrees Celsius. 00:01:23.490 --> 00:01:24.980 Let's look at another example of how 00:01:24.980 --> 00:01:26.700 changing how we write the equation 00:01:26.700 --> 00:01:29.600 changes the value for the equilibrium constant. 00:01:29.600 --> 00:01:31.330 And we start by looking at 00:01:31.330 --> 00:01:33.560 the ionization of hydrofluoric acid, 00:01:33.560 --> 00:01:37.350 turning into an H plus ion and a fluoride anion. 00:01:37.350 --> 00:01:39.450 The K C value for this reaction is equal to 00:01:39.450 --> 00:01:41.700 6.8 times 10 to the negative fourth 00:01:41.700 --> 00:01:44.570 at 25 degrees Celsius. 00:01:44.570 --> 00:01:46.410 This time we're going to multiply everything through 00:01:46.410 --> 00:01:47.800 by a factor of two. 00:01:47.800 --> 00:01:50.120 So it would be two H F 00:01:50.120 --> 00:01:54.790 turning into two H plus, plus two F minus. 00:01:54.790 --> 00:01:56.210 Our goal is to find the value 00:01:56.210 --> 00:01:58.920 for the new equilibrium constant K C. 00:01:58.920 --> 00:02:01.609 And since we multiplied everything through 00:02:01.609 --> 00:02:02.830 by a factor of two, 00:02:02.830 --> 00:02:05.070 we're going to take the old equilibrium constant 00:02:05.070 --> 00:02:08.220 and we're going to raise it to the second power. 00:02:08.220 --> 00:02:10.120 So this is K C is equal to 00:02:10.120 --> 00:02:15.050 6.8 times 10 to the negative fourth. 00:02:15.050 --> 00:02:16.893 And we're going to square that. 00:02:18.010 --> 00:02:23.010 So this is equal to 4.6 times 10 to the negative seventh 00:02:24.140 --> 00:02:27.010 at 25 degrees Celsius. 00:02:27.010 --> 00:02:30.050 So since we multiplied through by a factor of two, 00:02:30.050 --> 00:02:32.940 we raised the equilibrium constant to the second power. 00:02:32.940 --> 00:02:35.490 If we had multiplied through by a factor of three, 00:02:35.490 --> 00:02:37.820 we would have raised the old equilibrium constant 00:02:37.820 --> 00:02:39.602 to the third power. 00:02:39.602 --> 00:02:42.800 Let's use the two reactions we've just looked at here. 00:02:42.800 --> 00:02:46.640 So we have oxalic acid turning into H plus and oxalate. 00:02:46.640 --> 00:02:50.010 And the other one we looked at was hydrofluoric acid 00:02:50.010 --> 00:02:52.830 ionizing to turn into H plus and F minus. 00:02:52.830 --> 00:02:54.270 Let's use those two reactions 00:02:54.270 --> 00:02:56.610 to calculate the equilibrium constant 00:02:56.610 --> 00:02:58.470 for a third reaction. 00:02:58.470 --> 00:03:00.450 And this third reaction shows oxalate 00:03:00.450 --> 00:03:02.550 reacting with hydrofluoric acid 00:03:02.550 --> 00:03:06.800 to form oxalic acid and the fluoride anion. 00:03:06.800 --> 00:03:09.260 Our first step is to get these first two reactions 00:03:09.260 --> 00:03:11.360 to look more like the third reaction. 00:03:11.360 --> 00:03:15.051 For example, we have oxalic acid on the reactant side, 00:03:15.051 --> 00:03:17.120 in the first reaction here, 00:03:17.120 --> 00:03:19.500 and oxalic acid is on the product side 00:03:19.500 --> 00:03:21.230 for the third reaction. 00:03:21.230 --> 00:03:23.240 Also notice, oxalate is on the product side 00:03:23.240 --> 00:03:25.170 for the first and over here for the third, 00:03:25.170 --> 00:03:26.860 it's on the reactant side. 00:03:26.860 --> 00:03:29.610 So we need to reverse the first reaction 00:03:29.610 --> 00:03:32.060 to make it look more like the third one. 00:03:32.060 --> 00:03:34.060 Here we have that reaction reversed. 00:03:34.060 --> 00:03:37.136 And since we reversed the reaction, we would have to take 00:03:37.136 --> 00:03:40.210 the inverse of the equilibrium constant, 00:03:40.210 --> 00:03:42.610 which we did in the earlier example. 00:03:42.610 --> 00:03:45.530 So let's go ahead and cross out this first one here, 00:03:45.530 --> 00:03:48.215 because we're not going to need it anymore. 00:03:48.215 --> 00:03:50.700 And let's move on to our next reaction 00:03:50.700 --> 00:03:53.510 and try to make it look more like the third one. 00:03:53.510 --> 00:03:55.100 So here we have hydrofluoric acid 00:03:55.100 --> 00:03:58.210 turning into H plus and F minus. 00:03:58.210 --> 00:04:00.140 And for our third reaction, 00:04:00.140 --> 00:04:02.650 we have hydrofluoric acid on the reactant side, 00:04:02.650 --> 00:04:05.040 but notice there is a coefficient of two 00:04:05.040 --> 00:04:06.400 in the balanced equation, 00:04:06.400 --> 00:04:09.970 and we have only a one in how we have it written. 00:04:09.970 --> 00:04:12.720 So we need to multiply everything through 00:04:12.720 --> 00:04:14.609 by a factor of two 00:04:14.609 --> 00:04:19.609 to give us two H F and two H plus and two F minus. 00:04:21.630 --> 00:04:23.270 And since we multiplied everything through 00:04:23.270 --> 00:04:25.040 by a factor of two, 00:04:25.040 --> 00:04:27.760 we would need to square the equilibrium constant 00:04:27.760 --> 00:04:29.310 for the original reaction. 00:04:29.310 --> 00:04:32.670 And once again, this is from our previous example. 00:04:32.670 --> 00:04:35.583 So we already know that equilibrium constant value. 00:04:36.430 --> 00:04:38.970 Next, let's add these two reactions together 00:04:38.970 --> 00:04:43.140 and make sure they give us the third reaction. 00:04:43.140 --> 00:04:46.726 So we have all of our reactants on one side 00:04:46.726 --> 00:04:51.726 and we have all of our products on the other side here. 00:04:52.000 --> 00:04:53.830 And we look for things that are the same 00:04:53.830 --> 00:04:56.050 on both the reactants and the products side. 00:04:56.050 --> 00:04:58.170 Well, there's two H plus on the left 00:04:58.170 --> 00:05:00.050 and there's two H plus on the right, 00:05:00.050 --> 00:05:01.700 so that would cancel out. 00:05:01.700 --> 00:05:04.640 And now, notice that we would have 00:05:04.640 --> 00:05:09.640 oxalate plus two H F going to oxalic acid plus two F minus. 00:05:13.620 --> 00:05:15.610 So adding our two reactions together 00:05:15.610 --> 00:05:18.800 does give us our third reaction. 00:05:18.800 --> 00:05:21.980 Once we've confirmed that adding our two reactions together 00:05:21.980 --> 00:05:24.350 gives us our third reaction, 00:05:24.350 --> 00:05:26.240 we can use the equilibrium constants 00:05:26.240 --> 00:05:27.670 for the first two reactions 00:05:27.670 --> 00:05:30.950 to figure out the equilibrium constant for the third. 00:05:30.950 --> 00:05:32.370 So the equilibrium constant 00:05:32.370 --> 00:05:34.960 for this first reaction we're gonna call K one, 00:05:34.960 --> 00:05:37.720 for the second reaction we're gonna call K two, 00:05:37.720 --> 00:05:39.090 and the equilibrium constant 00:05:39.090 --> 00:05:41.260 that we're trying to find for the third reaction 00:05:41.260 --> 00:05:42.653 we'll call K C. 00:05:43.530 --> 00:05:47.150 To find the equilibrium constant for the third reaction K C, 00:05:47.150 --> 00:05:49.530 we need to multiply the equilibrium constants 00:05:49.530 --> 00:05:51.720 for the first two reactions together. 00:05:51.720 --> 00:05:56.720 So K C is equal to K one times K two. 00:05:58.040 --> 00:06:00.240 We've already calculated the value for our K one 00:06:00.240 --> 00:06:02.298 from an earlier example, 00:06:02.298 --> 00:06:03.650 it was 2.6 times 10 to the fifth, 00:06:03.650 --> 00:06:05.880 and we also calculated K two, 00:06:05.880 --> 00:06:09.830 and it was 4.6 times 10 to the negative seven. 00:06:09.830 --> 00:06:10.870 So to find K C, 00:06:10.870 --> 00:06:12.980 the equilibrium constant for the third reaction, 00:06:12.980 --> 00:06:15.230 we simply multiply those two together 00:06:15.230 --> 00:06:20.130 and we get that K C is equal to point one two 00:06:20.130 --> 00:06:23.410 at 25 degrees Celsius. 00:06:23.410 --> 00:06:25.350 So whenever you add reactions together 00:06:25.350 --> 00:06:26.930 to get a new reaction, 00:06:26.930 --> 00:06:29.280 to find the equilibrium constant for the new reaction, 00:06:29.280 --> 00:06:32.206 you simply need to multiply the equilibrium constants 00:06:32.206 --> 00:06:34.853 of the reactions that you added together.

Magnitude of the equilibrium constant

https://www.youtube.com/watch?v=CoRawW0_Kns

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en

WEBVTT Kind: captions Language: en 00:00:00.080 --> 00:00:02.070 - [Instructor] The magnitude of the equilibrium constant 00:00:02.070 --> 00:00:04.480 tells us the relative amounts of products 00:00:04.480 --> 00:00:06.840 and reactants at equilibrium. 00:00:06.840 --> 00:00:09.270 For example, let's look at a hypothetical reaction 00:00:09.270 --> 00:00:13.110 where gas A turns into gas B. 00:00:13.110 --> 00:00:14.350 And for the first example, 00:00:14.350 --> 00:00:17.640 let's say that gas A is represented by a red sphere 00:00:17.640 --> 00:00:20.740 and gas B is represented by a blue sphere. 00:00:20.740 --> 00:00:22.900 And down here, we have a particulate diagram 00:00:22.900 --> 00:00:26.910 showing an equilibrium mixture of our hypothetical reaction. 00:00:26.910 --> 00:00:29.160 Let's write the equilibrium constant expression 00:00:29.160 --> 00:00:31.440 for this hypothetical reaction. 00:00:31.440 --> 00:00:34.310 So we're gonna write Kc is equal to, 00:00:34.310 --> 00:00:36.820 and we think products over reactants. 00:00:36.820 --> 00:00:38.600 So our product is B. 00:00:38.600 --> 00:00:41.520 So this can be the concentration of B. 00:00:41.520 --> 00:00:44.200 And since the coefficient is a one in the balanced equation, 00:00:44.200 --> 00:00:47.000 it's the concentration of B raised to the first power 00:00:47.000 --> 00:00:50.530 divided by the concentration of our reactant, which is A. 00:00:50.530 --> 00:00:52.540 And A in the balanced equation 00:00:52.540 --> 00:00:54.070 also has a coefficient of one. 00:00:54.070 --> 00:00:57.160 So this is the concentration of A raised to the first power. 00:00:57.160 --> 00:01:00.910 If we assume that each particle in our particulate diagram 00:01:00.910 --> 00:01:04.510 represents 0.1 moles of a substance, 00:01:04.510 --> 00:01:08.610 and the volume is one liter, 00:01:08.610 --> 00:01:13.160 we can calculate the concentration of both A and B. 00:01:13.160 --> 00:01:16.950 For example, for B, there are five blue spheres. 00:01:16.950 --> 00:01:21.360 So that'll be five times 0.1 moles or 0.5 moles. 00:01:21.360 --> 00:01:23.510 So for the concentration of B, 00:01:23.510 --> 00:01:28.510 we have 0.5 moles divided by a volume of one liter. 00:01:28.650 --> 00:01:32.670 So 0.5 divided by one is 0.5 molar. 00:01:32.670 --> 00:01:34.060 So we can go ahead and plug that in 00:01:34.060 --> 00:01:36.120 for our concentration of B. 00:01:36.120 --> 00:01:38.730 It's 0.5 molar. 00:01:38.730 --> 00:01:41.440 Next, we can do the same thing for A. 00:01:41.440 --> 00:01:44.430 There are also five red spheres. 00:01:44.430 --> 00:01:48.270 And so therefore the concentration of A is also 0.5 molar. 00:01:48.270 --> 00:01:49.180 So we can plug that 00:01:49.180 --> 00:01:52.240 into our equilibrium constant expression. 00:01:52.240 --> 00:01:54.410 0.5 divided by 0.5 is equal to one. 00:01:54.410 --> 00:01:57.080 So therefore, Kc, the equilibrium constant 00:01:57.080 --> 00:02:00.060 is equal to one at whatever temperature we have 00:02:00.060 --> 00:02:02.130 for our hypothetical reaction. 00:02:02.130 --> 00:02:05.130 So our equilibrium constant Kc is equal to one. 00:02:05.130 --> 00:02:08.800 And we saw in our particulate diagram at equilibrium, 00:02:08.800 --> 00:02:12.170 we have equal amounts of reactants and products. 00:02:12.170 --> 00:02:13.720 Therefore just by knowing the value 00:02:13.720 --> 00:02:15.530 for the equilibrium constant, 00:02:15.530 --> 00:02:17.757 we know about the relative amounts of reactants 00:02:17.757 --> 00:02:20.620 and products at equilibrium. 00:02:20.620 --> 00:02:22.680 Let's look at another hypothetical reaction, 00:02:22.680 --> 00:02:25.990 which also has gas A turning into gas B. 00:02:25.990 --> 00:02:30.310 However, this time gas A is green and gas B is red. 00:02:30.310 --> 00:02:33.250 And let's calculate the equilibrium constant Kc 00:02:33.250 --> 00:02:34.570 for this reaction. 00:02:34.570 --> 00:02:36.620 And once again, our particulate diagram 00:02:36.620 --> 00:02:38.970 shows an equilibrium mixture. 00:02:38.970 --> 00:02:40.960 So Kc is equal to the concentration of B 00:02:40.960 --> 00:02:42.917 over the concentration of A. 00:02:42.917 --> 00:02:46.540 And it's a lot faster to simply count our particles. 00:02:46.540 --> 00:02:50.420 So for B, B is red, we have one red particle here, 00:02:50.420 --> 00:02:53.400 so we can go ahead and put in one. 00:02:53.400 --> 00:02:57.740 And then for gas A, we have one, two, three, four, 00:02:57.740 --> 00:03:01.260 five, six, seven, eight, nine, 10 particles. 00:03:01.260 --> 00:03:04.700 So one divided by 10 is equal to 0.1. 00:03:04.700 --> 00:03:08.530 So Kc is equal to 0.1 for this hypothetical reaction 00:03:08.530 --> 00:03:10.100 at a certain temperature. 00:03:10.100 --> 00:03:13.120 So the magnitude of the equilibrium constant tells us 00:03:13.120 --> 00:03:16.840 about the reaction mixture at equilibrium. 00:03:16.840 --> 00:03:19.950 For this reaction, Kc is equal to 0.1. 00:03:19.950 --> 00:03:22.660 So K is less than one. 00:03:22.660 --> 00:03:24.930 And if we think about what that means, 00:03:24.930 --> 00:03:27.320 K is equal to products over reactants. 00:03:27.320 --> 00:03:28.910 So if K is less than one, 00:03:28.910 --> 00:03:32.120 that means we have a smaller number in the numerator 00:03:32.120 --> 00:03:34.080 and a larger number in the denominator, 00:03:34.080 --> 00:03:37.610 which means there are more reactants than products 00:03:37.610 --> 00:03:39.530 at equilibrium. 00:03:39.530 --> 00:03:41.520 Let's look at another hypothetical reaction 00:03:41.520 --> 00:03:43.750 where gas A turns into gas B. 00:03:43.750 --> 00:03:47.370 This time gas A is yellow and gas B is blue. 00:03:47.370 --> 00:03:49.360 If we look at our particulate diagram, 00:03:49.360 --> 00:03:52.100 showing our reaction mixture at equilibrium, 00:03:52.100 --> 00:03:57.100 there are 10 blue particles and only one yellow particle. 00:03:58.240 --> 00:04:02.070 So when plug into our equilibrium constant expression, 00:04:02.070 --> 00:04:04.980 this time it's going to be 10 over one. 00:04:04.980 --> 00:04:09.460 Therefore the equilibrium constant Kc is equal to 10 00:04:09.460 --> 00:04:12.920 for this particular reaction at a certain temperature. 00:04:12.920 --> 00:04:15.410 Once again, the magnitude of the equilibrium constant 00:04:15.410 --> 00:04:18.270 tells us something about the reaction mixture 00:04:18.270 --> 00:04:19.730 at equilibrium. 00:04:19.730 --> 00:04:23.260 For this hypothetical reaction, Kc is equal to 10. 00:04:23.260 --> 00:04:25.830 So K is greater than one. 00:04:25.830 --> 00:04:27.480 And when K is greater than one, 00:04:27.480 --> 00:04:29.720 once again, we have products over reactants. 00:04:29.720 --> 00:04:33.520 So the numerator must be larger than the denominator, 00:04:33.520 --> 00:04:35.410 which means we have a lot more products 00:04:35.410 --> 00:04:38.190 than reactants at equilibrium. 00:04:38.190 --> 00:04:40.610 Let's look at the reaction of carbon monoxide 00:04:40.610 --> 00:04:44.030 and chlorine gas to form phosgene. 00:04:44.030 --> 00:04:46.530 At 100 degrees Celsius, the equilibrium constant 00:04:46.530 --> 00:04:50.653 for this reaction is 4.56 times 10 to the ninth. 00:04:51.830 --> 00:04:55.460 Since the equilibrium constant K is greater than one, 00:04:55.460 --> 00:04:57.870 we know there are more products than reactants 00:04:57.870 --> 00:04:59.310 at equilibrium. 00:04:59.310 --> 00:05:01.410 And with the extremely large value for K, 00:05:01.410 --> 00:05:02.800 like 10 to the ninth, 00:05:02.800 --> 00:05:04.300 we could even assume this reaction 00:05:04.300 --> 00:05:06.610 essentially goes to completion. 00:05:06.610 --> 00:05:09.240 For the reaction of hydrogen gas and iodine gas 00:05:09.240 --> 00:05:13.010 to form hydrogen iodine, the equilibrium constant Kc 00:05:13.010 --> 00:05:17.120 is equal to 51 at 448 degrees Celsius. 00:05:17.120 --> 00:05:20.540 Since the equilibrium constant is relatively close to one. 00:05:20.540 --> 00:05:21.850 This means at equilibrium, 00:05:21.850 --> 00:05:24.920 we have appreciable amounts of both our reactants 00:05:24.920 --> 00:05:27.010 and our products. 00:05:27.010 --> 00:05:30.010 Let's look at the reaction of nitrogen gas plus oxygen gas 00:05:30.010 --> 00:05:32.700 plus bromine gas to form NOBr. 00:05:32.700 --> 00:05:36.370 At 298 Kelvin, the equilibrium constant for this reaction 00:05:36.370 --> 00:05:40.490 is 9.5 times 10 to the negative 31st. 00:05:40.490 --> 00:05:43.960 Since the equilibrium constant K is less than one, 00:05:43.960 --> 00:05:45.260 we know at equilibrium, 00:05:45.260 --> 00:05:48.200 there are more reactants than there are products. 00:05:48.200 --> 00:05:50.360 And with an extremely small K value, 00:05:50.360 --> 00:05:53.518 like 9.5 times 10 to the negative 31st, 00:05:53.518 --> 00:05:56.720 this reaction barely proceeds at all. 00:05:56.720 --> 00:05:58.080 So at equilibrium, you're gonna have 00:05:58.080 --> 00:06:00.770 almost all nitrogen, oxygen and bromine 00:06:00.770 --> 00:06:02.833 and very little NOBr.

Worked examples: Calculating equilibrium constants

https://www.youtube.com/watch?v=5HZbCNg9mIw

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WEBVTT Kind: captions Language: en 00:00:00.220 --> 00:00:03.200 - [Instructor] An equilibrium constant can be calculated 00:00:03.200 --> 00:00:06.130 from experimentally measured concentrations 00:00:06.130 --> 00:00:08.930 or partial pressures of reactants and products 00:00:08.930 --> 00:00:09.963 at equilibrium. 00:00:10.800 --> 00:00:12.700 As an example, let's look at the reaction 00:00:12.700 --> 00:00:16.040 where N2O4 in the gaseous state turns into 2NO2 00:00:16.930 --> 00:00:18.820 also in the gaseous state. 00:00:18.820 --> 00:00:20.490 And let's say we do an experiment 00:00:20.490 --> 00:00:23.920 and we allow this reaction to come to equilibrium 00:00:23.920 --> 00:00:26.888 and the temperature is 100 degrees Celsius. 00:00:26.888 --> 00:00:28.650 And at equilibrium, 00:00:28.650 --> 00:00:33.650 the concentration of NO2 0.017 molar 00:00:33.710 --> 00:00:38.710 and the concentration of N2O4 is 0.00140 molar. 00:00:40.350 --> 00:00:43.590 To calculate the equilibrium constant for this reaction 00:00:43.590 --> 00:00:46.210 at 100 degrees Celsius, we first need to write 00:00:46.210 --> 00:00:48.950 the equilibrium constant expression. 00:00:48.950 --> 00:00:51.170 We can write the equilibrium constant expression 00:00:51.170 --> 00:00:53.170 by using the balanced equation. 00:00:53.170 --> 00:00:55.410 We start by writing the equilibrium constant, 00:00:55.410 --> 00:00:57.110 which is symbolized by K. 00:00:57.110 --> 00:00:58.950 And since we're dealing with concentrations, 00:00:58.950 --> 00:01:00.760 we're calculating Kc. 00:01:00.760 --> 00:01:04.690 And Kc is equal to, we do products over reactants. 00:01:04.690 --> 00:01:08.740 So this would be the concentration of NO2. 00:01:08.740 --> 00:01:12.490 And since there is a coefficient of two in front of NO2, 00:01:12.490 --> 00:01:15.994 this is the concentration of NO2 raised to the second power 00:01:15.994 --> 00:01:19.823 divided by the concentration of our reactant, N2O4. 00:01:21.650 --> 00:01:24.630 And since there's an implied one in front of N2O4, 00:01:24.630 --> 00:01:28.530 this is the concentration of N2O4 raised to the first power. 00:01:28.530 --> 00:01:31.470 Next, we plug in our equilibrium concentrations. 00:01:31.470 --> 00:01:36.470 So the equilibrium concentration of NO2 is 0.0172. 00:01:36.770 --> 00:01:38.130 So let's plug that in. 00:01:38.130 --> 00:01:42.463 So this is equal to 0.0172 squared 00:01:45.084 --> 00:01:48.643 divided by the equilibrium concentration of N2O4, 00:01:49.780 --> 00:01:51.553 which was 0.00140. 00:01:52.750 --> 00:01:54.920 So we plug that in as well. 00:01:54.920 --> 00:01:59.480 So 0.00140. 00:01:59.480 --> 00:02:03.033 When we solve this, we get that Kc is equal to 0.211, 00:02:05.460 --> 00:02:08.660 and this is at 100 degrees Celsius. 00:02:08.660 --> 00:02:10.960 It's important to always give the temperature 00:02:10.960 --> 00:02:13.960 when you're giving a value for an equilibrium constant, 00:02:13.960 --> 00:02:17.110 because an equilibrium constant is only constant 00:02:17.110 --> 00:02:20.930 for a particular reaction at a particular temperature. 00:02:20.930 --> 00:02:22.470 And it's also important to note 00:02:22.470 --> 00:02:26.090 that the equilibrium constant doesn't have any units. 00:02:26.090 --> 00:02:29.850 So we would just say that Kc is equal to 0.211 00:02:29.850 --> 00:02:34.100 at 100 degrees Celsius for this particular reaction. 00:02:34.100 --> 00:02:36.080 Let's calculate the equilibrium constant 00:02:36.080 --> 00:02:37.720 for another reaction. 00:02:37.720 --> 00:02:42.000 In this reaction, carbon dioxide reacts with hydrogen gas 00:02:42.000 --> 00:02:45.410 to produce carbon monoxide and H2O. 00:02:45.410 --> 00:02:47.920 And since everything is in the gaseous state, 00:02:47.920 --> 00:02:50.960 experimentally, it's easier to work with partial pressures 00:02:50.960 --> 00:02:53.150 than it is to work with concentrations. 00:02:53.150 --> 00:02:55.490 So instead of calculating Kc, 00:02:55.490 --> 00:02:59.110 we're gonna calculate Kp or the p stands for pressure. 00:02:59.110 --> 00:03:03.840 So we're trying to find Kp at 500 Kelvin for this reaction. 00:03:03.840 --> 00:03:07.080 To help us find Kp, we're going to use an ICE table 00:03:07.080 --> 00:03:10.150 where I stands for the initial partial pressure 00:03:10.150 --> 00:03:12.070 in atmospheres, C stands for the 00:03:12.070 --> 00:03:15.020 change in the partial pressure, also in atmospheres 00:03:15.020 --> 00:03:18.070 and E is the equilibrium partial pressure. 00:03:18.070 --> 00:03:20.640 Let's say that a mixture of carbon dioxide, 00:03:20.640 --> 00:03:22.940 hydrogen gas and H2O are placed 00:03:22.940 --> 00:03:25.150 in a previously evacuated flask 00:03:25.150 --> 00:03:29.320 and allowed to come to equilibrium at 500 Kelvin. 00:03:29.320 --> 00:03:32.510 And let's say the initial measured partial pressures 00:03:32.510 --> 00:03:36.540 are 4.10 atmospheres for carbon dioxide, 00:03:36.540 --> 00:03:40.160 1.80 atmospheres for hydrogen gas 00:03:40.160 --> 00:03:44.090 and 3.20 atmospheres for H2O. 00:03:44.090 --> 00:03:46.420 And since we didn't add any carbon monoxide 00:03:46.420 --> 00:03:48.320 in the beginning, the initial partial pressure 00:03:48.320 --> 00:03:49.888 of that would be zero. 00:03:49.888 --> 00:03:52.850 And after the reaction comes to equilibrium, 00:03:52.850 --> 00:03:55.440 we measure the partial pressure of H2O 00:03:55.440 --> 00:03:58.440 to be 3.40 atmospheres. 00:03:58.440 --> 00:04:01.885 So that's why we have 3.40 in the equilibrium parts 00:04:01.885 --> 00:04:04.780 on the ICE table under H2O. 00:04:04.780 --> 00:04:09.150 So the initial partial pressure of H2O is 3.20 atmospheres 00:04:09.150 --> 00:04:13.370 and the equilibrium partial pressure is 3.40. 00:04:13.370 --> 00:04:16.790 So H2O has increased in partial pressure. 00:04:16.790 --> 00:04:19.080 We can go ahead in here and write plus X 00:04:19.080 --> 00:04:21.750 for an increase in the partial pressure of H2O 00:04:21.750 --> 00:04:26.750 and 3.20 plus X must be equal to 3.40. 00:04:28.920 --> 00:04:33.080 So X is equal to 0.20. 00:04:33.080 --> 00:04:37.050 So the partial pressure of water increased by 0.20. 00:04:37.050 --> 00:04:40.115 And we could either write plus X in here on our ICE table, 00:04:40.115 --> 00:04:43.193 or we could just write plus 0.20. 00:04:45.290 --> 00:04:47.710 Now that we know that change in the partial pressure 00:04:47.710 --> 00:04:50.900 for H2O, we can use this information 00:04:50.900 --> 00:04:53.350 to fill out the rest of our ICE table. 00:04:53.350 --> 00:04:56.270 For example, the mole ratio of carbon monoxide 00:04:56.270 --> 00:04:58.970 to H2O is 1:1. 00:04:58.970 --> 00:05:02.810 So if we gained plus 0.20 for H2O, 00:05:02.810 --> 00:05:07.810 we're also gonna gain plus 0.20 for carbon monoxide. 00:05:08.010 --> 00:05:10.120 And if we're gaining for our two products here, 00:05:10.120 --> 00:05:12.410 the net reaction is moving to the right 00:05:12.410 --> 00:05:14.270 to increase the amount of products, 00:05:14.270 --> 00:05:16.990 which means we're losing reactants. 00:05:16.990 --> 00:05:19.460 And we can figure out by how much by looking 00:05:19.460 --> 00:05:21.340 at the mole ratios again. 00:05:21.340 --> 00:05:22.900 So for both of our reactants, 00:05:22.900 --> 00:05:26.460 we have ones as coefficients in the balanced equation. 00:05:26.460 --> 00:05:29.310 So if it's plus X for both of our products, 00:05:29.310 --> 00:05:32.400 it must be minus X for both of our reactants. 00:05:32.400 --> 00:05:36.877 And since X is 0.20, it'd be minus 0.20 00:05:38.000 --> 00:05:39.960 for the change in the partial pressure 00:05:39.960 --> 00:05:42.143 for both of our reactants. 00:05:43.250 --> 00:05:46.830 Therefore the equilibrium partial pressure of carbon dioxide 00:05:46.830 --> 00:05:51.670 would be 4.10 minus 0.20, which is 3.90 00:05:53.720 --> 00:05:58.720 and for H2, it'd be 1.80 minus 0.20, which is 1.60. 00:06:00.750 --> 00:06:03.650 And for carbon monoxide, we started off with zero 00:06:03.650 --> 00:06:06.260 and we gained positive 0.20. 00:06:06.260 --> 00:06:11.020 Therefore the equilibrium partial pressure is 0.20. 00:06:11.020 --> 00:06:13.900 So as the net reaction moved to the right, 00:06:13.900 --> 00:06:16.140 we lost some of our reactants 00:06:16.140 --> 00:06:18.300 and we gained some of our products 00:06:18.300 --> 00:06:20.870 until the reaction reached equilibrium 00:06:20.870 --> 00:06:23.920 and we got our equilibrium partial pressures. 00:06:23.920 --> 00:06:27.310 In our equilibrium, the rate of the forward reaction 00:06:28.410 --> 00:06:30.300 is equal to the rate of the reverse reaction 00:06:30.300 --> 00:06:33.040 and therefore these equilibrium partial pressures 00:06:33.040 --> 00:06:35.130 remain constant. 00:06:35.130 --> 00:06:37.470 Now that we know our equilibrium partial pressures, 00:06:37.470 --> 00:06:41.040 we're ready to calculate the equilibrium constant Kp. 00:06:41.040 --> 00:06:43.900 So we need to write an equilibrium constant expression. 00:06:43.900 --> 00:06:48.460 So Kp is equal to, we think about products over reactants. 00:06:48.460 --> 00:06:49.820 And for our products, 00:06:49.820 --> 00:06:53.270 we would have the partial pressure of carbon monoxide. 00:06:53.270 --> 00:06:56.100 And since the coefficient is a one 00:06:56.100 --> 00:06:59.010 in front of carbon monoxide in the balanced equation, 00:06:59.010 --> 00:07:01.410 it would be the partial pressure of carbon monoxide 00:07:01.410 --> 00:07:04.530 raised to the first power times the partial pressure 00:07:04.530 --> 00:07:07.440 of our other product, which is H2O. 00:07:07.440 --> 00:07:09.830 And once again, the coefficient is a one. 00:07:09.830 --> 00:07:13.300 So that's the partial pressure raised to the first power. 00:07:13.300 --> 00:07:15.470 All of this is divided by, 00:07:15.470 --> 00:07:17.300 we think about our reactants next, 00:07:17.300 --> 00:07:19.270 and they both have coefficients of one 00:07:19.270 --> 00:07:20.770 in the balanced equation. 00:07:20.770 --> 00:07:24.940 So it would be the partial pressure of carbon dioxide 00:07:24.940 --> 00:07:29.940 times the partial pressure of hydrogen gas. 00:07:30.780 --> 00:07:34.090 The partial pressures in our equilibrium constant expression 00:07:34.090 --> 00:07:36.370 are the equilibrium partial pressures, 00:07:36.370 --> 00:07:38.266 which we can get from the ICE table. 00:07:38.266 --> 00:07:41.590 So the equilibrium partial pressure of carbon monoxide 00:07:41.590 --> 00:07:46.590 is 0.20, the equilibrium partial pressure of H2O is 3.40. 00:07:48.070 --> 00:07:50.270 We can plug in the equilibrium partial pressures 00:07:50.270 --> 00:07:53.369 for carbon dioxide and the equilibrium partial pressure 00:07:53.369 --> 00:07:56.030 for hydrogen gas as well. 00:07:56.030 --> 00:07:58.240 And here we have the equilibrium partial pressures 00:07:58.240 --> 00:08:01.230 plugged into our equilibrium constant expression. 00:08:01.230 --> 00:08:02.600 And when we solve this, 00:08:02.600 --> 00:08:06.767 we get that Kp is equal to 0.11 at 500 Kelvin.

Writing equilibrium constant and reaction quotient expressions

https://www.youtube.com/watch?v=lG8tCeNzEjY

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en

WEBVTT Kind: captions Language: en 00:00:00.480 --> 00:00:02.820 - [Instructor] The equilibrium constant is symbolized 00:00:02.820 --> 00:00:07.410 by the letter K, and equilibrium constant tells us 00:00:07.410 --> 00:00:10.110 about the relative concentrations of reactants 00:00:10.110 --> 00:00:12.590 and products at equilibrium. 00:00:12.590 --> 00:00:14.822 Let's say we have a hypothetical reaction 00:00:14.822 --> 00:00:19.822 where reactants A and B turn into products C and D. 00:00:20.070 --> 00:00:21.510 And in the balanced equation, 00:00:21.510 --> 00:00:23.840 the lowercase letters are the coefficients. 00:00:23.840 --> 00:00:26.380 So we have a lowercase a, a lowercase b, 00:00:26.380 --> 00:00:29.230 lowercase c and lowercase d as coefficients 00:00:29.230 --> 00:00:31.020 in our balanced equation. 00:00:31.020 --> 00:00:33.530 If we were to write an equilibrium constant expression 00:00:33.530 --> 00:00:35.420 for this hypothetical reaction, 00:00:35.420 --> 00:00:37.700 we'd start by writing the equilibrium constant K 00:00:37.700 --> 00:00:39.760 and then we have a subscript c here 00:00:39.760 --> 00:00:41.670 because we're dealing with concentrations 00:00:41.670 --> 00:00:43.930 in our equilibrium constant expression. 00:00:43.930 --> 00:00:47.330 And the equilibrium constant Kc is equal to, 00:00:47.330 --> 00:00:51.180 and in the numerator, we have the concentrations 00:00:51.180 --> 00:00:53.420 of our two products multiplied together. 00:00:53.420 --> 00:00:55.200 And the concentration of each product 00:00:55.200 --> 00:00:58.810 is raised to the power of the coefficient. 00:00:58.810 --> 00:00:59.940 In the denominator, 00:00:59.940 --> 00:01:03.260 we have the concentrations of the two reactants 00:01:03.260 --> 00:01:06.720 multiplied by each other and raised to the power, 00:01:06.720 --> 00:01:08.400 each concentration is raised to the power 00:01:08.400 --> 00:01:11.220 of the coefficient in the balanced equation. 00:01:11.220 --> 00:01:13.351 It's important to emphasize that the concentrations 00:01:13.351 --> 00:01:14.184 that we're plugging 00:01:14.184 --> 00:01:16.730 into our equilibrium constant expression 00:01:16.730 --> 00:01:19.550 are equilibrium concentrations. 00:01:19.550 --> 00:01:22.020 And when we plug in our equilibrium concentrations 00:01:22.020 --> 00:01:24.430 into our equilibrium constant expression, 00:01:24.430 --> 00:01:27.800 we get a value for the equilibrium constant K. 00:01:27.800 --> 00:01:30.960 And K is constant for a particular reaction 00:01:30.960 --> 00:01:32.940 at a certain temperature. 00:01:32.940 --> 00:01:35.360 Let's write an equilibrium constant expression 00:01:35.360 --> 00:01:36.890 for the following reaction, 00:01:36.890 --> 00:01:39.450 which shows the synthesis of ammonia 00:01:39.450 --> 00:01:41.910 from nitrogen and hydrogen, 00:01:41.910 --> 00:01:43.940 and everything is in the gaseous state. 00:01:43.940 --> 00:01:46.910 We start by writing the equilibrium constant Kc, 00:01:46.910 --> 00:01:49.180 c because we're dealing with concentrations, 00:01:49.180 --> 00:01:51.700 and we start with our product, which is ammonia. 00:01:51.700 --> 00:01:55.340 So we write the concentration of ammonia 00:01:55.340 --> 00:01:58.490 and we raise the concentration of ammonia 00:01:58.490 --> 00:02:00.370 to the power of the coefficient 00:02:00.370 --> 00:02:02.400 in the balanced equation, which is a two. 00:02:02.400 --> 00:02:06.600 So this is the concentration of ammonia to the second power. 00:02:06.600 --> 00:02:10.150 Then, in the denominator, we think about our reactants. 00:02:10.150 --> 00:02:11.730 So we have nitrogen. 00:02:11.730 --> 00:02:14.620 So we write the concentration of nitrogen. 00:02:14.620 --> 00:02:17.620 And since the coefficient is a one in the balanced equation, 00:02:17.620 --> 00:02:20.700 that'd be the concentration of nitrogen to the first power 00:02:20.700 --> 00:02:23.540 multiplied by the concentration of our other reactants, 00:02:23.540 --> 00:02:24.880 which is hydrogens. 00:02:24.880 --> 00:02:26.650 We write in here H2. 00:02:26.650 --> 00:02:28.500 And because there's a coefficient of three 00:02:28.500 --> 00:02:29.910 in the balanced equation, 00:02:29.910 --> 00:02:34.140 we raise the concentration of hydrogen to the third power. 00:02:34.140 --> 00:02:35.940 For gases, it's often more convenient 00:02:35.940 --> 00:02:37.820 to measure partial pressures 00:02:37.820 --> 00:02:40.340 instead of measuring concentrations. 00:02:40.340 --> 00:02:44.750 So let's say that A, B, C and D are all gases. 00:02:44.750 --> 00:02:47.780 We could write an equilibrium constant expression 00:02:47.780 --> 00:02:52.260 using partial pressures instead of concentrations. 00:02:52.260 --> 00:02:54.720 And if we did that, instead of writing Kc, 00:02:54.720 --> 00:02:58.290 we would write Kp where p stands for pressure. 00:02:58.290 --> 00:03:03.200 And Kc and Kp usually have different values from each other. 00:03:03.200 --> 00:03:05.560 So if we go back to our previous reaction 00:03:05.560 --> 00:03:07.580 where everything was in the gaseous state, 00:03:07.580 --> 00:03:10.020 we could write a Kp expression. 00:03:10.020 --> 00:03:13.130 So we would write Kp is equal to, 00:03:13.130 --> 00:03:15.540 we think about products over reactants. 00:03:15.540 --> 00:03:19.020 So this would be the partial pressure of our product, 00:03:19.020 --> 00:03:23.770 ammonia, raised to the second power 00:03:23.770 --> 00:03:27.600 divided by the partial pressure of nitrogen 00:03:27.600 --> 00:03:32.000 raised to the first power times the partial pressure 00:03:32.000 --> 00:03:37.000 of hydrogen raised to the third power. 00:03:37.620 --> 00:03:39.360 For the synthesis of ammonia, 00:03:39.360 --> 00:03:41.650 everything was in the gaseous state. 00:03:41.650 --> 00:03:44.660 And when all substances, reactants and products 00:03:44.660 --> 00:03:46.390 are in the same phase, 00:03:46.390 --> 00:03:50.460 we call this a homogeneous equilibrium. 00:03:50.460 --> 00:03:52.850 When the substances are in different phases, 00:03:52.850 --> 00:03:56.630 we call it a heterogeneous equilibrium. 00:03:56.630 --> 00:03:59.860 For example, in the decomposition of calcium carbonate 00:03:59.860 --> 00:04:03.060 to turn into calcium oxide and carbon dioxide, 00:04:03.060 --> 00:04:07.450 calcium carbonate is a solid and calcium oxide is a solid, 00:04:07.450 --> 00:04:09.210 but carbon dioxide is a gas. 00:04:09.210 --> 00:04:12.910 So we have substances in different phases. 00:04:12.910 --> 00:04:15.310 When we write an equilibrium constant expression 00:04:15.310 --> 00:04:18.090 for a heterogeneous equilibrium, 00:04:18.090 --> 00:04:20.980 we leave pure solids and pure liquids 00:04:20.980 --> 00:04:23.740 out of the equilibrium constant expression. 00:04:23.740 --> 00:04:26.160 So if we write an equilibrium constant expression 00:04:26.160 --> 00:04:28.850 for the decomposition of calcium carbonate, 00:04:28.850 --> 00:04:31.350 let's write a Kc expression first here. 00:04:31.350 --> 00:04:33.620 So we write Kc is equal to, 00:04:33.620 --> 00:04:36.850 and we think about products over reactants. 00:04:36.850 --> 00:04:40.030 For products, we have carbon dioxide in the gaseous state. 00:04:40.030 --> 00:04:41.670 So it's okay to include that 00:04:41.670 --> 00:04:43.710 in our equilibrium constant expression. 00:04:43.710 --> 00:04:46.880 So we write the concentration of CO2. 00:04:46.880 --> 00:04:49.560 And since the coefficient is a one in the balanced equation, 00:04:49.560 --> 00:04:51.400 this would be the concentration of CO2 00:04:51.400 --> 00:04:53.280 raised to the first power. 00:04:53.280 --> 00:04:55.090 Our other products is a solid. 00:04:55.090 --> 00:04:56.150 So we're gonna leave that 00:04:56.150 --> 00:04:59.450 out of our equilibrium constant expression. 00:04:59.450 --> 00:05:03.510 And for our reactant calcium carbonate, that's also a solid, 00:05:03.510 --> 00:05:06.600 so that's also left out of our expression. 00:05:06.600 --> 00:05:09.670 If we were to write a Kp expression here, 00:05:09.670 --> 00:05:12.640 we would include the partial pressure of our gas, 00:05:12.640 --> 00:05:14.180 which is carbon dioxide. 00:05:14.180 --> 00:05:16.470 So this would be the partial pressure of carbon dioxide 00:05:16.470 --> 00:05:17.720 to the first power. 00:05:17.720 --> 00:05:19.920 And once again, we would leave the two solids 00:05:19.920 --> 00:05:23.600 out of our equilibrium constant expression. 00:05:23.600 --> 00:05:25.200 The reason why we leave pure solids 00:05:25.200 --> 00:05:28.150 and pure liquids out of equilibrium constant expressions 00:05:28.150 --> 00:05:31.690 for heterogeneous equilibria is because the concentration 00:05:31.690 --> 00:05:35.930 of a pure solid or a pure liquid remains constant over time. 00:05:35.930 --> 00:05:38.120 So it doesn't help us to include it 00:05:38.120 --> 00:05:40.530 in our equilibrium expression. 00:05:40.530 --> 00:05:43.420 Finally, let's talk about the reaction quotient, 00:05:43.420 --> 00:05:46.680 which is symbolized by the letter Q. 00:05:46.680 --> 00:05:48.560 A Q expression has the same form 00:05:48.560 --> 00:05:50.800 as an equilibrium constant expression. 00:05:50.800 --> 00:05:54.120 And Q tells us the relative concentrations of reactants 00:05:54.120 --> 00:05:57.950 and products at any moment in time. 00:05:57.950 --> 00:06:02.290 And just like we could write a Kc or a Kp expression, 00:06:02.290 --> 00:06:05.960 we could write a Qc or a Qp expression. 00:06:05.960 --> 00:06:09.940 Let's go back to our reaction for the synthesis of ammonia 00:06:09.940 --> 00:06:12.880 from nitrogen gas and hydrogen gas. 00:06:12.880 --> 00:06:16.410 Notice how the Qc expression has the same form 00:06:16.410 --> 00:06:18.660 as the Kc expression. 00:06:18.660 --> 00:06:21.560 The difference is, for the Kc expression, 00:06:21.560 --> 00:06:25.430 all of our concentrations are equilibrium concentrations. 00:06:25.430 --> 00:06:27.500 So I could put an eq here 00:06:27.500 --> 00:06:32.233 for the concentrations of ammonia, nitrogen, and hydrogen. 00:06:33.170 --> 00:06:34.780 So for the Kc expression, 00:06:34.780 --> 00:06:37.240 it's only equilibrium concentrations, 00:06:37.240 --> 00:06:39.430 but for the Qc expression, 00:06:39.430 --> 00:06:42.850 it's the concentrations at any moment in time. 00:06:42.850 --> 00:06:46.150 So that moment in time might be at equilibrium 00:06:46.150 --> 00:06:49.630 or it might not be at equilibrium. 00:06:49.630 --> 00:06:54.450 If Qc is equal to KC, the reaction is at equilibrium, 00:06:54.450 --> 00:06:59.450 but if Qc is greater than Kc, or if Qc is less than Kc, 00:06:59.600 --> 00:07:02.233 the reaction is not at equilibrium.

Direction of reversible reactions

https://www.youtube.com/watch?v=9u914ckHlUI

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en

WEBVTT Kind: captions Language: en 00:00:00.190 --> 00:00:02.660 - [Instructor] As an example of a reversible reaction, 00:00:02.660 --> 00:00:04.900 let's look at the hypothetical reaction 00:00:04.900 --> 00:00:09.900 where diatomic gas X2 turns into its individual atoms X, 00:00:10.050 --> 00:00:11.340 and it would turn into two of them. 00:00:11.340 --> 00:00:13.650 So X2 goes to 2X. 00:00:13.650 --> 00:00:18.160 The forward reaction is X2 turning into 2X. 00:00:18.160 --> 00:00:23.160 And the reverse reaction is 2X combining to form X2. 00:00:23.210 --> 00:00:26.930 And let's say the X2 is a reddish brown gas. 00:00:26.930 --> 00:00:30.850 If we assume that both the forward and the reverse reactions 00:00:30.850 --> 00:00:32.690 are elementary reactions, 00:00:32.690 --> 00:00:34.570 we can actually write the rate law 00:00:34.570 --> 00:00:36.350 from the balanced equation. 00:00:36.350 --> 00:00:38.330 So for the forward reaction, 00:00:38.330 --> 00:00:41.680 let's go ahead and write the rate of the forward reaction 00:00:41.680 --> 00:00:45.320 is equal to the rate constant for the forward reaction, 00:00:45.320 --> 00:00:49.340 which we will symbolize as K with the subscript F, 00:00:49.340 --> 00:00:51.560 times the concentration of, 00:00:51.560 --> 00:00:53.880 if we're going in the forward direction, 00:00:53.880 --> 00:00:56.190 the reactants would be X2. 00:00:56.190 --> 00:00:59.490 So times the concentration of X2. 00:00:59.490 --> 00:01:02.930 And since we have a coefficient of 1 in front of X2, 00:01:02.930 --> 00:01:04.313 for this elementary reaction, 00:01:04.313 --> 00:01:07.083 this would be raised to the first power. 00:01:07.940 --> 00:01:11.090 Next we can write the rate law for the reverse reaction. 00:01:11.090 --> 00:01:13.320 So the rate of the reverse reaction 00:01:13.320 --> 00:01:15.040 is equal to the rate constant, 00:01:15.040 --> 00:01:17.300 and we'll put in a subscript R here. 00:01:17.300 --> 00:01:19.840 So that's the rate constant for the reverse reaction. 00:01:19.840 --> 00:01:24.710 And in the reverse reaction, 2X combines to form X2. 00:01:24.710 --> 00:01:28.190 So this would be times the concentration of X. 00:01:28.190 --> 00:01:30.870 And since we have a two as our coefficient, 00:01:30.870 --> 00:01:34.743 we need to raise the concentration of X to the second power. 00:01:35.980 --> 00:01:38.130 Next, let's look at these particulate diagrams 00:01:38.130 --> 00:01:41.520 and think about what happens for the forward reaction. 00:01:41.520 --> 00:01:43.210 So we start at time is equal to zero, 00:01:43.210 --> 00:01:45.060 and we start with only X2. 00:01:45.060 --> 00:01:47.680 So here are five particles of X2. 00:01:47.680 --> 00:01:49.130 If we wait 10 seconds, 00:01:49.130 --> 00:01:51.260 now we've gone from five particles of X2 00:01:51.260 --> 00:01:54.120 to only three particles of X2. 00:01:54.120 --> 00:01:57.010 So overall, two of those particles of X2 00:01:57.010 --> 00:01:58.720 have turned into X. 00:01:58.720 --> 00:02:01.180 And so there are four particles of X 00:02:01.180 --> 00:02:03.480 in this second particular diagram. 00:02:03.480 --> 00:02:04.990 We wait another 10 seconds 00:02:04.990 --> 00:02:07.120 for a total of time is equal to 20 seconds. 00:02:07.120 --> 00:02:09.980 And we've gone from three particles of X2 00:02:09.980 --> 00:02:13.240 to only two particles of X2, 00:02:13.240 --> 00:02:16.050 and we've increased in the particles of X. 00:02:16.050 --> 00:02:19.650 So now we're up to six particles of X. 00:02:19.650 --> 00:02:22.650 So the concentration of X2 has decreased. 00:02:22.650 --> 00:02:27.650 We went from five particles of X2 to three particles of X2 00:02:28.220 --> 00:02:30.530 to only two particles of X2. 00:02:30.530 --> 00:02:34.240 And if we look at the rate law for the forward reaction, 00:02:34.240 --> 00:02:37.500 the rate of the forward reaction is proportional 00:02:37.500 --> 00:02:40.470 to the concentration of X2. 00:02:40.470 --> 00:02:43.760 So if the concentration of X2 decreases, 00:02:43.760 --> 00:02:48.450 the rate of the forward reaction also decreases. 00:02:48.450 --> 00:02:49.810 We can see the same concept 00:02:49.810 --> 00:02:53.560 if we look at a graph of rate versus time. 00:02:53.560 --> 00:02:56.100 So if we look at this line right here, 00:02:56.100 --> 00:02:59.250 we're starting on a certain rate for the forward reaction. 00:02:59.250 --> 00:03:01.730 And as the concentration of X2 decreases, 00:03:01.730 --> 00:03:04.760 we can see the rate of the reaction decrease. 00:03:04.760 --> 00:03:07.130 So the rate of the reaction stops decreasing 00:03:07.130 --> 00:03:10.430 when we get to time is equal to 20 seconds. 00:03:10.430 --> 00:03:13.960 Next, let's think about the rate of the reverse reaction. 00:03:13.960 --> 00:03:15.850 Well, when time is equal to zero, 00:03:15.850 --> 00:03:18.450 the rate of the reverse reaction is zero. 00:03:18.450 --> 00:03:21.410 And that's because when we start out, we have only X2, 00:03:21.410 --> 00:03:23.330 we don't have any X present. 00:03:23.330 --> 00:03:25.760 So the reverse reaction doesn't happen. 00:03:25.760 --> 00:03:28.430 But as soon as some of that X2 turns into X, 00:03:28.430 --> 00:03:31.400 it's possible for the reverse reaction to happen. 00:03:31.400 --> 00:03:34.310 And as we increase in the amount of X, 00:03:34.310 --> 00:03:37.444 and we look at our rate law here for the reverse reaction, 00:03:37.444 --> 00:03:40.590 as we increase in the concentration of X, 00:03:40.590 --> 00:03:44.530 the rate of the reverse reaction should increase as well. 00:03:44.530 --> 00:03:46.120 And so that's why we see, 00:03:46.120 --> 00:03:49.550 that's why we see the rate of the reverse reaction increase 00:03:49.550 --> 00:03:52.220 as time increases. 00:03:52.220 --> 00:03:54.760 So as the forward reaction is happening, 00:03:54.760 --> 00:03:59.490 the reverse reaction is also occurring at the same time. 00:03:59.490 --> 00:04:00.820 However, we don't really see that 00:04:00.820 --> 00:04:03.120 when we look at our particular diagram. 00:04:03.120 --> 00:04:04.410 In our particular diagrams, 00:04:04.410 --> 00:04:07.520 we see a net conversion of X2 into 2X, 00:04:07.520 --> 00:04:10.030 for example, looking from the first particular diagram 00:04:10.030 --> 00:04:11.410 to the second, 00:04:11.410 --> 00:04:13.950 we see that two particles of X2 00:04:13.950 --> 00:04:17.130 have turned into four particles of X. 00:04:17.130 --> 00:04:20.600 And going from the second diagram to the third diagram, 00:04:20.600 --> 00:04:23.170 we see that another particle of X2 00:04:23.170 --> 00:04:24.700 has turned into 2X. 00:04:24.700 --> 00:04:27.860 And therefore we have six particles of X 00:04:27.860 --> 00:04:30.780 at time is equal to 20 seconds. 00:04:30.780 --> 00:04:34.050 So since we see a net conversion of reactants to products 00:04:34.050 --> 00:04:35.770 in our particular diagram, 00:04:35.770 --> 00:04:38.840 the rate of the forward reaction must be greater 00:04:38.840 --> 00:04:41.250 than the rate of the reverse reaction. 00:04:41.250 --> 00:04:42.490 And we can see that. 00:04:42.490 --> 00:04:46.460 So before time is equal to 20 seconds here, 00:04:46.460 --> 00:04:48.720 if we look at our rates, let's just pick, 00:04:48.720 --> 00:04:51.391 for example, time is equal to 10 seconds, 00:04:51.391 --> 00:04:53.140 for the forward reaction, 00:04:53.140 --> 00:04:57.473 there's a higher rate than for the reverse reaction. 00:04:58.683 --> 00:05:00.428 So at times equal to 20 seconds, 00:05:00.428 --> 00:05:03.030 the rate of the forward reaction 00:05:03.030 --> 00:05:06.480 becomes equal to the rate of the reverse reaction. 00:05:06.480 --> 00:05:09.070 So here's the line on our graph, 00:05:09.070 --> 00:05:11.010 where the rates become equal, 00:05:11.010 --> 00:05:14.480 and also notice the rates become constant at this point. 00:05:14.480 --> 00:05:16.520 And when the rate of the forward reaction 00:05:16.520 --> 00:05:19.070 is equal to the rate of the reverse reaction, 00:05:19.070 --> 00:05:21.220 the reaction has reached equilibrium. 00:05:21.220 --> 00:05:23.330 So to the right of the dotted line, 00:05:23.330 --> 00:05:25.830 the reaction is at equilibrium. 00:05:25.830 --> 00:05:27.780 And to the left of the dotted line, 00:05:27.780 --> 00:05:31.573 the reaction is not at equilibrium. 00:05:34.210 --> 00:05:36.550 And since the rate of the forward reaction 00:05:36.550 --> 00:05:39.983 is equal to the rate of the reverse reaction at equilibrium, 00:05:39.983 --> 00:05:41.900 X2 is turning into 2X 00:05:41.900 --> 00:05:45.180 at the same rate that 2X is turning back into X2. 00:05:45.180 --> 00:05:47.690 Therefore, the concentrations of X2 00:05:47.690 --> 00:05:50.480 and X at equilibrium remain constant. 00:05:50.480 --> 00:05:53.260 And we can see that when we look at the particular diagrams 00:05:53.260 --> 00:05:54.710 where time is equal to 20 seconds, 00:05:54.710 --> 00:05:56.780 and time is equal to 30 seconds. 00:05:56.780 --> 00:06:01.780 So both of these particular diagrams have two X2 particles 00:06:01.960 --> 00:06:05.733 and six X particles. 00:06:06.660 --> 00:06:09.530 Let's look at a summary of what the rates of the forward 00:06:09.530 --> 00:06:11.110 and reverse reaction mean 00:06:11.110 --> 00:06:13.590 in terms of reactants and products. 00:06:13.590 --> 00:06:16.280 If the rate of the forward reaction 00:06:16.280 --> 00:06:19.450 is greater than the rate of the reverse reaction, 00:06:19.450 --> 00:06:21.290 that means there's a net conversion 00:06:21.290 --> 00:06:23.170 of reactants to products. 00:06:23.170 --> 00:06:25.070 So therefore over time, 00:06:25.070 --> 00:06:26.970 the amount of reactants would decrease 00:06:26.970 --> 00:06:29.193 and the amount of products would increase. 00:06:30.040 --> 00:06:32.880 Eventually, the rate of the forward reaction 00:06:32.880 --> 00:06:36.200 becomes equal to the rate of the reverse reaction. 00:06:36.200 --> 00:06:38.860 And that means the reaction is at equilibrium 00:06:38.860 --> 00:06:40.430 and there's no net change 00:06:40.430 --> 00:06:42.820 in the amounts of reactants or products. 00:06:42.820 --> 00:06:45.640 So reactants are turning into products at the same rate 00:06:45.640 --> 00:06:48.850 the products are turning back into reactants. 00:06:48.850 --> 00:06:51.610 And then finally, if the rate of the reverse reaction 00:06:51.610 --> 00:06:54.710 is greater than the rate of the forward reaction, 00:06:54.710 --> 00:06:57.430 there's a net conversion of products to reactants. 00:06:57.430 --> 00:07:00.890 So products are turning into reactants 00:07:00.890 --> 00:07:04.150 faster than reactants are turning into products. 00:07:04.150 --> 00:07:06.240 And the example that we looked at, 00:07:06.240 --> 00:07:08.270 the rate of the forward reaction 00:07:08.270 --> 00:07:10.910 was greater than the rate of the reverse reaction. 00:07:10.910 --> 00:07:12.770 And eventually the rates became equal 00:07:12.770 --> 00:07:15.100 and the reaction reached equilibrium. 00:07:15.100 --> 00:07:17.750 If we had looked at an example of this third case here 00:07:17.750 --> 00:07:19.920 where the rate of the reverse reaction is greater 00:07:19.920 --> 00:07:22.300 than the rate of the forward reaction, 00:07:22.300 --> 00:07:24.530 eventually the two rates would become equal 00:07:24.530 --> 00:07:27.393 and this reaction would reach equilibrium too.

Dynamic equilibrium

https://www.youtube.com/watch?v=zaHM74k9Z1w

vtt

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en

WEBVTT Kind: captions Language: en 00:00:00.650 --> 00:00:02.940 - [Instructor] To illustrate the concept of equilibrium, 00:00:02.940 --> 00:00:04.970 let's say that we have a beaker 00:00:04.970 --> 00:00:08.840 and we put some water into our beaker. 00:00:08.840 --> 00:00:13.840 And also we make sure that our beaker has a lid on it. 00:00:14.220 --> 00:00:17.210 Some of those water molecules are going to evaporate 00:00:17.210 --> 00:00:19.160 and turn into a gas. 00:00:19.160 --> 00:00:22.050 And eventually once we have enough gaseous water, 00:00:22.050 --> 00:00:24.650 some of the gaseous water is going to condense 00:00:24.650 --> 00:00:27.610 and turn back into liquid water. 00:00:27.610 --> 00:00:29.250 To represent these two processes, 00:00:29.250 --> 00:00:31.120 we can show liquid water on the left 00:00:31.120 --> 00:00:33.370 and gaseous water on the right. 00:00:33.370 --> 00:00:35.220 So in the forward process, 00:00:35.220 --> 00:00:37.630 liquid water turns into gaseous water. 00:00:37.630 --> 00:00:39.460 And this forward arrow here 00:00:39.460 --> 00:00:42.490 represents the process of vaporization. 00:00:42.490 --> 00:00:45.490 And when gaseous water turns back into liquid water, 00:00:45.490 --> 00:00:47.930 that's represented by this arrow here on the bottom, 00:00:47.930 --> 00:00:51.240 and so that's the process of condensation. 00:00:51.240 --> 00:00:53.430 Since we start with liquid water, 00:00:53.430 --> 00:00:55.960 at first, the rate of vaporization 00:00:55.960 --> 00:00:59.100 is greater than the rate of condensation. 00:00:59.100 --> 00:01:01.860 But eventually we reach a point where the rate of 00:01:01.860 --> 00:01:05.800 vaporization is equal to the rate of condensation. 00:01:05.800 --> 00:01:09.340 And when that happens, if you're turning liquid water 00:01:09.340 --> 00:01:11.470 into gaseous water at the same rate, 00:01:11.470 --> 00:01:14.890 you're turning gaseous water back into liquid water, 00:01:14.890 --> 00:01:18.330 the number of water molecules in a liquid and gaseous state 00:01:18.330 --> 00:01:20.420 would remain constant. 00:01:20.420 --> 00:01:23.570 So when the rate of vaporization is equal to the rate of 00:01:23.570 --> 00:01:27.700 condensation, we've reached a state of equilibrium. 00:01:27.700 --> 00:01:31.720 And this is a dynamic equilibrium because if we zoom in and 00:01:31.720 --> 00:01:35.170 we look at this, water molecules are being converted 00:01:35.170 --> 00:01:38.220 from the liquid state to the gaseous state all the time 00:01:38.220 --> 00:01:40.970 and molecules are going from the gaseous state back to the 00:01:40.970 --> 00:01:42.190 liquid state all the time. 00:01:42.190 --> 00:01:44.890 However, since the rates are equal, 00:01:44.890 --> 00:01:46.670 the number of molecules in the liquid 00:01:46.670 --> 00:01:48.660 and gaseous state remained constant. 00:01:48.660 --> 00:01:51.340 And if we look at it from a macroscopic point of view, 00:01:51.340 --> 00:01:54.640 the level of water wouldn't change at all. 00:01:54.640 --> 00:01:57.660 Now let's apply this concept of dynamic equilibrium 00:01:57.660 --> 00:02:00.440 to a hypothetical chemical reaction. 00:02:00.440 --> 00:02:04.140 In our hypothetical reaction X2 which is a reddish brown 00:02:04.140 --> 00:02:09.140 gas, decomposes into its individual atoms to form 2X 00:02:09.740 --> 00:02:12.610 and individual atoms are colorless. 00:02:12.610 --> 00:02:17.610 So in the forward reaction, we're going from X2 to 2X. 00:02:18.414 --> 00:02:20.960 So X2 is decomposing to 2X. 00:02:20.960 --> 00:02:25.140 And in the reverse reaction, the two atoms of X 00:02:25.140 --> 00:02:28.870 are combining together to form X2. 00:02:28.870 --> 00:02:32.040 When we have a forward reaction and a reverse reaction 00:02:32.040 --> 00:02:35.190 by convention, we say, what's on the left side, 00:02:35.190 --> 00:02:39.760 are the reactants, and what's on the right side, 00:02:39.760 --> 00:02:41.150 are the products. 00:02:41.150 --> 00:02:45.023 And by using these terms, we can avoid confusion. 00:02:46.000 --> 00:02:48.990 Let's say that we start our reaction with only reactants. 00:02:48.990 --> 00:02:53.040 So only X2 is present in this first container here. 00:02:53.040 --> 00:02:57.250 And there are five particles of X2. 00:02:57.250 --> 00:03:01.060 If every particle represents 0.1 moles, 00:03:01.060 --> 00:03:03.580 since we have five particles of X2, 00:03:03.580 --> 00:03:07.230 we have 0.5 moles of X2. 00:03:07.230 --> 00:03:10.325 And let's say, this is a one liter container. 00:03:10.325 --> 00:03:14.290 0.5 divided by one would be 0.5 molar. 00:03:14.290 --> 00:03:19.290 So the initial concentration of gas, X2 is 0.5 molar. 00:03:21.290 --> 00:03:24.250 And since we don't have any of the X, 00:03:24.250 --> 00:03:27.220 there are no white dots in this box, right? 00:03:27.220 --> 00:03:30.723 The initial concentration of X would be zero molar. 00:03:32.170 --> 00:03:34.748 So let me just write that in here, 0M. 00:03:34.748 --> 00:03:36.330 Next we wait 10 seconds. 00:03:36.330 --> 00:03:38.077 So we start off at time's equal to zero seconds, 00:03:38.077 --> 00:03:40.650 and now we're at time's equal to 10 seconds. 00:03:40.650 --> 00:03:41.860 And now we can see, there are 00:03:41.860 --> 00:03:46.400 three particles of X2 in our box. 00:03:46.400 --> 00:03:49.260 And so that would be 0.3M, 00:03:49.260 --> 00:03:51.940 so let's go ahead and write 0.3M in 00:03:51.940 --> 00:03:53.490 for our concentration. 00:03:53.490 --> 00:03:56.370 And now we have some particles of X. 00:03:56.370 --> 00:04:00.230 There are one, two, three, four particles. 00:04:00.230 --> 00:04:04.060 And once again, if each particle represents 0.1 moles 00:04:04.060 --> 00:04:08.133 that's 0.4 moles of X divided by one or 0.4M. 00:04:11.930 --> 00:04:13.310 We wait another 10 seconds, 00:04:13.310 --> 00:04:15.940 so when time is equal to 20 seconds, 00:04:15.940 --> 00:04:19.130 now there are two particles of X2 00:04:19.130 --> 00:04:23.820 and one, two, three, four, five, six particles of X. 00:04:23.820 --> 00:04:28.820 So now the concentrations are 0.2M for X2 and 0.6M for X. 00:04:33.640 --> 00:04:36.900 We wait another 10 seconds for a total of 30 seconds. 00:04:36.900 --> 00:04:40.330 And there are still two particles of X2 00:04:40.330 --> 00:04:43.170 and six particles of X. 00:04:43.170 --> 00:04:46.000 And so the concentrations after 30 seconds, 00:04:46.000 --> 00:04:49.160 the concentration of X2 is 0.2M and of X is 0.6M. 00:04:54.350 --> 00:04:57.728 Notice how the concentration of X2 went from 0.5M 00:04:57.728 --> 00:05:02.728 to 0.3M to 0.2, and then it was also 0.2 after 30 seconds. 00:05:03.430 --> 00:05:07.860 So it became constant when time is equal to 20 seconds. 00:05:07.860 --> 00:05:10.410 The concentration of X went from zero to 0.4 to 0.6 00:05:11.760 --> 00:05:15.560 and then it was also 0.6 after 30 seconds. 00:05:15.560 --> 00:05:18.650 So the concentrations became constant when time is equal to 00:05:18.650 --> 00:05:22.440 20 seconds, which means the reaction reached equilibrium 00:05:22.440 --> 00:05:24.160 after 20 seconds. 00:05:24.160 --> 00:05:26.770 So at time is equal to zero was not at equilibrium. 00:05:26.770 --> 00:05:29.900 When time is equal to 10 seconds, it was not at equilibrium. 00:05:29.900 --> 00:05:32.220 Only when time was equal to 20 seconds, 00:05:32.220 --> 00:05:33.700 did it reach equilibrium. 00:05:33.700 --> 00:05:37.470 And that equilibrium, the rate of the forward reaction 00:05:37.470 --> 00:05:40.530 is equal to the rate of the reverse reaction. 00:05:40.530 --> 00:05:44.440 And if that's true, then X2 is being turned into 2X 00:05:44.440 --> 00:05:49.080 at the same rate that 2X is being turned back into X2. 00:05:49.080 --> 00:05:53.490 And if those rates are equal, the concentrations of X2 00:05:53.490 --> 00:05:56.893 and X at equilibrium would remain constant. 00:05:57.800 --> 00:05:59.280 We can see the same concept, 00:05:59.280 --> 00:06:03.440 if you look at a graph of concentration versus time. 00:06:03.440 --> 00:06:06.840 Concentration of X2 starts at 0.5M when time is 00:06:06.840 --> 00:06:10.270 equal to zero seconds, and then drops to 0.3M 00:06:10.270 --> 00:06:11.400 after 10 seconds. 00:06:11.400 --> 00:06:14.220 And after 20 seconds, it's at 0.2M 00:06:14.220 --> 00:06:16.480 and then stays constant after that. 00:06:16.480 --> 00:06:19.650 For the concentration of X, we start out at 0M, 00:06:19.650 --> 00:06:22.560 we increase 2.4 after 10 seconds, 00:06:22.560 --> 00:06:25.640 then we're at 0.6 and then we are constant. 00:06:25.640 --> 00:06:27.660 So if you think about a line, 00:06:27.660 --> 00:06:31.280 if we just draw a dash line here at 20 seconds. 00:06:31.280 --> 00:06:34.014 That's the dividing line between on the left, 00:06:34.014 --> 00:06:36.750 where we're not at equilibrium. 00:06:36.750 --> 00:06:40.000 And so the concentrations are always changing. 00:06:40.000 --> 00:06:42.290 And then to the right of that dotted line, 00:06:42.290 --> 00:06:44.260 we are at equilibrium where 00:06:44.260 --> 00:06:47.000 the concentrations remain constant. 00:06:47.000 --> 00:06:51.873 So the equilibrium concentration of X2 gas is equal to 00:06:51.873 --> 00:06:56.220 0.2M and the equilibrium concentration 00:06:56.220 --> 00:07:00.470 of X gas is equal to 0.6M. 00:07:00.470 --> 00:07:02.850 Finally, let's use these particular diagrams 00:07:02.850 --> 00:07:04.450 to think about what we would see 00:07:04.450 --> 00:07:06.630 at a macroscopic level as the 00:07:06.630 --> 00:07:08.980 reaction proceeds to equilibrium. 00:07:08.980 --> 00:07:10.520 And the first particulate diagram 00:07:10.520 --> 00:07:12.752 we see only red particles. 00:07:12.752 --> 00:07:17.752 So only our reactants X2 are present in the beginning. 00:07:18.010 --> 00:07:20.097 However, as time goes on, the number of red particles 00:07:20.097 --> 00:07:23.620 decreases from five in the first particular diagram 00:07:23.620 --> 00:07:26.650 to three in the second to two 00:07:26.650 --> 00:07:30.140 and then the number stays at two because remember we reach 00:07:30.140 --> 00:07:33.500 equilibrium after 20 seconds. 00:07:33.500 --> 00:07:36.360 So what we would see from a macroscopic point of view, 00:07:36.360 --> 00:07:40.223 is we'd start out with a reaction vessel that is a darker 00:07:40.223 --> 00:07:44.340 brown red, and then it would be a lighter brown red. 00:07:44.340 --> 00:07:46.670 And then finally, when we reach equilibrium 00:07:46.670 --> 00:07:48.180 and even lighter brown red, 00:07:48.180 --> 00:07:51.680 and it would stay that same light brown red, 00:07:51.680 --> 00:07:54.207 because we've reached equilibrium and the concentrations 00:07:54.207 --> 00:07:58.400 of reactants and products remain constant at equilibrium. 00:07:58.400 --> 00:08:02.520 Even though our reactants are turning into our products. 00:08:02.520 --> 00:08:05.360 Our products are turning back into our reactants at the same 00:08:05.360 --> 00:08:07.440 rate and therefore the concentrations 00:08:07.440 --> 00:08:10.143 of both reactants and products are constant.

Biodiversity

https://www.youtube.com/watch?v=TZ-S9sc6HYM

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en

WEBVTT Kind: captions Language: en 00:00:00.170 --> 00:00:01.280 - [Instructor] Today, we're going to talk 00:00:01.280 --> 00:00:02.990 about biodiversity. 00:00:02.990 --> 00:00:05.270 So biodiversity as you might've guessed 00:00:05.270 --> 00:00:09.310 comes from two words, biological and diversity. 00:00:09.310 --> 00:00:11.460 And essentially, it's the variations 00:00:11.460 --> 00:00:15.060 or the diversity present between living things. 00:00:15.060 --> 00:00:18.020 Now, I grew up in the sunny state of Arizona. 00:00:18.020 --> 00:00:19.170 And at first glance, 00:00:19.170 --> 00:00:21.060 there doesn't seem to be much variety 00:00:21.060 --> 00:00:22.720 in such a hot and dry place. 00:00:22.720 --> 00:00:24.970 I mean, it's the desert, right? 00:00:24.970 --> 00:00:26.210 But if we take a closer look 00:00:26.210 --> 00:00:28.530 at a place such as the Sonoran Desert, 00:00:28.530 --> 00:00:30.090 and here's a picture, 00:00:30.090 --> 00:00:32.790 we would find all kinds of different living things 00:00:32.790 --> 00:00:37.750 like saguaro cacti, jackrabbits, tarantulas, 00:00:37.750 --> 00:00:39.710 silver-haired bats and the roadrunner, 00:00:39.710 --> 00:00:41.830 which is my personal favorite. 00:00:41.830 --> 00:00:45.910 So we can say that the desert is actually quite biodiverse. 00:00:45.910 --> 00:00:47.340 So in other words, 00:00:47.340 --> 00:00:52.150 it's home to a large variety of living things. 00:00:52.150 --> 00:00:54.940 Since there are both big and small differences 00:00:54.940 --> 00:00:56.580 between different living things, 00:00:56.580 --> 00:01:00.350 we can think of biodiversity on three different levels. 00:01:00.350 --> 00:01:02.530 So the first and smallest scale level 00:01:02.530 --> 00:01:04.750 is going to be genetic biodiversity, 00:01:04.750 --> 00:01:07.310 which is just the genetic variation 00:01:07.310 --> 00:01:09.170 within a group of organisms. 00:01:09.170 --> 00:01:12.700 And a really great example of this kind of biodiversity 00:01:12.700 --> 00:01:14.650 is the rock pocket mouse. 00:01:14.650 --> 00:01:16.450 And I know, that's a mouthful of words to say. 00:01:16.450 --> 00:01:17.860 But it's really cute. 00:01:17.860 --> 00:01:19.540 I mean, look at this picture. 00:01:19.540 --> 00:01:22.230 So this is a species that can actually be found right here 00:01:22.230 --> 00:01:23.270 in the Sonoran Desert. 00:01:23.270 --> 00:01:26.070 And the cool thing about this mouse species 00:01:26.070 --> 00:01:28.180 is that there exists both tan 00:01:28.180 --> 00:01:30.380 and black-colored rock pocket mice. 00:01:30.380 --> 00:01:32.240 So even though both colored rodents 00:01:32.240 --> 00:01:34.100 come from the same species, 00:01:34.100 --> 00:01:35.580 there are genetic biodiversity 00:01:35.580 --> 00:01:38.590 creates mice with completely different traits. 00:01:38.590 --> 00:01:39.910 And what's even more awesome 00:01:39.910 --> 00:01:42.530 is the fact that these colors, tan and black, 00:01:42.530 --> 00:01:45.660 have come about due to natural selection. 00:01:45.660 --> 00:01:46.840 So scientists have found 00:01:46.840 --> 00:01:49.450 that there are more black-colored rock pocket mice 00:01:49.450 --> 00:01:53.280 in places with dark black-ish lava rock. 00:01:53.280 --> 00:01:55.680 While there are more tan-colored rock pocket mice 00:01:55.680 --> 00:01:57.570 in lighter colored sands. 00:01:57.570 --> 00:02:00.150 And this makes sense because if you think about it, 00:02:00.150 --> 00:02:04.090 the mice are hunted from above by predators like birds. 00:02:04.090 --> 00:02:07.860 So having a coat color that blends into their surroundings 00:02:07.860 --> 00:02:10.810 would increase their chances of survival. 00:02:10.810 --> 00:02:14.650 So having this kind of genetic biodiversity is really great, 00:02:14.650 --> 00:02:16.060 because the rock pocket mice 00:02:16.060 --> 00:02:19.570 can use many different colored landscapes as habitats. 00:02:19.570 --> 00:02:23.110 And if let's say the landscape color were to change 00:02:23.110 --> 00:02:26.710 and suddenly become dominated by dark lava rock, 00:02:26.710 --> 00:02:28.700 then this population of mice 00:02:28.700 --> 00:02:30.450 would have the genetic diversity 00:02:30.450 --> 00:02:33.550 that it needs to adapt to this change. 00:02:33.550 --> 00:02:36.550 So the second level is species biodiversity, 00:02:36.550 --> 00:02:41.140 which is the variety of species in a particular area. 00:02:41.140 --> 00:02:43.160 So going back to the Sonoran Desert, 00:02:43.160 --> 00:02:46.560 there are all kinds of mammals and birds, 00:02:46.560 --> 00:02:48.060 plants, and insects. 00:02:48.060 --> 00:02:50.960 But what I think is even more mind boggling 00:02:50.960 --> 00:02:53.670 is the sheer number of bat species there are. 00:02:53.670 --> 00:02:57.040 I mean, we have silver-haired bats, as I mentioned earlier. 00:02:57.040 --> 00:03:01.470 But also bats like spotted bats, Western red bats, 00:03:01.470 --> 00:03:05.180 and even this kind of bat called Peter's ghost-faced bat, 00:03:05.180 --> 00:03:07.880 which I think is a really awesome name. 00:03:07.880 --> 00:03:09.540 And you're probably thinking, okay, Abby, 00:03:09.540 --> 00:03:10.860 this is really interesting. 00:03:10.860 --> 00:03:12.440 But so what? 00:03:12.440 --> 00:03:14.320 Well, like genetic biodiversity, 00:03:14.320 --> 00:03:17.780 species biodiversity is super duper important. 00:03:17.780 --> 00:03:20.520 Because having a lot of different species 00:03:20.520 --> 00:03:23.730 means that more roles can be filled in an ecosystem, 00:03:23.730 --> 00:03:26.170 which makes the ecosystem healthier. 00:03:26.170 --> 00:03:27.760 So moving on to the final level, 00:03:27.760 --> 00:03:30.000 we have ecosystem biodiversity. 00:03:30.000 --> 00:03:34.150 And this is just the variety of ecosystems on the planet. 00:03:34.150 --> 00:03:36.620 Now we've been talking a lot about the Sonoran Desert. 00:03:36.620 --> 00:03:39.910 And a desert is actually a type of ecosystem on earth. 00:03:39.910 --> 00:03:40.830 And if you recall, 00:03:40.830 --> 00:03:43.960 ecosystems are made up of both living things 00:03:43.960 --> 00:03:45.760 and their physical environment. 00:03:45.760 --> 00:03:48.990 So in this case, living things in the desert ecosystem 00:03:48.990 --> 00:03:52.280 could include things like rattlesnakes, scorpions, 00:03:52.280 --> 00:03:53.690 and cacti. 00:03:53.690 --> 00:03:57.470 Well, non-living things could include rock formations 00:03:57.470 --> 00:03:58.890 or a sand dunes. 00:03:58.890 --> 00:04:01.240 And in addition to desert ecosystems, 00:04:01.240 --> 00:04:05.250 earth has forest ecosystems, coral reef ecosystems, 00:04:05.250 --> 00:04:07.790 all kinds of different ecosystems. 00:04:07.790 --> 00:04:10.310 And having all these different kinds of ecosystems 00:04:10.310 --> 00:04:13.490 is extremely critical for our survival. 00:04:13.490 --> 00:04:16.770 Because the diversity of ecosystems on earth 00:04:16.770 --> 00:04:21.180 provide us humans with important resources and services. 00:04:21.180 --> 00:04:23.630 So without ecosystem biodiversity, 00:04:23.630 --> 00:04:24.730 our quality of life, 00:04:24.730 --> 00:04:28.860 and in fact, our very survival could be at risk. 00:04:28.860 --> 00:04:32.120 So as you can tell, every level of biodiversity 00:04:32.120 --> 00:04:35.323 is incredibly important for unique reasons. 00:04:36.160 --> 00:04:38.640 Now, biodiversity doesn't remain constant. 00:04:38.640 --> 00:04:40.490 So we can think of biodiversity 00:04:40.490 --> 00:04:43.010 as something that can change over time. 00:04:43.010 --> 00:04:44.690 Now, the important thing to remember 00:04:44.690 --> 00:04:47.970 is that speciation increases biodiversity 00:04:47.970 --> 00:04:51.260 while extinction decreases biodiversity. 00:04:51.260 --> 00:04:53.630 So let's talk about speciation first. 00:04:53.630 --> 00:04:57.640 So speciation happens when one species evolves into two 00:04:57.640 --> 00:05:00.210 or more species over time. 00:05:00.210 --> 00:05:01.170 And you might have heard 00:05:01.170 --> 00:05:03.420 or seen something that looks like this. 00:05:03.420 --> 00:05:05.870 And I'll draw it out right here to the right. 00:05:05.870 --> 00:05:09.050 And this is just a very simple speciation model 00:05:09.050 --> 00:05:11.170 or a phylogenetic tree. 00:05:11.170 --> 00:05:13.350 So we have here a common ancestor 00:05:13.350 --> 00:05:17.130 that branches off into different species over time. 00:05:17.130 --> 00:05:20.620 This arrow representing time or T for short. 00:05:20.620 --> 00:05:22.910 And it's speciation events like these 00:05:22.910 --> 00:05:25.170 that have led to the biodiversity 00:05:25.170 --> 00:05:27.590 that we see on earth today. 00:05:27.590 --> 00:05:29.690 On the flip side, we have extinction 00:05:29.690 --> 00:05:32.240 which causes biodiversity to decrease. 00:05:32.240 --> 00:05:36.490 And what we mean by this word extinction is that a species 00:05:36.490 --> 00:05:40.100 or population of living things dies off completely. 00:05:40.100 --> 00:05:42.750 So notice that it can happen on different levels too. 00:05:42.750 --> 00:05:45.400 For example, if a population goes extinct, 00:05:45.400 --> 00:05:48.500 then we no longer have the genetic variants in the gene pool 00:05:48.500 --> 00:05:52.460 or the set of genes for a particular species. 00:05:52.460 --> 00:05:55.400 But if a species went extinct entirely, 00:05:55.400 --> 00:05:57.180 then that species is no longer there 00:05:57.180 --> 00:06:01.590 to fulfill its unique role or its ecological niche. 00:06:01.590 --> 00:06:03.230 So you might've heard of the dodo bird, 00:06:03.230 --> 00:06:05.510 which is a species that actually went extinct 00:06:05.510 --> 00:06:08.340 way back in the 1600s. 00:06:08.340 --> 00:06:09.850 So, some of the major causes 00:06:09.850 --> 00:06:11.500 behind the dodo birds extinction 00:06:11.500 --> 00:06:14.860 include over hunting, habitat loss 00:06:14.860 --> 00:06:18.240 and competition with some newly introduced species. 00:06:18.240 --> 00:06:20.480 And unfortunately, the last dodo bird 00:06:20.480 --> 00:06:24.260 was reportedly killed in 1681. 00:06:24.260 --> 00:06:25.910 And even though we can't enjoy 00:06:25.910 --> 00:06:28.010 the presence of dodo birds anymore, 00:06:28.010 --> 00:06:29.440 learning about this extinct bird 00:06:29.440 --> 00:06:32.150 actually brings up a really good question. 00:06:32.150 --> 00:06:34.690 That is, what effect do humans have 00:06:34.690 --> 00:06:36.760 on biodiversity right now? 00:06:36.760 --> 00:06:38.040 Well, it's sad to say, 00:06:38.040 --> 00:06:41.850 but humans are actually causing biodiversity to decrease. 00:06:41.850 --> 00:06:45.710 Things like climate change, habitat destruction, 00:06:45.710 --> 00:06:47.810 and overexploiting resources 00:06:47.810 --> 00:06:51.160 have caused a huge loss of biodiversity. 00:06:51.160 --> 00:06:53.190 And if you remember having biodiversity 00:06:53.190 --> 00:06:55.320 is really critical to our survival, 00:06:55.320 --> 00:06:57.720 which is a big, oh, oh. 00:06:57.720 --> 00:06:59.370 In fact, species extinction 00:06:59.370 --> 00:07:03.840 is now occurring at a rate of 100 to 1000 times faster 00:07:03.840 --> 00:07:07.040 than the background rate detected in fossil records. 00:07:07.040 --> 00:07:08.680 And because the extinction rate 00:07:08.680 --> 00:07:11.400 is much bigger than the speciation rate, 00:07:11.400 --> 00:07:16.400 the result is an overall loss or decrease in biodiversity. 00:07:16.680 --> 00:07:18.850 So today we learned about biodiversity, 00:07:18.850 --> 00:07:21.800 which is the variety of life present on earth. 00:07:21.800 --> 00:07:25.170 We talked about three different levels of biodiversity 00:07:25.170 --> 00:07:28.270 moving from genetic biodiversity to species 00:07:28.270 --> 00:07:30.660 and then ecosystem biodiversity. 00:07:30.660 --> 00:07:33.720 And we also talked about how biodiversity isn't stagnant 00:07:33.720 --> 00:07:35.040 or fixed in place. 00:07:35.040 --> 00:07:38.270 We have speciation events that can increase biodiversity 00:07:38.270 --> 00:07:41.770 and extinctions that decrease biodiversity. 00:07:41.770 --> 00:07:43.940 And finally, we learned that human actions 00:07:43.940 --> 00:07:45.800 are threatening biodiversity, 00:07:45.800 --> 00:07:48.640 as we currently have a greater extinction rate 00:07:48.640 --> 00:07:50.603 than speciation rate.

Formation of biomolecules

https://www.youtube.com/watch?v=9RUSMa-UIcQ

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https://www.youtube.com/api/timedtext?v=9RUSMa-UIcQ&ei=4VWUZfm7NYCrp-oPqtKyOA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245329&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=B6BDCED3842D06224A263AE4AE535DA2B163224E.B084F087499E5AF069E75D98319547792075B4F2&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.320 --> 00:00:04.740 - [Sal] So all organisms need food to survive. 00:00:04.740 --> 00:00:07.820 Now, for some of you, this might be pretty obvious. 00:00:07.820 --> 00:00:10.080 You realize what might happen to your body 00:00:10.080 --> 00:00:11.940 if you don't get food. 00:00:11.940 --> 00:00:16.380 You might realize that you need that food for both energy 00:00:17.240 --> 00:00:22.090 and you need that to actually build up your actual body. 00:00:22.090 --> 00:00:26.270 So you need it for matter as well. 00:00:26.270 --> 00:00:27.810 But some of you might be thinking, 00:00:27.810 --> 00:00:29.240 all right, I have a mouth. 00:00:29.240 --> 00:00:31.670 I understand where the food goes. 00:00:31.670 --> 00:00:33.270 I also understand what's left over 00:00:33.270 --> 00:00:34.530 when I'm done with the food. 00:00:34.530 --> 00:00:38.070 And so I must have extracted some energy and matter from it, 00:00:38.070 --> 00:00:40.430 but you just said all organisms, Sal, 00:00:40.430 --> 00:00:42.230 and I'm looking outside of a window 00:00:42.230 --> 00:00:45.500 and I see a tree and a tree does not have a mouth. 00:00:45.500 --> 00:00:46.750 A tree is an organism. 00:00:46.750 --> 00:00:48.700 How does it get its food? 00:00:48.700 --> 00:00:51.130 And the answer you might already realize 00:00:51.130 --> 00:00:51.963 is that the tree 00:00:51.963 --> 00:00:55.970 can make its own food through photosynthesis. 00:00:55.970 --> 00:00:58.360 We've seen this in other videos. 00:00:58.360 --> 00:01:03.360 You have carbon dioxide in the air and you have water. 00:01:04.030 --> 00:01:07.840 And the presence of energy in the form of sunlight, 00:01:07.840 --> 00:01:12.840 through the process of photosynthesis, would produce glucose 00:01:14.150 --> 00:01:17.270 and molecular oxygen as a byproduct. 00:01:17.270 --> 00:01:19.700 And you can count the various carbons 00:01:19.700 --> 00:01:21.610 that are in this dark gray color. 00:01:21.610 --> 00:01:24.350 Oxygen's in this red color and hydrogen's here. 00:01:24.350 --> 00:01:27.050 And you can see that everything all adds up. 00:01:27.050 --> 00:01:30.790 You have one, two, three, four, five six carbons here 00:01:30.790 --> 00:01:33.200 in the six carbon dioxide molecules. 00:01:33.200 --> 00:01:34.570 And then you could see in this glucose, 00:01:34.570 --> 00:01:38.330 you have one, two, three, four, five, six carbons. 00:01:38.330 --> 00:01:39.730 I encourage you to pause the video 00:01:39.730 --> 00:01:41.980 and make sure that the oxygens are all accounted for 00:01:41.980 --> 00:01:45.500 between the glucose molecule and this molecular oxygen, 00:01:45.500 --> 00:01:48.250 and that the hydrogens are all accounted for. 00:01:48.250 --> 00:01:51.330 And so a plant can produce its own food, 00:01:51.330 --> 00:01:52.400 and then when they need it, 00:01:52.400 --> 00:01:54.840 they can metabolize that food 00:01:54.840 --> 00:01:57.760 through a process of respiration. 00:01:57.760 --> 00:01:59.910 It's really important to realize 00:01:59.910 --> 00:02:03.170 that respiration does not just occur in your 00:02:03.170 --> 00:02:04.280 and my bodies. 00:02:04.280 --> 00:02:07.450 Even organisms like plants need to break down the food 00:02:07.450 --> 00:02:11.180 that they produced if they wanna use it for energy. 00:02:11.180 --> 00:02:12.580 Now, the question you might have is, 00:02:12.580 --> 00:02:14.090 where is the energy here? 00:02:14.090 --> 00:02:16.600 Just as it takes energy to rearrange these atoms 00:02:16.600 --> 00:02:20.290 and molecules into glucose, under the right conditions, 00:02:20.290 --> 00:02:22.320 through a metabolic pathway, 00:02:22.320 --> 00:02:24.050 you can go the other way around. 00:02:24.050 --> 00:02:27.810 And chemically, that will release energy, which even a plant 00:02:27.810 --> 00:02:30.530 which isn't running around, it isn't doing jumping jacks, 00:02:30.530 --> 00:02:31.710 it isn't talking. 00:02:31.710 --> 00:02:34.310 It needs energy just to be alive. 00:02:34.310 --> 00:02:37.980 All living things need energy in order to exist, 00:02:37.980 --> 00:02:41.420 in order to maintain their cells, in order to reproduce. 00:02:41.420 --> 00:02:43.901 So many of you all have probably heard the term 00:02:43.901 --> 00:02:46.970 carbohydrate when we're thinking about food 00:02:46.970 --> 00:02:49.650 or when we're thinking about an energy context. 00:02:49.650 --> 00:02:52.690 And it's important to note that this glucose molecule here 00:02:52.690 --> 00:02:55.080 is an example of a carbohydrate. 00:02:55.080 --> 00:02:57.210 It's not the only example of it, 00:02:57.210 --> 00:02:58.960 but one question might be, 00:02:58.960 --> 00:03:01.088 why is it even called a carbohydrate? 00:03:01.088 --> 00:03:03.430 Well, when you break down the different parts, 00:03:03.430 --> 00:03:05.860 it seems like it would involve carbon. 00:03:05.860 --> 00:03:07.720 Right over here, you have the carbo part. 00:03:07.720 --> 00:03:09.370 And it seems like it's somehow 00:03:09.370 --> 00:03:11.910 involving water: carbohydrate. 00:03:11.910 --> 00:03:13.690 And that's because early chemists, 00:03:13.690 --> 00:03:15.970 they didn't actually understand the structure 00:03:15.970 --> 00:03:18.320 of a carbohydrate the way that we do now. 00:03:18.320 --> 00:03:21.000 All they saw was the ratio between the carbons, 00:03:21.000 --> 00:03:22.800 the hydrogens and the oxygens. 00:03:22.800 --> 00:03:23.890 That for every carbon. 00:03:23.890 --> 00:03:25.820 So let's say there are N carbons, 00:03:25.820 --> 00:03:28.800 there's going to be twice as many hydrogens, 00:03:28.800 --> 00:03:31.870 and N oxygens as well. 00:03:31.870 --> 00:03:33.400 So one thing, one way to think about it, 00:03:33.400 --> 00:03:36.470 for every single carbon, you have an H2O. 00:03:36.470 --> 00:03:38.220 So every carbon, you have a water. 00:03:38.220 --> 00:03:40.050 So that's why early chemists, 00:03:40.050 --> 00:03:43.350 when they saw this ratio, they called it carbohydrates. 00:03:43.350 --> 00:03:45.420 Glucose is a very simple carbohydrate, 00:03:45.420 --> 00:03:48.350 but you can make up chains of things like glucose 00:03:48.350 --> 00:03:51.380 to build up more complex carbohydrates. 00:03:51.380 --> 00:03:52.920 Now it's important to realize 00:03:52.920 --> 00:03:56.600 that these types of molecules aren't just used for energy. 00:03:56.600 --> 00:03:59.480 They can also be used for matter. 00:03:59.480 --> 00:04:01.620 The more that you study biochemistry, 00:04:01.620 --> 00:04:03.780 you're going to see a lot of different molecules 00:04:03.780 --> 00:04:05.430 that are made up of these building blocks 00:04:05.430 --> 00:04:07.740 of carbons, oxygens, and hydrogens. 00:04:07.740 --> 00:04:10.130 And sometimes you might even recognize structures 00:04:10.130 --> 00:04:11.710 that look a little bit like glucose, 00:04:11.710 --> 00:04:13.900 or look like things that have been put together 00:04:13.900 --> 00:04:16.160 from some of these basic building blocks. 00:04:16.160 --> 00:04:19.320 For example, this molecule right over here 00:04:19.320 --> 00:04:22.450 is known as thymidine monophosphate, 00:04:22.450 --> 00:04:23.870 and it's a fancy name. 00:04:23.870 --> 00:04:25.850 But you can look at the building blocks here. 00:04:25.850 --> 00:04:28.940 The monophosphate, you have a phosphate group 00:04:28.940 --> 00:04:30.290 right over here. 00:04:30.290 --> 00:04:33.880 Thymidine comes from this part of the molecule. 00:04:33.880 --> 00:04:36.610 It's sometimes known as a nitrogenous base 00:04:36.610 --> 00:04:39.600 because it has nitrogen that you see in blue over here. 00:04:39.600 --> 00:04:42.430 And then right over here, connecting the pieces, 00:04:42.430 --> 00:04:46.730 you have a five carbon sugar: Ribose. 00:04:46.730 --> 00:04:48.670 Now glucose is a six carbon sugar, 00:04:48.670 --> 00:04:50.650 Ribose is a five carbon sugar, 00:04:50.650 --> 00:04:52.280 but there are metabolic pathways 00:04:52.280 --> 00:04:53.960 where you can go from five carbon sugars 00:04:53.960 --> 00:04:56.280 to six carbon sugars, and back and forth. 00:04:56.280 --> 00:04:57.210 And what's interesting 00:04:57.210 --> 00:04:59.894 about things like thymidine monophosphate 00:04:59.894 --> 00:05:02.760 is it is a building block for something 00:05:02.760 --> 00:05:06.420 that is very, very, very important: DNA. 00:05:06.420 --> 00:05:08.510 Thymidine monophosphate is a nucleotide 00:05:08.510 --> 00:05:10.390 with a nitrogenous base thymine. 00:05:10.390 --> 00:05:12.610 You put a bunch of these nucleotides together, 00:05:12.610 --> 00:05:16.130 not all of them have a nitrogenous base of thymine here, 00:05:16.130 --> 00:05:19.290 but they form this double helix structure that we study 00:05:19.290 --> 00:05:21.720 in a lot of depth in many, many other videos. 00:05:21.720 --> 00:05:24.270 And you might already realize that DNA 00:05:24.270 --> 00:05:26.360 is the molecular basis of inheritance. 00:05:26.360 --> 00:05:31.000 We could not be who we are without these types of molecules. 00:05:31.000 --> 00:05:32.890 Now, what's also interesting is how do these 00:05:32.890 --> 00:05:35.740 different constituent molecules rearrange themselves, 00:05:35.740 --> 00:05:39.130 even in the presence of energy, to make other molecules? 00:05:39.130 --> 00:05:42.259 Or to get energy, how do they re-rearrange themselves 00:05:42.259 --> 00:05:44.040 to release that energy? 00:05:44.040 --> 00:05:47.700 And all of these metabolic pathways are facilitated 00:05:47.700 --> 00:05:50.220 by what are known as enzymes. 00:05:50.220 --> 00:05:53.370 And just to give an example of an enzyme 00:05:53.370 --> 00:05:56.470 this big thing here is commonly known 00:05:56.470 --> 00:05:58.210 as the Rubisco enzyme. 00:05:58.210 --> 00:05:59.430 Now you don't have to know its name 00:05:59.430 --> 00:06:01.140 at this point in your careers, 00:06:01.140 --> 00:06:03.870 but this is one of the enzymes in the metabolic pathways 00:06:03.870 --> 00:06:06.650 that's able to take carbon dioxide 00:06:06.650 --> 00:06:08.530 and attach it to another molecule 00:06:08.530 --> 00:06:12.840 that eventually can get us to forming a glucose molecule. 00:06:12.840 --> 00:06:15.450 And what happens here is the various constituents 00:06:15.450 --> 00:06:18.170 attach to different parts of these enzymes. 00:06:18.170 --> 00:06:20.590 And these enzymes change their shape as they attach 00:06:20.590 --> 00:06:23.330 to certain things and they can jam things together. 00:06:23.330 --> 00:06:25.170 They can synthesize other molecules 00:06:25.170 --> 00:06:27.840 or they can even help to break them apart. 00:06:27.840 --> 00:06:29.510 And it all comes full circle 00:06:29.510 --> 00:06:32.690 because the enzymes themselves, these are proteins. 00:06:32.690 --> 00:06:34.830 These are made up of amino acids, 00:06:34.830 --> 00:06:36.728 which themselves are made up 00:06:36.728 --> 00:06:39.610 of a lot of these building block molecules 00:06:39.610 --> 00:06:41.981 that contain your carbons, your oxygens, 00:06:41.981 --> 00:06:44.300 and your hydrogens in them. 00:06:44.300 --> 00:06:46.050 I'll let you go now. 00:06:46.050 --> 00:06:47.440 But the important thing is to realize 00:06:47.440 --> 00:06:49.600 is that we have whole universes occurring 00:06:49.600 --> 00:06:51.800 in our cells, and that all of these molecules 00:06:51.800 --> 00:06:54.850 and biological systems are connected in different ways. 00:06:54.850 --> 00:06:56.150 And you have a whole series 00:06:56.150 --> 00:06:58.370 of metabolic pathways that are facilitated 00:06:58.370 --> 00:07:03.150 by enzymes that take one set of things and step-by-step, 00:07:03.150 --> 00:07:06.798 put them together or break them apart 00:07:06.798 --> 00:07:11.410 in order to do all of the different biological functions 00:07:11.410 --> 00:07:14.163 that we know are necessary for life.

Photosynthesis

https://www.youtube.com/watch?v=Yxm-WMYEpHg

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https://www.youtube.com/api/timedtext?v=Yxm-WMYEpHg&ei=4FWUZee7ItuhhcIPnJqfmAY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245328&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=7981BA537DF70D18857525DC3570BFC31E94ABA1.182B3B7796F5ED7B7A85850957A8255B974581ED&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.740 --> 00:00:02.530 - Hey everybody, Dr. Sammy here, 00:00:02.530 --> 00:00:04.460 your friendly neighborhood entomologist. 00:00:04.460 --> 00:00:08.173 And today, we're gonna talk about photosynthesis. 00:00:09.420 --> 00:00:10.990 There's very little life on this planet 00:00:10.990 --> 00:00:13.570 that could exist without photosynthesis. 00:00:13.570 --> 00:00:15.430 It is the prerequisite for pretty much 00:00:15.430 --> 00:00:17.190 everything you see around you. 00:00:17.190 --> 00:00:20.020 It's how you get from the intangible light of the sun 00:00:20.020 --> 00:00:22.860 to physical bodies like those of humans are 00:00:22.860 --> 00:00:25.180 hungry, hungry caterpillars. 00:00:25.180 --> 00:00:27.963 But what does photosynthesis actually mean? 00:00:28.860 --> 00:00:31.340 I hear people say all the time, that photosynthesis 00:00:31.340 --> 00:00:36.340 is the process by which plants make sugar from light. 00:00:38.260 --> 00:00:39.950 And it almost seems like magic. 00:00:39.950 --> 00:00:42.110 Light is not a substance. 00:00:42.110 --> 00:00:44.500 It is not made up of the molecular building blocks 00:00:44.500 --> 00:00:45.940 that compose all matter. 00:00:45.940 --> 00:00:47.920 And thus, it doesn't have mass. 00:00:47.920 --> 00:00:51.320 You can fill a room with light and never run out of space. 00:00:51.320 --> 00:00:55.100 So how could you possibly make something physical out of it? 00:00:55.100 --> 00:00:57.170 Well, you can't. 00:00:57.170 --> 00:01:01.930 So, instead light is a form of energy, 00:01:01.930 --> 00:01:04.313 energy being the capacity to do work. 00:01:06.940 --> 00:01:09.430 So, this is where it's helpful to know word origin. 00:01:09.430 --> 00:01:13.800 The word photosynthesis is made up of two Greek words. 00:01:13.800 --> 00:01:18.800 It literally means light and to put together, that's right. 00:01:19.030 --> 00:01:20.750 In addition to being an entomologist, 00:01:20.750 --> 00:01:22.870 a double a little bit in etymology, just to make sure 00:01:22.870 --> 00:01:25.090 I'm maximally confusing to people. 00:01:25.090 --> 00:01:28.880 Anyway, you're literally using light to drive reactions 00:01:28.880 --> 00:01:32.890 that combine ingredients into new products, a form of work. 00:01:32.890 --> 00:01:37.060 So you're not turning light itself into material sugar. 00:01:37.060 --> 00:01:41.360 You're taking matter that already exists in the form of 00:01:41.360 --> 00:01:43.700 six molecules of carbon dioxide, 00:01:43.700 --> 00:01:45.780 six molecules of liquid water, 00:01:45.780 --> 00:01:49.800 and using the energy of the sun to power a reaction 00:01:49.800 --> 00:01:52.400 that combines them into a new substance 00:01:52.400 --> 00:01:55.203 with molecular oxygen as a byproduct. 00:01:56.310 --> 00:01:57.610 Think of it this way. 00:01:57.610 --> 00:02:01.360 When you bake a cake, you don't say that you made cake 00:02:01.360 --> 00:02:02.680 from heat. 00:02:02.680 --> 00:02:04.680 It would be more accurate to say 00:02:04.680 --> 00:02:09.680 that you took flour, eggs, sugar, and butter 00:02:10.150 --> 00:02:13.123 and used heat to combine them into something new. 00:02:14.790 --> 00:02:16.370 So sticking with our analogy, 00:02:16.370 --> 00:02:19.700 the ingredients for photosynthesis are just carbon dioxide 00:02:19.700 --> 00:02:22.453 from the atmosphere and water from the ground, 00:02:23.690 --> 00:02:26.580 with light to do the heavy lifting. 00:02:26.580 --> 00:02:30.960 Carbon dioxide and water are put together 00:02:30.960 --> 00:02:34.030 to produce a carbohydrate which literally means 00:02:34.030 --> 00:02:35.143 water and carbon. 00:02:36.000 --> 00:02:39.150 As you can see with this typical carbohydrate molecule, 00:02:39.150 --> 00:02:43.130 glucose, your carbon is attached to the same atoms 00:02:43.130 --> 00:02:46.320 that compose water, two hydrogens and oxygen. 00:02:46.320 --> 00:02:49.820 This carbohydrate has more chemical energy 00:02:49.820 --> 00:02:54.350 or bond energy than the molecules of water or carbon dioxide 00:02:54.350 --> 00:02:56.200 that served as ingredients. 00:02:56.200 --> 00:02:59.930 Thus, the energy at the end of the process is much greater 00:02:59.930 --> 00:03:03.170 than the energy that the ingredients had at the outset. 00:03:03.170 --> 00:03:06.790 This means that photosynthesis is a sort of useful reaction 00:03:06.790 --> 00:03:10.630 that stores energy like a big biochemical solar cell. 00:03:10.630 --> 00:03:13.820 We call these endergonic reactions. 00:03:13.820 --> 00:03:15.250 And you might be able to see already 00:03:15.250 --> 00:03:18.040 why they will be so valuable in biological systems. 00:03:18.040 --> 00:03:20.790 You take molecules that don't have much energy 00:03:20.790 --> 00:03:23.640 and use them to produce something that can do work. 00:03:23.640 --> 00:03:25.500 And the kitchen where all this work happens 00:03:25.500 --> 00:03:26.770 is the chloroplast. 00:03:26.770 --> 00:03:29.201 Literally, the green maker. 00:03:29.201 --> 00:03:32.040 The chloroplast is a little organ or organelle 00:03:32.040 --> 00:03:33.920 present in some plant cells. 00:03:33.920 --> 00:03:37.890 And it's what makes plant cells and ultimately plants green. 00:03:37.890 --> 00:03:41.720 And this is because the chloroplasts bear the green pigment 00:03:41.720 --> 00:03:42.553 chlorophyll. 00:03:44.080 --> 00:03:46.490 Now the same way that the word photosynthesis 00:03:46.490 --> 00:03:49.410 can be broken down neatly into its two base words. 00:03:49.410 --> 00:03:52.200 The process itself can be separated into two 00:03:52.200 --> 00:03:53.590 neat little segments. 00:03:53.590 --> 00:03:55.910 I like to think of them as a charging step, 00:03:55.910 --> 00:03:59.260 where energy from light is converted into chemical energy 00:03:59.260 --> 00:04:02.270 and synthesis step where the energy is used to do the work 00:04:02.270 --> 00:04:04.540 of actually synthesizing the end product, 00:04:04.540 --> 00:04:06.113 typically a carbohydrate. 00:04:07.160 --> 00:04:09.610 So the first segment requires the direct input of light 00:04:09.610 --> 00:04:13.590 and as such is referred to as the light dependent reactions. 00:04:13.590 --> 00:04:15.800 These happened in a section of the chloroplast 00:04:15.800 --> 00:04:17.280 called the thylakoids. 00:04:17.280 --> 00:04:18.960 They form these neat little pouches. 00:04:18.960 --> 00:04:21.780 The inside of which is called the lumen and the outside 00:04:21.780 --> 00:04:23.060 called the stroma. 00:04:23.060 --> 00:04:25.210 When a photon of light ends it's eight minute journey 00:04:25.210 --> 00:04:27.670 from the surface of the sun to the surface of a leaf 00:04:27.670 --> 00:04:29.590 its energy is absorbed by the chlorophyll 00:04:29.590 --> 00:04:31.810 embedded in the thylakoid membrane. 00:04:31.810 --> 00:04:35.600 This energy powers a pump which literally charges the inside 00:04:35.600 --> 00:04:39.183 of the thylakoids like a battery by moving the ions inside. 00:04:40.150 --> 00:04:43.760 As the charge builds up, the energy can be used to do work. 00:04:43.760 --> 00:04:47.040 But the next segment of this process happens outside 00:04:47.040 --> 00:04:49.760 of the thylakoids in the stroma. 00:04:49.760 --> 00:04:51.580 So they get that energy where it's needed. 00:04:51.580 --> 00:04:53.480 The thylakoids transfers the energy 00:04:53.480 --> 00:04:57.700 to a molecule called ADP, adenosine diphosphate. 00:04:57.700 --> 00:04:59.350 By adding another phosphate bond 00:04:59.350 --> 00:05:02.443 and making it ATP, adenosine triphosphate. 00:05:03.320 --> 00:05:04.912 Now you might've heard of this one. 00:05:04.912 --> 00:05:08.052 It's often called the energy currency of the cell. 00:05:08.052 --> 00:05:10.960 Pretty much wherever energy is needed for cell to do work 00:05:10.960 --> 00:05:12.209 ATP is involved. 00:05:12.209 --> 00:05:16.160 You might have also heard that the energy is stored inside 00:05:16.160 --> 00:05:20.230 of the phosphate bonds, and that breaking them 00:05:20.230 --> 00:05:22.500 releases the energy. 00:05:22.500 --> 00:05:25.710 But try to remember that these bonds depicted by the lines 00:05:25.710 --> 00:05:28.410 in these diagrams are just a convention. 00:05:28.410 --> 00:05:30.960 It represents an adherence of these atoms together 00:05:30.960 --> 00:05:34.290 via attraction, and it shows you where they adhere. 00:05:34.290 --> 00:05:36.983 But the energy isn't actually in the bond, 00:05:38.060 --> 00:05:41.550 the attraction between the atoms builds up potential energy 00:05:41.550 --> 00:05:45.120 like a rubber band that's being pulled really tightly. 00:05:45.120 --> 00:05:48.680 When you let it go, it could hit a paper cup 00:05:48.680 --> 00:05:51.240 and do the work of displacing that cup. 00:05:51.240 --> 00:05:53.070 The rubber band then falls to the ground 00:05:53.070 --> 00:05:56.820 in a low energy state because it's energy has been released. 00:05:56.820 --> 00:05:59.120 So when you put the energy into a system 00:05:59.120 --> 00:06:02.020 great enough to overcome the attraction between the atoms 00:06:02.020 --> 00:06:05.270 and force them apart, thereby breaking the bonds, 00:06:05.270 --> 00:06:08.480 atoms or in this case a whole phosphate group 00:06:08.480 --> 00:06:11.150 can go spiraling off taking what was 00:06:11.150 --> 00:06:12.710 potential energy with it. 00:06:12.710 --> 00:06:14.593 If it hits something that it can bond with 00:06:14.593 --> 00:06:17.300 that energy is released as the bond is formed 00:06:17.300 --> 00:06:21.050 and can be used to do work such as the magic of ATP 00:06:21.050 --> 00:06:23.860 is described as energetic because it's easy 00:06:23.860 --> 00:06:27.160 to break the bond between it and the last phosphate group. 00:06:27.160 --> 00:06:29.820 Meaning you don't have to put much energy in 00:06:29.820 --> 00:06:31.843 but you get a ton of energy out. 00:06:33.450 --> 00:06:35.130 So back to the thylakoids. 00:06:35.130 --> 00:06:37.250 Those photons of light were able to net us 00:06:37.250 --> 00:06:40.300 a highly energetic ATP molecule. 00:06:40.300 --> 00:06:42.310 But the next segment of photosynthesis 00:06:42.310 --> 00:06:44.210 is gonna need some electrons too. 00:06:44.210 --> 00:06:45.990 That light energy is used to do the work 00:06:45.990 --> 00:06:48.650 of loading up a mobile electron carrier with electrons 00:06:48.650 --> 00:06:50.200 and a proton. 00:06:50.200 --> 00:06:52.740 This carrier is called NADP plus. 00:06:52.740 --> 00:06:54.710 And when it's got a full load to take to the next 00:06:54.710 --> 00:06:57.313 set of reactions, it's called NADPH, 00:06:58.810 --> 00:07:01.100 and that's pretty much the light dependent reactions 00:07:01.100 --> 00:07:02.160 in a nutshell. 00:07:02.160 --> 00:07:06.210 The only other thing you should probably remember is that, 00:07:06.210 --> 00:07:09.430 well, this is where all the oxygen in your lungs comes from. 00:07:09.430 --> 00:07:11.870 So, you know, no big deal. 00:07:11.870 --> 00:07:14.940 When chlorophyll gets excited by that photon of light 00:07:15.890 --> 00:07:17.980 it turns into a real bully. 00:07:17.980 --> 00:07:20.260 The work it's doing creates such a powerful 00:07:20.260 --> 00:07:22.890 electro-chemical imbalance and the chlorophyll 00:07:22.890 --> 00:07:25.560 balances the equation by just stealing an electron 00:07:25.560 --> 00:07:26.610 from water. 00:07:26.610 --> 00:07:29.550 This causes water to fall apart releasing its oxygen 00:07:29.550 --> 00:07:31.583 which the plant just lets go of. 00:07:32.580 --> 00:07:34.350 So now that we've taken care of the section 00:07:34.350 --> 00:07:37.430 that's dependent on light, let's discuss the section 00:07:37.430 --> 00:07:38.263 that isn't. 00:07:39.750 --> 00:07:43.420 The light independent reactions or the Calvin cycle 00:07:43.420 --> 00:07:45.770 occur in the stroma of the chloroplast. 00:07:45.770 --> 00:07:48.000 And this is where the earth shattering 00:07:48.000 --> 00:07:51.140 chemical reaction occurs that allows for all life 00:07:51.140 --> 00:07:52.060 on this planet. 00:07:52.060 --> 00:07:55.660 The fixation of gaseous carbon or inorganic carbon 00:07:55.660 --> 00:07:58.193 into carbon chains, organic carbon. 00:07:59.120 --> 00:08:01.330 And this is so important because fixation 00:08:01.330 --> 00:08:03.053 doesn't just happen on its own. 00:08:03.980 --> 00:08:07.370 CO2 in the atmosphere doesn't form organic chains or sugars. 00:08:07.370 --> 00:08:10.010 When it bumps into more CO2 in the atmosphere 00:08:10.010 --> 00:08:13.390 it requires the help of enzymes and energetic molecules 00:08:13.390 --> 00:08:15.140 made by living organisms. 00:08:15.140 --> 00:08:17.780 It's why when we first landed a rover on Mars 00:08:17.780 --> 00:08:19.870 we immediately started looking for the evidence 00:08:19.870 --> 00:08:21.320 of organic molecules. 00:08:21.320 --> 00:08:25.660 It will be evidence that something is or was living there. 00:08:25.660 --> 00:08:28.680 In the light independent reactions a plant enzyme 00:08:28.680 --> 00:08:33.020 fixes carbon dioxide from the air into a chain of carbon. 00:08:33.020 --> 00:08:35.750 So ATP and NADPH which were produced 00:08:35.750 --> 00:08:38.720 in the light dependent reactions, provide the energy 00:08:38.720 --> 00:08:42.490 and the electrons to create two energetic reactive molecules 00:08:42.490 --> 00:08:44.120 that can be combined to make glucose 00:08:44.120 --> 00:08:46.443 or other useful molecules. 00:08:47.360 --> 00:08:49.770 And the beauty of it is that the byproducts 00:08:49.770 --> 00:08:54.770 of the light independent reactions, ADP and NADP plus 00:08:55.270 --> 00:08:58.540 are shuttled off from the stroma back to the thylakoids 00:08:58.540 --> 00:09:00.920 for more light dependent reactions 00:09:00.920 --> 00:09:03.140 where they can be recharged and recycled 00:09:03.140 --> 00:09:04.343 for use again later. 00:09:05.350 --> 00:09:09.300 And, wow, photosynthesis. 00:09:09.300 --> 00:09:11.710 The takeaway here is that photosynthesis 00:09:11.710 --> 00:09:16.020 allows you to go from the intangible energy of the sun 00:09:16.020 --> 00:09:17.930 to the stored chemical energy 00:09:17.930 --> 00:09:20.780 that life on this planet is based on. 00:09:20.780 --> 00:09:23.510 The sun's energy is converted to the chemical energy 00:09:23.510 --> 00:09:26.500 of a carbohydrate molecule in the chloroplast. 00:09:26.500 --> 00:09:28.680 That molecule can later be broken down 00:09:28.680 --> 00:09:31.610 in most of that energy reclaimed either by the plant 00:09:31.610 --> 00:09:33.410 or by creatures that eat that plant. 00:09:34.300 --> 00:09:37.810 All life on earth is carbon-based. 00:09:37.810 --> 00:09:40.780 And every single molecule of that carbon 00:09:40.780 --> 00:09:44.230 once existed in the atmosphere in gaseous form 00:09:44.230 --> 00:09:47.660 as carbon dioxide until some enterprising plant 00:09:47.660 --> 00:09:52.090 or microorganism synthesized it into something you can use. 00:09:52.090 --> 00:09:54.910 And while they were added many of them filled the atmosphere 00:09:54.910 --> 00:09:57.470 with the oxygen that we all need to breathe. 00:09:57.470 --> 00:10:00.000 So the next time you see a plant, 00:10:00.000 --> 00:10:01.967 shake its leaf and say thank you.

Organization of multicellular organisms

https://www.youtube.com/watch?v=8pWMifyG5bU

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https://www.youtube.com/api/timedtext?v=8pWMifyG5bU&ei=5VWUZcnXBsG-mLAP65eU4Ag&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=71C792F3A6FBD24E9A2F07127EEB0CF97B2929A4.B46C05655E38887530A60304B40C4CDFF481B814&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.310 --> 00:00:01.143 - [Instructor] In this video, 00:00:01.143 --> 00:00:03.180 we're gonna take a journey in life 00:00:03.180 --> 00:00:07.150 and we're gonna start with the smallest scale of life 00:00:07.150 --> 00:00:11.210 that is indisputably life, and that is the cell. 00:00:11.210 --> 00:00:13.390 Now, the reason why I qualified that a little bit 00:00:13.390 --> 00:00:16.940 is some people debate whether viruses are living or not, 00:00:16.940 --> 00:00:18.580 'cause they have certain aspects. 00:00:18.580 --> 00:00:22.430 Viruses can reproduce, they do have genetic information 00:00:22.430 --> 00:00:25.070 but they need other living forms to reproduce, 00:00:25.070 --> 00:00:27.660 in particular they need other cells. 00:00:27.660 --> 00:00:29.352 But even though we imagined cells 00:00:29.352 --> 00:00:34.060 to be these very small microscopic things, 00:00:34.060 --> 00:00:37.406 they are in and of themselves almost an entire world, 00:00:37.406 --> 00:00:39.821 and we go into depth into that into other videos 00:00:39.821 --> 00:00:41.410 in Khan Academy. 00:00:41.410 --> 00:00:43.057 But the fact that every cell in your body, 00:00:43.057 --> 00:00:45.770 except for a few like red blood cells 00:00:45.770 --> 00:00:48.960 have all of your genetic information in there. 00:00:48.960 --> 00:00:52.246 All of those 3 billion base pairs of DNA 00:00:52.246 --> 00:00:55.630 that make you a human being 00:00:55.630 --> 00:00:58.349 in and of itself is mind-boggling. 00:00:58.349 --> 00:01:01.460 But then the fact that the cell specializes 00:01:01.460 --> 00:01:02.863 so that not all cells are the same 00:01:02.863 --> 00:01:05.027 even though they have that same genetic information, 00:01:05.027 --> 00:01:08.210 they somehow know what type of cell to be 00:01:08.210 --> 00:01:10.570 that's even more interesting. 00:01:10.570 --> 00:01:13.705 So we could start at the most basic building block 00:01:13.705 --> 00:01:16.079 in your body or really any organism's body 00:01:16.079 --> 00:01:18.860 and that's specialized cells. 00:01:18.860 --> 00:01:21.533 So what you're seeing here is a big cluster of neurons 00:01:21.533 --> 00:01:24.190 which are dyed here in the red, 00:01:24.190 --> 00:01:27.758 and I believe these blues show their actual nuclei 00:01:27.758 --> 00:01:30.300 where they have their genetic information. 00:01:30.300 --> 00:01:32.880 And then dyed in green, you have what are called 00:01:32.880 --> 00:01:35.807 neuroglia cells which are other types of cells 00:01:35.807 --> 00:01:39.870 that are inside the human brain mainly to support neurons. 00:01:39.870 --> 00:01:42.520 Most of what we believe is thought 00:01:42.520 --> 00:01:45.170 occurs through triggering neurons, 00:01:45.170 --> 00:01:46.870 which then trigger other neurons 00:01:46.870 --> 00:01:50.290 and form cascades of these electro-chemical signals 00:01:50.290 --> 00:01:53.400 which we're just starting to understand. 00:01:53.400 --> 00:01:57.520 But this is just one little small fraction of a human brain. 00:01:57.520 --> 00:01:59.680 A human brain, for example we'll have 00:01:59.680 --> 00:02:04.130 on the order of 80 to 90 billion neurons. 00:02:04.130 --> 00:02:05.625 And for every one of those neurons 00:02:05.625 --> 00:02:08.740 depending on what part of the brain you're talking about, 00:02:08.740 --> 00:02:11.870 you're talking about five to 10 neuroglia cells. 00:02:11.870 --> 00:02:14.810 So you're talking about many hundreds of billions of cells 00:02:14.810 --> 00:02:17.960 just in one human brain. 00:02:17.960 --> 00:02:19.990 But then if we were to zoom out a little bit 00:02:19.990 --> 00:02:21.837 and you take a bunch of these specialized cells 00:02:21.837 --> 00:02:25.120 working together or at least near each other, 00:02:25.120 --> 00:02:26.623 you have tissue. 00:02:27.600 --> 00:02:31.280 And so as I said before, this is a zoomed in view 00:02:31.280 --> 00:02:35.060 of neural tissue in particular of brain tissue. 00:02:35.060 --> 00:02:37.710 And then if you zoom out a little bit more, 00:02:37.710 --> 00:02:41.400 the tissue makes up organs. 00:02:41.400 --> 00:02:44.770 And if we're thinking about neural tissue like this, 00:02:44.770 --> 00:02:47.680 we can imagine that it makes up the brain 00:02:47.680 --> 00:02:49.660 which is an organ. 00:02:49.660 --> 00:02:53.930 And then organs build up to systems. 00:02:53.930 --> 00:02:56.410 And right over here, you have a picture 00:02:56.410 --> 00:02:59.620 of the nervous system of which the brain is apart. 00:02:59.620 --> 00:03:00.453 You also have the spinal cord, 00:03:00.453 --> 00:03:01.936 and then you also have all of the nerves 00:03:01.936 --> 00:03:03.660 that go throughout the body. 00:03:03.660 --> 00:03:05.713 So we have a system. 00:03:06.580 --> 00:03:09.170 And then you put all of the systems together 00:03:09.170 --> 00:03:13.060 and you get the actual organism, 00:03:13.060 --> 00:03:16.470 which of course you can somewhat visualize right over here 00:03:16.470 --> 00:03:18.290 where you can see all of these different organs 00:03:18.290 --> 00:03:22.150 and organ systems put together to create who we are. 00:03:22.150 --> 00:03:24.796 And just to connect to the organism with the cells, 00:03:24.796 --> 00:03:27.460 that basic building block of life. 00:03:27.460 --> 00:03:30.210 If you are a average size human being, 00:03:30.210 --> 00:03:35.210 you likely have 30 to 40 trillion cells in your body. 00:03:35.910 --> 00:03:37.570 And if that isn't mind blowing enough 00:03:37.570 --> 00:03:38.910 and it is just an estimate, 00:03:38.910 --> 00:03:40.460 it's estimated that there's as many 00:03:40.460 --> 00:03:45.190 as 100 trillion bacteria in your body. 00:03:45.190 --> 00:03:49.500 And so even though you think you are just "an individual" 00:03:49.500 --> 00:03:51.532 you are a universe of living things 00:03:51.532 --> 00:03:53.430 in these complex systems. 00:03:53.430 --> 00:03:55.560 And it's an interesting question of why 00:03:55.560 --> 00:03:56.930 and right over here. 00:03:56.930 --> 00:03:59.410 We know that organisms interact with each other. 00:03:59.410 --> 00:04:01.810 We know that they interact with their environment 00:04:01.810 --> 00:04:04.211 just as each of our nerve cells might not appreciate 00:04:04.211 --> 00:04:08.350 that they are one of 86 billion in dissension mind, 00:04:08.350 --> 00:04:11.710 maybe we ourselves as organisms don't appreciate 00:04:11.710 --> 00:04:13.620 that we too are building blocks 00:04:13.620 --> 00:04:15.553 of maybe something even larger.

Genes, proteins, and cells

https://www.youtube.com/watch?v=_iVu3g_S05I

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https://www.youtube.com/api/timedtext?v=_iVu3g_S05I&ei=4VWUZargAu-sxN8P28aG6AM&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245329&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=18AA64D4F2B0C7FA39B4AA2E86430D9C25BFE651.78F80CC2275635517302C67D5C0F6ADC275070&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.500 --> 00:00:02.010 - [Instructor] So when I was younger, 00:00:02.010 --> 00:00:03.550 around seven or eight years old, 00:00:03.550 --> 00:00:06.450 I used to have a beta fish named Bob, 00:00:06.450 --> 00:00:09.670 and he happened to be a blue-colored fish. 00:00:09.670 --> 00:00:12.670 Now, I've always wondered how he got his color. 00:00:12.670 --> 00:00:15.940 So for example, were his parents also blue? 00:00:15.940 --> 00:00:18.440 Does he have any siblings that were blue? 00:00:18.440 --> 00:00:21.530 So today, let's try to answer in simple terms 00:00:21.530 --> 00:00:24.430 how Bob gets his blue color. 00:00:24.430 --> 00:00:27.030 And we'll start by defining genes. 00:00:27.030 --> 00:00:30.040 Genes. What are genes? 00:00:30.040 --> 00:00:33.140 Well, genes are basic hereditary units 00:00:33.140 --> 00:00:34.620 that are well, first of all, 00:00:34.620 --> 00:00:37.110 they are passed down from parent to offspring. 00:00:37.110 --> 00:00:41.530 So I'll write passed down parent to offspring. 00:00:41.530 --> 00:00:43.690 And they also contain information 00:00:43.690 --> 00:00:46.220 about an organism's traits. 00:00:46.220 --> 00:00:51.220 Contain info about an organism's traits. 00:00:51.310 --> 00:00:53.810 So it would only make sense that since Bob 00:00:53.810 --> 00:00:57.260 is a blue-colored fish, that he must have a parent 00:00:57.260 --> 00:01:00.300 or ancestor who is also blue-colored. 00:01:00.300 --> 00:01:03.920 Now, a single kind of gene can have many different versions, 00:01:03.920 --> 00:01:07.030 and we call these versions alleles. 00:01:07.030 --> 00:01:08.420 Alleles. 00:01:08.420 --> 00:01:11.823 And these are just different variations of a single gene. 00:01:12.720 --> 00:01:14.460 So for example, there may be a gene 00:01:14.460 --> 00:01:17.930 that provides information about beta fish coloration. 00:01:17.930 --> 00:01:22.930 So let me draw an arrow from Bob to, we'll say, color gene. 00:01:23.500 --> 00:01:28.500 And one allele of this gene may lead to a blue-colored fish. 00:01:29.110 --> 00:01:32.910 So I'll write here blue color. 00:01:32.910 --> 00:01:36.950 And another allele may lead to green coloration. 00:01:36.950 --> 00:01:41.530 And I'll write here, next to blue color, green color. 00:01:41.530 --> 00:01:44.790 But physically, what are genes, exactly? 00:01:44.790 --> 00:01:49.410 Well, genes are part of DNA or deoxyribonucleic acid. 00:01:49.410 --> 00:01:51.110 And I know, it's a really big word. 00:01:51.110 --> 00:01:53.060 So let me write it out for us. 00:01:53.060 --> 00:01:57.120 DNA. Deoxyribonucleic acid. 00:01:57.120 --> 00:01:57.953 DNA. 00:01:59.550 --> 00:02:02.220 So DNA is a macromolecule, 00:02:02.220 --> 00:02:05.980 or a really big and complex molecule. 00:02:05.980 --> 00:02:10.760 So when you hear macro, just think big or complex molecule. 00:02:10.760 --> 00:02:14.530 And you can also think of DNA like a giant cookbook 00:02:14.530 --> 00:02:17.890 of genetic information, because that's what it is. 00:02:17.890 --> 00:02:21.473 It is a cookbook of genetic information. 00:02:23.230 --> 00:02:25.890 So here is a close-up sketch of DNA. 00:02:25.890 --> 00:02:29.220 And what I'm showing here is that DNA consists 00:02:29.220 --> 00:02:32.530 of subunits called nucleotides. 00:02:32.530 --> 00:02:33.930 Nucleotides. 00:02:33.930 --> 00:02:38.930 And these are represented by As, Ts, Cs, and Gs. 00:02:40.860 --> 00:02:43.810 Now genes are like the individual recipes. 00:02:43.810 --> 00:02:46.390 You can think of them as the recipes found 00:02:46.390 --> 00:02:48.700 inside your DNA cookbook. 00:02:48.700 --> 00:02:51.420 So they are specific segments of nucleotides 00:02:51.420 --> 00:02:54.090 within the long DNA molecule. 00:02:54.090 --> 00:02:57.350 So a gene could be from here to here. 00:02:57.350 --> 00:03:00.970 And we think of this as a recipe. 00:03:00.970 --> 00:03:03.920 But what do these gene recipes make anyways? 00:03:03.920 --> 00:03:06.690 Well, many genes encode proteins, 00:03:06.690 --> 00:03:09.330 which are made of long sequences 00:03:09.330 --> 00:03:11.710 or chains of amino acids. 00:03:11.710 --> 00:03:13.520 So, I'll write here under proteins 00:03:13.520 --> 00:03:17.620 that proteins are made of amino acid chains. 00:03:17.620 --> 00:03:19.430 And genes provide instructions 00:03:19.430 --> 00:03:21.370 for how to create these chains. 00:03:21.370 --> 00:03:24.770 So thinking back to how genes are like recipes 00:03:24.770 --> 00:03:27.030 because each gene has a different set 00:03:27.030 --> 00:03:30.440 of quote-unquote, "ingredients", or in this case, 00:03:30.440 --> 00:03:32.450 a different nucleotide sequence, 00:03:32.450 --> 00:03:36.500 different genes would therefore encode different proteins. 00:03:36.500 --> 00:03:38.050 So proteins in this example, 00:03:38.050 --> 00:03:40.630 if you want to continue on with this analogy, 00:03:40.630 --> 00:03:42.650 proteins would be like the finished products 00:03:42.650 --> 00:03:44.030 of your recipes. 00:03:44.030 --> 00:03:46.410 Remember how I mentioned that alleles 00:03:46.410 --> 00:03:49.500 are different versions of a specific gene? 00:03:49.500 --> 00:03:53.020 Well, different alleles have different nucleotide sequences 00:03:53.020 --> 00:03:57.740 and are therefore likely to also encode different proteins. 00:03:57.740 --> 00:03:59.980 So I'll write under here, different alleles, 00:03:59.980 --> 00:04:02.820 and draw an arrow to different proteins. 00:04:02.820 --> 00:04:05.300 So different genes and different alleles 00:04:05.300 --> 00:04:07.680 can give us different proteins. 00:04:07.680 --> 00:04:11.370 So of blue fish like Bob, likely has a distinct allele 00:04:11.370 --> 00:04:15.460 that encodes proteins, which give him his natural blue color 00:04:15.460 --> 00:04:17.260 or his natural blue shade. 00:04:17.260 --> 00:04:20.230 Now, an organism has many different genes, 00:04:20.230 --> 00:04:23.010 which means an organism has the capacity 00:04:23.010 --> 00:04:25.920 to produce many different kinds of proteins. 00:04:25.920 --> 00:04:28.800 And proteins serve all kinds of functions, 00:04:28.800 --> 00:04:33.600 some of which include growth, sending messages, 00:04:33.600 --> 00:04:36.210 oh, also catalyzing chemical reactions, 00:04:36.210 --> 00:04:38.440 if you have heard of enzymes before. 00:04:38.440 --> 00:04:41.640 So, I'll write catalyzing chemical reactions 00:04:42.650 --> 00:04:44.303 and providing structure. 00:04:45.430 --> 00:04:47.270 Ultimately, it's the activity 00:04:47.270 --> 00:04:49.440 of these different kinds of proteins 00:04:49.440 --> 00:04:52.320 that help determine an organism's physical traits, 00:04:52.320 --> 00:04:54.680 just like how proteins can determine 00:04:54.680 --> 00:04:56.883 the color of Bob the beta fish. 00:04:57.810 --> 00:04:59.190 Now, you might be wondering too, 00:04:59.190 --> 00:05:01.760 what genes do in the big picture. 00:05:01.760 --> 00:05:04.930 For example, not all of the cells in Bob the beta fish 00:05:04.930 --> 00:05:06.670 function in the same way, right? 00:05:06.670 --> 00:05:08.660 You could say that there are different types 00:05:08.660 --> 00:05:11.600 of cells that perform specific functions. 00:05:11.600 --> 00:05:14.830 And this is known as cell specialization. 00:05:14.830 --> 00:05:18.560 So cell specialization is when different cells specialize 00:05:18.560 --> 00:05:20.290 in different functions. 00:05:20.290 --> 00:05:22.160 And each type of specialized cell 00:05:22.160 --> 00:05:25.430 contains a unique combination of proteins 00:05:25.430 --> 00:05:27.910 that give the cell its specialized function 00:05:27.910 --> 00:05:29.140 within an organism. 00:05:29.140 --> 00:05:31.930 So I'll write here that cell specialization 00:05:31.930 --> 00:05:35.260 comes from unique combos of proteins. 00:05:35.260 --> 00:05:36.640 And it's important to note 00:05:36.640 --> 00:05:40.640 that every cell in an organism contains the same genes. 00:05:40.640 --> 00:05:41.970 So going back to Bob here, 00:05:41.970 --> 00:05:44.530 we can say that all his cells contain 00:05:44.530 --> 00:05:47.210 the same DNA and genetic information, 00:05:47.210 --> 00:05:50.740 whether it's on his tail or in his eye, 00:05:50.740 --> 00:05:52.470 but what makes their functions different 00:05:52.470 --> 00:05:55.350 is which subset of genes are expressed 00:05:55.350 --> 00:05:59.460 or used to build proteins in the different cell types. 00:05:59.460 --> 00:06:03.360 So, I'll write here, all of the cells in Bob the beta fish 00:06:03.360 --> 00:06:06.280 have the same DNA and genes, 00:06:06.280 --> 00:06:09.163 just different protein combinations. 00:06:10.070 --> 00:06:13.100 So today, we learned about genes which are passed down 00:06:13.100 --> 00:06:14.440 from parent to offspring, 00:06:14.440 --> 00:06:18.230 just like how Bob the blue beta fish got his color, 00:06:18.230 --> 00:06:21.950 and they contain information about organisms' traits. 00:06:21.950 --> 00:06:24.890 So genes are parts of longer DNA molecules 00:06:24.890 --> 00:06:28.980 and they consist of specific segments of nucleotide bases. 00:06:28.980 --> 00:06:33.520 So thinking back to our cookbook and recipe analogy, right? 00:06:33.520 --> 00:06:36.780 Genes can encode all kinds of different proteins. 00:06:36.780 --> 00:06:39.150 And it's this unique set of proteins 00:06:39.150 --> 00:06:42.370 within each cell that gives the cell its specific function 00:06:42.370 --> 00:06:43.500 within the organism. 00:06:43.500 --> 00:06:46.490 As we talked about with Bob the beta fish, 00:06:46.490 --> 00:06:50.330 and all of the cells in Bob having the same DNA and genes, 00:06:50.330 --> 00:06:51.873 just different proteins.

Greenhouse effect and greenhouse gases

https://www.youtube.com/watch?v=YpfxiDktSoI

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https://www.youtube.com/api/timedtext?v=YpfxiDktSoI&ei=41WUZeLoPOD5vdIPg_yQuAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245332&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=6FD0810A390C65E0EFAE027A26DD9D6284C588F0.62E609846918608F50F275A384CC1D24C8808572&key=yt8&lang=en&name=Default&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:00.370 --> 00:00:01.203 - [Instructor] In this video, 00:00:01.203 --> 00:00:04.400 we're gonna talk about the greenhouse effect 00:00:04.400 --> 00:00:06.790 and also the greenhouse gases, 00:00:06.790 --> 00:00:09.680 which cause the greenhouse effect. 00:00:09.680 --> 00:00:12.070 Now let's just start with a basic idea. 00:00:12.070 --> 00:00:15.880 Imagine if earth had no atmosphere, what would happen? 00:00:15.880 --> 00:00:16.880 Well, you have the sun, 00:00:16.880 --> 00:00:20.130 which is on average, 93 million miles away. 00:00:20.130 --> 00:00:22.700 It's sending electromagnetic radiation 00:00:22.700 --> 00:00:24.750 our way to the surface of the earth. 00:00:24.750 --> 00:00:27.080 We're actually getting a very, very small fraction 00:00:27.080 --> 00:00:30.570 of the total electromagnetic radiation of the sun. 00:00:30.570 --> 00:00:33.570 And then that would heat up the surface of the earth. 00:00:33.570 --> 00:00:36.330 Now, what I have always found mind-blowing 00:00:36.330 --> 00:00:38.330 is anything with temperature 00:00:38.330 --> 00:00:40.850 will emit electromagnetic radiation. 00:00:40.850 --> 00:00:42.950 And so it's emitting some of that energy, 00:00:42.950 --> 00:00:44.930 it's losing some of that energy 00:00:44.930 --> 00:00:47.320 to electromagnetic radiation. 00:00:47.320 --> 00:00:50.750 So the surface would be releasing that 00:00:50.750 --> 00:00:53.950 and it would go out into space. 00:00:53.950 --> 00:00:57.450 But now let's introduce the idea of an atmosphere. 00:00:57.450 --> 00:00:58.930 And in particular, we're gonna think 00:00:58.930 --> 00:01:00.403 about our lower atmosphere, 00:01:00.403 --> 00:01:02.760 which starts at the surface and goes up 00:01:02.760 --> 00:01:05.160 to about five to nine miles in altitude, 00:01:05.160 --> 00:01:07.282 often known as the troposphere. 00:01:07.282 --> 00:01:10.840 Now the troposphere has molecules in it, 00:01:10.840 --> 00:01:14.030 has gases in it like carbon dioxide, 00:01:14.030 --> 00:01:17.140 like water vapor, like methane. 00:01:17.140 --> 00:01:21.320 Others include nitrous oxide and chlorofluorocarbons. 00:01:21.320 --> 00:01:22.990 You don't have to know the chemical formula 00:01:22.990 --> 00:01:24.240 of all of these things, 00:01:24.240 --> 00:01:27.310 but what's interesting about these gases 00:01:27.310 --> 00:01:31.230 that are in the lower atmosphere is that they can absorb 00:01:31.230 --> 00:01:33.410 some of those electromagnetic waves 00:01:33.410 --> 00:01:35.680 that the surface of the earth is emitting. 00:01:35.680 --> 00:01:38.130 So some of that energy will make it out to space, 00:01:38.130 --> 00:01:40.740 but some of that energy will then be absorbed 00:01:40.740 --> 00:01:43.020 by these molecules, by these gases, 00:01:43.020 --> 00:01:45.850 and then they will emit some of that back to earth. 00:01:45.850 --> 00:01:48.150 And so that's why it's called a greenhouse effect 00:01:48.150 --> 00:01:49.550 because of all of that energy 00:01:49.550 --> 00:01:51.510 that might have been sent out to space 00:01:51.510 --> 00:01:54.590 if you didn't have an atmosphere, not all of it is. 00:01:54.590 --> 00:01:57.080 Some of it is reabsorbed by the atmosphere, 00:01:57.080 --> 00:01:59.610 which then sends it back down to the surface 00:01:59.610 --> 00:02:01.750 and that process can go on and on and on. 00:02:01.750 --> 00:02:05.930 You can imagine that it's trapping some of the energy. 00:02:05.930 --> 00:02:09.280 Now, the greenhouse effect and greenhouse gases, 00:02:09.280 --> 00:02:12.088 the ones that I just listed, they're oftentimes associated 00:02:12.088 --> 00:02:15.550 with man-made climate change and global warming. 00:02:15.550 --> 00:02:17.720 And they are, for good reason, 00:02:17.720 --> 00:02:18.920 but it's important to realize 00:02:18.920 --> 00:02:22.570 that we actually need some base level greenhouse effect 00:02:22.570 --> 00:02:25.870 just for earth to be habitable in the way that it is. 00:02:25.870 --> 00:02:29.080 Without greenhouse gases, earth's surface 00:02:29.080 --> 00:02:32.730 would be about negative 18 degrees Celsius, 00:02:32.730 --> 00:02:35.326 which is the same as zero degrees Fahrenheit, 00:02:35.326 --> 00:02:37.225 which I think most of y'all would recognize 00:02:37.225 --> 00:02:39.920 is very, very, very cold 00:02:39.920 --> 00:02:42.940 relative to what the actual averages are, 00:02:42.940 --> 00:02:47.940 which are 15 degrees Celsius or 59 degrees Fahrenheit. 00:02:48.920 --> 00:02:50.570 And of course, these are average temperatures, 00:02:50.570 --> 00:02:53.291 but 59 is a nice refreshing brisk day, 00:02:53.291 --> 00:02:56.870 not a frigid day like zero degrees Fahrenheit. 00:02:56.870 --> 00:02:57.930 And of course, these are averages. 00:02:57.930 --> 00:03:01.920 It fluctuates around this on time of year and where you are. 00:03:01.920 --> 00:03:02.900 But this has makes it clear 00:03:02.900 --> 00:03:05.270 that we do need these greenhouse gases 00:03:05.270 --> 00:03:07.920 to keep the earth reasonably warm. 00:03:07.920 --> 00:03:09.020 Now, the problem is, 00:03:09.020 --> 00:03:11.660 is if the concentration of these greenhouse gases 00:03:11.660 --> 00:03:15.490 go out of equilibrium, become unusually high, 00:03:15.490 --> 00:03:18.860 and it does look like that is indeed happening. 00:03:18.860 --> 00:03:20.639 This right over here is a chart, 00:03:20.639 --> 00:03:22.750 and the way that we're able to figure that out 00:03:22.750 --> 00:03:25.340 is by taking ice samples and rock samples 00:03:25.340 --> 00:03:27.980 and looking into our past or the geologic record 00:03:27.980 --> 00:03:30.020 of how much carbon dioxide there has been 00:03:30.020 --> 00:03:31.990 over the last 800,000 years. 00:03:31.990 --> 00:03:35.210 And 800,000 years is a very long time period. 00:03:35.210 --> 00:03:37.100 Modern human beings have only been around 00:03:37.100 --> 00:03:40.090 for 200,000 or 300,000 years. 00:03:40.090 --> 00:03:43.370 And what you can see is the concentration of carbon dioxide 00:03:43.370 --> 00:03:46.870 has roughly fluctuated between about 200 parts per million 00:03:46.870 --> 00:03:48.880 and about 300 parts per million, 00:03:48.880 --> 00:03:51.290 at least over the duration of this chart. 00:03:51.290 --> 00:03:54.240 But in recent times, we've gone well beyond that. 00:03:54.240 --> 00:03:57.810 We've almost gone double that average right over here. 00:03:57.810 --> 00:03:59.319 And this is actually the highest levels 00:03:59.319 --> 00:04:03.430 we've seen in three million years. 00:04:03.430 --> 00:04:05.920 That's important to realize that carbon dioxide 00:04:05.920 --> 00:04:08.760 makes up a small percentage of our atmosphere. 00:04:08.760 --> 00:04:10.220 In fact, all of these greenhouse gases 00:04:10.220 --> 00:04:12.090 make up a small percentage. 00:04:12.090 --> 00:04:16.510 78% of the troposphere is in nitrogen, 21% is oxygen. 00:04:16.510 --> 00:04:19.740 The last 1% is things like argon, 00:04:19.740 --> 00:04:23.130 water vapor, carbon dioxide, methane. 00:04:23.130 --> 00:04:25.050 So even this small amount, 00:04:25.050 --> 00:04:27.540 when the concentration increases dramatically 00:04:27.540 --> 00:04:29.300 can have a huge effect. 00:04:29.300 --> 00:04:31.630 Now you might say, "Hey, we were at these levels 00:04:31.630 --> 00:04:34.120 three million years ago roughly, 00:04:34.120 --> 00:04:36.696 maybe this is just some type of cycle 00:04:36.696 --> 00:04:38.780 that we're seeing on earth." 00:04:38.780 --> 00:04:41.720 And to recognize it, this is indeed manmade, 00:04:41.720 --> 00:04:44.170 we just have to look at a chart like this. 00:04:44.170 --> 00:04:45.940 This tells us two things. 00:04:45.940 --> 00:04:48.850 This tells us annual emissions in this blue line, 00:04:48.850 --> 00:04:51.660 this blue curve, and then the total emission 00:04:51.660 --> 00:04:53.920 or the total concentration in the atmosphere, 00:04:53.920 --> 00:04:56.870 because about one of the things about greenhouse gas 00:04:56.870 --> 00:04:58.780 like carbon dioxide, when it's emitted, 00:04:58.780 --> 00:05:00.360 it doesn't just disappear. 00:05:00.360 --> 00:05:02.907 And we can see if we go to the pre-industrial revolution 00:05:02.907 --> 00:05:05.910 or the early stages of the industrial revolution, 00:05:05.910 --> 00:05:09.770 CO2 emissions were pretty low, pretty close to zero, 00:05:09.770 --> 00:05:11.660 at least on this scale right over here. 00:05:11.660 --> 00:05:14.030 There might've been some basic CO2 emissions, 00:05:14.030 --> 00:05:16.750 people had fires and stoves and things like that. 00:05:16.750 --> 00:05:19.070 But then as the industrial revolution came into play 00:05:19.070 --> 00:05:21.440 and we started using fossil fuels more and more 00:05:21.440 --> 00:05:24.610 to fuel transportation and factories and other things, 00:05:24.610 --> 00:05:26.460 our emissions have gone up dramatically. 00:05:26.460 --> 00:05:28.870 And this coincides with the total concentration 00:05:28.870 --> 00:05:33.180 going well above that 800,000 year average. 00:05:33.180 --> 00:05:35.900 So it's important to keep in mind, the greenhouse effect 00:05:35.900 --> 00:05:38.870 is needed to some degree, but the problem is, 00:05:38.870 --> 00:05:41.130 is when the concentration of greenhouse gases 00:05:41.130 --> 00:05:44.950 like carbon dioxide go well beyond their historic averages, 00:05:44.950 --> 00:05:46.880 which can over time warm the earth 00:05:46.880 --> 00:05:50.230 and even a few degrees centigrade of warming the earth 00:05:50.230 --> 00:05:52.940 can have huge consequences on our environment 00:05:52.940 --> 00:05:54.400 and on our weather. 00:05:54.400 --> 00:05:56.830 I'll throw out one last idea just for kicks 00:05:56.830 --> 00:05:59.180 because even though it's called the greenhouse effect, 00:05:59.180 --> 00:06:01.990 it's actually not how actual greenhouses work. 00:06:01.990 --> 00:06:04.400 The greenhouse effect, as we just described it, 00:06:04.400 --> 00:06:07.526 is really based on this idea that things 00:06:07.526 --> 00:06:09.950 are getting the infrared radiation. 00:06:09.950 --> 00:06:12.610 The electromagnetic radiation is getting reabsorbed, 00:06:12.610 --> 00:06:15.440 which then gets reemitted back to the surface 00:06:15.440 --> 00:06:16.630 and vice right versa. 00:06:16.630 --> 00:06:17.640 In a real greenhouse, 00:06:17.640 --> 00:06:20.640 you can imagine that it is made out of glass. 00:06:20.640 --> 00:06:24.030 What's happening is the sunlight can come in 00:06:24.030 --> 00:06:25.720 and it's warming the surface 00:06:25.720 --> 00:06:29.260 and it's warming the air inside the greenhouse, 00:06:29.260 --> 00:06:32.180 and then that air is not allowed to circulate. 00:06:32.180 --> 00:06:34.880 So if the greenhouse had a little hole 00:06:34.880 --> 00:06:36.070 at the top right over here, 00:06:36.070 --> 00:06:38.040 that hot air would be allowed to go out 00:06:38.040 --> 00:06:40.530 and circulate with the cool air up here, 00:06:40.530 --> 00:06:42.420 but the air isn't allowed to mix, 00:06:42.420 --> 00:06:44.920 and so the air gets hotter and hotter and hotter. 00:06:44.920 --> 00:06:46.590 It actually turns out that the glass 00:06:46.590 --> 00:06:49.220 can let the electromagnetic radiation out 00:06:49.220 --> 00:06:51.470 unlike greenhouse gases. 00:06:51.470 --> 00:06:54.890 So it is a different actual physical process, 00:06:54.890 --> 00:06:57.793 but you can see where people try to create the metaphor.

Cell specialization

https://www.youtube.com/watch?v=TdZr_ucEhgo

vtt

https://www.youtube.com/api/timedtext?v=TdZr_ucEhgo&ei=4VWUZf6qL9iNp-oP3eGVsA0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245329&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=6B0FE1682530C8457163B91D5EAD74E3484AA8D5.937C62C4FFE368D7DB9BFC3CF188107DDBD6B0D9&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:02.040 --> 00:00:04.080 - [Instructor] Ah, the basic building blocks 00:00:04.080 --> 00:00:06.823 of all living things, cells. 00:00:09.620 --> 00:00:12.040 These incredible packages of organelles 00:00:12.040 --> 00:00:13.940 and subcellular components 00:00:13.940 --> 00:00:17.030 carry out a variety of functions in the body, 00:00:17.030 --> 00:00:19.003 like taking in nutrients, 00:00:20.950 --> 00:00:22.823 converting them into energy, 00:00:24.540 --> 00:00:27.040 and working with other cells to produce things 00:00:27.040 --> 00:00:28.223 that the body needs. 00:00:29.550 --> 00:00:33.253 Each cell is essentially like its own little mini factory, 00:00:34.270 --> 00:00:36.860 with complex processes occurring within the cell 00:00:36.860 --> 00:00:38.623 to carry out specific functions. 00:00:39.790 --> 00:00:43.170 Okay, so when we zoom into the cell to figure out 00:00:43.170 --> 00:00:46.690 how exactly these cell processes are carried out, 00:00:46.690 --> 00:00:48.260 one of the star players 00:00:48.260 --> 00:00:53.133 is a class of biological macromolecules known as proteins. 00:00:57.910 --> 00:00:58.810 Proteins carry out 00:00:58.810 --> 00:01:01.660 many incredibly important tasks in the cell, 00:01:01.660 --> 00:01:05.110 such as providing structural support, 00:01:05.110 --> 00:01:07.490 aiding in chemical reactions 00:01:10.730 --> 00:01:13.283 and even building or repairing the cell. 00:01:16.020 --> 00:01:20.600 We can imagine proteins as a chain of amino acids, 00:01:20.600 --> 00:01:23.800 kind of think of them as like beads on a bracelet 00:01:24.870 --> 00:01:29.870 that fold and twist into distinct three-dimensional shapes. 00:01:31.370 --> 00:01:33.490 The structure of a protein 00:01:34.610 --> 00:01:38.670 along with the chemical properties of its amino acid, 00:01:38.670 --> 00:01:40.623 evidently, determine its function. 00:01:45.290 --> 00:01:47.950 Does it form a round, globular sphere 00:01:47.950 --> 00:01:51.080 that can attach and interact with other compounds? 00:01:51.080 --> 00:01:54.320 Or does it twist into long and narrow strands 00:01:54.320 --> 00:01:56.343 that can provide structural support? 00:01:57.230 --> 00:01:59.600 The huge variety of structures 00:01:59.600 --> 00:02:01.450 that proteins can take on 00:02:01.450 --> 00:02:03.850 leads to the wide range of cellular functions 00:02:03.850 --> 00:02:05.050 that they can carry out. 00:02:06.330 --> 00:02:09.050 Okay, so now that we've talked about proteins, 00:02:09.050 --> 00:02:10.640 let's zoom back out 00:02:10.640 --> 00:02:13.780 to analyze how different types of cells come together 00:02:13.780 --> 00:02:17.013 to carry out a variety of functions in an organism. 00:02:18.420 --> 00:02:21.903 This is where cell specialization comes into play, 00:02:25.290 --> 00:02:28.700 which is the process by which a cell takes on 00:02:28.700 --> 00:02:31.620 a specific structure and function. 00:02:31.620 --> 00:02:34.610 So to better help understand this concept, 00:02:34.610 --> 00:02:37.730 let's consider a movie theater analogy 00:02:37.730 --> 00:02:40.900 where the movie theater is your body. 00:02:40.900 --> 00:02:44.870 There is the cashier that handles all the money, 00:02:44.870 --> 00:02:49.870 the snack vendor who handles out popcorn and snacks 00:02:49.950 --> 00:02:51.870 and even the ticket operator 00:02:51.870 --> 00:02:54.590 who directs you to the proper screenings. 00:02:54.590 --> 00:02:55.870 In this analogy, 00:02:55.870 --> 00:02:59.370 each person has their own distinct functions 00:02:59.370 --> 00:03:00.400 in what they handle, 00:03:00.400 --> 00:03:03.660 like money or popcorn or tickets. 00:03:03.660 --> 00:03:04.890 In a similar way, 00:03:04.890 --> 00:03:07.860 the body is also composed of specialized cells 00:03:07.860 --> 00:03:09.023 with unique roles, 00:03:10.530 --> 00:03:14.823 such as red blood cells that carry oxygen in the blood, 00:03:16.330 --> 00:03:20.230 muscle cells that contract and relax 00:03:20.230 --> 00:03:24.130 or even nerve cells that carry signaling messages 00:03:24.130 --> 00:03:25.130 throughout the body. 00:03:26.170 --> 00:03:29.863 Now, remember how I told you about proteins before? 00:03:30.810 --> 00:03:33.630 Well, cell specialization is largely based 00:03:33.630 --> 00:03:38.630 on which proteins are present or absent in the cell. 00:03:40.410 --> 00:03:44.350 It is a cell's unique combination of proteins 00:03:46.250 --> 00:03:48.750 that determine which functions can be carried out. 00:03:52.900 --> 00:03:54.930 But no cell works alone, 00:03:54.930 --> 00:03:57.363 because teamwork makes the dream work. 00:03:58.470 --> 00:04:00.740 Groups of specialized cells 00:04:02.660 --> 00:04:05.820 that carry out specific functions for the organism 00:04:05.820 --> 00:04:08.053 are organized into tissues. 00:04:11.890 --> 00:04:15.220 Looking back at our movie theater analogy, 00:04:15.220 --> 00:04:18.360 there are multiple people within each department 00:04:18.360 --> 00:04:21.713 that work together to help the theater function efficiently. 00:04:22.990 --> 00:04:26.020 Similarly, our specialized cells work together 00:04:26.020 --> 00:04:28.903 as tissues to help the organism function. 00:04:29.810 --> 00:04:32.733 The red blood cells make up the blood, 00:04:33.930 --> 00:04:34.930 a connective tissue 00:04:34.930 --> 00:04:37.770 that moves important substances throughout the body. 00:04:37.770 --> 00:04:40.750 The muscle cells help make up muscle tissue 00:04:43.180 --> 00:04:44.790 which helps the body move 00:04:44.790 --> 00:04:48.440 and neurons or nerve cells make up nervous tissue 00:04:50.220 --> 00:04:52.653 that helps the organism process information. 00:04:53.610 --> 00:04:57.910 So what are the key takeaways about cell specialization? 00:04:57.910 --> 00:05:02.010 Number one, cells are the fundamental unit of life. 00:05:02.010 --> 00:05:03.520 They're the smallest structural 00:05:03.520 --> 00:05:06.370 and functional unit of an organism. 00:05:06.370 --> 00:05:11.170 Number two, proteins help carry out cell processes. 00:05:11.170 --> 00:05:13.810 Number three, specialized cells carry out 00:05:13.810 --> 00:05:16.160 specific functions in an organism. 00:05:16.160 --> 00:05:18.020 Think of the movie theater analogy 00:05:18.020 --> 00:05:21.490 where each person has its own specific role. 00:05:21.490 --> 00:05:24.270 And number four, groups of specialized cells 00:05:24.270 --> 00:05:25.900 come together as tissues 00:05:25.900 --> 00:05:28.120 to carry out one or more specific functions 00:05:28.120 --> 00:05:29.083 for the organism.

Taking and visualizing powers of a complex number

https://www.youtube.com/watch?v=VZmnLQ3CMbE

vtt

https://www.youtube.com/api/timedtext?v=VZmnLQ3CMbE&ei=5VWUZYf-CqT6mLAPppCGoAc&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=915C3F1C3B0C6C4268860B518300D60D2CE7BE4E.A771B351AA43F23963EAA4AAB6256E4DC1A73BBF&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.450 --> 00:00:02.940 - We're told to consider the complex number z is equal 00:00:02.940 --> 00:00:05.950 to -1 plus i times the square root of 3. 00:00:05.950 --> 00:00:09.500 Find z to the fourth in polar and rectangular form. 00:00:09.500 --> 00:00:12.250 So pause this video and see if you can figure that out. 00:00:13.090 --> 00:00:14.910 All right, now let's work through this together. 00:00:14.910 --> 00:00:19.320 So first let's just think about what the modulus of z is. 00:00:19.320 --> 00:00:22.240 We know that the modulus is going to be equal to the 00:00:22.240 --> 00:00:25.480 square root of the real part squared 00:00:25.480 --> 00:00:28.940 plus the square root of 3, plus the imaginary part squared. 00:00:28.940 --> 00:00:33.040 So it is going to be -1 squared 00:00:33.040 --> 00:00:36.800 plus square root of 3 squared, 00:00:36.800 --> 00:00:39.400 which is going to be equal to 1 plus 3. 00:00:39.400 --> 00:00:43.300 So principal root of 4, which is equal to 2. 00:00:43.300 --> 00:00:44.910 Now the next interesting question is, 00:00:44.910 --> 00:00:47.540 what is the argument of z? 00:00:47.540 --> 00:00:49.300 And the reason why I'm even going through this is 00:00:49.300 --> 00:00:50.850 once we put it into polar form, 00:00:50.850 --> 00:00:53.720 it's going to be a lot easier to both visualize 00:00:53.720 --> 00:00:56.640 what it means to take the various exponents of it. 00:00:56.640 --> 00:00:59.850 And then we can convert back into rectangular form. 00:00:59.850 --> 00:01:04.850 And so let us, let me draw another complex plane here. 00:01:05.790 --> 00:01:07.130 Imaginary axis. 00:01:07.130 --> 00:01:10.410 That is my real axis. 00:01:10.410 --> 00:01:14.100 And if I were to plot z, it would look something like this. 00:01:14.100 --> 00:01:16.160 We have -1 in the real direction. 00:01:16.160 --> 00:01:18.530 So that might be -1 there. 00:01:18.530 --> 00:01:22.940 And we have square root of 3 in the imaginary direction, 00:01:22.940 --> 00:01:24.120 square root of 3. 00:01:24.120 --> 00:01:27.950 So our point z is right over here 00:01:27.950 --> 00:01:32.090 and we know the distance from the origin, the modulus, 00:01:32.090 --> 00:01:35.630 we know that this distance right over here is 2. 00:01:35.630 --> 00:01:38.050 We know that this distance right over here 00:01:38.050 --> 00:01:40.000 is square root of 3. 00:01:40.000 --> 00:01:44.540 And we know that this distance right over here is 1. 00:01:44.540 --> 00:01:46.650 And so you might immediately recognize this 00:01:46.650 --> 00:01:50.470 as a 30-60-90 triangle because in a 30-60-90 triangle, 00:01:50.470 --> 00:01:53.680 the short side is half of the hypotenuse, 00:01:53.680 --> 00:01:55.840 and the long side is the square root of 3 00:01:55.840 --> 00:01:57.400 times the short side. 00:01:57.400 --> 00:02:00.450 So we know that this is a 60-degree angle. 00:02:00.450 --> 00:02:02.820 We know that this is a 30-degree angle. 00:02:02.820 --> 00:02:04.460 And the reason why that helps us, 00:02:04.460 --> 00:02:06.340 sorry, it's hard to see that 30 degree. 00:02:06.340 --> 00:02:07.830 The reason why that helps us is 00:02:07.830 --> 00:02:09.540 if this is 60 degrees, 00:02:09.540 --> 00:02:14.540 we know that the argument here must be 120 degrees. 00:02:14.970 --> 00:02:19.610 So the arg of z, the argument of z, is 120 degrees. 00:02:19.610 --> 00:02:20.880 And so just like that 00:02:20.880 --> 00:02:24.210 we can now think about z in polar form. 00:02:24.210 --> 00:02:26.360 So let me write it right over here. 00:02:26.360 --> 00:02:31.360 We can write that z is equal to its modulus, 2, 00:02:31.370 --> 00:02:34.153 times the cosine of 120 degrees, 00:02:36.486 --> 00:02:39.569 plus i times the sine of 120 degrees. 00:02:41.660 --> 00:02:44.990 And we could also visualize z now over here. 00:02:44.990 --> 00:02:48.030 So its modulus is 2. 00:02:48.030 --> 00:02:53.030 So that's halfway to 4, and its argument is 120 degrees. 00:02:53.820 --> 00:02:56.450 So it would put us right over here. 00:02:56.450 --> 00:03:00.140 This is where z is. 00:03:00.140 --> 00:03:02.933 Now, what would z squared be? 00:03:03.830 --> 00:03:05.700 Well, when you multiply complex numbers 00:03:05.700 --> 00:03:08.740 and you've represented them in polar form, 00:03:08.740 --> 00:03:11.570 we know that you would multiply the moduli, 00:03:11.570 --> 00:03:13.690 so it would then be 2 squared. 00:03:13.690 --> 00:03:15.500 So it'd be 4 right over here. 00:03:15.500 --> 00:03:18.220 And then you would add the arguments. 00:03:18.220 --> 00:03:21.560 So you would essentially rotate z by another 120 degrees 00:03:21.560 --> 00:03:23.000 'cause you're multiplying it by z. 00:03:23.000 --> 00:03:26.680 So it's going to be cosine of 240 degrees 00:03:26.680 --> 00:03:31.680 plus i sine of 240 degrees. 00:03:31.800 --> 00:03:34.440 Once again, 2 times 2 is equal to 4. 00:03:34.440 --> 00:03:38.290 120 degrees plus another 120 degrees is 240 degrees. 00:03:38.290 --> 00:03:41.240 And so now where would z squared sit? 00:03:41.240 --> 00:03:46.140 Well, its argument is 240 degrees and its modulus is 4. 00:03:46.140 --> 00:03:50.000 So now it is twice as far from the origin. 00:03:50.000 --> 00:03:51.930 And now let's think about what, 00:03:51.930 --> 00:03:53.520 I'll do this in a new color, 00:03:53.520 --> 00:03:56.563 what z to the third power is going to be equal to. 00:03:57.420 --> 00:04:00.180 Well, that's going to be z squared times z again. 00:04:00.180 --> 00:04:03.350 So we're gonna multiply 2 times this modulus. 00:04:03.350 --> 00:04:06.130 So that's going to be equal to 8 times, 00:04:06.130 --> 00:04:09.330 and then we're going to rotate z squared by 120 degrees. 00:04:09.330 --> 00:04:14.330 So cosine of 360 degrees 00:04:14.870 --> 00:04:19.113 plus i sine of 360 degrees. 00:04:20.900 --> 00:04:25.270 And so that's going to put us at 8 for our modulus. 00:04:25.270 --> 00:04:27.670 And 360 degrees is the same thing as zero degrees. 00:04:27.670 --> 00:04:29.190 So we are right over here. 00:04:29.190 --> 00:04:32.260 So this is z to the third power. 00:04:32.260 --> 00:04:34.520 And I think, you know where this is going. 00:04:34.520 --> 00:04:37.480 What is z to the fourth power going to be? 00:04:37.480 --> 00:04:39.410 Let me move my screen down a little bit 00:04:39.410 --> 00:04:41.490 so I have a little more real estate. 00:04:41.490 --> 00:04:43.060 z to the 4th. 00:04:43.060 --> 00:04:45.260 Well, I'm just gonna take this modulus here 00:04:45.260 --> 00:04:47.684 since I'm going to multiply z to the third times z, 00:04:47.684 --> 00:04:51.810 I'm gonna multiply that modulus times 2 to get to 16. 00:04:51.810 --> 00:04:55.370 And then I'm going to add another 120 degrees. 00:04:55.370 --> 00:04:58.800 Well, I could write cosine of 480 degrees, 00:04:58.800 --> 00:05:01.760 or 360 degrees is the same thing as zero degrees. 00:05:01.760 --> 00:05:05.710 So this I could say is zero degrees. This is zero degrees. 00:05:05.710 --> 00:05:09.473 So if I add 120 to that, I get cosine of 120 degrees. 00:05:11.504 --> 00:05:13.754 Plus i sine of 120 degrees. 00:05:15.910 --> 00:05:19.580 So my argument is back to being at 120 degrees, 00:05:19.580 --> 00:05:21.790 but now my modulus is 16. 00:05:21.790 --> 00:05:23.740 So there's 4, 8, 12, 16, 00:05:23.740 --> 00:05:25.840 this outer circle right over here. 00:05:25.840 --> 00:05:29.910 I am right over there with z to the fourth. 00:05:29.910 --> 00:05:30.910 So we're almost done. 00:05:30.910 --> 00:05:34.450 We've just represented z to the fourth in polar form. 00:05:34.450 --> 00:05:37.780 Now we just have to think about it in rectangular form. 00:05:37.780 --> 00:05:39.570 Now, lucky for us, 00:05:39.570 --> 00:05:42.020 we already know what cosine of 120 degrees is 00:05:42.020 --> 00:05:45.430 and sine of 120 degrees is. 00:05:45.430 --> 00:05:48.160 It is, we can construct if we want 00:05:48.160 --> 00:05:52.410 another 30-60-90 triangle right over here. 00:05:52.410 --> 00:05:56.930 So the hypotenuse here has length 16. 00:05:56.930 --> 00:05:59.000 The short side is going to be 1/2 of that. 00:05:59.000 --> 00:06:00.320 So it has length 8. 00:06:00.320 --> 00:06:02.620 And then the long side is gonna be square root of 3 00:06:02.620 --> 00:06:03.680 times the short side. 00:06:03.680 --> 00:06:06.170 So it's going to be 8 square roots of 3. 00:06:06.170 --> 00:06:08.180 So if we wanted to write z to the fourth 00:06:08.180 --> 00:06:10.690 in rectangular form, 00:06:10.690 --> 00:06:13.223 it would be the real part is -8. 00:06:14.270 --> 00:06:19.270 Plus i times 8 square roots of 3, and we're done.

Indoor air pollutants

https://www.youtube.com/watch?v=nqRPRwO79kg

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https://www.youtube.com/api/timedtext?v=nqRPRwO79kg&ei=5VWUZcqeD9WNmLAP6pCFoAg&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=4874ECE50FEA59FBB7974C172687C417175BE284.7CEE2769DE496242B3A660010DFBAC81A67600F3&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.530 --> 00:00:03.260 - [Instructor] Let's talk about indoor air pollution. 00:00:03.260 --> 00:00:05.950 I remember when I first heard about indoor air pollution 00:00:05.950 --> 00:00:08.320 in my AP environmental science class, 00:00:08.320 --> 00:00:10.370 I was a little confused. 00:00:10.370 --> 00:00:12.280 When I used to think of pollution, 00:00:12.280 --> 00:00:14.420 I would think of images like this 00:00:14.420 --> 00:00:15.850 or this. 00:00:15.850 --> 00:00:18.380 But pollution is often invisible, 00:00:18.380 --> 00:00:21.540 and it isn't just restricted to the outdoors. 00:00:21.540 --> 00:00:25.170 Indoor air pollutants can lead to serious health conditions 00:00:25.170 --> 00:00:26.930 and even death. 00:00:26.930 --> 00:00:29.030 But there are ways to identify 00:00:29.030 --> 00:00:31.400 and prevent indoor air pollution. 00:00:31.400 --> 00:00:33.360 Let's take a look at an example. 00:00:33.360 --> 00:00:34.730 This is Ava. 00:00:34.730 --> 00:00:37.560 And let's say Ava, like many other people in the world, 00:00:37.560 --> 00:00:40.210 spends the majority of her time indoors. 00:00:40.210 --> 00:00:44.350 She works, sleeps, cooks, and eats in her home. 00:00:44.350 --> 00:00:46.950 And lately, she's been experiencing headaches 00:00:46.950 --> 00:00:48.630 and she's been coughing. 00:00:48.630 --> 00:00:51.620 A possible culprit is indoor air pollution. 00:00:51.620 --> 00:00:54.730 So, where could the indoor air pollution be coming from? 00:00:54.730 --> 00:00:57.010 Let's take a look at Ava's house. 00:00:57.010 --> 00:00:58.640 Indoor air pollution can come from 00:00:58.640 --> 00:01:00.610 many different kinds of sources, 00:01:00.610 --> 00:01:03.550 both human-made and natural. 00:01:03.550 --> 00:01:05.650 Ava's furniture, paneling, and carpets 00:01:05.650 --> 00:01:08.300 could be releasing volatile organic compounds, 00:01:08.300 --> 00:01:11.130 which are often written down as VOCs, 00:01:11.130 --> 00:01:12.960 and they basically include substances 00:01:12.960 --> 00:01:17.080 that form gases at room temperature, like formaldehyde. 00:01:17.080 --> 00:01:19.030 Formaldehyde is used as an adhesive 00:01:19.030 --> 00:01:21.110 in building materials and upholstery, 00:01:21.110 --> 00:01:22.990 and it can get into the air. 00:01:22.990 --> 00:01:25.560 This is the same stuff that is used to embalm 00:01:25.560 --> 00:01:27.620 and preserve dead bodies. 00:01:27.620 --> 00:01:30.700 In other words, it's not pleasant to breathe in. 00:01:30.700 --> 00:01:33.210 The United States Environmental Protection Agency 00:01:33.210 --> 00:01:35.140 says that formaldehyde is one of the four 00:01:35.140 --> 00:01:38.320 most dangerous air pollutants in the country. 00:01:38.320 --> 00:01:41.630 Formaldehyde can make you dizzy and nauseated. 00:01:41.630 --> 00:01:45.610 And if you're exposed to enough of it, it can kill you. 00:01:45.610 --> 00:01:48.880 Indoor air pollutants can also come from Ava's house itself 00:01:48.880 --> 00:01:53.080 in the form of particulates, which are teeny-tiny particles 00:01:53.080 --> 00:01:56.120 so small that they can stay suspended in the air. 00:01:56.120 --> 00:01:58.320 And they can be really dangerous 00:01:58.320 --> 00:02:00.650 because they can travel deep into the lungs 00:02:00.650 --> 00:02:02.480 and damage cells. 00:02:02.480 --> 00:02:05.460 One such particulate is asbestos. 00:02:05.460 --> 00:02:09.260 It's actually, in my opinion, a really cool substance. 00:02:09.260 --> 00:02:11.060 It's an electrical insulator, 00:02:11.060 --> 00:02:14.210 and it's fireproof and it's acid-proof. 00:02:14.210 --> 00:02:16.880 And there was a time when many houses were being built 00:02:16.880 --> 00:02:19.150 with asbestos in its building insulation, 00:02:19.150 --> 00:02:20.830 flooring, and roofing. 00:02:20.830 --> 00:02:24.060 It's actually a natural material that's mined, 00:02:24.060 --> 00:02:27.630 and it has light fluffy fibers that can be woven into cloth. 00:02:27.630 --> 00:02:30.460 To me, it always looked like unicorn hair. 00:02:30.460 --> 00:02:33.400 But really, it's more like evil unicorn hair. 00:02:33.400 --> 00:02:36.080 Each fiber can break into microscopic pieces 00:02:36.080 --> 00:02:37.580 that could scar your lungs 00:02:37.580 --> 00:02:41.050 and lead to lung cancer and other lung diseases. 00:02:41.050 --> 00:02:42.640 Particulates in Ava's house 00:02:42.640 --> 00:02:44.650 could also be coming from the paint. 00:02:44.650 --> 00:02:47.920 Lead paint in Ava's house could begin to chip away 00:02:47.920 --> 00:02:50.920 and could suspend small particles in the air. 00:02:50.920 --> 00:02:53.320 These particles could cause lead poisoning, 00:02:53.320 --> 00:02:56.730 which can cause headaches and nerve and brain damage. 00:02:56.730 --> 00:02:59.430 The use of lead and asbestos has been restricted 00:02:59.430 --> 00:03:01.400 by many governments around the world, 00:03:01.400 --> 00:03:04.363 but these materials can still be found in older buildings. 00:03:05.480 --> 00:03:08.050 Another way inside air can become polluted 00:03:08.050 --> 00:03:11.393 is through combustion, which is to say burning stuff. 00:03:12.240 --> 00:03:14.910 Combustion can cause a wide variety of pollutants 00:03:14.910 --> 00:03:18.670 that can irritate lungs, including carbon monoxide, 00:03:18.670 --> 00:03:22.823 nitrogen oxides, sulfur dioxide, and particulates. 00:03:24.520 --> 00:03:27.900 One way that carbon monoxide can build up inside a building 00:03:27.900 --> 00:03:30.370 is from a poorly-maintained furnace. 00:03:30.370 --> 00:03:33.440 Carbon monoxide molecules have a secret weapon. 00:03:33.440 --> 00:03:35.760 They can trick the proteins in your blood 00:03:35.760 --> 00:03:37.740 into thinking they're oxygen. 00:03:37.740 --> 00:03:39.743 This can cause asphyxiation, 00:03:40.840 --> 00:03:43.520 which is when the body cannot get enough oxygen, 00:03:43.520 --> 00:03:45.470 and it can be deadly. 00:03:45.470 --> 00:03:48.270 Ava should make sure that her furnace is well maintained 00:03:48.270 --> 00:03:50.110 and serviced regularly. 00:03:50.110 --> 00:03:51.860 Combustion-related air pollution 00:03:51.860 --> 00:03:53.950 doesn't just come from furnaces, though. 00:03:53.950 --> 00:03:56.350 In less economically-developed countries, 00:03:56.350 --> 00:03:59.860 families sometimes use open fires for heating and cooking. 00:03:59.860 --> 00:04:04.360 People often burn wood, peat, and even coal inside homes 00:04:04.360 --> 00:04:06.690 without the necessary ventilation to circulate 00:04:06.690 --> 00:04:08.970 and dilute the pollutants in the air. 00:04:08.970 --> 00:04:10.630 Combustion-related air pollution 00:04:10.630 --> 00:04:13.570 can also come from poorly ventilated fireplaces 00:04:13.570 --> 00:04:14.923 and tobacco smoke. 00:04:16.150 --> 00:04:17.910 So, what could Eva do to reduce 00:04:17.910 --> 00:04:19.960 combustion-related air pollution? 00:04:19.960 --> 00:04:22.580 Well, she could open the windows 00:04:22.580 --> 00:04:24.760 and she could use fans to mix the smoke 00:04:24.760 --> 00:04:27.000 and pollution with outside air. 00:04:27.000 --> 00:04:30.030 Ava should also install a carbon monoxide detector 00:04:30.030 --> 00:04:32.130 to make sure that the air is safe. 00:04:32.130 --> 00:04:34.750 A carbon monoxide detector could also detect leaks 00:04:34.750 --> 00:04:36.993 from natural gas stoves or heaters. 00:04:37.930 --> 00:04:41.220 It's also possible that Ava's house has natural pollutants, 00:04:41.220 --> 00:04:44.550 like mold, dust, or even radon. 00:04:44.550 --> 00:04:47.140 Mold is a type of microscopic fungus 00:04:47.140 --> 00:04:49.420 that's always floating around in the air, 00:04:49.420 --> 00:04:52.950 and breathing it in can cause itchy eyes, runny noses, 00:04:52.950 --> 00:04:55.440 and it can trigger asthma attacks. 00:04:55.440 --> 00:04:58.200 And mold loves moisture. 00:04:58.200 --> 00:05:00.100 So, when the air is really humid, 00:05:00.100 --> 00:05:03.520 there will be more mold spores, which is good for the molds, 00:05:03.520 --> 00:05:05.400 but bad for us. 00:05:05.400 --> 00:05:08.240 And what areas of the house tend to have the most moisture? 00:05:08.240 --> 00:05:10.190 The bathrooms and the kitchen, 00:05:10.190 --> 00:05:12.980 so these areas especially need to have windows 00:05:12.980 --> 00:05:15.960 that can be opened or air fans that can draw moisture 00:05:15.960 --> 00:05:18.530 and the molds outta the house. 00:05:18.530 --> 00:05:22.300 Another kind of natural indoor air pollutant is radon, 00:05:22.300 --> 00:05:25.360 which is a radioactive noble gas. 00:05:25.360 --> 00:05:27.110 It's produced by the natural decay 00:05:27.110 --> 00:05:29.443 of radioactive rocks in the ground. 00:05:30.550 --> 00:05:33.380 Let's say that Ava's basement has some cracks 00:05:33.380 --> 00:05:35.530 in the foundation and the walls. 00:05:35.530 --> 00:05:37.550 That would mean that when radon seeps upward 00:05:37.550 --> 00:05:40.210 through the soil, it could enter these cracks 00:05:40.210 --> 00:05:42.920 and be stuck inside Ava's house. 00:05:42.920 --> 00:05:45.730 Breathing in too much radon could damage lung tissue 00:05:45.730 --> 00:05:47.930 and even lead to lung cancer, 00:05:47.930 --> 00:05:50.000 depending on where the house is built. 00:05:50.000 --> 00:05:52.150 Some places, like Ava's house, 00:05:52.150 --> 00:05:54.940 are more likely to have radon than others. 00:05:54.940 --> 00:05:57.970 Ava can prevent radon exposure by sealing up the cracks 00:05:57.970 --> 00:05:59.890 in the foundation of her house 00:05:59.890 --> 00:06:02.460 and by ventilating her basement. 00:06:02.460 --> 00:06:06.050 Some indoor air pollution requires professional remediation. 00:06:06.050 --> 00:06:08.330 To get rid of asbestos, for example, 00:06:08.330 --> 00:06:11.030 an accredited asbestos abatement specialist 00:06:11.030 --> 00:06:14.300 basically shows up in a biohazard suit like this. 00:06:14.300 --> 00:06:16.500 That's how dangerous it is. 00:06:16.500 --> 00:06:18.000 For other situations, 00:06:18.000 --> 00:06:21.810 there's a simple solution to Ava's problem: ventilation. 00:06:21.810 --> 00:06:23.050 Ava could open windows 00:06:23.050 --> 00:06:26.200 on the opposing sides of her home just a crack. 00:06:26.200 --> 00:06:28.000 The outdoor air would flow in, 00:06:28.000 --> 00:06:30.090 dilute the indoor air pollutants, 00:06:30.090 --> 00:06:32.080 and carry them outta the house. 00:06:32.080 --> 00:06:35.260 This natural ventilation allows the air to circulate 00:06:35.260 --> 00:06:38.400 and reduces the buildup of indoor air pollutants. 00:06:38.400 --> 00:06:40.850 There are a lot of ways that indoor air pollutants 00:06:40.850 --> 00:06:43.650 could sneak into Ava's house, from nature, 00:06:43.650 --> 00:06:47.050 from combustion, and from human-made items. 00:06:47.050 --> 00:06:48.820 Being aware of these sources can help Ava 00:06:48.820 --> 00:06:50.490 make sure that the air inside her house 00:06:50.490 --> 00:06:52.653 is fresh, clean, and healthy.

Human impact on aquatic environments

https://www.youtube.com/watch?v=nZjMRq9tev8

vtt

https://www.youtube.com/api/timedtext?v=nZjMRq9tev8&ei=5VWUZeTtFL-ShcIPp9KnoAo&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=C0391BCA17FC11E1913FE3D9918AB5CFD554E795.618164E9D693CF972420124505BF39369A6861CD&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.230 --> 00:00:01.200 - [Narrator] When you go to the beach 00:00:01.200 --> 00:00:04.630 and you look at the ocean, it oftentimes might look fine. 00:00:04.630 --> 00:00:05.910 But as we'll see in this video, 00:00:05.910 --> 00:00:09.210 we, human beings have been stressing aquatic environments. 00:00:09.210 --> 00:00:12.850 And if we're not careful, we might completely ruin them. 00:00:12.850 --> 00:00:17.850 For example, this is what a healthy coral reef looks like. 00:00:18.160 --> 00:00:21.500 And coral are fascinating organisms. 00:00:21.500 --> 00:00:25.050 You can view them as little animals that are fixed in place 00:00:25.050 --> 00:00:28.280 because they're releasing this calcium carbonate 00:00:28.280 --> 00:00:30.430 the same things that eggshells are made of 00:00:30.430 --> 00:00:31.680 that they're fixed too 00:00:31.680 --> 00:00:34.550 and then that builds the coral reef. 00:00:34.550 --> 00:00:37.640 But they can exist, they have their homeostasis 00:00:37.640 --> 00:00:39.900 in a particular temperature range 00:00:39.900 --> 00:00:43.140 and a given amount of various chemicals 00:00:43.140 --> 00:00:45.150 that are in the water. 00:00:45.150 --> 00:00:47.270 Now we know that human beings 00:00:47.270 --> 00:00:49.300 were causing the climate to warm, 00:00:49.300 --> 00:00:52.570 and that's all also causing ocean temperatures to warm 00:00:52.570 --> 00:00:54.640 and as ocean temperatures warm, 00:00:54.640 --> 00:00:59.320 many of our coral reefs are not able to be as sustainable. 00:00:59.320 --> 00:01:03.410 For example, this is a less healthy coral reef. 00:01:03.410 --> 00:01:07.020 Some of the coral is still alive and seemingly doing okay, 00:01:07.020 --> 00:01:10.790 but in other places, you see what is known as bleaching, 00:01:10.790 --> 00:01:12.830 where the reef is now white 00:01:12.830 --> 00:01:15.030 and that's because in those situations, 00:01:15.030 --> 00:01:17.380 the coral is essentially dying off. 00:01:17.380 --> 00:01:19.010 Now you might say, well, this is unfortunate 00:01:19.010 --> 00:01:21.490 because coral reefs are very beautiful 00:01:21.490 --> 00:01:23.760 and now it is less beautiful. 00:01:23.760 --> 00:01:25.550 But what are the other impacts? 00:01:25.550 --> 00:01:28.240 Well, all sorts of organisms and animals 00:01:28.240 --> 00:01:31.000 live in coral reefs, get their food from it, 00:01:31.000 --> 00:01:32.150 get their shelter from it 00:01:32.150 --> 00:01:33.780 and if the coral reef start to die off, 00:01:33.780 --> 00:01:35.630 then the animals are going to die off. 00:01:35.630 --> 00:01:37.490 On top of that, these coral reefs 00:01:37.490 --> 00:01:40.700 that the coral are essentially building as they live, 00:01:40.700 --> 00:01:43.200 they prevent erosion on the coastlines. 00:01:43.200 --> 00:01:45.580 So one change that affects one organism 00:01:45.580 --> 00:01:47.180 or one part of an ecosystem 00:01:47.180 --> 00:01:49.260 can have a lot of follow on effects 00:01:49.260 --> 00:01:52.080 on other parts of the ecosystem. 00:01:52.080 --> 00:01:54.130 Another perhaps more obvious way 00:01:54.130 --> 00:01:57.580 that we've been not being nice to aquatic environments 00:01:57.580 --> 00:01:59.820 is things like oil spills. 00:01:59.820 --> 00:02:01.650 And you've probably seen this on the news 00:02:01.650 --> 00:02:03.220 when you have major oil spills, 00:02:03.220 --> 00:02:05.350 they tend to be pretty disturbing images, 00:02:05.350 --> 00:02:07.820 but this right here is a bird that is covered in oil. 00:02:07.820 --> 00:02:10.420 And you can imagine when a bird is covered in oil, 00:02:10.420 --> 00:02:12.250 it's not going to be able to fly, 00:02:12.250 --> 00:02:13.760 it's not going to be able to swim, 00:02:13.760 --> 00:02:15.450 it's not going to be able to have food 00:02:15.450 --> 00:02:18.110 and in a lot of cases, it is likely to die. 00:02:18.110 --> 00:02:20.350 And just as we described at the coral reef, 00:02:20.350 --> 00:02:22.030 this doesn't just affect the bird, 00:02:22.030 --> 00:02:25.610 it affects the entire ecosystem, including human beings. 00:02:25.610 --> 00:02:27.500 And this is just the effect on a bird, 00:02:27.500 --> 00:02:28.630 it has affects on the fish, 00:02:28.630 --> 00:02:31.770 it has effects on the just natural balance 00:02:31.770 --> 00:02:33.833 that occurs in that aquatic environment. 00:02:34.670 --> 00:02:38.380 Now, another idea that is less talked about 00:02:38.380 --> 00:02:42.240 is this notion of oceanic dead zones. 00:02:42.240 --> 00:02:46.210 And this right over here is a picture of the gulf coast 00:02:46.210 --> 00:02:48.010 right off the coast of Louisiana, 00:02:48.010 --> 00:02:51.420 I was actually born right around there. 00:02:51.420 --> 00:02:55.300 And what it shows is every year, this hypoxic zone, 00:02:55.300 --> 00:02:59.210 which is a zone of low oxygen levels in the water form 00:02:59.210 --> 00:03:01.350 off the coast, and this shows how bad it is. 00:03:01.350 --> 00:03:04.890 The red areas are the really bad, very low oxygen. 00:03:04.890 --> 00:03:06.990 Sometimes you might forget, and you might say, okay, 00:03:06.990 --> 00:03:10.080 for all of us who live in the land or live on the land 00:03:10.080 --> 00:03:12.140 or in the air, we breathe oxygen, 00:03:12.140 --> 00:03:15.010 but organisms in the water need oxygen as well. 00:03:15.010 --> 00:03:17.270 Oxygen that has been dissolved in the water. 00:03:17.270 --> 00:03:19.430 And what's interesting is why this forms, 00:03:19.430 --> 00:03:21.720 it's actually a little bit counter-intuitive. 00:03:21.720 --> 00:03:24.640 It turns out that chemicals from human runoff, 00:03:24.640 --> 00:03:25.710 especially fertilizer. 00:03:25.710 --> 00:03:27.230 So you might not realize it, 00:03:27.230 --> 00:03:29.460 but this is the Delta of the Mississippi river. 00:03:29.460 --> 00:03:31.860 And that has run off farm runoff 00:03:31.860 --> 00:03:35.650 from as far north as Minnesota and Chicago and sewage 00:03:35.650 --> 00:03:39.010 and that farm runoff and that fertilizer and that sewage, 00:03:39.010 --> 00:03:41.350 as it comes into the Gulf, 00:03:41.350 --> 00:03:44.180 it actually promotes algae formation. 00:03:44.180 --> 00:03:45.470 And you might say, well, that's good. 00:03:45.470 --> 00:03:48.890 Some more life is growing, but so much algae gets formed 00:03:48.890 --> 00:03:52.570 and when that algae finally dies and it gets decomposed, 00:03:52.570 --> 00:03:55.110 the decomposers actually use the oxygen. 00:03:55.110 --> 00:03:58.130 Remember, when you are actually trying to metabolize things, 00:03:58.130 --> 00:04:01.180 you're using oxygen in order to extract that energy. 00:04:01.180 --> 00:04:04.710 And so the oxygen in the ocean in that area gets depleted, 00:04:04.710 --> 00:04:06.180 and then you have a situation 00:04:06.180 --> 00:04:08.460 where almost nothing can live 00:04:08.460 --> 00:04:10.480 in these zones right over here. 00:04:10.480 --> 00:04:12.450 And this is just a sample of the things 00:04:12.450 --> 00:04:14.270 that we are doing to our aquatic environment. 00:04:14.270 --> 00:04:15.460 And there's other things, 00:04:15.460 --> 00:04:17.760 there's elemental sources of mercury 00:04:17.760 --> 00:04:19.460 that we throw into aquatic environments, 00:04:19.460 --> 00:04:21.660 and it makes the water highly toxic. 00:04:21.660 --> 00:04:24.670 There's obviously other forms of trash, pollution, 00:04:24.670 --> 00:04:25.840 that we've put into the water 00:04:25.840 --> 00:04:27.210 but this is just to give you a sense 00:04:27.210 --> 00:04:29.210 and to start giving you an appreciation 00:04:29.210 --> 00:04:31.410 about how imbalanced everything is 00:04:31.410 --> 00:04:35.270 and how in one part of the country say in Chicago 00:04:35.270 --> 00:04:37.870 waste that's going into the Mississippi River 00:04:37.870 --> 00:04:41.113 can affect aquatic environments, thousands of miles away.

Dividing complex numbers in polar form

https://www.youtube.com/watch?v=lyWaNZ1ERMw

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https://www.youtube.com/api/timedtext?v=lyWaNZ1ERMw&ei=5VWUZby6HOW4mLAP6sWjqAg&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=17EF6E7759197FFC78B04BF48BF799729DDA5E5F.1AE623BBC8E6CA6533EB7DDF18002A5BB8A284FC&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.330 --> 00:00:02.620 - [Narrator] So we are given these two complex numbers 00:00:02.620 --> 00:00:03.453 and we want to know 00:00:03.453 --> 00:00:07.010 what W sub one divided by W sub two is. 00:00:07.010 --> 00:00:09.760 So pause this video and see if you can figure that out. 00:00:10.620 --> 00:00:13.760 All right, now let's work through this together. 00:00:13.760 --> 00:00:15.760 So the form that they've written this in 00:00:15.760 --> 00:00:18.080 it actually makes it pretty straightforward 00:00:18.080 --> 00:00:20.270 to spot the modulus 00:00:20.270 --> 00:00:23.670 and the argument of each of these complex numbers. 00:00:23.670 --> 00:00:25.920 The modulus of W sub one 00:00:25.920 --> 00:00:28.780 we can see out here is equal to eight. 00:00:28.780 --> 00:00:30.180 And the argument 00:00:30.180 --> 00:00:31.870 of W sub one 00:00:31.870 --> 00:00:32.703 we can see 00:00:32.703 --> 00:00:34.960 is four Pi over three 00:00:34.960 --> 00:00:36.760 if we're thinking in terms of radians. 00:00:36.760 --> 00:00:41.360 So four Pi over three radians, and then similarly 00:00:41.360 --> 00:00:43.620 for W sub two 00:00:43.620 --> 00:00:45.060 its modulus 00:00:45.060 --> 00:00:46.570 is equal to 00:00:46.570 --> 00:00:47.403 two 00:00:47.403 --> 00:00:49.620 and its argument 00:00:49.620 --> 00:00:50.810 is equal to 00:00:50.810 --> 00:00:53.163 seven Pi over six. 00:00:54.230 --> 00:00:55.800 Seven 00:00:55.800 --> 00:00:56.633 Pi 00:00:57.590 --> 00:00:59.250 over six. 00:00:59.250 --> 00:01:00.730 Now, in many videos 00:01:00.730 --> 00:01:02.320 we have talked about when you multiply 00:01:02.320 --> 00:01:04.220 one complex number by another 00:01:04.220 --> 00:01:05.780 you're essentially transforming it. 00:01:05.780 --> 00:01:08.680 So you are going to scale the modulus of one 00:01:08.680 --> 00:01:10.030 by the modulus of the other. 00:01:10.030 --> 00:01:12.500 And you're going to rotate the argument of one 00:01:12.500 --> 00:01:14.780 by the argument of the other, I guess you could say 00:01:14.780 --> 00:01:16.930 you're going to add the angles. 00:01:16.930 --> 00:01:18.840 So another way to think about it is 00:01:18.840 --> 00:01:23.840 if you have the modulus of W sub one divided by W sub two. 00:01:23.840 --> 00:01:26.810 Well then you're just going to divide these moduli here. 00:01:26.810 --> 00:01:29.550 So this is just going to be eight over two 00:01:29.550 --> 00:01:31.380 which is equal to four. 00:01:31.380 --> 00:01:33.340 And then the argument 00:01:33.340 --> 00:01:34.610 of 00:01:34.610 --> 00:01:35.460 W 00:01:35.460 --> 00:01:36.660 sub one 00:01:36.660 --> 00:01:38.900 over W sub two. 00:01:38.900 --> 00:01:42.360 This is, you could imagine you're starting at W sub one 00:01:42.360 --> 00:01:46.260 and then you are going to rotate it clockwise 00:01:46.260 --> 00:01:48.540 by W sub two's argument. 00:01:48.540 --> 00:01:51.330 And so this is going to be four Pi 00:01:51.330 --> 00:01:52.310 over three 00:01:52.310 --> 00:01:53.300 minus 00:01:53.300 --> 00:01:54.970 seven Pi 00:01:54.970 --> 00:01:55.920 over six. 00:01:55.920 --> 00:01:57.230 And let's see what this is going to be. 00:01:57.230 --> 00:02:00.440 If we have a common denominator four Pi over three 00:02:00.440 --> 00:02:04.010 is the same thing as eight Pi over six 00:02:04.010 --> 00:02:06.080 minus seven Pi 00:02:06.080 --> 00:02:07.260 over six 00:02:07.260 --> 00:02:10.550 which is going to be equal to Pi over six. 00:02:10.550 --> 00:02:12.340 And so we could write this, 00:02:12.340 --> 00:02:13.410 the 00:02:13.410 --> 00:02:14.560 quotient 00:02:14.560 --> 00:02:17.110 W one divided by W two 00:02:17.110 --> 00:02:18.260 is going to be equal to 00:02:18.260 --> 00:02:19.700 if we wanted to write it in this form 00:02:19.700 --> 00:02:22.140 its modulus is equal to four. 00:02:22.140 --> 00:02:23.720 It's going to be four times 00:02:23.720 --> 00:02:24.930 cosine 00:02:24.930 --> 00:02:26.870 of Pi over six 00:02:26.870 --> 00:02:27.890 plus 00:02:27.890 --> 00:02:28.740 i 00:02:28.740 --> 00:02:30.330 times sine 00:02:30.330 --> 00:02:32.090 of Pi 00:02:32.090 --> 00:02:33.960 over six. 00:02:33.960 --> 00:02:36.950 Now cosine of Pi over six, we can figure out 00:02:36.950 --> 00:02:40.280 Pi over six is the same thing as a 30 degree angle. 00:02:40.280 --> 00:02:42.270 And so the cosine of that 00:02:42.270 --> 00:02:44.160 is square root of three over two 00:02:44.160 --> 00:02:46.060 square root three over two. 00:02:46.060 --> 00:02:48.930 And the sine of Pi over six 00:02:48.930 --> 00:02:52.120 we know from our 30, 60, 90 triangles 00:02:52.120 --> 00:02:53.720 is going to be one half. 00:02:53.720 --> 00:02:55.380 So this is one half. 00:02:55.380 --> 00:02:56.980 And so if you distribute this four 00:02:56.980 --> 00:02:57.990 this is going to be equal to 00:02:57.990 --> 00:02:59.960 four times square root of three over two 00:02:59.960 --> 00:03:02.290 is two square roots of three 00:03:02.290 --> 00:03:04.520 and then four times one half is two. 00:03:04.520 --> 00:03:05.890 So plus 00:03:05.890 --> 00:03:06.980 two 00:03:06.980 --> 00:03:07.900 i 00:03:07.900 --> 00:03:09.453 and we are done.

Multiplying complex numbers in polar form

https://www.youtube.com/watch?v=VkdXztTFsvM

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https://www.youtube.com/api/timedtext?v=VkdXztTFsvM&ei=5VWUZY7_G8Gyp-oP_ZOw4Aw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=2DC465D35C8D145F8DF600FAB74837F4DF3BDBCB.4E9F3038030B5A55460C1DAA0081E45538E44091&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.420 --> 00:00:02.480 - [Instructor] We're given two different complex numbers 00:00:02.480 --> 00:00:05.310 here, and we want to figure out what is the product? 00:00:05.310 --> 00:00:08.610 Pause this video and see if you can figure that out. 00:00:08.610 --> 00:00:11.910 All right, now let's work on this together. 00:00:11.910 --> 00:00:14.550 So we know from the form that it's written here 00:00:14.550 --> 00:00:18.810 that the modulus of w sub one is equal to three. 00:00:18.810 --> 00:00:20.620 And we know that the argument 00:00:20.620 --> 00:00:24.683 of w sub one is equal to 330 degrees. 00:00:25.700 --> 00:00:27.830 And by the same line of reasoning, we know 00:00:27.830 --> 00:00:31.610 that the modulus of w sub two is equal to two. 00:00:31.610 --> 00:00:35.650 And that the argument of w sub two 00:00:35.650 --> 00:00:37.720 is going to be equal to, 00:00:37.720 --> 00:00:42.300 we can see that right over here, 120 degrees. 00:00:42.300 --> 00:00:44.250 Now, when you multiply complex numbers 00:00:44.250 --> 00:00:46.930 you could view as one transforming the other. 00:00:46.930 --> 00:00:49.150 We've seen this in multiple examples. 00:00:49.150 --> 00:00:52.600 So let's imagine that we are transforming w two 00:00:52.600 --> 00:00:55.200 by multiplying it by w one. 00:00:55.200 --> 00:00:56.740 So what is going to happen? 00:00:56.740 --> 00:00:58.200 Well, let me write it here. 00:00:58.200 --> 00:01:03.200 So what's the resulting modulus of w one times w two? 00:01:03.230 --> 00:01:05.620 Well, we're just going to scale up w two's modulus 00:01:05.620 --> 00:01:07.410 by w one's modulus. 00:01:07.410 --> 00:01:09.320 Or essentially we're just going to multiply the two. 00:01:09.320 --> 00:01:12.850 So this is going to be equal to six, three times two. 00:01:12.850 --> 00:01:17.850 And then the argument of w sub one times w sub two, 00:01:18.210 --> 00:01:21.800 if we start at w sub two's argument, 120 degrees 00:01:21.800 --> 00:01:25.120 and then we rotate it by w sub one's argument, 00:01:25.120 --> 00:01:27.270 well then you're going to add these two angles, 00:01:27.270 --> 00:01:30.350 that gets you to 450 degrees. 00:01:30.350 --> 00:01:32.380 So this is equal to 450 degrees, 00:01:32.380 --> 00:01:34.320 which is more than a complete rotation. 00:01:34.320 --> 00:01:36.130 And so if we wanted to give it an angle 00:01:36.130 --> 00:01:38.860 between zero and 360 degrees, 00:01:38.860 --> 00:01:41.260 if we just subtract 360 from that, 00:01:41.260 --> 00:01:45.140 that is going to be equal to 90 degrees. 00:01:45.140 --> 00:01:48.060 And so we can rewrite this here, 00:01:48.060 --> 00:01:49.890 or we can rewrite the product 00:01:49.890 --> 00:01:54.200 as w sub one times w sub two is equal 00:01:54.200 --> 00:01:59.200 to its modulus six times cosine of its argument, 90 degrees. 00:02:01.170 --> 00:02:05.980 Plus i times sine of its argument. 00:02:05.980 --> 00:02:10.210 Now we know what the cosine and sine of 90 degrees is. 00:02:10.210 --> 00:02:13.830 Cosine of 90 degrees is equal to zero 00:02:13.830 --> 00:02:16.700 and sine of 90 degrees is equal to one. 00:02:16.700 --> 00:02:18.590 So all of the simplifies quite nicely. 00:02:18.590 --> 00:02:21.430 All you're left with is a six times I. 00:02:21.430 --> 00:02:26.123 So this is equal to six i, and we are done.

Multiplying complex numbers graphically example: -1-i

https://www.youtube.com/watch?v=ebEwF4kb6pI

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https://www.youtube.com/api/timedtext?v=ebEwF4kb6pI&ei=5VWUZbHlGuKAp-oP3dKD4Ao&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=A769F761C0C6C55CBB7C49DD6D43028F4CE4528D.59D81FE9228481530F3DD5C980A7487A56E0F904&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.150 --> 00:00:02.420 - [Instructor] We are told, suppose we multiply 00:00:02.420 --> 00:00:06.380 a complex number Z by -1-i. 00:00:07.460 --> 00:00:09.440 So this is Z right over here. 00:00:09.440 --> 00:00:13.740 Which point represents the product of Z and -1-I? 00:00:16.550 --> 00:00:19.150 Pause this video and see if you can figure that out. 00:00:20.320 --> 00:00:23.350 All right, now let's work through this together. 00:00:23.350 --> 00:00:26.490 So the way I think about this is, 00:00:26.490 --> 00:00:30.660 when you multiply by a complex number, 00:00:30.660 --> 00:00:33.680 you are going to rotate by the argument 00:00:33.680 --> 00:00:35.260 of that complex number. 00:00:35.260 --> 00:00:38.640 And you're going to scale the modulus of Z 00:00:38.640 --> 00:00:42.070 by the modulus of this complex number. 00:00:42.070 --> 00:00:44.160 Now, let me just think about that a little bit. 00:00:44.160 --> 00:00:47.163 So I'm gonna draw another complex plane here. 00:00:48.240 --> 00:00:50.770 And so this is my real axis, 00:00:50.770 --> 00:00:55.053 this is my imaginary axis, right over here. 00:00:56.740 --> 00:00:58.790 And -1-I, so that's -1 and then minus 1i. 00:01:03.580 --> 00:01:05.480 So it would go right over there. 00:01:05.480 --> 00:01:07.870 It would be that right over here. 00:01:07.870 --> 00:01:10.390 And so let's think about two things. 00:01:10.390 --> 00:01:12.710 Let's think about what its argument is, 00:01:12.710 --> 00:01:16.090 and let's think about what it's modulus is. 00:01:16.090 --> 00:01:20.810 So its argument is going to be this angle right over here. 00:01:20.810 --> 00:01:23.090 And you might already recognize 00:01:23.090 --> 00:01:26.190 that if this has a length of one, 00:01:26.190 --> 00:01:28.340 if this has a length of one, 00:01:28.340 --> 00:01:29.360 or another way of thinking about, 00:01:29.360 --> 00:01:34.360 this has a length of one, this is a 45, 45, 90 triangle. 00:01:35.290 --> 00:01:37.340 So this is 45 degrees 00:01:37.340 --> 00:01:40.400 but then of course you have this 180 before that. 00:01:40.400 --> 00:01:45.400 So that's going to be 180 plus 45, is a 225 degree argument. 00:01:47.260 --> 00:01:52.260 So the argument here is going to be equal to 225 degrees. 00:01:53.740 --> 00:01:55.230 So when you multiply by this, 00:01:55.230 --> 00:01:58.740 you are going to rotate by 225 degrees. 00:01:58.740 --> 00:02:02.380 So let's see this is going to be rotating by 180 degrees 00:02:02.380 --> 00:02:04.370 and then another 45. 00:02:04.370 --> 00:02:06.467 So if you just rotate it by that, 00:02:06.467 --> 00:02:08.743 you would end up right over here. 00:02:09.890 --> 00:02:12.740 Now we also are going to scale the modulus. 00:02:12.740 --> 00:02:16.350 And you can see two choices that scale that modulus. 00:02:16.350 --> 00:02:19.090 And so we know it's going to be choice A or choice B 00:02:19.090 --> 00:02:22.300 because choices C or D you'd have to rotate more 00:02:22.300 --> 00:02:23.630 to get over there. 00:02:23.630 --> 00:02:24.960 And so to think about that, 00:02:24.960 --> 00:02:27.270 we have to just think about the modulus 00:02:27.270 --> 00:02:31.040 of -1-i, this point right over here 00:02:31.040 --> 00:02:35.130 and then just scale up this modulus by that same amount. 00:02:35.130 --> 00:02:36.910 Well, the modulus is just the distance 00:02:36.910 --> 00:02:39.000 from zero in the complex plane. 00:02:39.000 --> 00:02:41.380 So it's going to be this distance right over here. 00:02:41.380 --> 00:02:43.130 And you could use the Pythagorean theorem 00:02:43.130 --> 00:02:46.370 to know that this squared, if you call this C, 00:02:46.370 --> 00:02:49.750 C squared is equal to one squared plus one squared 00:02:49.750 --> 00:02:51.630 or C squared is equal to two 00:02:51.630 --> 00:02:54.540 or C is equal to the square root of two. 00:02:54.540 --> 00:02:56.840 So that's the modulus right over here. 00:02:56.840 --> 00:03:00.670 Modulus is equal to square root of two 00:03:00.670 --> 00:03:03.550 which is approximately, it's a little bit more than 1.4. 00:03:03.550 --> 00:03:05.550 So let's just call it approximately 1.4. 00:03:06.950 --> 00:03:10.720 So not only going to rotate by 225 degrees, 00:03:10.720 --> 00:03:12.640 we're going to scale the modulus, 00:03:12.640 --> 00:03:14.440 the distance from the origin by 1.4. 00:03:15.347 --> 00:03:18.630 So it looks like it's three units 00:03:18.630 --> 00:03:20.550 from the origin right over here. 00:03:20.550 --> 00:03:22.480 If you multiply that by 1.4, 00:03:22.480 --> 00:03:27.120 three times 1.4 is about four, or it is exactly 4.2. 00:03:27.120 --> 00:03:31.030 So 4.2 of these units is one, two, three, four, 00:03:31.030 --> 00:03:34.000 a little bit further, you get right over here 00:03:34.000 --> 00:03:36.273 to choice B and we're done.

Multiplying complex numbers graphically example: -3i

https://www.youtube.com/watch?v=fqwR6RNPJgc

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https://www.youtube.com/api/timedtext?v=fqwR6RNPJgc&ei=5VWUZZKyJp--mLAPruG14A4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=D01D652FD535E234BE2FB7E89777F8CBF0EFF1E0.53DE2F99A551DE2AEB725AF2C7D9EC187739F336&key=yt8&lang=en&name=Default&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:00.000 --> 00:00:02.460 - [Instructor] Suppose we multiply a complex number Z 00:00:02.460 --> 00:00:04.130 by negative three i, 00:00:04.130 --> 00:00:06.600 and there shows Z right over here 00:00:06.600 --> 00:00:09.540 Plot the point that represents the product of Z 00:00:09.540 --> 00:00:11.563 and negative three i. 00:00:11.563 --> 00:00:15.308 I so pause this video and see if you can work through that. 00:00:15.308 --> 00:00:17.800 All right, now let's do it step-by-step. 00:00:17.800 --> 00:00:20.290 First I wanna think about what would, 00:00:20.290 --> 00:00:22.123 where would three Z be? 00:00:23.340 --> 00:00:28.050 Well, three Z would have the same angle as Z, 00:00:28.050 --> 00:00:30.010 but it's absolute value 00:00:30.010 --> 00:00:32.920 or it's modulus would be three times larger. 00:00:32.920 --> 00:00:34.440 So you'd be going in this direction 00:00:34.440 --> 00:00:35.610 but it'd be three times further. 00:00:35.610 --> 00:00:37.860 So that's one times it's modulus. 00:00:37.860 --> 00:00:38.802 That's two times it's modulus. 00:00:38.802 --> 00:00:40.580 That's three times it's modulus 00:00:40.580 --> 00:00:42.800 or it's three times it's absolute value. 00:00:42.800 --> 00:00:45.820 So three Z would be right over here. 00:00:45.820 --> 00:00:50.060 Now what about negative three Z? 00:00:50.060 --> 00:00:51.917 Well, if you multiply it by a negative, 00:00:51.917 --> 00:00:53.385 it's just going to flip it around. 00:00:53.385 --> 00:00:56.920 You can think about it as flipping it at 180 degrees 00:00:56.920 --> 00:00:59.730 but it's going to have the same modulus. 00:00:59.730 --> 00:01:01.400 So instead of being right over here, 00:01:01.400 --> 00:01:02.540 at three in this direction, 00:01:02.540 --> 00:01:05.190 it's going to be one, two, three in this direction, 00:01:05.190 --> 00:01:06.023 right over here. 00:01:06.023 --> 00:01:08.860 So that is negative three Z. 00:01:08.860 --> 00:01:11.100 And now perhaps most interestingly, 00:01:11.100 --> 00:01:13.830 what happens when you multiply it by i? 00:01:13.830 --> 00:01:17.940 So if we have negative three i times Z, 00:01:17.940 --> 00:01:21.010 now which is exactly what they want us to figure out. 00:01:21.010 --> 00:01:23.000 Well, let's think about what happens if you had one 00:01:23.000 --> 00:01:24.830 and if you multiply that by i. 00:01:24.830 --> 00:01:26.900 So one times i becomes one i. 00:01:26.900 --> 00:01:28.930 So it goes over there. 00:01:28.930 --> 00:01:32.060 What if you then took one i and multiplied it by i? 00:01:32.060 --> 00:01:33.750 Well, then you have negative one. 00:01:33.750 --> 00:01:35.625 What if you took negative one and you multiplied it by i? 00:01:35.625 --> 00:01:40.360 Well, then now you have negative one i. 00:01:40.360 --> 00:01:42.790 So notice every time we multiply by i, 00:01:42.790 --> 00:01:45.550 we are rotating by 90 degrees. 00:01:45.550 --> 00:01:47.669 So over here, if we take negative three Z 00:01:47.669 --> 00:01:52.390 and multiply it by i, you're just going to rotate 90 degrees 00:01:52.390 --> 00:01:55.040 and you're going to get right over there. 00:01:55.040 --> 00:01:58.930 So this is negative three i times Z, 00:01:58.930 --> 00:02:01.373 which is exactly what we were looking for.

Converting a complex number from polar to rectangular form

https://www.youtube.com/watch?v=auywa7dydAk

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WEBVTT Kind: captions Language: en 00:00:00.200 --> 00:00:03.840 - [Instructor] We are told, consider the complex number Z, 00:00:03.840 --> 00:00:06.900 which is equal to the square root of 17 00:00:06.900 --> 00:00:11.900 times cosine of 346 degrees plus I sine of 346 degrees. 00:00:13.530 --> 00:00:17.290 And they ask us to plot Z in the complex plane below. 00:00:17.290 --> 00:00:20.520 If necessary round the points coordinates 00:00:20.520 --> 00:00:22.290 to the nearest integer. 00:00:22.290 --> 00:00:23.830 So, I encourage you to pause this video 00:00:23.830 --> 00:00:24.663 and at least think about 00:00:24.663 --> 00:00:29.130 where we would likely plot this complex number. 00:00:29.130 --> 00:00:29.963 All right. 00:00:29.963 --> 00:00:31.410 Now let's work through it together. 00:00:31.410 --> 00:00:32.900 So when you look at it like this, 00:00:32.900 --> 00:00:34.910 you can see that what's being attempted 00:00:34.910 --> 00:00:39.910 is a conversion from polar form to rectangular form. 00:00:40.240 --> 00:00:42.820 And if we're thinking about polar form, 00:00:42.820 --> 00:00:47.100 we can think about the angle of this complex number, 00:00:47.100 --> 00:00:49.840 which is clearly 346 degrees. 00:00:49.840 --> 00:00:51.870 And so, 346 degrees 00:00:52.850 --> 00:00:57.180 is about 14 degrees short of a full circle. 00:00:57.180 --> 00:01:01.880 So, it would get us probably something around there. 00:01:01.880 --> 00:01:06.580 And then we also see what the magnitude 00:01:06.580 --> 00:01:09.500 or the modulus of the complex number is right over here. 00:01:09.500 --> 00:01:11.000 Square root of 17. 00:01:11.000 --> 00:01:14.470 Square root of 17 is a little bit more than four 00:01:14.470 --> 00:01:16.250 'cause four squared is 16. 00:01:16.250 --> 00:01:17.630 So if we go in this direction, 00:01:17.630 --> 00:01:22.043 let's see, that's gonna be about one, two, three, four. 00:01:23.060 --> 00:01:24.630 We're gonna go right about there. 00:01:24.630 --> 00:01:26.880 So, if I were to just guess 00:01:26.880 --> 00:01:28.580 where this is going to put us, 00:01:28.580 --> 00:01:30.920 it's going to put us right around here, 00:01:30.920 --> 00:01:34.450 right around four minus one I. 00:01:34.450 --> 00:01:36.040 But let's actually (indistinct) get a calculator out 00:01:36.040 --> 00:01:40.830 and see if this evaluates to roughly four minus one I. 00:01:40.830 --> 00:01:45.830 So for the real part, let's go 346 degrees. 00:01:46.250 --> 00:01:49.200 And we're gonna take the cosine of it. 00:01:49.200 --> 00:01:50.650 And then we're gonna multiply that 00:01:50.650 --> 00:01:52.940 times the square root of 17. 00:01:52.940 --> 00:01:57.940 So times 17 square root, a little over four, 00:01:58.020 --> 00:01:59.340 which is equal to that. 00:01:59.340 --> 00:02:00.173 Actually, yes. 00:02:00.173 --> 00:02:02.390 The real part does look almost exactly four. 00:02:02.390 --> 00:02:05.050 Especially, if we are rounding to the nearest integer. 00:02:05.050 --> 00:02:06.620 It's a little bit more than four. 00:02:06.620 --> 00:02:09.200 And now let's do the imaginary part. 00:02:09.200 --> 00:02:13.100 So we have 346 degrees. 00:02:13.100 --> 00:02:15.400 And we're gonna take the sine of it. 00:02:15.400 --> 00:02:18.800 And we're going to multiply that times the square root of 17 00:02:18.800 --> 00:02:23.800 times 17 square root, which is equal to, yep. 00:02:24.090 --> 00:02:25.840 If we were round to the nearest integer, 00:02:25.840 --> 00:02:27.960 it's about negative one. 00:02:27.960 --> 00:02:30.820 So, we get to this point right over here, 00:02:30.820 --> 00:02:35.820 which is approximately four minus I. 00:02:36.330 --> 00:02:37.963 And we are done.

Energy Conservation

https://www.youtube.com/watch?v=GSc5zo4WjJs

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WEBVTT Kind: captions Language: en 00:00:00.290 --> 00:00:01.123 - [Narrator] In this video, 00:00:01.123 --> 00:00:04.360 we're going to talk about energy conservation 00:00:04.360 --> 00:00:09.310 or trying to save or lower the amount of energy that we use. 00:00:09.310 --> 00:00:11.350 Now, a lot of y'all might already have a sense 00:00:11.350 --> 00:00:12.320 that that is a good thing 00:00:12.320 --> 00:00:13.950 while others of you might say, 00:00:13.950 --> 00:00:17.710 hey why can't I just use as much energy as possible? 00:00:17.710 --> 00:00:20.060 Why should I try to use less energy? 00:00:20.060 --> 00:00:22.690 And there's several answers to that question. 00:00:22.690 --> 00:00:26.410 The number one reason is energy has a huge impact 00:00:26.410 --> 00:00:27.730 on the environment. 00:00:27.730 --> 00:00:29.510 Most of the energy we produce. 00:00:29.510 --> 00:00:31.270 And we'll talk about that in a little bit 00:00:31.270 --> 00:00:35.270 has the by-product of emitting greenhouse gases 00:00:35.270 --> 00:00:38.270 which are contributing to global warming. 00:00:38.270 --> 00:00:41.300 And then on top of that, it has implications to society 00:00:41.300 --> 00:00:43.140 the infrastructure around us. 00:00:43.140 --> 00:00:45.380 And it has frankly impacts on your 00:00:45.380 --> 00:00:48.760 and your family's pocketbooks because energy costs money. 00:00:48.760 --> 00:00:51.830 So let's first think about the household. 00:00:51.830 --> 00:00:53.600 So this chart right over here, 00:00:53.600 --> 00:00:57.600 which is from the U.S. Energy Information Administration 00:00:57.600 --> 00:01:02.230 says residential site electricity consumption by end use. 00:01:02.230 --> 00:01:03.780 And I want to stress that it says 00:01:03.780 --> 00:01:06.710 electricity consumption because electricity consumption 00:01:06.710 --> 00:01:10.870 is not the only energy consumption in a household. 00:01:10.870 --> 00:01:12.880 Depending on where you are it might be around 00:01:12.880 --> 00:01:15.330 50 or 60% of consumption 00:01:15.330 --> 00:01:18.190 but natural gas is another major source. 00:01:18.190 --> 00:01:20.400 And in some cases even petroleum. 00:01:20.400 --> 00:01:21.620 But when you look over here, 00:01:21.620 --> 00:01:26.220 the major uses of energy in a household are air conditioning 00:01:26.220 --> 00:01:27.980 and that's going to be especially pronounced 00:01:27.980 --> 00:01:31.290 if you live in a hot and humid part of the country 00:01:31.290 --> 00:01:32.650 or part of the world. 00:01:32.650 --> 00:01:33.730 You have space heating 00:01:33.730 --> 00:01:35.970 which is what most of us associate with heaters. 00:01:35.970 --> 00:01:37.460 And then you have water heating. 00:01:37.460 --> 00:01:39.200 Many of y'all probably don't appreciate 00:01:39.200 --> 00:01:42.080 when you take those long hot showers 00:01:42.080 --> 00:01:44.810 that it took energy to warm up that water. 00:01:44.810 --> 00:01:45.880 And then you have lighting. 00:01:45.880 --> 00:01:48.300 You have appliances after that. 00:01:48.300 --> 00:01:51.430 So as you as an individual wanted to conserve energy 00:01:51.430 --> 00:01:55.920 it makes a lot of sense to look at things like this. 00:01:55.920 --> 00:01:58.210 And so if you wanna conserve energy at home 00:01:58.210 --> 00:02:01.070 use less air conditioning if you can. 00:02:01.070 --> 00:02:03.560 Use less heating if you can. 00:02:03.560 --> 00:02:06.720 Take shorter showers or maybe not as hot showers. 00:02:06.720 --> 00:02:09.550 And showers not only have the energy consumption 00:02:09.550 --> 00:02:10.970 from heating the water 00:02:10.970 --> 00:02:12.990 but it also has the energy consumption that's happening 00:02:12.990 --> 00:02:15.740 at the water treatment plant to clean your water 00:02:15.740 --> 00:02:18.150 and to process your water that also takes energy. 00:02:18.150 --> 00:02:20.480 And right over here is a refrigerator. 00:02:20.480 --> 00:02:22.680 And if you have an older refrigerator 00:02:22.680 --> 00:02:24.600 or a less efficient refrigerator 00:02:24.600 --> 00:02:27.410 that's going to use a lot more energy to do the same work. 00:02:27.410 --> 00:02:29.880 And once again, it's not just impact on the climate. 00:02:29.880 --> 00:02:32.140 It's going to save you and your family money 00:02:32.140 --> 00:02:34.460 by using less energy. 00:02:34.460 --> 00:02:35.780 But as we'll see, 00:02:35.780 --> 00:02:38.066 energy consumption is not just a phenomenon 00:02:38.066 --> 00:02:40.270 inside of the house. 00:02:40.270 --> 00:02:43.360 What we see here is U.S. energy consumption 00:02:43.360 --> 00:02:46.570 by source and sector in 2020. 00:02:46.570 --> 00:02:49.100 And this is a little bit of a complex diagram 00:02:49.100 --> 00:02:51.690 but it's telling us a lot of information 00:02:51.690 --> 00:02:55.040 Here on the left it tells us our sources of energy. 00:02:55.040 --> 00:02:59.100 So 35% of the energy in the United States 00:02:59.100 --> 00:03:00.720 comes from petroleum. 00:03:00.720 --> 00:03:03.800 Then 34% from natural gas. 00:03:03.800 --> 00:03:05.720 Then 12% from renewable energy. 00:03:05.720 --> 00:03:08.780 That'd be things like wind power or solar power. 00:03:08.780 --> 00:03:13.130 Then you have 10% from coal and 9% from nuclear. 00:03:13.130 --> 00:03:14.880 And then not only does this tell us where 00:03:14.880 --> 00:03:16.340 the energy is coming from 00:03:16.340 --> 00:03:18.800 it's telling us how it is being used. 00:03:18.800 --> 00:03:21.670 So we could see 36% in the industrial sector. 00:03:21.670 --> 00:03:23.180 So that's all of the factories 00:03:23.180 --> 00:03:25.550 the manufacturing that produces all of the goods 00:03:25.550 --> 00:03:28.910 and services and raw materials that we have in society. 00:03:28.910 --> 00:03:32.100 35% of our energy in the United States is used 00:03:32.100 --> 00:03:34.350 for transportation, moving things around. 00:03:34.350 --> 00:03:37.770 Moving ourselves around, but also moving stuff around. 00:03:37.770 --> 00:03:40.410 And then 17% is residential. 00:03:40.410 --> 00:03:41.740 12% is commercial. 00:03:41.740 --> 00:03:43.420 So this would be things like 00:03:43.420 --> 00:03:45.570 the energy that the shopping mall all is using 00:03:45.570 --> 00:03:48.630 or the energy that's being used in an office building. 00:03:48.630 --> 00:03:51.370 This little gray and black box down here is interesting 00:03:51.370 --> 00:03:54.520 because it shows the role of electricity 00:03:54.520 --> 00:03:56.180 in this whole scheme. 00:03:56.180 --> 00:03:58.050 Sometimes something like petroleum 00:03:58.050 --> 00:04:02.490 might be directly used by say an industrial user, 00:04:02.490 --> 00:04:04.529 but sometimes that petroleum is then used 00:04:04.529 --> 00:04:07.350 for electricity generation 00:04:07.350 --> 00:04:09.550 which can then be used by these various sectors. 00:04:09.550 --> 00:04:13.570 Similarly, something like coal could be used directly 00:04:13.570 --> 00:04:16.360 or it could be used to produce electricity. 00:04:16.360 --> 00:04:18.770 Now one of the eye-popping things that I didn't appreciate 00:04:18.770 --> 00:04:20.550 until I saw this diagram 00:04:20.550 --> 00:04:23.740 are the electrical system energy losses, 00:04:23.740 --> 00:04:27.090 roughly 65% of the energy is lost. 00:04:27.090 --> 00:04:29.440 If you have a more efficient electrical grid 00:04:29.440 --> 00:04:31.550 or we know when to produce the electricity. 00:04:31.550 --> 00:04:34.030 So it more matches up with the demand. 00:04:34.030 --> 00:04:37.350 Then we can once again conserve energy as a society. 00:04:37.350 --> 00:04:39.030 And so that's why it's important to realize 00:04:39.030 --> 00:04:42.010 some people think, hey, if I'm using electricity 00:04:42.010 --> 00:04:43.860 or if I'm using an electric car 00:04:43.860 --> 00:04:46.770 that maybe has less impact on the environment. 00:04:46.770 --> 00:04:49.170 Well it depends where that electricity is coming from. 00:04:49.170 --> 00:04:51.400 An electric car actually gives us the option 00:04:51.400 --> 00:04:53.660 of not necessarily using petroleum. 00:04:53.660 --> 00:04:56.304 It gives us the option of potentially using renewable energy 00:04:56.304 --> 00:04:58.100 or nuclear energy. 00:04:58.100 --> 00:04:59.391 But that electricity could be coming from 00:04:59.391 --> 00:05:02.810 things that significantly impact the environment. 00:05:02.810 --> 00:05:04.490 But when you generally look at this, 00:05:04.490 --> 00:05:06.560 it tells us that as a society 00:05:06.560 --> 00:05:09.020 we have to think about things like 00:05:09.020 --> 00:05:10.912 investing in public transportation 00:05:10.912 --> 00:05:13.600 so that we conserve energy there. 00:05:13.600 --> 00:05:15.690 This right over here is a cement plant. 00:05:15.690 --> 00:05:17.470 Are there ways to produce these things 00:05:17.470 --> 00:05:20.190 that are more efficient that use less energy. 00:05:20.190 --> 00:05:22.750 And then there's regulations that might 00:05:22.750 --> 00:05:25.100 motivate us as individuals to 00:05:25.100 --> 00:05:27.970 say carpool or drive electric cars 00:05:27.970 --> 00:05:30.820 which once again, aren't necessarily going to be clean. 00:05:30.820 --> 00:05:33.000 It depends where that electricity comes from. 00:05:33.000 --> 00:05:36.740 But it gives us the option of using renewable sources. 00:05:36.740 --> 00:05:37.840 So I'll leave you there. 00:05:37.840 --> 00:05:40.480 Energy conservation is a complex topic 00:05:40.480 --> 00:05:42.420 but a very very very important one. 00:05:42.420 --> 00:05:43.938 But hopefully this gives you a start 00:05:43.938 --> 00:05:47.460 on how you can look at how energy is being used in the world 00:05:47.460 --> 00:05:50.110 where it comes from and how you can make a change 00:05:50.110 --> 00:05:53.030 both at the personal level, right over here. 00:05:53.030 --> 00:05:56.670 And as a member of our democracy at the societal level

Reduction of Air Pollutants

https://www.youtube.com/watch?v=s4OcGrPQqJ0

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WEBVTT Kind: captions Language: en 00:00:00.820 --> 00:00:01.950 - [Instructor] Hey there, friends. 00:00:01.950 --> 00:00:03.960 Today, we're gonna learn about air pollution. 00:00:03.960 --> 00:00:06.340 And to start off, we're going back in time 00:00:06.340 --> 00:00:08.990 to the small town of Donora, Pennsylvania, 00:00:08.990 --> 00:00:11.447 in October of 1948. 00:00:11.447 --> 00:00:13.930 (pensive harp music) 00:00:13.930 --> 00:00:16.480 Walking into this small industrial town, 00:00:16.480 --> 00:00:19.800 you can immediately sense that something is wrong. 00:00:19.800 --> 00:00:20.900 It's the middle of the day, 00:00:20.900 --> 00:00:24.130 but there's a thick yellowish smog everywhere, 00:00:24.130 --> 00:00:27.350 enveloping everything and even blocking out the sun. 00:00:27.350 --> 00:00:31.400 It's so dark that streetlights are on during the daytime. 00:00:31.400 --> 00:00:35.820 It stings your eyes and it's hard, even painful to breathe. 00:00:35.820 --> 00:00:38.890 What we're experiencing is the Donora Death Fog, 00:00:38.890 --> 00:00:41.100 one of the worst air pollution disasters 00:00:41.100 --> 00:00:42.860 in the United States. 00:00:42.860 --> 00:00:47.300 Donora was an industrial town full of steel plants and mills 00:00:47.300 --> 00:00:49.000 which released toxic emissions, 00:00:49.000 --> 00:00:52.010 such as hydrogen fluoride and sulfur dioxide 00:00:52.010 --> 00:00:54.850 when processing steel and other metals. 00:00:54.850 --> 00:00:56.700 Normally, these poisonous gases 00:00:56.700 --> 00:00:58.960 would disperse into the atmosphere. 00:00:58.960 --> 00:01:02.250 But this time, there was a temperature inversion, 00:01:02.250 --> 00:01:03.940 which caused a blanket of warm air 00:01:03.940 --> 00:01:06.700 to cover a layer of colder air near the surface 00:01:06.700 --> 00:01:08.453 and ride over Donora. 00:01:09.630 --> 00:01:11.400 Consequently, the toxic emissions 00:01:11.400 --> 00:01:14.400 were essentially trapped under the warm air. 00:01:14.400 --> 00:01:15.870 Over the course of several days 00:01:15.870 --> 00:01:19.360 from October 26th to October 31st, 00:01:19.360 --> 00:01:22.210 these toxic emissions had accumulated so much 00:01:22.210 --> 00:01:25.760 that half of the 14,000 people living in Donora 00:01:25.760 --> 00:01:29.490 suffered from respiratory problems and 20 people died. 00:01:29.490 --> 00:01:32.320 Relief only came when the steel mills were shut down 00:01:32.320 --> 00:01:34.683 and a rainstorm alleviated the smog. 00:01:36.230 --> 00:01:38.950 But following the deadly Donora smog, 00:01:38.950 --> 00:01:41.360 the public began to realize just how dangerous 00:01:41.360 --> 00:01:43.770 and life-threatening air pollution could be, 00:01:43.770 --> 00:01:46.380 and citizens demanded change. 00:01:46.380 --> 00:01:48.850 Donora became a turning point in US history 00:01:48.850 --> 00:01:51.780 and was a start of the clean air movement. 00:01:51.780 --> 00:01:54.810 The Air Pollution Control Act of 1955 00:01:54.810 --> 00:01:57.940 was the first piece of US federal legislation 00:01:57.940 --> 00:02:00.790 involving air pollution and provided funds 00:02:00.790 --> 00:02:03.810 for research about air pollution. 00:02:03.810 --> 00:02:07.640 Then, in 1963, the Clean Air Act was passed, 00:02:07.640 --> 00:02:11.240 the first federal legislation to control air pollution, 00:02:11.240 --> 00:02:15.440 and later expanded in 1970, which resulted in the creation 00:02:15.440 --> 00:02:19.300 of the US Environmental Protection Agency, the EPA, 00:02:19.300 --> 00:02:21.160 to develop and enforce regulations 00:02:21.160 --> 00:02:22.780 to protect the general public 00:02:22.780 --> 00:02:25.810 from exposure to major outdoor air pollutants. 00:02:25.810 --> 00:02:30.810 The Clean Air Act was expanded in 1977 and again in 1990. 00:02:30.890 --> 00:02:33.550 And throughout its nearly-60-year history, 00:02:33.550 --> 00:02:36.570 our air quality has drastically improved 00:02:36.570 --> 00:02:38.743 and pollutants have dropped sharply. 00:02:39.950 --> 00:02:44.300 Since 1990, major air pollutants such as carbon monoxide, 00:02:44.300 --> 00:02:47.240 nitrogen oxides, sulfur dioxide, 00:02:47.240 --> 00:02:50.360 and volatile organic compounds have greatly decreased, 00:02:50.360 --> 00:02:52.700 and that's just since 1990. 00:02:52.700 --> 00:02:55.550 These four main air pollutants that I highlighted 00:02:55.550 --> 00:02:59.160 are largely released as emissions from burning fossil fuels, 00:02:59.160 --> 00:03:01.220 which comes from driving vehicles 00:03:01.220 --> 00:03:03.400 and operating coal-fired power plants 00:03:03.400 --> 00:03:05.590 and other industrial facilities. 00:03:05.590 --> 00:03:09.020 So, as we've started to drive more efficient vehicles 00:03:09.020 --> 00:03:12.670 and obtain more energy from clean renewable sources, 00:03:12.670 --> 00:03:15.560 we've decreased the amount of fossil fuels that we use. 00:03:15.560 --> 00:03:18.090 And in turn, we've reduced emissions 00:03:18.090 --> 00:03:21.263 from fossil fuels and associated air pollutants. 00:03:22.670 --> 00:03:25.710 But how does the Clean Air Act work exactly? 00:03:25.710 --> 00:03:27.140 How do we clean the air 00:03:27.140 --> 00:03:30.310 and limit emissions of harmful pollutants? 00:03:30.310 --> 00:03:33.190 Clean Air Act regulations implemented by the EPA 00:03:33.190 --> 00:03:36.140 have led to new technologies that help to limit emissions 00:03:36.140 --> 00:03:38.290 and remove pollutants from the air. 00:03:38.290 --> 00:03:40.240 In particular, many of these technologies 00:03:40.240 --> 00:03:41.620 help to reduce air pollution 00:03:41.620 --> 00:03:45.020 from coal-burning power plants and vehicles. 00:03:45.020 --> 00:03:47.250 Each of these pollution-control technologies 00:03:47.250 --> 00:03:50.260 functions to remove harmful components out of emissions 00:03:50.260 --> 00:03:53.000 and release a less harmful substance. 00:03:53.000 --> 00:03:55.870 In the last decade or so, you've also probably noticed 00:03:55.870 --> 00:03:59.120 more and more electric vehicles on the road. 00:03:59.120 --> 00:04:01.100 Improving the fuel economy of vehicles 00:04:01.100 --> 00:04:04.010 and even using battery-powered electric vehicles 00:04:04.010 --> 00:04:06.670 can reduce the need to burn as much gasoline, 00:04:06.670 --> 00:04:10.010 thereby reducing emissions and giving us cleaner air. 00:04:10.010 --> 00:04:11.950 A good example is the growing demand 00:04:11.950 --> 00:04:14.730 for hybrid and purely electric vehicles. 00:04:14.730 --> 00:04:16.390 Here we have a simplified figure 00:04:16.390 --> 00:04:18.280 that explains the sources of energy 00:04:18.280 --> 00:04:19.890 for different types of vehicles 00:04:19.890 --> 00:04:21.800 and their respective emissions. 00:04:21.800 --> 00:04:24.220 On the left, we have conventional vehicles, 00:04:24.220 --> 00:04:27.680 which rely on fossil fuels such as gasoline or diesel 00:04:27.680 --> 00:04:30.030 and, when driven, produce lots of emissions 00:04:30.030 --> 00:04:33.080 like carbon dioxide and air pollutants. 00:04:33.080 --> 00:04:35.240 Hybrid and plug-in hybrid vehicles 00:04:35.240 --> 00:04:37.930 are similar to conventional vehicles 00:04:37.930 --> 00:04:40.500 in that they have an internal combustion engine, 00:04:40.500 --> 00:04:42.670 but they also have an electric motor 00:04:42.670 --> 00:04:45.080 which uses energy stored in batteries. 00:04:45.080 --> 00:04:48.410 These batteries can be charged by regenerative braking 00:04:48.410 --> 00:04:50.580 or, in the case of plug-in hybrids, 00:04:50.580 --> 00:04:53.940 just by using a wall outlet or other charging equipment. 00:04:53.940 --> 00:04:56.110 In turn, because these hybrid cars 00:04:56.110 --> 00:04:58.240 are partially fueled by batteries, 00:04:58.240 --> 00:05:01.620 they produce fewer emissions than a conventional car. 00:05:01.620 --> 00:05:04.780 Finally, vehicles that rely solely on electricity, 00:05:04.780 --> 00:05:08.510 known as battery electric vehicles or BEVs, 00:05:08.510 --> 00:05:10.530 can use an alternative electricity source 00:05:10.530 --> 00:05:12.050 so that there's no emissions 00:05:12.050 --> 00:05:14.530 at the source of the electricity. 00:05:14.530 --> 00:05:17.090 What else can we do to ensure cleaner air? 00:05:17.090 --> 00:05:19.360 We can reduce our reliance on fossil fuels 00:05:19.360 --> 00:05:22.220 and instead invest in cleaner renewable resources 00:05:22.220 --> 00:05:26.890 to generate electricity such as geothermal, wind, and solar. 00:05:26.890 --> 00:05:29.440 And we can make decisions in our day-to-day lives 00:05:29.440 --> 00:05:31.160 to reduce or prevent air pollution 00:05:31.160 --> 00:05:34.460 by using less energy and alternative fuels. 00:05:34.460 --> 00:05:37.880 For example, walking, biking, or using mass transportation 00:05:37.880 --> 00:05:40.550 can reduce the need to burn fossil fuels. 00:05:40.550 --> 00:05:42.190 And there are plenty of other ways 00:05:42.190 --> 00:05:44.010 to reduce our electrical needs; 00:05:44.010 --> 00:05:47.540 in particular, using more energy-efficient appliances. 00:05:47.540 --> 00:05:49.830 For example, think of the LED bulb, 00:05:49.830 --> 00:05:53.870 which uses 75% less energy than incandescent lighting. 00:05:53.870 --> 00:05:55.750 So, even switching out bulbs 00:05:55.750 --> 00:05:57.560 in the lights around your house or apartment 00:05:57.560 --> 00:05:59.363 can make a huge difference. 00:06:01.070 --> 00:06:04.790 But there's still many other places out there like Donora, 00:06:04.790 --> 00:06:07.040 and oftentimes folks living in cities 00:06:07.040 --> 00:06:10.810 with heavy air pollution literally can't afford to leave. 00:06:10.810 --> 00:06:13.120 There's still much work to be done. 00:06:13.120 --> 00:06:16.010 In Donora, though, there's the Donora Smog Museum, 00:06:16.010 --> 00:06:19.640 which has the tagline "Clean air started here." 00:06:19.640 --> 00:06:22.160 The terrible incident suffered by Donora's community 00:06:22.160 --> 00:06:24.290 played a huge and pivotal role 00:06:24.290 --> 00:06:25.860 in opening the eyes of Americans 00:06:25.860 --> 00:06:29.100 to the hazards of air pollution and spurred political action 00:06:29.100 --> 00:06:31.090 that's carried forth through today 00:06:31.090 --> 00:06:33.650 and will continue into the future. 00:06:33.650 --> 00:06:36.373 Let's all take a deep breath and be glad that we can.

Nuclear Power Generation

https://www.youtube.com/watch?v=E7UIonbL4FU

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en

WEBVTT Kind: captions Language: en 00:00:01.300 --> 00:00:02.610 - [Professor] Hey there, friends. 00:00:02.610 --> 00:00:05.520 Today, we're going to learn about nuclear power, 00:00:05.520 --> 00:00:08.800 and to do so, we're gonna visit my home state. 00:00:08.800 --> 00:00:10.710 Idaho? That's right. 00:00:10.710 --> 00:00:14.390 Land of the potatoes and also nuclear power? 00:00:14.390 --> 00:00:15.860 If you've driven through Idaho, 00:00:15.860 --> 00:00:17.350 there's a good chance that you passed 00:00:17.350 --> 00:00:19.960 by a quaint small town called Arco 00:00:19.960 --> 00:00:22.260 where you'll find the restaurant Pickle's Place, 00:00:22.260 --> 00:00:24.520 home to the Atomic Burger. 00:00:24.520 --> 00:00:29.520 Wait, a radioactive burger? Sounds a little disturbing. 00:00:30.320 --> 00:00:32.610 Actually, Arco became the first city in the world 00:00:32.610 --> 00:00:35.090 to be powered by nuclear energy. 00:00:35.090 --> 00:00:37.430 And of course, Arco became the first city 00:00:37.430 --> 00:00:40.620 to serve Atomic Burgers, grilled and seared 00:00:40.620 --> 00:00:43.053 to perfection using nuclear energy. 00:00:43.890 --> 00:00:46.470 But what's going on under that grill? 00:00:46.470 --> 00:00:48.440 Are they using glowing green rocks 00:00:48.440 --> 00:00:50.880 to make those delicious Atomic Burgers? 00:00:50.880 --> 00:00:51.913 Let's find out. 00:00:54.270 --> 00:00:56.750 Nuclear power plants often look ominous 00:00:56.750 --> 00:00:58.470 and a little bit scary, 00:00:58.470 --> 00:01:00.330 but they produce power the same way 00:01:00.330 --> 00:01:02.550 most other power plants do. 00:01:02.550 --> 00:01:05.720 Simply put, they boil water to create steam, 00:01:05.720 --> 00:01:08.710 which spins turbines to produce energy. 00:01:08.710 --> 00:01:12.100 Most nuclear power plants use light water reactors 00:01:12.100 --> 00:01:13.930 to generate electricity, 00:01:13.930 --> 00:01:16.513 which are made up of five basic parts. 00:01:17.480 --> 00:01:20.120 First off, we have the core of the reactor 00:01:20.120 --> 00:01:22.650 where fuel rods are inserted. 00:01:22.650 --> 00:01:24.780 Next up, we have the containment shell 00:01:24.780 --> 00:01:28.610 that encases the reactor and the spent fuel rods. 00:01:28.610 --> 00:01:30.670 Within there, we have supply of water, 00:01:30.670 --> 00:01:32.900 which is boiled to reduce steam. 00:01:32.900 --> 00:01:35.720 That steam then rotates a turbine attached 00:01:35.720 --> 00:01:38.890 to a generator which produces electricity. 00:01:38.890 --> 00:01:41.480 This act of turning an electric generator 00:01:41.480 --> 00:01:45.060 is actually the same process that's used for coal, 00:01:45.060 --> 00:01:49.030 gas, geothermal, hydro power, and wind power. 00:01:49.030 --> 00:01:52.170 No matter how complex the electricity generation system, 00:01:52.170 --> 00:01:54.660 that all boils down to the same idea, 00:01:54.660 --> 00:01:56.410 basically turning a wheel, 00:01:56.410 --> 00:02:00.120 one of the oldest agricultural-era human inventions, 00:02:00.120 --> 00:02:02.500 and that's what makes electricity. 00:02:02.500 --> 00:02:06.090 Finally, we have excess steam or water vapor, 00:02:06.090 --> 00:02:07.510 which is the only direct emission 00:02:07.510 --> 00:02:09.470 from nuclear power generation. 00:02:09.470 --> 00:02:13.393 Easy as pie, right? Well, it's actually pretty complex. 00:02:14.650 --> 00:02:18.050 So how is the water heated exactly? 00:02:18.050 --> 00:02:21.140 Nuclear energy isn't as easy as lighting up a grill, 00:02:21.140 --> 00:02:23.120 and it requires us to go down 00:02:23.120 --> 00:02:26.490 to the smallest unit of matter, the atom. 00:02:26.490 --> 00:02:28.960 Here, we get our energy at the atomic level, 00:02:28.960 --> 00:02:31.360 but it's not from the atom alone. 00:02:31.360 --> 00:02:34.720 No, to gain energy, we need to split the atom. 00:02:34.720 --> 00:02:37.130 This process is called fission, which occurs 00:02:37.130 --> 00:02:39.680 when neutrons are fired at an atom, 00:02:39.680 --> 00:02:41.560 causing it to split into separate atoms 00:02:41.560 --> 00:02:43.610 of other smaller elements. 00:02:43.610 --> 00:02:46.230 This split produces a huge amount of energy, 00:02:46.230 --> 00:02:48.400 which is largely converted to heat, 00:02:48.400 --> 00:02:51.310 which boils the water and produces steam. 00:02:51.310 --> 00:02:53.160 However, we need a special kind 00:02:53.160 --> 00:02:55.160 of atom for fission to happen, 00:02:55.160 --> 00:02:58.623 and most nuclear reactors use uranium-235. 00:03:00.290 --> 00:03:03.200 Wait, why uranium-235 though? 00:03:03.200 --> 00:03:06.470 Well, first, uranium-235 is big, 00:03:06.470 --> 00:03:10.770 not a triple 1/4 pounder big but big on the atomic scale. 00:03:10.770 --> 00:03:13.830 In the atomic world, this is known as being heavy. 00:03:13.830 --> 00:03:18.600 Secondly, uranium-235 is unstable because it's not only big, 00:03:18.600 --> 00:03:21.700 but it's also an isotope, meaning it has a different number 00:03:21.700 --> 00:03:24.620 of neutrons than the more common form of uranium, 00:03:24.620 --> 00:03:28.710 which is uranium-238, which has three more neutrons. 00:03:28.710 --> 00:03:33.530 This makes uranium-235 unstable or fissile like fission, 00:03:33.530 --> 00:03:36.120 which means it can be split by a neutron, 00:03:36.120 --> 00:03:40.350 thereby producing other elements, energy, and more neutrons. 00:03:40.350 --> 00:03:44.140 Those produced neutrons crash into other U-235 atoms, 00:03:44.140 --> 00:03:47.060 splitting them and causing a chain reaction, 00:03:47.060 --> 00:03:49.163 which is what makes nuclear energy work. 00:03:50.070 --> 00:03:52.850 This chain reaction is really important to note 00:03:52.850 --> 00:03:56.010 because it's what makes a nuclear power plant so different 00:03:56.010 --> 00:04:00.240 from its, well, more destructive cousin, the atomic bomb. 00:04:00.240 --> 00:04:04.380 In atomic bombs, the same process of nuclear fission is used 00:04:04.380 --> 00:04:07.450 except that it's a fast destructive, runaway, 00:04:07.450 --> 00:04:09.490 and uncontrolled reaction that results 00:04:09.490 --> 00:04:11.910 in massively powerful explosions, 00:04:11.910 --> 00:04:13.510 not something that we would want 00:04:13.510 --> 00:04:15.523 to happen in a nuclear power plant. 00:04:16.490 --> 00:04:18.330 Now a little goes a long way 00:04:18.330 --> 00:04:20.400 when it comes to nuclear fission. 00:04:20.400 --> 00:04:23.400 The fuel is actually composed of tiny pellets 00:04:23.400 --> 00:04:28.070 of uranium-235, each the size of a pencil eraser, 00:04:28.070 --> 00:04:32.160 but each also has the equivalent energy of a ton of coal. 00:04:32.160 --> 00:04:34.440 Yes, a literal ton. 00:04:34.440 --> 00:04:37.190 These pellets are packed together to form fuel rods, 00:04:37.190 --> 00:04:39.580 which are bunched into fuel assemblies 00:04:39.580 --> 00:04:42.200 and then placed in the nuclear reactor. 00:04:42.200 --> 00:04:44.810 Nuclear fusion is therefore really powerful 00:04:44.810 --> 00:04:48.810 and can generate a lot of heat from very little material. 00:04:48.810 --> 00:04:51.390 But to keep temperatures from getting too hot, 00:04:51.390 --> 00:04:53.380 which would cause a nuclear meltdown, 00:04:53.380 --> 00:04:57.210 and no, I'm not talking about melty cheese, unfortunately, 00:04:57.210 --> 00:05:00.320 the reactor is therefore cooled with water. 00:05:00.320 --> 00:05:03.170 When more heat is generated by the nuclear reactor 00:05:03.170 --> 00:05:05.410 than can be removed by the cooling system, 00:05:05.410 --> 00:05:08.120 or water in the case of nuclear reactors, 00:05:08.120 --> 00:05:11.340 the fuel rods can get so hot that they could start to melt 00:05:11.340 --> 00:05:13.090 and fall to the bottom of the reactor 00:05:13.090 --> 00:05:15.100 and potentially melt through and escape 00:05:15.100 --> 00:05:16.940 into the surrounding environment. 00:05:16.940 --> 00:05:18.950 That's called a nuclear meltdown, 00:05:18.950 --> 00:05:21.980 and that's also why in part, the reactor is surrounded 00:05:21.980 --> 00:05:24.480 by a containment shell of thick steel and concrete, 00:05:24.480 --> 00:05:27.550 which keeps radioactive materials from escaping. 00:05:27.550 --> 00:05:29.833 We don't wanna have any radioactive burgers. 00:05:30.750 --> 00:05:33.550 But fuel rods don't last forever. 00:05:33.550 --> 00:05:35.660 After three to six years in a reactor, 00:05:35.660 --> 00:05:38.730 fuel rods can't sustain the fission reaction effectively 00:05:38.730 --> 00:05:41.510 anymore and become highly radioactive. 00:05:41.510 --> 00:05:44.450 In turn, they need to be carefully removed and stored. 00:05:44.450 --> 00:05:46.950 But what to do with nuclear waste? 00:05:46.950 --> 00:05:49.130 The problem with spent nuclear fuel 00:05:49.130 --> 00:05:51.170 is that it's really radioactive. 00:05:51.170 --> 00:05:54.640 These leftover radioactive materials can persist in the air, 00:05:54.640 --> 00:05:57.930 soil, and water for thousands and thousands of years 00:05:57.930 --> 00:06:00.140 and damage the DNA of living organisms, 00:06:00.140 --> 00:06:02.990 causing cancer and other health conditions. 00:06:02.990 --> 00:06:05.550 For a while, actually quite a long time 00:06:05.550 --> 00:06:08.980 from 1946 to 1993, to be exact, 00:06:08.980 --> 00:06:12.290 many countries just dumped radioactive nuclear waste 00:06:12.290 --> 00:06:13.640 into the ocean. 00:06:13.640 --> 00:06:17.380 This was consequently banned, and you can imagine why. 00:06:17.380 --> 00:06:20.030 Instead, nuclear waste can be buried, 00:06:20.030 --> 00:06:22.280 but there's problems with that, too. 00:06:22.280 --> 00:06:24.400 Nuclear waste can still leak into soil 00:06:24.400 --> 00:06:27.060 and water if it isn't properly contained. 00:06:27.060 --> 00:06:30.930 So where do we safely bury it? Well, nowhere really. 00:06:30.930 --> 00:06:33.110 Radioactive spent fuel is stored all 00:06:33.110 --> 00:06:35.600 over the world in various containment systems, 00:06:35.600 --> 00:06:38.120 but none of them are truly longterm. 00:06:38.120 --> 00:06:41.210 Alternatively, spent fuel rods can also be recycled 00:06:41.210 --> 00:06:43.810 and reprocessed where unused uranium 00:06:43.810 --> 00:06:46.480 is separated from spent nuclear fuel. 00:06:46.480 --> 00:06:51.110 However, this reprocessing is quite expensive and dangerous. 00:06:51.110 --> 00:06:53.660 Reprocessing is often much more expensive 00:06:53.660 --> 00:06:56.730 than storing or disposing of spent nuclear fuel, 00:06:56.730 --> 00:06:59.070 and it still results in a substantial amount 00:06:59.070 --> 00:07:01.060 of leftover radioactive materials 00:07:01.060 --> 00:07:02.993 that still need to be disposed of. 00:07:04.440 --> 00:07:06.230 There's no perfect solution when it comes 00:07:06.230 --> 00:07:08.050 to energy production, though. 00:07:08.050 --> 00:07:10.030 Any kind of electricity production 00:07:10.030 --> 00:07:12.780 has its own benefits and drawbacks. 00:07:12.780 --> 00:07:16.280 But Pickle Place's Atomic Burger is quite perfect, 00:07:16.280 --> 00:07:17.743 and I think I'll eat one now.

Student tips for using course mastery on Khan Academy

https://www.youtube.com/watch?v=yqdGJ-_0AGg

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https://www.youtube.com/api/timedtext?v=yqdGJ-_0AGg&ei=5VWUZfTYEavjxN8PtLSwgA0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=2BCDA25DDDFD29DA6E5151AD700478DF8577101D.8FA129B74B295D0848F5B20864A1FFC2A8F3C970&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:03.200 --> 00:00:06.050 - Hi, I'm Shannon from Khan Academy, 00:00:06.050 --> 00:00:08.240 and I want to show you how to make the most 00:00:08.240 --> 00:00:09.483 of your learning time. 00:00:10.490 --> 00:00:12.570 First, make sure you're logged in 00:00:12.570 --> 00:00:14.410 to your Khan Academy account 00:00:14.410 --> 00:00:17.793 by checking for your name in the upper right hand corner. 00:00:19.550 --> 00:00:21.410 Now on the left side, 00:00:21.410 --> 00:00:22.860 you should see your classes 00:00:22.860 --> 00:00:25.640 where your teacher has given you a mastery goal, 00:00:25.640 --> 00:00:27.253 or assignment to work on. 00:00:28.520 --> 00:00:32.150 Click on the tab that says Course mastery 00:00:32.150 --> 00:00:34.543 to view the goal your teacher set for you. 00:00:35.520 --> 00:00:37.450 Now that you're viewing your goal, 00:00:37.450 --> 00:00:40.360 let's talk about the top five things to know 00:00:40.360 --> 00:00:43.913 about working towards Course mastery on Khan Academy. 00:00:45.020 --> 00:00:48.180 Number one, your course mastery placement 00:00:48.180 --> 00:00:51.010 is a big goal from your teacher. 00:00:51.010 --> 00:00:52.860 On your learner home page, 00:00:52.860 --> 00:00:55.170 you can see the progress you've already made 00:00:55.170 --> 00:00:57.110 towards your mastery goal, 00:00:57.110 --> 00:00:59.720 and the due date your teacher set. 00:00:59.720 --> 00:01:01.710 But you can always work ahead, 00:01:01.710 --> 00:01:03.393 or view past goals. 00:01:05.180 --> 00:01:06.670 Click on the mastery goal 00:01:06.670 --> 00:01:09.113 to be taken to the course homepage. 00:01:10.940 --> 00:01:14.010 Here, you'll see the units that make up the course, 00:01:14.010 --> 00:01:17.453 as well as your progress towards mastery on each unit. 00:01:18.530 --> 00:01:21.010 As you make progress on each unit, 00:01:21.010 --> 00:01:24.373 you'll see the purple bar fill from left to right. 00:01:25.940 --> 00:01:28.900 It's helpful to regularly check your unit progress, 00:01:28.900 --> 00:01:31.350 so you know if you're on track to meet your goal. 00:01:33.090 --> 00:01:36.280 Click into a unit to view your current mastery level 00:01:36.280 --> 00:01:39.313 for each skill on the left-hand side. 00:01:40.430 --> 00:01:41.890 With every skill, 00:01:41.890 --> 00:01:43.300 try to get the crown, 00:01:43.300 --> 00:01:46.003 and move your mastery level to Mastered. 00:01:46.890 --> 00:01:51.343 Number two, mastery takes time and practice. 00:01:52.600 --> 00:01:55.840 As you practice skills and answer questions, 00:01:55.840 --> 00:01:59.260 your mastery level for each skill will go up, 00:01:59.260 --> 00:02:01.110 if you answered correctly, 00:02:01.110 --> 00:02:03.453 or down, if you miss questions. 00:02:04.550 --> 00:02:07.040 If you want to make progress more quickly, 00:02:07.040 --> 00:02:08.803 try a Mastery challenge. 00:02:09.660 --> 00:02:12.760 Mastery challenges allow you to strengthen the skills 00:02:12.760 --> 00:02:16.343 you've already practiced in just six questions. 00:02:17.650 --> 00:02:20.000 Mastery is not easily earned, 00:02:20.000 --> 00:02:21.870 and that's intentional! 00:02:21.870 --> 00:02:24.710 But putting in the work to achieve mastery 00:02:24.710 --> 00:02:27.323 will prove you've earned and learned it. 00:02:28.400 --> 00:02:31.360 Remember, you can retry an exercise 00:02:31.360 --> 00:02:33.170 as many times as you'd like 00:02:33.170 --> 00:02:35.782 until you earn a score you're happy with. 00:02:35.782 --> 00:02:36.615 (stars twinkling) 00:02:36.615 --> 00:02:39.629 Struggles and mistakes are what helps your brain grow. 00:02:39.629 --> 00:02:41.920 (congratulatory sound) 00:02:41.920 --> 00:02:45.023 Number three, follow the blue button. 00:02:46.490 --> 00:02:49.590 Your big goal has a lot of skills you can practice 00:02:49.590 --> 00:02:50.973 at your own pace. 00:02:51.870 --> 00:02:54.250 To make sure you stay on track, 00:02:54.250 --> 00:02:57.053 look for the blue buttons throughout the course. 00:02:58.140 --> 00:03:00.110 The blue buttons will always guide you 00:03:00.110 --> 00:03:01.993 to what you should work on next. 00:03:03.350 --> 00:03:06.863 Number four, if you're stuck, take a hint. 00:03:08.030 --> 00:03:11.320 It's normal to feel stuck when you're learning new skills. 00:03:11.320 --> 00:03:13.503 The important thing is you don't give up. 00:03:14.410 --> 00:03:17.600 Take a hint to get step-by-step instructions 00:03:17.600 --> 00:03:20.680 to the specific question you're working on. 00:03:20.680 --> 00:03:23.280 And write them down so you can reference them later. 00:03:24.940 --> 00:03:27.000 You can also try watching a video, 00:03:27.000 --> 00:03:29.133 or reading an article on the skill. 00:03:30.240 --> 00:03:31.410 Once you've done that, 00:03:31.410 --> 00:03:33.743 you're ready to retry the exercise. 00:03:34.970 --> 00:03:36.810 And if you're still stuck, 00:03:36.810 --> 00:03:39.750 reach out to your classmate, a teacher, 00:03:39.750 --> 00:03:41.605 or a family member for support. 00:03:41.605 --> 00:03:43.900 (stars twinkling) 00:03:43.900 --> 00:03:46.813 Number five, the sky's the limit! 00:03:48.000 --> 00:03:51.810 Remember, you have the potential to succeed. 00:03:51.810 --> 00:03:54.510 Keep trying, keep making mistakes, 00:03:54.510 --> 00:03:57.140 and keep asking for help when you need it. 00:03:57.140 --> 00:03:59.393 There is no limit to what you can learn.

Student tips for completing assignments on Khan Academy

https://www.youtube.com/watch?v=dT_-uESRLqw

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en

WEBVTT Kind: captions Language: en 00:00:03.200 --> 00:00:06.050 - Hi, I'm Shannon from Khan Academy, 00:00:06.050 --> 00:00:08.240 and I wanna show you how to make the most 00:00:08.240 --> 00:00:10.000 of your learning time. 00:00:10.000 --> 00:00:11.910 First, make sure you're logged into 00:00:11.910 --> 00:00:14.850 your Khan Academy account by checking for your name 00:00:14.850 --> 00:00:17.500 in the upper right-hand corner. 00:00:17.500 --> 00:00:19.270 If you are not logged in, 00:00:19.270 --> 00:00:21.200 you won't be able to view your assignments 00:00:21.200 --> 00:00:22.500 and any progress you make 00:00:22.500 --> 00:00:24.520 won't be counted towards your classes. 00:00:24.520 --> 00:00:26.440 Now, on the left-hand side, 00:00:26.440 --> 00:00:27.870 you should see your classes 00:00:27.870 --> 00:00:30.700 where your teacher has given you a mastery goal 00:00:30.700 --> 00:00:32.363 or an assignment to work on. 00:00:33.500 --> 00:00:36.430 Click on the tab that says assignments 00:00:36.430 --> 00:00:38.373 to view assignments from your teacher. 00:00:40.050 --> 00:00:41.730 On the assignments tab, 00:00:41.730 --> 00:00:46.470 you can see all your upcoming and past due assignments. 00:00:46.470 --> 00:00:48.180 Assignments with the nearest due date 00:00:48.180 --> 00:00:51.121 will appear at the very top of the list. 00:00:51.121 --> 00:00:53.480 Now that you're ready to go, 00:00:53.480 --> 00:00:56.280 let's talk about the top five things to know 00:00:56.280 --> 00:00:58.720 about assignments on Khan Academy. 00:00:58.720 --> 00:01:01.943 Number one, types of assignments. 00:01:03.180 --> 00:01:05.480 There are two main types of assignments 00:01:05.480 --> 00:01:08.230 you can receive on Khan Academy. 00:01:08.230 --> 00:01:12.163 The first type is practice, noted with the star symbol. 00:01:13.240 --> 00:01:18.120 Practice can be an exercise, quiz, or unit test. 00:01:18.120 --> 00:01:20.770 When you complete practice on Khan Academy, 00:01:20.770 --> 00:01:23.693 you receive instant feedback after every answer. 00:01:24.730 --> 00:01:27.460 Answer all questions to complete the assignment 00:01:27.460 --> 00:01:28.593 and receive a score. 00:01:31.330 --> 00:01:36.110 The second type is instruction, noted with the play symbol 00:01:36.110 --> 00:01:37.483 and the paper symbol. 00:01:38.550 --> 00:01:42.253 Instruction can be in the form of videos or articles. 00:01:43.300 --> 00:01:46.360 Videos and articles can help you learn new skills 00:01:46.360 --> 00:01:48.373 or review things you learned in class. 00:01:49.810 --> 00:01:51.590 All videos have subtitles, 00:01:51.590 --> 00:01:54.080 so you can easily follow along. 00:01:54.080 --> 00:01:57.950 Number two, if you're stuck, take a hint. 00:01:57.950 --> 00:02:00.780 It's normal to feel stuck when you're learning new skills, 00:02:00.780 --> 00:02:03.700 but the important thing is you don't give up. 00:02:03.700 --> 00:02:06.520 Take a hint to get step-by-step instructions 00:02:06.520 --> 00:02:08.833 to the specific question you're working on. 00:02:09.700 --> 00:02:12.060 We recommend that you write these hints down 00:02:12.060 --> 00:02:13.713 so you can reference them later. 00:02:15.230 --> 00:02:17.180 You can also try watching a video 00:02:17.180 --> 00:02:19.420 or reading an article on this skill, 00:02:19.420 --> 00:02:22.120 even if it hasn't been assigned to you. 00:02:22.120 --> 00:02:24.140 You'll find the videos and articles 00:02:24.140 --> 00:02:26.460 related to each skill you're practicing 00:02:26.460 --> 00:02:30.560 by clicking this stuck or get help link. 00:02:30.560 --> 00:02:31.660 Once you've done that, 00:02:31.660 --> 00:02:34.373 you're ready to retry the practice exercise. 00:02:35.920 --> 00:02:37.460 And if you're still stuck, 00:02:37.460 --> 00:02:41.080 reach out to a classmate, teacher, or a family member 00:02:41.080 --> 00:02:41.913 for support. 00:02:44.060 --> 00:02:47.830 Number three, be patient and persistent. 00:02:47.830 --> 00:02:49.710 Not happy with the score you received 00:02:49.710 --> 00:02:51.660 on a practice exercise? 00:02:51.660 --> 00:02:54.070 Trust me, we've all been there. 00:02:54.070 --> 00:02:56.290 But remember, on Khan Academy, 00:02:56.290 --> 00:02:58.200 you can always retry assignments 00:02:58.200 --> 00:03:00.890 until you earn a that you are proud of. 00:03:00.890 --> 00:03:03.103 Click try again to retry. 00:03:08.690 --> 00:03:10.480 You can also return to assignments 00:03:10.480 --> 00:03:11.880 where the due dates have passed 00:03:11.880 --> 00:03:13.433 to try to improve your score. 00:03:14.540 --> 00:03:17.430 Struggles and mistakes are what helps your brain grow, 00:03:17.430 --> 00:03:18.380 so keep persisting. 00:03:19.860 --> 00:03:23.120 Number four, check your progress. 00:03:23.120 --> 00:03:24.810 On the left-hand side, 00:03:24.810 --> 00:03:28.520 select progress under my account. 00:03:28.520 --> 00:03:30.520 From here, you can review 00:03:30.520 --> 00:03:33.780 all of the activity on Khan Academy. 00:03:33.780 --> 00:03:38.130 This includes exercises you completed, videos you watched, 00:03:38.130 --> 00:03:39.783 or articles you read. 00:03:41.210 --> 00:03:43.260 Depending on your learning goals, 00:03:43.260 --> 00:03:44.270 you may wanna filter 00:03:44.270 --> 00:03:47.823 to view a certain type of content or activity. 00:03:49.180 --> 00:03:51.330 And for each practice you completed, 00:03:51.330 --> 00:03:53.960 you can view your current mastery level, 00:03:53.960 --> 00:03:56.910 the number of questions you answered correctly, 00:03:56.910 --> 00:03:59.533 and how much time you spent on the activity. 00:04:01.120 --> 00:04:03.620 It's helpful to regularly check your progress, 00:04:03.620 --> 00:04:06.790 so you know if you're on track to meet your goals. 00:04:06.790 --> 00:04:11.070 Number five, make Khan Academy your own. 00:04:11.070 --> 00:04:13.640 And finally, adjust Khan Academy 00:04:13.640 --> 00:04:16.630 to work for your unique learning style. 00:04:16.630 --> 00:04:21.190 You can slow down, speed up, or rewind the videos 00:04:21.190 --> 00:04:22.203 when you need to. 00:04:23.400 --> 00:04:26.930 You can learn in the language you're most comfortable with. 00:04:26.930 --> 00:04:29.830 Khan Academy is available in over 50 languages 00:04:29.830 --> 00:04:32.100 and you can easily switch between languages 00:04:32.100 --> 00:04:33.313 in your settings. 00:04:34.810 --> 00:04:36.160 And if you prefer, 00:04:36.160 --> 00:04:38.870 you can complete assignments on a smartphone. 00:04:38.870 --> 00:04:43.163 Just download the Khan Academy app on Android or iOS. 00:04:44.930 --> 00:04:48.870 And remember, you have the potential to succeed. 00:04:48.870 --> 00:04:51.730 Keep trying, keep making mistakes, 00:04:51.730 --> 00:04:54.320 and keep asking for help when you need it. 00:04:54.320 --> 00:04:56.533 There is no limit to what you can learn.

Impacts of Urbanization

https://www.youtube.com/watch?v=2C7F_2OpRT8

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en

WEBVTT Kind: captions Language: en 00:00:00.150 --> 00:00:01.000 - [Instructor] In this video, 00:00:01.000 --> 00:00:03.160 we're going to talk about cities. 00:00:03.160 --> 00:00:06.410 Today, more than 50% of the world's population 00:00:06.410 --> 00:00:09.900 lives in a city, and this percentage grows every year 00:00:09.900 --> 00:00:12.360 because of something called urbanization. 00:00:12.360 --> 00:00:15.460 Urbanization is the creation and growth of cities, 00:00:15.460 --> 00:00:18.010 human settlements with higher population densities 00:00:18.010 --> 00:00:19.440 than rural areas. 00:00:19.440 --> 00:00:21.750 The high population density in cities 00:00:21.750 --> 00:00:23.560 has a lot of advantages. 00:00:23.560 --> 00:00:25.320 Because people are closer together, 00:00:25.320 --> 00:00:28.530 it's easier to coordinate community efforts like recycling, 00:00:28.530 --> 00:00:31.793 public education, and public transportation. 00:00:33.320 --> 00:00:35.500 Urban residents also tend to have better access 00:00:35.500 --> 00:00:38.573 to things like medical care and family planning. 00:00:39.710 --> 00:00:42.450 However, the high population density of cities 00:00:42.450 --> 00:00:45.530 also can have a negative impact on the environment. 00:00:45.530 --> 00:00:47.190 You can imagine, with so many people 00:00:47.190 --> 00:00:49.110 living close together in one space, 00:00:49.110 --> 00:00:52.170 it can cause a strain on the natural resources of the area 00:00:52.170 --> 00:00:54.760 and generate more air and water pollution. 00:00:54.760 --> 00:00:56.520 Cities can use sustainable planning 00:00:56.520 --> 00:00:58.130 to counter these effects. 00:00:58.130 --> 00:01:00.240 Urban planners often talk about the ratio 00:01:00.240 --> 00:01:03.400 between green space and gray space in a city, 00:01:03.400 --> 00:01:07.570 where the green spaces are vegetation and the gray spaces 00:01:07.570 --> 00:01:10.790 are things like buildings and paved surfaces, 00:01:10.790 --> 00:01:13.720 like roads and sidewalks and parking lots. 00:01:13.720 --> 00:01:15.940 These gray spaces may be necessary, 00:01:15.940 --> 00:01:17.900 but cities should be planned thoughtfully 00:01:17.900 --> 00:01:19.730 to add some green spaces too. 00:01:19.730 --> 00:01:23.100 When gray spaces dominate a cityscape too much, 00:01:23.100 --> 00:01:24.760 the paved surfaces prevent water 00:01:24.760 --> 00:01:27.520 from seeping into the ground, so when it rains, 00:01:27.520 --> 00:01:29.730 it creates large amounts of runoff. 00:01:29.730 --> 00:01:30.900 Having nowhere to go, 00:01:30.900 --> 00:01:33.690 the water travels across paved surfaces, 00:01:33.690 --> 00:01:35.570 picking up pollutants along the way. 00:01:35.570 --> 00:01:38.220 All this runoff can lead to flash flooding. 00:01:38.220 --> 00:01:39.420 Adding more green spaces 00:01:39.420 --> 00:01:41.010 can help prevent this from happening 00:01:41.010 --> 00:01:42.960 because plants soak up water. 00:01:42.960 --> 00:01:45.260 There are many creative ways that urban planners 00:01:45.260 --> 00:01:47.580 can add green spaces to our environment. 00:01:47.580 --> 00:01:50.500 Some buildings in cities grow plants on their roofs. 00:01:50.500 --> 00:01:52.260 They're called green roofs. 00:01:52.260 --> 00:01:55.960 The plants soak up the rain and snow melt that hits the roof 00:01:55.960 --> 00:01:58.040 and prevents it from going into the street. 00:01:58.040 --> 00:02:01.150 Also, plants create shade and help cool things down, 00:02:01.150 --> 00:02:02.670 so buildings with green roofs 00:02:02.670 --> 00:02:05.340 wouldn't have to spend as much energy on air conditioning. 00:02:05.340 --> 00:02:07.400 Green spaces can also help the environment 00:02:07.400 --> 00:02:09.730 by allowing for groundwater recharge, 00:02:09.730 --> 00:02:12.580 which is allowing more water to sink back into the ground 00:02:12.580 --> 00:02:14.710 and replenish the aquifer. 00:02:14.710 --> 00:02:17.300 Some cities do this by building rain gardens, 00:02:17.300 --> 00:02:19.420 which are gardens specifically designed 00:02:19.420 --> 00:02:21.610 to filter water back into the ground, 00:02:21.610 --> 00:02:23.070 the plants and the rain gardens 00:02:23.070 --> 00:02:24.970 help filter pollutants in the runoff 00:02:24.970 --> 00:02:25.860 and prevent the pollutants 00:02:25.860 --> 00:02:28.230 from contaminating water ecosystems. 00:02:28.230 --> 00:02:30.280 Cities can also help with groundwater recharge 00:02:30.280 --> 00:02:33.250 by building permeable surfaces in their cities. 00:02:33.250 --> 00:02:34.970 Parking lots can be constructed 00:02:34.970 --> 00:02:37.920 with permeable paving to reduce runoff. 00:02:37.920 --> 00:02:39.850 Groundwater recharge can help prevent something 00:02:39.850 --> 00:02:43.223 that happens in coastal cities called saltwater intrusion. 00:02:44.950 --> 00:02:49.050 Let's say that this city right here is a city on the coast, 00:02:49.050 --> 00:02:51.300 so it's really close to the ocean. 00:02:51.300 --> 00:02:54.180 And let's say this city decides to build a pump right here 00:02:54.180 --> 00:02:57.820 to access the freshwater in the aquifer below it. 00:02:57.820 --> 00:02:59.660 But the city has a lot of people in it 00:02:59.660 --> 00:03:01.000 and it needs a lot of water. 00:03:01.000 --> 00:03:03.820 The city might end up taking more and more freshwater 00:03:03.820 --> 00:03:05.890 at a rate too fast for the freshwater 00:03:05.890 --> 00:03:07.820 in the aquifer to replenish itself. 00:03:07.820 --> 00:03:09.080 Pulling up so much fresh water 00:03:09.080 --> 00:03:12.580 can create a space for the saltwater in the nearby ocean 00:03:12.580 --> 00:03:14.230 to come and take its place. 00:03:14.230 --> 00:03:15.600 The saltwater from the ocean 00:03:15.600 --> 00:03:17.730 would contaminate the freshwater supply, 00:03:17.730 --> 00:03:19.540 making it unpotable. 00:03:19.540 --> 00:03:21.710 But if this city decided to increase 00:03:21.710 --> 00:03:24.740 the amount of permeable surfaces and green spaces, 00:03:24.740 --> 00:03:27.220 then when it rains, the water will be able 00:03:27.220 --> 00:03:29.430 to seep back into the ground 00:03:29.430 --> 00:03:31.130 and it would refill the empty space 00:03:31.130 --> 00:03:33.550 the pump has created with freshwater. 00:03:33.550 --> 00:03:34.760 This would mean that freshwater 00:03:34.760 --> 00:03:36.570 would continually be replacing the water 00:03:36.570 --> 00:03:38.140 that was taken out by the pump, 00:03:38.140 --> 00:03:40.790 so saltwater intrusion couldn't happen. 00:03:40.790 --> 00:03:44.490 Adding more green spaces could also help with air pollution 00:03:44.490 --> 00:03:48.630 because plants absorb carbon dioxide and produce oxygen. 00:03:48.630 --> 00:03:51.730 Cities produce most of the world's air pollution. 00:03:51.730 --> 00:03:54.350 Part of this is because of urban sprawl. 00:03:54.350 --> 00:03:56.130 As cities become more populated, 00:03:56.130 --> 00:03:59.800 they expand into suburban areas and even exurban areas, 00:03:59.800 --> 00:04:01.830 the areas past the suburbs. 00:04:01.830 --> 00:04:03.360 All of that expansion leads people 00:04:03.360 --> 00:04:05.250 to live farther away from their jobs. 00:04:05.250 --> 00:04:08.700 This causes people to spend a lot more time in their cars 00:04:08.700 --> 00:04:10.610 on their long commutes to work. 00:04:10.610 --> 00:04:14.380 And these long lines of cars of people traveling to work 00:04:14.380 --> 00:04:17.330 emit a lot of carbon dioxide. 00:04:17.330 --> 00:04:19.820 Carbon dioxide is a greenhouse gas, 00:04:19.820 --> 00:04:22.470 which means it retains heat from the sun, 00:04:22.470 --> 00:04:24.660 and it contributes to climate change. 00:04:24.660 --> 00:04:26.930 But carbon dioxide isn't the only kind 00:04:26.930 --> 00:04:29.370 of air pollution that cities produce. 00:04:29.370 --> 00:04:30.890 Emissions from motor vehicles 00:04:30.890 --> 00:04:33.130 and industrial facilities in cities 00:04:33.130 --> 00:04:36.370 can create something called photochemical smog. 00:04:36.370 --> 00:04:39.280 This is the kind of hazy, smoky pollution 00:04:39.280 --> 00:04:42.310 that you can sometimes see in the horizon line of cities 00:04:42.310 --> 00:04:45.680 and it can be really harmful to eyes and lungs. 00:04:45.680 --> 00:04:48.480 So, how could urban planners solve this problem? 00:04:48.480 --> 00:04:50.320 Cities could cluster grocery stores, 00:04:50.320 --> 00:04:52.500 offices, and homes closer together, 00:04:52.500 --> 00:04:54.300 building up rather than out, 00:04:54.300 --> 00:04:57.080 making buildings taller rather than wider. 00:04:57.080 --> 00:04:59.570 This would mean that people wouldn't have to travel as far 00:04:59.570 --> 00:05:00.610 to get where they need, 00:05:00.610 --> 00:05:03.030 so people could rely on walking or biking 00:05:03.030 --> 00:05:05.310 rather than fossil-fuel-consuming vehicles. 00:05:05.310 --> 00:05:07.390 Cities can reduce vehicle emissions 00:05:07.390 --> 00:05:08.980 by making public transportation 00:05:08.980 --> 00:05:10.900 more convenient and affordable. 00:05:10.900 --> 00:05:12.200 This would mean that fewer people 00:05:12.200 --> 00:05:14.890 would need to use their own cars to get around 00:05:14.890 --> 00:05:17.440 and it would cut down on traffic. 00:05:17.440 --> 00:05:19.600 Also, cities could build more paths, 00:05:19.600 --> 00:05:21.480 sidewalks, and bike lanes 00:05:21.480 --> 00:05:22.920 so people wouldn't have to rely 00:05:22.920 --> 00:05:25.810 on fossil-fuel-consuming vehicles to get around. 00:05:25.810 --> 00:05:27.510 Encouraging walking and biking 00:05:27.510 --> 00:05:30.220 could also improve the health of city residents. 00:05:30.220 --> 00:05:32.640 Cities can use clean forms of energy, 00:05:32.640 --> 00:05:36.320 like solar or wind power, to reduce air pollution. 00:05:36.320 --> 00:05:39.130 They could also use fossil fuels more effectively 00:05:39.130 --> 00:05:41.640 through energy co-generation, 00:05:41.640 --> 00:05:43.270 which is a way of using the heat 00:05:43.270 --> 00:05:45.770 given off by the burning of fossil fuels, 00:05:45.770 --> 00:05:47.850 not just the electricity. 00:05:47.850 --> 00:05:50.600 By creating goals and sustainable plans, 00:05:50.600 --> 00:05:53.350 cities can help reduce air and water pollution 00:05:53.350 --> 00:05:56.270 and increase the wellbeing of their residents. 00:05:56.270 --> 00:05:59.060 The city of Espoo in Finland, for example, 00:05:59.060 --> 00:06:03.240 was recognized as a world leader in energy sustainability 00:06:03.240 --> 00:06:07.400 for its goal to become carbon neutral by the year 2030. 00:06:07.400 --> 00:06:10.620 Cities are centers for economic development, innovation, 00:06:10.620 --> 00:06:13.540 social and cultural diversity, and jobs. 00:06:13.540 --> 00:06:14.940 With some urban planning, 00:06:14.940 --> 00:06:17.913 cities can become centers for sustainability too.

Impacts of Agricultural Practices

https://www.youtube.com/watch?v=dbEtcjNxGVQ

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https://www.youtube.com/api/timedtext?v=dbEtcjNxGVQ&ei=5VWUZdrVHpChp-oP9cKL6AM&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=A61D2511D93E3F793B6F9EB10CE2B67940068604.8D891B2C7440848316C904372EABEC4755EA4D41&key=yt8&lang=en&name=Default&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:00.310 --> 00:00:01.470 - [Voiceover] Hey there. 00:00:01.470 --> 00:00:05.180 Today I'm gonna cover the impacts of agricultural practices. 00:00:05.180 --> 00:00:08.800 And to do so, I'm gonna take you through my morning ritual. 00:00:08.800 --> 00:00:12.260 It sounds weird, but my bowl of multigrain Cheerios, 00:00:12.260 --> 00:00:16.180 and rice milk, and relaxing in my super comfy pajamas, 00:00:16.180 --> 00:00:19.610 they're all connected to intensive agricultural practices, 00:00:19.610 --> 00:00:23.770 in particular tilling, slash and burn farming, 00:00:23.770 --> 00:00:25.123 and fertilizers. 00:00:27.960 --> 00:00:30.710 So let's take a closer look at the Cheerios in my bowl. 00:00:32.350 --> 00:00:35.430 Most of my cereal here is actually composed of grains 00:00:35.430 --> 00:00:39.280 like oats, corn and barley, and growing grains 00:00:39.280 --> 00:00:41.700 typically starts with tilling in good quality soil 00:00:41.700 --> 00:00:43.053 with lots of nutrients. 00:00:44.310 --> 00:00:45.970 But what is tilling? 00:00:45.970 --> 00:00:49.310 In short, tilling is the process of turning soil, 00:00:49.310 --> 00:00:51.980 and it's really useful because loosening the soil 00:00:51.980 --> 00:00:54.470 allows farmers to easily control weeds 00:00:54.470 --> 00:00:56.730 and other pests at the surface of the soil, 00:00:56.730 --> 00:00:59.730 and it helps them to prepare the soil for seeding. 00:00:59.730 --> 00:01:02.890 Sometimes in areas that have been heavily farmed, 00:01:02.890 --> 00:01:05.540 the soil can become compacted over time. 00:01:05.540 --> 00:01:07.900 Here, tilling can help to break down the soil 00:01:07.900 --> 00:01:11.120 into smaller pieces, called soil aggregates, 00:01:11.120 --> 00:01:13.373 and allow for easier crop planting. 00:01:15.110 --> 00:01:18.260 Now, tillage has been done for thousands of years, 00:01:18.260 --> 00:01:20.180 but it's changed a lot from the past 00:01:20.180 --> 00:01:23.050 when we used human labor and big draft animals 00:01:23.050 --> 00:01:26.370 to till small fields, usually once per year. 00:01:26.370 --> 00:01:29.550 Nowadays, in the era of industrialized agriculture, 00:01:29.550 --> 00:01:32.390 large-scale farmers use heavy mechanized equipment 00:01:32.390 --> 00:01:35.893 that can till thousands of acres multiple times per year. 00:01:37.470 --> 00:01:42.210 Okay, we're growing more so we're tilling more, a lot more. 00:01:42.210 --> 00:01:46.640 And while soil tillage can help to loosen and aerate soil, 00:01:46.640 --> 00:01:49.070 what do you think happens when heavy machinery 00:01:49.070 --> 00:01:51.420 passes over the land and tears up the surface 00:01:51.420 --> 00:01:53.143 multiple times per year? 00:01:54.610 --> 00:01:57.800 This practice of repeated intensive tilling 00:01:57.800 --> 00:02:00.300 compacts the lower layers of the soil 00:02:00.300 --> 00:02:02.650 and loosens the top soil to the point that 00:02:02.650 --> 00:02:07.490 it loses the ability to hold water and nutrients in place. 00:02:07.490 --> 00:02:10.570 Tillage also reduces any leftover crop residue 00:02:10.570 --> 00:02:11.750 like plant stocks. 00:02:11.750 --> 00:02:14.410 So, in turn, the exposed soil surface 00:02:14.410 --> 00:02:17.320 becomes really vulnerable to wind and rain, 00:02:17.320 --> 00:02:19.900 because nothing is really holding the soil down 00:02:19.900 --> 00:02:21.770 or providing cover. 00:02:21.770 --> 00:02:23.513 So, what happens next? 00:02:24.950 --> 00:02:27.980 Loose soil can start to collect in surface runoff 00:02:27.980 --> 00:02:30.520 and become displaced through erosion. 00:02:30.520 --> 00:02:33.300 When this happens, soil, organic matter, 00:02:33.300 --> 00:02:36.673 and nutrients are literally washed or blown away. 00:02:37.720 --> 00:02:39.440 Who would have thought that these Cheerios 00:02:39.440 --> 00:02:41.940 were doing so much damage? 00:02:41.940 --> 00:02:43.790 But, there's some good news here too. 00:02:45.150 --> 00:02:47.730 Low till or no till farming alternatives 00:02:47.730 --> 00:02:50.640 can alleviate some of these problems thankfully. 00:02:50.640 --> 00:02:54.620 In the Palouse, a huge agricultural area in the Western US 00:02:54.620 --> 00:02:57.290 where a lot of grains, just like the ones in my cereal, 00:02:57.290 --> 00:02:58.160 are grown. 00:02:58.160 --> 00:03:02.140 No till farming is really important because the fields are, 00:03:02.140 --> 00:03:04.590 well, they're really steep and hilly 00:03:04.590 --> 00:03:07.390 and wind and rain can cause a lot of erosion 00:03:07.390 --> 00:03:10.020 when the soil is heavily tilled. 00:03:10.020 --> 00:03:13.340 By not tilling the fields, farmers can prevent soil erosion, 00:03:13.340 --> 00:03:16.480 and, more importantly, from my perspective, 00:03:16.480 --> 00:03:20.320 make sure that they can grow lots of grains for my Cheerios 00:03:20.320 --> 00:03:23.183 But wait, cereals aren't complete without milk. 00:03:25.240 --> 00:03:27.140 Now, I personally like rice milk 00:03:27.140 --> 00:03:29.020 because I'm lactose sensitive, 00:03:29.020 --> 00:03:31.360 and rice is commonly grown in temperate 00:03:31.360 --> 00:03:34.200 and tropical regions, oftentimes in areas where 00:03:34.200 --> 00:03:37.963 soil quality isn't the best and nutrients are lacking. 00:03:38.820 --> 00:03:42.500 So, how do farmers get nutrients back into the soil? 00:03:42.500 --> 00:03:45.090 And, more importantly, how do they grow rice 00:03:45.090 --> 00:03:47.300 for my rice milk at breakfast? 00:03:47.300 --> 00:03:50.530 Well, they often use slash and burn farming. 00:03:50.530 --> 00:03:54.330 So, like the name, forest plots are slashed or cut, 00:03:54.330 --> 00:03:56.700 left to dry and then burned. 00:03:56.700 --> 00:03:59.440 The ash left over from the burning fertilizes the soil. 00:03:59.440 --> 00:04:01.733 But, it's only a temporary benefit. 00:04:02.920 --> 00:04:05.030 After about three to five years, 00:04:05.030 --> 00:04:06.900 the productivity of slashed and burned plots 00:04:06.900 --> 00:04:09.780 goes down really quickly due to the loss of nutrients 00:04:09.780 --> 00:04:12.170 and as weeds start to grow again. 00:04:12.170 --> 00:04:14.840 When this happens, farmers simply abandon the field, 00:04:14.840 --> 00:04:18.090 move over to a new area and repeat the process. 00:04:18.090 --> 00:04:20.970 But it can take decades for these plots to recover 00:04:20.970 --> 00:04:23.370 once they've been slashed, burned, and farmed, 00:04:23.370 --> 00:04:25.913 and this practice can become a vicious cycle. 00:04:26.770 --> 00:04:29.770 In the Amazon, for example, people in rural areas 00:04:29.770 --> 00:04:32.210 rely on slash-and-burn so that they can make money 00:04:32.210 --> 00:04:35.040 selling the crops they grow or create open pastures 00:04:35.040 --> 00:04:36.810 for animals to graze. 00:04:36.810 --> 00:04:40.480 In the process, thousands of acres are burned each year. 00:04:40.480 --> 00:04:43.110 And as these trees and plants burn, 00:04:43.110 --> 00:04:45.360 enormous amounts of greenhouse gases, 00:04:45.360 --> 00:04:47.850 mainly carbon dioxide are produced, 00:04:47.850 --> 00:04:49.870 which contribute to climate change. 00:04:49.870 --> 00:04:53.260 And, sometimes too, fires may not be well-managed 00:04:53.260 --> 00:04:57.823 and can burn out of control causing huge, costly wildfires. 00:04:59.180 --> 00:05:00.840 But it's not all doom and gloom 00:05:00.840 --> 00:05:02.960 when it comes to my tasty rice milk. 00:05:02.960 --> 00:05:04.890 There are alternatives to slash and burn 00:05:04.890 --> 00:05:08.180 which include applying animal fertilizer like manure 00:05:08.180 --> 00:05:10.770 on used plots to add nutrients back to the soil 00:05:10.770 --> 00:05:14.270 or using alley cropping in which trees or other vegetation 00:05:14.270 --> 00:05:16.530 are planted between crops to help keep nutrients 00:05:16.530 --> 00:05:18.023 and moisture in the soil. 00:05:18.990 --> 00:05:21.380 All right, by now I've eaten my breakfast 00:05:21.380 --> 00:05:25.140 and I'm relaxing in my super comfy cotton pajamas, 00:05:25.140 --> 00:05:27.640 but turns out that cotton is actually 00:05:27.640 --> 00:05:29.410 a really finicky crop to grow 00:05:29.410 --> 00:05:32.240 and it requires a lot of fertilizers. 00:05:32.240 --> 00:05:34.780 And fertilizers help plants grow, 00:05:34.780 --> 00:05:36.863 but that's not a bad thing, right? 00:05:37.730 --> 00:05:38.740 And it's not. 00:05:38.740 --> 00:05:41.860 In fact, we've been using fertilizers for a millennia. 00:05:41.860 --> 00:05:45.030 For thousands of years, people have used natural fertilizers 00:05:45.030 --> 00:05:47.780 to replenish or increase nutrients in the soil 00:05:47.780 --> 00:05:49.420 and promote plant growth. 00:05:49.420 --> 00:05:51.960 Natural fertilizers meant that farmers use things 00:05:51.960 --> 00:05:55.740 like leftover crops, manure, wood ash, ground bones, 00:05:55.740 --> 00:05:58.933 fish or fish parts, and bird and bat poop. 00:06:00.170 --> 00:06:03.080 In the early 1800s though, scientists discovered 00:06:03.080 --> 00:06:05.430 that nitrogen, phosphorus, and potassium 00:06:05.430 --> 00:06:07.130 were key to plant growth. 00:06:07.130 --> 00:06:09.620 And in time, many farmers began to switch 00:06:09.620 --> 00:06:12.430 from natural fertilizers to artificial fertilizers 00:06:12.430 --> 00:06:14.830 with higher concentrations of these nutrients 00:06:14.830 --> 00:06:17.130 which greatly increased crop yields. 00:06:17.130 --> 00:06:18.850 In other words, farmers could grow more, 00:06:18.850 --> 00:06:21.913 and that means more cotton and more comfy pajamas. 00:06:22.990 --> 00:06:25.130 But, there's a catch. 00:06:25.130 --> 00:06:27.050 Now that farmers are growing more crops 00:06:27.050 --> 00:06:29.250 than any time in history, we've learned 00:06:29.250 --> 00:06:31.360 that there are impacts to using large amounts 00:06:31.360 --> 00:06:33.800 of very concentrated fertilizers. 00:06:33.800 --> 00:06:36.730 Applying too much fertilizer can pollute runoff water 00:06:36.730 --> 00:06:40.870 with excess fertilizer and pollute local surface waters. 00:06:40.870 --> 00:06:43.270 As nutrient rich materials like fertilizer 00:06:43.270 --> 00:06:46.100 make their way into nearby rivers, lakes, and oceans, 00:06:46.100 --> 00:06:49.210 they can cause major problems in the balance of nutrients 00:06:49.210 --> 00:06:51.660 in marine ecosystems. 00:06:51.660 --> 00:06:53.860 When too many nutrients from fertilizers 00:06:53.860 --> 00:06:57.320 saturate a body of water, called eutrophication, 00:06:57.320 --> 00:07:00.270 these nutrients feed the rapid growth of algae. 00:07:00.270 --> 00:07:02.410 In turn, these massive algae blooms 00:07:02.410 --> 00:07:04.870 can suck up all the oxygen in bodies of water 00:07:04.870 --> 00:07:08.200 and lead to enormous fish die-offs called dead zones. 00:07:08.200 --> 00:07:11.160 Quite a morbid situation and very different 00:07:11.160 --> 00:07:13.403 from the happy unicorns of my pajamas. 00:07:14.250 --> 00:07:17.050 But, there's ways to reduce the amount of fertilizers 00:07:17.050 --> 00:07:18.720 released into waterways. 00:07:18.720 --> 00:07:22.100 Farmers can limit the amount of fertilizer that they apply 00:07:22.100 --> 00:07:25.500 or use compost, which is decomposed organic material, 00:07:25.500 --> 00:07:28.640 as a fertilizer which tends to have lower and safer levels 00:07:28.640 --> 00:07:30.840 of nitrates and phosphates. 00:07:30.840 --> 00:07:32.370 And, there you have it. 00:07:32.370 --> 00:07:35.060 Common agricultural practices and their impacts 00:07:35.060 --> 00:07:38.003 in a nutshell, or really, a bowl of cereal.

Global wind patterns

https://www.youtube.com/watch?v=xp50_ixPOhY

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https://www.youtube.com/api/timedtext?v=xp50_ixPOhY&ei=5VWUZcqUF_nDmLAP9dCqkAk&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=A323E9088CA64B9139BBDB899CFCC5825DB9FF69.5962D77FCA061C1B4A0782C69EF969C4CB8BB7BB&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.340 --> 00:00:01.570 - [Instructor] Today, we're going to talk 00:00:01.570 --> 00:00:03.623 about Global Wind Patterns. 00:00:04.730 --> 00:00:08.860 Wind determines more than just the best places to fly kite. 00:00:08.860 --> 00:00:12.200 Global wind patterns help control where it rains. 00:00:12.200 --> 00:00:15.020 What kinds of species can survive in an area 00:00:15.020 --> 00:00:18.720 and even where tropical rainforests and deserts are located. 00:00:18.720 --> 00:00:20.610 In other words, global wind patterns 00:00:20.610 --> 00:00:22.460 are really important to life. 00:00:22.460 --> 00:00:25.610 And one of the reasons we have global wind patterns at all, 00:00:25.610 --> 00:00:28.220 is actually because of the sun. 00:00:28.220 --> 00:00:30.660 Sunlight shines on the earth like this. 00:00:30.660 --> 00:00:34.100 As you can see, the sunlight hits the equator directly 00:00:34.100 --> 00:00:38.330 but, the light hits the North and South poles at an angle 00:00:38.330 --> 00:00:42.430 kind of skimming the surface like this. 00:00:42.430 --> 00:00:45.060 So the equator is getting direct sunlight 00:00:46.290 --> 00:00:49.163 and the poles are only getting indirect sunlight. 00:00:50.500 --> 00:00:51.920 With all the direct sunlight, 00:00:51.920 --> 00:00:54.580 the air on the equator gets really hot. 00:00:54.580 --> 00:00:57.290 And the air around the poles doesn't heat up as much 00:00:57.290 --> 00:01:00.120 because it's only getting indirect sunlight. 00:01:00.120 --> 00:01:01.570 And you may be thinking, 00:01:01.570 --> 00:01:04.000 what does this all have to do with airflow? 00:01:04.000 --> 00:01:05.880 Let's take a closer look at the equator 00:01:05.880 --> 00:01:07.530 to see what's happening. 00:01:07.530 --> 00:01:10.100 So imagine you're standing on a piece of land 00:01:10.100 --> 00:01:11.580 near the equator. 00:01:11.580 --> 00:01:14.600 When the direct sunlight hits the equator, 00:01:14.600 --> 00:01:18.610 the hot air near the ground begins to rise up 00:01:18.610 --> 00:01:20.720 because hot air rises. 00:01:20.720 --> 00:01:24.750 All the direct sunlight also causes evaporation to increase, 00:01:24.750 --> 00:01:27.120 which means that this air that's rising up right here 00:01:27.120 --> 00:01:29.530 is both hot and moist. 00:01:29.530 --> 00:01:31.960 But once all this hot moist air reaches 00:01:31.960 --> 00:01:33.400 a high enough altitude, 00:01:33.400 --> 00:01:36.660 it begins to expand and cool down. 00:01:36.660 --> 00:01:40.540 The water vapor in the air begins to condense into clouds 00:01:40.540 --> 00:01:43.193 and it eventually falls as rain around the equator. 00:01:44.370 --> 00:01:47.630 The air which is now cool and holds less moisture, 00:01:47.630 --> 00:01:51.930 sinks down to the ground because cold air sinks 00:01:51.930 --> 00:01:53.270 and the cycle repeats 00:01:53.270 --> 00:01:56.973 with the hot moist air rising and the cool dry air falling. 00:01:58.560 --> 00:02:00.710 And the fact that the hot air rises up, 00:02:00.710 --> 00:02:02.750 it means that this area right here 00:02:02.750 --> 00:02:04.803 is an area of low pressure. 00:02:07.430 --> 00:02:09.130 Because the air is rising up, 00:02:09.130 --> 00:02:11.210 it creates a space for cooler air 00:02:11.210 --> 00:02:15.250 in surrounding areas to move in and take its place. 00:02:15.250 --> 00:02:18.340 And over here, where the cool air is coming down 00:02:18.340 --> 00:02:21.120 to the ground, that's an area of high pressure 00:02:22.640 --> 00:02:25.640 because all of that cool dry air is coming down 00:02:25.640 --> 00:02:28.030 and pushing the air below it away. 00:02:28.030 --> 00:02:31.380 This cyclical movement of air create something 00:02:31.380 --> 00:02:35.363 called a convection cell. 00:02:37.520 --> 00:02:39.540 If the earth wasn't spinning, 00:02:39.540 --> 00:02:42.930 we would just have one convection cell in each hemisphere 00:02:42.930 --> 00:02:45.300 where the air would heat up at the equator, 00:02:45.300 --> 00:02:48.940 move up towards the poles and sink down. 00:02:48.940 --> 00:02:50.900 And in the 18th century, 00:02:50.900 --> 00:02:52.420 this was how some scientists 00:02:52.420 --> 00:02:54.760 believe global wind patterns worked. 00:02:54.760 --> 00:02:57.210 But, because the earth is spinning, 00:02:57.210 --> 00:03:01.490 the earth's rotation pushes air masses from East to West. 00:03:01.490 --> 00:03:04.700 This movement of air creates a clockwise pattern 00:03:04.700 --> 00:03:07.870 in the Northern hemisphere and a counter-clockwise pattern 00:03:07.870 --> 00:03:09.720 in the Southern hemisphere. 00:03:09.720 --> 00:03:11.763 This is called the Coriolis effect. 00:03:12.770 --> 00:03:16.070 And this movement of air because of the earth's spin, 00:03:16.070 --> 00:03:19.670 causes us to actually get three convection cells 00:03:19.670 --> 00:03:21.550 in each hemisphere. 00:03:21.550 --> 00:03:24.000 These two, the two closest to the equator 00:03:24.000 --> 00:03:26.910 are called the hadley cells. 00:03:26.910 --> 00:03:28.710 They're between the equator 00:03:28.710 --> 00:03:32.040 and the 30 degree latitude marks in both hemispheres. 00:03:32.040 --> 00:03:35.510 These next two are called the ferrel cells 00:03:35.510 --> 00:03:37.500 or the temperate cells. 00:03:37.500 --> 00:03:40.890 And these are located between a 30 and 60 degree marks 00:03:40.890 --> 00:03:43.550 in both the Northern and Southern hemispheres. 00:03:43.550 --> 00:03:46.200 And lastly, we have the polar cells 00:03:46.200 --> 00:03:49.730 which as you can probably guess are right by the poles. 00:03:49.730 --> 00:03:53.720 So we have polar cells up here at the North pole 00:03:53.720 --> 00:03:58.210 and we also have polar cells down here at the South pole. 00:03:58.210 --> 00:04:02.470 And these convection cells create prevailing winds 00:04:02.470 --> 00:04:05.840 that move heat and moisture around the earth. 00:04:05.840 --> 00:04:08.040 Let's take a look at what happens in the bottom half 00:04:08.040 --> 00:04:09.470 of each convection cell, 00:04:09.470 --> 00:04:11.520 the parts closer to the ground. 00:04:11.520 --> 00:04:13.730 Because these parts are closer to us, 00:04:13.730 --> 00:04:16.560 we experience the air movement as wind. 00:04:16.560 --> 00:04:20.030 So on the bottom of this convection cell, the hadley cells, 00:04:20.030 --> 00:04:23.430 the cold air is moving towards the equator. 00:04:23.430 --> 00:04:25.420 So that means that the prevailing winds 00:04:25.420 --> 00:04:28.300 would also be moving towards the equator. 00:04:28.300 --> 00:04:31.500 Winds are named after where they come from. 00:04:31.500 --> 00:04:35.030 So, these two winds are called the Northeast 00:04:35.030 --> 00:04:36.650 and Southeast trade winds 00:04:36.650 --> 00:04:39.480 because they come from the Northeast and Southeast 00:04:39.480 --> 00:04:41.560 and they move West. 00:04:41.560 --> 00:04:44.450 And the bottom halves of the ferrel convection cells, 00:04:44.450 --> 00:04:46.900 take cool air from a 30 degree line 00:04:46.900 --> 00:04:50.610 and pull it towards the 60 degree latitude line. 00:04:50.610 --> 00:04:52.920 This creates the westerlies. 00:04:52.920 --> 00:04:55.270 And they're called the westerlies because they pull air 00:04:55.270 --> 00:04:57.230 from the West to the East 00:04:57.230 --> 00:04:59.770 and the bottom halves of the polar convection cells 00:04:59.770 --> 00:05:01.920 take the cool air from the poles 00:05:01.920 --> 00:05:04.570 and sweep it to the 60 degree latitude lines 00:05:04.570 --> 00:05:07.480 and this creates the Easterlies winds. 00:05:07.480 --> 00:05:10.300 It's important to remember that everything in this diagram 00:05:10.300 --> 00:05:12.650 is just an overall model. 00:05:12.650 --> 00:05:15.150 Global wind patterns are even more complicated 00:05:15.150 --> 00:05:18.490 because water covered areas and land covered areas 00:05:18.490 --> 00:05:21.250 absorb solar energy differently. 00:05:21.250 --> 00:05:23.550 These prevailing wind patterns distribute heat 00:05:23.550 --> 00:05:26.980 and precipitation unevenly between the tropics, temperate 00:05:26.980 --> 00:05:28.970 and polar regions of the earth. 00:05:28.970 --> 00:05:33.830 And this uneven distribution creates different biomes. 00:05:33.830 --> 00:05:37.360 And this helps determine what species can survive where. 00:05:37.360 --> 00:05:40.620 The tropical rainforest will be in the low pressure areas 00:05:40.620 --> 00:05:42.400 near the equator. 00:05:42.400 --> 00:05:44.980 And right here, between the polar and ferrel cells 00:05:44.980 --> 00:05:47.270 is another area of low pressure. 00:05:47.270 --> 00:05:51.220 Just like near the equator, hot moist air rises here 00:05:51.220 --> 00:05:54.600 causing more precipitation in the surrounding areas. 00:05:54.600 --> 00:05:56.970 So along this longitudinal line, 00:05:56.970 --> 00:05:59.410 you'll find many coniferous forests 00:05:59.410 --> 00:06:02.550 that thrive because of all that precipitation. 00:06:02.550 --> 00:06:04.910 And there's a high pressure area right here 00:06:04.910 --> 00:06:06.940 where the cool dry air sinks down 00:06:06.940 --> 00:06:08.900 so there's not as much precipitation. 00:06:08.900 --> 00:06:10.410 The air is drier here. 00:06:10.410 --> 00:06:14.040 You'll find many deserts along this line and this line. 00:06:14.040 --> 00:06:15.880 So even though convection cells 00:06:15.880 --> 00:06:18.240 and prevailing winds are invisible, 00:06:18.240 --> 00:06:20.853 the ways they shape the environment are not.

El Niño and La Niña

https://www.youtube.com/watch?v=U2-ACg2kbTA

vtt

https://www.youtube.com/api/timedtext?v=U2-ACg2kbTA&ei=5VWUZevUE_Stp-oPwdexuAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=5B01CF1E1169AAC6850E99D6F8557CA57ACAB42F.8A3D1A0993BD6DE3DA6779E1DEE7C64A67AC0063&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.000 --> 00:00:02.720 - [Instructor] Every few years you might hear about El Nino 00:00:02.720 --> 00:00:03.553 in the news. 00:00:03.553 --> 00:00:06.140 And this also might come with powerful images 00:00:06.140 --> 00:00:10.179 of flooding and rainfall, but it is not just a storm. 00:00:10.179 --> 00:00:13.470 It's actually a climate pattern that takes place 00:00:13.470 --> 00:00:15.623 in the Pacific Ocean. 00:00:16.680 --> 00:00:17.830 And we'll get a little bit more 00:00:17.830 --> 00:00:21.130 into what that actually means. 00:00:21.130 --> 00:00:23.826 Now, fun fact about how El Nino is named 00:00:23.826 --> 00:00:25.800 is that a long time ago, 00:00:25.800 --> 00:00:28.760 South American fishermen noticed one December 00:00:28.760 --> 00:00:31.810 that the Pacific Ocean was actually warmer 00:00:31.810 --> 00:00:33.090 than it normally is. 00:00:33.090 --> 00:00:35.470 And this brought about an abundance of fish. 00:00:35.470 --> 00:00:38.770 So because they were grateful, they named this event 00:00:38.770 --> 00:00:41.550 El Nino to correlate with the commemoration 00:00:41.550 --> 00:00:45.560 of Christ in this part of the world during Christmas time. 00:00:45.560 --> 00:00:47.940 So that's a little trivia you can keep in your back pocket, 00:00:47.940 --> 00:00:52.330 but back to what we hear about El Nino in the news, 00:00:52.330 --> 00:00:56.031 you might also notice that it isn't talked about 00:00:56.031 --> 00:00:59.571 every single year and that's because El Nino comes around 00:00:59.571 --> 00:01:04.220 every two to seven years on average. 00:01:04.220 --> 00:01:07.240 And scientists are still not sure exactly 00:01:07.240 --> 00:01:10.540 what triggers El Nino, but they know what signs to look 00:01:10.540 --> 00:01:15.450 for once it is approaching or once we're in that event. 00:01:15.450 --> 00:01:16.740 So even though it's true 00:01:16.740 --> 00:01:21.020 that El Nino can bring about heavy rainfall and flooding, 00:01:21.020 --> 00:01:24.280 it can also cause severe drought. 00:01:24.280 --> 00:01:28.880 So it's really important to note that different regions 00:01:28.880 --> 00:01:33.880 around the world experience different effects of El Nino. 00:01:37.120 --> 00:01:39.810 And we'll see a few examples of those. 00:01:39.810 --> 00:01:43.205 So sometimes an El Nino year is actually, 00:01:43.205 --> 00:01:45.160 a little bit helpful and might bring 00:01:45.160 --> 00:01:46.980 about some much needed rain, 00:01:46.980 --> 00:01:50.796 but other times bigger and more severe El Nino events 00:01:50.796 --> 00:01:55.796 can bring devastating weather events all across the globe. 00:01:56.470 --> 00:01:58.130 So we can look at the different effects 00:01:58.130 --> 00:02:01.470 by looking at the biggest El Nino on record, 00:02:01.470 --> 00:02:04.720 during the 1997 to 1998 season. 00:02:04.720 --> 00:02:07.450 And we can see how different regions were affected. 00:02:07.450 --> 00:02:09.940 So for example, in California, 00:02:09.940 --> 00:02:13.380 we saw very destructive mudslides. 00:02:13.380 --> 00:02:16.433 In Ecuador, there was heavy rainfall. 00:02:19.286 --> 00:02:22.740 And in Indonesia there was actually 00:02:22.740 --> 00:02:25.873 extreme drought and fire. 00:02:30.170 --> 00:02:33.440 And overall, this El Nino event was really large 00:02:33.440 --> 00:02:35.720 and very destructive, there are estimates 00:02:35.720 --> 00:02:40.720 of about $36 billion in damage to infrastructure. 00:02:41.300 --> 00:02:43.050 So you might be asking yourself, 00:02:43.050 --> 00:02:46.950 how can the same climate pattern caused such drastically 00:02:46.950 --> 00:02:49.490 different effects around the globe? 00:02:49.490 --> 00:02:51.220 So to understand this better, 00:02:51.220 --> 00:02:53.250 we have to look at what a quote unquote 00:02:53.250 --> 00:02:55.340 normal year looks like. 00:02:55.340 --> 00:02:57.570 So this is a map of the Pacific Ocean. 00:02:57.570 --> 00:03:01.735 And normally there are winds that are pushing warm water 00:03:01.735 --> 00:03:03.620 towards the West. 00:03:03.620 --> 00:03:07.118 So towards Asia and the Pacific islands 00:03:07.118 --> 00:03:11.493 and warm water accumulates on this side, 00:03:12.860 --> 00:03:14.280 on the other side on the East side, 00:03:14.280 --> 00:03:16.080 near Central and South America, 00:03:16.080 --> 00:03:19.468 we have an accumulation of cool water. 00:03:19.468 --> 00:03:23.983 And this is also due to a process called upwelling, 00:03:25.290 --> 00:03:29.880 where cold water from the bottom of the ocean is pushed up. 00:03:29.880 --> 00:03:32.340 And during these conditions in general, 00:03:32.340 --> 00:03:35.403 you would see less rain on this side. 00:03:39.726 --> 00:03:42.809 Or you'll see more rain on this side. 00:03:44.940 --> 00:03:47.610 Okay, so now let's look at what changes during 00:03:47.610 --> 00:03:49.460 an El Nino year. 00:03:49.460 --> 00:03:50.830 So during an El Nino year, 00:03:50.830 --> 00:03:54.642 you still have trade winds pushing water West, 00:03:54.642 --> 00:03:58.775 but they're much weaker and that's the key here, 00:03:58.775 --> 00:04:01.320 that your trade winds are weakened, 00:04:01.320 --> 00:04:04.300 which means less water is pushed 00:04:04.300 --> 00:04:07.820 towards Asia and the Pacific islands. 00:04:07.820 --> 00:04:10.457 So compared to a normal year, 00:04:10.457 --> 00:04:15.050 you actually have cooler water that starts to accumulate 00:04:15.050 --> 00:04:17.720 on this side of the Pacific Ocean. 00:04:17.720 --> 00:04:19.630 Meanwhile, on the other side, 00:04:19.630 --> 00:04:23.680 since less water is being pushed away from Central 00:04:23.680 --> 00:04:28.590 and South America, you get more warm water accumulating. 00:04:28.590 --> 00:04:31.570 So now you can think back to those South American fishermen 00:04:31.570 --> 00:04:34.980 and why they were experiencing warm water 00:04:34.980 --> 00:04:38.150 along their coasts during an El Nino year. 00:04:38.150 --> 00:04:40.090 So this map does a really great job 00:04:40.090 --> 00:04:42.090 of showing the temperature gradient, 00:04:42.090 --> 00:04:45.436 and how this warm water is staying along the coast 00:04:45.436 --> 00:04:49.760 of Central and South America during an El Nino year. 00:04:49.760 --> 00:04:54.760 Meanwhile, you have cold water accumulating towards Asia. 00:04:55.460 --> 00:04:58.430 And this causes a lot of changes 00:04:58.430 --> 00:05:00.470 that affect weather patterns. 00:05:00.470 --> 00:05:03.480 So in the US the Pacific jet stream 00:05:03.480 --> 00:05:06.193 moves South of its neutral position. 00:05:07.140 --> 00:05:08.970 And because of that, we start to see 00:05:08.970 --> 00:05:13.360 that the Northern US and Canada are actually warmer 00:05:13.360 --> 00:05:15.910 and drier than they usually are. 00:05:15.910 --> 00:05:19.200 The US Gulf Coast and Southwest regions 00:05:19.200 --> 00:05:21.170 of the US are wetter, 00:05:21.170 --> 00:05:24.460 and are there more increased risk for flooding. 00:05:24.460 --> 00:05:29.210 And South Asia and the Pacific islands experience warmer 00:05:29.210 --> 00:05:32.259 and drier conditions that lead to drought 00:05:32.259 --> 00:05:36.460 on the other side of the Pacific Ocean. 00:05:36.460 --> 00:05:40.150 So now you start to see how one event El Nino, 00:05:40.150 --> 00:05:44.880 this one climate pattern can cause both flooding 00:05:44.880 --> 00:05:47.150 on one side of the Pacific Ocean, 00:05:47.150 --> 00:05:50.870 while also causing drought on the other side. 00:05:50.870 --> 00:05:53.820 And finally, we'll end with La Nina, 00:05:53.820 --> 00:05:56.190 because you might hear this come up too 00:05:56.190 --> 00:05:57.711 when talking about El Nino. 00:05:57.711 --> 00:06:00.310 So La Nina is essentially the opposite. 00:06:00.310 --> 00:06:02.020 So if you've got El Nino down, 00:06:02.020 --> 00:06:04.090 you can start to understand what happens 00:06:04.090 --> 00:06:06.680 during La Nina event. 00:06:06.680 --> 00:06:08.030 So in this situation, 00:06:08.030 --> 00:06:13.030 our trade winds are getting stronger than our normal years. 00:06:13.970 --> 00:06:15.970 So they're pushing water West, 00:06:15.970 --> 00:06:18.780 but they're doing so even more 00:06:18.780 --> 00:06:20.910 than we normally would expect. 00:06:20.910 --> 00:06:24.520 So this warm water that accumulates 00:06:24.520 --> 00:06:28.440 towards Asia actually accumulates more West 00:06:28.440 --> 00:06:32.320 than it would during our normal years. 00:06:32.320 --> 00:06:35.970 And on the other side, we have even more upwelling 00:06:37.280 --> 00:06:40.610 causing this cold water to accumulate. 00:06:40.610 --> 00:06:45.220 So the next time you hear El Nino or LA Nina in the news, 00:06:45.220 --> 00:06:46.860 you'll know that they're not just talking 00:06:46.860 --> 00:06:50.050 about a really big storm, but they're actually talking 00:06:50.050 --> 00:06:53.072 about climate patterns that can affect weather 00:06:53.072 --> 00:06:55.223 all across the globe.

Soil Texture Triangle

https://www.youtube.com/watch?v=Bn9ul9lxIdg

vtt

https://www.youtube.com/api/timedtext?v=Bn9ul9lxIdg&ei=5VWUZdL_KY_YxN8PkJqK2A0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=E671E17F223E01808D2F404000138B6B5C7BD041.D95926EF6113E1C26882319F4BED5721BC99A7D4&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.670 --> 00:00:04.150 - [Narrator] Today, we're going to talk about soil. 00:00:04.150 --> 00:00:05.450 And you've probably noticed 00:00:05.450 --> 00:00:07.830 that there are many different kinds of soils. 00:00:07.830 --> 00:00:09.160 The soil near a beach 00:00:09.160 --> 00:00:10.730 looks and feels very different 00:00:10.730 --> 00:00:12.620 than the soil in a forest. 00:00:12.620 --> 00:00:14.190 And part of the reason for that difference 00:00:14.190 --> 00:00:17.620 is something called soil texture. 00:00:17.620 --> 00:00:19.470 Soil texture. 00:00:19.470 --> 00:00:21.290 So when soil is formed, 00:00:21.290 --> 00:00:24.140 different types of rock break down 00:00:24.140 --> 00:00:27.690 because of the wind and the rain and the weather. 00:00:27.690 --> 00:00:31.200 And they become differently sized particles. 00:00:31.200 --> 00:00:33.740 And the combination of these differently sized particles 00:00:33.740 --> 00:00:35.810 creates soil texture. 00:00:35.810 --> 00:00:37.420 So these differently sized particles 00:00:37.420 --> 00:00:39.740 can be broken down into three groups. 00:00:39.740 --> 00:00:41.483 We have sand. 00:00:43.744 --> 00:00:44.577 We have silt. 00:00:46.100 --> 00:00:47.803 And we have clay. 00:00:48.760 --> 00:00:52.370 Sand is made up of the larger and heavier particles of soil. 00:00:52.370 --> 00:00:55.430 Sand is around two millimeters. 00:00:55.430 --> 00:01:00.350 Two millimeters to 0.05 millimeters. 00:01:00.350 --> 00:01:03.720 Which is a 20th of a millimeter in diameter. 00:01:03.720 --> 00:01:06.553 And if it feels very gritty to touch. 00:01:08.570 --> 00:01:12.310 Clay on the other hand is the smallest particles of soil. 00:01:12.310 --> 00:01:16.680 It's around 0.002 millimeters and the smaller. 00:01:16.680 --> 00:01:18.640 That's a 500th of a millimeter. 00:01:18.640 --> 00:01:20.083 So really tiny. 00:01:20.930 --> 00:01:22.650 If you rubbed clay between your fingers, 00:01:22.650 --> 00:01:23.690 it would feel smooth, 00:01:23.690 --> 00:01:24.720 and you wouldn't be able to feel 00:01:24.720 --> 00:01:26.950 the individual particles of soil. 00:01:26.950 --> 00:01:29.870 And unlike sand, you can't even see clay's particles 00:01:29.870 --> 00:01:30.883 with a naked eye. 00:01:31.840 --> 00:01:35.240 Most people are familiar with sandy soils and clay soils, 00:01:35.240 --> 00:01:37.550 but silt is right in between. 00:01:37.550 --> 00:01:40.810 It's made of a particles between the size of clay and sand. 00:01:40.810 --> 00:01:45.410 So, it's between 0.05 millimeters 00:01:45.410 --> 00:01:48.283 and 0.002 millimeters. 00:01:50.610 --> 00:01:53.240 So when you think of silt imagine baking flour. 00:01:53.240 --> 00:01:55.180 It's powdery. 00:01:55.180 --> 00:01:56.013 Powdery. 00:01:58.280 --> 00:02:01.210 And it can be carried easily by wind and water. 00:02:01.210 --> 00:02:02.840 But soil is more complicated 00:02:02.840 --> 00:02:05.160 than just these three soil types. 00:02:05.160 --> 00:02:07.810 If you grabbed a handful of soil from outside your home, 00:02:07.810 --> 00:02:10.880 it would probably be a combination of all three soil types, 00:02:10.880 --> 00:02:13.010 along with some organic material. 00:02:13.010 --> 00:02:14.870 A soil's unique texture, 00:02:14.870 --> 00:02:18.040 that is it's combination of silt, sand, and clay, 00:02:18.040 --> 00:02:19.840 affects how plants will grow. 00:02:19.840 --> 00:02:22.560 And so gardeners often wanna know about their soil textures. 00:02:22.560 --> 00:02:25.210 They know what kinds of plants to grow in that area. 00:02:25.210 --> 00:02:26.600 This is where a helpful diagram 00:02:26.600 --> 00:02:29.610 called the soil texture triangle comes in. 00:02:29.610 --> 00:02:31.713 The soil texture triangle. 00:02:32.910 --> 00:02:35.660 The triangle allows us to place any soil sample 00:02:35.660 --> 00:02:38.670 into one of 12 different soil texture categories. 00:02:38.670 --> 00:02:40.850 The 12 different categories are broken up 00:02:40.850 --> 00:02:43.150 based on the percentage of silt, clay 00:02:43.150 --> 00:02:44.780 and sand in the soil. 00:02:44.780 --> 00:02:45.790 To see how this works, 00:02:45.790 --> 00:02:47.640 let's do an example. 00:02:47.640 --> 00:02:50.973 Let's say a gardener determined that a soil was 30% sand, 00:02:52.270 --> 00:02:53.513 40% silt, 00:02:54.430 --> 00:02:56.253 and 30% clay. 00:02:57.310 --> 00:03:00.080 So, how can we figure out what kind of soil have? 00:03:00.080 --> 00:03:03.570 First, we need to pick a side of the triangle to start with. 00:03:03.570 --> 00:03:05.070 I'll start with sand. 00:03:05.070 --> 00:03:07.160 We know we have 30% sand. 00:03:07.160 --> 00:03:09.800 So, we go along the side of the triangle, 00:03:09.800 --> 00:03:12.620 until we find that 30% mark, right here. 00:03:12.620 --> 00:03:14.630 We want to draw a line from this point, 00:03:14.630 --> 00:03:15.540 to the side of the triangle, 00:03:15.540 --> 00:03:17.790 that this arrow is pointing to. 00:03:17.790 --> 00:03:20.070 The arrow is pointing towards the clay side. 00:03:20.070 --> 00:03:23.570 So we draw the line from the 30% sand mark, 00:03:23.570 --> 00:03:26.120 through the triangle, to the clay side. 00:03:26.120 --> 00:03:26.973 Like this. 00:03:27.850 --> 00:03:29.080 And by drawing this line, 00:03:29.080 --> 00:03:31.190 we already know that our soil sample 00:03:31.190 --> 00:03:34.030 will fall into a category along this line. 00:03:34.030 --> 00:03:39.000 So, it could be so silt loam, loam, clay loam, or clay. 00:03:39.000 --> 00:03:41.020 But to figure out exactly where it is, 00:03:41.020 --> 00:03:42.810 let's draw another line. 00:03:42.810 --> 00:03:44.220 Let's do silt. 00:03:44.220 --> 00:03:46.153 The soil has 40% silt. 00:03:47.030 --> 00:03:48.900 We'll draw our line through the triangle 00:03:48.900 --> 00:03:51.263 towards the sand side because of this arrow. 00:03:52.690 --> 00:03:54.750 So again, we do the same thing. 00:03:54.750 --> 00:03:58.030 We go along the percent silt's line. 00:03:58.030 --> 00:04:00.160 We find the 40% mark. 00:04:00.160 --> 00:04:03.270 And we're gonna draw a line from 40% 00:04:03.270 --> 00:04:05.063 to the other side of the triangle. 00:04:06.020 --> 00:04:07.920 And you can see when we draw that line, 00:04:07.920 --> 00:04:11.323 these two lines intersect at the clay loam point. 00:04:12.300 --> 00:04:13.150 But to double-check, 00:04:13.150 --> 00:04:18.150 let's draw a line from the 30% clay mark, 00:04:19.300 --> 00:04:23.590 to the other side of the triangle, like this. 00:04:23.590 --> 00:04:25.470 As you can see, these three lines, 00:04:25.470 --> 00:04:27.300 all intersected at the same point, 00:04:27.300 --> 00:04:28.603 right here in clay loam. 00:04:30.020 --> 00:04:31.943 So we know that we have clay loam. 00:04:32.970 --> 00:04:36.250 Notice that the percentage of clay, silt and sand, 00:04:36.250 --> 00:04:38.860 all add up to a total of 100%. 00:04:38.860 --> 00:04:40.960 We could have picked any two of these lines, 00:04:40.960 --> 00:04:43.090 just two, to find our answer. 00:04:43.090 --> 00:04:45.810 We don't need to draw three lines every time. 00:04:45.810 --> 00:04:47.610 Let's do another example. 00:04:47.610 --> 00:04:50.370 Let's say our gardener has another plot, 00:04:50.370 --> 00:04:53.593 with soil that contains 58% sand, 00:04:54.560 --> 00:04:55.873 27% silt, 00:04:57.770 --> 00:04:59.693 and 15% clay. 00:05:00.890 --> 00:05:05.060 So, I'm gonna go along the percent sand part and find 58%. 00:05:05.060 --> 00:05:07.400 So that's pretty close to 60. 00:05:07.400 --> 00:05:09.490 And I'm gonna draw a line from 58%, 00:05:09.490 --> 00:05:11.740 to the other side of the triangle. 00:05:11.740 --> 00:05:13.260 Sometimes the example question, 00:05:13.260 --> 00:05:15.630 doesn't give us round numbers to work with. 00:05:15.630 --> 00:05:16.520 So when that's the case, 00:05:16.520 --> 00:05:18.270 I like to use a ruler or a straight edge, 00:05:18.270 --> 00:05:21.000 just to make sure that my lines are in the correct spot. 00:05:21.000 --> 00:05:24.500 Next I'm gonna use the 15% clay. 00:05:24.500 --> 00:05:27.160 So I find the 15% clay mark, 00:05:27.160 --> 00:05:30.810 and we draw a line for 15% to the other side of the triangle 00:05:30.810 --> 00:05:33.750 following the direction of the lines within the triangle. 00:05:33.750 --> 00:05:36.660 And we can see that our two lines 00:05:36.660 --> 00:05:39.470 intersect at this point. 00:05:39.470 --> 00:05:41.690 So, we have sandy loam. 00:05:41.690 --> 00:05:46.690 The gardener has these two plots clay loam and sandy loam. 00:05:48.550 --> 00:05:50.540 So when the gardener looks at her two plots, 00:05:50.540 --> 00:05:52.160 she might wanna take into account 00:05:52.160 --> 00:05:53.670 the different kinds of soil. 00:05:53.670 --> 00:05:56.230 Certain kinds of plants do better in clay loam soil 00:05:56.230 --> 00:05:58.970 while others would do better in the sandy loam soil. 00:05:58.970 --> 00:06:00.960 This is partly because clay soils, 00:06:00.960 --> 00:06:04.050 hold onto moisture for longer than sandy soils do. 00:06:04.050 --> 00:06:06.520 So the plants in the gardeners sandy loam plot, 00:06:06.520 --> 00:06:08.610 would do better if they had longer root systems 00:06:08.610 --> 00:06:10.800 to access water deeper in the ground. 00:06:10.800 --> 00:06:14.330 Or if they stored water inside themselves to access later. 00:06:14.330 --> 00:06:16.690 The gardener could use a gardening book or the internet 00:06:16.690 --> 00:06:18.340 to see what kinds of plants would do best 00:06:18.340 --> 00:06:20.320 in each of her two plots. 00:06:20.320 --> 00:06:23.353 Thanks for watching and happy soil identification.

Hess's law

https://www.youtube.com/watch?v=QU3zVec5I_M

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https://www.youtube.com/api/timedtext?v=QU3zVec5I_M&ei=5VWUZdD9GNm-mLAPrqqEgA4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=482E3D5D68285B4BECAF4EF0CC9AB25197940059.216A2F393A1B9670800BE7CAB0D4239E3AD2A848&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.710 --> 00:00:03.040 - [Instructor] Hess's law states that the overall change 00:00:03.040 --> 00:00:06.270 in enthalpy for a chemical reaction is equal to the sum 00:00:06.270 --> 00:00:09.333 of the enthalpy changes for each step. 00:00:09.333 --> 00:00:12.020 And this is independent of the path taken. 00:00:12.020 --> 00:00:14.900 So it doesn't matter what set of reactions you use. 00:00:14.900 --> 00:00:18.660 If you add up those reactions and they equal the reaction 00:00:18.660 --> 00:00:19.891 that you're trying to find, 00:00:19.891 --> 00:00:22.270 you can also sum the enthalpies 00:00:22.270 --> 00:00:25.090 to find the enthalpy change for the reaction. 00:00:25.090 --> 00:00:27.440 As an example, let's say we're trying to find the change 00:00:27.440 --> 00:00:31.490 in enthalpy for the reaction of carbon with hydrogen gas 00:00:31.490 --> 00:00:35.210 to form C2H2, which is acetylene. 00:00:35.210 --> 00:00:38.080 We can calculate the change in enthalpy for the formation 00:00:38.080 --> 00:00:42.600 of acetylene using these three reactions below. 00:00:42.600 --> 00:00:45.460 Our approach will involve looking at these three reactions 00:00:45.460 --> 00:00:48.440 and comparing them to the original reaction 00:00:48.440 --> 00:00:50.940 to see if we need to change anything. 00:00:50.940 --> 00:00:53.690 For example, if we look at reaction one, 00:00:53.690 --> 00:00:55.480 there's one mole of acetylene 00:00:55.480 --> 00:00:57.900 on the left side of the equation. 00:00:57.900 --> 00:01:00.980 And if we compare that to the original reaction, 00:01:00.980 --> 00:01:03.670 there's one mole of acetylene on the right side 00:01:03.670 --> 00:01:04.870 of the equation. 00:01:04.870 --> 00:01:07.480 So we need to reverse equation one 00:01:07.480 --> 00:01:11.240 to make it look more like our original reaction. 00:01:11.240 --> 00:01:14.480 To save time, I've gone ahead and reversed equation one. 00:01:14.480 --> 00:01:16.840 So you can see, I did that down here. 00:01:16.840 --> 00:01:19.820 Looking at the original equation for equation one, 00:01:19.820 --> 00:01:21.160 here where the products 00:01:21.160 --> 00:01:24.730 and now we've made those products the reactants. 00:01:24.730 --> 00:01:28.110 And what were the reactants over here for equation one 00:01:28.110 --> 00:01:30.960 have now become the products. 00:01:30.960 --> 00:01:33.520 The change in enthalpy for equation one 00:01:33.520 --> 00:01:38.520 is -1,299.6 kilojoules per mole of reaction. 00:01:38.740 --> 00:01:41.780 Kilojoules per mole reaction just means how the reaction 00:01:41.780 --> 00:01:43.670 is written in the balanced equation. 00:01:43.670 --> 00:01:46.940 And since we reversed equation one, 00:01:46.940 --> 00:01:50.240 we also need to reverse the sign for Delta H. 00:01:50.240 --> 00:01:52.800 So instead of this being a negative, 00:01:52.800 --> 00:01:53.840 instead of this being a negative, 00:01:53.840 --> 00:01:56.350 we're gonna go ahead and change this into a positive. 00:01:56.350 --> 00:02:00.000 And also let's go ahead and cross out the first equation. 00:02:00.000 --> 00:02:03.290 So we don't get confused. 00:02:03.290 --> 00:02:05.010 Next, we look at equation two 00:02:05.010 --> 00:02:07.360 and we compare it to our original. 00:02:07.360 --> 00:02:10.540 For equation two there's one mole of solid carbon 00:02:10.540 --> 00:02:14.608 on the left side and looking at our original reaction, 00:02:14.608 --> 00:02:18.200 there's two moles of carbon on the left side. 00:02:18.200 --> 00:02:21.670 So to get equation two, to look like our original equation 00:02:21.670 --> 00:02:25.710 we need to multiply everything through, by a factor of two. 00:02:25.710 --> 00:02:29.080 So we're gonna multiply everything in equation two 00:02:29.080 --> 00:02:30.623 by a factor of two. 00:02:31.630 --> 00:02:33.540 To save some time, I have gone ahead and written out 00:02:33.540 --> 00:02:34.373 what we would get. 00:02:34.373 --> 00:02:37.330 We would get two carbons plus two O2s 00:02:37.330 --> 00:02:39.850 goes to 2CO2. 00:02:39.850 --> 00:02:42.170 The change in the enthalpy for the formation 00:02:42.170 --> 00:02:44.350 of one mole of CO2 00:02:44.350 --> 00:02:49.050 was -393.5 kilojoules per mole of reaction. 00:02:49.050 --> 00:02:52.250 But now we're forming two moles of CO2. 00:02:52.250 --> 00:02:55.070 And since we multiplied the equation through 00:02:55.070 --> 00:02:56.860 by a factor of two, we also need 00:02:56.860 --> 00:02:58.910 to multiply the change in enthalpy 00:02:58.910 --> 00:03:01.200 by a factor of two as well. 00:03:01.200 --> 00:03:03.810 And also let's go ahead and cross out 00:03:03.810 --> 00:03:04.859 this first version here 00:03:04.859 --> 00:03:08.333 because now we're forming two moles of CO2. 00:03:09.170 --> 00:03:10.670 Next, we look at equation three 00:03:10.670 --> 00:03:12.870 and we can see there's one mole of hydrogen gas 00:03:12.870 --> 00:03:14.700 on the left side of the equation 00:03:14.700 --> 00:03:16.770 which matches the original reaction 00:03:16.770 --> 00:03:20.040 which also has one mole of hydrogen gas on the left side. 00:03:20.040 --> 00:03:22.840 So we don't need to do anything to equation three. 00:03:22.840 --> 00:03:25.020 And since we're not doing anything to the equation, 00:03:25.020 --> 00:03:26.750 we're also not gonna do anything 00:03:26.750 --> 00:03:28.980 to the change in the enthalpy. 00:03:28.980 --> 00:03:33.980 So it's gonna stay -285.8 kilojoules per mole of reaction. 00:03:34.270 --> 00:03:37.210 Next we add up all of our reactants and products. 00:03:37.210 --> 00:03:42.210 So we have two CO2 plus H2O plus 2C plus 2O2 00:03:46.060 --> 00:03:48.830 plus H2 plus one half O2. 00:03:48.830 --> 00:03:52.460 So those are all written down here for our reactants. 00:03:52.460 --> 00:03:56.210 And then for the products, let me just change colors here. 00:03:56.210 --> 00:03:59.400 We have C2H2 plus 5O2 00:04:00.470 --> 00:04:04.410 plus 2CO2 plus H2O. 00:04:04.410 --> 00:04:08.623 And so those are written over here for the products. 00:04:09.530 --> 00:04:11.310 Next we see what we can cancel out. 00:04:11.310 --> 00:04:13.220 There's 2CO2 on the left side 00:04:13.220 --> 00:04:15.250 and there's 2CO2 on the right side. 00:04:15.250 --> 00:04:16.540 So those cancel out. 00:04:16.540 --> 00:04:19.990 There's one water on the left and one water on the right. 00:04:19.990 --> 00:04:23.598 And there's 2O2s plus one half O2 00:04:23.598 --> 00:04:28.430 which is 2.5O2s or five halves O2s. 00:04:28.430 --> 00:04:32.020 So the oxygen's cancel out on both sides as well. 00:04:32.020 --> 00:04:37.020 And we can see we're left with two carbons plus one hydrogen 00:04:37.390 --> 00:04:40.680 goes to form one C2H2 00:04:40.680 --> 00:04:45.560 which is the same as our original equation. 00:04:45.560 --> 00:04:48.250 Since we were able to add up our equations 00:04:48.250 --> 00:04:50.310 and get the overall equation, 00:04:50.310 --> 00:04:51.640 according to Hess's law, 00:04:51.640 --> 00:04:55.310 we should also be able to add the changes in enthalpies 00:04:55.310 --> 00:04:57.483 for these steps to get the change in the enthalpy 00:04:57.483 --> 00:05:00.380 for the overall reaction. 00:05:00.380 --> 00:05:01.980 If we look at the changes in enthalpy 00:05:01.980 --> 00:05:06.910 for the individual steps, we had +1299.6 00:05:06.910 --> 00:05:08.090 for the first equation. 00:05:08.090 --> 00:05:09.470 And so that's up here. 00:05:09.470 --> 00:05:13.560 For the second equation we had negative 393.5 times two, 00:05:13.560 --> 00:05:14.450 which is -787. 00:05:15.640 --> 00:05:19.570 And for our third equation, we had -285.8. 00:05:19.570 --> 00:05:22.120 So that's -285.8. 00:05:22.120 --> 00:05:23.360 When we add everything together 00:05:23.360 --> 00:05:28.360 we get +226.8 kilojoules per mole of reaction. 00:05:29.600 --> 00:05:32.570 So for the formation of one mole of acetylene 00:05:32.570 --> 00:05:35.920 from two moles of carbon and one mole of hydrogen 00:05:35.920 --> 00:05:38.380 the change in enthalpy for this reaction 00:05:38.380 --> 00:05:43.163 is equal to +226.8 kilojoules per mole of reaction.

Complex numbers with the same modulus (absolute value)

https://www.youtube.com/watch?v=uB5QDraFefs

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en

WEBVTT Kind: captions Language: en 00:00:00.200 --> 00:00:02.890 - [Instructor] We are asked, which of these complex numbers 00:00:02.890 --> 00:00:06.520 has a modulus of 13? 00:00:06.520 --> 00:00:08.430 And just as a bit of a hint, 00:00:08.430 --> 00:00:12.520 when we're talking about the modulus of a complex number, 00:00:12.520 --> 00:00:15.510 we're really just talking about its absolute value. 00:00:15.510 --> 00:00:18.410 Or if we were to plot it in the complex plane, 00:00:18.410 --> 00:00:20.100 which is what we have right over here, 00:00:20.100 --> 00:00:23.290 what is its distance from the origin? 00:00:23.290 --> 00:00:26.650 So really you need to find which of these complex numbers 00:00:26.650 --> 00:00:31.650 has a distance of 13 from the origin in the complex plane. 00:00:31.940 --> 00:00:34.540 Pause this video and see if you can figure that out. 00:00:35.640 --> 00:00:38.240 All right, now let's work through this together. 00:00:38.240 --> 00:00:40.640 Now one might jump out at you immediately 00:00:40.640 --> 00:00:43.880 that's going to have a distance of 13 from the origin. 00:00:43.880 --> 00:00:46.350 If this is the origin right over here, 00:00:46.350 --> 00:00:49.180 we see that if we go exactly 13 units down 00:00:49.180 --> 00:00:52.500 we have this point right over here, negative 13i. 00:00:52.500 --> 00:00:55.117 So immediately right out of the gate, I say, 00:00:55.117 --> 00:00:58.270 "Okay, that complex number has a modulus of 13," 00:00:58.270 --> 00:01:00.450 but is that the only one? 00:01:00.450 --> 00:01:03.290 Well, we can actually visualize all of the complex numbers 00:01:03.290 --> 00:01:06.770 that have a modulus of 13 by drawing a circle 00:01:06.770 --> 00:01:09.840 with the radius 13 centered at the origin. 00:01:09.840 --> 00:01:12.160 So let's do that. 00:01:12.160 --> 00:01:14.630 And we can see that it contains 00:01:14.630 --> 00:01:17.000 the first complex number that we looked for, 00:01:17.000 --> 00:01:20.900 but it also seems to have included in it 00:01:20.900 --> 00:01:24.160 this one right over here, and we can verify 00:01:24.160 --> 00:01:27.360 that the modulus right over here is going to be 13. 00:01:27.360 --> 00:01:29.910 We can just use the Pythagorean theorem. 00:01:29.910 --> 00:01:33.730 So this distance right over here is 12. 00:01:33.730 --> 00:01:36.920 And this distance right over here is 5. 00:01:36.920 --> 00:01:38.840 And so we just need to figure out 00:01:38.840 --> 00:01:41.600 the hypotenuse right over here. 00:01:41.600 --> 00:01:43.910 And so we know that the hypotenuse 00:01:43.910 --> 00:01:47.660 is going to be the square root of 5 squared 00:01:47.660 --> 00:01:49.240 plus 12 squared, 00:01:49.240 --> 00:01:53.637 which is equal to the square root of 25 plus 144, 00:01:54.530 --> 00:01:57.670 which is equal to the square root of 169, 00:01:57.670 --> 00:02:01.050 which indeed does equal 13. 00:02:01.050 --> 00:02:03.160 So I like that choice as well. 00:02:03.160 --> 00:02:05.760 And we can see visually that none of these other points 00:02:05.760 --> 00:02:08.600 that they already plotted sit on that circle. 00:02:08.600 --> 00:02:10.710 So they don't have a modulus of 13. 00:02:10.710 --> 00:02:13.860 If we wanted to come up with some other interesting points, 00:02:13.860 --> 00:02:17.640 we could instead of having negative 5 plus 12i, 00:02:17.640 --> 00:02:20.270 we could have negative 5 minus 12i. 00:02:20.270 --> 00:02:21.770 It would get us right over there. 00:02:21.770 --> 00:02:24.480 And that would have a modulus of 13. 00:02:24.480 --> 00:02:27.310 And notice, when you have your complex conjugate, 00:02:27.310 --> 00:02:29.260 it has the same modulus. 00:02:29.260 --> 00:02:30.760 Or you could go the other way around. 00:02:30.760 --> 00:02:35.510 Instead of negative 5 plus 12i, you could have 5 plus 12i. 00:02:35.510 --> 00:02:38.130 That also would have a modulates of 13. 00:02:38.130 --> 00:02:41.240 Or you could have 5 minus 12i. 00:02:41.240 --> 00:02:43.730 That also would have a modulus of 13. 00:02:43.730 --> 00:02:45.670 Now there's an infinite number of points, 00:02:45.670 --> 00:02:47.590 any of these points on the circle, 00:02:47.590 --> 00:02:50.323 that will have a modulus of 13.

Factoring polynomials using complex numbers

https://www.youtube.com/watch?v=9NhTVXhfqzU

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https://www.youtube.com/api/timedtext?v=9NhTVXhfqzU&ei=5VWUZYvWI4rVhcIP652h-AU&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=284557ABF9D3DDA954B25302F475B7C87319E5F6.B5AC8BE209B71D098AB368E42BB97DB2C4D1A5AB&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.340 --> 00:00:01.830 - [Instructor] We're told that Amat tried 00:00:01.830 --> 00:00:05.570 to write x to the fourth plus 5x squared plus 4 00:00:05.570 --> 00:00:08.490 as a product of linear factors. 00:00:08.490 --> 00:00:10.140 This is his work. 00:00:10.140 --> 00:00:12.400 And then they'd tell us all of the steps that he did. 00:00:12.400 --> 00:00:14.180 And then they say, in what step 00:00:14.180 --> 00:00:17.330 did Amat make his first mistake? 00:00:17.330 --> 00:00:20.203 So pause this video and see if you can figure that out. 00:00:21.360 --> 00:00:24.190 All right, now let's work through this together. 00:00:24.190 --> 00:00:26.530 So we're starting with x to the fourth plus 10x squared 00:00:26.530 --> 00:00:27.760 plus 9. 00:00:27.760 --> 00:00:30.220 And it looks like Amat tried to factor that 00:00:30.220 --> 00:00:33.570 into x squared plus 9 times x squared plus 1. 00:00:33.570 --> 00:00:35.920 And this indeed does make sense, 00:00:35.920 --> 00:00:38.100 because if we said that let's say, 00:00:38.100 --> 00:00:41.770 u is equal to x squared, 00:00:41.770 --> 00:00:43.840 we could rewrite this right over here 00:00:43.840 --> 00:00:48.097 as u squared plus 10u plus 9. 00:00:49.960 --> 00:00:51.640 The whole reason why you would do this is 00:00:51.640 --> 00:00:54.580 so that you could write this higher order expression 00:00:54.580 --> 00:00:57.130 in terms of a second degree expression. 00:00:57.130 --> 00:00:59.090 And then we've learned how to factor things like 00:00:59.090 --> 00:01:00.020 this many times. 00:01:00.020 --> 00:01:00.980 We look, we say, okay, 00:01:00.980 --> 00:01:03.570 what two numbers when I add them I get 10, 00:01:03.570 --> 00:01:05.620 and when I multiply them I get nine, 00:01:05.620 --> 00:01:07.270 and it would be nine and one? 00:01:07.270 --> 00:01:08.480 And so you could write this 00:01:08.480 --> 00:01:13.480 as u plus 9 times u plus 1. 00:01:13.800 --> 00:01:15.700 And of course, if u is equal to x squared, 00:01:15.700 --> 00:01:20.700 this would be x squared plus 9 times x squared plus 1. 00:01:21.220 --> 00:01:23.730 Which is exactly what Amat has right over here. 00:01:23.730 --> 00:01:27.120 So step one is looking great. 00:01:27.120 --> 00:01:30.670 All right, now let's think about what Amat did in step two. 00:01:30.670 --> 00:01:33.400 They didn't do anything to x squared plus 9 00:01:33.400 --> 00:01:34.380 but it looks like they tried 00:01:34.380 --> 00:01:37.490 to further factor x squared plus 1. 00:01:37.490 --> 00:01:39.240 And this does seem right. 00:01:39.240 --> 00:01:41.560 We just have to remind ourselves just 00:01:41.560 --> 00:01:43.610 as you have a difference of squares 00:01:43.610 --> 00:01:45.620 if you're dealing with non-complex numbers, 00:01:45.620 --> 00:01:47.540 so we could rewrite this right over here 00:01:47.540 --> 00:01:52.200 as x plus a times x minus a. 00:01:52.200 --> 00:01:54.840 We could have a sum of squares 00:01:54.840 --> 00:01:57.200 if we're thinking about complex numbers. 00:01:57.200 --> 00:02:02.200 This is going to be x plus ai times x minus ai. 00:02:04.760 --> 00:02:08.170 And in this situation while the x is x 00:02:08.170 --> 00:02:10.880 and then our a would be one. 00:02:10.880 --> 00:02:13.760 So we're going to have x plus i, 00:02:13.760 --> 00:02:17.960 x plus i times x minus i . 00:02:17.960 --> 00:02:20.770 So step two is looking great. 00:02:20.770 --> 00:02:22.640 And now let's go do step three. 00:02:22.640 --> 00:02:24.150 So in step three, 00:02:24.150 --> 00:02:28.120 no change to this part of the expression. 00:02:28.120 --> 00:02:29.900 And it looks like Amat is trying 00:02:29.900 --> 00:02:33.280 to factor x squared plus 9 based on the same principle. 00:02:33.280 --> 00:02:35.480 Now x squared plus 9 is the same thing 00:02:35.480 --> 00:02:38.690 as x squared plus 3 squared. 00:02:38.690 --> 00:02:41.100 So if you use this exact same idea here, 00:02:41.100 --> 00:02:46.100 if you factored it should be x plus 3i times x minus 3i. 00:02:48.420 --> 00:02:49.920 But what we see over here 00:02:49.920 --> 00:02:52.340 is Amat took the square root of three, 00:02:52.340 --> 00:02:54.090 instead of just having a three here. 00:02:54.090 --> 00:02:56.450 Amat treated it instead of having a nine here 00:02:56.450 --> 00:02:58.790 as if we actually had a three 00:02:58.790 --> 00:03:01.730 so they made a little bit of an error there. 00:03:01.730 --> 00:03:06.730 So this is the step where Amat makes his first mistake 00:03:07.800 --> 00:03:08.633 and we're done.

Expected payoff example: protection plan

https://www.youtube.com/watch?v=mKPeuVjPDo0

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https://www.youtube.com/api/timedtext?v=mKPeuVjPDo0&ei=5VWUZZGVJv2dp-oPkJaXeA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=0ECD94EABFB43094F74E96F6DFFC4EC37F2253A1.53798958353851084D5A21AB517E55B03BE62BE2&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.160 --> 00:00:01.790 - [Instructor] We're told that an electronic store 00:00:01.790 --> 00:00:05.000 gives customers the option of purchasing a protection plan 00:00:05.000 --> 00:00:06.720 when customers buy a new television. 00:00:06.720 --> 00:00:08.330 That's actually quite common. 00:00:08.330 --> 00:00:10.990 The customer pays $80 for the plan and, 00:00:10.990 --> 00:00:14.180 if their television is damaged or stops working, 00:00:14.180 --> 00:00:17.420 the store will replace it for no additional charge. 00:00:17.420 --> 00:00:21.010 The store knows that 2% of customers who buy this plan 00:00:21.010 --> 00:00:22.426 end up needing a replacement 00:00:22.426 --> 00:00:25.860 that costs the store $1,200 each. 00:00:25.860 --> 00:00:28.685 Here's a table that summarizes the possible outcomes 00:00:28.685 --> 00:00:31.269 from the store's perspective. 00:00:31.269 --> 00:00:34.490 Let X represent the store's net gain 00:00:34.490 --> 00:00:36.580 from one of these plans. 00:00:36.580 --> 00:00:39.680 Calculate the expected net gain. 00:00:39.680 --> 00:00:41.850 So pause this video, see if you can have a go at that 00:00:41.850 --> 00:00:43.700 before we work through this together. 00:00:44.860 --> 00:00:46.780 So we have the two scenarios here. 00:00:46.780 --> 00:00:49.060 The first scenario is that the store does need 00:00:49.060 --> 00:00:51.230 to replace the TV because something happens 00:00:51.230 --> 00:00:53.660 and so it's gonna cost $1,200 to the store. 00:00:53.660 --> 00:00:56.300 But remember they got $80 for the protection plan. 00:00:56.300 --> 00:00:59.900 So you have a net gain of negative $1,120 00:00:59.900 --> 00:01:01.520 from the store's perspective. 00:01:01.520 --> 00:01:02.540 There's the other scenario, 00:01:02.540 --> 00:01:03.960 which is more favorable for the store, 00:01:03.960 --> 00:01:07.330 which is a customer does not need a replacement TV, 00:01:07.330 --> 00:01:09.360 so that has no cost and so their net gain 00:01:09.360 --> 00:01:11.520 is just the $80 for the plan. 00:01:11.520 --> 00:01:13.370 So to figure out the expected net gain, 00:01:13.370 --> 00:01:14.750 we just have to figure out the probabilities 00:01:14.750 --> 00:01:17.400 of each of these and take the weighted average of them. 00:01:17.400 --> 00:01:19.050 So what's the probability that they will have 00:01:19.050 --> 00:01:20.640 to replace the TV? 00:01:20.640 --> 00:01:22.930 Well, we know 2% of customers who buy this plan 00:01:22.930 --> 00:01:24.640 end up needing a replacement. 00:01:24.640 --> 00:01:26.840 So we could say this is two over 100 00:01:26.840 --> 00:01:29.300 or maybe I'll write it as 0.02. 00:01:29.300 --> 00:01:32.170 This is the probability of X. 00:01:32.170 --> 00:01:36.570 And then the probability of not needing a replacement, 0.98. 00:01:36.570 --> 00:01:40.040 And so our expected net gain 00:01:40.040 --> 00:01:43.110 is going to be equal to the probability 00:01:43.110 --> 00:01:46.840 of needing replacement times the net gain of a replacement. 00:01:46.840 --> 00:01:50.687 So it's going to be times -$1,120. 00:01:53.720 --> 00:01:56.150 And then we're gonna have plus the probability 00:01:56.150 --> 00:01:58.980 of not needing replacement, which is 0.98 00:02:00.380 --> 00:02:01.870 times the net gain there. 00:02:01.870 --> 00:02:04.333 So that is $80. 00:02:05.690 --> 00:02:10.140 So we have 0.02x-1,120 is equal to that. 00:02:13.890 --> 00:02:15.850 And to that, we're gonna add, 00:02:15.850 --> 00:02:19.397 I'll open parentheses, 0.98x80, close parentheses, 00:02:23.170 --> 00:02:27.980 is going to be equal to $56. 00:02:27.980 --> 00:02:30.910 So this is equal to $56. 00:02:30.910 --> 00:02:33.410 And now you understand why the stores 00:02:33.410 --> 00:02:35.533 like to sell these replacement plans.

Expected payoff example: lottery ticket

https://www.youtube.com/watch?v=Ay1bVzqTKzg

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https://www.youtube.com/api/timedtext?v=Ay1bVzqTKzg&ei=5VWUZe31KuOip-oP1PWsiA4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=7654965341D18CE7F159559D8D8F7DC0CF5F63E9.D8DB599A363EE2AD730A079955C37FE2507607EC&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.200 --> 00:00:01.870 - [Instructor] We're told a Pick 4 lottery game 00:00:01.870 --> 00:00:05.120 involves drawing four numbered balls from separate bins, 00:00:05.120 --> 00:00:08.530 each containing balls labeled from 0 to 9. 00:00:08.530 --> 00:00:11.500 So there are 10,000 possible selections in total. 00:00:11.500 --> 00:00:12.333 For example, you could get 00:00:12.333 --> 00:00:14.530 a 0, a 0, a 0 and a 0, 00:00:14.530 --> 00:00:16.670 a 0, a 0, a 0 and a 1, 00:00:16.670 --> 00:00:21.080 all the way up to 9,999, four nines. 00:00:21.080 --> 00:00:23.890 Players can choose to play a straight bet, where the player 00:00:23.890 --> 00:00:27.470 wins if they match all four digits in the correct order. 00:00:27.470 --> 00:00:32.470 The lottery pays $4,500 on a successful $1 straight bet. 00:00:33.520 --> 00:00:38.520 Let X represent a player's net gain on a $1 straight bet. 00:00:39.580 --> 00:00:42.860 Calculate the expected net gain. 00:00:42.860 --> 00:00:46.310 And they say, hint, the expected net gain can be negative. 00:00:46.310 --> 00:00:47.610 So why don't you pause this video 00:00:47.610 --> 00:00:50.363 and see if you can calculate the expected net gain? 00:00:51.780 --> 00:00:52.613 All right. 00:00:52.613 --> 00:00:54.570 So there's a couple of ways that we can approach this. 00:00:54.570 --> 00:00:57.940 One way is to just think about the two different outcomes. 00:00:57.940 --> 00:01:01.430 There's a scenario where you win with your straight bet. 00:01:01.430 --> 00:01:04.420 There's a scenario where you lose with your straight bet. 00:01:04.420 --> 00:01:05.253 Now let's think about 00:01:05.253 --> 00:01:09.090 the net gain in either one of those scenarios. 00:01:09.090 --> 00:01:12.420 The scenario where you win, you pay $1, 00:01:12.420 --> 00:01:16.950 we know it's a $1 straight bet, and you get $4,500. 00:01:16.950 --> 00:01:18.750 So what's the net gain? 00:01:18.750 --> 00:01:22.160 So it's going to be $4,500 minus one. 00:01:22.160 --> 00:01:24.887 So your net gain is going to be $4,499. 00:01:29.040 --> 00:01:32.020 Now what about the net gain in the situation that you lose? 00:01:32.020 --> 00:01:33.880 Well, in the situation that you lose, 00:01:33.880 --> 00:01:35.280 you just lose a dollar. 00:01:35.280 --> 00:01:40.280 So this is just going to be negative $1 right over here. 00:01:40.860 --> 00:01:41.693 Now let's think about 00:01:41.693 --> 00:01:44.230 the probabilities of each of these situations. 00:01:44.230 --> 00:01:47.550 So the probability, so the probability of a win 00:01:47.550 --> 00:01:49.770 we know is 1 in 10,000, 00:01:49.770 --> 00:01:52.800 1 in 10,000. 00:01:52.800 --> 00:01:55.010 And what's the probability of a loss? 00:01:55.010 --> 00:01:58.540 Well, that's going to be 9,999 00:01:58.540 --> 00:02:00.990 out of 10,000. 00:02:00.990 --> 00:02:02.984 And then our expected net gain is just 00:02:02.984 --> 00:02:05.320 going to be the weighted average of these two. 00:02:05.320 --> 00:02:08.960 So I could write our expected net gain 00:02:08.960 --> 00:02:12.940 is going to be 4,499 00:02:12.940 --> 00:02:15.780 times the probability of that, 1 in 10,000 00:02:16.810 --> 00:02:19.030 plus negative 1 times this, 00:02:19.030 --> 00:02:20.020 so that I could just write that 00:02:20.020 --> 00:02:25.020 as minus 9,999 over 10,000. 00:02:26.250 --> 00:02:28.730 And so this is going to be equal to, let's see, 00:02:28.730 --> 00:02:30.470 it's going to be 4,499 00:02:31.730 --> 00:02:35.120 minus 9,999, 00:02:35.120 --> 00:02:38.970 all of that over 10,000. 00:02:38.970 --> 00:02:43.380 And let's see, this is going to be equal to 00:02:43.380 --> 00:02:45.943 negative 5,500 over 10,000, 00:02:47.028 --> 00:02:50.607 negative 5,500 over 10,000, 00:02:51.570 --> 00:02:55.410 which is the same thing as negative 55 over 100, 00:02:55.410 --> 00:02:56.880 or I could write it this way. 00:02:56.880 --> 00:03:00.150 This is equal to negative .55. 00:03:00.150 --> 00:03:03.780 I could write it this way, 0.55. 00:03:03.780 --> 00:03:07.010 So that's one way to calculate the expected net gain. 00:03:07.010 --> 00:03:09.070 Another way to approach it is to say, all right, 00:03:09.070 --> 00:03:11.280 what if we were to get 10,000 tickets? 00:03:11.280 --> 00:03:15.280 What is our expected net gain on the 10,000 tickets? 00:03:15.280 --> 00:03:19.460 Well, we would pay $10,000 00:03:21.120 --> 00:03:22.970 and we would expect to win once. 00:03:22.970 --> 00:03:25.830 It's not a guarantee, but we would expect to win once. 00:03:25.830 --> 00:03:27.150 So expect 00:03:29.250 --> 00:03:32.570 4,500 in payout. 00:03:32.570 --> 00:03:36.600 And so you would then, let's see, you would have a net gain 00:03:37.840 --> 00:03:41.860 of, it would be negative $5,500, 00:03:41.860 --> 00:03:45.240 negative $5,500. 00:03:45.240 --> 00:03:48.910 Now this is the net gain when you do 10,000 tickets. 00:03:48.910 --> 00:03:52.130 Now, if you wanted to find the expected net gain per ticket 00:03:52.130 --> 00:03:53.890 you would then just divide by 10,000. 00:03:53.890 --> 00:03:54.780 And if you did that, 00:03:54.780 --> 00:03:58.760 you would get exactly what we just calculated the other way. 00:03:58.760 --> 00:04:00.950 So any way you try to approach this 00:04:00.950 --> 00:04:02.693 this is not a great bet.

Interpreting expected value

https://www.youtube.com/watch?v=SNIW7MmCdhA

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https://www.youtube.com/api/timedtext?v=SNIW7MmCdhA&ei=5VWUZey-ItbUxN8P_tq54AY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=C69322E1C4E1C02FBDC85F26D31867E4D2053226.25811F4F9863015D102430F52C499749ABA4DBD2&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.400 --> 00:00:03.380 - [Instructor] We're told a certain lottery ticket costs $2 00:00:03.380 --> 00:00:05.057 and the back of the ticket says, 00:00:05.057 --> 00:00:08.020 "The overall odds of winning a prize with this ticket 00:00:08.020 --> 00:00:09.500 are one to 50, 00:00:09.500 --> 00:00:13.530 and the expected return for this ticket is $0.95." 00:00:13.530 --> 00:00:16.650 Which interpretations of the expected value are correct? 00:00:16.650 --> 00:00:19.300 Choose all answers that apply. 00:00:19.300 --> 00:00:21.570 Pause this video, have a go at that. 00:00:21.570 --> 00:00:24.370 All right, now let's go through each of these choices. 00:00:24.370 --> 00:00:26.160 So choice A says the probability 00:00:26.160 --> 00:00:31.160 that one of these tickets wins a prize is 0.95 on average. 00:00:32.870 --> 00:00:34.940 Well, I see where they're getting that 0.95. 00:00:34.940 --> 00:00:36.490 They're getting it from right over here, 00:00:36.490 --> 00:00:38.460 but that's not the probability that you're winning, 00:00:38.460 --> 00:00:40.400 that's the expected return. 00:00:40.400 --> 00:00:42.830 The probability that you win is much lower. 00:00:42.830 --> 00:00:44.750 If the odds are one to 50, 00:00:44.750 --> 00:00:49.340 that means that the probability of winning is one to 51. 00:00:49.340 --> 00:00:52.490 So it's a much lower probability than this right over here. 00:00:52.490 --> 00:00:54.530 So definitely rule that out. 00:00:54.530 --> 00:00:58.443 Someone who buys this ticket is most likely to win $0.95. 00:01:00.390 --> 00:01:03.890 That is not necessarily the case either. 00:01:03.890 --> 00:01:08.660 We don't know what the different outcomes are for the prize. 00:01:08.660 --> 00:01:10.760 It's very likely that there's no outcome 00:01:10.760 --> 00:01:13.420 for that prize where you win exactly $0.95. 00:01:13.420 --> 00:01:16.610 Instead, there's likely to be outcomes that are much larger 00:01:16.610 --> 00:01:18.550 than that with very low probabilities, 00:01:18.550 --> 00:01:20.440 and then when you take the weighted average 00:01:20.440 --> 00:01:22.310 of all of the outcomes, 00:01:22.310 --> 00:01:25.690 then you get an expected return of $0.95. 00:01:25.690 --> 00:01:29.490 So it's actually maybe even impossible to win exactly $0.95. 00:01:29.490 --> 00:01:31.410 So I would rule that out. 00:01:31.410 --> 00:01:33.080 If we looked at many of these tickets, 00:01:33.080 --> 00:01:37.523 the average return would be about $0.95 per ticket. 00:01:38.370 --> 00:01:40.260 That one feels pretty interesting, 00:01:40.260 --> 00:01:42.460 'cause we're looking at many of these tickets. 00:01:42.460 --> 00:01:46.720 And so across many of them, you would expect to, on average, 00:01:46.720 --> 00:01:49.430 get the expected return as your return. 00:01:49.430 --> 00:01:51.720 And so this is what we are seeing here. 00:01:51.720 --> 00:01:53.370 The average return would be about that. 00:01:53.370 --> 00:01:54.600 It would be approximately that. 00:01:54.600 --> 00:01:56.140 So I like that choice. 00:01:56.140 --> 00:01:58.980 That is a good interpretation of expected value. 00:01:58.980 --> 00:02:00.070 And then choice D, 00:02:00.070 --> 00:02:03.000 if 1,000 people each bought one of these tickets, 00:02:03.000 --> 00:02:08.000 they'd expect a net gain of about $950 in total. 00:02:09.100 --> 00:02:10.810 This one is tempting. 00:02:10.810 --> 00:02:14.540 Instead of net gain, if it just said return, 00:02:14.540 --> 00:02:15.710 this would make a lot of sense. 00:02:15.710 --> 00:02:18.540 In fact, it would be completely consistent with choice C. 00:02:18.540 --> 00:02:21.250 If you have 1,000 people, that would be many tickets, 00:02:21.250 --> 00:02:23.500 and if on average, if their average return 00:02:23.500 --> 00:02:25.560 is about $0.95 per ticket, 00:02:25.560 --> 00:02:29.210 then their total return would be about $950, 00:02:29.210 --> 00:02:32.550 but they didn't write return here, they wrote net gain. 00:02:32.550 --> 00:02:36.640 Net gain would be how much you get minus how much you paid. 00:02:36.640 --> 00:02:38.320 And 1,000 people would have to pay, 00:02:38.320 --> 00:02:41.530 if they each got a ticket, would pay $2,000. 00:02:41.530 --> 00:02:43.210 So they would pay 2,000. 00:02:43.210 --> 00:02:46.590 They would expect a return of $950. 00:02:46.590 --> 00:02:50.690 Their net gain would actually be negative $1,050. 00:02:50.690 --> 00:02:53.113 So we would rule that one out as well.

Probability distributions from empirical data

https://www.youtube.com/watch?v=wztjEa7893c

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https://www.youtube.com/api/timedtext?v=wztjEa7893c&ei=5VWUZafEI9bAhcIPuuG72AI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=E9F9BC002B98F5042F0B6103327E1D6A524DE96E.BD68CB7E14DDB663F65B23F9B9F72F080258E0B8&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.270 --> 00:00:02.140 - [Instructor] We're told that Jada owns a restaurant 00:00:02.140 --> 00:00:05.200 where customers can make their orders using an app. 00:00:05.200 --> 00:00:07.130 She decides to offer a discount 00:00:07.130 --> 00:00:09.610 on appetizers to attract more customers. 00:00:09.610 --> 00:00:12.130 And she's curious about the probability 00:00:12.130 --> 00:00:16.690 that a customer orders a large number of appetizers. 00:00:16.690 --> 00:00:19.070 Jada tracked how many appetizers 00:00:19.070 --> 00:00:22.950 were in each of the past 500 orders. 00:00:22.950 --> 00:00:24.810 All right, so the number of appetizers, 00:00:24.810 --> 00:00:27.760 so 40 out of the 500 ordered zero appetizers, 00:00:27.760 --> 00:00:32.030 and for example, 120 out of the 500 ordered three appetizers 00:00:32.030 --> 00:00:33.600 and so on and so forth. 00:00:33.600 --> 00:00:37.990 Let X represent the number of appetizers in a random order. 00:00:37.990 --> 00:00:40.560 Based on these results, construct an approximate 00:00:40.560 --> 00:00:43.230 probability distribution of X. 00:00:43.230 --> 00:00:45.530 Pause this video and see if you can have a go at this 00:00:45.530 --> 00:00:47.093 before we do this together. 00:00:48.030 --> 00:00:51.640 All right, so they're telling us an approximate 00:00:51.640 --> 00:00:54.700 probability distribution, because we don't know 00:00:54.700 --> 00:00:56.560 the actual probability. 00:00:56.560 --> 00:00:58.180 We can't get into people's minds 00:00:58.180 --> 00:01:00.530 and figure out the probability that the neurons fire 00:01:00.530 --> 00:01:03.840 in exactly the right way to order appetizers. 00:01:03.840 --> 00:01:06.880 But what we can do is look at past results, 00:01:06.880 --> 00:01:09.260 empirical data right over here 00:01:09.260 --> 00:01:11.860 to approximate the distribution. 00:01:11.860 --> 00:01:14.810 So what we can do is look at the last 500, 00:01:14.810 --> 00:01:16.390 and for each of the outcomes 00:01:16.390 --> 00:01:19.710 think about what fraction of the last 500 had that outcome. 00:01:19.710 --> 00:01:21.910 And that will be our approximation. 00:01:21.910 --> 00:01:24.610 And so the outcomes here, we could have zero appetizers, 00:01:24.610 --> 00:01:29.610 one, two, three, four, five, or six. 00:01:29.680 --> 00:01:32.356 Now the approximate probability 00:01:32.356 --> 00:01:36.280 of zero appetizers is going to be 40 over 500, 00:01:36.280 --> 00:01:40.140 which is the same thing as four over 50, 00:01:40.140 --> 00:01:44.160 which is the same thing as two over 25. 00:01:44.160 --> 00:01:47.410 So I'll write two 25th right over there. 00:01:47.410 --> 00:01:49.997 The probability of one appetizer, 00:01:49.997 --> 00:01:52.090 well, that's going to be 90, the over 500, 00:01:52.090 --> 00:01:55.390 which is the same thing as nine over 50. 00:01:55.390 --> 00:01:58.450 And I think that's already in lowest terms. 00:01:58.450 --> 00:02:03.450 Then 160 over 500 is the same thing as 16 over 50, 00:02:05.580 --> 00:02:08.773 which is the same thing as eight over 25. 00:02:10.546 --> 00:02:11.700 And we just keep going. 00:02:11.700 --> 00:02:16.700 120 out of 500 is the same thing as 12 out of 50, 00:02:17.000 --> 00:02:19.263 or six out of 25. 00:02:20.444 --> 00:02:25.444 Six out of 25, and then 50 out of 500. 00:02:25.750 --> 00:02:27.510 Well, that's one out of every 10. 00:02:27.510 --> 00:02:29.900 So I'll just write it like that. 00:02:29.900 --> 00:02:34.900 30 out of 500 is the same thing as three out of 50. 00:02:36.120 --> 00:02:38.370 I'll just write it like that. 00:02:38.370 --> 00:02:40.950 And that last but not least, 10 out of 500 00:02:40.950 --> 00:02:43.303 is the same thing as one in 50. 00:02:44.150 --> 00:02:45.330 And we're done. 00:02:45.330 --> 00:02:47.420 We have just constructed an approximate 00:02:47.420 --> 00:02:50.853 probability distribution for our random variable X.

Theoretical probability distribution example: multiplication

https://www.youtube.com/watch?v=2jExPaoTrQE

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https://www.youtube.com/api/timedtext?v=2jExPaoTrQE&ei=5VWUZfD2I9SAp-oPxpKHqAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=CA3AB67E27B98B1D63F13B13EEF354E406A2C754.2EF90C0A44B6AB36C734AF59AF4A7759114E113D&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.290 --> 00:00:02.180 - [Instructor] We're told that Kai goes to a restaurant 00:00:02.180 --> 00:00:04.417 that advertises a promotion saying, 00:00:04.417 --> 00:00:07.910 "1 in 5 customers get a free dessert!" 00:00:07.910 --> 00:00:12.050 Suppose Kai goes to the restaurant twice in a given week, 00:00:12.050 --> 00:00:15.330 and each time he has a 1/5 probability 00:00:15.330 --> 00:00:17.520 of getting a free dessert. 00:00:17.520 --> 00:00:21.400 Let X represent the number of free desserts 00:00:21.400 --> 00:00:23.980 Kai gets in his two trips. 00:00:23.980 --> 00:00:28.560 Construct the theoretical probability distribution of X. 00:00:28.560 --> 00:00:30.220 All right, so pause this video 00:00:30.220 --> 00:00:31.670 and see if you can work through this 00:00:31.670 --> 00:00:33.333 before we do it together. 00:00:34.260 --> 00:00:36.380 All right, so first let's just think about 00:00:36.380 --> 00:00:38.930 the possible values that X could take on. 00:00:38.930 --> 00:00:40.950 This is the number of free desserts he gets, 00:00:40.950 --> 00:00:42.630 and he visits twice. 00:00:42.630 --> 00:00:44.760 So there's some world in which 00:00:44.760 --> 00:00:46.610 he doesn't get any free desserts. 00:00:46.610 --> 00:00:50.090 So that's 0 in his two visits. 00:00:50.090 --> 00:00:51.340 Maybe on one of the visits, 00:00:51.340 --> 00:00:53.410 he gets a dessert and the other one he doesn't, 00:00:53.410 --> 00:00:55.170 and maybe in both of his visits 00:00:55.170 --> 00:00:57.420 he actually is able to get a free dessert. 00:00:57.420 --> 00:00:58.810 So he's going to have someplace 00:00:58.810 --> 00:01:02.310 from 0 to 2 free desserts in a given week. 00:01:02.310 --> 00:01:03.143 So we just have to figure out 00:01:03.143 --> 00:01:05.420 the probability of each of these. 00:01:05.420 --> 00:01:08.630 So let's first of all think about the probability, 00:01:08.630 --> 00:01:10.010 let me write it over here, 00:01:10.010 --> 00:01:13.710 the probability that capital X is equal to 0 00:01:13.710 --> 00:01:15.600 is going to be equal to what? 00:01:15.600 --> 00:01:16.960 Well, that's going to be the probability 00:01:16.960 --> 00:01:20.200 that he doesn't get a dessert on both days. 00:01:20.200 --> 00:01:21.460 And it's important to realize 00:01:21.460 --> 00:01:23.110 that these are independent events. 00:01:23.110 --> 00:01:24.387 It's not like the restaurant's gonna say, 00:01:24.387 --> 00:01:26.080 "Oh, if you didn't get a dessert on one day, 00:01:26.080 --> 00:01:27.530 you're more likely to get another the other day," 00:01:27.530 --> 00:01:29.850 or, "Somehow if you got it on a previous day, 00:01:29.850 --> 00:01:31.700 you're less likely than another day." 00:01:31.700 --> 00:01:33.610 That they are independent events. 00:01:33.610 --> 00:01:36.730 So the probability of not getting it on any one day 00:01:36.730 --> 00:01:39.540 is four out of five, 00:01:39.540 --> 00:01:42.600 and the probability of not getting it on two of the days, 00:01:42.600 --> 00:01:43.640 I would just multiply them 00:01:43.640 --> 00:01:45.810 because they are independent events. 00:01:45.810 --> 00:01:49.400 So four over five times four over five. 00:01:49.400 --> 00:01:52.210 So the probability that X is equal to 0 00:01:52.210 --> 00:01:57.210 is going to be 16/25, 16 over 25. 00:01:57.490 --> 00:02:02.490 Now what about the probability that X is equal to 1? 00:02:03.090 --> 00:02:05.040 What is this going to be? 00:02:05.040 --> 00:02:08.290 Well, there are two scenarios over here. 00:02:08.290 --> 00:02:12.670 There's one scenario where let's say on day one 00:02:12.670 --> 00:02:14.370 he does not get the dessert, 00:02:14.370 --> 00:02:17.940 and on day two he does get the dessert. 00:02:17.940 --> 00:02:20.490 But then of course there's the other scenario 00:02:20.490 --> 00:02:23.430 where on day one he gets the dessert, 00:02:23.430 --> 00:02:28.070 and then on day two he doesn't get the dessert. 00:02:28.070 --> 00:02:29.670 These are the two scenarios 00:02:29.670 --> 00:02:31.770 where he's going to get X equals 1. 00:02:31.770 --> 00:02:34.280 And so if we add these together, let's see. 00:02:34.280 --> 00:02:39.095 4/5 times 1/5, this is going to be 4/25, 00:02:39.095 --> 00:02:43.200 and then this is going to be 4/25 again. 00:02:43.200 --> 00:02:45.070 And you add these two together, 00:02:45.070 --> 00:02:48.890 you're going to get 8/25. 00:02:48.890 --> 00:02:50.050 And then last but not least, 00:02:50.050 --> 00:02:51.670 and actually we could figure out this last one 00:02:51.670 --> 00:02:55.270 by subtracting 16 and 8 from 25, 00:02:55.270 --> 00:02:57.180 which would actually give us 1/25th, 00:02:57.180 --> 00:02:58.450 but let's just write this out. 00:02:58.450 --> 00:03:01.480 The probability that X equals 2, 00:03:01.480 --> 00:03:03.710 this is the probability that he gets a dessert on both days. 00:03:03.710 --> 00:03:06.790 So 1/5 chance on day one 00:03:06.790 --> 00:03:09.080 and 1/5 chance on the second day. 00:03:09.080 --> 00:03:12.730 So 1/5 times 1/5 is 1/25. 00:03:12.730 --> 00:03:14.380 And you can do a reality check here. 00:03:14.380 --> 00:03:16.090 These all need to add up to one. 00:03:16.090 --> 00:03:17.830 And they do indeed add up to one. 00:03:17.830 --> 00:03:20.396 16 plus 8 plus 1 is 25. 00:03:20.396 --> 00:03:22.940 So 25/25 is what they all add up to. 00:03:22.940 --> 00:03:23.773 And we're done.

Theoretical probability distribution example: tables

https://www.youtube.com/watch?v=hA8VmhkKEJo

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https://www.youtube.com/api/timedtext?v=hA8VmhkKEJo&ei=5VWUZfu2K4COp-oP-PebwAY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=9D00BCBA06105636862D87BD5C8573FB996AD348.E74557E60B973C131C650F6AC31C1FE8B2CBBD0A&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.360 --> 00:00:01.560 - [Instructor] We're told that a board game 00:00:01.560 --> 00:00:05.450 has players roll two 3-sided dice, these exist 00:00:05.450 --> 00:00:07.030 and actually I looked it up, they do exist 00:00:07.030 --> 00:00:08.280 and they're actually fascinating. 00:00:08.280 --> 00:00:12.120 And subtract the numbers showing on the faces. 00:00:12.120 --> 00:00:16.290 The game only looks at non-negative differences. 00:00:16.290 --> 00:00:20.490 For example, if a player rolls a one and a three, 00:00:20.490 --> 00:00:22.660 the difference is two. 00:00:22.660 --> 00:00:26.690 Let D represent the difference in a given roll. 00:00:26.690 --> 00:00:30.760 Construct the theoretical probability distribution of D. 00:00:30.760 --> 00:00:32.840 So pause this video and see if you can have a go at that 00:00:32.840 --> 00:00:35.410 before we work through it together. 00:00:35.410 --> 00:00:38.520 All right, now let's work through it together. 00:00:38.520 --> 00:00:40.630 So let's just think about all of the scenarios 00:00:40.630 --> 00:00:42.370 for the two die. 00:00:42.370 --> 00:00:44.560 So let me draw a little table here. 00:00:44.560 --> 00:00:46.220 So let me do it like that 00:00:46.220 --> 00:00:49.250 and let me do it like this. 00:00:49.250 --> 00:00:52.870 And then let me put a little divider right over here. 00:00:52.870 --> 00:00:56.790 And for this top, this is going to be die one 00:00:56.790 --> 00:00:59.010 and then this is going to be die two. 00:00:59.010 --> 00:01:03.660 Die one can take on one, two, or three 00:01:03.660 --> 00:01:08.250 and die two could be one, two, or three. 00:01:08.250 --> 00:01:12.103 And so let me finish making this a bit of a table here. 00:01:13.850 --> 00:01:16.180 And what we wanna do is look at the difference 00:01:16.180 --> 00:01:17.890 but the non-negative difference. 00:01:17.890 --> 00:01:21.480 So we'll always subtract the lower die from the higher die. 00:01:21.480 --> 00:01:22.870 So what's the difference here? 00:01:22.870 --> 00:01:24.460 Well, this is going to be zero. 00:01:24.460 --> 00:01:26.590 If I roll a one and a one. 00:01:26.590 --> 00:01:28.420 Now, what if I roll a two and a one? 00:01:28.420 --> 00:01:30.990 Well, here the difference is going to be two minus one, 00:01:30.990 --> 00:01:32.040 which is one. 00:01:32.040 --> 00:01:35.630 Here the difference is three minus one, which is two. 00:01:35.630 --> 00:01:37.450 Now what about right over here? 00:01:37.450 --> 00:01:41.370 Well, here the higher die is two the lower one is one, 00:01:41.370 --> 00:01:42.230 right over here. 00:01:42.230 --> 00:01:45.600 So two minus one is one, 00:01:45.600 --> 00:01:47.900 two minus two is zero. 00:01:47.900 --> 00:01:50.240 And now this is gonna be the higher roll, 00:01:50.240 --> 00:01:52.670 die one is gonna have the high roll in this scenario. 00:01:52.670 --> 00:01:54.570 Three minus two is one. 00:01:54.570 --> 00:01:59.080 And then right over here, three minus one is two. 00:01:59.080 --> 00:02:00.960 Now die one rolls a two, 00:02:00.960 --> 00:02:03.090 die two rolls a three. 00:02:03.090 --> 00:02:05.960 Die three is higher, three minus two is one. 00:02:05.960 --> 00:02:08.350 And then three minus three is zero. 00:02:08.350 --> 00:02:10.470 So we've come up with all of the scenarios 00:02:10.470 --> 00:02:13.440 and we can see that we're either gonna end up with a zero 00:02:13.440 --> 00:02:16.900 or one or a two when we look at the positive difference. 00:02:16.900 --> 00:02:21.900 So there's a scenario of getting a zero, a one or a two. 00:02:22.330 --> 00:02:24.570 Those are the different differences 00:02:24.570 --> 00:02:26.140 that we could actually get. 00:02:26.140 --> 00:02:28.780 And so let's think about the probability of each of them. 00:02:28.780 --> 00:02:31.780 What's the probability that the difference is zero. 00:02:31.780 --> 00:02:36.010 Well, we can see that one, two, three 00:02:36.010 --> 00:02:38.940 of the nine equally likely outcomes, 00:02:38.940 --> 00:02:40.580 result in a difference of zero. 00:02:40.580 --> 00:02:44.670 So it's gonna be three out of nine or one-third. 00:02:44.670 --> 00:02:49.670 What about a difference of, let me use the blue, one? 00:02:49.900 --> 00:02:54.300 Well, we could see there are one, two, three, four 00:02:54.300 --> 00:02:56.450 of the nine scenarios have that. 00:02:56.450 --> 00:02:59.000 So there is a four ninths probability. 00:02:59.000 --> 00:03:02.070 And then last but not least a difference of two. 00:03:02.070 --> 00:03:05.470 Well, there's two out of the nine scenarios that have that. 00:03:05.470 --> 00:03:09.408 So there is a two ninths probability right over there. 00:03:09.408 --> 00:03:10.390 And we're done. 00:03:10.390 --> 00:03:13.090 We've constructed the theoretical probability distribution 00:03:13.090 --> 00:03:13.953 of D.

Interpreting general multiplication rule

https://www.youtube.com/watch?v=yTJawPRzCzI

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https://www.youtube.com/api/timedtext?v=yTJawPRzCzI&ei=5VWUZZiFJPuJp-oPmPWP-A0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=247827DB31AA70BA45C1A2514FCDCA3C6E0E8075.D29868C2F69F6097F43FF5238BA6C22A37F8871C&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.230 --> 00:00:01.610 - [Instructor] We're told that two contestants 00:00:01.610 --> 00:00:03.990 are finalists in a cooking competition. 00:00:03.990 --> 00:00:06.530 For the final round, each of them spin a wheel 00:00:06.530 --> 00:00:09.950 to determine what star ingredient must be in their dish. 00:00:09.950 --> 00:00:11.420 I guess the primary ingredient, 00:00:11.420 --> 00:00:12.820 and we can see it could be chard, spinach, 00:00:12.820 --> 00:00:16.720 romaine lettuce, I'm guessing, cabbage, arugula, or kale. 00:00:16.720 --> 00:00:19.510 And so then they give us these different types of events, 00:00:19.510 --> 00:00:21.860 or at least the symbols for these different types of events, 00:00:21.860 --> 00:00:23.040 and then give us their meaning. 00:00:23.040 --> 00:00:27.800 So K-sub 1 means, the first contestant lands on kale, 00:00:27.800 --> 00:00:32.800 K-sub 2 means, the second contestant lands on kale, 00:00:32.800 --> 00:00:36.030 K-sub 1 with this superscript C, 00:00:36.030 --> 00:00:37.730 which you could view as complement. 00:00:37.730 --> 00:00:39.480 So K-sub 1 one complement, 00:00:39.480 --> 00:00:44.350 the first contestant does not land on kale. 00:00:44.350 --> 00:00:46.970 So it's the complement of this one right over here. 00:00:46.970 --> 00:00:49.010 And then K-sub 2 complement, 00:00:49.010 --> 00:00:52.670 would be that the second contestant does not land on kale. 00:00:52.670 --> 00:00:56.440 So the not of K-sub 2 right over here. 00:00:56.440 --> 00:00:59.570 Using the general multiplication rule, 00:00:59.570 --> 00:01:02.530 express symbolically the probability 00:01:02.530 --> 00:01:06.700 that neither contestant lands on kale. 00:01:06.700 --> 00:01:09.550 So pause this video and see if you can have a go at this. 00:01:10.480 --> 00:01:14.760 All right, so the general multiplication rule 00:01:14.760 --> 00:01:16.610 is just saying this notion 00:01:16.610 --> 00:01:21.610 that the probability of two events, A and B, 00:01:22.810 --> 00:01:26.740 is going to be equal to the probability of, 00:01:26.740 --> 00:01:31.740 let's say A given B, times the probability of B. 00:01:33.020 --> 00:01:34.820 Now, if they're independent events, 00:01:34.820 --> 00:01:36.740 if the probability of A occurring 00:01:36.740 --> 00:01:40.930 does not depend in any way on whether B occurred or not, 00:01:40.930 --> 00:01:45.160 then this would simplify to this probability of A given B, 00:01:45.160 --> 00:01:47.100 would just become the probability of A. 00:01:47.100 --> 00:01:48.850 And so if you have two independent events, 00:01:48.850 --> 00:01:50.570 you would just multiply their probability. 00:01:50.570 --> 00:01:51.940 So that's just all they're talking about, 00:01:51.940 --> 00:01:53.870 the general multiplication rule. 00:01:53.870 --> 00:01:54.860 But let me express 00:01:54.860 --> 00:01:57.270 what they're actually asking us to express. 00:01:57.270 --> 00:02:01.770 The probability that neither contestant lands on kale. 00:02:01.770 --> 00:02:03.790 So that means that this is going to happen, 00:02:03.790 --> 00:02:06.360 the first contestant does not land on kale, 00:02:06.360 --> 00:02:07.800 and this is going to happen, 00:02:07.800 --> 00:02:10.990 the second contestant does not land on kale. 00:02:10.990 --> 00:02:12.630 So I could write it this way. 00:02:12.630 --> 00:02:17.320 The probability that K-sub 1 complement 00:02:17.320 --> 00:02:22.320 and K-sub 2 complement, and I could write it this way. 00:02:24.470 --> 00:02:25.960 This is going to be equal to, 00:02:25.960 --> 00:02:28.650 we know that these are independent events 00:02:28.650 --> 00:02:31.530 because if the first contestant gets kale 00:02:31.530 --> 00:02:32.740 or whatever they get it, 00:02:32.740 --> 00:02:34.600 it doesn't get taken out of the running 00:02:34.600 --> 00:02:35.730 for the second contestant. 00:02:35.730 --> 00:02:38.410 The second contestant still has an equal probability 00:02:38.410 --> 00:02:40.080 of getting or not getting kale, 00:02:40.080 --> 00:02:42.780 regardless of what happened for the first contestant. 00:02:42.780 --> 00:02:44.290 So that means we're just in the situation 00:02:44.290 --> 00:02:45.930 where we multiply these probabilities. 00:02:45.930 --> 00:02:50.930 So that's gonna be the probability of K-sub 1 complement, 00:02:51.060 --> 00:02:56.060 times the probability of K-sub 2 complement. 00:02:56.560 --> 00:02:58.570 All right, now let's do part two. 00:02:58.570 --> 00:03:00.380 Interpret what each part 00:03:00.380 --> 00:03:03.860 of this probability statement represents. 00:03:03.860 --> 00:03:05.010 So I encourage you like always, 00:03:05.010 --> 00:03:07.210 pause this video and try to figure that out. 00:03:08.090 --> 00:03:11.950 All right, so first let's think about what is going on here. 00:03:11.950 --> 00:03:13.340 So this is saying, 00:03:13.340 --> 00:03:18.340 the probability that this is K-sub 1 complement. 00:03:19.130 --> 00:03:22.600 So the first contestant does not land on kale. 00:03:22.600 --> 00:03:27.600 So first, first contestant does not get kale, 00:03:33.790 --> 00:03:36.920 and, I'll write and in caps, 00:03:36.920 --> 00:03:39.600 and second contestant does get kale. 00:03:39.600 --> 00:03:44.130 And second does get kale. 00:03:45.490 --> 00:03:47.670 So that's what this left-hand is saying. 00:03:47.670 --> 00:03:51.190 And now they say that that is going to be equal to, 00:03:51.190 --> 00:03:52.330 so this part right over here, 00:03:52.330 --> 00:03:57.330 probability that the first contestant does not get kale. 00:03:58.270 --> 00:04:03.270 Probability that first does not get kale, 00:04:06.930 --> 00:04:10.440 times, right over here. 00:04:10.440 --> 00:04:12.450 And the second part right over here 00:04:12.450 --> 00:04:17.450 is the probability that the second contestant gets kale, 00:04:18.600 --> 00:04:22.170 given that the first contestant does not get kale. 00:04:22.170 --> 00:04:27.130 So probability that the second gets kale, 00:04:29.130 --> 00:04:32.920 given, that's what this vertical line right over here means, 00:04:32.920 --> 00:04:35.110 it means given, shorthand for given. 00:04:35.110 --> 00:04:36.780 Given, I wrote it up there too. 00:04:36.780 --> 00:04:41.780 Given that first does not get kale. 00:04:46.350 --> 00:04:49.300 And we're done, we've just explained what is going on here.

General multiplication rule example: independent events

https://www.youtube.com/watch?v=OqbkCYy37hI

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https://www.youtube.com/api/timedtext?v=OqbkCYy37hI&ei=5VWUZbChHvahp-oPlPa_-Ac&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=A73E43CE5DA0AFFE7B486EB1CB9EA9489993AA51.E2C95CEC55D25E68A25093FF1A0FF875E83E4C2D&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.300 --> 00:00:01.310 - [Instructor] We're told that Maya and Doug 00:00:01.310 --> 00:00:03.870 are finalists in a crafting competition. 00:00:03.870 --> 00:00:06.400 For the final round, each of them spin a wheel 00:00:06.400 --> 00:00:10.510 to determine what star material must be in their craft. 00:00:10.510 --> 00:00:14.640 Maya and Doug both want to get silk as their star material. 00:00:14.640 --> 00:00:17.990 Maya will spin first, followed by Doug. 00:00:17.990 --> 00:00:22.410 What is the probability that neither contestant gets silk? 00:00:22.410 --> 00:00:24.600 Pause this video and think through this on your own 00:00:24.600 --> 00:00:26.450 before we work through this together. 00:00:27.630 --> 00:00:30.960 All right, so first let's think about what they're asking. 00:00:30.960 --> 00:00:33.890 They want to figure out the probability that neither 00:00:33.890 --> 00:00:35.970 gets silk, so I'm gonna write this in shorthand. 00:00:35.970 --> 00:00:40.723 So I'm going to use MNS for Maya no silk. 00:00:41.610 --> 00:00:45.260 And we're also thinking about Doug 00:00:45.260 --> 00:00:47.990 not being able to pick silk. 00:00:47.990 --> 00:00:51.810 So Maya no silk and Doug no silk. 00:00:51.810 --> 00:00:54.200 So we know that this could be viewed 00:00:54.200 --> 00:00:59.200 as the probability that Maya doesn't get silk. 00:00:59.420 --> 00:01:02.740 She, after all does get to spin this wheel first, 00:01:02.740 --> 00:01:04.350 and then we can multiply that 00:01:04.350 --> 00:01:09.350 by the probability that Doug doesn't get silk, 00:01:09.350 --> 00:01:13.860 Doug no silk, given that Maya did not get silk. 00:01:13.860 --> 00:01:16.030 Maya no silk. 00:01:16.030 --> 00:01:19.390 Now it's important to think about whether Doug's probability 00:01:19.390 --> 00:01:24.032 is independent or dependent on whether Maya got silk or not. 00:01:24.032 --> 00:01:28.060 So let's remember Maya will spin first, but it's not like 00:01:28.060 --> 00:01:30.430 if she picks silk, that somehow silk 00:01:30.430 --> 00:01:31.890 is taken out of the running. 00:01:31.890 --> 00:01:33.160 In fact, no matter what she picks, 00:01:33.160 --> 00:01:34.660 it's not taken out of the running. 00:01:34.660 --> 00:01:36.580 Doug will then spin it again. 00:01:36.580 --> 00:01:39.230 And so these are really two independent events, 00:01:39.230 --> 00:01:42.090 and so the probability that Doug doesn't get silk 00:01:42.090 --> 00:01:44.160 given that Maya doesn't get silk, 00:01:44.160 --> 00:01:45.400 this is going to be the same thing 00:01:45.400 --> 00:01:48.790 as the probability that just Doug doesn't get silk. 00:01:48.790 --> 00:01:51.550 It doesn't matter what happens to Maya. 00:01:51.550 --> 00:01:53.930 And so what are each of these? 00:01:53.930 --> 00:01:55.300 Well, this is all going to be equal 00:01:55.300 --> 00:01:57.850 to the probability that Maya does not get silk. 00:01:57.850 --> 00:02:00.630 There's six pieces or six options 00:02:00.630 --> 00:02:02.330 of this wheel right over here. 00:02:02.330 --> 00:02:06.730 Five of them entail her not getting silk on her spin. 00:02:06.730 --> 00:02:09.080 So five over six. 00:02:09.080 --> 00:02:12.350 And then similarly, when Doug goes to spin this wheel 00:02:12.350 --> 00:02:13.870 there are six possibilities. 00:02:13.870 --> 00:02:17.480 Five of them are showing that he does not get silk, 00:02:17.480 --> 00:02:18.680 Doug no silk. 00:02:18.680 --> 00:02:23.480 So times 5/6, which is of course going to be equal 00:02:23.480 --> 00:02:27.423 to 25/36, and we're done.

General multiplication rule example: dependent events

https://www.youtube.com/watch?v=iyQNk090FGM

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https://www.youtube.com/api/timedtext?v=iyQNk090FGM&ei=5VWUZdjqKv6PmLAP8bqzqAQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=8939A89FCB9530A43547BE6E8789BA48B18A4707.27234CF6AD6C61A7DC7AAC0633AAE251A8F1158A&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.330 --> 00:00:02.390 - [Instructor] We're told that Maya and Doug are finalists 00:00:02.390 --> 00:00:04.170 in a crafting competition. 00:00:04.170 --> 00:00:06.080 For the final round, each of them 00:00:06.080 --> 00:00:08.705 will randomly select a card without replacement 00:00:08.705 --> 00:00:11.560 that will reveal what the star material 00:00:11.560 --> 00:00:13.460 must be in their craft. 00:00:13.460 --> 00:00:15.480 Here are the available cards. 00:00:15.480 --> 00:00:17.870 So I guess the star material is the primary material 00:00:17.870 --> 00:00:20.540 they need to use in this competition. 00:00:20.540 --> 00:00:25.540 Maya and Doug both want to get silk as their star material. 00:00:25.810 --> 00:00:29.630 Maya will draw first, followed by Doug. 00:00:29.630 --> 00:00:34.570 What is the probability that neither contestant draws silk? 00:00:34.570 --> 00:00:36.480 Pause this video and see if you can work through that 00:00:36.480 --> 00:00:38.330 before we work through this together. 00:00:39.980 --> 00:00:41.940 All right, now let's work through this together. 00:00:41.940 --> 00:00:45.730 So the probably that neither contestant draws silk. 00:00:45.730 --> 00:00:47.490 So that would be, I'll just write it another way, 00:00:47.490 --> 00:00:52.310 the probability that, I'll write MNS for Maya no silk. 00:00:52.310 --> 00:00:57.310 So Maya no silk and Doug no silk. 00:01:00.190 --> 00:01:02.029 That's just another way of saying, what is the probability 00:01:02.029 --> 00:01:04.740 that neither contestant draws silk? 00:01:04.740 --> 00:01:09.540 And so this is going to be equivalent to the probability 00:01:09.540 --> 00:01:14.540 that Maya does not get silk, Maya no silk, right over here, 00:01:15.010 --> 00:01:20.010 times the probability that Doug doesn't get silk, 00:01:20.800 --> 00:01:24.160 given that Maya did not get silk. 00:01:24.160 --> 00:01:26.060 Given Maya no silk. 00:01:26.060 --> 00:01:28.340 This line right over, this vertical line, 00:01:28.340 --> 00:01:30.560 this is shorthand for given. 00:01:30.560 --> 00:01:32.840 And so let's calculate each of these. 00:01:32.840 --> 00:01:35.020 So this is going to be equal to the probability 00:01:35.020 --> 00:01:36.420 that Maya gets no silk. 00:01:36.420 --> 00:01:40.030 She picked first there's six options out of here. 00:01:40.030 --> 00:01:44.680 Five of them are not silk, so it is five over six. 00:01:44.680 --> 00:01:47.030 And then the probability that Doug does not get silk, 00:01:47.030 --> 00:01:49.690 given that Maya did not get silk. 00:01:49.690 --> 00:01:52.212 So Maya did not get silk, then that means 00:01:52.212 --> 00:01:55.070 that silk is still in the mix, 00:01:55.070 --> 00:01:57.490 but there's only five possibilities left 00:01:57.490 --> 00:01:59.290 because Maya picked one of them, 00:01:59.290 --> 00:02:01.810 and four of them are not silk. 00:02:01.810 --> 00:02:03.800 There's still silk as an option. 00:02:03.800 --> 00:02:05.900 And it's important to recognize that the probability 00:02:05.900 --> 00:02:09.130 that Doug gets no silk is dependent 00:02:09.130 --> 00:02:12.080 on whether Maya got silk or not. 00:02:12.080 --> 00:02:14.950 So it's very important to have this given right over here. 00:02:14.950 --> 00:02:16.930 If these were independent events, 00:02:16.930 --> 00:02:19.770 if Maya picked and then put her card back in 00:02:19.770 --> 00:02:21.840 and then Doug were to pick separately, 00:02:21.840 --> 00:02:24.080 then the probability that Doug gets no silk, 00:02:24.080 --> 00:02:26.620 given that Maya got no silk, would be the same thing, 00:02:26.620 --> 00:02:29.438 as a probability that Doug gets no silk regardless 00:02:29.438 --> 00:02:30.930 of what Maya was doing. 00:02:30.930 --> 00:02:35.220 And so this will end up becoming four over six 00:02:35.220 --> 00:02:37.713 which is the same thing as two thirds.

Probability with combinations example: choosing cards

https://www.youtube.com/watch?v=Tp-SjgG11G8

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en

WEBVTT Kind: captions Language: en 00:00:00.160 --> 00:00:02.560 - We're told that a standard deck of 52 playing cards 00:00:02.560 --> 00:00:06.700 includes four aces, four kings, and 44 other cards. 00:00:06.700 --> 00:00:09.950 Suppose that Luis randomly draws four cards 00:00:09.950 --> 00:00:11.940 without replacement. 00:00:11.940 --> 00:00:15.310 What is the probability that Luis gets two aces 00:00:15.310 --> 00:00:18.490 and two kings, in any order? 00:00:18.490 --> 00:00:19.340 So like always, 00:00:19.340 --> 00:00:22.040 pause this video and see if you can work through this. 00:00:23.110 --> 00:00:23.943 All right. 00:00:23.943 --> 00:00:25.410 Now, to figure out this probability, 00:00:25.410 --> 00:00:26.243 we can think about this, 00:00:26.243 --> 00:00:29.110 it's going to be the number of, 00:00:29.110 --> 00:00:31.430 let's call them draws 00:00:31.430 --> 00:00:36.200 with exactly two aces and two kings, 00:00:36.200 --> 00:00:41.200 two aces and two kings. 00:00:42.220 --> 00:00:43.053 And that's going to be 00:00:43.053 --> 00:00:47.700 over the total number of possible draws of four cards. 00:00:47.700 --> 00:00:48.700 So number 00:00:49.560 --> 00:00:54.130 of possible draws 00:00:55.760 --> 00:00:59.300 of four cards. 00:00:59.300 --> 00:01:00.300 Now, for many of y'all, 00:01:00.300 --> 00:01:01.920 this bottom, the denominator here, 00:01:01.920 --> 00:01:03.980 might be a little bit easier to think about. 00:01:03.980 --> 00:01:06.760 We know that there's 52 total cards, 00:01:06.760 --> 00:01:09.030 of which we are choosing four. 00:01:09.030 --> 00:01:11.330 So we could say 52 choose four, 00:01:11.330 --> 00:01:13.270 and that will tell us the total number 00:01:13.270 --> 00:01:15.610 of possible draws of four cards. 00:01:15.610 --> 00:01:17.700 How many combinations of four cards can we get 00:01:17.700 --> 00:01:19.880 when we're picking from 52? 00:01:19.880 --> 00:01:23.053 But the top here might be a little bit more of a stumper. 00:01:24.170 --> 00:01:28.040 We can think we have exactly two spots for aces, 00:01:28.040 --> 00:01:30.260 so we're choosing two aces 00:01:30.260 --> 00:01:32.400 out of how many possible aces? 00:01:32.400 --> 00:01:34.120 Well, there's four total aces. 00:01:34.120 --> 00:01:38.020 So if we say four choose two, 00:01:38.020 --> 00:01:40.270 this is the total number of ways, 00:01:40.270 --> 00:01:41.500 when you don't care about order, 00:01:41.500 --> 00:01:45.040 that you can have two out of your four aces picked. 00:01:45.040 --> 00:01:45.873 And then separately, 00:01:45.873 --> 00:01:47.690 we can use similar logic to say, 00:01:47.690 --> 00:01:52.260 all right, there's also four choose two ways 00:01:52.260 --> 00:01:55.940 of picking two kings out of four possible kings. 00:01:55.940 --> 00:01:57.870 And now the total number of draws 00:01:57.870 --> 00:02:00.080 with two aces and two kings, 00:02:00.080 --> 00:02:02.950 this is going to be the product of these two. 00:02:02.950 --> 00:02:05.540 And if you're wondering why you can just multiply it, 00:02:05.540 --> 00:02:06.430 think about it. 00:02:06.430 --> 00:02:09.970 For every scenario that you have these two aces, 00:02:09.970 --> 00:02:12.720 you have four choose two scenarios 00:02:12.720 --> 00:02:15.230 of which kings you're dealing with. 00:02:15.230 --> 00:02:17.700 So you would take the product of them. 00:02:17.700 --> 00:02:19.520 And we've already done many examples 00:02:19.520 --> 00:02:21.610 of computing combinatorics like this, 00:02:21.610 --> 00:02:22.660 so I will leave you there. 00:02:22.660 --> 00:02:24.810 If you're so motivated, I encourage you to be, 00:02:24.810 --> 00:02:27.533 you can actually calculate this value.

Probability with combinations example: choosing groups

https://www.youtube.com/watch?v=gy8E0-wf4a0

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en

WEBVTT Kind: captions Language: en 00:00:00.420 --> 00:00:01.410 - [Instructor] We're told that Kyra works 00:00:01.410 --> 00:00:03.930 on a team of 13 total people. 00:00:03.930 --> 00:00:06.470 Her manager is randomly selecting three members 00:00:06.470 --> 00:00:09.760 from her team to represent the company at a conference. 00:00:09.760 --> 00:00:12.430 What is the probability that Kyra is chosen 00:00:12.430 --> 00:00:13.810 for the conference? 00:00:13.810 --> 00:00:15.360 Pause this video and see if you can have a go 00:00:15.360 --> 00:00:18.270 at this before we work through this together. 00:00:18.270 --> 00:00:20.250 All right, now let's work through this together. 00:00:20.250 --> 00:00:22.640 So we wanna figure out this probability. 00:00:22.640 --> 00:00:24.600 And so one way to think about it is, 00:00:24.600 --> 00:00:27.440 what are the number of ways that Kyra can be 00:00:27.440 --> 00:00:31.245 on a team or the number of possible teams, 00:00:31.245 --> 00:00:34.463 teams with Kyra, 00:00:36.390 --> 00:00:41.390 and then over the total number of possible teams, 00:00:41.400 --> 00:00:46.203 total number of possible teams. 00:00:48.760 --> 00:00:52.200 And if this little hint gets you even more inspired. 00:00:52.200 --> 00:00:53.500 If you weren't able to do it the first time, 00:00:53.500 --> 00:00:55.230 I encourage you to try to pause it again 00:00:55.230 --> 00:00:56.980 and then work through it. 00:00:56.980 --> 00:00:59.740 All right, now I will continue to continue. 00:00:59.740 --> 00:01:01.590 So first let me do the denominator here. 00:01:01.590 --> 00:01:04.650 What are the total possible number of teams? 00:01:04.650 --> 00:01:05.660 Some of y'all might've found 00:01:05.660 --> 00:01:07.900 that a little bit easier to figure out. 00:01:07.900 --> 00:01:10.810 Well, we know that we're choosing from 13 people 00:01:10.810 --> 00:01:12.880 and we're picking three of them 00:01:12.880 --> 00:01:14.700 and we don't care about order. 00:01:14.700 --> 00:01:17.180 It's not like we're saying someone's gonna be president 00:01:17.180 --> 00:01:18.013 of the team, 00:01:18.013 --> 00:01:19.120 someone's gonna be vice-president 00:01:19.120 --> 00:01:20.580 and someone's gonna be treasurer. 00:01:20.580 --> 00:01:23.250 We just say there are three people in the team. 00:01:23.250 --> 00:01:26.680 And so this is a situation where out of 13, 00:01:26.680 --> 00:01:30.210 we are choosing three people. 00:01:30.210 --> 00:01:32.160 Now, what are the total number of teams, 00:01:32.160 --> 00:01:35.660 possible teams that could have Kyra in it? 00:01:35.660 --> 00:01:38.330 Well, one way to think about it is if we know 00:01:38.330 --> 00:01:41.600 that Kyra is on a team, then the possibilities are 00:01:41.600 --> 00:01:44.310 who's gonna be the other two people on the team, 00:01:44.310 --> 00:01:45.920 and who are the possible candidates 00:01:45.920 --> 00:01:47.260 for the other two people? 00:01:47.260 --> 00:01:49.260 Well, if Kyra is already on the team 00:01:49.260 --> 00:01:51.970 then there's a possible 12 people to pick from. 00:01:51.970 --> 00:01:53.810 So there's 12 people to choose from 00:01:53.810 --> 00:01:55.680 for those other two slots. 00:01:55.680 --> 00:01:57.430 And so we're gonna choose two. 00:01:57.430 --> 00:01:58.510 And once again, we don't care 00:01:58.510 --> 00:02:00.920 about the order with which we are choosing them. 00:02:00.920 --> 00:02:04.290 So once again, it is gonna be a combination. 00:02:04.290 --> 00:02:05.960 And then we can just go ahead 00:02:05.960 --> 00:02:09.690 and calculate each of these combinations here. 00:02:09.690 --> 00:02:11.890 What is 12 choose two? 00:02:11.890 --> 00:02:13.890 Well, there's 12 possible people 00:02:13.890 --> 00:02:16.370 for that first nine Kyra's seat. 00:02:16.370 --> 00:02:18.750 And then there would be 11 people there 00:02:18.750 --> 00:02:21.270 for that other non Kyra's spot. 00:02:21.270 --> 00:02:22.640 And of course it's a combination. 00:02:22.640 --> 00:02:24.550 We don't care what order we picked it in. 00:02:24.550 --> 00:02:27.580 And so there are two ways to get these two people. 00:02:27.580 --> 00:02:28.750 We could say two factorial 00:02:28.750 --> 00:02:31.250 but that's just the same thing as two or two times one. 00:02:31.250 --> 00:02:33.210 And then the denominator here. 00:02:33.210 --> 00:02:37.250 For that first spot, there's 13 people to pick from , 00:02:37.250 --> 00:02:39.410 then in that second spot, there are 12. 00:02:39.410 --> 00:02:42.240 Then in that third spot, there are 11. 00:02:42.240 --> 00:02:45.210 And then once again, we don't care about order, 00:02:45.210 --> 00:02:47.530 three factorial ways to arrange three people. 00:02:47.530 --> 00:02:50.540 So I could write three times two, 00:02:50.540 --> 00:02:52.800 and for kicks I could write one right over here, 00:02:52.800 --> 00:02:55.030 and then we can, let's go down here. 00:02:55.030 --> 00:02:57.150 This is gonna be equal to my numerator 00:02:57.150 --> 00:03:00.700 over here is gonna be six times 11. 00:03:00.700 --> 00:03:05.700 And then my denominator is going to be 12 divided by six 00:03:06.560 --> 00:03:08.180 right over here is two. 00:03:08.180 --> 00:03:13.180 So it's gonna be 13 times 11 times two. 00:03:14.410 --> 00:03:16.700 Just to be clear, I divided both the denominator 00:03:16.700 --> 00:03:18.520 and this numerator over here 00:03:18.520 --> 00:03:21.160 by six to get two right over there. 00:03:21.160 --> 00:03:23.530 Now this cancels with that. 00:03:23.530 --> 00:03:26.790 And then if we divide the numerator and denominator by two, 00:03:26.790 --> 00:03:28.380 this is gonna be three here. 00:03:28.380 --> 00:03:29.580 This is gonna be one. 00:03:29.580 --> 00:03:33.120 And so we are left with a probability 00:03:33.120 --> 00:03:37.623 of 3/13 that Kyra is chosen for the conference.

Representing systems of any number of equations with matrices

https://www.youtube.com/watch?v=lVle8hJTZAk

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en

WEBVTT Kind: captions Language: en 00:00:00.170 --> 00:00:01.070 - [Instructor] In a previous video, 00:00:01.070 --> 00:00:02.240 we saw that if you have a system 00:00:02.240 --> 00:00:04.900 of three equations with three unknowns like this, 00:00:04.900 --> 00:00:08.360 you can represent it as a matrix vector equation, 00:00:08.360 --> 00:00:10.520 where this matrix right over here 00:00:10.520 --> 00:00:13.740 is a three-by-three matrix. 00:00:13.740 --> 00:00:16.180 That is essentially a coefficient matrix. 00:00:16.180 --> 00:00:18.300 It has all of the coefficients of the Xs, 00:00:18.300 --> 00:00:21.350 the Ys, and the Zs as its various columns. 00:00:21.350 --> 00:00:24.660 and then you're going to multiply that times this vector, 00:00:24.660 --> 00:00:27.070 which is really the vector of the unknown variables, 00:00:27.070 --> 00:00:29.570 and this is a three-by-one vector. 00:00:29.570 --> 00:00:30.830 And then you would result 00:00:30.830 --> 00:00:35.450 in this other three-by-one vector, which is a vector 00:00:35.450 --> 00:00:40.050 that contains these constant terms right over here. 00:00:40.050 --> 00:00:42.160 What we're gonna do in this video is recognize 00:00:42.160 --> 00:00:44.570 that you can generalize this phenomenon. 00:00:44.570 --> 00:00:46.340 It's not just true with a system 00:00:46.340 --> 00:00:48.580 of three equations with three unknowns. 00:00:48.580 --> 00:00:52.490 It actually generalizes to N equations with N unknowns. 00:00:52.490 --> 00:00:55.600 But just to appreciate that that is indeed the case, 00:00:55.600 --> 00:00:59.610 let us look at a system of two equations with two unknowns. 00:00:59.610 --> 00:01:04.020 So let's say you had 2x plus y is equal to nine, 00:01:04.020 --> 00:01:09.020 and we had 3x minus y is equal to five. 00:01:09.230 --> 00:01:10.630 I encourage you, pause this video 00:01:10.630 --> 00:01:12.610 and think about how that would be represented 00:01:12.610 --> 00:01:15.253 as a matrix vector equation. 00:01:16.520 --> 00:01:19.520 All right, now let's work on this together. 00:01:19.520 --> 00:01:23.710 So this is a system of two equations with two unknowns. 00:01:23.710 --> 00:01:27.010 So the matrix that represents the coefficients is going 00:01:27.010 --> 00:01:29.390 to be a two-by-two matrix 00:01:29.390 --> 00:01:31.680 and then that's going to be multiplied 00:01:31.680 --> 00:01:35.200 by a vector that represents the unknown variables. 00:01:35.200 --> 00:01:37.090 We have two unknown variables over here. 00:01:37.090 --> 00:01:40.120 So this is going to be a two-by-one vector, 00:01:40.120 --> 00:01:42.910 and then that's going to be equal to a vector 00:01:42.910 --> 00:01:45.520 that represents the constants on the right-hand side, 00:01:45.520 --> 00:01:46.810 and obviously we have two of those. 00:01:46.810 --> 00:01:49.430 So that's going to be a two-by-one vector as well. 00:01:49.430 --> 00:01:51.870 And then we can do exactly what we did 00:01:51.870 --> 00:01:54.410 in that previous example in a previous video. 00:01:54.410 --> 00:01:58.850 The coefficients on the X terms, two and three, 00:01:58.850 --> 00:02:02.570 and then we have the coefficients on the Y terms. 00:02:02.570 --> 00:02:04.530 This would be a positive one 00:02:04.530 --> 00:02:06.930 and then this would be a negative one. 00:02:06.930 --> 00:02:09.570 And then we multiply it times the vector 00:02:09.570 --> 00:02:12.490 of the variables, X, Y, 00:02:12.490 --> 00:02:14.700 and then last but not least, 00:02:14.700 --> 00:02:19.200 you have this nine and this five over here, nine and five. 00:02:19.200 --> 00:02:22.040 And I encourage you and multiply this out. 00:02:22.040 --> 00:02:25.390 Multiply this matrix times this vector. 00:02:25.390 --> 00:02:28.560 And when you do that and you still set up this equality, 00:02:28.560 --> 00:02:30.620 you're going to see that it essentially turns 00:02:30.620 --> 00:02:32.770 into this exact same system 00:02:32.770 --> 00:02:36.010 of two equations and two unknowns. 00:02:36.010 --> 00:02:37.650 Now, what's interesting about this is 00:02:37.650 --> 00:02:40.730 that we see a generalizable form. 00:02:40.730 --> 00:02:43.860 In general, you can represent a system 00:02:43.860 --> 00:02:47.890 of N equations and N unknowns in the form. 00:02:47.890 --> 00:02:51.740 Sum N-by-N matrix A, 00:02:51.740 --> 00:02:53.400 N by N, 00:02:53.400 --> 00:02:58.340 times sum N-by-one vector X. 00:02:58.340 --> 00:02:59.523 This isn't just the variable X. 00:02:59.523 --> 00:03:03.950 This is a vector X that has N dimensions to it. 00:03:03.950 --> 00:03:08.610 So times sum N-by-one vector X is going 00:03:08.610 --> 00:03:13.530 to be equal to sum N-by-one vector B. 00:03:14.480 --> 00:03:17.360 These are the letters that people use by convention. 00:03:17.360 --> 00:03:19.030 This is going to be N by one. 00:03:19.030 --> 00:03:21.890 And so you can see in these different scenarios. 00:03:21.890 --> 00:03:24.850 In that first one, this is a three-by-three matrix. 00:03:24.850 --> 00:03:26.870 We could call that A, 00:03:26.870 --> 00:03:29.580 and then we could call this the vector X, 00:03:29.580 --> 00:03:31.760 and then we could call this the vector B. 00:03:31.760 --> 00:03:32.850 Now in that second scenario, 00:03:32.850 --> 00:03:34.800 we could call this the matrix A, 00:03:34.800 --> 00:03:36.850 we could call this the vector X, 00:03:36.850 --> 00:03:39.320 and then we could call this the vector B, 00:03:39.320 --> 00:03:42.820 but we can generalize that to N dimensions. 00:03:42.820 --> 00:03:44.760 And as I talked about in the previous video, 00:03:44.760 --> 00:03:47.880 what's interesting about this is you could think about, 00:03:47.880 --> 00:03:49.890 for example, in this system of two equations 00:03:49.890 --> 00:03:52.300 with two unknowns, as all right, I have a line here, 00:03:52.300 --> 00:03:53.490 I have a line here, 00:03:53.490 --> 00:03:57.640 and X and Y represent the intersection of those lines. 00:03:57.640 --> 00:03:58.850 But when you represent it this way, 00:03:58.850 --> 00:04:00.580 you could also imagine it as saying, 00:04:00.580 --> 00:04:04.470 okay, I have some unknown vector in the coordinate plane 00:04:04.470 --> 00:04:07.460 and I'm transforming it using this matrix 00:04:07.460 --> 00:04:10.030 to get this vector nine five. 00:04:10.030 --> 00:04:12.410 And so I have to figure out what vector, 00:04:12.410 --> 00:04:15.510 when transformed in this way, gets us to nine five, 00:04:15.510 --> 00:04:18.160 and we also thought about it in the three-by-three case. 00:04:18.160 --> 00:04:20.950 What three-dimensional vector, when transformed in this way, 00:04:20.950 --> 00:04:23.360 gets us to this vector right over here? 00:04:23.360 --> 00:04:24.910 And so that hints, 00:04:24.910 --> 00:04:27.410 that foreshadows where we might be able to go. 00:04:27.410 --> 00:04:30.530 If we can unwind this transformation somehow, 00:04:30.530 --> 00:04:33.850 then we can figure out what these unknown vectors are. 00:04:33.850 --> 00:04:37.270 And if we can do it in two dimensions or three dimensions, 00:04:37.270 --> 00:04:40.010 why not be able to do it in N dimensions? 00:04:40.010 --> 00:04:41.980 Which you'll see is actually very useful 00:04:41.980 --> 00:04:43.650 if you ever become a data scientist, 00:04:43.650 --> 00:04:45.220 or you go into computer science, 00:04:45.220 --> 00:04:47.670 or if you go into computer graphics of some kind.

Probability with permutations & combinations example: taste testing

https://www.youtube.com/watch?v=C6gQZ7qKtdM

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https://www.youtube.com/api/timedtext?v=C6gQZ7qKtdM&ei=5VWUZaPiG4_YxN8Plc-K6AQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=3902749BC808D138091AE242970BD068198DF69E.9954A4236EAED0F00FDF6A9F47A45EA00E06D57E&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.210 --> 00:00:02.080 - [Instructor] We're told that Samara is setting up 00:00:02.080 --> 00:00:05.130 an olive tasting competition for a festival. 00:00:05.130 --> 00:00:07.630 From 15 distinct varieties, 00:00:07.630 --> 00:00:11.160 Samara will choose three different olive oils 00:00:11.160 --> 00:00:12.870 and blend them together. 00:00:12.870 --> 00:00:14.990 A contestant will taste the blend 00:00:14.990 --> 00:00:18.870 and try to identify which three of the 15 varieties 00:00:18.870 --> 00:00:20.860 were used to make it. 00:00:20.860 --> 00:00:24.020 Assume that a contestant can't taste any difference 00:00:24.020 --> 00:00:26.050 and is randomly guessing. 00:00:26.050 --> 00:00:27.560 What is the probability 00:00:27.560 --> 00:00:29.730 that a contestant correctly guesses 00:00:29.730 --> 00:00:33.090 which three varieties were used? 00:00:33.090 --> 00:00:35.840 So pause this video and see if you can think about that. 00:00:35.840 --> 00:00:37.330 And if you can just come up with the expression, 00:00:37.330 --> 00:00:38.640 you don't have to compute it. 00:00:38.640 --> 00:00:41.493 That is probably good enough, at least for our purposes. 00:00:42.420 --> 00:00:45.420 All right, now let's work through this together. 00:00:45.420 --> 00:00:48.040 So we know several things here. 00:00:48.040 --> 00:00:51.410 We have 15 distinct varieties 00:00:51.410 --> 00:00:55.860 and we are choosing three of those varieties. 00:00:55.860 --> 00:00:57.080 And anytime we're talking 00:00:57.080 --> 00:00:59.470 about probability and combinatorics, 00:00:59.470 --> 00:01:02.230 it's always interesting to say, "Does order matter? 00:01:02.230 --> 00:01:04.440 Does it matter what order that Samara 00:01:04.440 --> 00:01:07.590 is picking those three from the 15?" 00:01:07.590 --> 00:01:09.000 It doesn't look like it matters. 00:01:09.000 --> 00:01:10.610 It looks like we just have to think 00:01:10.610 --> 00:01:11.740 about what three they are. 00:01:11.740 --> 00:01:15.390 It doesn't matter what order either she picked them in, 00:01:15.390 --> 00:01:18.410 or the order in which the contestant guesses them in. 00:01:18.410 --> 00:01:20.290 And so if you think about the total number 00:01:20.290 --> 00:01:24.540 of ways of picking three things from a group of 15, 00:01:24.540 --> 00:01:28.913 you could write that as 15, choose three. 00:01:30.700 --> 00:01:33.020 Once again, this is just shorthand notation 00:01:33.020 --> 00:01:35.610 for how many combinations are there, 00:01:35.610 --> 00:01:38.630 so you can pick three things from a group of 15? 00:01:38.630 --> 00:01:40.657 So some of you might have been tempted to say, 00:01:40.657 --> 00:01:43.730 "Hey, let me think about permutations here. 00:01:43.730 --> 00:01:46.340 And I have 15 things. 00:01:46.340 --> 00:01:49.180 And from that, I wanna figure out how many ways 00:01:49.180 --> 00:01:54.160 can I pick three things that actually has order mattering?" 00:01:54.160 --> 00:01:56.010 But this would be the situation where we're talking 00:01:56.010 --> 00:01:58.960 about the contestant actually having to maybe guess 00:01:58.960 --> 00:02:02.120 in the same order in which the varieties 00:02:02.120 --> 00:02:04.710 were originally blended, or something like that, 00:02:04.710 --> 00:02:05.543 but we're not doing that, 00:02:05.543 --> 00:02:08.400 we just care about getting the right three varieties. 00:02:08.400 --> 00:02:11.930 So this will tell us the total number of ways 00:02:11.930 --> 00:02:14.930 that you can pick three out of 15. 00:02:14.930 --> 00:02:16.170 And so what's the probability 00:02:16.170 --> 00:02:17.870 that the contestant correctly guesses 00:02:17.870 --> 00:02:19.430 which three varieties were used? 00:02:19.430 --> 00:02:21.610 Well, the contestant is going to be guessing one 00:02:21.610 --> 00:02:24.450 out of the possible number of scenarios here. 00:02:24.450 --> 00:02:29.450 So the probability would be one over 15, choose three. 00:02:30.200 --> 00:02:31.860 And if you wanted to compute this, 00:02:31.860 --> 00:02:35.310 this would be equal to one over, 00:02:35.310 --> 00:02:37.930 now, how many ways can you pick three things from 15? 00:02:37.930 --> 00:02:39.530 And of course there is a formula here, 00:02:39.530 --> 00:02:41.470 but I always like to reason through it. 00:02:41.470 --> 00:02:44.450 Well, you could say, "All right, if there's three slots, 00:02:44.450 --> 00:02:46.910 there's 15 different varieties 00:02:46.910 --> 00:02:49.260 that could've gone into that first slot, 00:02:49.260 --> 00:02:52.310 and then there's 14 that could go into that second slot, 00:02:52.310 --> 00:02:55.720 and then there's 13 that can go into that third slot." 00:02:55.720 --> 00:02:56.760 But then we have to remember 00:02:56.760 --> 00:02:59.600 that it doesn't matter what order we pick them in. 00:02:59.600 --> 00:03:02.730 So how many ways can you rearrange three things? 00:03:02.730 --> 00:03:04.700 Well, it would be three factorial, 00:03:04.700 --> 00:03:08.050 or three times two times one. 00:03:08.050 --> 00:03:12.580 So this would be the same thing as three times two times one 00:03:12.580 --> 00:03:17.580 over 15 times 14 times 13. 00:03:17.900 --> 00:03:19.190 See, I can simplify this, 00:03:19.190 --> 00:03:22.330 divide numerator and denominator by two, 00:03:22.330 --> 00:03:25.850 divide numerator and denominator by three. 00:03:25.850 --> 00:03:30.423 This is going to be equal to one over 35 times 13. 00:03:32.030 --> 00:03:37.030 This is going to be one over 350 plus 105, which is 455. 00:03:42.090 --> 00:03:44.103 And we are done.

Vector word problem: resultant force

https://www.youtube.com/watch?v=Ga1Di84GcgM

vtt

https://www.youtube.com/api/timedtext?v=Ga1Di84GcgM&ei=5VWUZc7pKsW2mLAPi5umqA4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=C5B144EA765B1056F778C47F9AB1B7CA904B8B76.165AAB36BAC1D130A75C333A2BC0E26689F699DD&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.480 --> 00:00:01.730 - [Instructor] We're told that a metal ball 00:00:01.730 --> 00:00:04.290 lies on a flat, horizontal surface. 00:00:04.290 --> 00:00:07.800 It is attracted by two magnets placed around it. 00:00:07.800 --> 00:00:12.800 We're told that the first magnets force on the ball is 5 N. 00:00:14.220 --> 00:00:17.700 We're then told the second magnets force on the ball 00:00:17.700 --> 00:00:20.858 is 3 N in a direction 00:00:20.858 --> 00:00:25.650 that is 100 degree rotation from the first magnets force. 00:00:25.650 --> 00:00:27.080 And we can see that drawn here. 00:00:27.080 --> 00:00:29.950 This is the first magnets force, it's 5 N. 00:00:29.950 --> 00:00:34.110 And then the second magnets force is 3 N 00:00:34.110 --> 00:00:37.130 at 100 degree angle, 100 degree rotation 00:00:37.130 --> 00:00:39.460 from the first magnets force. 00:00:39.460 --> 00:00:42.040 Now they're asking us a few interesting questions. 00:00:42.040 --> 00:00:46.000 What is the combined strength of the magnets' pulls? 00:00:46.000 --> 00:00:47.110 And then they also say, 00:00:47.110 --> 00:00:49.930 what is the direction of the magnets' combined pulls, 00:00:49.930 --> 00:00:53.350 relative to the direction of the first magnets pull? 00:00:53.350 --> 00:00:56.340 So I encourage you, pause this video 00:00:56.340 --> 00:00:58.250 and have a go at this on your own 00:00:58.250 --> 00:01:00.100 before we work through this together. 00:01:01.290 --> 00:01:03.440 All right, now let's work through this together. 00:01:03.440 --> 00:01:05.120 So they're really saying, 00:01:05.120 --> 00:01:07.210 if I take the sum of these two vectors, 00:01:07.210 --> 00:01:09.450 what is gonna be the resultant force vector? 00:01:09.450 --> 00:01:10.810 What is going to be the magnitude 00:01:10.810 --> 00:01:12.457 of that result in force vector? 00:01:12.457 --> 00:01:16.690 And what is its direction going to be? 00:01:16.690 --> 00:01:18.340 There's two ways we could approach this. 00:01:18.340 --> 00:01:20.440 We could break it down each of these vectors 00:01:20.440 --> 00:01:22.260 into their respective components, 00:01:22.260 --> 00:01:24.330 and then add the respect of components, 00:01:24.330 --> 00:01:26.750 and then from that figure out what the magnitude 00:01:26.750 --> 00:01:27.583 and direction is. 00:01:27.583 --> 00:01:29.160 And we do that in other videos, 00:01:29.160 --> 00:01:31.430 or we could take the geometric approach. 00:01:31.430 --> 00:01:32.680 So that's what we're gonna do here. 00:01:32.680 --> 00:01:34.030 And to help us with that, 00:01:34.030 --> 00:01:36.770 we're gonna use what we've called the Parallelogram rule 00:01:36.770 --> 00:01:38.730 which is really the same idea 00:01:38.730 --> 00:01:42.030 as the head to tail addition of vectors. 00:01:42.030 --> 00:01:44.030 I can take the 3 N vector, 00:01:44.030 --> 00:01:45.510 I can shift it over, 00:01:45.510 --> 00:01:48.750 so its tails at the head of the 5 N vector. 00:01:48.750 --> 00:01:51.930 It would look something like this. 00:01:51.930 --> 00:01:54.590 So this is 3 N right over there. 00:01:54.590 --> 00:01:57.100 And then I could also go the other way around. 00:01:57.100 --> 00:01:59.000 I could take the 3 N vector first, 00:01:59.000 --> 00:02:01.330 and take the tail of the 5 N vector 00:02:01.330 --> 00:02:06.270 at the head of the 3 N vector and shift it like this. 00:02:06.270 --> 00:02:08.230 You can add an either direction 00:02:09.100 --> 00:02:10.940 and either way you look at it, 00:02:10.940 --> 00:02:13.590 when you start at the tails and you get to the head 00:02:13.590 --> 00:02:15.360 of the second vector, 00:02:15.360 --> 00:02:17.920 you're going to have a resultant force 00:02:17.920 --> 00:02:19.130 that looks like this, 00:02:19.130 --> 00:02:22.390 which is the diagonal of this parallelogram. 00:02:22.390 --> 00:02:24.600 So there we go. 00:02:24.600 --> 00:02:28.900 And let me just call that our force vector right over there. 00:02:28.900 --> 00:02:32.190 So if we can figure out the length of this line 00:02:32.190 --> 00:02:33.680 of this diagonal right over here, 00:02:33.680 --> 00:02:37.200 that would be the magnitude of this force vector. 00:02:37.200 --> 00:02:38.950 Now, how can we do that? 00:02:38.950 --> 00:02:40.290 Well, let's just think geometrically 00:02:40.290 --> 00:02:44.100 what else we can figure out about what's going on over here? 00:02:44.100 --> 00:02:45.810 This is a parallelogram. 00:02:45.810 --> 00:02:49.000 So if this is 100 degree angle right over here, 00:02:49.000 --> 00:02:53.850 this angle right over here is also going to be 100 degree. 00:02:53.850 --> 00:02:57.420 And we also know that these two opposite angles 00:02:57.420 --> 00:02:59.930 are also gonna have the same measure right over here. 00:02:59.930 --> 00:03:02.850 And we also know that the sum of all of the angles 00:03:02.850 --> 00:03:06.850 in a quadrilateral are going to be 360 degrees. 00:03:06.850 --> 00:03:08.930 So these two make up 200 degrees, 00:03:08.930 --> 00:03:11.610 we have 160 degrees left that have to be split 00:03:11.610 --> 00:03:13.290 between that one and that one. 00:03:13.290 --> 00:03:15.310 So we know that this is 80 degrees, 00:03:15.310 --> 00:03:17.700 and we know that this is 80 degrees. 00:03:17.700 --> 00:03:19.320 Well, how does that help us? 00:03:19.320 --> 00:03:21.730 So we know the length of this brown side, 00:03:21.730 --> 00:03:24.950 we know the length of this side right over here. 00:03:24.950 --> 00:03:26.860 We know the angle between them, 00:03:26.860 --> 00:03:28.030 and what we're trying to do, 00:03:28.030 --> 00:03:31.310 is figure out the length of the side opposite this angle, 00:03:31.310 --> 00:03:33.200 opposite this 80 degree angle. 00:03:33.200 --> 00:03:37.370 And some of you might remember the Law of Cosines here. 00:03:37.370 --> 00:03:39.880 And the Law of Cosines I always imagine it 00:03:39.880 --> 00:03:42.950 as an adaptation of the Pythagorean theorem, 00:03:42.950 --> 00:03:46.340 so that we can deal with non-right triangles. 00:03:46.340 --> 00:03:49.080 And the Law of Cosines will tell us 00:03:49.080 --> 00:03:52.380 that the magnitude, I'll just write it over here, 00:03:52.380 --> 00:03:54.770 the magnitude of this vector, 00:03:54.770 --> 00:03:57.130 which is the length of this diagonal, 00:03:57.130 --> 00:04:01.650 is going to be equal to the square root of, 00:04:01.650 --> 00:04:04.830 we're going to have this side squared, 00:04:04.830 --> 00:04:07.910 so let me write 3 squared, 00:04:07.910 --> 00:04:09.730 plus this side squared, 00:04:09.730 --> 00:04:13.580 plus 5 squared minus 00:04:13.580 --> 00:04:15.863 2 times this side. 00:04:16.910 --> 00:04:19.800 So times 3 times that side. 00:04:19.800 --> 00:04:24.800 So times 5 times the cosine of 80 degrees. 00:04:27.620 --> 00:04:30.450 And so let's get our calculator out to calculate that. 00:04:30.450 --> 00:04:34.510 I'll start with taking the cosine of 80 degrees, 00:04:34.510 --> 00:04:38.370 then I'm just gonna multiply that times looks like 30. 00:04:38.370 --> 00:04:41.890 So times 30 is equal to that. 00:04:41.890 --> 00:04:44.610 Let's put it a little negative there. 00:04:44.610 --> 00:04:49.610 And then to that, I'm going to add 25 and 9, which is 34. 00:04:51.200 --> 00:04:55.280 So plus 34 is equal to that. 00:04:55.280 --> 00:04:57.380 And now I just take the square root of all of that. 00:04:57.380 --> 00:04:59.770 And they tell us to round our answer to the nearest tenths. 00:04:59.770 --> 00:05:03.960 So I can round this to approximately 5.4. 00:05:03.960 --> 00:05:08.423 So this is approximately 5.4 N. 00:05:10.343 --> 00:05:11.820 Now they say, what is the direction 00:05:11.820 --> 00:05:13.140 of the magnets combined poles 00:05:13.140 --> 00:05:16.470 relative to the direction of the first magnets pole? 00:05:16.470 --> 00:05:18.070 So really what we want to do, 00:05:18.070 --> 00:05:21.340 is figure out this angle right over here, 00:05:21.340 --> 00:05:23.040 let's call that data. 00:05:23.040 --> 00:05:25.780 Well, we know what the length of the side opposite is. 00:05:25.780 --> 00:05:29.060 So maybe we could use the Law of Sines. 00:05:29.060 --> 00:05:30.650 The Law of Sines would tell us 00:05:30.650 --> 00:05:33.630 that the sine of theta 00:05:33.630 --> 00:05:36.210 over the length of the side opposite to it 00:05:36.210 --> 00:05:39.630 is going to be equal to, let's pick another angle we know, 00:05:39.630 --> 00:05:41.180 sine of this angle. 00:05:41.180 --> 00:05:44.310 Sine of 80 degrees 00:05:44.310 --> 00:05:47.100 over the length of the side opposite to it. 00:05:47.100 --> 00:05:50.230 And so this is approximately 5.4. 00:05:50.230 --> 00:05:52.390 And so if we wanna solve a for theta, 00:05:52.390 --> 00:05:54.970 we can multiply both sides by 3. 00:05:54.970 --> 00:05:57.950 So we're going to get sine of theta, 00:05:57.950 --> 00:06:00.620 I'll just stay in this purple color for simplicity, 00:06:00.620 --> 00:06:05.620 is equal to 3 times sine of 80 degrees 00:06:07.170 --> 00:06:09.510 divided by 5.4. 00:06:09.510 --> 00:06:11.580 And then we could say that theta 00:06:11.580 --> 00:06:16.450 is equal to the inverse sine of all of this business. 00:06:16.450 --> 00:06:21.107 Three sine of 80 degrees over 5.4. 00:06:22.310 --> 00:06:26.380 So we're going to take 80 degrees, take the sine of it, 00:06:26.380 --> 00:06:29.860 we're going to multiply that by 3, 00:06:29.860 --> 00:06:33.740 divide that by 5.4, that equals that, 00:06:33.740 --> 00:06:37.780 and then I'm going to take the inverse sine of all of that. 00:06:37.780 --> 00:06:41.210 And they want us to round to the nearest integer. 00:06:41.210 --> 00:06:44.860 So that's approximately 33 degrees. 00:06:44.860 --> 00:06:46.220 When you do the Law of Sines, 00:06:46.220 --> 00:06:49.120 it's possible that you're also dealing with an obtuse angle. 00:06:49.120 --> 00:06:51.070 And when you do all of this, you get the acute one, 00:06:51.070 --> 00:06:53.010 and then you would have to make an adjustment. 00:06:53.010 --> 00:06:54.640 But that's not what we're dealing here, 00:06:54.640 --> 00:06:58.020 so we know that this data is approximately 00:06:58.020 --> 00:06:59.840 equal to 33 degrees. 00:06:59.840 --> 00:07:02.340 So we know the magnitude of the force 00:07:02.340 --> 00:07:04.240 and we know that it forms an angle 00:07:04.240 --> 00:07:06.330 of approximately 33 degrees 00:07:06.330 --> 00:07:10.003 with the direction of the force of that first magnet.

Vector word problem: resultant velocity

https://www.youtube.com/watch?v=Neon4dpvmUk

vtt

https://www.youtube.com/api/timedtext?v=Neon4dpvmUk&ei=5VWUZZyuJoygp-oP5fKH0A8&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=3268E322AA007AC3FF8DBB35239037AABC21C7AA.AD94C16B97E6B80737D35B11C38707FF17DF1903&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.350 --> 00:00:02.260 - [Instructor] We're told a boat is traveling 00:00:02.260 --> 00:00:05.900 at a speed of 26 kilometers per hour in a direction 00:00:05.900 --> 00:00:09.940 that is a 300 degree rotation from East. 00:00:09.940 --> 00:00:12.630 At a certain point it encounters a current 00:00:12.630 --> 00:00:16.070 at a speed of 15 kilometers per hour in a direction 00:00:16.070 --> 00:00:19.720 that is a 25 degree rotation from East. 00:00:19.720 --> 00:00:22.380 Answer two questions about the boat's velocity 00:00:22.380 --> 00:00:24.470 after it meets the current. 00:00:24.470 --> 00:00:25.850 Alright, the first question is, 00:00:25.850 --> 00:00:29.110 what is the boat's speed after it meets the current? 00:00:29.110 --> 00:00:31.390 And it says, round your answer to the nearest 10th. 00:00:31.390 --> 00:00:34.140 You can round intermediate values to the nearest 100th. 00:00:34.140 --> 00:00:37.070 And what is the direction of the boat's velocity 00:00:37.070 --> 00:00:38.670 after it meets the current? 00:00:38.670 --> 00:00:40.730 And they say the same, well, they actually here it say, 00:00:40.730 --> 00:00:42.500 round your answer to the nearest integer 00:00:42.500 --> 00:00:45.570 and you can round intermediate values to the nearest 100th. 00:00:45.570 --> 00:00:47.170 So like always pause this video 00:00:47.170 --> 00:00:49.020 and see if you can work through this. 00:00:50.070 --> 00:00:52.120 All right, now let's work on this together. 00:00:52.120 --> 00:00:54.230 So first let's visualize each of these vectors. 00:00:54.230 --> 00:00:57.610 We have this vector 26 kilometers per hour in a direction 00:00:57.610 --> 00:01:00.270 that is a 300 degree rotation from East. 00:01:00.270 --> 00:01:04.110 And we have this vector 15 kilometers per hour 00:01:04.110 --> 00:01:08.210 in a direction that is a 25 degree rotation from East. 00:01:08.210 --> 00:01:12.120 And so let me draw some axes here. 00:01:12.120 --> 00:01:16.990 So let's say that is my Y-axis. 00:01:16.990 --> 00:01:21.990 And then let's say that this over here is my X-axis. 00:01:22.000 --> 00:01:26.140 And then that first vector 300 degree rotation from East, 00:01:26.140 --> 00:01:28.660 East is in the positive X direction. 00:01:28.660 --> 00:01:33.170 This would be 90 degrees, 180 degrees, 270 degrees. 00:01:33.170 --> 00:01:35.390 I'm going counter-clockwise cause that's the convention 00:01:35.390 --> 00:01:37.280 for a positive angle. 00:01:37.280 --> 00:01:40.490 And then we'd go a little bit past 270, 00:01:40.490 --> 00:01:44.810 we would go right, right over there. 00:01:44.810 --> 00:01:48.240 And the magnitude of this vector is 26 kilometers per hour. 00:01:48.240 --> 00:01:50.970 I'll just write a 26 right over there. 00:01:50.970 --> 00:01:54.230 And then this other vector which is the current 00:01:54.230 --> 00:01:56.060 15 kilometers per hour in a direction 00:01:56.060 --> 00:01:58.230 that is a 25 degree rotation from East. 00:01:58.230 --> 00:02:02.880 So 25 degree rotation might be something like this 00:02:02.880 --> 00:02:04.340 and it's going to be shorter. 00:02:04.340 --> 00:02:06.720 It's 15 kilometers per hour. 00:02:06.720 --> 00:02:10.540 So, it's going to be roughly about that long. 00:02:10.540 --> 00:02:12.780 I'm obviously just approximating it 00:02:12.780 --> 00:02:15.860 and I'll just write 15 there for its magnitude. 00:02:15.860 --> 00:02:19.410 So we can visualize what the boat's speed 00:02:19.410 --> 00:02:21.770 and direction it is after it meets the current. 00:02:21.770 --> 00:02:24.350 It's going to be the sum of these two vectors. 00:02:24.350 --> 00:02:27.480 And so if we wanted to sum these two vectors 00:02:27.480 --> 00:02:30.640 we could put the tail of one at the head of the other. 00:02:30.640 --> 00:02:33.440 And so let's shift this blue vector down here. 00:02:33.440 --> 00:02:36.260 So it's at the head of the red vector. 00:02:36.260 --> 00:02:38.310 So it would be something like this. 00:02:38.310 --> 00:02:43.010 And so our resulting speed after it meets the current 00:02:43.010 --> 00:02:44.790 would look something like this. 00:02:44.790 --> 00:02:48.360 We've seen this in many other videos so far 00:02:48.360 --> 00:02:50.410 but we don't want to just figure it out visually. 00:02:50.410 --> 00:02:52.640 We want to actually figure out its actual speed 00:02:52.640 --> 00:02:54.430 which would be the magnitude of this vector 00:02:54.430 --> 00:02:55.900 and its actual direction. 00:02:55.900 --> 00:02:57.030 So what is the angle? 00:02:57.030 --> 00:02:58.620 And we could say it as a positive angle. 00:02:58.620 --> 00:03:01.410 So what the rotation, the positive rotation 00:03:01.410 --> 00:03:05.230 from the positive X-axis or from due East. 00:03:05.230 --> 00:03:07.530 So to do that, what I'm going to do is represent each 00:03:07.530 --> 00:03:11.430 of our original vectors in terms of their components. 00:03:11.430 --> 00:03:14.060 And so this red vector up here 00:03:14.060 --> 00:03:17.370 and we've done this multiple times explaining the intuition. 00:03:17.370 --> 00:03:20.680 It's X component is going to be its magnitude 00:03:20.680 --> 00:03:23.920 26 times the cosine of this angle, 00:03:23.920 --> 00:03:26.980 cosine of 300 degrees. 00:03:26.980 --> 00:03:31.290 And it's Y component is going to be 26 times the sine 00:03:31.290 --> 00:03:33.280 of 300 degrees. 00:03:33.280 --> 00:03:35.760 If that's unfamiliar to you, I encourage you to review it 00:03:35.760 --> 00:03:37.600 in other videos where we first introduced 00:03:37.600 --> 00:03:40.030 the notion of components, it comes straight out 00:03:40.030 --> 00:03:43.930 of our unit circle definition of trig functions. 00:03:43.930 --> 00:03:47.040 And similarly, this vector right over here 00:03:47.040 --> 00:03:49.710 it's X component is going to be its magnitude times 00:03:49.710 --> 00:03:53.150 the cosine of 25 degrees. 00:03:53.150 --> 00:03:58.150 And it's Y component is going to be 15 times the sine 00:03:58.250 --> 00:04:00.740 of 25 degrees. 00:04:00.740 --> 00:04:03.070 And now when we have it expressed this way, 00:04:03.070 --> 00:04:05.870 if we want to have the resulting vector, 00:04:05.870 --> 00:04:08.490 let's call the resulting vector S 00:04:08.490 --> 00:04:10.970 for maybe the resulting speed. 00:04:10.970 --> 00:04:13.970 Its components are going to be the sum of each of these. 00:04:13.970 --> 00:04:16.120 So we can write it over here. 00:04:16.120 --> 00:04:19.860 Vector S is going to be equal to 00:04:19.860 --> 00:04:22.550 it's going to be the X component of this red vector 00:04:22.550 --> 00:04:23.970 of our original speed vector. 00:04:23.970 --> 00:04:28.880 So, 26 cosine of 300 degrees 00:04:28.880 --> 00:04:31.710 plus the X component of the current. 00:04:31.710 --> 00:04:36.710 So, 15 times cosine of 25 degrees and then the Y components. 00:04:37.340 --> 00:04:40.230 Once again, I add the corresponding Y components 00:04:40.230 --> 00:04:45.080 26 sine of 300 degrees 00:04:45.080 --> 00:04:50.080 plus 15 sine of 25 degrees. 00:04:50.290 --> 00:04:52.170 And now we could use a calculator to figure out 00:04:52.170 --> 00:04:55.170 what these are, to say what these approximately are. 00:04:55.170 --> 00:04:57.780 So first the X component, we're going to take 00:04:57.780 --> 00:05:02.513 the cosine of 300 degrees, times 26, 00:05:03.380 --> 00:05:06.400 plus I'll open a parenthesis here. 00:05:06.400 --> 00:05:11.343 We're going to take the cosine of 25 degrees, times 15, 00:05:12.280 --> 00:05:14.210 close our parentheses. 00:05:14.210 --> 00:05:19.210 And that is equal to 26.59 if I round to the nearest 100th. 00:05:20.689 --> 00:05:23.960 26.59. 00:05:23.960 --> 00:05:25.960 And now let's do the Y component. 00:05:25.960 --> 00:05:30.960 We have the sine of 300 degrees, times 26, 00:05:32.190 --> 00:05:34.960 plus I'll open parentheses, 00:05:34.960 --> 00:05:39.960 the sine of 25 degrees times 15, close parentheses, 00:05:40.480 --> 00:05:45.480 is equal to negative 16.18 to round to the nearest 100th. 00:05:45.720 --> 00:05:48.690 Negative 16.18. 00:05:48.690 --> 00:05:52.200 And let's just make sure that this makes intuitive sense. 00:05:52.200 --> 00:05:54.460 So, 26.59. 00:05:54.460 --> 00:05:56.610 So we're going to go forward in this direction 00:05:56.610 --> 00:05:59.340 26.59 on the X direction. 00:05:59.340 --> 00:06:03.340 And then we go negative 16.18 in the Y direction. 00:06:03.340 --> 00:06:06.890 So this does seem to match our intuition 00:06:06.890 --> 00:06:09.260 when we tried to look at this visually. 00:06:09.260 --> 00:06:11.650 So we now have the X and Y components 00:06:11.650 --> 00:06:13.410 of the resulting vector 00:06:13.410 --> 00:06:14.720 but that's not what they're asking for. 00:06:14.720 --> 00:06:17.500 They're asking for the speed which would be the magnitude 00:06:17.500 --> 00:06:19.670 of this vector right over here. 00:06:19.670 --> 00:06:24.430 And so I could write the magnitude of that vector 00:06:24.430 --> 00:06:26.360 which is going to be its speed. 00:06:26.360 --> 00:06:28.510 We'll just use the Pythagorean theorem here. 00:06:28.510 --> 00:06:31.750 It is going to be the square root of this squared 00:06:31.750 --> 00:06:33.620 plus this squared, because once again 00:06:33.620 --> 00:06:35.100 this forms a right triangle here. 00:06:35.100 --> 00:06:37.340 And we review this in other videos, 00:06:37.340 --> 00:06:41.457 it's going to be the square root of 26.59 squared 00:06:42.780 --> 00:06:47.780 plus negative 16.18 squared which is approximately equal to, 00:06:48.810 --> 00:06:50.706 they want us to round to the nearest 10th, 00:06:50.706 --> 00:06:55.706 26.59 squared plus. 00:06:57.070 --> 00:06:58.600 And it doesn't matter that there's a negative here 00:06:58.600 --> 00:06:59.433 cause I'm squaring it. 00:06:59.433 --> 00:07:04.370 So I'll just write 16.18 squared is equal to that. 00:07:06.770 --> 00:07:09.500 And then we want to take the square root of that. 00:07:09.500 --> 00:07:13.693 We get 31 point, if we round to the nearest 10th, 31.1. 00:07:15.670 --> 00:07:20.360 So it's approximately 31.1 and we'll write the units here, 00:07:20.360 --> 00:07:25.240 kilometers per hour is the speed the boat's speed 00:07:25.240 --> 00:07:27.340 after it meets the current. 00:07:27.340 --> 00:07:29.720 And now the second question is what is the direction 00:07:29.720 --> 00:07:33.500 of the boat's velocity after it meets the current? 00:07:33.500 --> 00:07:35.870 Well, one way to think about it is 00:07:35.870 --> 00:07:38.250 if we look at this angle right over here 00:07:38.250 --> 00:07:40.360 which would tell us the direction 00:07:40.360 --> 00:07:44.220 the tangent of that angle, theta, let me write this down. 00:07:44.220 --> 00:07:47.070 Tangent of that angle theta. 00:07:47.070 --> 00:07:48.950 We know your tangent is your change in Y 00:07:48.950 --> 00:07:50.050 over your change in X. 00:07:50.050 --> 00:07:51.660 You can even view it as the slope 00:07:51.660 --> 00:07:53.570 of this vector right over here. 00:07:53.570 --> 00:07:56.130 We know what our changes in X or Y are. 00:07:56.130 --> 00:07:58.100 Those are X and Y components. 00:07:58.100 --> 00:07:59.460 So it's going to be our change in Y 00:07:59.460 --> 00:08:04.063 which is negative 16.18, over 26.59, 00:08:06.870 --> 00:08:08.320 our change in X. 00:08:08.320 --> 00:08:10.520 And so to solve for theta, we could say 00:08:10.520 --> 00:08:15.320 that theta will be equal to the inverse tangent. 00:08:15.320 --> 00:08:16.930 And we'll have to think about this for a second 00:08:16.930 --> 00:08:19.650 because this might not get us the exact theta that we want 00:08:19.650 --> 00:08:21.160 because the inverse tangent function 00:08:21.160 --> 00:08:24.030 is going to give us something between positive 90 degrees 00:08:24.030 --> 00:08:25.830 and negative 90 degrees. 00:08:25.830 --> 00:08:27.740 But the number we want, actually it looks 00:08:27.740 --> 00:08:31.811 like it's going to be between 270 and 360 degrees 00:08:31.811 --> 00:08:33.650 because we're doing a, 00:08:33.650 --> 00:08:35.630 we want to think about a positive rotation 00:08:35.630 --> 00:08:38.380 instead of a negative one but let's just try to evaluate it. 00:08:38.380 --> 00:08:40.530 The inverse tan of this, 00:08:40.530 --> 00:08:45.073 of negative 16.18, over 26.59. 00:08:47.284 --> 00:08:50.010 16.18 negative 00:08:50.910 --> 00:08:55.910 divided by 26.59 is equal to this. 00:08:57.100 --> 00:09:00.840 And now I am going to take the inverse tangent of that. 00:09:00.840 --> 00:09:05.690 And that gets us negative 31 degrees, which makes sense. 00:09:05.690 --> 00:09:06.740 This looks intuitive sense 00:09:06.740 --> 00:09:09.230 that if you were to do a clockwise rotation 00:09:09.230 --> 00:09:11.660 which would be a negative angle from the positive X-axis 00:09:11.660 --> 00:09:14.290 it looks like what we drew, but let's just go 00:09:14.290 --> 00:09:15.880 with the convention of everything else here. 00:09:15.880 --> 00:09:17.680 And let's try to have a positive angle. 00:09:17.680 --> 00:09:21.070 So what we can do is add 360 degrees to that 00:09:21.070 --> 00:09:22.570 to make a full rotation around. 00:09:22.570 --> 00:09:24.710 And we essentially have the equivalent angle. 00:09:24.710 --> 00:09:29.480 So let's add 360 to that to get that right over there. 00:09:29.480 --> 00:09:31.170 So if we round to the nearest integer 00:09:31.170 --> 00:09:35.840 we're looking at approximately 329 degrees. 00:09:35.840 --> 00:09:40.840 So theta is approximately 329 degrees. 00:09:41.110 --> 00:09:43.680 So here, when I said theta is equal to this I could write 00:09:43.680 --> 00:09:48.680 theta is going to be equal to this plus 360 degrees. 00:09:49.140 --> 00:09:52.110 Now what's interesting is, I was able to add 360 degrees 00:09:52.110 --> 00:09:54.120 to get to the exact same place. 00:09:54.120 --> 00:09:57.350 If we had a situation where our angle was actually 00:09:57.350 --> 00:09:59.310 this angle right over here not the situation 00:09:59.310 --> 00:10:00.270 that we actually dealt with, 00:10:00.270 --> 00:10:02.170 but if it was in the second quadrant, 00:10:02.170 --> 00:10:04.070 we would have gotten this theta. 00:10:04.070 --> 00:10:06.510 And we would have had to be able to realize that, 00:10:06.510 --> 00:10:08.030 hey we're dealing with the second quadrant 00:10:08.030 --> 00:10:09.420 that has the same slope. 00:10:09.420 --> 00:10:11.080 So instead of adding 360 degrees 00:10:11.080 --> 00:10:12.950 we would have added 180 degrees. 00:10:12.950 --> 00:10:16.153 And we've also covered that in other videos as well.

Help Khan Academy supercharge learning

https://www.youtube.com/watch?v=T7QKMp6AS90

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https://www.youtube.com/api/timedtext?v=T7QKMp6AS90&ei=5VWUZbuUI_rKp-oPkpmMiAM&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=51A03A3A69D9D19C99E24B3E5FB9C58A17AD0368.71AC76459953149BB63129D9125DB98DCB3BCD7F&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.320 --> 00:00:01.153 - Hi everyone. 00:00:01.153 --> 00:00:02.870 Sal Khan here from Khan Academy 00:00:02.870 --> 00:00:05.400 which you probably know is a not-for-profit 00:00:05.400 --> 00:00:08.130 with a mission of providing a free world-class education 00:00:08.130 --> 00:00:09.810 for anyone anywhere. 00:00:09.810 --> 00:00:12.450 And not for profit means no one owns Khan Academy, 00:00:12.450 --> 00:00:13.560 we are a public charity. 00:00:13.560 --> 00:00:15.870 You own as much of Khan Academy as I do. 00:00:15.870 --> 00:00:17.970 It's there for the public good. 00:00:17.970 --> 00:00:19.030 And the way that we're able 00:00:19.030 --> 00:00:21.330 to provide this service, the videos, 00:00:21.330 --> 00:00:24.450 the software, the exercises for free 00:00:24.450 --> 00:00:27.180 to tens of millions of learners around the world is 00:00:27.180 --> 00:00:29.120 because of philanthropic donations 00:00:29.120 --> 00:00:31.570 from generous folks like yourself. 00:00:31.570 --> 00:00:33.060 So if you're in a position to do so, 00:00:33.060 --> 00:00:35.900 please think about making a donation to Khan Academy. 00:00:35.900 --> 00:00:39.870 As you probably know, in 2020 the world leaned more heavily 00:00:39.870 --> 00:00:41.880 on Khan Academy than ever before 00:00:41.880 --> 00:00:44.760 to keep the learning going during the pandemic. 00:00:44.760 --> 00:00:46.690 We had 12 billion learning minutes 00:00:46.690 --> 00:00:47.950 on the platform last year. 00:00:47.950 --> 00:00:50.130 6 billion students coming on their own. 00:00:50.130 --> 00:00:52.310 6 billion is hundreds of thousands 00:00:52.310 --> 00:00:53.910 of teachers getting their students to do it 00:00:53.910 --> 00:00:57.220 as part of the distance learning classroom experience 00:00:57.220 --> 00:00:58.350 during the pandemic. 00:00:58.350 --> 00:01:00.820 And as now we get, hopefully, to the light 00:01:00.820 --> 00:01:02.900 at the end of the tunnel on the pandemic 00:01:02.900 --> 00:01:04.760 and hopefully over the next couple 00:01:04.760 --> 00:01:07.290 of months things are able to normalize, 00:01:07.290 --> 00:01:10.240 it's even more important that we don't let up 00:01:10.240 --> 00:01:12.460 because when all of the stuff has been thrown up 00:01:12.460 --> 00:01:14.210 into the air, we, one, need to make sure 00:01:14.210 --> 00:01:16.540 that as it falls things don't break. 00:01:16.540 --> 00:01:19.720 Students have accrued gaps in their learning. 00:01:19.720 --> 00:01:23.450 Teachers need more support, families need more support. 00:01:23.450 --> 00:01:27.110 And also there's a unique window of time right now, 00:01:27.110 --> 00:01:30.240 over the next six to 12 months, where as things come back 00:01:30.240 --> 00:01:32.000 to earth, not only should they not break 00:01:32.000 --> 00:01:33.550 but there's an opportunity to make sure 00:01:33.550 --> 00:01:36.160 that the post-pandemic world could even be better 00:01:36.160 --> 00:01:38.030 than the pre-pandemic world. 00:01:38.030 --> 00:01:41.180 A world where every student is able to fill in their gaps, 00:01:41.180 --> 00:01:42.790 learn at their own time and space 00:01:42.790 --> 00:01:45.280 and really reach whatever that potential is, 00:01:45.280 --> 00:01:48.300 that every teacher has the tools to personalize 00:01:48.300 --> 00:01:50.770 for their students and feels optimally informed 00:01:50.770 --> 00:01:53.180 and optimally empowered and every parent feels 00:01:53.180 --> 00:01:55.220 like there are resources regardless of your zip code, 00:01:55.220 --> 00:01:56.530 regardless of your income 00:01:56.530 --> 00:01:59.570 where your children can reach their potential. 00:01:59.570 --> 00:02:02.020 So I hope you can consider making a donation. 00:02:02.020 --> 00:02:03.120 In particular, we're hoping 00:02:03.120 --> 00:02:06.630 to get at least 12,000 new donors of any level. 00:02:06.630 --> 00:02:08.830 Any donation makes a huge difference 00:02:08.830 --> 00:02:10.610 to become part of this mission 00:02:10.610 --> 00:02:14.713 to provide a free world-class education for anyone anywhere.

Virtual Mindfulness Retreat with Khan Academy and Headspace

https://www.youtube.com/watch?v=LDlIq56e6IY

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en

WEBVTT Kind: captions Language: en 00:00:00.700 --> 00:00:04.850 - The intention for today's hour is really just to relax 00:00:04.850 --> 00:00:08.050 just to unwind, not a lot of information coming at you, 00:00:08.050 --> 00:00:10.290 just embodied practices. 00:00:10.290 --> 00:00:13.040 And I know that a lot of you probably have commitments 00:00:13.040 --> 00:00:15.440 at home right now, maybe kids coming in. 00:00:15.440 --> 00:00:18.500 And so really just do what you can 00:00:18.500 --> 00:00:20.160 and take those breaks as needed 00:00:20.160 --> 00:00:23.090 and really there's no pressure to do this 00:00:23.090 --> 00:00:24.193 in any certain way. 00:00:25.510 --> 00:00:29.820 So we're just gonna start with something that helps us relax 00:00:29.820 --> 00:00:33.170 and that's stretching and a little bit of massage. 00:00:33.170 --> 00:00:35.862 So Jeremy, if you wouldn't mind, we're gonna just have 00:00:35.862 --> 00:00:38.283 some background music while we do this. 00:00:39.320 --> 00:00:44.040 And wherever you are, if you're seated at a chair 00:00:44.040 --> 00:00:46.130 or I'm sitting on the floor right now 00:00:46.130 --> 00:00:48.350 'cause that helps me relax a little more 00:00:48.350 --> 00:00:50.360 but wherever you are, if you're comfortable 00:00:50.360 --> 00:00:53.130 you can just close your eyes just to shut off 00:00:53.130 --> 00:00:55.913 some stimuli that's unneeded right now. 00:01:10.270 --> 00:01:12.980 It might be nighttime or evening where you are, 00:01:12.980 --> 00:01:17.980 so it might be cooler to just take a moment to notice 00:01:18.960 --> 00:01:21.327 the temperature and the environment around you. 00:01:32.059 --> 00:01:35.060 And then just take a moment to notice the feeling 00:01:35.060 --> 00:01:38.193 of your body on the chair, the surface that you're on, 00:01:45.420 --> 00:01:47.420 and notice where your hands are resting, 00:01:52.200 --> 00:01:55.150 maybe notice if you're leaning forward or leaning back, 00:01:55.150 --> 00:01:58.600 or if there's any adjustment that you'd like to make. 00:01:58.600 --> 00:02:00.533 So there's more ease in your body. 00:02:07.644 --> 00:02:12.227 And we're gonna set the intention of letting go of time 00:02:14.186 --> 00:02:15.870 and that sounds like a giant task, 00:02:15.870 --> 00:02:18.610 but we'll see if we can touch 00:02:18.610 --> 00:02:20.430 into this sense of timelessness 00:02:21.270 --> 00:02:23.420 as if we had all the time in the world 00:02:25.080 --> 00:02:27.113 to let go and unwind. 00:02:32.340 --> 00:02:35.000 Just take a moment to connect with your breath 00:02:37.700 --> 00:02:40.104 breathing in through your nose 00:02:40.104 --> 00:02:42.370 (inhales) 00:02:42.370 --> 00:02:44.210 and then breathing out a little slower 00:02:44.210 --> 00:02:45.773 through your nose or mouth, 00:02:56.500 --> 00:03:00.140 just letting our nervous systems start to unwind 00:03:09.660 --> 00:03:12.490 and then just let your breath find its own natural 00:03:12.490 --> 00:03:17.070 easy rhythm, letting go of any effort 00:03:26.002 --> 00:03:30.410 and then just notice any sensations in your body. 00:03:30.410 --> 00:03:35.410 So you might be hungry, you might be full, 00:03:40.210 --> 00:03:42.530 you might notice your eyes are fluttering 00:03:47.620 --> 00:03:50.150 and just as we notice these sensations 00:03:50.150 --> 00:03:54.080 we don't have to try and change them or make them different 00:03:54.080 --> 00:03:58.940 but we can just kind of greet them with curiosity 00:03:58.940 --> 00:04:02.140 and acceptance, and that this is how things are right now 00:04:02.140 --> 00:04:03.023 and that's okay. 00:04:05.270 --> 00:04:07.650 If you happen to notice pain 00:04:07.650 --> 00:04:11.090 it might be difficult to stay with that sensation 00:04:11.090 --> 00:04:14.410 so you can just touch into any sensation of pain 00:04:14.410 --> 00:04:17.283 and then find a place in your body where there's ease. 00:04:19.680 --> 00:04:22.200 And sometimes we really have to search for ease 00:04:22.200 --> 00:04:24.800 maybe it's at the tip of our big toe 00:04:24.800 --> 00:04:29.800 or one of our thumbs, but somewhere in the body 00:04:29.880 --> 00:04:33.403 usually there's a contact with ease. 00:04:38.120 --> 00:04:41.803 So just allowing us to be as we are in this moment, 00:04:43.350 --> 00:04:46.523 less than perfect, just as we are. 00:04:51.350 --> 00:04:54.880 And we're just going to add in a little bit of movement here 00:04:54.880 --> 00:04:57.560 just to stretch our body gently 00:04:57.560 --> 00:05:00.400 and if any of these movements don't work for you 00:05:00.400 --> 00:05:01.910 you can just stay resting 00:05:01.910 --> 00:05:03.823 and whatever position is comfortable. 00:05:04.710 --> 00:05:07.820 So if you'd like to open your eyes to see these stretches 00:05:07.820 --> 00:05:09.927 you can, or you can keep your eyes closed 00:05:09.927 --> 00:05:13.083 and just follow my verbal instructions. 00:05:14.070 --> 00:05:16.990 So go ahead and anchor your hand to a spot next to you 00:05:16.990 --> 00:05:18.760 maybe you're holding onto a chair 00:05:18.760 --> 00:05:23.760 or your fingertips are resting on a couch or a desk 00:05:23.830 --> 00:05:27.040 and then we're gonna lift our left arm up, 00:05:27.040 --> 00:05:30.260 lift our left fingertips toward the sky 00:05:30.260 --> 00:05:32.850 and just feel that length all the way down 00:05:32.850 --> 00:05:34.623 from your fingertips to your hip. 00:05:36.450 --> 00:05:40.680 And then we're just gonna slowly reach over to our right 00:05:40.680 --> 00:05:44.210 you can look down that is nice on your neck 00:05:44.210 --> 00:05:48.340 or you can look up and I'm just keeping my eyes closed 00:05:48.340 --> 00:05:51.720 just to stay embodied, just to stay with 00:05:51.720 --> 00:05:54.760 the pleasant sensations of stretch 00:05:54.760 --> 00:05:56.863 and loosening up the muscles. 00:05:58.820 --> 00:06:02.310 Take a deep breath here feeling your lungs expand 00:06:05.100 --> 00:06:06.473 and a deep breath out, 00:06:11.710 --> 00:06:14.210 and then reaching back up to the center 00:06:15.420 --> 00:06:18.227 and then just letting that left arm float down 00:06:20.324 --> 00:06:24.050 and just pause to notice different sensations 00:06:24.050 --> 00:06:25.600 between the side we stretched 00:06:25.600 --> 00:06:27.253 and the one we haven't stretched. 00:06:32.417 --> 00:06:35.010 *Subtle tingling or heat 00:06:36.720 --> 00:06:40.513 maybe more space in your diaphragm to breathe. 00:06:44.820 --> 00:06:46.520 And then we'll go to the other side 00:06:46.520 --> 00:06:50.260 so just lifting that right arm up toward the sky 00:06:50.260 --> 00:06:54.890 reaching the fingertips up, anchoring your left hand 00:06:54.890 --> 00:06:58.173 then reach up and over toward your left, 00:06:59.140 --> 00:07:02.010 feeling that stretch from your right hip all the way 00:07:02.010 --> 00:07:04.080 through to your right fingertips 00:07:04.080 --> 00:07:07.510 and again, you can look down, you can look up 00:07:12.455 --> 00:07:17.372 and just take a few deep breaths feeling your lungs expand. 00:07:24.380 --> 00:07:27.610 And then on your next in breath just reaching back up 00:07:30.230 --> 00:07:35.110 and as you exhale slowly letting that right arm drift down 00:07:39.200 --> 00:07:41.230 again, just pausing here 00:07:42.710 --> 00:07:45.740 noticing if you feel a little different than when we started 00:07:48.260 --> 00:07:52.713 maybe you can feel the nervous systems start to unwind. 00:07:55.800 --> 00:07:57.520 And if not, that's okay too, 00:07:57.520 --> 00:08:02.080 sometimes our nervous system kind of holds on tight 00:08:02.080 --> 00:08:07.080 to what's going on or what kind of day we had. 00:08:07.280 --> 00:08:08.533 So that's okay too. 00:08:12.000 --> 00:08:15.350 And then you can open your eyes and just follow along 00:08:15.350 --> 00:08:18.440 we're just going to do a few movements 00:08:18.440 --> 00:08:21.610 to give ourselves some love, give our muscles some love. 00:08:21.610 --> 00:08:23.560 So just go ahead, if it's comfortable for you 00:08:23.560 --> 00:08:26.750 and you don't have any injuries just go ahead and squeeze, 00:08:26.750 --> 00:08:30.880 gentle squeeze here, right in your shoulder blades. 00:08:30.880 --> 00:08:32.710 This is where we hold a lot of tension 00:08:32.710 --> 00:08:37.270 and then down the arm all the way down to your forearm 00:08:38.549 --> 00:08:41.343 and give a little squeezed her hand and your fingers, 00:08:43.870 --> 00:08:47.133 and let's just do that one more time going back up, 00:08:48.100 --> 00:08:53.100 squeezing and then again, pause, 00:08:53.330 --> 00:08:56.950 just to notice the difference between the side we squeezed 00:08:56.950 --> 00:08:58.423 and the side we haven't yet. 00:09:07.214 --> 00:09:08.610 A head, it's all about awareness, right? 00:09:08.610 --> 00:09:13.610 We're just bringing awareness to ease so we can identify it. 00:09:13.760 --> 00:09:15.160 And then let's go to the other side 00:09:15.160 --> 00:09:18.320 so just giving it a nice squeeze to that shoulder blade 00:09:20.310 --> 00:09:22.900 anywhere you feel tight you can stay in that area 00:09:22.900 --> 00:09:26.320 if you'd like and then you can just work your way down 00:09:26.320 --> 00:09:31.320 the arm, giving a squeeze to the shoulder and the forearm 00:09:32.760 --> 00:09:34.683 all the way down to your hand. 00:09:36.310 --> 00:09:39.190 Maybe just look down at your hand and your fingers 00:09:39.190 --> 00:09:41.850 thinking about all the things that they do, 00:09:41.850 --> 00:09:46.760 they type, and they hug, cook dinner, wash dishes 00:09:46.760 --> 00:09:47.713 just giving love. 00:09:51.730 --> 00:09:55.350 And then again, just pausing and seeing 00:09:55.350 --> 00:09:58.223 if you can really feel into the sensations, 00:10:00.080 --> 00:10:05.080 maybe tingling, maybe lightness, maybe some ease emerged 00:10:16.304 --> 00:10:18.230 then we're gonna do one last movement. 00:10:18.230 --> 00:10:20.650 So we're gonna make fists with our hands like this, 00:10:20.650 --> 00:10:22.880 you can tuck the thumbs and do your fingers 00:10:23.820 --> 00:10:26.553 and just hold the squeeze for a few moments. 00:10:27.810 --> 00:10:31.290 Noticing what happens in the mind is we tense up 00:10:33.240 --> 00:10:35.940 this is kind of representative of how we sometimes 00:10:35.940 --> 00:10:38.800 hold on to things or resist what's happening 00:10:40.560 --> 00:10:43.461 and then take a deep breath in through your nose. 00:10:43.461 --> 00:10:44.900 (inhales) 00:10:44.900 --> 00:10:47.980 And as you breathe out, just release the hands 00:10:50.020 --> 00:10:55.020 opening the palms and this hand gesture kind of 00:10:55.960 --> 00:11:00.030 represents letting go or accepting, 00:11:00.030 --> 00:11:04.633 or being open and curious, notice how different that is. 00:11:10.979 --> 00:11:12.340 And then the last movement we'll do 00:11:12.340 --> 00:11:15.080 if you wanna open your eyes to look is just 00:11:15.080 --> 00:11:17.050 imagine you're kind of hugging a tree 00:11:17.990 --> 00:11:19.550 and relax the shoulders down. 00:11:19.550 --> 00:11:21.800 We don't want them to be up toward your ears. 00:11:23.630 --> 00:11:26.510 And imagine this gesture, this posture is like 00:11:26.510 --> 00:11:31.360 we're holding whatever we encounter in this next 40 minutes 00:11:31.360 --> 00:11:32.873 together with kindness, 00:11:34.070 --> 00:11:38.340 whether someone interrupts our session or we don't feel it, 00:11:38.340 --> 00:11:41.790 we don't feel what we think we're supposed to be feeling. 00:11:41.790 --> 00:11:46.083 See if we can hold all of that with kindness and compassion. 00:11:50.870 --> 00:11:55.870 And then we can rest our hands back down, opening our eyes. 00:12:00.980 --> 00:12:05.870 So hopefully the nervous system has started to enter 00:12:05.870 --> 00:12:09.590 what we call the rest and digest mode of operating, right? 00:12:09.590 --> 00:12:12.170 Our parasympathetic nervous system. 00:12:12.170 --> 00:12:15.420 The next few activities we're gonna do are all about that. 00:12:15.420 --> 00:12:18.400 All about activating our parasympathetic nervous system 00:12:18.400 --> 00:12:22.030 which is the opposite of our sympathetic nervous system 00:12:22.030 --> 00:12:24.820 which is our, what we call fight or flight, right? 00:12:24.820 --> 00:12:27.440 When we're stressed or we're trying to get things done 00:12:27.440 --> 00:12:30.980 or we're rushed, we wanna move into the other way, 00:12:30.980 --> 00:12:32.880 the opposite of that. 00:12:32.880 --> 00:12:36.400 So I have a little exercise, 00:12:36.400 --> 00:12:39.700 which is about thinking about teachers 00:12:39.700 --> 00:12:41.060 that you've had in your life. 00:12:41.060 --> 00:12:43.290 And these don't have to be teachers you've had at school 00:12:43.290 --> 00:12:46.000 but any kind of teacher that's left an impression on you, 00:12:46.000 --> 00:12:50.870 someone that has taught you something or connected with you 00:12:50.870 --> 00:12:53.700 someone with whom you felt really seen. 00:12:53.700 --> 00:12:55.960 And I want you just to reflect on that 00:12:55.960 --> 00:13:00.060 and maybe a few people could share what comes up 00:13:00.060 --> 00:13:01.650 when you think about that. 00:13:01.650 --> 00:13:03.790 And as you're thinking, as you're processing, 00:13:03.790 --> 00:13:06.940 I'll just share a person I thought of. 00:13:06.940 --> 00:13:09.740 She was my 11th grade science teacher 00:13:09.740 --> 00:13:13.530 and I was really disenchanted with school, 00:13:13.530 --> 00:13:16.880 at the time, I felt like I wasn't able to really focus on 00:13:16.880 --> 00:13:18.160 what I wanted to focus on, 00:13:18.160 --> 00:13:20.870 which was neuroscience and psychology. 00:13:20.870 --> 00:13:22.850 And she happened to go to UCLA 00:13:22.850 --> 00:13:25.180 which at the time was my dream school 00:13:25.180 --> 00:13:27.310 and she gave me a textbook from one 00:13:27.310 --> 00:13:30.070 of her neuroscience classes that she still had 00:13:30.070 --> 00:13:32.320 and she told me that at any time I could go sit 00:13:32.320 --> 00:13:35.670 in *on classes at UCLA. 00:13:35.670 --> 00:13:39.816 And I was so appreciative of that because first of all, 00:13:39.816 --> 00:13:43.040 she took my interests seriously, 00:13:43.040 --> 00:13:45.630 she gave me a book that was special to her 00:13:45.630 --> 00:13:49.440 and she let me know that I could pursue my interests 00:13:49.440 --> 00:13:50.830 and I didn't have to wait for it, 00:13:50.830 --> 00:13:53.080 I could go sit in on a lecture and be part of it 00:13:53.080 --> 00:13:56.650 even though I hadn't graduated high school yet. 00:13:56.650 --> 00:14:01.050 So that was a teacher that I'll never forget, 00:14:01.050 --> 00:14:03.910 I'll remember her for the rest of my life. 00:14:03.910 --> 00:14:06.200 So is there anyone who'd like to share 00:14:06.200 --> 00:14:09.200 just a story of a teacher you can share verbally, 00:14:09.200 --> 00:14:11.270 I would love to hear your voices, 00:14:11.270 --> 00:14:12.620 and I totally understand too, 00:14:12.620 --> 00:14:15.520 if you you're not able to share verbally 00:14:15.520 --> 00:14:17.710 'cause of what's going on at home. 00:14:17.710 --> 00:14:21.080 But yeah, who's a teacher in your life that you've had 00:14:21.080 --> 00:14:23.900 maybe it's even a teacher who inspired you to go 00:14:23.900 --> 00:14:28.063 into this profession and yeah- 00:14:28.063 --> 00:14:30.180 - [Jeremy] We're getting a number of great responses 00:14:30.180 --> 00:14:32.970 in the question section, which is awesome. 00:14:32.970 --> 00:14:36.500 Carrie says her ninth to 12th grade music teacher 00:14:36.500 --> 00:14:38.480 was totally present and authentic, 00:14:38.480 --> 00:14:41.180 such an encouragement to all students. 00:14:41.180 --> 00:14:43.620 And Katie says, fifth grade teacher that I worked 00:14:43.620 --> 00:14:45.140 with when I was a para, 00:14:45.140 --> 00:14:47.623 I still think about him and my classroom plans. 00:14:48.590 --> 00:14:51.600 And then just to get a couple of folks to share live, 00:14:51.600 --> 00:14:53.140 I see some folks have raised their hands 00:14:53.140 --> 00:14:56.053 so I'm gonna actually unmute crystal Davis here. 00:14:58.948 --> 00:15:02.193 So crystal, if you like, you are now live with Sam. 00:15:03.810 --> 00:15:04.743 - Hi crystal. 00:15:08.040 --> 00:15:09.170 - Sometimes it can take a little while 00:15:09.170 --> 00:15:11.206 for the audio to kick it. 00:15:11.206 --> 00:15:13.350 Crystal, feel free to share, 00:15:13.350 --> 00:15:16.093 I'm also gonna go over to Theresa here. 00:15:18.460 --> 00:15:20.670 Theresa, if you're there, feel free to share. 00:15:20.670 --> 00:15:22.740 - [Crystal] Okay, they said I was muted, 00:15:22.740 --> 00:15:24.080 can you hear me now? 00:15:24.080 --> 00:15:25.650 - Yeah, we can hear you- - Pretty great, yeah. 00:15:25.650 --> 00:15:27.150 - [Crystal] Okay, awesome. 00:15:27.150 --> 00:15:29.990 So for me, it's my fifth grade teacher, 00:15:29.990 --> 00:15:31.280 her name was Gail Lidy, 00:15:31.280 --> 00:15:33.710 I remember her like it was yesterday. 00:15:33.710 --> 00:15:35.420 I had a very challenging childhood, 00:15:35.420 --> 00:15:36.270 I'll just leave it at that 00:15:36.270 --> 00:15:38.290 to make it comfortable for everyone 00:15:38.290 --> 00:15:40.690 but she showed me that she loved me 00:15:40.690 --> 00:15:43.320 even though I felt so uncomfortable. 00:15:43.320 --> 00:15:45.190 And the biggest thing of all is that 00:15:45.190 --> 00:15:46.640 I was actually able to meet her 00:15:46.640 --> 00:15:49.410 after I finally got my degree six years ago 00:15:49.410 --> 00:15:51.520 and I did my student teaching 00:15:51.520 --> 00:15:55.290 and she told me how proud of me that she was. 00:15:55.290 --> 00:15:57.760 And so I got a chance to have that last conversation 00:15:57.760 --> 00:15:59.630 with her before she passed away, 00:15:59.630 --> 00:16:02.820 so just very super precious. 00:16:02.820 --> 00:16:04.540 That's my share. 00:16:04.540 --> 00:16:05.683 - That's amazing. 00:16:06.780 --> 00:16:07.880 Thank you for sharing that 00:16:07.880 --> 00:16:11.300 and I heard that the word love really resonated with me. 00:16:11.300 --> 00:16:13.170 Like you know when you're loved, 00:16:13.170 --> 00:16:18.170 you know when you're cherished by someone, 00:16:18.540 --> 00:16:22.270 even if it is a teacher, that kind of connection 00:16:23.160 --> 00:16:27.000 can translate or transcend to love 00:16:27.000 --> 00:16:29.300 and that's why we're here, 00:16:29.300 --> 00:16:30.850 I would say that's why we're here on earth. 00:16:30.850 --> 00:16:33.410 You know, we're here to connect, we're here to love. 00:16:33.410 --> 00:16:36.040 So it's an amazing connection you had 00:16:36.040 --> 00:16:39.780 and so great that you were able to meet with her 00:16:39.780 --> 00:16:42.590 and let her know what impact she had on you 00:16:42.590 --> 00:16:44.140 because when we're kids 00:16:44.140 --> 00:16:46.040 and my husband just told me this too, he's like 00:16:46.040 --> 00:16:48.020 we never get thank yous. 00:16:48.020 --> 00:16:49.700 You know, the only thing he's that come 00:16:49.700 --> 00:16:52.700 or when maybe when the students are older 00:16:52.700 --> 00:16:55.660 and they realize, you know, how important and 00:16:56.520 --> 00:16:58.320 what sacrifices teachers made 00:16:58.320 --> 00:17:01.690 and what effort they brought forth to the classroom. 00:17:01.690 --> 00:17:04.883 So, yeah and I'm sure it meant a lot to her. 00:17:07.050 --> 00:17:10.310 - Thank you so much Crystal for that amazing share. 00:17:10.310 --> 00:17:11.660 Let's go over to Theresa. 00:17:11.660 --> 00:17:14.050 Theresa, I'm gonna unmute your line, 00:17:14.050 --> 00:17:15.786 are you still there? 00:17:15.786 --> 00:17:17.240 - [Theresa] Yes. 00:17:17.240 --> 00:17:20.630 So I had a college professor in Algebra 00:17:20.630 --> 00:17:23.130 that was a very difficult class. 00:17:23.130 --> 00:17:26.210 It was my third algebra class trying to take just to get 00:17:26.210 --> 00:17:28.860 through so I could even take a college Algebra course 00:17:30.050 --> 00:17:33.400 was feeling very overwhelmed and quite frankly 00:17:33.400 --> 00:17:35.853 very stupid when it came to Algebra. 00:17:36.760 --> 00:17:41.760 And he told me that even if you don't get it one way 00:17:43.817 --> 00:17:45.920 and showed me that if you don't get it one way 00:17:45.920 --> 00:17:48.410 there's always another way to solve a problem 00:17:49.530 --> 00:17:51.100 which is something that I've taken, 00:17:51.100 --> 00:17:52.650 not just when it comes to math 00:17:52.650 --> 00:17:54.800 but I've taken it into life as well. 00:17:54.800 --> 00:17:57.310 That there's multiple ways to solve our problems 00:17:58.500 --> 00:18:01.230 and don't just stick with that one message. 00:18:01.230 --> 00:18:03.310 That's something that I brought into my teaching as well 00:18:03.310 --> 00:18:05.760 because I do the same thing with my students now, 00:18:07.050 --> 00:18:08.250 if they can't get it one way 00:18:08.250 --> 00:18:10.670 then let's look at it a different way. 00:18:10.670 --> 00:18:12.440 And it's been very successful 00:18:12.440 --> 00:18:15.070 and has been a good thing for me to learn. 00:18:15.070 --> 00:18:17.720 So I'm very thankful to Mr. Cherry, 00:18:17.720 --> 00:18:19.853 who was an algebra teacher in college. 00:18:20.770 --> 00:18:23.420 - Wow, that's a beautiful story 00:18:23.420 --> 00:18:25.350 and the fact that he was able to connect it 00:18:25.350 --> 00:18:27.000 or you are connecting it to life 00:18:27.000 --> 00:18:31.710 just that when we hit a roadblock we have this capacity 00:18:31.710 --> 00:18:34.330 to come up with creative solutions 00:18:34.330 --> 00:18:37.400 or ways of looking at problems differently 00:18:37.400 --> 00:18:40.100 and not just by ourselves, but even asking for help 00:18:40.100 --> 00:18:41.323 and being able to ask our teachers 00:18:41.323 --> 00:18:46.173 to help us look at something a new way, that's beautiful. 00:18:47.360 --> 00:18:49.070 - Thank you so much. 00:18:49.070 --> 00:18:50.170 - [Theresa] Thank you. 00:18:51.840 --> 00:18:54.173 - So I would love for us to, 00:18:55.266 --> 00:18:57.010 you might not know this practice 00:18:57.010 --> 00:18:59.940 but it's called loving kindness meditation, 00:18:59.940 --> 00:19:04.720 and for the skeptics out there, it can sound kind of wooey 00:19:04.720 --> 00:19:09.170 because we're sending kind wishes to someone else. 00:19:09.170 --> 00:19:11.360 And it's not that we're in any kind 00:19:11.360 --> 00:19:15.360 of delusional state that we think we can send wishes 00:19:15.360 --> 00:19:16.960 and they'll somehow be received, 00:19:18.065 --> 00:19:19.300 we're doing it for ourselves. 00:19:19.300 --> 00:19:23.720 There's been a lot of research on this form of meditation 00:19:23.720 --> 00:19:27.413 and the research has shown that it connects us to, 00:19:28.290 --> 00:19:29.880 well it has a lot of health benefits, 00:19:29.880 --> 00:19:32.227 reduces inflammation, increases 00:19:32.227 --> 00:19:34.390 and boosts our immune system. 00:19:34.390 --> 00:19:38.500 But it also puts us in touch with the neurocircuitry 00:19:38.500 --> 00:19:40.370 that makes us wanna connect, 00:19:40.370 --> 00:19:42.300 that helps us attune to each other 00:19:42.300 --> 00:19:45.810 and helps us to be present for another human being 00:19:45.810 --> 00:19:48.232 and to be present for ourselves. 00:19:48.232 --> 00:19:51.560 So it's really a selfish exercise 00:19:51.560 --> 00:19:53.320 even though we're doing something 00:19:54.590 --> 00:19:57.170 that sounds really benevolent for others 00:19:57.170 --> 00:19:58.790 but it also makes you want, 00:19:58.790 --> 00:20:01.320 sometimes it stimulates this desire 00:20:01.320 --> 00:20:04.250 to reach out to the person or to write 00:20:04.250 --> 00:20:07.170 a letter of thank you or to really just 00:20:07.170 --> 00:20:09.430 be with them in a deeper way. 00:20:09.430 --> 00:20:11.560 So we don't know exactly what it's gonna do 00:20:11.560 --> 00:20:15.053 but we go into it knowing it will be in some way beneficial. 00:20:16.240 --> 00:20:19.360 So I would love for us to send some kind wishes 00:20:19.360 --> 00:20:22.520 to the teachers that you thought of, 00:20:22.520 --> 00:20:26.060 the teacher that came to mind and I'll lead you through. 00:20:26.060 --> 00:20:28.610 It's a very brief practice, I'll lead you through it. 00:20:28.610 --> 00:20:30.590 And just keep this person in mind 00:20:30.590 --> 00:20:33.264 and keep the feeling alive in you, 00:20:33.264 --> 00:20:37.490 the feeling of connection of being seen of 00:20:39.127 --> 00:20:44.127 that freedom that you had when you 00:20:44.410 --> 00:20:46.110 from your algebra teacher, when you learned 00:20:46.110 --> 00:20:47.840 that you could look at something a different way 00:20:47.840 --> 00:20:49.730 or that there were multiple ways of solving it 00:20:49.730 --> 00:20:54.700 how that kind of opened it, like on tied the knot 00:20:54.700 --> 00:20:57.370 that you may have felt when you didn't get something. 00:20:57.370 --> 00:20:59.220 So just keeping that feeling alive 00:20:59.220 --> 00:21:01.570 and we can close our eyes, 00:21:01.570 --> 00:21:04.090 if we're comfortable, you can always rest your eyes 00:21:04.090 --> 00:21:05.230 on a spot in front of you, 00:21:05.230 --> 00:21:07.080 if you don't like to close your eyes. 00:21:08.920 --> 00:21:11.363 And let's just bring that teacher to mind. 00:21:15.120 --> 00:21:18.207 You can imagine that they're in front of you or next to you, 00:21:19.640 --> 00:21:24.200 maybe recall what they look like, their demeanor 00:21:33.110 --> 00:21:36.270 and just recall the moment of connection 00:21:36.270 --> 00:21:38.113 that you had with this teacher. 00:21:39.840 --> 00:21:42.130 And if no teacher came to mind, you can just think 00:21:42.130 --> 00:21:44.943 of someone you like or love, someone in your life. 00:21:52.890 --> 00:21:56.200 And if you'd like, you can put a hand on your heart 00:21:56.200 --> 00:21:58.073 but you definitely don't have to. 00:21:59.550 --> 00:22:02.440 And I'm just gonna say some words of kindness out loud 00:22:02.440 --> 00:22:05.270 and just imagine that you're sending this kindness, 00:22:05.270 --> 00:22:08.943 you're genuinely wishing for this person to feel this way. 00:22:11.690 --> 00:22:13.083 May you be happy, 00:22:17.010 --> 00:22:19.183 may you know you're appreciated, 00:22:22.930 --> 00:22:26.660 may you be free from pain, may you be healthy. 00:22:34.290 --> 00:22:38.010 I'm just taking a breath to reconnect to that feeling 00:22:38.010 --> 00:22:40.933 of appreciation you have for this person. 00:22:43.930 --> 00:22:45.283 May you be happy, 00:22:47.980 --> 00:22:49.373 may you be safe, 00:22:52.040 --> 00:22:54.423 may you be healthy and at ease. 00:22:57.056 --> 00:22:59.223 (sighing) 00:23:04.560 --> 00:23:05.750 And while we're here 00:23:05.750 --> 00:23:09.020 just bringing to mind your own efforts 00:23:09.020 --> 00:23:11.570 you can place your hand down if it's getting tired. 00:23:12.590 --> 00:23:16.150 But bringing to mind your efforts to show 00:23:16.150 --> 00:23:18.763 up when you don't feel like it sometimes, 00:23:20.540 --> 00:23:23.040 your efforts to be on time, 00:23:23.040 --> 00:23:27.110 to be present your genuine desire to show care 00:23:31.290 --> 00:23:36.060 and concern for your students, for your family 00:23:39.060 --> 00:23:43.393 all of that effort and goodness that you show to the world. 00:23:51.380 --> 00:23:53.020 And if exceptions come up, 00:23:53.020 --> 00:23:54.250 if you start thinking about, 00:23:54.250 --> 00:23:57.380 well not always like that, that's okay too. 00:23:57.380 --> 00:23:59.660 Just notice what the mind does, 00:23:59.660 --> 00:24:01.940 what the mind shows you greet it with, 00:24:01.940 --> 00:24:05.640 remember that holding with compassion and kindness 00:24:08.840 --> 00:24:12.150 and we're just gonna send some kind wishes to ourselves. 00:24:12.150 --> 00:24:13.780 We're gonna use the word you, 00:24:13.780 --> 00:24:17.080 just so we have that little bit of space 00:24:17.080 --> 00:24:22.080 where our loving kind self wishing it to our vulnerable, 00:24:22.860 --> 00:24:24.823 maybe sometimes hesitant self. 00:24:27.190 --> 00:24:30.420 So may you be happy, may you be safe, 00:24:39.170 --> 00:24:41.343 may you be free from pain, 00:24:44.870 --> 00:24:46.623 may you have ease. 00:24:52.056 --> 00:24:54.620 (inhales) Taking a deep breath in (exhales) 00:24:54.620 --> 00:24:57.800 and out just to reconnect with your efforts 00:24:59.065 --> 00:25:00.103 and your goodness. 00:25:02.370 --> 00:25:07.370 May you be happy, may you be safe, 00:25:11.380 --> 00:25:13.553 may you know you're appreciated, 00:25:17.610 --> 00:25:19.613 I may you feel at ease. 00:25:23.890 --> 00:25:27.630 I'm just staying with the feeling of ease 00:25:27.630 --> 00:25:32.113 that maybe has bubbled up from this exercise, 00:25:33.020 --> 00:25:36.833 we're holding whatever's here, skepticism. 00:25:38.480 --> 00:25:41.250 May be you feeling of difficulty or hesitation, 00:25:41.250 --> 00:25:44.707 holding that with kindness, not resisting it (sighs). 00:25:49.125 --> 00:25:54.125 And when you feel ready, you can open your eyes 00:25:57.210 --> 00:26:00.490 and I just wanna open it up again to the group to see 00:26:00.490 --> 00:26:04.350 if anyone would like to share what that was like for you. 00:26:04.350 --> 00:26:06.880 Often, when we do this loving kindness practice 00:26:06.880 --> 00:26:10.030 for the first time, it can be, 00:26:10.030 --> 00:26:12.820 we have what we call the inner heckler, right? 00:26:12.820 --> 00:26:15.310 We have this voice inner voice kind of heckling us 00:26:15.310 --> 00:26:16.650 and saying, what are you doing? 00:26:16.650 --> 00:26:17.623 What is this? 00:26:18.520 --> 00:26:19.930 But maybe you've tried it before, 00:26:19.930 --> 00:26:21.790 maybe it just felt really natural to you. 00:26:21.790 --> 00:26:25.500 So yeah, just feel free to jump in 00:26:25.500 --> 00:26:27.993 and share what that experience was like for you. 00:26:31.230 --> 00:26:33.750 And again, we can do it verbally or in chat. 00:26:33.750 --> 00:26:35.854 I love hearing your voices, 00:26:35.854 --> 00:26:38.970 that's my preference, but whatever you can do 00:26:43.360 --> 00:26:44.210 - [Jeremy] Christy says, 00:26:44.210 --> 00:26:47.373 I'm always surprised of this exercise can bring me to tears. 00:26:49.540 --> 00:26:50.990 And Abby said, it made me wanna 00:26:50.990 --> 00:26:53.073 call my guiding teacher and say, thanks. 00:26:55.580 --> 00:26:57.950 And then I know there were a couple 00:26:57.950 --> 00:26:59.250 of folks who'd wanna share online, 00:26:59.250 --> 00:27:01.063 so I'm gonna go with Carol first. 00:27:02.180 --> 00:27:04.310 So Carol, I've got open up your line, 00:27:04.310 --> 00:27:07.690 if you wanna unmute yourself, feel free to say hi to Sam. 00:27:07.690 --> 00:27:09.110 - [Carol] Hello, can you hear me? 00:27:09.110 --> 00:27:10.843 - Yeah, hi Carol. 00:27:10.843 --> 00:27:12.923 - [Carol] Hi, for me, 00:27:14.780 --> 00:27:18.440 I tell my students to do these things 00:27:18.440 --> 00:27:20.590 and I don't always tell myself to. 00:27:20.590 --> 00:27:25.590 And so to hear you, as the teacher telling me 00:27:25.830 --> 00:27:30.060 as the student to just send myself love, 00:27:30.060 --> 00:27:34.160 it's that kind of switch that I think 00:27:34.160 --> 00:27:36.043 teachers need more often. 00:27:37.681 --> 00:27:39.480 So thank you. 00:27:39.480 --> 00:27:41.550 - You're welcome, it's so true. 00:27:41.550 --> 00:27:42.780 We get in the habit 00:27:42.780 --> 00:27:46.780 of wanting to give advice, give tools 00:27:46.780 --> 00:27:49.893 and then we're we forget to do it for ourselves, right? 00:27:51.332 --> 00:27:52.770 And we need teachers, right? 00:27:52.770 --> 00:27:55.570 Just like we never stopped needing teachers 00:27:55.570 --> 00:27:57.433 and helpers as we get older. 00:28:01.760 --> 00:28:04.050 - [Jeremy] And let's go next to Elizabeth and Elizabeth 00:28:04.050 --> 00:28:06.053 feel free to share your story with Sam. 00:28:06.930 --> 00:28:08.080 - [Elizabeth] Sure, hi. 00:28:10.079 --> 00:28:11.280 I always think it's awesome because 00:28:11.280 --> 00:28:13.390 I have so many people I wanna thank 00:28:13.390 --> 00:28:15.960 and I'm grateful for and appreciate, 00:28:15.960 --> 00:28:18.810 but I can't remember your name 00:28:18.810 --> 00:28:21.920 but you said it brings you to tears. 00:28:21.920 --> 00:28:25.540 And that's what always surprises me when I do stuff 00:28:25.540 --> 00:28:30.170 like this, you'd come in and like, it happens 00:28:30.170 --> 00:28:33.260 and mine is, my brother passed away at age 48, 00:28:33.260 --> 00:28:34.880 a couple of years ago. 00:28:34.880 --> 00:28:38.400 And we had a teacher we both had in sixth grade 00:28:38.400 --> 00:28:41.600 and she ended up moving across the street from my mom. 00:28:41.600 --> 00:28:46.210 And 15 years later, she had a picture 00:28:46.210 --> 00:28:49.113 that my brother had given her like a school picture. 00:28:50.760 --> 00:28:51.920 And she told the story. 00:28:51.920 --> 00:28:53.350 She said, he asked, well 00:28:53.350 --> 00:28:54.900 do you want a school picture of me? 00:28:54.900 --> 00:28:56.610 And she said, well, of course I do. 00:28:56.610 --> 00:28:59.780 And so the next day he came in with an eight by 10 00:28:59.780 --> 00:29:03.010 and this is like in the mid early eighties. 00:29:03.010 --> 00:29:07.030 So she kept it and had it when she moved 00:29:07.030 --> 00:29:09.870 in across the street from my mom, like 15 years later. 00:29:09.870 --> 00:29:13.800 And so it just was a pretty amazing story that she 00:29:13.800 --> 00:29:16.480 was able to tell us, you know, at my brother's funeral. 00:29:16.480 --> 00:29:19.150 And like she made a difference then 00:29:19.150 --> 00:29:21.950 and continue to, you know. 00:29:21.950 --> 00:29:25.633 - Wow, what a special woman. 00:29:27.810 --> 00:29:29.233 Thank you for sharing that. 00:29:33.750 --> 00:29:35.470 - [Jeremy] And Sam, do we have time for one more share? 00:29:35.470 --> 00:29:37.020 - Yes, we do. 00:29:37.020 --> 00:29:40.960 - [Jeremy] Let's go over to the Jamie (mumbles) actually. 00:29:40.960 --> 00:29:43.560 So Jamie, I'm gonna unmute your line, 00:29:43.560 --> 00:29:45.793 feel free to open it up yourself. 00:29:49.140 --> 00:29:49.973 Hi, Jamie. 00:29:53.380 --> 00:29:55.530 It looks like you might have a little trouble with Jamie. 00:29:55.530 --> 00:29:56.680 - Oh, okay. 00:29:56.680 --> 00:29:59.500 - [Jeremy] Let me try with Jennifer here. 00:29:59.500 --> 00:30:00.453 Hold on one second. 00:30:02.700 --> 00:30:03.900 Jennifer, are you there? 00:30:11.098 --> 00:30:13.100 Hi, Jennifer, how are you? 00:30:13.100 --> 00:30:15.040 - [Jennifer] I'm good, how are you? 00:30:15.040 --> 00:30:17.230 - Great, thank you so much for joining? 00:30:17.230 --> 00:30:19.030 - [Jennifer] Yeah, again like everyone 00:30:19.030 --> 00:30:21.210 I'm echoing just the sentiments of tears 00:30:21.210 --> 00:30:22.760 and I was surprised 00:30:22.760 --> 00:30:25.270 with how my other senses really connected 00:30:25.270 --> 00:30:27.570 to when I was thinking about my fourth grade teacher, 00:30:27.570 --> 00:30:29.820 going back to that, I could 00:30:29.820 --> 00:30:31.090 just like it was in the classroom 00:30:31.090 --> 00:30:33.400 and I could hear his laugh and see his smile 00:30:33.400 --> 00:30:35.800 and it just felt so good. 00:30:35.800 --> 00:30:37.160 And I made the connection that, 00:30:37.160 --> 00:30:38.340 what made him so special, 00:30:38.340 --> 00:30:40.930 he was always a hundred percent present 00:30:40.930 --> 00:30:42.410 with us as his students. 00:30:42.410 --> 00:30:46.563 And that gift of being present is not forgotten. 00:30:47.460 --> 00:30:52.460 - Yeah, wow, that's incredible. 00:30:52.640 --> 00:30:56.020 The presence it's love, you know 00:30:56.020 --> 00:30:59.150 it's the ultimate show of care 00:30:59.150 --> 00:31:01.940 when you're present and it's hard, 00:31:01.940 --> 00:31:04.840 it gets harder to develop in these times. 00:31:04.840 --> 00:31:09.840 So when we need this time to stop and just reflect 00:31:11.000 --> 00:31:14.490 and I think we need people to bring us into it, 00:31:14.490 --> 00:31:16.650 we need all the tools we can get 00:31:16.650 --> 00:31:20.590 to pull us in to this time, to connect with each other, 00:31:20.590 --> 00:31:23.500 to commune, to share stories. 00:31:23.500 --> 00:31:26.610 So it's so precious and when we do it, I don't know. 00:31:26.610 --> 00:31:28.830 It sounds like a lot of you are expressing this too 00:31:28.830 --> 00:31:31.260 but when we do it, we remember how good it is, 00:31:31.260 --> 00:31:33.590 how good it feels and we wanna do it again 00:31:33.590 --> 00:31:34.530 and again and again. 00:31:34.530 --> 00:31:38.700 So yeah, it's just rediscovering the importance 00:31:38.700 --> 00:31:41.070 of community and connection, lovely. 00:31:45.050 --> 00:31:48.100 I wanna take us into our next exercise 00:31:48.100 --> 00:31:51.630 which is all about all awe, A-W-E. 00:31:51.630 --> 00:31:53.760 I feel like we don't use that word very often 00:31:53.760 --> 00:31:55.780 because these moments of awe 00:31:56.770 --> 00:31:59.123 for most of us happen infrequently. 00:32:00.431 --> 00:32:04.300 A moment of awe to me and how it's been defined 00:32:04.300 --> 00:32:07.880 by the research is these moments that leave you 00:32:07.880 --> 00:32:11.430 or inspire this feeling of surprise 00:32:11.430 --> 00:32:14.660 and you kind of feel small 00:32:14.660 --> 00:32:19.365 because the thing feels so big in a good way. 00:32:19.365 --> 00:32:20.570 You feels you feel expansive, 00:32:20.570 --> 00:32:24.620 you feel inspired, you feel, 00:32:24.620 --> 00:32:27.930 you're completely absorbed in that moment, 00:32:27.930 --> 00:32:29.480 you're not thinking about anything else 00:32:29.480 --> 00:32:32.410 because the thing you're witnessing is so beautiful 00:32:32.410 --> 00:32:35.690 or so touching or so magnificent, right? 00:32:35.690 --> 00:32:39.650 So these are moments of awe and reflecting, 00:32:39.650 --> 00:32:41.080 there's a lot of research, and I can share 00:32:41.080 --> 00:32:44.924 with you the research literature that the authors 00:32:44.924 --> 00:32:46.750 there's a woman actually in Canada 00:32:46.750 --> 00:32:49.570 who does this and these amazing workshops on awe 00:32:49.570 --> 00:32:53.475 but she's found in the research that awe 00:32:53.475 --> 00:32:56.160 is really good for our mental health, 00:32:56.160 --> 00:32:59.420 it stimulates our vagus nerve which again 00:32:59.420 --> 00:33:01.730 activates our parasympathetic nervous system 00:33:01.730 --> 00:33:06.730 and helps us to actually feel like we're suspended in time. 00:33:07.420 --> 00:33:10.420 We lose track of time, we're in this flow state, 00:33:10.420 --> 00:33:12.010 so it's really good for us. 00:33:12.010 --> 00:33:14.400 So I wanna introduce you to this activity 00:33:14.400 --> 00:33:17.160 so that you can put yourself in that state of awe 00:33:17.160 --> 00:33:19.160 and then hopefully find a lot 00:33:19.160 --> 00:33:22.606 of extraordinary moments in the ordinariness 00:33:22.606 --> 00:33:25.600 of life, of day-to-day life. 00:33:25.600 --> 00:33:27.610 So what I'm gonna have you do is 00:33:27.610 --> 00:33:29.920 and you can do this again with your eyes open or closed. 00:33:29.920 --> 00:33:32.980 I like to shut my eyes just to close out 00:33:32.980 --> 00:33:37.490 all the visual stimuli but we're going to think 00:33:37.490 --> 00:33:40.063 of a moment of awe that we've had. 00:33:41.150 --> 00:33:46.150 So think of a moment when you were surprised 00:33:47.680 --> 00:33:52.257 and suspended for a moment in this feeling of amazement, 00:33:55.470 --> 00:34:00.210 this feeling of connection, maybe it was a connection 00:34:01.402 --> 00:34:06.170 with a person or a moment of witnessing a child being born 00:34:06.170 --> 00:34:10.890 or seeing a wedding or someone's, 00:34:10.890 --> 00:34:15.530 some experience where you felt really present 00:34:15.530 --> 00:34:17.543 and in awe of the beauty. 00:34:22.220 --> 00:34:23.580 I'm gonna give you a few moments 00:34:23.580 --> 00:34:26.800 in silence just to allow this memory 00:34:26.800 --> 00:34:29.643 to bubble up without pushing it or rushing it. 00:34:39.061 --> 00:34:43.644 And as a moment comes to mind, see if you can expand it 00:34:44.480 --> 00:34:48.760 by noticing the details, the sensory details. 00:34:48.760 --> 00:34:51.440 So if there were people involved, 00:34:51.440 --> 00:34:53.280 remembering what people were there 00:34:56.290 --> 00:34:58.040 maybe even as you take a deep breath 00:34:58.040 --> 00:35:01.723 you can remember the smell, the sense in the air. 00:35:05.730 --> 00:35:08.093 Maybe you can remember the sounds, 00:35:13.110 --> 00:35:17.260 maybe you remember the temperature, if it was a hot day 00:35:17.260 --> 00:35:20.663 or a cold day, neutral. 00:35:25.470 --> 00:35:30.470 And then remember what you saw the colors, the shapes, 00:35:32.020 --> 00:35:36.703 the facial expressions, the movements. 00:35:43.940 --> 00:35:46.960 Then we're gonna connect with a movement 00:35:46.960 --> 00:35:49.963 that represents this moment of awe. 00:35:51.680 --> 00:35:55.350 So maybe it's your arms open, or if you were 00:35:57.808 --> 00:36:01.110 holding something, you might put your arms in that position. 00:36:01.110 --> 00:36:03.720 So just find a movement in your body right now 00:36:06.552 --> 00:36:08.323 or a gesture that represents this moment of awe. 00:36:13.200 --> 00:36:16.520 And just hold that movement or posture 00:36:16.520 --> 00:36:20.940 for few breaths, helping you connect to that feeling 00:36:23.350 --> 00:36:24.650 to let it sock in (sighs). 00:36:34.540 --> 00:36:37.145 Maybe you notice the thoughts that you had 00:36:37.145 --> 00:36:38.495 at the time or the emotions 00:36:43.800 --> 00:36:46.170 and you can place your hands back down 00:36:49.800 --> 00:36:53.380 and just let that feeling simmer 00:36:53.380 --> 00:36:55.343 for just a few more moments. 00:37:15.370 --> 00:37:17.630 Then when you're ready, you can 00:37:19.641 --> 00:37:21.991 if your eyes are closed, you can open your eyes 00:37:24.343 --> 00:37:25.293 coming back to the space that we have together. 00:37:27.150 --> 00:37:31.350 And I wanted to share when I did this exercise 00:37:31.350 --> 00:37:34.130 for the first time, what moment of awe came to mind? 00:37:34.130 --> 00:37:39.130 And I think Jeremy has a photograph to pull up to show you. 00:37:41.580 --> 00:37:45.450 It was, when I was 15 00:37:45.450 --> 00:37:47.530 I went to Europe for the first time 00:37:47.530 --> 00:37:52.203 and I saw the Coliseum, oh that's a beautiful picture. 00:37:53.140 --> 00:37:54.750 Yeah and I looked at it 00:37:54.750 --> 00:37:56.840 and I heard the tour guide starting 00:37:56.840 --> 00:37:59.579 to talk about the history and 00:37:59.579 --> 00:38:03.030 some major points they were making stuck out 00:38:03.030 --> 00:38:05.800 like how old it was and the events 00:38:05.800 --> 00:38:08.123 that used to take place inside. 00:38:09.178 --> 00:38:12.633 But what got me was how old it was just, 00:38:12.633 --> 00:38:16.400 how it withstood all this time passing 00:38:16.400 --> 00:38:19.030 and all these different events in history 00:38:19.030 --> 00:38:22.170 and it just stood there and it remained. 00:38:22.170 --> 00:38:24.600 And my gesture was this, 00:38:24.600 --> 00:38:26.300 you can't see my hands in the shot 00:38:27.336 --> 00:38:29.760 but like, wow, Kind of pushed back by it 00:38:29.760 --> 00:38:32.470 by how huge it was and how beautiful. 00:38:32.470 --> 00:38:35.720 And I also just felt so lucky to be there, 00:38:35.720 --> 00:38:39.310 I felt so lucky to witness this living piece 00:38:39.310 --> 00:38:42.560 of history that I had learned about and in a textbook 00:38:42.560 --> 00:38:43.860 and now I got to see it. 00:38:43.860 --> 00:38:46.400 So that was my moment of awe 00:38:46.400 --> 00:38:50.070 and I just, I really did lose track of time, 00:38:50.070 --> 00:38:51.400 I think I was there for hours 00:38:51.400 --> 00:38:53.433 but it felt like only minutes went by. 00:38:54.980 --> 00:38:57.480 So I would love to hear from you again 00:38:57.480 --> 00:39:00.630 just what was your moment of awe, what came up for you? 00:39:00.630 --> 00:39:05.620 What sensory details did you recall? 00:39:05.620 --> 00:39:09.090 And what did it feel like to just stay in 00:39:09.090 --> 00:39:11.150 that feeling of awe? 00:39:11.150 --> 00:39:14.746 There's a lot of research showing that it takes 30 seconds 00:39:14.746 --> 00:39:19.130 to a minute for a positive emotion to really register 00:39:19.130 --> 00:39:22.590 in our system for us to kind of what they say 00:39:22.590 --> 00:39:24.970 or what they call hard-wiring happiness so 00:39:24.970 --> 00:39:29.050 that our default state becomes more likely 00:39:29.050 --> 00:39:32.350 to look for the good, to look for what we're grateful for, 00:39:32.350 --> 00:39:34.450 to recall these moments of awe. 00:39:34.450 --> 00:39:36.960 It becomes more of a habit over time, 00:39:36.960 --> 00:39:39.088 if we really let it simmer 00:39:39.088 --> 00:39:41.670 and stay with us for longer periods of time. 00:39:41.670 --> 00:39:43.750 So even talking about it 00:39:43.750 --> 00:39:46.380 and listening to each other is helping us 00:39:46.380 --> 00:39:50.070 to really develop that hard wiring for the good 00:39:50.070 --> 00:39:52.380 looking for the good, so we can be fully present 00:39:52.380 --> 00:39:55.740 with each other as we're listening to these stories of awe. 00:39:55.740 --> 00:39:57.630 And if you'd like, you can share your gesture 00:39:57.630 --> 00:40:01.750 and I can try to replicate it and we can all do it together 00:40:01.750 --> 00:40:03.180 so that we're mirroring each other 00:40:03.180 --> 00:40:05.800 and really maybe even touching deeply 00:40:05.800 --> 00:40:08.123 into empathizing with your experience. 00:40:09.380 --> 00:40:10.933 So who would like to share? 00:40:11.890 --> 00:40:13.430 - [Jeremy] We heard a number of great ones 00:40:13.430 --> 00:40:15.970 in the question section, Sam needs to talk 00:40:15.970 --> 00:40:18.593 about sunset on the rim of the Grand Canyon. 00:40:19.512 --> 00:40:21.600 Jackline talked about the first time I was out 00:40:21.600 --> 00:40:24.860 on the ocean without being able to see the shore. 00:40:24.860 --> 00:40:26.700 You met said the moment of owe 00:40:26.700 --> 00:40:28.480 that really struck her when she was presented 00:40:28.480 --> 00:40:30.270 with her daughter for the first time 00:40:30.270 --> 00:40:32.483 and she looked in her daughter's eyes 00:40:32.483 --> 00:40:33.970 and has had a connection with her, 00:40:33.970 --> 00:40:35.920 this lasted all the way into adulthood. 00:40:38.800 --> 00:40:42.160 - It's an incredible moment, I'm not a parent yet 00:40:42.160 --> 00:40:46.470 but I imagine that moment of, I mean overwhelming awe 00:40:46.470 --> 00:40:49.033 when your child is placed in your arms. 00:40:50.310 --> 00:40:53.500 And sometimes we'll do these exercises with parents 00:40:53.500 --> 00:40:56.020 and kids and have them look into each other's eyes 00:40:56.020 --> 00:40:58.230 and talk about the color of their eyes. 00:40:58.230 --> 00:41:00.410 Like I noticed you have a little ring 00:41:00.410 --> 00:41:02.775 around your eye and it's light brown or whatever it is 00:41:02.775 --> 00:41:06.770 but for me witnessing those moments of connection 00:41:06.770 --> 00:41:10.730 and seeing parents just their whole face light up 00:41:10.730 --> 00:41:13.810 as their child, children are looking into their eyes 00:41:13.810 --> 00:41:16.239 is for me, that's a moment of awe 00:41:16.239 --> 00:41:21.080 that I happily get to relive a lot of the time. 00:41:21.080 --> 00:41:24.033 So yeah, just thank you for sharing that. 00:41:25.590 --> 00:41:27.603 Let's invite some folks that join live. 00:41:28.690 --> 00:41:30.460 Let's start with Michelle. 00:41:30.460 --> 00:41:32.810 So Michelle, I'm gonna unmute your line, 00:41:32.810 --> 00:41:34.710 feel free to do the same on your side. 00:41:38.860 --> 00:41:40.510 Michelle, are you there? 00:41:40.510 --> 00:41:42.360 - [Michelle] Yes, I am. 00:41:42.360 --> 00:41:43.770 - Hi Michelle. 00:41:43.770 --> 00:41:46.653 - [Michelle] Hi, this is amazing. 00:41:47.990 --> 00:41:51.720 The first time I held each of my children 00:41:51.720 --> 00:41:56.720 and it washed over me the enormity of 00:41:59.040 --> 00:42:01.800 how my life would change after that 00:42:02.860 --> 00:42:06.300 life as I had known, it would never be the same 00:42:06.300 --> 00:42:11.300 and in ways I couldn't even at that time imagine, 00:42:14.595 --> 00:42:18.928 so that, that was awesome (laughs). 00:42:24.250 --> 00:42:27.780 - Beautiful, I got chills all through my head 00:42:27.780 --> 00:42:29.933 and back as you were describing that. 00:42:31.650 --> 00:42:33.300 - [Michelle] Took my breath away. 00:42:38.430 --> 00:42:39.263 - That's beautiful. 00:42:39.263 --> 00:42:40.830 Was there a movement that came to mind? 00:42:40.830 --> 00:42:42.602 Was it like holding? 00:42:42.602 --> 00:42:47.602 - [Michelle] Yes, it was as I cradled them in my arms 00:42:48.590 --> 00:42:50.963 and looked down at their little faces. 00:42:54.610 --> 00:42:58.040 And that was so many years ago they're adults now, 00:42:58.040 --> 00:43:03.040 but I can cherish that moment once again, 00:43:04.570 --> 00:43:06.630 so I appreciate this. 00:43:06.630 --> 00:43:10.280 I hadn't thought about that in many, many years, 00:43:10.280 --> 00:43:12.920 so thank you for that. 00:43:12.920 --> 00:43:15.593 - Yeah, you're so welcome, thank you for sharing. 00:43:17.100 --> 00:43:22.100 Sometimes I look into, I work with children quite a bit 00:43:22.300 --> 00:43:25.100 and I look into their eyes as if I was their parent. 00:43:25.100 --> 00:43:28.130 Like I try to connect with them with that kind of love 00:43:28.130 --> 00:43:31.850 and unconditional positive regard, 00:43:31.850 --> 00:43:33.610 if you wanna use therapy terms 00:43:33.610 --> 00:43:35.880 but that feeling of connection. 00:43:35.880 --> 00:43:37.870 And we can kind of do that 00:43:37.870 --> 00:43:39.747 with other people, even strangers, we can look 00:43:39.747 --> 00:43:44.190 at them through the eyes of a mother or a sister friend 00:43:44.190 --> 00:43:49.190 and really connect to each other on this level. 00:43:49.770 --> 00:43:51.330 You know, we have that capacity 00:43:51.330 --> 00:43:53.920 reminds me of the algebra teacher, 00:43:53.920 --> 00:43:57.860 kind of asking us to look at the problem of disconnection 00:43:57.860 --> 00:44:01.273 or loneliness and maybe one solution is to recall 00:44:03.010 --> 00:44:05.060 that feeling of love that we have 00:44:05.060 --> 00:44:07.380 for people in our lives and to maybe look 00:44:07.380 --> 00:44:10.003 at each other through that lens. 00:44:12.589 --> 00:44:15.033 - [Jeremy] Let's go over to Lindsay. 00:44:15.880 --> 00:44:18.270 Lindsay, I'm gonna unmute your line, 00:44:18.270 --> 00:44:20.080 feel free to connect with Sam. 00:44:20.080 --> 00:44:21.770 - [Lindsay] Hi there, can you guys hear me? 00:44:21.770 --> 00:44:23.103 - Yeah, hi Lindsey. 00:44:24.400 --> 00:44:28.000 - Thank you so much Sam, I appreciate this, this is awesome. 00:44:28.000 --> 00:44:29.870 So for me, it was also when I was traveling 00:44:29.870 --> 00:44:31.180 and I think there was a few of us 00:44:31.180 --> 00:44:34.160 said we were somewhere far, far away from home. 00:44:34.160 --> 00:44:36.910 So I was in *** me and (mumbles) a couple of years ago 00:44:36.910 --> 00:44:39.000 and I was watching the sunrise 00:44:39.000 --> 00:44:42.400 over hundreds of thousands of pagodas 00:44:42.400 --> 00:44:45.810 and it literally just seemed like time stopped 00:44:45.810 --> 00:44:47.890 and the sun took forever to come up, 00:44:47.890 --> 00:44:49.840 the transformation of the sky. 00:44:49.840 --> 00:44:52.990 And so that was really nice to kind of go back 00:44:52.990 --> 00:44:54.230 and revisit that I haven't looked 00:44:54.230 --> 00:44:55.430 at those photos in forever 00:44:55.430 --> 00:44:58.380 but I was sitting on the pagoda, looking at the sunrise, 00:44:58.380 --> 00:45:00.990 I was right there back in that moment. 00:45:00.990 --> 00:45:04.410 And I try not to focus too much on the past, 00:45:04.410 --> 00:45:07.970 try to focus on the present and maybe the next day 00:45:07.970 --> 00:45:09.660 or the next week but this was a nice reminder 00:45:09.660 --> 00:45:12.590 that it is okay to kind of go back to the past 00:45:12.590 --> 00:45:15.010 and think of those, those moments of awe, 00:45:15.010 --> 00:45:17.113 so thank you for that so much. 00:45:18.116 --> 00:45:19.690 - Oh, thank you. 00:45:19.690 --> 00:45:21.430 Again, I got chills for my whole body. 00:45:21.430 --> 00:45:24.240 I could see them as these pagodas 00:45:24.240 --> 00:45:26.190 as you were describing them in my mind. 00:45:27.170 --> 00:45:28.922 You know, when we listened to stories 00:45:28.922 --> 00:45:31.430 each other's stories, we're giving gifts. 00:45:31.430 --> 00:45:33.560 That's how I see stories where we're really 00:45:33.560 --> 00:45:37.970 gifting each other with these pleasant experiences 00:45:37.970 --> 00:45:40.340 because our brain doesn't really know the difference 00:45:40.340 --> 00:45:43.080 between living it, like being there again 00:45:43.080 --> 00:45:45.400 and being here now, 00:45:45.400 --> 00:45:47.980 the only differences are our motor cortex 00:45:47.980 --> 00:45:51.280 which is activated when we're actually physically moving. 00:45:51.280 --> 00:45:53.900 But our brain, you know, going back to the past 00:45:53.900 --> 00:45:57.714 and really reliving some of these moments is so good. 00:45:57.714 --> 00:46:01.880 The researcher who I love and I've done these workshops with 00:46:01.880 --> 00:46:04.730 she talks about it like a multivitamin, you know, 00:46:04.730 --> 00:46:06.510 the benefits you're giving yourself 00:46:06.510 --> 00:46:09.040 by recalling the past and sharing stories 00:46:09.040 --> 00:46:13.410 and listening to stories is medicinal in so many ways. 00:46:13.410 --> 00:46:18.410 So yeah, I encourage you all to really go there often. 00:46:19.110 --> 00:46:21.670 and every time we go there, we're reinforcing it, 00:46:21.670 --> 00:46:25.550 so it's easier to drop in, before bed, 00:46:25.550 --> 00:46:27.780 if you're having a hard time falling asleep, 00:46:27.780 --> 00:46:31.540 go to the pagodas, go to your child, 00:46:31.540 --> 00:46:35.430 looking into their eyes, go to the Coliseum, 00:46:35.430 --> 00:46:39.870 wherever it is, but touch into it because it's this pool 00:46:39.870 --> 00:46:44.870 of nourishment and it really, you know changes. 00:46:46.600 --> 00:46:50.090 I think hopefully you all feel a shift, 00:46:50.090 --> 00:46:51.960 our nervous system has settled down 00:46:51.960 --> 00:46:55.430 and if I was just listening to these stories 00:46:55.430 --> 00:46:58.040 after a busy day and didn't ground myself 00:46:58.040 --> 00:47:01.090 I wouldn't have gotten all these wonderful sensations 00:47:01.090 --> 00:47:04.435 of tingling and warmth and the visions of your stories. 00:47:04.435 --> 00:47:09.435 So we need that rest, we need that stillness 00:47:09.760 --> 00:47:14.110 and rejuvenation to then access each other more intimately 00:47:14.110 --> 00:47:17.463 and have empathy this deep, deep embodied empathy. 00:47:19.110 --> 00:47:21.710 So that's what I would love you to take away 00:47:21.710 --> 00:47:25.850 the time has flown by, but that it, 00:47:25.850 --> 00:47:30.820 to take away looking for the good, it's a practice, 00:47:30.820 --> 00:47:34.740 looking around and even just the space you're in right now 00:47:34.740 --> 00:47:39.630 and appreciating, I have photos of my family up here 00:47:39.630 --> 00:47:41.010 and all these wonderful memories 00:47:41.010 --> 00:47:44.283 but just looking for the good and finding 00:47:44.283 --> 00:47:47.449 extraordinariness in the ordinary 00:47:47.449 --> 00:47:50.590 especially in your students, and maybe even try looking 00:47:50.590 --> 00:47:55.120 at them through the eyes of a parent 00:47:55.120 --> 00:48:00.110 or that feeling of unconditional positive regard, even, 00:48:00.110 --> 00:48:02.490 especially I should say the most difficult students 00:48:02.490 --> 00:48:06.910 who challenge us in so many ways, but yeah 00:48:06.910 --> 00:48:11.690 also teach her some really beautiful lessons. 00:48:11.690 --> 00:48:15.250 So it has just been such a pleasure to be with you 00:48:15.250 --> 00:48:19.630 I really hope that you feel refreshed, rejuvenated 00:48:19.630 --> 00:48:22.880 that the tank is a little bit more full 00:48:22.880 --> 00:48:25.709 and we can just close our eyes for one last moment 00:48:25.709 --> 00:48:30.709 and just connect there, how many people on this cal? 00:48:30.720 --> 00:48:35.720 Over 400, 600, 650 people that we can't see, 00:48:35.749 --> 00:48:39.750 we heard some voices, but we know they're there. 00:48:39.750 --> 00:48:44.750 All of us educator, teaching, learning from our students 00:48:45.030 --> 00:48:48.690 giving of ourselves, we're part of this community 00:48:49.610 --> 00:48:52.000 even though we can't see each other, we're here 00:48:54.797 --> 00:48:58.710 to just taking a moment to send these kind wishes 00:48:58.710 --> 00:49:02.140 to our whole community, can think of these beams 00:49:02.140 --> 00:49:05.030 of light going out from our home all the way around 00:49:05.030 --> 00:49:08.300 the world and at least the country, 00:49:08.300 --> 00:49:11.513 but we can include the world community of teachers. 00:49:13.290 --> 00:49:18.290 May we all find joy, may we all have as many moments 00:49:23.210 --> 00:49:26.303 of awe as possible in our lives. 00:49:30.080 --> 00:49:32.283 May we all know we're not alone, 00:49:35.720 --> 00:49:39.913 may we all feel connected and safe and healthy. 00:49:43.124 --> 00:49:46.050 (exhales) 00:49:46.050 --> 00:49:47.290 And you've sent that out 00:49:47.290 --> 00:49:50.200 and now just take a few breaths receiving 00:49:50.200 --> 00:49:53.720 from the 650 people that have sent it to you 00:49:53.720 --> 00:49:58.720 just breathing in all the good that's been sent your way 00:50:00.340 --> 00:50:04.683 and letting it land in your body, your mind, your heart. 00:50:14.510 --> 00:50:18.400 Letting this feeling of connection and ease, linger 00:50:23.070 --> 00:50:26.173 and just taking it into the rest of your evening. 00:50:27.500 --> 00:50:31.140 Maybe the next person you make eye contact 00:50:31.140 --> 00:50:35.300 with just sending them kindness through your eyes 00:50:41.070 --> 00:50:45.870 and can open our eyes and for all the timekeepers, 00:50:45.870 --> 00:50:47.900 it's exactly five o'clock. 00:50:47.900 --> 00:50:50.570 So I wanna honor and respect your time 00:50:51.450 --> 00:50:55.690 and just say thank you so much to our whole team, 00:50:55.690 --> 00:50:57.840 to Jeremy and Alice and Leno 00:50:57.840 --> 00:51:01.950 for putting this all together and all of you for coming. 00:51:01.950 --> 00:51:05.210 It's been a pleasure and I hope we can do this again soon. 00:51:05.210 --> 00:51:06.463 Really genuinely. 00:51:11.580 --> 00:51:13.160 - Thank you so much, Sam. 00:51:13.160 --> 00:51:15.825 I just wanted to end on this note from Michelle 00:51:15.825 --> 00:51:16.790 which I thought was so perfect. 00:51:16.790 --> 00:51:19.400 Michelle says thank you for this brief respite 00:51:19.400 --> 00:51:20.517 from the rigors of the day to day. 00:51:20.517 --> 00:51:23.080 And we all know that teaching has those rigors, 00:51:23.080 --> 00:51:24.520 it was a precious gift. 00:51:24.520 --> 00:51:27.530 Maybe you also find joy and time that you can share 00:51:27.530 --> 00:51:29.470 with others in this wonderful practice. 00:51:29.470 --> 00:51:31.370 So thank you all for joining today. 00:51:31.370 --> 00:51:34.110 Thank you, Sam, for sharing your gifts with us 00:51:34.110 --> 00:51:36.180 and we wish you all a wonderful spring. 00:51:36.180 --> 00:51:37.013 - Thank you. 00:51:38.240 --> 00:51:39.163 Bye everybody.

Subtracting vectors with parallelogram rule

https://www.youtube.com/watch?v=4mRDJf6tShs

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https://www.youtube.com/api/timedtext?v=4mRDJf6tShs&ei=5VWUZam0KabCmLAP_4ediAM&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=A3F8E2A10BE29DE522BCF7E1BE9A5FA512CC4085.B414A17448B429D1EC58AF9A5603934011CE9D97&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.170 --> 00:00:01.430 - [Instructor] In this video, we're gonna think 00:00:01.430 --> 00:00:04.140 about what it means to subtract vectors, 00:00:04.140 --> 00:00:06.620 especially in the context of what we talked about 00:00:06.620 --> 00:00:08.760 as the parallelogram rule. 00:00:08.760 --> 00:00:11.880 So let's say we want to start with vector a 00:00:11.880 --> 00:00:15.360 and from that we want to subtract vector b. 00:00:15.360 --> 00:00:18.300 And we have vectors a and b depicted here. 00:00:18.300 --> 00:00:19.740 What do you think this is going to be? 00:00:19.740 --> 00:00:22.210 What do you think is going to be the resulting vector? 00:00:22.210 --> 00:00:24.110 Pause this video and think about that. 00:00:25.060 --> 00:00:25.893 All right. 00:00:25.893 --> 00:00:30.893 Now the key thing to realize is a minus b is the same thing 00:00:31.360 --> 00:00:36.360 as vector a plus the negative of vector b. 00:00:38.720 --> 00:00:42.000 Now, what is the negative of vector b look like? 00:00:42.000 --> 00:00:43.850 Well, that's going to be a vector 00:00:43.850 --> 00:00:46.980 that has the exact same magnitude as vector b 00:00:46.980 --> 00:00:49.500 but just in the opposite direction. 00:00:49.500 --> 00:00:53.320 For example this vector right over here 00:00:53.320 --> 00:00:55.043 would be the vector -b. 00:00:56.501 --> 00:00:57.960 Now we just have to think about 00:00:57.960 --> 00:01:01.230 what is vector a plus the vector -b? 00:01:01.230 --> 00:01:03.400 Well, there's two ways of thinking about that. 00:01:03.400 --> 00:01:05.200 I could put the tails of both of them 00:01:05.200 --> 00:01:08.010 at the same starting point, might as well do the origin. 00:01:08.010 --> 00:01:10.650 So let me draw -b over here. 00:01:10.650 --> 00:01:15.650 So we know the vector -b looks like that. 00:01:16.160 --> 00:01:18.410 So one way that you are probably familiar 00:01:18.410 --> 00:01:22.630 is you have vector a and then what you do is you take a copy 00:01:22.630 --> 00:01:24.690 or you could think of shifting vector b 00:01:24.690 --> 00:01:27.940 so its tail starts at the head of vector a. 00:01:27.940 --> 00:01:32.940 And if you did that, it would look like this. 00:01:33.100 --> 00:01:35.060 It would look like this. 00:01:35.060 --> 00:01:36.553 This is also the vector -b. 00:01:38.060 --> 00:01:42.553 And then the sum of vector a and vector -b 00:01:44.210 --> 00:01:48.570 is going to be going from the tail of vector a 00:01:48.570 --> 00:01:50.802 to the head of vector -b. 00:01:50.802 --> 00:01:55.510 So this would be the result, right over here. 00:01:55.510 --> 00:01:58.860 Which you could view as the sum of a plus -b 00:01:58.860 --> 00:02:01.650 or the difference of vectors a and b 00:02:01.650 --> 00:02:04.070 or vector a minus vector b. 00:02:04.070 --> 00:02:05.230 Now, if we wanna think about it 00:02:05.230 --> 00:02:07.400 in terms of the parallelogram rule, 00:02:07.400 --> 00:02:09.504 we could take another copy of vector a 00:02:09.504 --> 00:02:14.380 and put it so that it's tail's at the head of this -b 00:02:14.380 --> 00:02:18.220 and then we would get it right over here 00:02:18.220 --> 00:02:20.400 and we are forming the parallelogram. 00:02:20.400 --> 00:02:22.340 And then the resulting vector 00:02:22.340 --> 00:02:24.960 is the diagonal of the parallelogram. 00:02:24.960 --> 00:02:26.900 And this just helps us appreciate 00:02:26.900 --> 00:02:28.510 that we could start with -b 00:02:28.510 --> 00:02:31.540 and then add vector a to that. 00:02:31.540 --> 00:02:33.510 Or we could start with vector a 00:02:33.510 --> 00:02:35.810 and then add -b to that. 00:02:35.810 --> 00:02:39.370 But either way you get this white vector right over here 00:02:39.370 --> 00:02:44.370 which we can view as the vector a minus vector b, 00:02:44.620 --> 00:02:45.453 and we're done.

Adding vectors in magnitude and direction form

https://www.youtube.com/watch?v=yFPfO_eHJdY

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https://www.youtube.com/api/timedtext?v=yFPfO_eHJdY&ei=5VWUZaX1JeiAp-oP_Z-awAw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=D5CFCCD6ACD1FCEC4C5641415FF17E971FDFB5A3.43C87059E6EF65E2C7B2FB4C7C646BC9A757999E&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.910 --> 00:00:03.030 - [Instructor] We're told that vector a has magnitude four 00:00:03.030 --> 00:00:06.340 and direction 170 degrees from the positive x-axis. 00:00:06.340 --> 00:00:09.180 Vector b has magnitude three and direction 240 degrees 00:00:09.180 --> 00:00:10.960 from the positive x-axis. 00:00:10.960 --> 00:00:15.490 Find the magnitude and direction of vector a plus vector b. 00:00:15.490 --> 00:00:18.510 So pause this video and see if you can have a go at that. 00:00:18.510 --> 00:00:19.960 All right, now let's work through this together. 00:00:19.960 --> 00:00:21.390 And the way that I'm going to approach it, 00:00:21.390 --> 00:00:24.410 I'm going to represent each vector in component form. 00:00:24.410 --> 00:00:26.270 And then I'm going to add the corresponding components. 00:00:26.270 --> 00:00:28.650 And from that, I'll try to figure out the magnitude 00:00:28.650 --> 00:00:30.550 and the direction of the sum. 00:00:30.550 --> 00:00:33.660 So vector a, what is its x-component? 00:00:33.660 --> 00:00:36.160 Well, the change in x here, there's multiple ways 00:00:36.160 --> 00:00:39.150 that you could try to do this using trigonometry. 00:00:39.150 --> 00:00:42.390 But we've reviewed this or gone over this in other videos. 00:00:42.390 --> 00:00:43.830 The simplest way to think about it is 00:00:43.830 --> 00:00:46.057 our change in x here is going to be the length. 00:00:46.057 --> 00:00:49.620 And we know vector a has magnitude four times the cosine 00:00:49.620 --> 00:00:51.160 of the angle that the vector makes 00:00:51.160 --> 00:00:54.910 with the positive x-axis, cosine of 170 degrees. 00:00:54.910 --> 00:00:56.730 And so that's our x-component right over here, 00:00:56.730 --> 00:01:00.530 four times cosine of 170 degrees. 00:01:00.530 --> 00:01:03.140 And then what's our y-component? 00:01:03.140 --> 00:01:06.670 Well, our y-component is going to be this change in y here. 00:01:06.670 --> 00:01:08.870 And as we've reviewed in other videos, 00:01:08.870 --> 00:01:11.470 that's going to be the length times the sine 00:01:11.470 --> 00:01:13.810 of the angle we make with a positive x-axis, 00:01:13.810 --> 00:01:17.800 sine of 170 degrees. 00:01:17.800 --> 00:01:19.760 And we can maybe use a calculator in a bit 00:01:19.760 --> 00:01:22.130 to get approximations for these values. 00:01:22.130 --> 00:01:25.750 But then we can do the exact same thing for vector b. 00:01:25.750 --> 00:01:30.670 Vector b here is going to be, by the same logic, 00:01:30.670 --> 00:01:34.290 it's x-component is going to be the length of the vector, 00:01:34.290 --> 00:01:36.230 and it would be three. 00:01:36.230 --> 00:01:37.220 They tell us that. 00:01:37.220 --> 00:01:39.930 So it's going to be three times the cosine 00:01:39.930 --> 00:01:43.290 of this angle, 240 degrees. 00:01:43.290 --> 00:01:46.990 And then the y-component is going to be the length 00:01:46.990 --> 00:01:50.200 of our vector three times the sine 00:01:50.200 --> 00:01:53.650 of 240 degrees. 00:01:53.650 --> 00:01:56.270 Now, when we wanna take the sum of the two vectors, 00:01:56.270 --> 00:01:57.103 let me write it here, 00:01:57.103 --> 00:02:01.350 vector a plus vector b, 00:02:01.350 --> 00:02:04.490 I can just add the corresponding components. 00:02:04.490 --> 00:02:08.520 This is going to be equal to four cosine 00:02:08.520 --> 00:02:13.520 of 170 degrees plus three cosine 00:02:14.130 --> 00:02:16.160 of 240 degrees. 00:02:16.160 --> 00:02:20.137 And then the y-component is going to be four sine 00:02:21.880 --> 00:02:26.880 of 170 degrees plus three sine 00:02:27.860 --> 00:02:30.750 of 240 degrees. 00:02:30.750 --> 00:02:34.360 And so let me get my calculator out to evaluate these. 00:02:34.360 --> 00:02:36.350 We say 170 degrees. 00:02:36.350 --> 00:02:41.350 We take the cosine times four, that equals this. 00:02:41.720 --> 00:02:43.500 And then we're going to add to that. 00:02:43.500 --> 00:02:45.430 I'll open parentheses. 00:02:45.430 --> 00:02:47.240 We'll take the cosine of 240. 00:02:47.240 --> 00:02:52.240 240 cosine times three, close parentheses, 00:02:53.000 --> 00:02:54.840 is equal to this, negative, 00:02:54.840 --> 00:02:58.210 approximately negative 5.44. 00:02:58.210 --> 00:03:03.210 So this is approximately negative 5.44. 00:03:03.690 --> 00:03:07.120 And then if we were to take 170 degrees, 00:03:07.120 --> 00:03:10.740 take the sine of it, multiply it by four. 00:03:10.740 --> 00:03:13.670 And then to that, I'm going to open parentheses. 00:03:13.670 --> 00:03:17.310 I'm gonna take 240 degrees, take the sine, 00:03:17.310 --> 00:03:20.950 multiply that times three, close my parentheses. 00:03:20.950 --> 00:03:23.800 That is going to be equal to approximately negative 1.90. 00:03:25.250 --> 00:03:28.773 So this is approximately negative 1.90. 00:03:30.280 --> 00:03:32.460 And this is consistent with our intuition. 00:03:32.460 --> 00:03:35.590 If the sum has both negative components, 00:03:35.590 --> 00:03:37.240 that means it's going to be in the third quadrant. 00:03:37.240 --> 00:03:39.000 And if I were to do the head to tail method 00:03:39.000 --> 00:03:41.120 of adding vectors, if I were to take vector b 00:03:41.120 --> 00:03:44.080 and I were to put it right over here, 00:03:44.080 --> 00:03:45.890 we see that the resulting vector, 00:03:45.890 --> 00:03:48.960 the sum will sit in the third quadrant. 00:03:48.960 --> 00:03:51.540 It makes sense that our x and y-components 00:03:51.540 --> 00:03:53.120 would indeed be negative. 00:03:53.120 --> 00:03:54.270 Now, the question didn't ask 00:03:54.270 --> 00:03:56.070 just to find the components of the sum. 00:03:56.070 --> 00:03:59.210 It asked to find the magnitude and the direction 00:03:59.210 --> 00:04:00.980 of the resulting sum. 00:04:00.980 --> 00:04:02.320 And so to do that, 00:04:02.320 --> 00:04:05.730 we just have to use a little bit more of our trigonometry 00:04:05.730 --> 00:04:07.880 and actually a little bit of our geometry. 00:04:07.880 --> 00:04:12.210 For example, our change in x is this value right over here 00:04:12.210 --> 00:04:14.350 as we go from the tail to the tip. 00:04:14.350 --> 00:04:16.650 It's negative 5.44. 00:04:16.650 --> 00:04:19.260 If we're just thinking in terms of length right over here, 00:04:19.260 --> 00:04:23.563 the absolute value, this side would have length 5.44. 00:04:24.880 --> 00:04:27.040 And then same way, you are changing y, its negative, 00:04:27.040 --> 00:04:29.080 we're going down in y. 00:04:29.080 --> 00:04:30.992 But if we were just thinking in terms of a triangle, 00:04:30.992 --> 00:04:33.533 the length on this side of a triangle is 1.90. 00:04:35.570 --> 00:04:38.500 And we can see from the Pythagorean theorem 00:04:38.500 --> 00:04:42.400 that the length of our hypotenuse, 00:04:42.400 --> 00:04:44.670 which is the same thing as the magnitude of this vector, 00:04:44.670 --> 00:04:47.400 squared is going to be equal to the sum of the squares 00:04:47.400 --> 00:04:48.710 of these two sides. 00:04:48.710 --> 00:04:50.280 Or another way of thinking about it is, 00:04:50.280 --> 00:04:53.560 the length of this vector, the magnitude of this vector, 00:04:53.560 --> 00:04:56.140 which we can write as a magnitude of vector a 00:04:56.140 --> 00:05:00.070 plus vector b is going to be equal to, 00:05:00.070 --> 00:05:01.660 or I should say approximately equal to 00:05:01.660 --> 00:05:04.350 since we're already approximating these values, 00:05:04.350 --> 00:05:09.350 the principal root of 5.44 squared. 00:05:09.620 --> 00:05:10.660 And that's 'cause I'm just thinking 00:05:10.660 --> 00:05:12.750 about the absolute length of the side. 00:05:12.750 --> 00:05:14.330 I could also think about a change in x. 00:05:14.330 --> 00:05:16.770 But if I had a negative 5.44 and I square that, 00:05:16.770 --> 00:05:18.440 that would still become positive. 00:05:18.440 --> 00:05:23.000 And then I'll have plus 1.90 squared. 00:05:23.000 --> 00:05:25.330 And I can get our calculator out for that. 00:05:25.330 --> 00:05:26.860 This is going to be 00:05:26.860 --> 00:05:31.860 approximately equal to 5.44 squared 00:05:32.490 --> 00:05:37.000 plus 1.9 squared, 00:05:37.000 --> 00:05:38.330 is equal to that. 00:05:38.330 --> 00:05:40.120 Take the square root of that. 00:05:40.120 --> 00:05:43.533 It's approximately equal to 5.76, 00:05:45.540 --> 00:05:50.010 5.76, which is going to be our magnitude. 00:05:50.010 --> 00:05:51.880 And then to figure out the direction, 00:05:51.880 --> 00:05:54.610 so we essentially want to figure out 00:05:54.610 --> 00:05:57.210 this angle right over here. 00:05:57.210 --> 00:06:01.340 You might recognize that the tangent of this angle, 00:06:01.340 --> 00:06:02.720 theta right over here, 00:06:02.720 --> 00:06:05.790 should be equal to, and I'll do approximately equal to 00:06:05.790 --> 00:06:07.810 since we're using these approximations, 00:06:07.810 --> 00:06:11.280 our change in y over our change in x. 00:06:11.280 --> 00:06:12.190 So negative 1.90 00:06:14.020 --> 00:06:17.920 over negative 5.44, 00:06:17.920 --> 00:06:20.070 or we could say that theta 00:06:20.070 --> 00:06:24.640 is going to be approximately equal to the inverse tangent 00:06:24.640 --> 00:06:26.150 of negative 1.90 00:06:29.100 --> 00:06:33.980 over negative 5.44. 00:06:33.980 --> 00:06:35.500 And we're gonna see in a second 00:06:35.500 --> 00:06:37.560 whether this is actually going to get us the answer 00:06:37.560 --> 00:06:38.393 that we want. 00:06:38.393 --> 00:06:39.890 So let's try this out. 00:06:39.890 --> 00:06:44.210 If we were to take 1.9 negative 00:06:44.210 --> 00:06:48.270 divided by 5.44 negative, 00:06:48.270 --> 00:06:49.770 that gets us that, which makes sense. 00:06:49.770 --> 00:06:52.100 Negative divided by a negative is a positive. 00:06:52.100 --> 00:06:55.490 And now let's try to take the inverse tangent of that. 00:06:55.490 --> 00:06:59.290 So here I press second, and then I'll do inverse tangent. 00:06:59.290 --> 00:07:00.123 So I'm getting 00:07:00.123 --> 00:07:04.740 19.25 degrees, approximately. 00:07:04.740 --> 00:07:05.650 So this is saying 00:07:05.650 --> 00:07:10.420 that this is approximately 19.25 degrees. 00:07:10.420 --> 00:07:13.263 And my question to you is, does that seem right? 00:07:14.490 --> 00:07:18.030 Well, 19.25 degrees would put us in the first quadrant. 00:07:18.030 --> 00:07:19.070 It would get us a vector 00:07:19.070 --> 00:07:22.630 that looks something like this. 00:07:22.630 --> 00:07:26.630 This would be 19.25 degrees. 00:07:26.630 --> 00:07:28.680 But clearly, that's not the vector we're talking about. 00:07:28.680 --> 00:07:31.120 We're talking about a vector in the third quadrant. 00:07:31.120 --> 00:07:33.190 And the reason why we got this result, 00:07:33.190 --> 00:07:34.900 is that when you take the inverse tangent 00:07:34.900 --> 00:07:36.290 on most calculators, 00:07:36.290 --> 00:07:37.920 it's going to give you an angle 00:07:37.920 --> 00:07:39.840 that's between negative 90 degrees 00:07:39.840 --> 00:07:41.690 and positive 90 degrees. 00:07:41.690 --> 00:07:43.300 While here we are at an angle 00:07:43.300 --> 00:07:45.370 that puts us out in the third quadrant. 00:07:45.370 --> 00:07:46.870 So we have to adjust. 00:07:46.870 --> 00:07:51.060 And to adjust here, we just have to add 180 degrees 00:07:51.060 --> 00:07:54.580 to get to the actual angle that we are talking about. 00:07:54.580 --> 00:07:57.920 So in our situation, the magnitude here 00:07:57.920 --> 00:08:01.200 is going to be approximately 5.76 00:08:03.690 --> 00:08:07.010 and then the direction is going to be 00:08:07.010 --> 00:08:12.010 approximately 19.25 plus 180 degrees, 00:08:12.100 --> 00:08:14.970 which is going to be 199.25 degrees. 00:08:17.820 --> 00:08:19.423 And now we are done.

Parallelogram rule for vector addition

https://www.youtube.com/watch?v=Mep0foZMOCg

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https://www.youtube.com/api/timedtext?v=Mep0foZMOCg&ei=5VWUZZ3HK_rYmLAPqeS6gAM&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=909F66E4B5BC31E27D0153AD4C9E49B1D84FDAD8.88D61B6A8F178DC74003C76537ACF7CF5AB14CC4&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.110 --> 00:00:01.460 - [Instructor] So we have two vectors here, 00:00:01.460 --> 00:00:03.280 vector A and vector B. 00:00:03.280 --> 00:00:05.110 And what we're gonna do in this video 00:00:05.110 --> 00:00:08.480 is think about what it means to add vectors. 00:00:08.480 --> 00:00:10.180 So for example, how could we think 00:00:10.180 --> 00:00:12.960 about what does it mean to take vector A 00:00:12.960 --> 00:00:15.630 and add to that vector B. 00:00:15.630 --> 00:00:19.265 And as we'll see, we'll get another third vector. 00:00:19.265 --> 00:00:23.490 And there's two ways that we can think about this visually. 00:00:23.490 --> 00:00:25.520 One way is to say, all right, 00:00:25.520 --> 00:00:30.120 if we want start with vector A and then add vector B to it, 00:00:30.120 --> 00:00:33.440 what we can do, let me take a copy of vector B 00:00:33.440 --> 00:00:36.970 and put its tail right at the head of vector A. 00:00:36.970 --> 00:00:38.810 Notice I have not changed the magnitude 00:00:38.810 --> 00:00:40.650 or the direction of vector B. 00:00:40.650 --> 00:00:43.850 If I did, I would actually be changing the vector. 00:00:43.850 --> 00:00:47.270 And when I do it like that, this defines a third vector 00:00:47.270 --> 00:00:49.680 which can be use the sum of a plus B. 00:00:49.680 --> 00:00:54.010 And the sum is going to start at the tail of vector A 00:00:54.010 --> 00:00:57.240 and end at the head of vector B here. 00:00:57.240 --> 00:00:58.200 So let me draw that. 00:00:58.200 --> 00:01:00.680 So it would look something like that. 00:01:00.680 --> 00:01:03.360 And we can call this right over here, vector C. 00:01:03.360 --> 00:01:08.360 So we could say A plus B is equal to vector C. 00:01:08.450 --> 00:01:09.970 Now we could have also thought about it 00:01:09.970 --> 00:01:11.540 the other way around. 00:01:11.540 --> 00:01:15.010 We could have said, let's start with vector B 00:01:15.010 --> 00:01:19.430 and then add vector A to that. 00:01:19.430 --> 00:01:22.220 So I'll start with the tail of vector B 00:01:22.220 --> 00:01:25.070 and then at the head of vector B, I'm going to put the tail 00:01:25.070 --> 00:01:26.250 of vector A. 00:01:26.250 --> 00:01:29.800 So it could look something like that. 00:01:29.800 --> 00:01:33.490 And then once again, the sum is going to have its tail 00:01:33.490 --> 00:01:34.830 at our starting point here 00:01:34.830 --> 00:01:37.440 and its head at our finishing point. 00:01:37.440 --> 00:01:39.780 Now, another way of thinking about it is 00:01:39.780 --> 00:01:42.460 we've just constructed a parallelogram 00:01:42.460 --> 00:01:45.490 with these two vectors by putting both 00:01:45.490 --> 00:01:48.450 of their tails together, and then by taking a copy 00:01:48.450 --> 00:01:51.420 of each of them and putting that copy's tail 00:01:51.420 --> 00:01:53.240 at the head of the other vector, 00:01:53.240 --> 00:01:55.400 you construct a parallelogram like this, 00:01:55.400 --> 00:01:57.830 and then the sum is going to be the diagonal 00:01:57.830 --> 00:01:59.440 of the parallelogram. 00:01:59.440 --> 00:02:02.900 But hopefully you appreciate this is the same exact idea. 00:02:02.900 --> 00:02:05.422 If you just add by putting the head to tail 00:02:05.422 --> 00:02:08.220 of the two vectors and you construct a triangle, 00:02:08.220 --> 00:02:10.470 the parallelogram just helps us appreciate 00:02:10.470 --> 00:02:12.140 that you can start with the yellow vector 00:02:12.140 --> 00:02:15.090 and then the blue vector or the blue vector first 00:02:15.090 --> 00:02:16.290 and then the yellow vector. 00:02:16.290 --> 00:02:19.353 But either way, the sum is going to be this vector C.

Cosine equation solution set in an interval

https://www.youtube.com/watch?v=0_4uTKBI99o

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WEBVTT Kind: captions Language: en 00:00:00.123 --> 00:00:02.550 - [Instructor] In a previous video, we established 00:00:02.550 --> 00:00:05.850 the entire solution set for the following equation. 00:00:05.850 --> 00:00:09.840 And we saw that all the x's that can satisfy this equation 00:00:09.840 --> 00:00:14.150 are a combination of these x's and these x's here. 00:00:14.150 --> 00:00:16.350 The reason why I'm referring to each of them 00:00:17.721 --> 00:00:21.390 as numerous xs is that for any integer value of n, 00:00:21.390 --> 00:00:22.950 you'll get another solution. 00:00:22.950 --> 00:00:26.480 For any integer value of n, you'll get another solution. 00:00:26.480 --> 00:00:27.860 What I wanna do in this video 00:00:27.860 --> 00:00:30.800 is to make things a little bit more concrete. 00:00:30.800 --> 00:00:33.450 And the way that we're going to do it is by exploring 00:00:33.450 --> 00:00:37.610 all of the x values that satisfy this equation 00:00:37.610 --> 00:00:40.120 that sit in the closed interval 00:00:40.120 --> 00:00:44.400 from negative pi over two to zero. 00:00:44.400 --> 00:00:46.030 So I encourage you like always, 00:00:46.030 --> 00:00:48.730 pause this video and have a go at it by yourself 00:00:48.730 --> 00:00:51.210 before we work through it together. 00:00:51.210 --> 00:00:54.240 All right, now let's work through this together. 00:00:54.240 --> 00:00:56.330 So the first helpful thing is 00:00:56.330 --> 00:00:58.500 we have these algebraic expressions. 00:00:58.500 --> 00:01:00.720 We have things written in terms of pi. 00:01:00.720 --> 00:01:04.530 Let's approximate them all in terms of decimals. 00:01:04.530 --> 00:01:07.950 So even pi over two, we can approximate that. 00:01:07.950 --> 00:01:10.570 Let's see, if pi is approximately 3.14, 00:01:11.968 --> 00:01:14.363 half of that is approximately 1.57, 00:01:15.420 --> 00:01:17.030 so we could say this is approximately 00:01:17.030 --> 00:01:22.030 the closed interval from -1.57 to zero. 00:01:24.160 --> 00:01:26.790 - 1.57 isn't exactly negative pi over two, 00:01:26.790 --> 00:01:28.550 but it'll hopefully be suitable 00:01:28.550 --> 00:01:31.040 for what we're trying to do here. 00:01:31.040 --> 00:01:32.710 And now let's see if we can write 00:01:32.710 --> 00:01:34.500 the different parts of these expressions, 00:01:34.500 --> 00:01:37.600 or at least approximate them as decimals. 00:01:37.600 --> 00:01:41.060 So this could be rewritten as x is approximately, 00:01:41.060 --> 00:01:44.210 if you were to take 1/8 times the inverse cosine 00:01:44.210 --> 00:01:46.720 of -1/6, I encourage you to verify this 00:01:46.720 --> 00:01:48.380 on your own on a calculator, 00:01:48.380 --> 00:01:51.003 you would get that that's approximately 0.22. 00:01:53.000 --> 00:01:57.153 And then pi over four is approximately 0.785. 00:02:00.810 --> 00:02:04.940 So this expression would be approximately 0.22 00:02:04.940 --> 00:02:09.940 minus 0.785 times n, 00:02:11.480 --> 00:02:13.140 where n could be any integer. 00:02:13.140 --> 00:02:15.330 And then this one over here on the right, 00:02:15.330 --> 00:02:18.960 let me do that in this yellow, x could be approximately 00:02:18.960 --> 00:02:23.960 equal to, well if this evaluates to approximately 0.22, 00:02:25.480 --> 00:02:26.910 then this is just the negative of it, 00:02:26.910 --> 00:02:30.630 so it's going to be -0.22. 00:02:30.630 --> 00:02:33.570 And then it's plus what approximately 00:02:33.570 --> 00:02:36.993 pi over four is, so 0.785n. 00:02:39.970 --> 00:02:42.660 And now what we could do is just try different n's 00:02:42.660 --> 00:02:45.170 and see if we're starting above or below this interval, 00:02:45.170 --> 00:02:47.220 and then see which of the x values actually 00:02:47.220 --> 00:02:49.230 fall in this interval. 00:02:49.230 --> 00:02:50.860 So let's just start here. 00:02:50.860 --> 00:02:53.620 If we just start at n equals zero, 00:02:53.620 --> 00:02:56.210 actually why don't I set up a little table here, 00:02:56.210 --> 00:02:59.730 we have n here and if we have the x value here, 00:02:59.730 --> 00:03:03.320 when n is zero, well, then you don't see this term, 00:03:03.320 --> 00:03:06.663 and you just get approximately 0.22. 00:03:08.920 --> 00:03:11.020 Now let's compare that to the interval. 00:03:11.020 --> 00:03:13.070 The upper bound of that interval is zero. 00:03:13.070 --> 00:03:14.710 So this does not sit in the interval. 00:03:14.710 --> 00:03:18.980 So this is too high and we would want to 00:03:18.980 --> 00:03:20.610 define the x's that sit in the interval. 00:03:20.610 --> 00:03:22.470 We wanna find lower values. 00:03:22.470 --> 00:03:26.740 So it's good that here, where you're subtracting 0.785, 00:03:26.740 --> 00:03:29.490 so I would use positive integer values of n 00:03:29.490 --> 00:03:32.930 to decrease this 0.22 here. 00:03:32.930 --> 00:03:37.690 So when n equals one, we would subtract 0.785 from that, 00:03:37.690 --> 00:03:39.800 and I'll round all of these to the hundredths place, 00:03:39.800 --> 00:03:44.213 and that would get us to -0.57, 00:03:45.860 --> 00:03:47.910 and that does sit in the interval. 00:03:47.910 --> 00:03:48.840 So this looks good. 00:03:48.840 --> 00:03:50.340 So this would be a solution 00:03:50.340 --> 00:03:52.770 in that interval right over here. 00:03:52.770 --> 00:03:54.580 And let's try n equals two. 00:03:54.580 --> 00:03:58.070 So we would subtract 0.785 again, 00:03:58.070 --> 00:04:01.273 and that would get us to -1.35, 00:04:05.130 --> 00:04:09.640 not 25, 35, and that also 00:04:09.640 --> 00:04:10.740 sits in the interval. 00:04:10.740 --> 00:04:14.410 It's larger than -1.57, so that looks good. 00:04:14.410 --> 00:04:19.030 Let's subtract 0.785 again, when n equals three, 00:04:19.030 --> 00:04:22.790 that would get us -2.14. 00:04:22.790 --> 00:04:24.440 Well, that's all of a sudden out of the interval 00:04:24.440 --> 00:04:26.860 because that's below the lower bound here. 00:04:26.860 --> 00:04:29.690 So this is too low. 00:04:29.690 --> 00:04:33.350 So using this expression, we've been able to find 00:04:33.350 --> 00:04:38.000 two x values that sit in the interval that we care about. 00:04:38.000 --> 00:04:41.870 Now let's use these x values right over here 00:04:41.870 --> 00:04:44.150 and I'll set up another table. 00:04:44.150 --> 00:04:47.750 So, let's see we have our n and then we have our x values. 00:04:47.750 --> 00:04:50.370 So let's start with n equals zero 'cause that's easy 00:04:50.370 --> 00:04:52.160 to compute, and then this term would go away, 00:04:52.160 --> 00:04:56.560 and we'd have -0.22, and that's actually 00:04:56.560 --> 00:04:58.370 in this interval here, it's below zero, 00:04:58.370 --> 00:05:02.660 it's larger than -1.57, so that one checks out. 00:05:02.660 --> 00:05:05.880 But now to really explore, we have to go in both directions. 00:05:05.880 --> 00:05:08.610 We have to increase it or decrease it. 00:05:08.610 --> 00:05:10.110 So if we wanted to increase it, 00:05:10.110 --> 00:05:12.760 we could have a situation where n equals one. 00:05:12.760 --> 00:05:16.740 So if n equals one, we're gonna add 0.785 to this. 00:05:16.740 --> 00:05:18.290 Now you immediately know that that's going to be 00:05:18.290 --> 00:05:22.130 a positive value, if you computed it, it'd be 0.57, 00:05:22.130 --> 00:05:26.600 which is larger than zero, so this is too high. 00:05:26.600 --> 00:05:30.180 So now we could try going lower than -0.22 00:05:30.180 --> 00:05:32.500 by having negative values of n. 00:05:32.500 --> 00:05:37.060 So if n is equal to -1, that means we're subtracting 0.785 00:05:37.060 --> 00:05:41.653 from this right over here which would get us to -1.01. 00:05:44.870 --> 00:05:47.780 Well, that one works out, so that's in our interval. 00:05:47.780 --> 00:05:50.570 And now let's subtract 0.785 again. 00:05:50.570 --> 00:05:53.020 So I'll have n equals -2. 00:05:53.020 --> 00:05:57.410 And so if I subtract 0.785 again, 00:05:57.410 --> 00:06:01.413 I could round that to -1.79, 00:06:02.270 --> 00:06:04.640 which is lower than -1.57, 00:06:04.640 --> 00:06:08.200 so it's out of our interval, so it's too low. 00:06:08.200 --> 00:06:11.050 So all of the x values that are in our interval 00:06:11.050 --> 00:06:15.020 that satisfy this equation are these two right over here. 00:06:15.020 --> 00:06:19.410 And this one and this one, 00:06:19.410 --> 00:06:20.923 and we are done.

Cosine equation algebraic solution set

https://www.youtube.com/watch?v=JoOfBpdaAiw

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https://www.youtube.com/api/timedtext?v=JoOfBpdaAiw&ei=5VWUZbb9M-2ghcIPut6P-AU&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=E8EA65FAE1E101774497F5A095EFAC72BEF49979.D6A7FABA5E8E03CE622DEB80E48175490BFA638C&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.380 --> 00:00:01.470 - [Lecturer] The goal of this video 00:00:01.470 --> 00:00:04.740 is to find the solution set for the following equation, 00:00:04.740 --> 00:00:06.760 negative six times the cosine 00:00:06.760 --> 00:00:10.640 of 8x plus four is equal to five. 00:00:10.640 --> 00:00:12.720 And like always, I encourage you to pause this video 00:00:12.720 --> 00:00:14.380 and see if you can have a go at this 00:00:14.380 --> 00:00:15.370 before we do it together. 00:00:15.370 --> 00:00:18.680 And a reminder, we want the entire solution set, 00:00:18.680 --> 00:00:20.123 not just one solution. 00:00:21.110 --> 00:00:23.210 All right, now let's work through this together. 00:00:23.210 --> 00:00:24.780 Some of you might recognize 00:00:24.780 --> 00:00:29.030 that it would be valuable to isolate the cosine of 8x, 00:00:29.030 --> 00:00:30.530 and a good way of doing that 00:00:30.530 --> 00:00:33.820 would be, first, to subtract four from both sides, 00:00:33.820 --> 00:00:35.560 and then that would get us 00:00:35.560 --> 00:00:39.600 negative six times cosine of 8x, 00:00:39.600 --> 00:00:41.080 I subtracted four from the left, 00:00:41.080 --> 00:00:42.620 so that four is going to be gone, 00:00:42.620 --> 00:00:45.720 and then if I subtract four from the five, 00:00:45.720 --> 00:00:47.630 I am going to get a one there. 00:00:47.630 --> 00:00:50.700 And now I can multiply both sides 00:00:50.700 --> 00:00:55.700 of this equation by negative 1/6, 00:00:56.550 --> 00:00:59.040 I just wanna have a one in front of the cosine, 00:00:59.040 --> 00:01:01.080 so negative 1/6. 00:01:01.080 --> 00:01:03.010 And so this is going to be one, 00:01:03.010 --> 00:01:06.390 so I'm just gonna have cosine of 8x 00:01:07.560 --> 00:01:10.660 is equal to negative 1/6. 00:01:10.660 --> 00:01:12.290 Now, if I just keep going, 00:01:12.290 --> 00:01:15.220 I could take the inverse cosine of negative 1/6, 00:01:15.220 --> 00:01:17.160 and whatever that is divided by eight, 00:01:17.160 --> 00:01:19.010 I would get a solution, 00:01:19.010 --> 00:01:20.710 but this is a good time to pause 00:01:20.710 --> 00:01:24.370 and to make sure that we are capturing all of the solutions. 00:01:24.370 --> 00:01:26.600 And I'll give us, or I'll refresh our memories 00:01:26.600 --> 00:01:28.910 with some identities. 00:01:28.910 --> 00:01:30.130 And to help with these identities, 00:01:30.130 --> 00:01:33.023 I like to draw a quick unit circle. 00:01:34.270 --> 00:01:38.600 So this is our x-axis, this is our y-axis, 00:01:38.600 --> 00:01:43.310 and so my quick hand-drawn unit circle 00:01:45.250 --> 00:01:47.140 might look something like this, (laughing) 00:01:47.140 --> 00:01:49.020 it's not that nice looking, 00:01:49.020 --> 00:01:51.240 but we wanna think about all of the angles 00:01:51.240 --> 00:01:54.360 that when I take the cosine, I get to negative 1/6. 00:01:54.360 --> 00:01:59.360 So negative 1/6 might be something like right over here. 00:01:59.400 --> 00:02:02.760 And so you can see that there might be an angle 00:02:02.760 --> 00:02:04.810 like this that would get us there, 00:02:04.810 --> 00:02:07.140 so let me draw that, draw the radius. 00:02:07.140 --> 00:02:11.610 We know the cosine of an angle is the x-coordinate 00:02:11.610 --> 00:02:15.210 of where that radius that's defined by that angle, 00:02:15.210 --> 00:02:17.710 where that radius intersects the unit circle. 00:02:17.710 --> 00:02:19.790 But we also see there's another place, 00:02:19.790 --> 00:02:23.010 if we essentially take the negative of that angle, 00:02:23.010 --> 00:02:25.420 we could go right over here 00:02:25.420 --> 00:02:27.490 and we would also get the same cosine. 00:02:27.490 --> 00:02:30.060 So we could go to the negative of the angle, go that way. 00:02:30.060 --> 00:02:31.430 And that's where we get the identity 00:02:31.430 --> 00:02:34.040 that cosine of negative theta 00:02:34.040 --> 00:02:37.450 is equal to cosine of theta. 00:02:37.450 --> 00:02:39.480 And so if cosine of 8x 00:02:39.480 --> 00:02:41.400 is equal to negative 1/6, 00:02:41.400 --> 00:02:43.550 using this identity, we also know 00:02:43.550 --> 00:02:45.760 that cosine of the negative of this 00:02:45.760 --> 00:02:48.480 will also be equal to negative 1/6. 00:02:48.480 --> 00:02:49.530 So let me write that down, 00:02:49.530 --> 00:02:51.690 cosine of negative 8x 00:02:52.650 --> 00:02:55.723 is also going to be equal to negative 1/6. 00:02:56.700 --> 00:02:59.730 Now, already we have expanded our solution set 00:02:59.730 --> 00:03:02.560 because this is going to give us another x-value 00:03:02.560 --> 00:03:05.370 that's going to get us the result that we want, 00:03:05.370 --> 00:03:06.720 but are we done? 00:03:06.720 --> 00:03:09.290 Well, the other thing to realize is, 00:03:09.290 --> 00:03:11.680 let's say I have some angle here, 00:03:11.680 --> 00:03:15.770 where if I take the cosine, I get to negative 1/6, 00:03:15.770 --> 00:03:17.750 but then if I had two pi again, 00:03:17.750 --> 00:03:19.000 I'm gonna get to the same place, 00:03:19.000 --> 00:03:21.480 and the cosine is, once again, going to be negative 1/6, 00:03:21.480 --> 00:03:23.260 and I could add two pi again, 00:03:23.260 --> 00:03:24.940 I could essentially add two pi 00:03:24.940 --> 00:03:27.820 an arbitrary integer number of times. 00:03:27.820 --> 00:03:32.340 So I could rewrite this right over here as cosine, 00:03:32.340 --> 00:03:33.530 instead of just 8x, 00:03:33.530 --> 00:03:38.120 it's 8x plus an integer multiple of two pi, 00:03:38.120 --> 00:03:41.270 that's also going to be equal to negative 1/6. 00:03:41.270 --> 00:03:43.820 And similarly for negative 8x, 00:03:43.820 --> 00:03:48.060 I could say cosine of negative 8x 00:03:48.060 --> 00:03:50.670 plus an integer multiple of two pi, 00:03:50.670 --> 00:03:54.620 and is going to be some integer in both of these situations, 00:03:54.620 --> 00:03:57.550 that's also going to get us to negative 1/6. 00:03:57.550 --> 00:03:59.130 And so now we can feel pretty good 00:03:59.130 --> 00:04:00.820 that we're capturing all of the solutions 00:04:00.820 --> 00:04:02.380 when we solve for x. 00:04:02.380 --> 00:04:03.213 So in both of these, 00:04:03.213 --> 00:04:05.630 now let's take the inverse cosine of negative 1/6 00:04:05.630 --> 00:04:08.420 in order to solve for x here. 00:04:08.420 --> 00:04:12.090 So if we were to take the inverse cosine of both sides, 00:04:12.090 --> 00:04:15.370 we could get that 8x plus two pi 00:04:15.370 --> 00:04:17.750 times some arbitrary integer n 00:04:17.750 --> 00:04:22.553 is equal to the inverse cosine of negative 1/6. 00:04:23.640 --> 00:04:25.560 And then now let's solve for x, 00:04:25.560 --> 00:04:28.770 we can subtract two pi n from both sides. 00:04:28.770 --> 00:04:33.770 So we could get 8x is equal to the inverse cosine 00:04:33.920 --> 00:04:38.920 of negative 1/6 minus two pi n. 00:04:39.270 --> 00:04:40.330 Now, it's interesting to note 00:04:40.330 --> 00:04:42.300 that the sign on this two pi n term 00:04:42.300 --> 00:04:43.730 actually doesn't matter so much, 00:04:43.730 --> 00:04:45.560 'cause n could be a negative integer, 00:04:45.560 --> 00:04:48.420 but I'll just stick with this negative two pi n. 00:04:48.420 --> 00:04:50.390 And so if we wanted to solve for x, 00:04:50.390 --> 00:04:52.410 we'd just divide both sides by eight, 00:04:52.410 --> 00:04:55.650 we get x is equal to 1/8 00:04:55.650 --> 00:04:59.890 times the inverse cosine of negative 1/6 00:04:59.890 --> 00:05:04.890 minus pi over four n. 00:05:05.080 --> 00:05:07.060 And now we can do the exact same thing 00:05:07.060 --> 00:05:10.690 in the other scenario, I'll call this the yellow scenario, 00:05:10.690 --> 00:05:12.070 where if I take the inverse cosine, 00:05:12.070 --> 00:05:15.940 I get negative 8x plus two pi, 00:05:15.940 --> 00:05:20.073 n is equal to the inverse cosine of negative 1/6. 00:05:21.060 --> 00:05:24.320 And now I can subtract two pi n from both sides, 00:05:24.320 --> 00:05:25.570 so I get negative 8x 00:05:26.590 --> 00:05:31.550 is equal to inverse cosine of negative 1/6 00:05:31.550 --> 00:05:34.330 minus two pi n. 00:05:34.330 --> 00:05:36.550 Now I can multiply both sides by negative 1/8, 00:05:36.550 --> 00:05:38.560 or divide both sides by negative eight, 00:05:38.560 --> 00:05:42.010 and I get x is equal to negative 1/8 00:05:42.010 --> 00:05:45.380 times the inverse cosine of negative 1/6 00:05:46.740 --> 00:05:50.303 plus pi over four n. 00:05:51.410 --> 00:05:53.660 So I will stop here for this video, 00:05:53.660 --> 00:05:56.610 where at least algebraically we know the solution set, 00:05:56.610 --> 00:05:58.460 and this is the complete solution set 00:06:01.447 --> 00:06:04.220 if you take the combination of both of these expressions. 00:06:04.220 --> 00:06:07.600 In a future video, we'll evaluate this with a calculator, 00:06:07.600 --> 00:06:09.790 and we'll think about the solutions 00:06:09.790 --> 00:06:12.083 that fit within a given interval.

Proof of the tangent angle sum and difference identities

https://www.youtube.com/watch?v=nUlElr4LXz8

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https://www.youtube.com/api/timedtext?v=nUlElr4LXz8&ei=5VWUZai1I8G3mLAPjc6c-AE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=B17B03BB3571EF337543E1ECDB42AF7458E21A95.CCD27137E0EA781423BAF430267B76A48C42C10A&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.150 --> 00:00:01.300 - [Instructor] In this video I'm going to assume 00:00:01.300 --> 00:00:02.550 that you already know a few things 00:00:02.550 --> 00:00:03.383 and we've covered this. 00:00:03.383 --> 00:00:04.690 We've proved this in other videos 00:00:04.690 --> 00:00:08.810 that sine of x plus y is equal to sine of x cosine y 00:00:08.810 --> 00:00:11.020 plus and then you swap the cosines and the sines, 00:00:11.020 --> 00:00:13.430 cosine of x sine y, 00:00:13.430 --> 00:00:18.430 and then cosine of x plus y is equal to cosine x cosine y 00:00:18.620 --> 00:00:20.770 minus sine x sin y. 00:00:20.770 --> 00:00:23.150 Once again, we've proven this in this in other videos 00:00:23.150 --> 00:00:25.820 and then there's some other properties we know of cosine 00:00:25.820 --> 00:00:27.610 and sine that we have looked at another video's. 00:00:27.610 --> 00:00:30.972 Cosine of -x is equal to cosine of x 00:00:30.972 --> 00:00:35.450 and that sine of negative x is equal to -sine of x. 00:00:35.450 --> 00:00:36.380 And that of course, 00:00:36.380 --> 00:00:38.540 the tangent of something is defined 00:00:38.540 --> 00:00:42.320 as a sine over cosine of that something. 00:00:42.320 --> 00:00:43.510 Now with that out of the way, 00:00:43.510 --> 00:00:44.980 I wanna come up with a formula 00:00:44.980 --> 00:00:48.380 for tangent of x plus y expressed just 00:00:48.380 --> 00:00:52.560 in terms of tangent of x and tangent of y. 00:00:52.560 --> 00:00:54.180 You can view it as the antilog 00:00:54.180 --> 00:00:56.663 for what we did up here for sine and cosine. 00:00:57.690 --> 00:00:59.850 Well, the immediate thing that you might recognize 00:00:59.850 --> 00:01:03.060 is that tangent of x plus y based on the definition 00:01:03.060 --> 00:01:08.060 of tangent is the same thing as sine of x plus y 00:01:08.210 --> 00:01:12.630 over cosine of x plus y. 00:01:12.630 --> 00:01:15.130 And what's that going to be equal to? 00:01:15.130 --> 00:01:17.350 Well, we know that sine of x plus y 00:01:17.350 --> 00:01:19.610 can be expressed this way. 00:01:19.610 --> 00:01:21.540 So let me write that down. 00:01:21.540 --> 00:01:26.447 So that's going to be sine of x cosine y 00:01:27.880 --> 00:01:32.120 plus cosine of x sine of y. 00:01:34.510 --> 00:01:36.230 And then, and actually, 00:01:36.230 --> 00:01:37.970 so that we can save a little bit of writing, 00:01:37.970 --> 00:01:40.240 I'm gonna awkwardly write, 00:01:40.240 --> 00:01:42.400 make the line down here, 00:01:42.400 --> 00:01:44.990 because we're gonna put something here in a second, 00:01:44.990 --> 00:01:46.100 but I think you'll get the idea. 00:01:46.100 --> 00:01:49.140 So there's going to be that over cosine of x plus y 00:01:49.140 --> 00:01:52.060 which is this expression when you just express it in terms 00:01:52.060 --> 00:01:55.620 of cosines of x and cosines of y and sines of x 00:01:55.620 --> 00:01:56.550 and sines of y. 00:01:56.550 --> 00:01:57.720 So let me write it here. 00:01:57.720 --> 00:02:02.720 So you're gonna have cosine of x cosine y minus sine of x, 00:02:07.100 --> 00:02:10.780 and then sine y. 00:02:10.780 --> 00:02:14.100 Now we wanna express everything in terms of tangents 00:02:14.100 --> 00:02:16.080 of xs and ys. 00:02:16.080 --> 00:02:18.530 And so it might make sense here to say, 00:02:18.530 --> 00:02:22.410 all right, well, we know tangent is sine over cosine. 00:02:22.410 --> 00:02:24.990 So what if we were to divide both the numerator 00:02:24.990 --> 00:02:27.970 and the denominator by some expression that can start 00:02:27.970 --> 00:02:30.110 to make the numerator and denominator express 00:02:30.110 --> 00:02:31.700 in terms of tangents. 00:02:31.700 --> 00:02:33.950 And I will cut a little bit to the chase here. 00:02:33.950 --> 00:02:35.400 So in the numerator, 00:02:35.400 --> 00:02:37.450 what I can do is, 00:02:37.450 --> 00:02:38.990 and I'm gonna do this just in the numerator, 00:02:38.990 --> 00:02:40.920 and then I'm gonna do it in the denominator as well. 00:02:40.920 --> 00:02:44.213 I'm gonna divide the numerator by cosine of x cosine y. 00:02:47.780 --> 00:02:49.990 And of course, I can't just divide the numerator 00:02:49.990 --> 00:02:51.570 by cosine of x cosine of y 00:02:51.570 --> 00:02:53.330 that would change the value of the, 00:02:53.330 --> 00:02:54.515 this rational expression. 00:02:54.515 --> 00:02:56.400 I have to do that to the denominator as well. 00:02:56.400 --> 00:02:58.830 So I know this is a very complex looking fraction here 00:02:58.830 --> 00:03:00.510 but it's going to simplify in a second. 00:03:00.510 --> 00:03:02.020 So I'm also going to divide the denominator 00:03:02.020 --> 00:03:06.180 by cosine of x cosine of y. 00:03:06.180 --> 00:03:08.550 And now let's see if we can simplify this 00:03:08.550 --> 00:03:10.520 in certain ways. 00:03:10.520 --> 00:03:13.860 In the numerator, we can see that this cosine y cancels 00:03:13.860 --> 00:03:15.340 with this cosine y. 00:03:15.340 --> 00:03:17.230 And so that first term becomes slightly 00:03:17.230 --> 00:03:18.220 in another color here. 00:03:18.220 --> 00:03:21.810 So this sine of x over cosine of x. 00:03:21.810 --> 00:03:23.200 And so the numerator, 00:03:23.200 --> 00:03:26.140 I can say this is going to be equal to sine of x 00:03:26.140 --> 00:03:30.200 over cosine of x is tangent of x. 00:03:30.200 --> 00:03:32.490 And then the second term here, 00:03:32.490 --> 00:03:35.540 we can see that this cosine of x cancels 00:03:35.540 --> 00:03:36.960 with this cosine of x . 00:03:36.960 --> 00:03:41.120 So we're left with sine of y over cosine of y, 00:03:41.120 --> 00:03:42.130 which is of course, 00:03:42.130 --> 00:03:43.620 tangent of y. 00:03:43.620 --> 00:03:47.500 So plus tangent of y 00:03:47.500 --> 00:03:49.860 and then all of that is going to be over, 00:03:49.860 --> 00:03:52.680 now we can look at the denominator. 00:03:52.680 --> 00:03:55.570 So this first term here, 00:03:55.570 --> 00:03:59.610 we can see the cosine of x cancels with the cosine of x 00:03:59.610 --> 00:04:03.380 and the cosine of y cancels out with the cosine of y. 00:04:03.380 --> 00:04:05.920 So you could view this first term here when you divide 00:04:05.920 --> 00:04:08.130 by this cosine of x cosine y, 00:04:08.130 --> 00:04:10.400 it just becomes one 00:04:10.400 --> 00:04:13.240 and then we're going to have the minus. 00:04:13.240 --> 00:04:15.850 And now this second term is interesting. 00:04:15.850 --> 00:04:19.090 We have sine of x over cosine of x, 00:04:19.090 --> 00:04:20.523 sine of y over cosine of y. 00:04:21.569 --> 00:04:24.800 So sine of x over cosine of x 00:04:24.800 --> 00:04:26.910 that over there is tangent of x, 00:04:26.910 --> 00:04:31.850 and then sine of y over cosine of y its tangent of y. 00:04:31.850 --> 00:04:36.850 So this is going to be tangent of x times tangent of y. 00:04:38.850 --> 00:04:39.710 And just like that, 00:04:39.710 --> 00:04:41.190 we have come up with an expression 00:04:41.190 --> 00:04:44.550 for tangent of x plus y that just deals 00:04:44.550 --> 00:04:48.400 with tangent of xs and tangent of ys. 00:04:48.400 --> 00:04:49.840 Now the next question you might say, 00:04:49.840 --> 00:04:52.950 well, all right, that's great for tangent of x plus y 00:04:52.950 --> 00:04:57.950 but what about tangent of x minus y? 00:04:58.250 --> 00:05:00.830 Well, here we just have to recognize a little bit 00:05:00.830 --> 00:05:02.690 of what we've seen before. 00:05:02.690 --> 00:05:03.730 Let me write it over here. 00:05:03.730 --> 00:05:08.670 Tangent of -x is equal to sine of -x 00:05:10.790 --> 00:05:13.830 over cosine of negative x 00:05:13.830 --> 00:05:15.990 and what's that going to be equal to? 00:05:15.990 --> 00:05:17.600 And I know I'm running out of space. 00:05:17.600 --> 00:05:20.120 This is going to be equal to sine of -x is the same thing 00:05:20.120 --> 00:05:24.360 as -sine of x, -sine of x, 00:05:24.360 --> 00:05:27.163 and then cosine of -x is just cosine of x. 00:05:28.490 --> 00:05:32.350 Well, this is just the negative of the tangent of x. 00:05:32.350 --> 00:05:37.340 So this is negative tangent of x. 00:05:37.340 --> 00:05:40.460 And the reason why that is useful 00:05:40.460 --> 00:05:43.020 is I can rewrite this as being, 00:05:43.020 --> 00:05:43.853 write here. 00:05:43.853 --> 00:05:48.710 This is the same thing as tangent of x plus -y. 00:05:50.950 --> 00:05:52.740 So everywhere we saw a y here, 00:05:52.740 --> 00:05:54.410 we can replace it with a -y. 00:05:54.410 --> 00:05:59.407 So this is going to be equal to tangent of x 00:06:00.500 --> 00:06:05.500 plus the tangent of -y, 00:06:05.510 --> 00:06:10.510 all of that over 1 minus the tangent of x 00:06:11.900 --> 00:06:14.313 times the tangent of -y. 00:06:15.520 --> 00:06:16.900 Well, we know the tangent of -y 00:06:16.900 --> 00:06:21.900 is the same thing as the negative tangent of y. 00:06:21.950 --> 00:06:24.070 And we know that over here as well. 00:06:24.070 --> 00:06:25.070 So this could just be, 00:06:25.070 --> 00:06:26.980 we could write the tangent of y here, 00:06:26.980 --> 00:06:30.010 and then the negative would turn this into a plus. 00:06:30.010 --> 00:06:33.290 And so just to write everything neatly, 00:06:33.290 --> 00:06:38.290 we know that also the tangent of x minus y 00:06:38.300 --> 00:06:43.300 can be rewritten as tangent of x minus tangent of y 00:06:46.010 --> 00:06:51.010 all of that over 1 plus tangent of x and tangent of y.

Interpreting statements about vectors

https://www.youtube.com/watch?v=eA2AW5F4N_M

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https://www.youtube.com/api/timedtext?v=eA2AW5F4N_M&ei=5VWUZbiMHrGJp-oPpviXkAQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=412521F97FEBA436407FAD54FBF0D4A07E19DA2D.3576153C6B156090009E015860D453D4929A142F&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.240 --> 00:00:01.770 - [Instructor] We're told that particles A and B 00:00:01.770 --> 00:00:03.780 are moving along a plane. 00:00:03.780 --> 00:00:06.750 Their velocities are represented by the vectors, 00:00:06.750 --> 00:00:09.490 vector a and vector b respectively. 00:00:09.490 --> 00:00:12.080 Which option best describes the meaning 00:00:12.080 --> 00:00:13.830 of the following statement. 00:00:13.830 --> 00:00:14.820 Choose one answer. 00:00:14.820 --> 00:00:17.340 So pause this video and try to work through this on your own 00:00:17.340 --> 00:00:19.190 before we work through this together. 00:00:20.090 --> 00:00:22.810 All right, now let's work through this together. 00:00:22.810 --> 00:00:26.190 So this is saying that the magnitude of vector a 00:00:26.190 --> 00:00:30.180 is equal to the magnitude of vector b. 00:00:30.180 --> 00:00:31.730 So we know that a vector is specified 00:00:31.730 --> 00:00:34.090 by both a magnitude and a direction. 00:00:34.090 --> 00:00:36.870 And this is just saying that the magnitudes are the same. 00:00:36.870 --> 00:00:41.870 So for example, vector a could look like this, 00:00:42.100 --> 00:00:44.850 and vector b could look like this. 00:00:44.850 --> 00:00:48.490 It could do something like that 00:00:48.490 --> 00:00:52.070 where it has the same magnitude and the same direction. 00:00:52.070 --> 00:00:55.700 Or vector b might be in a completely different direction. 00:00:55.700 --> 00:00:57.213 The magnitudes being equivalent just tells us 00:00:57.213 --> 00:00:59.920 that the length of these arrows are the same, 00:00:59.920 --> 00:01:02.320 but we could have different directions here. 00:01:02.320 --> 00:01:05.960 So what this tells me is that we have the same speed 00:01:05.960 --> 00:01:07.830 which is the magnitude of velocity, 00:01:07.830 --> 00:01:10.670 but not necessarily the same direction. 00:01:10.670 --> 00:01:12.200 Now let's look at the choices here. 00:01:12.200 --> 00:01:14.540 The first choice is that two particles move 00:01:14.540 --> 00:01:18.410 at the same speed and in the same direction. 00:01:18.410 --> 00:01:21.430 So we've already said that that's not necessarily the case. 00:01:21.430 --> 00:01:23.530 In order for choice A to be correct, 00:01:23.530 --> 00:01:26.330 they would essentially have to be equivalent vectors. 00:01:26.330 --> 00:01:29.130 Choice A would be the case if we said that vector a 00:01:29.130 --> 00:01:31.140 is equal to vector b, 00:01:31.140 --> 00:01:33.390 then they would have to have the same magnitude 00:01:33.390 --> 00:01:34.550 and the same direction, 00:01:34.550 --> 00:01:37.270 the same magnitude and the same direction. 00:01:37.270 --> 00:01:38.310 But that's not what they told us. 00:01:38.310 --> 00:01:40.520 They just told us that the magnitudes are the same, 00:01:40.520 --> 00:01:42.420 not necessarily the directions. 00:01:42.420 --> 00:01:44.130 So I'll rule that one out. 00:01:44.130 --> 00:01:45.950 The two particles move at the same speed, 00:01:45.950 --> 00:01:48.980 but not necessarily in the same direction. 00:01:48.980 --> 00:01:50.850 Yes, that's what we just talked about. 00:01:50.850 --> 00:01:51.990 They have the same speed, 00:01:51.990 --> 00:01:54.070 which is the magnitude of velocity, 00:01:54.070 --> 00:01:55.990 but they didn't tell us anything about the direction, 00:01:55.990 --> 00:01:57.300 just the magnitudes. 00:01:57.300 --> 00:02:00.250 So I like this choice, but let's look at choice C. 00:02:00.250 --> 00:02:03.040 The two particles move in the same direction, 00:02:03.040 --> 00:02:06.240 but not necessarily at the same speed. 00:02:06.240 --> 00:02:08.230 Well, we know they move at the same speed. 00:02:08.230 --> 00:02:09.910 That's what this is telling us. 00:02:09.910 --> 00:02:11.610 The magnitudes are the same. 00:02:11.610 --> 00:02:13.620 We just don't know anything about the direction. 00:02:13.620 --> 00:02:16.610 So I would rule this one out as well. 00:02:16.610 --> 00:02:18.890 In order for choice C to be the case, 00:02:18.890 --> 00:02:21.120 you would see something like this, 00:02:21.120 --> 00:02:24.160 maybe that is vector a right here, 00:02:24.160 --> 00:02:27.610 and then vector b would move in the same direction, 00:02:27.610 --> 00:02:29.880 but it would have a different magnitude. 00:02:29.880 --> 00:02:31.810 And here you would visualize the magnitude 00:02:31.810 --> 00:02:33.140 as the length of the arrow. 00:02:33.140 --> 00:02:34.560 But that's not what they told us. 00:02:34.560 --> 00:02:37.003 They told us this right over there.

Representing quantities with vectors

https://www.youtube.com/watch?v=TCOQzIONaz8

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https://www.youtube.com/api/timedtext?v=TCOQzIONaz8&ei=5VWUZYHgKre5mLAP-9Ka4AI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=71BCA7EC900A45534F28D91D544C4BA624F9BBF0.1CCF2775004D6D6723408664F1979C1E61C64CDC&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.380 --> 00:00:01.860 - [Instructor] We're told a powerful magnet 00:00:01.860 --> 00:00:05.410 is attracting a metal ball on a flat surface. 00:00:05.410 --> 00:00:07.530 The magnet is pulling the ball 00:00:07.530 --> 00:00:10.120 at a force of 15 Newtons, 00:00:10.120 --> 00:00:13.890 and the magnet is 20 degrees to the south 00:00:13.890 --> 00:00:18.280 from the eastward direction relative to the ball. 00:00:18.280 --> 00:00:21.500 Here are a few vectors where the magnitude of vector A 00:00:21.500 --> 00:00:23.510 is equal to the magnitude of vector C 00:00:23.510 --> 00:00:25.230 is equal to 15 Newtons, 00:00:25.230 --> 00:00:27.180 and the magnitude of vector B is equal to 00:00:27.180 --> 00:00:30.530 the magnitude of vector D which is equal to 20 Newtons. 00:00:30.530 --> 00:00:35.130 Which vectors can represent the force of the team's pull? 00:00:35.130 --> 00:00:36.580 All right, pause this video and see 00:00:36.580 --> 00:00:38.300 if you can think about that on your own 00:00:38.300 --> 00:00:40.370 before we do it together. 00:00:40.370 --> 00:00:41.810 All right, now let's do it together. 00:00:41.810 --> 00:00:43.180 So before I even look at this, 00:00:43.180 --> 00:00:44.740 I'm just gonna look at the description. 00:00:44.740 --> 00:00:47.520 It has a magnitude of 15 Newtons. 00:00:47.520 --> 00:00:48.660 If we're talking about a force, 00:00:48.660 --> 00:00:51.800 you can view it as a strength of 15 Newtons, 00:00:51.800 --> 00:00:54.760 and the magnet which is pulling on the ball 00:00:54.760 --> 00:00:56.890 is 20 degrees to the south 00:00:56.890 --> 00:00:59.730 from the eastward direction relative to the ball. 00:00:59.730 --> 00:01:01.800 So if this is the ball right over here, 00:01:01.800 --> 00:01:04.203 and if this is the eastward direction, 00:01:05.170 --> 00:01:07.570 it says that the magnet is 20 degrees to the south 00:01:07.570 --> 00:01:10.010 from the eastward direction relative to the ball. 00:01:10.010 --> 00:01:12.893 So the magnet would be in this direction, 00:01:13.780 --> 00:01:17.030 and this angle right over here is 20 degrees, 00:01:17.030 --> 00:01:19.130 and the magnet is pulling on the ball, 00:01:19.130 --> 00:01:22.280 so the vector would go in that direction towards the magnet, 00:01:22.280 --> 00:01:25.270 and we know it has a force of 15 Newtons, 00:01:25.270 --> 00:01:26.330 that's the magnitude. 00:01:26.330 --> 00:01:30.550 So it has to be a 15 Newton magnitude right over here. 00:01:30.550 --> 00:01:32.770 So when we look at the choices, 00:01:32.770 --> 00:01:34.810 choice A is interesting, 00:01:34.810 --> 00:01:36.590 at least the direction looks right. 00:01:36.590 --> 00:01:39.520 It looks like it's 20 degrees south of due east, 00:01:39.520 --> 00:01:43.360 and they also tell us that the magnitude of A is 15 Newtons. 00:01:43.360 --> 00:01:46.650 So I am liking A, now let's look at B. 00:01:46.650 --> 00:01:50.160 Well B looks 15 degrees south of due east, 00:01:50.160 --> 00:01:51.540 not 20 degrees south, 00:01:51.540 --> 00:01:52.860 so I will rule that out. 00:01:52.860 --> 00:01:56.330 And also B's magnitude is wrong, it's 20 Newtons. 00:01:56.330 --> 00:01:59.880 C, the magnitude is right, it's 15 Newtons, 00:01:59.880 --> 00:02:03.450 but the direction looks like 20 degrees north of due east. 00:02:03.450 --> 00:02:04.920 So I'll rule that one out. 00:02:04.920 --> 00:02:06.810 And last but not least, D, 00:02:06.810 --> 00:02:08.560 the direction is clearly wrong, 00:02:08.560 --> 00:02:10.850 it looks like 15 degrees north of due east, 00:02:10.850 --> 00:02:14.150 and its magnitude is 20 Newtons, not 15 Newtons, 00:02:14.150 --> 00:02:15.453 so I'd rule that one out. 00:02:16.450 --> 00:02:19.290 Now to be clear a vector is only defined 00:02:19.290 --> 00:02:21.040 by its magnitude and its direction, 00:02:21.040 --> 00:02:22.710 not by its starting point. 00:02:22.710 --> 00:02:24.120 So if I had some other vector 00:02:24.120 --> 00:02:26.550 that looked like this right over here, 00:02:26.550 --> 00:02:28.690 that had the same magnitude and direction 00:02:28.690 --> 00:02:31.920 if this was right over here, a 20 degree angle 00:02:31.920 --> 00:02:33.910 and it had a magnitude of 50 Newtons, 00:02:33.910 --> 00:02:36.420 then I would have selected this one as well. 00:02:36.420 --> 00:02:38.830 You can shift a vector around like this 00:02:38.830 --> 00:02:41.060 as long as it has the same magnitude 00:02:41.060 --> 00:02:42.560 and it has the same direction, 00:02:42.560 --> 00:02:44.433 it is an equivalent vector.

Introduction to vector components

https://www.youtube.com/watch?v=hJkKADcQWj0

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en

WEBVTT Kind: captions Language: en 00:00:00.630 --> 00:00:02.090 - [Instructor] In other videos, we have talked 00:00:02.090 --> 00:00:04.100 about how a vector can be completely defined 00:00:04.100 --> 00:00:06.880 by a magnitude and a direction, you need both. 00:00:06.880 --> 00:00:08.260 And here we have done that. 00:00:08.260 --> 00:00:09.880 We have said that the magnitude 00:00:09.880 --> 00:00:12.570 of vector a is equal to three units, 00:00:12.570 --> 00:00:15.190 these parallel lines here on both sides, 00:00:15.190 --> 00:00:17.170 it looks like a double absolute value. 00:00:17.170 --> 00:00:19.090 That means the magnitude of vector a. 00:00:19.090 --> 00:00:23.150 And you can also specify that visually by making sure 00:00:23.150 --> 00:00:26.200 that the length of this vector arrow is three units long. 00:00:26.200 --> 00:00:27.560 And we also have its direction. 00:00:27.560 --> 00:00:29.610 We see the direction of vector a is 30 degrees 00:00:29.610 --> 00:00:32.270 counter-clockwise of due East. 00:00:32.270 --> 00:00:34.860 Now in this video, we're gonna talk about other ways 00:00:34.860 --> 00:00:38.220 or another way to specify or to define a vector. 00:00:38.220 --> 00:00:41.050 And that's by using components. 00:00:41.050 --> 00:00:42.530 And the way that we're gonna do it is, 00:00:42.530 --> 00:00:44.100 we're gonna think about the tail 00:00:44.100 --> 00:00:47.300 of this vector and the head of this vector. 00:00:47.300 --> 00:00:50.450 And think about as we go from the tail to the head, 00:00:50.450 --> 00:00:53.990 what is our change in x? 00:00:53.990 --> 00:00:55.180 And we could see our change 00:00:55.180 --> 00:00:58.340 in x would be that right over there. 00:00:58.340 --> 00:01:00.980 We're going from this x value to this x value. 00:01:00.980 --> 00:01:05.370 And then what is going to be our change in y. 00:01:05.370 --> 00:01:07.980 And if we're going from down here to up here, 00:01:07.980 --> 00:01:12.310 our change in y, we can also specify like that. 00:01:12.310 --> 00:01:13.500 So let me label these. 00:01:13.500 --> 00:01:18.500 This is my change in x, and then this is my change in y. 00:01:19.060 --> 00:01:19.920 And if you think about it, 00:01:19.920 --> 00:01:22.780 if someone told you your change in x and change in y, 00:01:22.780 --> 00:01:25.390 you could reconstruct this vector right over here 00:01:25.390 --> 00:01:27.490 by starting here, having that change in x, 00:01:27.490 --> 00:01:31.200 then having the change in y and then defining where the tip 00:01:31.200 --> 00:01:34.740 of the vector would be relative to the tail. 00:01:34.740 --> 00:01:38.800 The notation for this is we would say that vector a 00:01:38.800 --> 00:01:42.870 is equal to, and we'll have parenthesis, 00:01:42.870 --> 00:01:46.290 and we'll have our change in x comma, change in y. 00:01:46.290 --> 00:01:47.780 And so if we wanted to get tangible 00:01:47.780 --> 00:01:50.340 for this particular vector right over here, 00:01:50.340 --> 00:01:53.550 we know the length of this vector is three. 00:01:53.550 --> 00:01:55.540 Its magnitude is three. 00:01:55.540 --> 00:01:58.350 We know that this is, since this is going due horizontally 00:01:58.350 --> 00:02:00.290 and then this is going straight up and down. 00:02:00.290 --> 00:02:02.420 This is a right triangle. 00:02:02.420 --> 00:02:05.170 And so we can use a little bit of geometry from the past. 00:02:05.170 --> 00:02:08.020 Don't worry if you need a little bit of a refresher on this, 00:02:08.020 --> 00:02:09.620 but we could use a little bit of geometry, 00:02:09.620 --> 00:02:11.490 or a little bit of trigonometry to establish, 00:02:11.490 --> 00:02:13.610 if we know this angle, if we know the length 00:02:13.610 --> 00:02:17.210 of this hypotenuse, that this side that's opposite 00:02:17.210 --> 00:02:20.180 the 30 degree angle is gonna be half the hypotenuse, 00:02:20.180 --> 00:02:22.020 so it's going to be 3/2. 00:02:22.020 --> 00:02:24.200 And that the change in x is going to be 00:02:24.200 --> 00:02:26.960 the square root of three times the 3/2. 00:02:26.960 --> 00:02:31.080 So it's going to be three, square roots of three over two. 00:02:31.080 --> 00:02:33.980 And so up here, we would write our x component 00:02:33.980 --> 00:02:37.680 is three times the square root of three over two. 00:02:37.680 --> 00:02:42.420 And we would write that the y component is 3/2. 00:02:42.420 --> 00:02:43.820 Now I know a lot of you might be thinking 00:02:43.820 --> 00:02:47.260 this looks a lot like coordinates in the coordinate plane, 00:02:47.260 --> 00:02:48.580 where this would be the x coordinate 00:02:48.580 --> 00:02:50.300 and this would be the y coordinate. 00:02:50.300 --> 00:02:51.970 But when you're dealing with vectors, 00:02:51.970 --> 00:02:54.610 that's not exactly the interpretation. 00:02:54.610 --> 00:02:57.000 It is the case that if the vector's tail 00:02:57.000 --> 00:03:00.860 were at the origin right over here, then its head 00:03:00.860 --> 00:03:04.670 would be at these coordinates on the coordinate plane. 00:03:04.670 --> 00:03:07.470 But we know that a vector is not defined 00:03:07.470 --> 00:03:10.180 by its position, by the position of the tail. 00:03:10.180 --> 00:03:12.200 I could shift this vector around wherever 00:03:12.200 --> 00:03:13.840 and it would still be the same vector. 00:03:13.840 --> 00:03:15.590 It can start wherever. 00:03:15.590 --> 00:03:19.000 So when you use this notation in a vector context, 00:03:19.000 --> 00:03:21.440 these aren't x coordinates and y coordinates. 00:03:21.440 --> 00:03:26.440 This is our change in x, and this is our change in y. 00:03:27.070 --> 00:03:28.480 Let me do one more example to show 00:03:28.480 --> 00:03:30.880 that we can actually go the other way. 00:03:30.880 --> 00:03:34.790 So let's say I defined some vector b, 00:03:34.790 --> 00:03:39.200 and let's say that its x component is square root of two. 00:03:39.200 --> 00:03:43.520 And let's say that its y component is square root of two. 00:03:43.520 --> 00:03:46.260 So let's think about what that vector would look like. 00:03:46.260 --> 00:03:49.380 So it would, if this is its tail, 00:03:49.380 --> 00:03:51.410 and its x component which is its change 00:03:51.410 --> 00:03:53.030 in x is square root of two. 00:03:53.030 --> 00:03:55.460 So it might look something like this. 00:03:55.460 --> 00:04:00.460 So that would be change in x is equal to square root of two. 00:04:00.800 --> 00:04:03.980 And then its y component would also be square root of two. 00:04:03.980 --> 00:04:07.230 So I could write our change in y over here 00:04:07.230 --> 00:04:08.970 is square root of two. 00:04:08.970 --> 00:04:12.850 And so the vector would look something like this. 00:04:12.850 --> 00:04:17.850 It would start here and then it would go over here, 00:04:18.580 --> 00:04:20.590 and we can use a little bit of geometry 00:04:20.590 --> 00:04:21.980 to figure out the magnitude 00:04:21.980 --> 00:04:24.260 and the direction of this vector. 00:04:24.260 --> 00:04:26.760 You can use the Pythagorean theorem to establish 00:04:26.760 --> 00:04:28.760 that this squared plus this squared 00:04:28.760 --> 00:04:30.410 is gonna be equal to that squared. 00:04:30.410 --> 00:04:32.380 And if you do that, you're going to get this 00:04:32.380 --> 00:04:34.510 having a length of two, which tells you 00:04:34.510 --> 00:04:39.370 that the magnitude of vector b is equal to two. 00:04:39.370 --> 00:04:42.420 And if you wanted to figure out this angle right over here, 00:04:42.420 --> 00:04:43.870 you could do a little bit of trigonometry 00:04:43.870 --> 00:04:46.110 or even a little bit of geometry recognizing 00:04:46.110 --> 00:04:49.500 that this is going to be a right angle right over here, 00:04:49.500 --> 00:04:52.130 and that this side and that side have the same length. 00:04:52.130 --> 00:04:53.410 So these are gonna be the same angles 00:04:53.410 --> 00:04:55.600 which are gonna be 45 degree angles. 00:04:55.600 --> 00:04:58.690 And so just like that, you could also specify the direction, 00:04:58.690 --> 00:05:02.770 45 degrees counter-clockwise of due East. 00:05:02.770 --> 00:05:05.360 So hopefully you appreciate that these are equivalent ways 00:05:05.360 --> 00:05:06.540 of representing a vector. 00:05:06.540 --> 00:05:08.950 You either can have a magnitude and a direction, 00:05:08.950 --> 00:05:10.200 or you can have your components 00:05:10.200 --> 00:05:12.350 and you can go back and forth between the two. 00:05:12.350 --> 00:05:15.283 And we'll get more practice of that in future videos.

Personal Pronouns

https://www.youtube.com/watch?v=qtkivKA4h9w

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https://www.youtube.com/api/timedtext?v=qtkivKA4h9w&ei=5VWUZen0NeyIp-oPv7K2kAI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=32C4353C078B5EBA68D24ECDEE02169DF4D359EF.D7356FBB963EAA7BAEF523B94FD15A5FCDC7649B&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.660 --> 00:00:02.190 - [Instructor] Hello, grammarians, 00:00:02.190 --> 00:00:04.140 let's talk about personal pronouns, 00:00:04.140 --> 00:00:07.040 but first let me lay some sentences on you. 00:00:07.040 --> 00:00:10.050 Jake and I baked a loaf of bread. 00:00:10.050 --> 00:00:12.450 We baked a loaf of bread. 00:00:12.450 --> 00:00:14.053 You can learn anything. 00:00:14.940 --> 00:00:17.140 My friends are cool. 00:00:17.140 --> 00:00:18.890 They are cool. 00:00:18.890 --> 00:00:20.650 Now I'm gonna circle a few of these words, 00:00:20.650 --> 00:00:23.120 so the ones I wrote in yellow, and point them out to you. 00:00:23.120 --> 00:00:24.650 I, we, 00:00:24.650 --> 00:00:27.540 you, my, they, 00:00:27.540 --> 00:00:29.130 these are personal pronouns. 00:00:29.130 --> 00:00:30.500 They're pronouns that change 00:00:30.500 --> 00:00:32.210 depending on how you're using them. 00:00:32.210 --> 00:00:34.780 On whether you're using them as the object of a sentence, 00:00:34.780 --> 00:00:37.580 as the subject to show ownership, and so on. 00:00:37.580 --> 00:00:39.220 But we'll get to those in later videos. 00:00:39.220 --> 00:00:40.730 For now, I'm gonna talk about 00:00:40.730 --> 00:00:44.290 the three basic types of English pronoun. 00:00:44.290 --> 00:00:47.100 Broadly speaking, there are pronouns about me, 00:00:47.100 --> 00:00:48.840 pronouns about you, 00:00:48.840 --> 00:00:52.190 and pronouns about something or someone else. 00:00:52.190 --> 00:00:54.610 This is an idea called grammatical person. 00:00:54.610 --> 00:00:56.950 Pronouns about me are first person, 00:00:56.950 --> 00:00:59.230 pronouns about you are second person, 00:00:59.230 --> 00:01:01.630 and pronouns about something or someone else 00:01:01.630 --> 00:01:03.410 are third person. 00:01:03.410 --> 00:01:06.750 So, when I say I love my dog, Phryne. 00:01:06.750 --> 00:01:09.450 I is a first person pronoun. 00:01:09.450 --> 00:01:10.283 In the sentence, 00:01:10.283 --> 00:01:13.720 she is an excellent dog, where she subs out for Phryne, 00:01:13.720 --> 00:01:16.860 she is a third person pronoun. 00:01:16.860 --> 00:01:18.420 Here is a picture of Phryne, 00:01:18.420 --> 00:01:21.040 I think we can all agree, she is perfect. 00:01:21.040 --> 00:01:21.873 Thank you. 00:01:21.873 --> 00:01:22.940 Okay. So what I want to do here 00:01:22.940 --> 00:01:24.330 is fill out this table 00:01:24.330 --> 00:01:25.970 with some of the basic pronouns 00:01:25.970 --> 00:01:27.730 we use to talk about ourselves 00:01:27.730 --> 00:01:30.560 divided between singular, that is one person, 00:01:30.560 --> 00:01:33.600 and plural, or more than one person. 00:01:33.600 --> 00:01:37.990 Some first person pronouns are I, me, my, and mine. 00:01:37.990 --> 00:01:39.780 But what if there's more than one of me? 00:01:39.780 --> 00:01:40.920 What if I'm part of a group 00:01:40.920 --> 00:01:43.280 and I wanna refer to that group? 00:01:43.280 --> 00:01:44.840 Well, then I'd use a plural pronoun 00:01:44.840 --> 00:01:48.290 like we, us, our, or ours. 00:01:48.290 --> 00:01:49.960 Second and third person are interesting 00:01:49.960 --> 00:01:52.750 because they have pronouns that pull double duty. 00:01:52.750 --> 00:01:53.850 In second person, 00:01:53.850 --> 00:01:56.950 the singular and plural are identical. 00:01:56.950 --> 00:02:00.940 Singular second person is you, your, and yours. 00:02:00.940 --> 00:02:02.760 And plural second person is the same, 00:02:02.760 --> 00:02:04.940 you, your, and yours. 00:02:04.940 --> 00:02:06.810 That is to say, it's the same whether or not 00:02:06.810 --> 00:02:09.850 you're referring to one person, here in singular, 00:02:09.850 --> 00:02:12.420 or many people, here in plural. 00:02:12.420 --> 00:02:14.570 Understanding whether you meet one you 00:02:14.570 --> 00:02:16.910 or a plural you depends on context. 00:02:16.910 --> 00:02:19.900 And it's usually very easy to figure out. 00:02:19.900 --> 00:02:22.870 Third person pronouns belong to the most crowded category 00:02:22.870 --> 00:02:25.460 because the world is full of things and people 00:02:25.460 --> 00:02:27.290 who are neither me nor you. 00:02:27.290 --> 00:02:30.750 For singular pronouns we have she, her, and hers, 00:02:30.750 --> 00:02:33.080 he, him, and his, 00:02:33.080 --> 00:02:35.210 they, them, their and theirs 00:02:35.210 --> 00:02:37.660 and it, and its. 00:02:37.660 --> 00:02:41.100 They, like you, can refer to both a single person 00:02:41.100 --> 00:02:42.140 or multiple people, 00:02:42.140 --> 00:02:45.570 but it doesn't specify a gender like she and he do. 00:02:45.570 --> 00:02:47.190 This is extremely useful. 00:02:47.190 --> 00:02:49.010 I'd take note that the word it 00:02:49.010 --> 00:02:50.950 only refers to inanimate objects 00:02:50.950 --> 00:02:52.820 and sometimes non-human animals, 00:02:52.820 --> 00:02:54.370 but never to people. 00:02:54.370 --> 00:02:55.690 A robot? Yes. 00:02:55.690 --> 00:02:57.303 A person? Not so much. 00:02:58.180 --> 00:02:59.940 Now, in the plural category, much simpler, 00:02:59.940 --> 00:03:03.930 we have they, them, their, and theirs. 00:03:03.930 --> 00:03:06.430 Similar to singular versus plural you 00:03:06.430 --> 00:03:09.220 singular versus plural they depends on context 00:03:09.220 --> 00:03:11.400 and it'll be obvious from the words around it 00:03:11.400 --> 00:03:12.900 which one you mean. 00:03:12.900 --> 00:03:15.280 This is a lot of information to swallow. 00:03:15.280 --> 00:03:17.410 Pause this if you need to practice saying 00:03:17.410 --> 00:03:19.210 the different pronouns aloud, 00:03:19.210 --> 00:03:22.420 do the exercises on the Khan Academy site or app. 00:03:22.420 --> 00:03:24.380 This is not the only time I'll be talking 00:03:24.380 --> 00:03:26.330 about personal pronouns in this course. 00:03:26.330 --> 00:03:28.480 So we have an opportunity to go deeper, 00:03:28.480 --> 00:03:30.660 especially if you want to know more about the history 00:03:30.660 --> 00:03:33.623 of the singular use of they, which is super cool. 00:03:34.490 --> 00:03:35.540 In the meantime, 00:03:35.540 --> 00:03:38.240 please enjoy this second image of my dog, Phryne, 00:03:38.240 --> 00:03:41.930 who is, again, as I said, a perfect dog. 00:03:41.930 --> 00:03:43.540 You can learn anything. 00:03:43.540 --> 00:03:44.373 David out.

Cosine, sine and tangent of π/6 and π/3

https://www.youtube.com/watch?v=Tt_ATh5mCGw

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https://www.youtube.com/api/timedtext?v=Tt_ATh5mCGw&ei=5VWUZZqNNN2UvdIP4vuo8AQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=D4BAB4B60650BCBCC5A6DB7B7851B95F3E76C497.6FB3C3E040F8CD0B4F232B07392A277D84BF9759&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.330 --> 00:00:01.290 - [Instructor] In this video, 00:00:01.290 --> 00:00:05.870 we're going to figure out what the sine, cosine and tangent 00:00:05.870 --> 00:00:07.640 of two very important angles are. 00:00:07.640 --> 00:00:10.000 Angles that you will see a lot in your trigonometric 00:00:10.000 --> 00:00:12.610 and just in general in your regular life. 00:00:12.610 --> 00:00:13.860 So these are the angles, 00:00:13.860 --> 00:00:17.720 pi over 3 radians and pi over 6 radians. 00:00:17.720 --> 00:00:21.540 And sometimes it's useful to visualize them as degrees. 00:00:21.540 --> 00:00:25.030 pi over 3, you might remember pi radians is 180 degrees, 00:00:25.030 --> 00:00:26.690 so you divide that by three, 00:00:26.690 --> 00:00:28.111 this is equivalent to 60 degrees. 00:00:28.111 --> 00:00:30.760 And once again, 180 degrees, 00:00:30.760 --> 00:00:32.990 which is the same thing as pi radians divided by six 00:00:32.990 --> 00:00:35.500 is the same thing as 30 degrees. 00:00:35.500 --> 00:00:38.470 Now, I'm going to do it using the unit circle definition 00:00:38.470 --> 00:00:40.430 of trig functions. 00:00:40.430 --> 00:00:41.480 But to help us there, 00:00:41.480 --> 00:00:43.160 I'm going to give us a little bit of a reminder 00:00:43.160 --> 00:00:45.010 of what some of you might be familiar with 00:00:45.010 --> 00:00:46.980 as 30, 60, 90 triangles, 00:00:46.980 --> 00:00:48.660 which I guess we could also call pi over six, 00:00:48.660 --> 00:00:51.250 pi over three, pi over two triangles. 00:00:51.250 --> 00:00:53.420 And so let me just draw one 00:00:53.420 --> 00:00:55.400 because this is going to be really helpful 00:00:55.400 --> 00:00:57.960 in establishing these trig functions 00:00:57.960 --> 00:01:00.520 using the unit circle definition. 00:01:00.520 --> 00:01:03.190 So let me draw a triangle here, 00:01:03.190 --> 00:01:04.180 it's hand drawn, 00:01:04.180 --> 00:01:06.190 so it's not as neat as it could be. 00:01:06.190 --> 00:01:08.840 So this right over here is a right angle, 00:01:08.840 --> 00:01:13.430 and let's say that this one is pi over three radians 00:01:13.430 --> 00:01:15.640 which is the same thing as 60 degrees, 00:01:15.640 --> 00:01:18.890 and this one over here is pi over six radians 00:01:18.890 --> 00:01:21.890 which is the same thing as 30 degrees. 00:01:21.890 --> 00:01:24.330 Now, let's also say that the longest side, 00:01:24.330 --> 00:01:26.453 the hypotenuse here has length one. 00:01:27.460 --> 00:01:29.850 Now, to help us think about what the other two sides are, 00:01:29.850 --> 00:01:32.650 what I'm going to do is flip this triangle over 00:01:32.650 --> 00:01:33.960 this side right over here, 00:01:33.960 --> 00:01:36.980 and essentially construct a mirror image. 00:01:36.980 --> 00:01:41.150 So because this right over here is a mirror image, 00:01:41.150 --> 00:01:42.780 we immediately know a few things. 00:01:42.780 --> 00:01:44.600 We know that this length right over here 00:01:44.600 --> 00:01:48.040 is going to be congruent to this length over here. 00:01:48.040 --> 00:01:51.530 And let me actually finish drawing the entire triangle, 00:01:51.530 --> 00:01:53.920 it's going to look something like this. 00:01:53.920 --> 00:01:56.350 And since once again, it's a reflection, 00:01:56.350 --> 00:01:59.120 this length over here is going to have length one, 00:01:59.120 --> 00:02:03.250 this is going to be pi over six radians, 00:02:03.250 --> 00:02:07.610 this is going to be pi over three radians. 00:02:07.610 --> 00:02:11.350 So what else do we know about this larger triangle now? 00:02:11.350 --> 00:02:13.470 Well, we know it's an equilateral triangle. 00:02:13.470 --> 00:02:15.610 All the angles, pi over three radians, 00:02:15.610 --> 00:02:16.443 pi over three radians 00:02:16.443 --> 00:02:18.080 and if you add two pi over sixes together, 00:02:18.080 --> 00:02:19.440 you're going to get pi over three as well, 00:02:19.440 --> 00:02:22.540 so it's a 60 degree, 60 degree, 60 degree triangle. 00:02:22.540 --> 00:02:24.790 And so all the sides are going to have the same length, 00:02:24.790 --> 00:02:27.000 so it's going to be one, one and one. 00:02:27.000 --> 00:02:29.500 And if these two sides are congruent 00:02:29.500 --> 00:02:30.960 of the smaller triangles, 00:02:30.960 --> 00:02:32.600 of the smaller right triangles, 00:02:32.600 --> 00:02:35.310 well then this right over here must be one half, 00:02:35.310 --> 00:02:39.770 and then this right over here must be one half as well. 00:02:39.770 --> 00:02:41.310 Now, that's going to be useful 00:02:41.310 --> 00:02:44.900 for figuring out what this length right over here 00:02:44.900 --> 00:02:45.960 is going to be. 00:02:45.960 --> 00:02:48.450 Because we have two right triangles, 00:02:48.450 --> 00:02:49.360 we could use either one, 00:02:49.360 --> 00:02:51.600 but if we just use this bottom right triangle here 00:02:51.600 --> 00:02:53.530 the Pythagorean theorem tells us 00:02:53.530 --> 00:02:56.700 that one half squared, let's call this B, 00:02:56.700 --> 00:02:58.550 so plus B squared, 00:02:58.550 --> 00:02:59.990 I'm just pattern matching, 00:02:59.990 --> 00:03:02.440 A squared plus B squared is equal to C squared 00:03:02.440 --> 00:03:04.660 where C is the length of the hypotenuse, 00:03:04.660 --> 00:03:06.650 is equal to one squared. 00:03:06.650 --> 00:03:11.270 And so we get that one fourth plus B squared 00:03:11.270 --> 00:03:12.580 is equal to one 00:03:12.580 --> 00:03:14.970 or subtracting one fourth from both sides. 00:03:14.970 --> 00:03:17.540 B squared is equal to three-fourths, 00:03:17.540 --> 00:03:19.580 and then taking the principle root of both sides, 00:03:19.580 --> 00:03:23.750 we get B is equal to the square root of three over two. 00:03:23.750 --> 00:03:25.460 So just like that, we have figured out 00:03:25.460 --> 00:03:28.870 what all the lengths of this 30, 60, 90 triangle are. 00:03:28.870 --> 00:03:33.240 So B here is equal to square root of three over two. 00:03:33.240 --> 00:03:35.200 Now, I said this would be useful as we go into 00:03:35.200 --> 00:03:38.150 the unit circle definitions of sine, cosine and tangent. 00:03:38.150 --> 00:03:39.910 And we're about to see why. 00:03:39.910 --> 00:03:42.790 So here I have two different unit circles, 00:03:42.790 --> 00:03:45.200 I'm going to use one for each of these angles. 00:03:45.200 --> 00:03:48.310 So first, let's think about pi over three radians. 00:03:48.310 --> 00:03:50.340 And so pi over three, 00:03:50.340 --> 00:03:53.513 would look something like this, 00:03:54.710 --> 00:03:58.940 this is pi over three radians. 00:03:58.940 --> 00:04:02.240 And the cosine and sine can be determined 00:04:02.240 --> 00:04:04.860 by the X and Y coordinates of this point 00:04:04.860 --> 00:04:08.200 where this radius intersects the actual unit circle. 00:04:08.200 --> 00:04:10.887 The coordinates here are going to be cosine 00:04:10.887 --> 00:04:13.006 of pi over three radians 00:04:13.006 --> 00:04:15.839 and sine of pi over three radians. 00:04:16.810 --> 00:04:18.400 Or another way to think about it is, 00:04:18.400 --> 00:04:20.780 I can set up a 30, 60, 90 triangle here, 00:04:20.780 --> 00:04:22.740 so I'm going to drop a perpendicular. 00:04:22.740 --> 00:04:26.370 This would be 90 degrees or pi over two radians. 00:04:26.370 --> 00:04:27.740 And then this angle over here, 00:04:27.740 --> 00:04:30.260 this is 60, this is 90, this is going to be 30, 00:04:30.260 --> 00:04:31.320 or another way of thinking about it, 00:04:31.320 --> 00:04:34.390 it's going to be pi over six radians. 00:04:34.390 --> 00:04:36.650 It's going to be just like one of these triangles here. 00:04:36.650 --> 00:04:38.260 And so the X coordinate, 00:04:38.260 --> 00:04:41.460 which is going to be the same thing as the cosine 00:04:41.460 --> 00:04:42.650 of pi over three, 00:04:42.650 --> 00:04:46.740 is going to be the length of this side, right over here. 00:04:46.740 --> 00:04:48.320 Well, what's that going to be? 00:04:48.320 --> 00:04:50.470 Well, when your hypotenuse is one, 00:04:50.470 --> 00:04:52.270 we know that the shorter side, 00:04:52.270 --> 00:04:56.280 the side opposite the pi over six radians, is one half. 00:04:56.280 --> 00:04:58.390 So just like that, we have been able to establish 00:04:58.390 --> 00:05:03.310 that cosine of pi over three radians is equal to one half. 00:05:03.310 --> 00:05:04.920 This right over here is one half, 00:05:04.920 --> 00:05:06.410 that is the X coordinate 00:05:06.410 --> 00:05:09.140 where this radius intersects the units circle. 00:05:09.140 --> 00:05:11.540 Now, what about the Y coordinate? 00:05:11.540 --> 00:05:13.970 What is sine of pi over three going to be? 00:05:13.970 --> 00:05:15.710 Well, the Y coordinate is the same thing 00:05:15.710 --> 00:05:18.200 as the length of this side, 00:05:18.200 --> 00:05:21.100 and once again, it goes back to being this triangle. 00:05:21.100 --> 00:05:22.800 If this is one, this is one half, 00:05:22.800 --> 00:05:24.390 this is one, this is one half, 00:05:24.390 --> 00:05:26.330 this other side is going to be square root 00:05:26.330 --> 00:05:27.540 of three over two. 00:05:27.540 --> 00:05:29.140 So sine of pi over three 00:05:29.140 --> 00:05:31.780 is going to be square root of three over two, 00:05:31.780 --> 00:05:32.840 so let me write that down. 00:05:32.840 --> 00:05:35.610 Sine of pi over three 00:05:35.610 --> 00:05:37.840 is equal to square root of three over two. 00:05:37.840 --> 00:05:39.250 And these are good ones to know. 00:05:39.250 --> 00:05:41.400 I never say really memorize things, 00:05:41.400 --> 00:05:43.770 it's always good to know how to derive things 00:05:43.770 --> 00:05:45.340 in case you forget. 00:05:45.340 --> 00:05:47.200 But if you have to memorize them 00:05:47.200 --> 00:05:49.320 I would highly recommend memorizing these, 00:05:49.320 --> 00:05:52.320 and then of course from these we can figure out the tangent. 00:05:52.320 --> 00:05:54.470 The tangent is just going to be the sine over the cosine, 00:05:54.470 --> 00:05:57.210 so let me write it down here. 00:05:57.210 --> 00:06:01.840 The tangent of pi over three 00:06:01.840 --> 00:06:03.350 is going to be the sine, 00:06:03.350 --> 00:06:05.950 which is square root of three over two, 00:06:05.950 --> 00:06:09.130 over the cosine which is one half, 00:06:09.130 --> 00:06:10.550 got a little squanchy down there, 00:06:10.550 --> 00:06:12.093 and so this is just going to be 00:06:12.093 --> 00:06:13.760 square root of three over two times two 00:06:13.760 --> 00:06:16.860 is just going to be square root of three. 00:06:16.860 --> 00:06:19.780 So now let's just use that same logic for pi over six. 00:06:19.780 --> 00:06:21.410 And in fact, I encourage you to pause this video 00:06:21.410 --> 00:06:23.353 and see if you can do that on your own. 00:06:24.430 --> 00:06:25.840 All right, now let's draw a radius 00:06:25.840 --> 00:06:30.190 that forms a pi over six radian angle 00:06:30.190 --> 00:06:32.340 with a positive X axis, might look like that. 00:06:32.340 --> 00:06:35.670 So if that's going to be pi over six radians, 00:06:35.670 --> 00:06:38.330 you might imagine it's interesting 00:06:38.330 --> 00:06:40.590 to drop a perpendicular here 00:06:40.590 --> 00:06:43.030 and see what type of triangle we've constructed. 00:06:43.030 --> 00:06:44.380 So this has length one, 00:06:44.380 --> 00:06:46.240 this is pi over six radians, 00:06:46.240 --> 00:06:47.990 this is a right angle. 00:06:47.990 --> 00:06:50.960 So this again, is going to follow the same pattern. 00:06:50.960 --> 00:06:54.610 This will be pi over three radians. 00:06:54.610 --> 00:06:57.150 And so the sides are exactly the exact same 00:06:57.150 --> 00:06:59.910 as this top blue triangle here. 00:06:59.910 --> 00:07:02.280 So we know that this length over here 00:07:02.280 --> 00:07:04.610 is going to be one half. 00:07:04.610 --> 00:07:07.480 We know that this length over here 00:07:07.480 --> 00:07:11.670 is going to be square root of three over two. 00:07:11.670 --> 00:07:13.910 And that's useful because that tells us 00:07:13.910 --> 00:07:15.110 the coordinates here. 00:07:15.110 --> 00:07:16.370 The coordinates here, 00:07:16.370 --> 00:07:18.480 the X coordinate of this point where the radius 00:07:18.480 --> 00:07:20.060 intersects the unit circle 00:07:20.060 --> 00:07:22.840 is square root of three over two, 00:07:22.840 --> 00:07:26.450 and then the Y coordinate is one half. 00:07:26.450 --> 00:07:28.360 And that immediately tells us the cosine 00:07:28.360 --> 00:07:30.390 and the sine of pi over six, 00:07:30.390 --> 00:07:31.770 let's just write it down. 00:07:31.770 --> 00:07:35.350 So this tells us that cosine of pi over six 00:07:35.350 --> 00:07:38.490 is equal to square root of three over two. 00:07:38.490 --> 00:07:41.930 And sine of pi over six 00:07:43.230 --> 00:07:45.540 is equal to one half. 00:07:45.540 --> 00:07:48.530 Notice, we actually just swap these two things around 00:07:48.530 --> 00:07:50.380 because now the angle that we're taking 00:07:50.380 --> 00:07:51.530 the sine or cosine of, 00:07:51.530 --> 00:07:54.870 is a different angle on a 30, 60, 90 triangle, 00:07:54.870 --> 00:07:57.900 but we're essentially utilizing the same side measure, 00:07:57.900 --> 00:07:59.090 just one way to think about it. 00:07:59.090 --> 00:08:01.340 And then what's the tangent going to be? 00:08:01.340 --> 00:08:02.730 I'll write it down here. 00:08:02.730 --> 00:08:06.250 The tangent of pi over six 00:08:06.250 --> 00:08:09.520 is going to be the sine over the cosine 00:08:09.520 --> 00:08:11.660 square root of three over two, 00:08:11.660 --> 00:08:14.320 and so that's going to be equal to one half 00:08:14.320 --> 00:08:17.580 times two over the square root of three, 00:08:17.580 --> 00:08:21.150 which is equal to one over the square root of three. 00:08:21.150 --> 00:08:24.210 Now some people sometimes don't like radicals 00:08:24.210 --> 00:08:25.043 in the denominator 00:08:25.043 --> 00:08:27.220 and so you can multiply the numerator and the denominator 00:08:27.220 --> 00:08:29.210 by square root of three if you like 00:08:29.210 --> 00:08:30.270 to get something like this, 00:08:30.270 --> 00:08:31.580 you multiply the numerator and denominator 00:08:31.580 --> 00:08:32.413 by square root of three 00:08:32.413 --> 00:08:34.660 you get square root of three over three, 00:08:34.660 --> 00:08:37.560 which is another way of writing tangent of pi over six. 00:08:37.560 --> 00:08:38.880 But either way, we're done, 00:08:38.880 --> 00:08:41.820 it's very useful to know the cosine, sine and tangent 00:08:41.820 --> 00:08:44.570 of both pi over three and pi over six. 00:08:44.570 --> 00:08:46.823 And now you also know how to derive it.

Rule of 70 to approximate population doubling time

https://www.youtube.com/watch?v=KBXec1ctCjg

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https://www.youtube.com/api/timedtext?v=KBXec1ctCjg&ei=5VWUZZHiIue_p-oPsJWeCA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=30EEC74951D749CC05F44C38296773C691B83B7B.46C655B2019186020123BA0F22BF069C30FFC449&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.230 --> 00:00:02.063 - [Instructor] When we're dealing 00:00:02.063 --> 00:00:03.280 with population growth rates 00:00:03.280 --> 00:00:05.050 an interesting question is, 00:00:05.050 --> 00:00:06.440 how long would it take 00:00:06.440 --> 00:00:10.650 for a given rate for the population to double? 00:00:10.650 --> 00:00:13.120 So we're gonna think about doubling time. 00:00:13.120 --> 00:00:15.640 Now if you were to actually calculate it precisely 00:00:15.640 --> 00:00:19.020 mathematically precisely, it gets a little bit mathy. 00:00:19.020 --> 00:00:20.960 You need to use a little bit of logarithms 00:00:20.960 --> 00:00:23.220 and you'll probably need a calculator, 00:00:23.220 --> 00:00:25.070 but I did that here in this spreadsheet 00:00:25.070 --> 00:00:27.130 by calculating the exact doubling time. 00:00:27.130 --> 00:00:29.790 So this is saying that if a population is growing 00:00:29.790 --> 00:00:31.800 at 1% a year, 00:00:31.800 --> 00:00:34.750 it's going to take almost 70 years 00:00:34.750 --> 00:00:36.700 for that population to double. 00:00:36.700 --> 00:00:41.250 But if that population is growing at 5% per year 00:00:41.250 --> 00:00:44.130 then it's going to take a little over 14 years 00:00:44.130 --> 00:00:46.240 for that population to double. 00:00:46.240 --> 00:00:48.790 If the population is growing at 10%, 00:00:48.790 --> 00:00:51.160 we know mathematically it's going to take a little bit 00:00:51.160 --> 00:00:54.400 over seven years for that population to double. 00:00:54.400 --> 00:00:56.840 Now, I was able to calculate this as I just mentioned 00:00:56.840 --> 00:00:59.420 using a little bit of fancy math, 00:00:59.420 --> 00:01:01.000 but what we see in this next column 00:01:01.000 --> 00:01:03.120 is there's actually a pretty 00:01:03.120 --> 00:01:06.490 easy way to approximate doubling time. 00:01:06.490 --> 00:01:08.840 And this is known as the rule of 70. 00:01:08.840 --> 00:01:12.480 And the rule of 70 is used in a lot of different areas, 00:01:12.480 --> 00:01:13.550 a lot of different subjects, 00:01:13.550 --> 00:01:15.780 people in finance would use it because once again, 00:01:15.780 --> 00:01:17.710 you're thinking about things growing 00:01:17.710 --> 00:01:19.190 at a certain percent every year, 00:01:19.190 --> 00:01:20.530 but you can also use it 00:01:20.530 --> 00:01:23.540 for things like population growth rates. 00:01:23.540 --> 00:01:26.120 So what we see with the rule of 70, 00:01:26.120 --> 00:01:27.580 and let me just write that down, 00:01:27.580 --> 00:01:32.580 rule of 70 is that you can approximate the doubling time 00:01:32.930 --> 00:01:34.960 by taking the number 70 00:01:34.960 --> 00:01:38.890 and dividing it by the, not actually the percentage, 00:01:38.890 --> 00:01:41.110 but just the number of the percentage. 00:01:41.110 --> 00:01:42.480 So for example, 00:01:42.480 --> 00:01:47.480 this right over here is 70 divided by this one here, 00:01:48.920 --> 00:01:50.980 which is equal to 70. 00:01:50.980 --> 00:01:55.330 And notice this 70 is pretty close to 69.7. 00:01:55.330 --> 00:01:56.760 If you wanted to figure out 00:01:56.760 --> 00:01:59.750 or you wanted to approximate the doubling time, 00:01:59.750 --> 00:02:02.800 if the population is growing at 7% a year, 00:02:02.800 --> 00:02:04.330 well what you would say is, all right, 00:02:04.330 --> 00:02:08.090 what is 70 divided by seven? 00:02:08.090 --> 00:02:09.280 Well, that is equal to 10. 00:02:09.280 --> 00:02:10.950 So this would be your approximation. 00:02:10.950 --> 00:02:13.440 And if you were to do it in a mathematically precise way, 00:02:13.440 --> 00:02:15.400 it would be 10.2. 00:02:15.400 --> 00:02:17.400 So if you're taking, 00:02:17.400 --> 00:02:20.000 say an AP environmental science course 00:02:20.000 --> 00:02:22.040 and they're asking you for the, 00:02:22.040 --> 00:02:24.400 how long it takes for something to double 00:02:24.400 --> 00:02:27.150 let's say a population that's growing 7% a year, 00:02:27.150 --> 00:02:31.563 they're probably expecting you to use the rule of 70. 00:02:32.830 --> 00:02:35.370 So let's say that we have a population 00:02:35.370 --> 00:02:40.370 that is growing at 14% per year, 00:02:41.390 --> 00:02:44.320 and that would actually be a very huge growth rate. 00:02:44.320 --> 00:02:45.970 What I want you to do is pause this video 00:02:45.970 --> 00:02:48.210 and approximate how long would it take 00:02:48.210 --> 00:02:50.853 for that population to double? 00:02:52.460 --> 00:02:54.240 All right, now let's work through this together. 00:02:54.240 --> 00:02:56.210 So as I mentioned, we're approximating, 00:02:56.210 --> 00:02:58.380 we don't have to do calculate the exact doubling time. 00:02:58.380 --> 00:03:00.170 So if we're approximating, 00:03:00.170 --> 00:03:05.170 it's going to be 70 divided by the rate of growth. 00:03:05.330 --> 00:03:09.603 So in this situation, this is going to be 70 divided by 14, 00:03:11.240 --> 00:03:13.600 which is equal to five. 00:03:13.600 --> 00:03:15.550 So if a population is growing at 14%, 00:03:15.550 --> 00:03:18.863 it'll take it roughly five years to double.

Worked example: Using bond enthalpies to calculate enthalpy of reaction

https://www.youtube.com/watch?v=TdBY-so0H0Q

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en

WEBVTT Kind: captions Language: en 00:00:00.510 --> 00:00:02.330 - [Educator] Bond enthalpies can be used 00:00:02.330 --> 00:00:05.100 to estimate the standard change in enthalpy 00:00:05.100 --> 00:00:07.490 for a chemical reaction. 00:00:07.490 --> 00:00:08.900 Let's use bond enthalpies 00:00:08.900 --> 00:00:11.740 to estimate the enthalpy of combustion 00:00:11.740 --> 00:00:13.100 of ethanol. 00:00:13.100 --> 00:00:14.510 Looking at our balanced equation, 00:00:14.510 --> 00:00:16.840 we have one mole of ethanol reacting 00:00:16.840 --> 00:00:18.920 with three moles of oxygen gas 00:00:18.920 --> 00:00:21.600 to produce two moles of carbon dioxide 00:00:21.600 --> 00:00:25.990 and three moles of water in the gaseous state. 00:00:25.990 --> 00:00:27.720 To find the standard change in enthalpy 00:00:27.720 --> 00:00:29.110 for this chemical reaction, 00:00:29.110 --> 00:00:31.980 we need to sum the bond enthalpies 00:00:31.980 --> 00:00:33.900 of the bonds that are broken. 00:00:33.900 --> 00:00:36.400 And from that, we subtract the sum 00:00:36.400 --> 00:00:38.670 of the bond enthalpies of the bonds 00:00:38.670 --> 00:00:41.550 that are formed in this chemical reaction. 00:00:41.550 --> 00:00:43.170 To figure out which bonds are broken 00:00:43.170 --> 00:00:44.450 and which bonds are formed, 00:00:44.450 --> 00:00:46.590 it's helpful to look at the dot structures 00:00:46.590 --> 00:00:48.130 for our molecules. 00:00:48.130 --> 00:00:51.180 So let's start with the ethanol molecule. 00:00:51.180 --> 00:00:52.770 We're gonna approach this problem first 00:00:52.770 --> 00:00:55.840 like we're breaking all of the bonds in these molecules. 00:00:55.840 --> 00:00:58.570 And we're also not gonna worry about units until the end, 00:00:58.570 --> 00:01:01.160 just to save some space on the screen. 00:01:01.160 --> 00:01:03.130 So looking at the ethanol molecule, 00:01:03.130 --> 00:01:06.820 we would need to break a carbon-carbon bond. 00:01:06.820 --> 00:01:08.530 So let's go ahead and write this down here. 00:01:08.530 --> 00:01:10.850 Right now, we're summing the bond enthalpies 00:01:10.850 --> 00:01:12.860 of the bonds that are broken. 00:01:12.860 --> 00:01:15.490 So we have one carbon-carbon bond. 00:01:15.490 --> 00:01:16.323 So let's write in here, 00:01:16.323 --> 00:01:20.590 the bond enthalpy for a carbon-carbon bond. 00:01:20.590 --> 00:01:24.630 Next, we have five carbon-hydrogen bonds 00:01:24.630 --> 00:01:26.390 that we need to break. 00:01:26.390 --> 00:01:27.640 So to this, 00:01:27.640 --> 00:01:30.140 we're going to write in here, a five, 00:01:30.140 --> 00:01:32.740 and then the bond enthalpy 00:01:32.740 --> 00:01:35.920 of a carbon-hydrogen bond. 00:01:35.920 --> 00:01:39.820 Next, we have to break a carbon-oxygen single bond. 00:01:39.820 --> 00:01:41.840 So we write a one, 00:01:41.840 --> 00:01:43.930 and then the bond enthalpy 00:01:43.930 --> 00:01:47.310 for a carbon-oxygen single bond. 00:01:47.310 --> 00:01:49.260 And then for this ethanol molecule, 00:01:49.260 --> 00:01:52.650 we also have an oxygen-hydrogen single bond. 00:01:52.650 --> 00:01:54.850 So we'll write in here, a one, 00:01:54.850 --> 00:01:56.880 and the bond enthalpy 00:01:56.880 --> 00:01:59.920 for an oxygen-hydrogen single bond. 00:01:59.920 --> 00:02:01.150 We saw in the balanced equation 00:02:01.150 --> 00:02:02.870 that one mole of ethanol reacts 00:02:02.870 --> 00:02:05.310 with three moles of oxygen gas. 00:02:05.310 --> 00:02:07.550 So to represent the three moles of oxygen gas, 00:02:07.550 --> 00:02:08.500 I've drawn in here, 00:02:08.500 --> 00:02:11.370 three molecules of O2. 00:02:11.370 --> 00:02:13.970 And we can see in each molecule of O2, 00:02:13.970 --> 00:02:16.160 there's an oxygen-oxygen double bond. 00:02:16.160 --> 00:02:20.610 So we would need to break three oxygen-oxygen double bonds. 00:02:20.610 --> 00:02:21.840 So to this, 00:02:21.840 --> 00:02:26.840 we're going to add a three times the bond enthalpy 00:02:27.140 --> 00:02:30.450 of an oxygen-oxygen double bond. 00:02:30.450 --> 00:02:34.650 And this now gives us the sum of the bond enthalpies 00:02:34.650 --> 00:02:37.360 for all the bonds that need to be broken. 00:02:37.360 --> 00:02:39.240 It takes energy to break a bond. 00:02:39.240 --> 00:02:41.340 So the summation of the bond enthalpies 00:02:41.340 --> 00:02:42.720 of the bonds that are broken 00:02:42.720 --> 00:02:45.230 is going to be a positive value. 00:02:45.230 --> 00:02:47.130 And since it takes energy to break bonds, 00:02:47.130 --> 00:02:49.580 energy is given off when bonds form. 00:02:49.580 --> 00:02:51.950 So next, we're gonna sum the bond enthalpies 00:02:51.950 --> 00:02:53.390 of the bonds that are formed. 00:02:53.390 --> 00:02:55.900 And notice we have this negative sign in here 00:02:55.900 --> 00:02:58.160 because this energy is given off. 00:02:58.160 --> 00:03:00.320 So we're gonna write a minus sign in here, 00:03:00.320 --> 00:03:02.910 and then we're gonna put some brackets 00:03:02.910 --> 00:03:05.620 because next we're going to sum the bond enthalpies 00:03:05.620 --> 00:03:07.580 of the bonds that are formed. 00:03:07.580 --> 00:03:08.550 In our balanced equation, 00:03:08.550 --> 00:03:12.250 we formed two moles of carbon dioxide. 00:03:12.250 --> 00:03:14.530 So to represent those two moles, 00:03:14.530 --> 00:03:15.400 I've drawn in here, 00:03:15.400 --> 00:03:18.620 two molecules of CO2. 00:03:18.620 --> 00:03:22.110 And we can see that in each molecule of CO2, 00:03:22.110 --> 00:03:27.110 we're going to form two carbon-oxygen double bonds. 00:03:27.540 --> 00:03:31.350 So that's a total of four carbon-oxygen double bonds. 00:03:31.350 --> 00:03:32.780 So down here, 00:03:32.780 --> 00:03:37.780 we're going to write a four times the bond enthalpy 00:03:38.020 --> 00:03:41.890 of a carbon-oxygen double bond. 00:03:41.890 --> 00:03:44.910 We also formed three moles of H2O. 00:03:44.910 --> 00:03:47.580 And in each molecule of water that's drawn here, 00:03:47.580 --> 00:03:51.950 we form two oxygen-hydrogen single bonds. 00:03:51.950 --> 00:03:54.290 And since we have three moles, 00:03:54.290 --> 00:03:58.340 we have a total of six oxygen-hydrogen single bonds. 00:03:58.340 --> 00:03:59.173 So to this, 00:03:59.173 --> 00:04:03.900 we're going to add six times the bond enthalpy 00:04:03.900 --> 00:04:07.623 of an oxygen-hydrogen single bond. 00:04:08.810 --> 00:04:11.120 The next step is to look up the bond enthalpies 00:04:11.120 --> 00:04:13.090 of all of these different bonds. 00:04:13.090 --> 00:04:14.640 For example, the bond enthalpy 00:04:14.640 --> 00:04:19.480 for a carbon-carbon single bond is about 348 kilojoules 00:04:19.480 --> 00:04:20.313 per mole. 00:04:20.313 --> 00:04:21.540 You might see a different value, 00:04:21.540 --> 00:04:23.620 if you look in a different textbook. 00:04:23.620 --> 00:04:26.640 However, we're gonna go with 348 kilojoules per mole 00:04:26.640 --> 00:04:28.350 for our calculation. 00:04:28.350 --> 00:04:29.450 And we're gonna multiply this 00:04:29.450 --> 00:04:32.960 by one mole of carbon-carbon single bonds. 00:04:32.960 --> 00:04:34.470 Next, we look up the bond enthalpy 00:04:34.470 --> 00:04:36.640 for our carbon-hydrogen single bond. 00:04:36.640 --> 00:04:39.840 And that's about 413 kilojoules per mole 00:04:39.840 --> 00:04:41.390 of carbon-hydrogen bonds. 00:04:41.390 --> 00:04:43.060 And we're multiplying this by five. 00:04:43.060 --> 00:04:45.560 And we continue with everything else 00:04:45.560 --> 00:04:47.820 for the summation of the the bond enthalpies 00:04:47.820 --> 00:04:49.560 of the bonds broken. 00:04:49.560 --> 00:04:51.000 When we do this, 00:04:51.000 --> 00:04:56.000 we get positive 4,719 kilojoules. 00:04:56.530 --> 00:04:57.670 Next, we do the same thing 00:04:57.670 --> 00:04:59.430 for the bond enthalpies 00:04:59.430 --> 00:05:00.590 of the bonds that are formed. 00:05:00.590 --> 00:05:01.450 So the bond enthalpy 00:05:01.450 --> 00:05:05.460 for our carbon-oxygen double bond is 799 kilojoules 00:05:05.460 --> 00:05:06.293 per mole, 00:05:06.293 --> 00:05:08.020 and we multiply that by four. 00:05:08.020 --> 00:05:10.760 The bonds enthalpy for an oxygen hydrogen single bond 00:05:10.760 --> 00:05:13.030 is 463 kilojoules per mole, 00:05:13.030 --> 00:05:14.940 and we multiply that by six. 00:05:14.940 --> 00:05:15.880 When we add these together, 00:05:15.880 --> 00:05:20.350 we get 5,974. 00:05:20.350 --> 00:05:23.210 So for the final standard change in enthalpy 00:05:23.210 --> 00:05:25.090 for our chemical reaction, 00:05:25.090 --> 00:05:30.090 it's positive 4,719 minus 5,974, 00:05:31.540 --> 00:05:36.540 which gives us negative 1,255 kilojoules. 00:05:37.280 --> 00:05:39.460 Notice that we got a negative value 00:05:39.460 --> 00:05:41.420 for the change in enthalpy. 00:05:41.420 --> 00:05:43.860 And that means the combustion of ethanol 00:05:43.860 --> 00:05:46.160 is an exothermic reaction. 00:05:46.160 --> 00:05:50.840 And 1,255 kilojoules of energy are given off 00:05:50.840 --> 00:05:52.880 for the combustion of one mole 00:05:52.880 --> 00:05:54.390 of ethanol. 00:05:54.390 --> 00:05:56.750 Also notice that the sum of the bond enthalpies 00:05:56.750 --> 00:05:57.950 of the bonds formed, 00:05:57.950 --> 00:06:01.040 which is 5,974, 00:06:01.040 --> 00:06:03.790 is greater than the sum of the bond enthalpies 00:06:03.790 --> 00:06:04.830 of the bonds broken, 00:06:04.830 --> 00:06:07.480 which is 4,719. 00:06:07.480 --> 00:06:10.270 And since we're subtracting a larger number 00:06:10.270 --> 00:06:11.500 from a smaller number, 00:06:11.500 --> 00:06:12.980 we get that negative sign 00:06:12.980 --> 00:06:15.240 for the change in enthalpy. 00:06:15.240 --> 00:06:17.160 If the sum of the bond enthalpies 00:06:17.160 --> 00:06:18.480 of the bonds that are broken, 00:06:18.480 --> 00:06:21.180 if this number is larger than the sum 00:06:21.180 --> 00:06:22.030 of the bond enthalpies 00:06:22.030 --> 00:06:23.120 of the bonds that have formed, 00:06:23.120 --> 00:06:24.700 we would've gotten a positive value 00:06:24.700 --> 00:06:26.480 for the change in enthalpy. 00:06:26.480 --> 00:06:29.950 And that would be true for an endothermic reaction. 00:06:29.950 --> 00:06:30.783 We did this problem, 00:06:30.783 --> 00:06:32.290 assuming that all of the bonds 00:06:32.290 --> 00:06:34.540 that we drew in our dots structures were broken 00:06:34.540 --> 00:06:35.470 and all of the bonds 00:06:35.470 --> 00:06:37.890 that we drew in the dot structures were formed. 00:06:37.890 --> 00:06:39.950 However, if we look closely to dots structures 00:06:39.950 --> 00:06:41.810 or just look closely to what we wrote here, 00:06:41.810 --> 00:06:46.810 we show breaking one oxygen-hydrogen single bonds over here, 00:06:47.640 --> 00:06:48.780 and we show the formation 00:06:48.780 --> 00:06:53.270 of six oxygen-hydrogen single bonds over here. 00:06:53.270 --> 00:06:55.320 So we could have just canceled out one 00:06:55.320 --> 00:06:57.840 of those oxygen-hydrogen single bonds. 00:06:57.840 --> 00:06:59.600 So we could have canceled this out. 00:06:59.600 --> 00:07:01.300 And instead of showing a six here, 00:07:01.300 --> 00:07:04.920 we could have written a five times the bond enthalpy 00:07:04.920 --> 00:07:08.520 of an oxygen-hydrogen single bond. 00:07:08.520 --> 00:07:10.630 We still would have ended up with the same answer 00:07:10.630 --> 00:07:14.250 of negative 1,255 kilojoules. 00:07:14.250 --> 00:07:15.910 So if you look at your dot structures, 00:07:15.910 --> 00:07:18.930 if you see a bond that's the same on the reactant side 00:07:18.930 --> 00:07:21.116 and the same on the product side, 00:07:21.116 --> 00:07:23.640 you don't have to show the breaking 00:07:23.640 --> 00:07:25.360 and forming of that bond. 00:07:25.360 --> 00:07:27.410 You can make the problem a little bit shorter, 00:07:27.410 --> 00:07:28.373 if you want to. 00:07:29.280 --> 00:07:31.400 Finally, let's show how we get our units. 00:07:31.400 --> 00:07:34.420 So to get kilojoules as your final answer, 00:07:34.420 --> 00:07:35.790 if we go back up to here, 00:07:35.790 --> 00:07:38.770 we wrote a one times 348. 00:07:38.770 --> 00:07:42.400 The one is referring to breaking one mole 00:07:42.400 --> 00:07:44.870 of carbon-carbon single bonds. 00:07:44.870 --> 00:07:46.690 And the 348, of course, 00:07:46.690 --> 00:07:48.160 is the bond enthalpy 00:07:48.160 --> 00:07:50.030 for a carbon-carbon single bond. 00:07:50.030 --> 00:07:54.340 So this was 348 kilojoules per one mole 00:07:54.340 --> 00:07:57.220 of carbon-carbon single bonds. 00:07:57.220 --> 00:07:59.140 When you multiply these two together, 00:07:59.140 --> 00:08:02.530 the moles of carbon-carbon single bonds cancels 00:08:02.530 --> 00:08:06.317 and this gives you 348 kilojoules. 00:08:06.317 --> 00:08:07.630 And so, that's how to end up 00:08:07.630 --> 00:08:11.270 with kilojoules as your final answer. 00:08:11.270 --> 00:08:14.880 You also might see kilojoules per mole of reaction 00:08:14.880 --> 00:08:16.880 as the units for this. 00:08:16.880 --> 00:08:19.580 And, kilojoules per mole reaction means 00:08:19.580 --> 00:08:21.510 how the reaction is written. 00:08:21.510 --> 00:08:22.670 So for the combustion 00:08:22.670 --> 00:08:24.780 of one mole of ethanol, 00:08:24.780 --> 00:08:29.720 1,255 kilojoules of energy are released. 00:08:29.720 --> 00:08:33.980 To get kilojoules per mole of reaction as our units, 00:08:33.980 --> 00:08:36.620 the balanced equation had a one as the coefficient 00:08:36.620 --> 00:08:38.630 in front of ethanol. 00:08:38.630 --> 00:08:41.070 Therefore, you're breaking one mole 00:08:41.070 --> 00:08:45.080 of carbon-carbon single bonds per one mole 00:08:45.080 --> 00:08:46.160 of reaction. 00:08:46.160 --> 00:08:48.950 So we can use this conversion factor. 00:08:48.950 --> 00:08:51.410 Now, when we multiply through the moles 00:08:51.410 --> 00:08:52.900 of carbon-carbon single bonds, 00:08:52.900 --> 00:08:57.900 cancel and this gives us 348 kilojoules per mole 00:08:59.000 --> 00:09:00.533 of reaction.

Bond enthalpies

https://www.youtube.com/watch?v=VxcqAIaO-cA

vtt

https://www.youtube.com/api/timedtext?v=VxcqAIaO-cA&ei=5VWUZeOOJJLXxN8Pz9idoA0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=4C1532868C4A8BEC4C1DF614961B372A98739079.1AA6FD6E7CCAC430663C4EF3220534A4C8D376C3&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.960 --> 00:00:03.300 - [Instructor] Bond enthalpy is the change in enthalpy 00:00:03.300 --> 00:00:06.200 or delta H for breaking a particular bond 00:00:06.200 --> 00:00:08.710 in one mole of a gaseous substance. 00:00:08.710 --> 00:00:11.040 If we think about the diatomic chlorine molecule, 00:00:11.040 --> 00:00:14.280 so Cl2, down here is a little picture of Cl2, 00:00:14.280 --> 00:00:16.540 each of the green spheres is a chlorine atom 00:00:16.540 --> 00:00:19.510 and they're bonded together by a single covalent bond. 00:00:19.510 --> 00:00:21.990 It would take energy to break this bond 00:00:21.990 --> 00:00:23.840 in diatomic chlorine gas 00:00:23.840 --> 00:00:26.570 and turn diatomic chlorine gas, Cl2 00:00:26.570 --> 00:00:29.210 in to two individual chlorine atoms. 00:00:29.210 --> 00:00:33.580 So we're going from Cl2 in the gaseous state to 2Cl. 00:00:33.580 --> 00:00:37.280 Bond enthalpy can be symbolized by the letters BE. 00:00:37.280 --> 00:00:41.100 So the bond enthalpy of the chlorine-chlorine single bond 00:00:41.100 --> 00:00:45.900 is equal to +242 kilojoules per mole. 00:00:45.900 --> 00:00:46.810 And what this means is, 00:00:46.810 --> 00:00:49.160 if we have one mole of chlorine-chlorine bonds, 00:00:49.160 --> 00:00:54.160 it takes +242 kilojoules of energy to break those bonds. 00:00:55.008 --> 00:00:57.150 Bond enthalpies are always positive 00:00:57.150 --> 00:01:00.370 because it takes energy to break bonds. 00:01:00.370 --> 00:01:04.270 Another name for bond enthalpy is bond dissociation energy. 00:01:04.270 --> 00:01:07.600 So you might see this symbolized as BDE 00:01:07.600 --> 00:01:10.800 or just simply the letter D. 00:01:10.800 --> 00:01:12.680 Bond enthalpies are often found 00:01:12.680 --> 00:01:15.965 in the appendices of chemistry textbooks. 00:01:15.965 --> 00:01:19.610 For example, we just saw the chlorine-chlorine single bond, 00:01:19.610 --> 00:01:23.290 the bond enthalpy is 242 kilojoules per mole. 00:01:23.290 --> 00:01:26.160 Whereas to break a carbon-carbon single bond 00:01:26.160 --> 00:01:29.300 takes 348 kilojoules of energy per mole 00:01:29.300 --> 00:01:31.620 of carbon-carbon single bonds. 00:01:31.620 --> 00:01:34.100 A carbon-carbon double bond has a bond enthalpy 00:01:34.100 --> 00:01:37.310 of 614 kilojoules per mole. 00:01:37.310 --> 00:01:40.320 Since the carbon-carbon double bond is stronger 00:01:40.320 --> 00:01:42.000 than a carbon-carbon single bond, 00:01:42.000 --> 00:01:44.910 it takes more energy to break the double bond. 00:01:44.910 --> 00:01:47.010 And that's why the carbon-carbon double bond 00:01:47.010 --> 00:01:50.320 has a high higher bond enthalpy. 00:01:50.320 --> 00:01:52.460 So the higher the value for the bond enthalpy, 00:01:52.460 --> 00:01:54.410 the stronger the bond. 00:01:54.410 --> 00:01:56.810 Notice that these are average bond enthalpies. 00:01:56.810 --> 00:01:58.400 So the average bond enthalpy 00:01:58.400 --> 00:02:00.480 for a carbon-carbon single bond 00:02:00.480 --> 00:02:03.530 is around 348 kilojoules per mole. 00:02:03.530 --> 00:02:05.920 You might see slightly different values for this, 00:02:05.920 --> 00:02:08.100 depending on which textbook you're looking in, 00:02:08.100 --> 00:02:10.700 but they're all pretty close to the same value. 00:02:10.700 --> 00:02:12.905 The reason why these are average bond enthalpies 00:02:12.905 --> 00:02:16.640 is because if we look at two different molecules down here, 00:02:16.640 --> 00:02:20.060 this is ethane on the left and propane on the right, 00:02:20.060 --> 00:02:23.250 if we break a carbon-carbon single bond in ethane, 00:02:23.250 --> 00:02:25.460 the bond enthalpy is slightly different 00:02:25.460 --> 00:02:28.580 from breaking a carbon-carbon single bond in propane. 00:02:28.580 --> 00:02:32.830 And that's why we use average bond enthalpies. 00:02:32.830 --> 00:02:35.740 We've already seen that it takes energy to break bonds. 00:02:35.740 --> 00:02:39.100 So to break the chlorine-chlorine single bond 00:02:39.100 --> 00:02:44.100 in diatomic chlorine gas takes +242 kilojoules per mole. 00:02:44.500 --> 00:02:46.320 If it takes energy to break bonds, 00:02:46.320 --> 00:02:49.890 that means energy is given off when bonds form. 00:02:49.890 --> 00:02:53.510 So when two individual chlorine gas atoms come together 00:02:53.510 --> 00:02:56.740 to form a chlorine-chlorine bond, 00:02:56.740 --> 00:02:58.270 so let's go into highlight that in here. 00:02:58.270 --> 00:03:02.020 So this bond is forming, energy is given off. 00:03:02.020 --> 00:03:06.310 The magnitude of energy is still 242 kilojoules per mole, 00:03:06.310 --> 00:03:08.700 however, now we have this negative sign in here 00:03:08.700 --> 00:03:12.660 to indicate the energy is given off when bonds form. 00:03:12.660 --> 00:03:14.660 Bond enthalpies can be used to estimate 00:03:14.660 --> 00:03:16.240 enthalpies of reactions. 00:03:16.240 --> 00:03:17.970 So to find the change in the enthalpy 00:03:17.970 --> 00:03:20.577 for a chemical reaction, you take the sum 00:03:20.577 --> 00:03:23.870 of the bond enthalpies of the bonds broken. 00:03:23.870 --> 00:03:26.120 And from that you subtract the sum 00:03:26.120 --> 00:03:29.400 of the bond enthalpies of the bonds formed. 00:03:29.400 --> 00:03:31.240 The minus sign is in there because energy 00:03:31.240 --> 00:03:33.920 is given off when bonds form. 00:03:33.920 --> 00:03:35.940 A good way to remember this equation 00:03:35.940 --> 00:03:40.020 is to remember that B comes before F in the alphabet. 00:03:40.020 --> 00:03:42.700 So B before F therefore it's bonds broken 00:03:42.700 --> 00:03:44.860 minus bonds formed. 00:03:44.860 --> 00:03:47.100 Let's use bond enthalpies to estimate 00:03:47.100 --> 00:03:50.400 the enthalpy of reaction for the following reaction here, 00:03:50.400 --> 00:03:54.380 methane with chlorine gas to form chloro methane 00:03:54.380 --> 00:03:56.950 and hydrogen chloride gas. 00:03:56.950 --> 00:03:59.040 It's often helpful to draw dot structures 00:03:59.040 --> 00:04:00.690 for these kinds of problems. 00:04:00.690 --> 00:04:03.330 If we look at the methane dot structure, 00:04:03.330 --> 00:04:07.670 we would need to break one carbon-hydrogen single bond 00:04:07.670 --> 00:04:09.350 in order to get to our products. 00:04:09.350 --> 00:04:13.220 We would also need to break a chlorine-chlorine single bond. 00:04:13.220 --> 00:04:18.130 Next, one of the chlorine goes over to the CH3 00:04:18.130 --> 00:04:19.200 to form CH3Cl. 00:04:19.200 --> 00:04:21.920 So therefore we are forming 00:04:21.920 --> 00:04:24.830 one carbon-chlorine single bonds, 00:04:24.830 --> 00:04:27.480 and the other Cl goes with the hydrogen. 00:04:27.480 --> 00:04:32.090 So we also need to form one hydrogen-chlorine single bond. 00:04:32.090 --> 00:04:35.150 The next step is to sum the bond enthalpies 00:04:35.150 --> 00:04:37.200 of the bonds broken. 00:04:37.200 --> 00:04:38.330 So let's think about this. 00:04:38.330 --> 00:04:40.430 For our reactants we're breaking bonds. 00:04:40.430 --> 00:04:42.170 So we have one mole of methane 00:04:42.170 --> 00:04:45.170 reacting with one mole of chlorine. 00:04:45.170 --> 00:04:48.840 And since we're breaking one carbon-hydrogen single bond 00:04:48.840 --> 00:04:51.000 for every one molecule of methane, 00:04:51.000 --> 00:04:53.670 since we have one mole of methane molecules, 00:04:53.670 --> 00:04:58.330 we're breaking one mole of carbon-hydrogen single bonds. 00:04:58.330 --> 00:05:00.070 Therefore we can write down here, 00:05:00.070 --> 00:05:03.660 one mole of carbon-hydrogen bonds, 00:05:03.660 --> 00:05:07.720 and the bond enthalpy for a carbon-hydrogen single bond 00:05:07.720 --> 00:05:12.323 is 413 kilojoules per mole. 00:05:14.510 --> 00:05:17.520 Since there's one chlorine-chlorine single bond 00:05:17.520 --> 00:05:19.540 for every Cl2 molecule, 00:05:19.540 --> 00:05:22.630 and we have one mole of chlorine molecules, 00:05:22.630 --> 00:05:26.120 we're breaking one mole of chlorine-chlorine single bonds. 00:05:26.120 --> 00:05:28.830 So to this we're gonna add one mole 00:05:28.830 --> 00:05:31.170 of chlorine-chlorine single bonds. 00:05:31.170 --> 00:05:34.860 And the bond enthalpy for a chlorine-chlorine single bond 00:05:34.860 --> 00:05:39.363 is 242 kilojoules per mole. 00:05:40.550 --> 00:05:43.700 Moles cancel out and we get that the sum 00:05:43.700 --> 00:05:46.050 of the bond enthalpies of the bonds broken 00:05:46.050 --> 00:05:51.050 is equal to 655 kilojoules. 00:05:51.240 --> 00:05:53.081 Next, we need to sum the bond enthalpies 00:05:53.081 --> 00:05:54.950 of the bonds formed. 00:05:54.950 --> 00:05:58.150 So we're forming one mole of chloro methane 00:05:58.150 --> 00:06:01.970 and one mole of hydrogen chloride gas. 00:06:01.970 --> 00:06:05.840 And since we're forming one carbon-chlorine single bond 00:06:05.840 --> 00:06:08.620 for every molecule of chloro methane, 00:06:08.620 --> 00:06:10.900 since we're forming one mole of chloral methane, 00:06:10.900 --> 00:06:15.070 we're reforming one mole of carbon-chlorine single bonds. 00:06:15.070 --> 00:06:17.640 So let's write down here, we're forming one mole 00:06:17.640 --> 00:06:21.014 of carbon-chlorine bonds and the bond enthalpy 00:06:21.014 --> 00:06:23.550 for a carbon-chlorine single bond 00:06:23.550 --> 00:06:28.550 is equal to 328 kilojoules per mole. 00:06:30.980 --> 00:06:34.112 And since we form one hydrogen-chlorine single bond 00:06:34.112 --> 00:06:36.760 for every molecule of hydrogen chloride, 00:06:36.760 --> 00:06:39.370 since we're making one mole of hydrogen chloride, 00:06:39.370 --> 00:06:43.310 we're forming one mole of hydrogen-chlorine single bonds. 00:06:43.310 --> 00:06:46.140 So to this we add one mole 00:06:46.140 --> 00:06:50.120 of hydrogen-chlorine single bonds and the bond enthalpy 00:06:50.120 --> 00:06:52.080 for a hydrogen-chlorine single bond 00:06:52.080 --> 00:06:57.063 is 431 kilojoules per mole. 00:06:58.660 --> 00:07:02.530 Moles cancel and we get that the sum of the bond enthalpies 00:07:02.530 --> 00:07:07.530 of the bonds formed is equal to 759 kilojoules. 00:07:09.690 --> 00:07:11.966 Next, we're ready to find the change in enthalpy 00:07:11.966 --> 00:07:14.720 for our chemical reaction. 00:07:14.720 --> 00:07:18.020 The sum of the enthalpy of the bonds broken, 00:07:18.020 --> 00:07:21.670 we found that was equal to 655 kilojoules. 00:07:21.670 --> 00:07:24.049 And from that we subtract the sum of the bond enthalpies 00:07:24.049 --> 00:07:29.049 of the bonds formed, which we found was 759 kilojoules. 00:07:29.420 --> 00:07:34.420 So 655 minus 759 gives -104 kilojoules. 00:07:37.430 --> 00:07:40.710 Sometimes we see kilojoules or kilojoules per mole 00:07:40.710 --> 00:07:43.330 or kilojoules per mole of reaction. 00:07:43.330 --> 00:07:45.460 Kilojoules per mole of reaction just means 00:07:45.460 --> 00:07:48.180 how the balanced equation is written. 00:07:48.180 --> 00:07:50.170 And let's see how we can look at the units 00:07:50.170 --> 00:07:52.570 to get kilojoules per mole of reaction 00:07:52.570 --> 00:07:54.680 when we do the calculations. 00:07:54.680 --> 00:07:56.190 If you go back to breaking 00:07:56.190 --> 00:08:00.200 the carbon hydrogen bond over here, 00:08:00.200 --> 00:08:03.780 we've seen there's one mole of carbon-hydrogen bonds 00:08:03.780 --> 00:08:06.950 that we need to break for how the equation is written. 00:08:06.950 --> 00:08:09.030 Therefore, we can write a conversion factor 00:08:09.030 --> 00:08:11.320 of one mole of carbon-hydrogen bonds 00:08:11.320 --> 00:08:14.160 per one mole of reaction as it's written. 00:08:14.160 --> 00:08:16.900 And then we multiply that by the bond enthalpy 00:08:16.900 --> 00:08:21.190 as 413 kilojoules per mole for a carbon-hydrogen bond, 00:08:21.190 --> 00:08:24.120 this cancels out moles of carbon-hydrogen bonds 00:08:24.120 --> 00:08:27.010 and this gives us kilojoules per mole of reaction 00:08:27.010 --> 00:08:28.480 as our units. 00:08:28.480 --> 00:08:30.530 So it's more time consuming to write it this way 00:08:30.530 --> 00:08:31.860 but we could do that for all 00:08:31.860 --> 00:08:33.540 of our different bond enthalpies 00:08:33.540 --> 00:08:35.450 to get kilojoules per mole of reaction 00:08:35.450 --> 00:08:37.600 for the units for our final answer. 00:08:37.600 --> 00:08:39.300 When everything is under standard conditions, 00:08:39.300 --> 00:08:41.400 we need to add a superscript of note. 00:08:41.400 --> 00:08:44.770 So this would be the standard change in enthalpy 00:08:44.770 --> 00:08:46.930 for a chemical reaction. 00:08:46.930 --> 00:08:48.740 So for the value we just calculated, 00:08:48.740 --> 00:08:52.520 - 140 kilojoules per mole of reaction, 00:08:52.520 --> 00:08:53.870 this is under standard of conditions. 00:08:53.870 --> 00:08:56.330 So this is actually the standard change in enthalpy 00:08:56.330 --> 00:08:58.850 for this chemical reaction. 00:08:58.850 --> 00:09:01.950 Remember that bond enthalpies are only averages. 00:09:01.950 --> 00:09:05.340 And so this value that we calculated is only an estimate 00:09:05.340 --> 00:09:07.050 for the standard change in enthalpy 00:09:07.050 --> 00:09:08.950 for this chemical reaction. 00:09:08.950 --> 00:09:11.350 A more accurate way of finding this standard change 00:09:11.350 --> 00:09:13.580 in enthalpy for a chemical reaction 00:09:13.580 --> 00:09:16.476 is to use standard enthalpies of formation. 00:09:16.476 --> 00:09:19.580 And when you use standard enthalpies of formation 00:09:19.580 --> 00:09:21.460 to find the standard change in enthalpy 00:09:21.460 --> 00:09:23.330 for this particular chemical reaction, 00:09:23.330 --> 00:09:28.330 you get -99.8 kilojoules per mole of reaction. 00:09:28.560 --> 00:09:32.413 So -104 is pretty close to -99.8.

Enthalpy of formation

https://www.youtube.com/watch?v=TNwGNHqwHxc

vtt

https://www.youtube.com/api/timedtext?v=TNwGNHqwHxc&ei=5VWUZfHKL9nWxN8PoKaHyAQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=4A8AB4333AAFA29804CECDF0856BAA3DB605E332.36D8FEA60FDC753A04CB806B1E2EB6899C52A6CD&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.910 --> 00:00:01.890 - [Instructor] Enthalpy of a formation 00:00:01.890 --> 00:00:03.830 refers to the change in enthalpy 00:00:03.830 --> 00:00:06.400 for the formation of one mole 00:00:06.400 --> 00:00:09.320 of a substance from the most stable form 00:00:09.320 --> 00:00:11.870 of its constituent elements. 00:00:11.870 --> 00:00:13.380 Change in enthalpy is symbolized 00:00:13.380 --> 00:00:17.060 by delta H and the f stands for formation. 00:00:17.060 --> 00:00:20.670 And the superscript nought refers to the fact 00:00:20.670 --> 00:00:23.440 that everything is under standard state conditions, 00:00:23.440 --> 00:00:25.600 which refers to atmospheric pressure 00:00:25.600 --> 00:00:28.830 of one atmosphere and a specified temperature 00:00:28.830 --> 00:00:33.240 that is usually 25 degrees Celsius. 00:00:33.240 --> 00:00:36.490 So when we're thinking about standard enthalpy of formation, 00:00:36.490 --> 00:00:38.100 we're thinking about the elements 00:00:38.100 --> 00:00:40.730 and the state that they exist under standard conditions. 00:00:40.730 --> 00:00:43.880 So the elements have to be in their standard states. 00:00:43.880 --> 00:00:48.420 So let's think about forming one mole of carbon dioxide. 00:00:48.420 --> 00:00:51.370 So carbon dioxide is composed of the elements 00:00:51.370 --> 00:00:53.680 carbon and oxygen. 00:00:53.680 --> 00:00:55.610 And under standard conditions, 00:00:55.610 --> 00:01:00.200 the most stable form of carbon is graphite. 00:01:00.200 --> 00:01:02.710 So we're gonna write carbon in the solid state 00:01:02.710 --> 00:01:05.590 and we're gonna write graphite over here. 00:01:05.590 --> 00:01:08.900 And next, when you think about the most stable form 00:01:08.900 --> 00:01:11.910 of oxygen under standard conditions. 00:01:11.910 --> 00:01:15.030 And so at one atmosphere, so atmospheric pressure 00:01:15.030 --> 00:01:17.590 and room temperature of 25 degrees Celsius, 00:01:17.590 --> 00:01:20.700 the most stable form of oxygen is oxygen gas. 00:01:20.700 --> 00:01:23.403 So we can go ahead and write in here O2. 00:01:24.490 --> 00:01:27.200 And since we're forming one mole of carbon dioxide 00:01:27.200 --> 00:01:29.920 from the elements that make up carbon dioxide 00:01:29.920 --> 00:01:33.620 in their most stable form under standard conditions, 00:01:33.620 --> 00:01:35.590 the change in enthalpy for this 00:01:35.590 --> 00:01:38.420 would be the standard enthalpy of formation. 00:01:38.420 --> 00:01:39.910 So we have our subscript f 00:01:39.910 --> 00:01:43.550 and our superscript nought indicate standard conditions. 00:01:43.550 --> 00:01:46.370 The change in enthalpy for the formation 00:01:46.370 --> 00:01:51.247 of one mole of CO2 is equal to negative 393.5 kilojoules 00:01:54.160 --> 00:01:56.673 per one mole of carbon dioxide. 00:01:57.620 --> 00:01:59.590 Let's look at some more equations showing the formation 00:01:59.590 --> 00:02:01.210 of one mole of a substance. 00:02:01.210 --> 00:02:02.930 For example, let's look at the equation 00:02:02.930 --> 00:02:06.030 showing the formation of one mole of water. 00:02:06.030 --> 00:02:08.490 So water is composed of hydrogen and oxygen 00:02:08.490 --> 00:02:10.720 and the most stable forms of those two elements 00:02:10.720 --> 00:02:14.170 under standard conditions are hydrogen gas and oxygen gas. 00:02:14.170 --> 00:02:17.590 And for the coefficients to make one mole of water, 00:02:17.590 --> 00:02:21.010 we need a 1/2 as our coefficient in front of O2. 00:02:21.010 --> 00:02:23.960 The standard change in enthalpy of formation 00:02:23.960 --> 00:02:26.270 for the formation of one mole of water 00:02:26.270 --> 00:02:30.720 is negative 285.8 kilojoules per mole. 00:02:30.720 --> 00:02:32.850 We can do the same thing for the formation of one mole 00:02:32.850 --> 00:02:34.800 of methane CH4. 00:02:34.800 --> 00:02:36.640 We already know that the most stable form 00:02:36.640 --> 00:02:38.210 of carbon is graphite 00:02:38.210 --> 00:02:41.470 and the most stable form of hydrogen is hydrogen gas. 00:02:41.470 --> 00:02:44.810 And the standard change in enthalpy of formation 00:02:44.810 --> 00:02:46.720 for the formation of one mole of methane 00:02:46.720 --> 00:02:50.770 is equal to negative 74.8 kilojoules per mole. 00:02:50.770 --> 00:02:55.160 Next, let's think about forming one mole of oxygen gas. 00:02:55.160 --> 00:02:56.850 Well, we're forming the oxygen gas 00:02:56.850 --> 00:02:59.050 from the most stable form of oxygen 00:02:59.050 --> 00:03:00.210 under standard conditions, 00:03:00.210 --> 00:03:04.070 which is also diatomic oxygen gas, O2. 00:03:04.070 --> 00:03:06.880 So we're not changing anything we're going from O2 to O2. 00:03:06.880 --> 00:03:09.900 And since there's no change, there's no change in enthalpy. 00:03:09.900 --> 00:03:11.700 Therefore, the standard enthalpy 00:03:11.700 --> 00:03:15.200 of formation is equal to zero. 00:03:15.200 --> 00:03:17.940 And this is true for the most stable form of any element. 00:03:17.940 --> 00:03:19.430 The standard enthalpy of formation 00:03:19.430 --> 00:03:22.120 of the most stable form of any element is zero 00:03:22.120 --> 00:03:24.830 since you'd be making it from itself. 00:03:24.830 --> 00:03:27.600 Standard enthalpies of formation and kilojoules per mole 00:03:27.600 --> 00:03:31.410 are often found in the appendices of many textbooks. 00:03:31.410 --> 00:03:33.380 And if you look in the appendix of a textbook, 00:03:33.380 --> 00:03:36.020 you'll see the standard enthalpy of formation 00:03:36.020 --> 00:03:40.740 for diatomic oxygen gas, O2, is equal to zero. 00:03:40.740 --> 00:03:44.760 Ozone, which is O3, also exists under standard conditions. 00:03:44.760 --> 00:03:48.190 However, it's not the most stable form of oxygen 00:03:48.190 --> 00:03:49.930 under standard conditions and therefore, 00:03:49.930 --> 00:03:54.930 its standard enthalpy formation is not zero, it's 142.3. 00:03:56.020 --> 00:03:58.550 Graphite is the most stable form of carbon 00:03:58.550 --> 00:03:59.550 under standard conditions. 00:03:59.550 --> 00:04:01.230 Therefore, it has a standard enthalpy 00:04:01.230 --> 00:04:03.720 of formation of zero, but of course, 00:04:03.720 --> 00:04:06.500 diamond also exists under standard conditions 00:04:06.500 --> 00:04:07.860 but it's not the most stable form. 00:04:07.860 --> 00:04:10.890 So its standard enthalpy formation is not zero, 00:04:10.890 --> 00:04:14.050 it's 1.88 kilojoules per mole. 00:04:14.050 --> 00:04:16.610 Enthalpies of formation can be used to calculate 00:04:16.610 --> 00:04:19.990 the change in enthalpy for a chemical reaction. 00:04:19.990 --> 00:04:22.560 We can do this by using the following equation. 00:04:22.560 --> 00:04:26.160 The standard change in enthalpy for a chemical reaction 00:04:26.160 --> 00:04:31.160 is equal to the sum of the standard enthalpies of formation 00:04:31.230 --> 00:04:34.880 of the products minus the sum of the standard enthalpies 00:04:34.880 --> 00:04:37.580 of formation of the reactants. 00:04:37.580 --> 00:04:39.810 Let's say our goal is to find the standard change 00:04:39.810 --> 00:04:42.960 in enthalpy for the following chemical reaction. 00:04:42.960 --> 00:04:44.360 So we have one mole of methane 00:04:44.360 --> 00:04:46.060 reacting with two moles of oxygen 00:04:46.060 --> 00:04:50.830 to form one mole of carbon dioxide and two moles of water. 00:04:50.830 --> 00:04:53.340 The first thing we need to do is sum 00:04:53.340 --> 00:04:57.090 all the standard enthalpies of formation of the products. 00:04:57.090 --> 00:04:59.550 So if we look at our two products over here 00:04:59.550 --> 00:05:02.760 and we'll start with one mole of carbon dioxide. 00:05:02.760 --> 00:05:04.460 So let's go ahead and write this down here. 00:05:04.460 --> 00:05:07.120 We have one mole of carbon dioxide 00:05:07.120 --> 00:05:10.760 and the standard molar enthalpy of carbon dioxide 00:05:10.760 --> 00:05:14.340 we've already seen as negative 393.5 kilojoules 00:05:14.340 --> 00:05:16.590 per mole of carbon dioxide. 00:05:16.590 --> 00:05:18.980 So we're gonna multiply one mole of carbon dioxide 00:05:18.980 --> 00:05:23.980 by negative 393.5 kilojoules per mole of carbon dioxide. 00:05:28.400 --> 00:05:31.490 Our other product is two moles of water. 00:05:31.490 --> 00:05:34.610 So we're going to add this to the other ones. 00:05:34.610 --> 00:05:37.530 We have two moles of H2O. 00:05:37.530 --> 00:05:40.040 And the standard enthalpy of formation of H2O 00:05:40.040 --> 00:05:42.460 is negative 285.8. 00:05:42.460 --> 00:05:43.530 So we're gonna multiply this 00:05:43.530 --> 00:05:48.530 by negative 285.8 kilojoules per mole. 00:05:51.600 --> 00:05:56.600 So moles cancel out and we get negative 393.5 kilojoules. 00:05:57.910 --> 00:05:59.990 And then for the other one, moles cancel out again. 00:05:59.990 --> 00:06:04.140 And this would be plus negative 571.6 kilojoules, 00:06:07.340 --> 00:06:12.340 which is equal to negative 965.1 kilojoules. 00:06:12.710 --> 00:06:15.550 So that's the sum of all 00:06:15.550 --> 00:06:20.540 of the standard enthalpies of formation of our products. 00:06:20.540 --> 00:06:23.170 Next, we need to sum the standard enthalpies 00:06:23.170 --> 00:06:24.920 of formation of our reactants. 00:06:24.920 --> 00:06:28.930 So the two reactants that we have are methane and oxygen 00:06:28.930 --> 00:06:31.730 and we have one mole of methane. 00:06:31.730 --> 00:06:33.810 So let's go ahead and write that in here. 00:06:33.810 --> 00:06:35.650 So we have one mole of methane. 00:06:35.650 --> 00:06:38.570 The standard molar enthalpy of formation of methane 00:06:38.570 --> 00:06:42.260 is negative 74.8 kilojoules per mole. 00:06:42.260 --> 00:06:43.430 So we're multiplying one mole 00:06:43.430 --> 00:06:48.430 by negative 74.8 kilojoules per mole. 00:06:50.860 --> 00:06:53.840 Our other reactant is oxygen. 00:06:53.840 --> 00:06:56.270 And we know that diatomic oxygen gas 00:06:56.270 --> 00:06:59.250 has a standard enthalpy of formation of zero. 00:06:59.250 --> 00:07:00.560 So we could go ahead and write this in 00:07:00.560 --> 00:07:01.560 just to show it. 00:07:01.560 --> 00:07:03.480 So we have two moles of oxygen 00:07:03.480 --> 00:07:06.113 but we're multiplying that number by zero. 00:07:07.300 --> 00:07:12.300 Some moles cancel and give us negative 74.8 kilojoules. 00:07:12.490 --> 00:07:14.530 And we're adding zero to that. 00:07:14.530 --> 00:07:18.300 So negative 74.8 kilojoules 00:07:18.300 --> 00:07:22.980 is the sum of all the standard enthalpies of formation 00:07:22.980 --> 00:07:24.547 of our reactants. 00:07:25.396 --> 00:07:28.020 So to find the standard change in enthalpy for our reaction, 00:07:28.020 --> 00:07:30.820 we take the summation of the enthalpies of formation 00:07:30.820 --> 00:07:35.040 of our products, which was negative 965.1 kilojoules. 00:07:35.040 --> 00:07:36.940 And from that, we subtract the sum 00:07:36.940 --> 00:07:39.160 of the standard enthalpies of formation of the reactants, 00:07:39.160 --> 00:07:43.220 which we found was negative 74.8 kilojoules. 00:07:43.220 --> 00:07:47.700 So negative 965.1 minus negative 74.8 00:07:47.700 --> 00:07:52.700 is equal to negative 890.3 kilojoules. 00:07:54.300 --> 00:07:56.130 For the unit, sometimes you see kilojoules, 00:07:56.130 --> 00:07:58.610 sometimes you see kilojoules per mole, 00:07:58.610 --> 00:08:01.830 and sometimes you see kilojoules per mole of reaction. 00:08:01.830 --> 00:08:04.080 And what kilojoules per mole a reaction means 00:08:04.080 --> 00:08:06.480 is how the balanced equation is written. 00:08:06.480 --> 00:08:07.530 For this balanced equation, 00:08:07.530 --> 00:08:10.790 we're showing the combustion of one mole of methane. 00:08:10.790 --> 00:08:12.570 So combusting one mole of methane 00:08:12.570 --> 00:08:17.510 releases 890.3 kilojoules of energy. 00:08:17.510 --> 00:08:19.670 So that's what kilojoules per mole of reaction 00:08:19.670 --> 00:08:20.573 is referring to. 00:08:21.650 --> 00:08:22.730 Let's go back to the step 00:08:22.730 --> 00:08:26.090 where we summed the standard enthalpies of formation 00:08:26.090 --> 00:08:28.880 of the products to see how we could actually get kilojoules 00:08:28.880 --> 00:08:32.460 per mole of reaction as our units. 00:08:32.460 --> 00:08:35.040 To do this, we need to use a conversion factor. 00:08:35.040 --> 00:08:36.500 For how the equation is written, 00:08:36.500 --> 00:08:39.340 we're producing one mole of carbon dioxide. 00:08:39.340 --> 00:08:41.510 So we can use as a conversion factor, 00:08:41.510 --> 00:08:46.510 there's one mole of carbon dioxide per one mole of reaction. 00:08:49.420 --> 00:08:51.800 We can do the same thing for our other product, 00:08:51.800 --> 00:08:52.930 which is water. 00:08:52.930 --> 00:08:54.500 For how the equation is written, 00:08:54.500 --> 00:08:56.790 we're forming two moles of water. 00:08:56.790 --> 00:09:01.440 So our conversion factor can be there are two moles of water 00:09:01.440 --> 00:09:04.803 for every one mole of reaction. 00:09:06.190 --> 00:09:09.030 Next, moles of carbon dioxide cancels out 00:09:09.030 --> 00:09:11.660 and moles of water cancel out. 00:09:11.660 --> 00:09:14.710 And this gives us kilojoules per mole of reaction 00:09:14.710 --> 00:09:16.350 as our units. 00:09:16.350 --> 00:09:17.730 It's a little more time-consuming 00:09:17.730 --> 00:09:19.310 to write out all the units this way. 00:09:19.310 --> 00:09:21.840 So often, it's faster to do it the first way 00:09:21.840 --> 00:09:23.933 and add in these units at the end.

Worked example: Measuring enthalpy of reaction using coffee-cup calorimetry

https://www.youtube.com/watch?v=pDCcRU5OGmA

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https://www.youtube.com/api/timedtext?v=pDCcRU5OGmA&ei=5VWUZbaYI6aJp-oPh-OQiAY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=DE25469684320004502432802BDB3CD22E42AB45.82DDC43799890C5D97266EFEA1330B360DA9BB52&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.680 --> 00:00:02.600 - [Instructor] A constant pressure calorimeter 00:00:02.600 --> 00:00:04.730 can be used to find the change in entropy 00:00:04.730 --> 00:00:06.980 for a chemical reaction. 00:00:06.980 --> 00:00:08.390 Let's look at the chemical reaction 00:00:08.390 --> 00:00:10.990 between an aqueous solution of silver nitrate 00:00:10.990 --> 00:00:13.210 and aqueous solution of sodium chloride 00:00:13.210 --> 00:00:15.470 to form a precipitate of silver chloride 00:00:15.470 --> 00:00:18.490 and an aqueous solution of sodium nitrate. 00:00:18.490 --> 00:00:20.880 Let's say we have 25.0 milliliters 00:00:20.880 --> 00:00:25.490 of a 0.100 molar solution of silver nitrate 00:00:25.490 --> 00:00:30.080 and 25.0 milliliters of a 0.100 molar solution 00:00:30.080 --> 00:00:31.760 of sodium chloride. 00:00:31.760 --> 00:00:34.010 Both solutions are initially at a temperature 00:00:34.010 --> 00:00:38.710 of 25.000 degrees Celsius. 00:00:38.710 --> 00:00:41.900 Next, we add our two solutions to our calorimeter 00:00:41.900 --> 00:00:44.270 which has made of two coffee cups. 00:00:44.270 --> 00:00:47.240 And since the top coffee cup is loose fitting, 00:00:47.240 --> 00:00:49.630 this reaction is under the constant pressure 00:00:49.630 --> 00:00:50.800 of the atmosphere. 00:00:50.800 --> 00:00:54.080 So this is constant pressure calorimetry. 00:00:54.080 --> 00:00:57.570 After the two aqueous solutions mix, the reaction occurs 00:00:57.570 --> 00:01:00.490 and we watch the thermometer in the calorimeter. 00:01:00.490 --> 00:01:04.240 In this case, the temperature of the solution increases 00:01:04.240 --> 00:01:05.860 and the final temperature, 00:01:05.860 --> 00:01:07.700 the highest one reached in our experiment 00:01:07.700 --> 00:01:12.100 is 25.781 degrees Celsius. 00:01:12.100 --> 00:01:14.130 So the change in the temperature of the solution 00:01:14.130 --> 00:01:15.670 would be the final temperature 00:01:15.670 --> 00:01:17.220 minus the initial temperature, 00:01:17.220 --> 00:01:22.217 which is 25.781 minus 25.0 00:01:24.510 --> 00:01:29.510 which is equal to positive 0.781 degrees Celsius. 00:01:32.350 --> 00:01:36.170 The total volume of solution would be 25 plus 25, 00:01:36.170 --> 00:01:41.170 which is equal to 50.0 milliliters of solution. 00:01:41.800 --> 00:01:43.640 If we assume the density of the solution 00:01:43.640 --> 00:01:45.840 is one gram per milliliter, 00:01:45.840 --> 00:01:50.840 50.0 milliliters is equal to 50.0 grams. 00:01:52.410 --> 00:01:55.580 Next we need to solve for the heat gained by the water 00:01:55.580 --> 00:01:59.600 and we can use the Q is equal to MC delta T equation. 00:01:59.600 --> 00:02:01.190 So we're solving for the heat, 00:02:01.190 --> 00:02:03.180 which is symbolized by Q, 00:02:03.180 --> 00:02:05.000 M is the mass of our solution 00:02:05.000 --> 00:02:07.530 which we saw was 50.0 grams, 00:02:07.530 --> 00:02:09.990 So we can plug that in. 00:02:09.990 --> 00:02:12.610 We can assume that the specific heat of the solution 00:02:12.610 --> 00:02:15.240 is the same as the specific heat of water, 00:02:15.240 --> 00:02:20.050 which is 4.18 joules per gram degrees Celsius 00:02:20.050 --> 00:02:22.170 and the change in the temperature of the solution 00:02:22.170 --> 00:02:24.940 was 0.781 degrees Celsius. 00:02:24.940 --> 00:02:27.593 So we can plug that in as well. 00:02:29.630 --> 00:02:33.900 Grams cancels out, degrees Celsius cancels out 00:02:33.900 --> 00:02:38.900 and we find that Q is equal to 00:02:39.040 --> 00:02:44.040 positive 1.63 times 10 to the second joules. 00:02:47.170 --> 00:02:51.030 The positive sign means that energy was gained by the water. 00:02:51.030 --> 00:02:52.680 But let's think about the distinction 00:02:52.680 --> 00:02:56.520 between system and surroundings. 00:02:56.520 --> 00:02:58.740 The system consists of the reactants 00:02:58.740 --> 00:03:01.320 and products for our particular reaction 00:03:01.320 --> 00:03:03.320 and the surroundings are everything else 00:03:03.320 --> 00:03:05.880 which includes the water. 00:03:05.880 --> 00:03:06.770 So in this case, 00:03:06.770 --> 00:03:09.660 since the temperature of the surroundings increased, right? 00:03:09.660 --> 00:03:11.830 We saw an increase in the temperature, 00:03:11.830 --> 00:03:13.110 that means that heat flowed 00:03:13.110 --> 00:03:15.880 from the system to the surroundings 00:03:15.880 --> 00:03:18.190 and so the surroundings increased in energy 00:03:18.190 --> 00:03:20.823 and that's what we see with this positive sign here. 00:03:21.940 --> 00:03:24.440 If we assume a perfect transfer of heat 00:03:24.440 --> 00:03:26.490 from the system to the surroundings, 00:03:26.490 --> 00:03:28.530 if the surroundings gained energy, 00:03:28.530 --> 00:03:31.210 that means the system lost energy. 00:03:31.210 --> 00:03:32.450 So if we're thinking about the heat 00:03:32.450 --> 00:03:34.830 transferred for the reaction, 00:03:34.830 --> 00:03:36.260 it's the same in magnitude, 00:03:36.260 --> 00:03:40.160 1.63 times 10 to the second joules. 00:03:40.160 --> 00:03:43.040 However, we need to put a negative sign in here 00:03:43.040 --> 00:03:47.223 which indicates that energy was given off by the reaction. 00:03:48.130 --> 00:03:50.590 The heat that's transferred under constant pressure 00:03:50.590 --> 00:03:52.730 is equal to the change in the entropy 00:03:52.730 --> 00:03:55.230 of the reaction, delta H. 00:03:55.230 --> 00:03:57.950 However, let's find the change in entropy, delta H 00:03:57.950 --> 00:04:01.270 in terms of kilojoules per mole 00:04:01.270 --> 00:04:05.120 of silver chloride for our units. 00:04:05.120 --> 00:04:07.010 Since silver chloride is one of our products, 00:04:07.010 --> 00:04:09.770 we first need to find moles of our reactants 00:04:09.770 --> 00:04:12.650 and we're gonna do that using the molarity equation 00:04:12.650 --> 00:04:13.770 which says that molarity 00:04:13.770 --> 00:04:16.540 is equal to moles divided by liters. 00:04:16.540 --> 00:04:19.290 For our silver nitrate solution, 00:04:19.290 --> 00:04:23.900 the concentration was 0.100 molar 00:04:23.900 --> 00:04:26.610 and trying to solve for moles so that's X. 00:04:26.610 --> 00:04:28.560 The volume of our silver nitrate solution 00:04:28.560 --> 00:04:30.710 was 25.0 milliliters 00:04:30.710 --> 00:04:35.310 which is 0.0250 liters. 00:04:35.310 --> 00:04:38.690 So we solve for X and we get 0.00250. 00:04:42.580 --> 00:04:46.140 So that's how many moles of silver nitrate 00:04:46.140 --> 00:04:48.010 that we started with 00:04:48.010 --> 00:04:50.100 and it's the exact same calculation 00:04:50.100 --> 00:04:52.230 for sodium chloride as well. 00:04:52.230 --> 00:04:56.913 So this is also how many moles of sodium chloride we have. 00:04:57.850 --> 00:05:00.800 Next, we go back to our balanced chemical equation 00:05:00.800 --> 00:05:03.290 and we can see we have coefficients of one 00:05:03.290 --> 00:05:04.610 in front of silver nitrate, 00:05:04.610 --> 00:05:05.820 in front of sodium chloride 00:05:05.820 --> 00:05:08.390 and in front of silver chloride. 00:05:08.390 --> 00:05:11.967 Therefore, we're also going to produce 0.00250 moles 00:05:14.930 --> 00:05:16.710 of silver chloride. 00:05:16.710 --> 00:05:18.530 Next we're gonna calculate the change 00:05:18.530 --> 00:05:21.870 in the entropy, delta H, for our reaction. 00:05:21.870 --> 00:05:23.400 The heat that was transferred 00:05:23.400 --> 00:05:28.400 was negative 1.63 times 10 to the second joules 00:05:30.060 --> 00:05:31.650 and we're gonna divide that 00:05:31.650 --> 00:05:33.690 by the moles of silver chloride 00:05:33.690 --> 00:05:38.690 which was 0.00250 moles of silver chlorides. 00:05:41.890 --> 00:05:45.447 This is equal to negative 65,200 joules 00:05:48.350 --> 00:05:51.050 per mole of silver chloride 00:05:51.050 --> 00:05:53.720 and we could convert that into kilojoules. 00:05:53.720 --> 00:05:58.720 And so this is equal to negative 65.2 kilojoules 00:05:59.550 --> 00:06:02.983 per mole of silver chloride. 00:06:04.930 --> 00:06:08.030 We could stop right here and give this as our final answer, 00:06:08.030 --> 00:06:09.020 but let's keep going 00:06:09.020 --> 00:06:13.200 and convert two kilojoules per mole of reaction. 00:06:13.200 --> 00:06:14.560 First, let's rewrite this. 00:06:14.560 --> 00:06:19.210 We have negative 65.2 kilojoules 00:06:19.210 --> 00:06:23.590 per mole of silver chlorides 00:06:23.590 --> 00:06:26.540 and what it means by kilojoules per mole of reaction 00:06:26.540 --> 00:06:28.590 is how the reaction is written 00:06:28.590 --> 00:06:31.660 in the balanced equation down here. 00:06:31.660 --> 00:06:33.440 So if we look at the balanced equation, 00:06:33.440 --> 00:06:37.010 there's one mole of silver chloride 00:06:37.010 --> 00:06:38.860 for how the reaction is written. 00:06:38.860 --> 00:06:41.370 So we can write a conversion factor 00:06:41.370 --> 00:06:45.340 of one mole of silver chloride 00:06:45.340 --> 00:06:49.520 per one mole of reaction 00:06:49.520 --> 00:06:52.490 and writing it this way for the conversion factor, 00:06:52.490 --> 00:06:55.850 the moles of silver chloride would cancel out 00:06:55.850 --> 00:07:00.850 and give us negative 65.2 kilojoules per mole of reaction. 00:07:06.100 --> 00:07:08.993 So this could also be our final answer. 00:07:11.540 --> 00:07:12.740 Finally, the negative sign 00:07:12.740 --> 00:07:15.350 means that we have an exothermic reaction. 00:07:15.350 --> 00:07:17.730 The reaction gave off energy 00:07:17.730 --> 00:07:19.720 and this value when you do 00:07:19.720 --> 00:07:21.750 a constant pressure calorimetry experiment 00:07:21.750 --> 00:07:24.520 is often a little bit lower than the actual value 00:07:24.520 --> 00:07:27.940 because in reality there's not always a perfect transfer 00:07:27.940 --> 00:07:31.610 of heat from the reaction to the water. 00:07:31.610 --> 00:07:35.063 Often some of the energy is lost to the environment.

Enthalpy of reaction

https://www.youtube.com/watch?v=cEzN33gfgVs

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https://www.youtube.com/api/timedtext?v=cEzN33gfgVs&ei=5VWUZev0Iu_6vdIPx6OPgA4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245333&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=A135A677D1D1028F2022E7CFB01460C9AAF0EEF1.3E3774F7D4F7DE1EE5D774178B32A785A27AEEB1&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.660 --> 00:00:01.580 - [Instructor] The change in enthalpy 00:00:01.580 --> 00:00:03.500 for a chemical reaction delta H, 00:00:03.500 --> 00:00:06.240 we could even write delta H of reaction in here 00:00:06.240 --> 00:00:08.230 is equal to the heat transferred 00:00:08.230 --> 00:00:11.100 during a chemical reaction at constant pressure. 00:00:11.100 --> 00:00:13.963 So delta H is equal to qp. 00:00:14.930 --> 00:00:17.100 Let's say we are performing a chemical reaction, 00:00:17.100 --> 00:00:20.710 an aqueous solution under constant atmospheric pressure. 00:00:20.710 --> 00:00:23.060 The reactants and products of that chemical reaction 00:00:23.060 --> 00:00:25.270 make up the system and everything else makes 00:00:25.270 --> 00:00:27.550 up the surroundings. 00:00:27.550 --> 00:00:30.800 When heat flows from the surroundings to the system, 00:00:30.800 --> 00:00:33.500 the system or the reaction absorbs heat 00:00:33.500 --> 00:00:35.690 and therefore the change in enthalpy 00:00:35.690 --> 00:00:38.010 is positive for the reaction. 00:00:38.010 --> 00:00:41.320 This is called an endothermic reaction. 00:00:41.320 --> 00:00:44.450 If heat flows from the system to the surroundings, 00:00:44.450 --> 00:00:46.730 the reaction gave off energy. 00:00:46.730 --> 00:00:48.290 Therefore the change in enthalpy 00:00:48.290 --> 00:00:50.030 for the reaction is negative 00:00:50.030 --> 00:00:52.853 and this is called an exothermic reaction. 00:00:53.790 --> 00:00:57.050 As an example of a reaction, let's look at the decomposition 00:00:57.050 --> 00:01:01.930 of hydrogen peroxide to form liquid water and oxygen gas. 00:01:01.930 --> 00:01:04.470 The change in the enthalpy for this reaction 00:01:04.470 --> 00:01:07.540 is equal to negative 196 kilojoules. 00:01:07.540 --> 00:01:11.330 The negative sign means the reaction is exothermic. 00:01:11.330 --> 00:01:14.420 And for the units, sometimes you might see kilojoules. 00:01:14.420 --> 00:01:18.320 Sometimes you might see kilojoules per mole, 00:01:18.320 --> 00:01:22.553 and sometimes you might see kilojoules per mole of reaction. 00:01:23.540 --> 00:01:25.150 What kilojoules per mole of reaction 00:01:25.150 --> 00:01:28.900 is referring to is how the equation is written. 00:01:28.900 --> 00:01:31.120 So if we look at this balanced equation, 00:01:31.120 --> 00:01:33.980 there's a two as a coefficient in front of hydrogen peroxide 00:01:33.980 --> 00:01:36.290 and therefore two moles of hydrogen peroxide 00:01:36.290 --> 00:01:39.470 are decomposing to form two moles of water 00:01:39.470 --> 00:01:42.080 and one mole of oxygen gas. 00:01:42.080 --> 00:01:45.770 So when two moles of hydrogen peroxide decompose, 00:01:45.770 --> 00:01:50.770 196 kilojoules of energy are given off. 00:01:50.810 --> 00:01:54.120 Next, let's calculate how much heat is released 00:01:54.120 --> 00:01:58.580 when 5.00 grams of hydrogen peroxide decomposes 00:01:58.580 --> 00:02:00.400 at a constant pressure. 00:02:00.400 --> 00:02:02.530 The first step is to find out how many moles 00:02:02.530 --> 00:02:04.680 of hydrogen peroxide that we have. 00:02:04.680 --> 00:02:07.020 So we take the mass of hydrogen peroxide 00:02:07.020 --> 00:02:08.490 which is five grams 00:02:08.490 --> 00:02:12.040 and we divide that by the molar mass of hydrogen peroxide 00:02:12.040 --> 00:02:15.530 which is 34.0 grams per mole. 00:02:15.530 --> 00:02:18.713 Grams cancels out and this gives us 0.147 00:02:21.570 --> 00:02:25.033 moles of hydrogen peroxide. 00:02:26.120 --> 00:02:29.770 Next, we take our negative 196 kilojoules 00:02:29.770 --> 00:02:32.430 per mole of reaction 00:02:32.430 --> 00:02:36.540 and we're gonna multiply this by a conversion factor. 00:02:36.540 --> 00:02:39.130 When we look at the balanced equation for how it's written, 00:02:39.130 --> 00:02:42.590 there are two moles of hydrogen peroxide. 00:02:42.590 --> 00:02:44.720 So for our conversion factor 00:02:44.720 --> 00:02:49.480 for every one mole of reaction as it is written, 00:02:49.480 --> 00:02:53.590 there are two moles of hydrogen peroxide. 00:02:53.590 --> 00:02:55.743 So two moles of H2O2. 00:02:58.450 --> 00:03:01.230 Now the of reaction will cancel out 00:03:01.230 --> 00:03:06.230 and this gives us negative 98.0 kilojoules 00:03:07.730 --> 00:03:12.033 per one mole of H2O2. 00:03:13.220 --> 00:03:15.560 So two moles of hydrogen peroxide 00:03:15.560 --> 00:03:18.810 would give off 196 kilojoules of energy. 00:03:18.810 --> 00:03:21.350 And one mole of hydrogen peroxide would give 00:03:21.350 --> 00:03:25.793 off half that amount or 98.0 kilojoules of energy. 00:03:26.690 --> 00:03:30.520 Next, we take our 0.147 moles of hydrogen peroxide. 00:03:30.520 --> 00:03:31.353 So let me just go ahead 00:03:31.353 --> 00:03:32.970 and write this down here really quickly. 00:03:32.970 --> 00:03:37.390 So we have 0.147 moles of H202. 00:03:38.710 --> 00:03:41.730 And remember, we're trying to calculate, 00:03:41.730 --> 00:03:44.930 we're trying to calculate the amount of heat 00:03:44.930 --> 00:03:46.410 that was released. 00:03:46.410 --> 00:03:51.153 So next we multiply that by negative 98.0 kilojoules 00:03:52.760 --> 00:03:57.760 per mole of H202, and moles of H2O2 will cancel out 00:04:00.910 --> 00:04:03.280 and this gives us our final answer. 00:04:03.280 --> 00:04:07.100 So the heat that was released when 5.00 grams 00:04:07.100 --> 00:04:10.440 of hydrogen peroxide decompose at constant pressure, 00:04:10.440 --> 00:04:15.430 this turns out to be equal to negative 14.4 kilojoules.

Heating curve for water

https://www.youtube.com/watch?v=MqAVc_XaIXQ

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en

WEBVTT Kind: captions Language: en 00:00:00.220 --> 00:00:02.700 - [Instructor] Let's look at the heating curve for water. 00:00:02.700 --> 00:00:05.360 A heating curve has temperature on the y-axis. 00:00:05.360 --> 00:00:07.730 In this case, we have it in degrees Celsius. 00:00:07.730 --> 00:00:10.550 And heat added on the x-axis, 00:00:10.550 --> 00:00:12.780 let's say it's in kilojoules. 00:00:12.780 --> 00:00:15.740 Let's say we have 18.0 grams of ice 00:00:15.740 --> 00:00:18.090 and our goal is to calculate the total heat 00:00:18.090 --> 00:00:21.460 necessary to convert that 18 grams of ice 00:00:21.460 --> 00:00:26.460 at -25 degrees Celsius to steam at 125 degrees Celsius. 00:00:28.880 --> 00:00:31.890 So we're starting with ice at -25 degrees Celsius 00:00:31.890 --> 00:00:34.010 and first we need to heat up the ice 00:00:34.010 --> 00:00:37.770 to zero degrees Celsius, which we know is the melting point. 00:00:37.770 --> 00:00:38.950 So on our heating curve, 00:00:38.950 --> 00:00:43.110 we're going from point A to point B. 00:00:43.110 --> 00:00:44.690 To calculate the heat necessary, 00:00:44.690 --> 00:00:49.690 we need to use the equation Q is equal to mc delta T, 00:00:50.350 --> 00:00:52.000 where q is the heat added, 00:00:52.000 --> 00:00:53.850 m is the mass of the ice. 00:00:53.850 --> 00:00:55.760 c is the specific heat of ice 00:00:55.760 --> 00:00:57.990 and delta T is the change in temperature, 00:00:57.990 --> 00:00:59.370 which is the final temperature 00:00:59.370 --> 00:01:01.520 minus the initial temperature. 00:01:01.520 --> 00:01:03.560 So we're trying to calculate q. 00:01:03.560 --> 00:01:08.147 We know the mass of our ice is 18.0 grams. 00:01:08.147 --> 00:01:10.740 The specific heat of ice 00:01:10.740 --> 00:01:15.740 is 2.03 joules per gram degrees Celsius. 00:01:18.420 --> 00:01:21.350 And for the change in temperature, it's final minus initial. 00:01:21.350 --> 00:01:24.898 So the final temperature would be zero degrees Celsius, 00:01:24.898 --> 00:01:27.030 initial is -25. 00:01:27.030 --> 00:01:32.030 So zero minus -25 gives us +25 degrees Celsius. 00:01:34.730 --> 00:01:38.130 So grams will cancel out, degrees Celsius cancels out. 00:01:38.130 --> 00:01:43.130 And this gives us q is equal to 9.1 times 10 00:01:44.580 --> 00:01:49.070 to the second joules to two significant figures 00:01:49.070 --> 00:01:54.070 or we could also write 0.91 kilojoules. 00:01:55.660 --> 00:01:57.620 Now that the ice is at zero degrees Celsius, 00:01:57.620 --> 00:01:58.850 we know what's going to melt. 00:01:58.850 --> 00:02:00.520 So we're gonna go from point B 00:02:00.520 --> 00:02:03.260 on the heating curve to point C. 00:02:03.260 --> 00:02:07.020 And to calculate how much heat is necessary to melt the ice, 00:02:07.020 --> 00:02:10.450 we need to know the heat of fusion of ice, 00:02:10.450 --> 00:02:14.200 which is equal to 6.01 kilojoules per mole. 00:02:18.213 --> 00:02:21.690 So we need to figure out how many moles of ice we have. 00:02:21.690 --> 00:02:24.890 After starting with 18.0 grams, 00:02:24.890 --> 00:02:27.050 we divide by the molar mass of H2O 00:02:27.050 --> 00:02:31.590 which is 18.0 grams per mole. 00:02:31.590 --> 00:02:35.370 And the grams will cancel and give us one mole. 00:02:35.370 --> 00:02:39.440 So we have 1.00 moles of ice 00:02:39.440 --> 00:02:43.520 and we multiply that by 6.01 kilojoules per mole 00:02:43.520 --> 00:02:48.040 and the moles cancel out and give us 6.01 kilojoules. 00:02:50.940 --> 00:02:53.790 Now that all the ice is melted, we have liquid water. 00:02:53.790 --> 00:02:54.940 And so on our heating curve, 00:02:54.940 --> 00:02:56.370 we're gonna heat that liquid water 00:02:56.370 --> 00:02:59.940 from zero degrees Celsius to 100 Celsius 00:02:59.940 --> 00:03:01.820 which is the boiling point of water. 00:03:01.820 --> 00:03:04.340 So we're going from point C to point D 00:03:04.340 --> 00:03:05.840 on the heating curve. 00:03:05.840 --> 00:03:07.430 To calculate the heat added, 00:03:07.430 --> 00:03:12.400 we use the Q is equal to mc delta T equation again. 00:03:12.400 --> 00:03:14.880 So we're solving for Q. 00:03:14.880 --> 00:03:18.450 The mass is still 18.0 grams 00:03:18.450 --> 00:03:20.060 but the specific heat now, 00:03:20.060 --> 00:03:21.250 since we have liquid water, 00:03:21.250 --> 00:03:24.020 we need to use the specific heat of liquid water, 00:03:24.020 --> 00:03:29.020 which is 4.18 joules per gram degrees Celsius. 00:03:29.720 --> 00:03:31.320 And for the change in temperature, 00:03:31.320 --> 00:03:34.489 the final temperature is 100. 00:03:34.489 --> 00:03:39.489 So 100 minus zero gives us +100 degrees Celsius. 00:03:42.870 --> 00:03:46.440 So grams cancel out, degrees Celsius cancels out 00:03:46.440 --> 00:03:50.447 and we find that Q is equal to 7.52 times 10 00:03:53.180 --> 00:03:57.113 to the third joules, let me just correct three there, 00:03:57.113 --> 00:04:00.470 7.52 times 10 to the third joules, 00:04:00.470 --> 00:04:03.177 which is equal to 7.52 kilojoules. 00:04:07.170 --> 00:04:09.170 Once we reached a point D in the heating curve, 00:04:09.170 --> 00:04:10.630 we're at the boiling point of water. 00:04:10.630 --> 00:04:12.680 So the heat that we add now is gonna go 00:04:12.680 --> 00:04:15.760 into turning the liquid water into gaseous water. 00:04:15.760 --> 00:04:18.730 So going from point D to point E, 00:04:18.730 --> 00:04:19.820 we're doing a phase change. 00:04:19.820 --> 00:04:23.240 We need to know the heat of vaporization of water, 00:04:23.240 --> 00:04:28.240 and that's equal to 40.7 kilojoules per mole. 00:04:31.030 --> 00:04:35.600 And we already know we have one mole of H2O. 00:04:35.600 --> 00:04:40.040 So one mole times 40.7 moles, the moles cancel 00:04:40.040 --> 00:04:45.040 and it takes 40.7 kilojoules of energy 00:04:45.740 --> 00:04:50.350 to convert the liquid water in to gaseous water or steam. 00:04:50.350 --> 00:04:52.130 Next we're gonna heat the gaseous water 00:04:52.130 --> 00:04:55.480 from 100 degrees Celsius to 125 degrees Celsius. 00:04:55.480 --> 00:04:59.050 So we're going from point E to point F on the heating curve. 00:04:59.050 --> 00:05:01.280 And to figure out how much heat we need to add, 00:05:01.280 --> 00:05:06.010 we use the Q is equal to mc delta T equation one more time. 00:05:06.010 --> 00:05:10.700 So we're solving for Q and we still have 18.0 grams. 00:05:10.700 --> 00:05:13.990 This time we need to use these specific heat of steam, 00:05:13.990 --> 00:05:18.990 which is 1.84 joules per gram degree Celsius. 00:05:20.570 --> 00:05:24.380 The change in temperature would be 125 minus 100 00:05:24.380 --> 00:05:28.780 or +25 degrees Celsius. 00:05:28.780 --> 00:05:31.540 So grams cancel, units cancel out 00:05:31.540 --> 00:05:36.540 and we get Q is equal to 8.3 times 10 to the second joules 00:05:40.130 --> 00:05:42.020 to two significant figures, 00:05:42.020 --> 00:05:46.823 which is equal to 0.83 kilojoules. 00:05:48.760 --> 00:05:50.700 Finally, we need to add everything up. 00:05:50.700 --> 00:05:53.960 So going from point A to point B in the heating curve. 00:05:53.960 --> 00:05:57.060 So think about just the X axis this time, all right? 00:05:57.060 --> 00:05:59.460 So going from point A to point B, 00:05:59.460 --> 00:06:04.460 we calculated that to be equal to 0.91 kilojoules. 00:06:05.370 --> 00:06:08.492 And then from point B to point C, 00:06:08.492 --> 00:06:13.480 we calculated that to be 6.01 kilojoules. 00:06:13.480 --> 00:06:18.480 From C to D, so this distance here was 7.52. 00:06:21.463 --> 00:06:24.990 From D to E, this was the big one here. 00:06:24.990 --> 00:06:29.047 This was equal to 40.7 kilojoules. 00:06:30.650 --> 00:06:32.760 And finally from E to F we calculated 00:06:32.760 --> 00:06:35.487 this was equal to 0.83 kilojoules. 00:06:39.010 --> 00:06:40.240 And when you add everything up 00:06:40.240 --> 00:06:44.870 this is equal to 56.0 kilojoules. 00:06:44.870 --> 00:06:47.360 So that's how much energy it takes 00:06:47.360 --> 00:06:52.360 to convert 18.0 grams of ice at -25 degrees Celsius 00:06:53.450 --> 00:06:58.270 to gaseous water at 125 degrees Celsius. 00:06:58.270 --> 00:06:59.970 Next, let's think about the slopes 00:06:59.970 --> 00:07:02.540 of the different lines on our heating curve. 00:07:02.540 --> 00:07:06.770 So let's look at the line going from B to C 00:07:06.770 --> 00:07:11.770 and also the line going from point D to point E. 00:07:12.170 --> 00:07:14.110 Both of these lines represent phase changes, 00:07:14.110 --> 00:07:16.280 going from point B to point C 00:07:16.280 --> 00:07:18.420 was going from a solid to a liquid 00:07:18.420 --> 00:07:20.370 and going from point D to E 00:07:20.370 --> 00:07:22.938 was going from a liquid to a gas. 00:07:22.938 --> 00:07:26.270 And since the slope of both of these lines is zero, 00:07:26.270 --> 00:07:29.930 that means as you add heat on the x-axis, 00:07:29.930 --> 00:07:32.207 the temperature doesn't change. 00:07:32.207 --> 00:07:34.070 So during a phase change, 00:07:34.070 --> 00:07:36.490 all the energy goes into disrupting 00:07:36.490 --> 00:07:39.660 the intermolecular forces that are present 00:07:39.660 --> 00:07:42.500 and they don't go into increasing the temperature. 00:07:42.500 --> 00:07:44.410 So there is no increase in temperature 00:07:44.410 --> 00:07:46.190 during a phase change. 00:07:46.190 --> 00:07:48.740 Think about going from point D to point E, 00:07:48.740 --> 00:07:52.160 this was converting our liquid water into gaseous water. 00:07:52.160 --> 00:07:54.470 So as the heat is being added, 00:07:54.470 --> 00:07:59.100 all that energy goes into breaking the intermolecular forces 00:07:59.100 --> 00:08:01.470 between water molecules and pulling apart 00:08:01.470 --> 00:08:02.790 those liquid water molecules 00:08:02.790 --> 00:08:06.400 and turning them into gaseous water molecules. 00:08:06.400 --> 00:08:09.190 So it's only after all of the liquid water molecules 00:08:09.190 --> 00:08:11.720 are converted into gaseous water molecules, 00:08:11.720 --> 00:08:14.000 that's when we see the temperature increase again. 00:08:14.000 --> 00:08:16.730 So talking about from point E to point F, 00:08:16.730 --> 00:08:19.160 everything is now in the gaseous state 00:08:19.160 --> 00:08:22.050 and then we see the increase in temperature. 00:08:22.050 --> 00:08:25.700 Finally, let's compare the slope of the line from A to B 00:08:25.700 --> 00:08:29.284 to the slope of the line from C to D. 00:08:29.284 --> 00:08:31.080 If we look at it, the slope of the line 00:08:31.080 --> 00:08:34.330 from A to B is a little bit steeper 00:08:34.330 --> 00:08:37.600 than the slope of the line from C to D. 00:08:37.600 --> 00:08:39.090 The reason for the different slopes 00:08:39.090 --> 00:08:41.671 has to do with the different specific heats. 00:08:41.671 --> 00:08:45.270 From A to B, we used the specific heat for ice 00:08:45.270 --> 00:08:49.060 which is 2.03 joules per gram degrees Celsius. 00:08:49.060 --> 00:08:51.840 From C to D in our calculation, 00:08:51.840 --> 00:08:53.910 we used the specific heat for water 00:08:53.910 --> 00:08:57.800 which is 4.1 joules per gram degrees Celsius. 00:08:57.800 --> 00:09:00.160 The higher the value for the specific heat, 00:09:00.160 --> 00:09:02.410 the more energy it takes to raise 00:09:02.410 --> 00:09:06.030 the temperature of a substance by a certain amount. 00:09:06.030 --> 00:09:07.660 So if we think about comparing these two, 00:09:07.660 --> 00:09:09.290 let's say we try to raise the temperature 00:09:09.290 --> 00:09:11.850 of ice by 25 degrees Celsius. 00:09:11.850 --> 00:09:15.460 So lets think about this distance here on the y-axis. 00:09:15.460 --> 00:09:18.560 We would have to put in only a small amount of energy 00:09:18.560 --> 00:09:20.700 to get ice to increase its temperature 00:09:20.700 --> 00:09:23.200 by 25 degrees Celsius. 00:09:23.200 --> 00:09:26.160 We think about that same temperature change on liquid water. 00:09:26.160 --> 00:09:28.820 So if we tried to increase the temperature of liquid water 00:09:28.820 --> 00:09:31.690 by that same amount, 25 degrees, 00:09:31.690 --> 00:09:34.280 we would have to put in more energy. 00:09:34.280 --> 00:09:37.440 So on the x-axis, we have to put in more energy 00:09:37.440 --> 00:09:40.020 to accomplish the same change in temperature. 00:09:40.020 --> 00:09:44.820 And that's because liquid water has a higher specific heat. 00:09:44.820 --> 00:09:47.100 Since it might be a little bit hard to see on that diagram, 00:09:47.100 --> 00:09:50.870 let's think about putting some heat into a substance here. 00:09:50.870 --> 00:09:52.590 So I'm gonna draw a horizontal line, 00:09:52.590 --> 00:09:54.050 and then we're trying to accomplish 00:09:54.050 --> 00:09:55.420 a certain temperature change. 00:09:55.420 --> 00:09:57.200 So I'll draw a vertical line. 00:09:57.200 --> 00:10:02.060 Those two give me a line with a slope. 00:10:02.060 --> 00:10:03.790 So let's say we're trying to accomplish 00:10:03.790 --> 00:10:06.060 the same change in temperature. 00:10:06.060 --> 00:10:09.182 So I'll draw this Y distance the same as before 00:10:09.182 --> 00:10:11.450 but we have a higher specific heat. 00:10:11.450 --> 00:10:14.190 So it takes more energy. 00:10:14.190 --> 00:10:17.370 Therefore this X distance is going to increase. 00:10:17.370 --> 00:10:19.740 And when we increase the X distance, 00:10:19.740 --> 00:10:23.230 we see that the slope decreases. 00:10:23.230 --> 00:10:26.780 So the greater the value for the specific heat, 00:10:26.780 --> 00:10:30.033 the lower the slope on the heating curve.

Enthalpy and phase changes

https://www.youtube.com/watch?v=jCKvxH5mXR4

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WEBVTT Kind: captions Language: en 00:00:00.030 --> 00:00:01.720 - [Instructor] Let's say that we have some solid water 00:00:01.720 --> 00:00:04.860 or ice and we want to melt the ice 00:00:04.860 --> 00:00:09.170 and turn the solid water into liquid water. 00:00:09.170 --> 00:00:10.900 This phase change of solid water 00:00:10.900 --> 00:00:13.010 to liquid water is called melting 00:00:13.010 --> 00:00:16.460 and it takes positive 6.01 kilojoules 00:00:16.460 --> 00:00:20.010 per one mole to melt ice. 00:00:20.010 --> 00:00:23.970 This change in enthalpy is symbolized by delta H 00:00:23.970 --> 00:00:27.090 with a subscript fus, which stands for fusion. 00:00:27.090 --> 00:00:29.913 So this is called the heat of fusion. 00:00:31.020 --> 00:00:32.580 Next let's think about the phase change 00:00:32.580 --> 00:00:35.630 of converting liquid water into gaseous water. 00:00:35.630 --> 00:00:38.040 This phase change is called vaporization 00:00:38.040 --> 00:00:40.810 and it also takes energy to convert liquid water 00:00:40.810 --> 00:00:42.860 into gaseous water. 00:00:42.860 --> 00:00:46.970 Specifically for water it takes 40.7 kilojoules 00:00:46.970 --> 00:00:50.700 per one mole of liquid water to vaporize it. 00:00:50.700 --> 00:00:54.490 And so this change in energy is called the enthalpy 00:00:54.490 --> 00:00:59.140 of vaporization or simply the heat of vaporization. 00:00:59.140 --> 00:01:01.600 Let's go back and think about the structure of ice. 00:01:01.600 --> 00:01:05.790 Ice has water molecules in a repeating crystal structure 00:01:05.790 --> 00:01:10.270 and the water molecules are held together by hydrogen bonds. 00:01:10.270 --> 00:01:12.690 So between these two water molecules here, 00:01:12.690 --> 00:01:16.810 when we add energy, we increase the freedom of motion, 00:01:16.810 --> 00:01:19.300 so over here is a picture of liquid water. 00:01:19.300 --> 00:01:22.100 So this is still held together by hydrogen bonds. 00:01:22.100 --> 00:01:23.850 These water molecules are still held together 00:01:23.850 --> 00:01:26.940 by hydrogen bonds but we no longer have a crystal structure. 00:01:26.940 --> 00:01:29.490 So we have increased freedom of motion 00:01:29.490 --> 00:01:33.690 and it takes energy to disrupt that crystal structure. 00:01:33.690 --> 00:01:35.680 And next, let's think about converting liquid water 00:01:35.680 --> 00:01:38.320 into gaseous water or steam. 00:01:38.320 --> 00:01:39.790 When water is in the gaseous state, 00:01:39.790 --> 00:01:42.240 there are no more intermolecular forces 00:01:42.240 --> 00:01:43.450 between the molecules. 00:01:43.450 --> 00:01:45.450 There's nothing holding them together. 00:01:45.450 --> 00:01:48.170 And so it takes a lot of energy to pull these 00:01:48.170 --> 00:01:49.600 two water molecules apart. 00:01:49.600 --> 00:01:52.610 It takes a lot of energy to overcome these hydrogen bonds. 00:01:52.610 --> 00:01:55.950 And that's the reason why we have such a large value 00:01:55.950 --> 00:01:58.730 for the heat of vaporization. 00:01:58.730 --> 00:02:01.703 So it takes a lot more energy to completely pull 00:02:01.703 --> 00:02:05.190 these molecules apart than it did to simply 00:02:05.190 --> 00:02:06.500 increase the freedom of motion. 00:02:06.500 --> 00:02:11.500 So 40.7 is a much bigger number than 6.01. 00:02:12.060 --> 00:02:15.070 If it takes positive 40.7 kilojoules per mole 00:02:15.070 --> 00:02:18.030 of energy to go from the liquid state to the gaseous state. 00:02:18.030 --> 00:02:19.090 If we go in reverse 00:02:19.090 --> 00:02:21.730 from the gaseous state back to the liquid state 00:02:21.730 --> 00:02:24.390 that same amount of energy is given off. 00:02:24.390 --> 00:02:28.930 So we can write 40.7 kilojoules per mole. 00:02:28.930 --> 00:02:31.110 However, since the energy is given off, 00:02:31.110 --> 00:02:33.870 we need to include a negative sign, 00:02:33.870 --> 00:02:35.130 going from the gaseous state 00:02:35.130 --> 00:02:37.900 to the liquid state is called condensation. 00:02:37.900 --> 00:02:39.410 So we could call this value 00:02:39.410 --> 00:02:42.260 of negative 40.7 kilojoules per mole, 00:02:42.260 --> 00:02:44.453 the heat of condensation. 00:02:45.470 --> 00:02:48.900 And if it takes positive 6.01 kilojoules per mole to go 00:02:48.900 --> 00:02:50.970 from the solid state to the liquid state. 00:02:50.970 --> 00:02:52.080 If we go in reverse 00:02:52.080 --> 00:02:54.880 from the liquid state back to the solid state 00:02:54.880 --> 00:02:59.880 we would give off 6.01 kilojoules per mole of energy. 00:03:00.690 --> 00:03:03.020 And so we need to write a negative sign here 00:03:03.020 --> 00:03:05.250 to indicate the energy is given off. 00:03:05.250 --> 00:03:08.770 When we go from a liquid to a solid, that's freezing. 00:03:08.770 --> 00:03:12.543 So this value is called the heat of freezing for water.

Constant-volume calorimetry

https://www.youtube.com/watch?v=1NueJQpqkuc

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WEBVTT Kind: captions Language: en 00:00:00.690 --> 00:00:03.170 - [Instructor] Calorimetry refers to the measurement 00:00:03.170 --> 00:00:04.370 of heat flow. 00:00:04.370 --> 00:00:07.060 And there are many different types of calorimeters. 00:00:07.060 --> 00:00:10.970 In this case, we're looking at a constant volume calorimeter 00:00:10.970 --> 00:00:14.460 which is also called a bomb calorimeter. 00:00:14.460 --> 00:00:17.190 Let's look at how a bomb calorimeter works. 00:00:17.190 --> 00:00:20.490 First, the sample to be combusted is placed 00:00:20.490 --> 00:00:23.220 in a container that has some oxygen. 00:00:23.220 --> 00:00:25.750 And then there's some ignition wires that go into 00:00:25.750 --> 00:00:29.690 this little container here, and the sample is ignited 00:00:29.690 --> 00:00:33.440 and heat is given off by the combustion reaction. 00:00:33.440 --> 00:00:37.460 So heat is being transferred from our sample 00:00:37.460 --> 00:00:41.460 to the water in our containers. 00:00:41.460 --> 00:00:43.530 Let me go ahead and draw some water in here. 00:00:43.530 --> 00:00:45.600 So imagine we have some water, 00:00:45.600 --> 00:00:49.560 and it's important to know that the container is rigid. 00:00:49.560 --> 00:00:54.560 So the walls are very, very solid and cannot move. 00:00:55.350 --> 00:00:57.550 There's also something to stir the water 00:00:57.550 --> 00:00:59.400 and since heat is being transferred 00:00:59.400 --> 00:01:02.110 from the combustion reaction to the water, 00:01:02.110 --> 00:01:04.480 the temperature of the water will increase 00:01:04.480 --> 00:01:07.980 which we can see on the thermometer. 00:01:07.980 --> 00:01:11.270 Now that we understand how a bomb calorimeter works, 00:01:11.270 --> 00:01:12.610 let's think about that heat 00:01:12.610 --> 00:01:15.420 that's being transferred from the combustion reaction 00:01:15.420 --> 00:01:19.980 to the water, so that heat is q. 00:01:19.980 --> 00:01:22.780 Let's go back to the first law of thermodynamics, 00:01:22.780 --> 00:01:25.220 which says that the change in the internal energy 00:01:25.220 --> 00:01:28.210 of the system is equal to q plus w, 00:01:28.210 --> 00:01:30.890 where q is the heat that's transferred 00:01:30.890 --> 00:01:33.840 and w is the work done. 00:01:33.840 --> 00:01:37.020 Let's say we do this combustion reaction in a container 00:01:37.020 --> 00:01:39.070 with a movable piston. 00:01:39.070 --> 00:01:41.830 And the combustion reaction is performed 00:01:41.830 --> 00:01:45.080 under the constant pressure of the atmosphere. 00:01:45.080 --> 00:01:47.971 So this time, when we do the combustion reaction, 00:01:47.971 --> 00:01:50.590 we will transfer some heat. 00:01:50.590 --> 00:01:53.640 So heat is being transferred from the combustion reaction 00:01:53.640 --> 00:01:57.060 and we would also produce some gases, which would push up 00:01:57.060 --> 00:02:00.200 on the piston and so the piston would move up, 00:02:00.200 --> 00:02:01.960 and since the piston's moving, 00:02:01.960 --> 00:02:06.130 work is being done by the combustion reaction. 00:02:06.130 --> 00:02:08.960 In this case, the heat that's transferred q 00:02:08.960 --> 00:02:11.090 is done under constant pressure, 00:02:11.090 --> 00:02:14.935 and so we can write qp here and by definition, 00:02:14.935 --> 00:02:17.395 the heat that's transferred at constant pressure, 00:02:17.395 --> 00:02:20.895 that's the change in the enthalpy delta H. 00:02:23.150 --> 00:02:25.399 So for this example with the container 00:02:25.399 --> 00:02:26.918 with the movable piston, 00:02:26.918 --> 00:02:29.385 when we did our combustion reaction, 00:02:29.385 --> 00:02:31.706 the heat that's transferred at constant pressure is equal 00:02:31.706 --> 00:02:35.401 to the enthalpy delta H, the change in enthalpy delta H, 00:02:35.401 --> 00:02:40.012 and as the gases expand and pushing the piston work is done. 00:02:40.012 --> 00:02:42.236 Let's compare the example with the movable piston 00:02:42.236 --> 00:02:44.293 to our bomb calorimeter. 00:02:44.293 --> 00:02:46.513 Our bomb calorimeter has rigid walls 00:02:46.513 --> 00:02:49.013 and therefore no work can be done. 00:02:49.013 --> 00:02:51.846 So the work done is equal to zero. 00:02:52.751 --> 00:02:55.841 When we plugged that into the first law of thermodynamics, 00:02:55.841 --> 00:03:00.038 we find that the change in the internal energy delta E 00:03:00.038 --> 00:03:02.955 is equal to the heat transferred q. 00:03:03.861 --> 00:03:06.617 And since this is a constant volume 00:03:06.617 --> 00:03:10.369 calorimeter right, the walls are rigid. 00:03:10.369 --> 00:03:12.536 We can write q sub v here. 00:03:13.406 --> 00:03:17.665 So this heat that's transferred from our combustion reaction 00:03:17.665 --> 00:03:20.782 in this case is not equal to the change in the enthalpy. 00:03:20.782 --> 00:03:25.782 It's equal to the change in the internal energy delta E. 00:03:26.209 --> 00:03:27.280 So the heat that's transferred 00:03:27.280 --> 00:03:30.210 at constant pressure is equal to the change 00:03:30.210 --> 00:03:32.180 in the enthalpy delta H, 00:03:32.180 --> 00:03:35.140 while the heat that's transferred at constant volume 00:03:35.140 --> 00:03:38.453 is equal to the change in the internal energy delta E. 00:03:39.330 --> 00:03:42.100 To do a constant volume calorimetry problem, 00:03:42.100 --> 00:03:44.650 we need to know the heat capacity of the calorimeter 00:03:44.650 --> 00:03:49.090 which is symbolized by C with a subscript cal. 00:03:49.090 --> 00:03:51.190 To find the heat capacity of the calorimeter, 00:03:51.190 --> 00:03:54.380 we need to combust something that we know the exact amount 00:03:54.380 --> 00:03:55.250 of heat for them. 00:03:55.250 --> 00:03:58.510 For example, if you combust exactly one gram 00:03:58.510 --> 00:04:00.170 of benzoic acid, 00:04:00.170 --> 00:04:05.170 you'll get 26.38 kilojoules released of energy. 00:04:05.500 --> 00:04:10.220 So let's say we have a 0.2350 gram sample of benzoic acid. 00:04:10.220 --> 00:04:13.080 And we put that in our calorimeter 00:04:13.080 --> 00:04:16.400 and we go ahead and combust the benzoic acid. 00:04:16.400 --> 00:04:18.610 And we find that the temperature increases 00:04:18.610 --> 00:04:22.793 by positive 1.642 degrees Celsius. 00:04:23.730 --> 00:04:25.870 To find the heat capacity for the calorimeter, 00:04:25.870 --> 00:04:27.210 first we take our known amount 00:04:27.210 --> 00:04:32.210 which is 26.38 kilojoules per gram 00:04:32.820 --> 00:04:35.820 and we multiply that by how much benzoic acid 00:04:35.820 --> 00:04:40.820 we used in our calorimeter which was 0.2350 grams. 00:04:41.430 --> 00:04:43.570 And so the grams will cancel out 00:04:43.570 --> 00:04:48.000 and this is equal to 6.199 kilojoules. 00:04:50.580 --> 00:04:53.090 Next we divide this by our temperature change 00:04:53.090 --> 00:04:56.467 which was positive 1.642 degrees Celsius. 00:04:59.070 --> 00:05:01.240 And this gives us the heat capacity 00:05:01.240 --> 00:05:06.240 of our calorimeter, which turns out to be 3.3775 kilojoules 00:05:11.590 --> 00:05:14.083 per degree Celsius. 00:05:15.060 --> 00:05:16.960 Now that we know the heat capacity 00:05:16.960 --> 00:05:19.340 for our specific calorimeter, 00:05:19.340 --> 00:05:21.610 we can use this value to calculate the heat 00:05:21.610 --> 00:05:24.500 of combustion for another substance. 00:05:24.500 --> 00:05:27.230 So the heat of combustion for another substance 00:05:27.230 --> 00:05:30.030 or just q would be equal to the negative 00:05:30.030 --> 00:05:32.650 of the heat capacity of the calorimeter 00:05:32.650 --> 00:05:35.740 times the change in the temperature of the water 00:05:35.740 --> 00:05:37.113 in that calorimeter. 00:05:38.320 --> 00:05:40.030 Let's say our goal is to calculate the heat 00:05:40.030 --> 00:05:43.760 of combustion of caffeine in kilojoules per mole. 00:05:43.760 --> 00:05:46.650 So we take 0.265 grams of caffeine. 00:05:46.650 --> 00:05:50.100 We put that in our calorimeter, we combust it 00:05:50.100 --> 00:05:52.510 and we find the temperature of the water increases 00:05:52.510 --> 00:05:57.510 by positive 1.525 degrees Celsius. 00:05:57.510 --> 00:06:00.530 So to calculate q, q is equal to the negative 00:06:00.530 --> 00:06:02.570 of the heat capacity of the calorimeter, 00:06:02.570 --> 00:06:07.570 which is 3.775 kilojoules per degree Celsius. 00:06:09.270 --> 00:06:11.580 And we multiply that by the temperature change 00:06:11.580 --> 00:06:15.880 which has 1.525 degrees Celsius. 00:06:15.880 --> 00:06:18.400 So degrees Celsius cancels out 00:06:18.400 --> 00:06:23.400 and this gives us negative 5.757 kilojoules. 00:06:26.830 --> 00:06:28.323 And technically this is the heat transfer 00:06:28.323 --> 00:06:32.640 to a constant volume so we could even write q sub v in here, 00:06:32.640 --> 00:06:35.130 and remember this is equal to the change 00:06:35.130 --> 00:06:37.970 in the internal energy of our system. 00:06:37.970 --> 00:06:41.690 So this is qv is equal to delta E. 00:06:41.690 --> 00:06:43.670 Since our goal is to find the heat of combustion 00:06:43.670 --> 00:06:46.270 of caffeine in kilojoules per mole, 00:06:46.270 --> 00:06:48.660 next we need to take our grams of caffeine, 00:06:48.660 --> 00:06:51.390 which is 0.265 grams, and divide that 00:06:51.390 --> 00:06:53.200 by the molar mass of caffeine. 00:06:53.200 --> 00:06:57.740 And so grams will cancel out and give us moles of caffeine. 00:06:57.740 --> 00:07:00.450 So this calculation is equal to 1.36 00:07:01.930 --> 00:07:06.930 times 10 to the negative third moles of caffeine. 00:07:08.800 --> 00:07:10.810 Now, all we have to do is divide our heat, 00:07:10.810 --> 00:07:15.810 which is negative 5.757 kilojoules by our moles 00:07:17.340 --> 00:07:22.340 of 1.36 times 10 to the negative third moles, 00:07:23.620 --> 00:07:26.830 to give us a final value of negative 4.23 00:07:28.289 --> 00:07:33.289 times 10 to the third kilojoules per mole 00:07:33.740 --> 00:07:36.713 with a negative sign, meaning heat is given off. 00:07:37.800 --> 00:07:39.490 So we can say that this is the value, 00:07:39.490 --> 00:07:41.890 this is the change in the internal energy 00:07:41.890 --> 00:07:44.920 for our reaction in kilojoules per mole. 00:07:44.920 --> 00:07:48.180 And often the change in enthalpy is about the same 00:07:48.180 --> 00:07:50.250 as the change in the internal energy. 00:07:50.250 --> 00:07:52.640 So we can say that this is approximately equal 00:07:52.640 --> 00:07:57.023 to the change in the enthalpy for the reaction as well.

Worked example: Measuring the energy content of foods using soda-can calorimetry

https://www.youtube.com/watch?v=1Zx71Il5jaM

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https://www.youtube.com/api/timedtext?v=1Zx71Il5jaM&ei=6VWUZeyCHui2mLAP48ehsA0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704245337&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=849CDBF96EA0863595B429434E82B9EF1A1B8BC1.AE7D0886476EDDB85EE8982D4E80A2A632FE9EE7&key=yt8&lang=en&name=Default&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.252 --> 00:00:01.450 - [Instructor] Calorimetry refers 00:00:01.450 --> 00:00:04.040 to the measurement of heat flow. 00:00:04.040 --> 00:00:05.550 And in this worked example, 00:00:05.550 --> 00:00:07.850 we're going to burn a marshmallow 00:00:07.850 --> 00:00:11.820 and find the energy content of the marshmallow. 00:00:11.820 --> 00:00:15.320 First, let's look at the setup for our soda can calorimeter. 00:00:15.320 --> 00:00:17.800 So our soda can has some water in it. 00:00:17.800 --> 00:00:19.970 So here's the water in our soda can. 00:00:19.970 --> 00:00:23.720 And then we also have a thermometer in the soda can 00:00:23.720 --> 00:00:26.950 to measure the change in the temperature of the water. 00:00:26.950 --> 00:00:29.560 If you take a stir rod and you put the stir rod 00:00:29.560 --> 00:00:31.840 through the tab on the soda can, 00:00:31.840 --> 00:00:35.290 you can attach the soda can to a stand. 00:00:35.290 --> 00:00:37.480 Next, we can put our marshmallow on a pin 00:00:37.480 --> 00:00:39.650 that's attached to a piece of cork. 00:00:39.650 --> 00:00:40.900 Before we start the experiment, 00:00:40.900 --> 00:00:43.360 we need to take the mass of the marshmallow 00:00:43.360 --> 00:00:44.470 with the cork and the pin, 00:00:44.470 --> 00:00:47.553 so we'll call that the initial mass. 00:00:47.553 --> 00:00:51.464 And we also need the initial temperature of the water. 00:00:51.464 --> 00:00:53.297 So we'll call that Ti. 00:00:54.990 --> 00:00:57.210 Next, we'd light the marshmallow on fire. 00:00:57.210 --> 00:01:00.190 As the marshmallow burns heat is given off 00:01:00.190 --> 00:01:04.290 and that heat is transferred to the water in the soda can. 00:01:04.290 --> 00:01:05.910 Therefore the water and the soda can 00:01:05.910 --> 00:01:07.520 will increase in temperature, 00:01:07.520 --> 00:01:10.370 which we can see on the thermometer. 00:01:10.370 --> 00:01:12.170 After the marshmallow burns for a little while, 00:01:12.170 --> 00:01:14.614 we can stop the burning process. 00:01:14.614 --> 00:01:17.700 And once we stop that, we wanna look at the thermometer 00:01:17.700 --> 00:01:20.576 for the maximum temperature reached. 00:01:20.576 --> 00:01:22.870 And when we find that maximum temperature 00:01:22.870 --> 00:01:24.230 we can go ahead and record it. 00:01:24.230 --> 00:01:26.230 So we have our final temperature 00:01:26.230 --> 00:01:28.350 and once the marshmallow with the cork and the pin 00:01:28.350 --> 00:01:31.313 cools down we can find our final mass. 00:01:33.430 --> 00:01:34.790 Let's say the initial mass 00:01:34.790 --> 00:01:39.130 of our marshmallow pin cork was 6.08 grams 00:01:39.130 --> 00:01:42.371 and the final mass was 6.00 grams. 00:01:42.371 --> 00:01:45.000 The initial temperature of the water in the soda can 00:01:45.000 --> 00:01:48.180 was 25.0 degrees Celsius and the final temperature 00:01:48.180 --> 00:01:52.080 was 30.0 degrees Celsius. 00:01:52.080 --> 00:01:57.080 Also, let's say that we started with 50.0 grams 00:01:57.240 --> 00:01:59.253 of water in the soda can. 00:02:00.360 --> 00:02:03.830 Let's calculate the heat gained by the water. 00:02:03.830 --> 00:02:07.850 To do that we can use the Q is equal to mc delta T equation 00:02:07.850 --> 00:02:11.527 where Q is equal to the heat transferred, 00:02:11.527 --> 00:02:16.527 m is the mass of the water which is 50.0 grams, 00:02:17.410 --> 00:02:20.160 c is the specific heat of water 00:02:20.160 --> 00:02:25.160 which is 4.18 joules per gram degrees Celsius, 00:02:27.153 --> 00:02:29.710 and delta T is the change in temperature 00:02:29.710 --> 00:02:31.780 which should be the final temperature 00:02:31.780 --> 00:02:35.640 minus the initial temperature of the water 00:02:35.640 --> 00:02:40.640 which is 30.0 minus 25.0 00:02:43.160 --> 00:02:48.160 which is equal to 5.0 degrees Celsius. 00:02:50.300 --> 00:02:53.240 Grams cancels out, degrees Celsius cancels out 00:02:53.240 --> 00:02:57.593 and we find that Q is equal to +1.0 times 10 00:03:00.973 --> 00:03:02.983 to the third joules. 00:03:02.983 --> 00:03:05.816 That's to two significant figures. 00:03:07.338 --> 00:03:09.304 The positive sign here means 00:03:09.304 --> 00:03:11.839 that heat was gained by the water 00:03:11.839 --> 00:03:15.154 which is why we saw an increase in the temperature. 00:03:15.154 --> 00:03:17.215 If we assume a perfect transfer of heat, 00:03:17.215 --> 00:03:19.394 so all the heat that was given off by the burning 00:03:19.394 --> 00:03:21.572 of the marshmallow was transferred to the water, 00:03:21.572 --> 00:03:25.342 if we think about Q for the burning of the marshmallow, 00:03:25.342 --> 00:03:27.592 it should be equal in magnitudes. 00:03:27.592 --> 00:03:30.341 So we can write Q is equal to, 00:03:30.341 --> 00:03:31.817 this time we're gonna write a negative 00:03:31.817 --> 00:03:34.886 since heat was given off by the burning of the marshmallow, 00:03:34.886 --> 00:03:37.636 1.0 times 10 to the third joules. 00:03:41.930 --> 00:03:43.940 So assuming a perfect transfer of heat, 00:03:43.940 --> 00:03:46.800 the magnitude of these two numbers is equal. 00:03:46.800 --> 00:03:49.260 However, there's no way that all of the heat was transferred 00:03:49.260 --> 00:03:51.320 from the combustion of the marshmallow 00:03:51.320 --> 00:03:53.954 to the soda can with our simple setup. 00:03:53.954 --> 00:03:56.200 So definitely not all of it was transferred. 00:03:56.200 --> 00:03:58.120 For example, some of it could have been lost 00:03:58.120 --> 00:03:59.743 simply to the environment. 00:04:00.820 --> 00:04:02.670 And since the soda can is open 00:04:02.670 --> 00:04:06.390 to the atmospheric pressure of the environment, 00:04:06.390 --> 00:04:08.960 so I'll go ahead and write atmospheric pressure in here, 00:04:08.960 --> 00:04:11.330 this soda can calorimeter is an example 00:04:11.330 --> 00:04:14.210 of constant pressure calorimetry. 00:04:14.210 --> 00:04:16.070 And since this is the heat 00:04:16.070 --> 00:04:18.360 that's transferred under constant pressure, 00:04:18.360 --> 00:04:20.970 I can go ahead and write QP here to indicate 00:04:20.970 --> 00:04:23.510 the heat transferred under constant pressure. 00:04:23.510 --> 00:04:27.863 That's the definition of the change in enthalpy delta H. 00:04:30.920 --> 00:04:34.240 So burning the marshmallow gave off energy 00:04:34.240 --> 00:04:36.850 which is an exothermic reaction, 00:04:36.850 --> 00:04:40.383 therefore, the sign for delta H is negative. 00:04:41.470 --> 00:04:46.240 Finally, let's relate this soda can calorimetry experiments 00:04:46.240 --> 00:04:49.010 to calories in everyday life. 00:04:49.010 --> 00:04:51.250 And so let's find the energy content 00:04:51.250 --> 00:04:54.870 of the marshmallow in calories per gram. 00:04:54.870 --> 00:04:58.370 A food calorie has a capital C 00:04:58.370 --> 00:05:02.100 and in chemistry, there's also a unit of energy 00:05:02.100 --> 00:05:05.260 with calorie with a lowercase C. 00:05:05.260 --> 00:05:07.925 So one food calorie with a capital C 00:05:07.925 --> 00:05:12.925 is equal to one kilocalorie or 1000 calories 00:05:13.260 --> 00:05:14.533 with a lowercase C. 00:05:15.830 --> 00:05:17.510 When we burned the marshmallow, 00:05:17.510 --> 00:05:20.360 we started off with a mass of 6.08 grams 00:05:20.360 --> 00:05:22.660 for the marshmallow pin cork. 00:05:22.660 --> 00:05:25.780 And the final mass was 6.00 grams, 00:05:25.780 --> 00:05:29.473 which means we burned 0.08 grams of marshmallows. 00:05:30.595 --> 00:05:31.610 And when we burned the marshmallow, 00:05:31.610 --> 00:05:32.830 we found there was a transfer 00:05:32.830 --> 00:05:37.830 of 1.0 times 10 to the third joules of energy. 00:05:38.460 --> 00:05:40.790 So first, let's convert that 00:05:40.790 --> 00:05:44.190 into calories with a lowercase C. 00:05:44.190 --> 00:05:46.950 So if we multiply by the conversion factor 00:05:46.950 --> 00:05:51.560 of there is one calorie with a lowercase C 00:05:51.560 --> 00:05:54.507 for every 4.184 joules. 00:05:57.530 --> 00:05:59.440 Joules will cancel out 00:05:59.440 --> 00:06:04.440 and this gives us 239 calories with a lowercase C. 00:06:08.040 --> 00:06:12.680 Next, let's convert 239 calories into food calories. 00:06:12.680 --> 00:06:16.760 So first let's take 239 calories, 00:06:16.760 --> 00:06:20.150 and we can multiply by the conversion factor of, 00:06:20.150 --> 00:06:24.430 there are 1000 calories with the lowercase C 00:06:24.430 --> 00:06:26.768 for every one kilocalorie. 00:06:26.768 --> 00:06:31.768 And that's gonna give us 0.239 kilocalories. 00:06:33.142 --> 00:06:35.780 And going back up here to our chart, 00:06:35.780 --> 00:06:40.020 remember, one kilocalorie is equal to one food calorie 00:06:40.020 --> 00:06:41.920 with a capital C. 00:06:41.920 --> 00:06:46.250 Therefore we have 0.239 kilocalories 00:06:46.250 --> 00:06:51.250 or 0.239 Calories with a capital C which is a food calorie. 00:06:54.340 --> 00:06:56.270 To find the answer content of the marshmallow 00:06:56.270 --> 00:06:59.170 in calories per gram, we just need to divide 00:06:59.170 --> 00:07:02.850 our food calories by how many grams of marshmallows we use 00:07:02.850 --> 00:07:06.060 which was 0.08 grams in our combustion. 00:07:06.060 --> 00:07:09.410 So dividing by 0.08 grams 00:07:09.410 --> 00:07:14.240 gives us approximately three calories per gram 00:07:14.240 --> 00:07:15.840 of marshmallows. 00:07:15.840 --> 00:07:16.700 And that's useful 00:07:16.700 --> 00:07:20.940 because let's say we had one serving size of marshmallows 00:07:20.940 --> 00:07:23.630 which is about 30 grams. 00:07:23.630 --> 00:07:25.688 So therefore, if we know the energy content 00:07:25.688 --> 00:07:27.891 is three calories per gram, 00:07:27.891 --> 00:07:31.690 we can simply multiply our four marshmallows 00:07:31.690 --> 00:07:36.660 approximately by three calories per gram 00:07:36.660 --> 00:07:38.660 and grams would cancel out. 00:07:38.660 --> 00:07:41.210 And we would find that those four marshmallows 00:07:41.210 --> 00:07:45.023 that we wanted to eat are about 90 food calories.