General Chemistry/Gases - Wikibooks, open books for an open world General Chemistry/Gases 3.2 Pressure and Collisions Characteristics of Gases[ edit ] Gases have a number of special characteristics that differentiate them from other states of matter. Here is a list of characteristics of gases: Characteristics of Gases Gases have neither definite shape nor definite volume. They expand to the size of their container. Gases are fluid, and flow easily. Gases have low density, unless compressed. Being made of tiny particles in a large, open space, gases are very compressible. Gases diffuse (mix and spread out) and effuse (travel through small holes). Standard Temperature and Pressure[ edit ] Wikipedia has related information at Standard conditions for temperature and pressure Standard Temperature and Pressure, or STP, is 0 °C and 1 atmosphere of pressure. Expressed in other units, STP is 273 K and 760 torr. The Kelvin and torr are useful units of temperature and pressure respectively that we will discuss later in the following sections. Avogadro's Law[ edit ] Amedeo Avogadro, the Italian chemist.

Avogadro's Law is named after him and his discoveries about the behavior of gases Wikipedia has related information at Avogadro's Law Avogadro's Law states that equal volumes of gases at the same temperature and pressure contain the same number of molecules. So both one mole of Xenon at STP (131.3 grams) and one mole of helium at STP (4.00 grams) take up 22.4 liters. Even 1 mole of air, which is a mixture of several gases, takes up 22.4 liters of volume. 22.4 L is the standard molar volume of a gas. [Avogadro's Law] s t {\displaystyle {\frac {p_{1}\cdot V_{1}}{T_{1}\cdot n_{1}}}={\frac {p_{2}\cdot V_{2}}{T_{2}\cdot n_{2}}}=const} where: p is the pressure of the gas T is the temperature of the gas has the same value for all gases, independent of the size or mass of the gas molecules. Pressure[ edit ] Gases exert pressure on their containers and all other objects. Pressure is measured as force per unit area.

A barometer is a device that measures pressure. There are a number of different units to measure pressure: torr, equal to millimeters of mercury (mm Hg): if a glass cylinder with no gas in it is placed in a dish of liquid mercury, the mercury will rise in the cylinder to a certain number of millimeters. atmosphere (atm), the pressure of air at sea level. pascal (Pa), equal to one newton (N) per square meter. A newton is the force necessary to accelerate one kilogram by one meter per second squared. You should know that 1 atm = 760 torr = 101.3 kPa. Wikipedia has related information at Ideal gas Gases are complicated things composed a large numbers of tiny particles zipping around at high speeds. There are a number of complex forces governing the interactions between molecules in the gas, which in turn affect the qualities of the gas as a whole. To get around these various complexities and to simplify our study, we will talk about ideal gases. An ideal gas is a simplified model of a gas that follows several strict rules and satisfies several limiting assumptions. Ideal gases can be perfectly modeled and predicted with a handful of equations.

Ideal gases follow, among others, these important rules: Rules of Ideal Gases The molecules that make up a gas are point masses, meaning they have no volume. Gas particles are spread out with very great distance between each molecule. Thus, intermolecular forces are essentially zero, meaning they neither attract nor repel each other. If collisions do occur between gas particles, these collisions are elastic, meaning there is no loss of kinetic (motion) energy. Gas molecules are in continuous random motion. Temperature is directly proportionate to kinetic energy. Note: Ideal gases never truly exist (because the nature of gases is so complicated), but gases are often close enough to an ideal gas that the equations still hold fairly accurate. Ideal Gas Law[ edit ] Ideal gases can be completely described using the ideal gas law: [Ideal Gas Law] Real Gases[ edit ] All real gases (or non-ideal gases) deviate from the ideal gas laws that we discussed above. These deviations can occur for several reasons: Real molecules have mass and volume. They are too big and no longer behave like ideal point masses Low volumes and high pressures cause molecules to be close enough for intermolecular forces. Polar molecules exaggerate the problem. Low temperature means low kinetic energy.

At lower temperatures, intermolecular forces become significant and cannot be ignored like they are in ideal gasses Other complicated factors may prevent ideal behavior. When these issues are present, gas molecules attract each other, and may even condense into a liquid. Gases act most like ideal gases when the molecules have low mass (small volume), are not polar, and are at high temperature and low pressure. Noble gases like Xenon or Argon act the most like ideal gases because they are mostly electrical neutral and non-interactive. Kinetic Molecular Theory[ edit ] This theory describes why gases exhibit their properties. It only applies accurately to ideal gases. Because there is no such thing as an ideal gas, the Kinetic Molecular Theory can only approximate gas behavior. It is still very useful to chemists. Wikipedia has related information at Kinetic theory The Kinetic Molecular Theory explains the pressure, temperature, kinetic energy, and speed of gases and their molecules. See Wikipedia for the exact equations of the Kinetic Molecular Theory, as well as detailed explanations. What is most important is understanding the general concepts, not the specific equations. Kinetic Energy and Temperature[ edit ] Kinetic energy is the mechanical, or movement, energy.

It is given by the equation: K {\displaystyle v_{rms}} is the average velocity Explained with words, kinetic energy is dependent on the product of a particle's mass and its velocity squared. The more kinetic energy, the faster a particle moves. Conversely, the faster a particle moves, the more kinetic energy it has. The Kinetic Molecular Theory states that kinetic energy and temperature are directly proportionate. Thus, a double in temperature will result in a double in kinetic energy and an increase in velocity by a factor of 1.4 (the square root of 2, see the KE equation). This means that the higher the temperature of a gas, the faster the individual particles in that gas are moving. A hotter gas has more kinetic energy than a colder gas. If two gases are at the same temperature, they will have the same kinetic energy. The lighter-massed gas will have a higher average speed for its particles at the same energy level. It is important to know that gas temperature must be measured in kelvin. Zero degrees Celsius is 273 kelvin. One Celsius degree is equal to one kelvin, but the kelvin scale has water's freezing point at 273 and boiling point at 373.

It is necessary to use kelvin because temperatures must always be positive when using the Kinetic Molecular Theory. Question for the reader A gas's temperature is increased from 20 °C to 40° C. What factor does its kinetic energy increase? Velocity? Keep in mind that gases are all about averages. For a temperature increase, there will be an average increase in the kinetic energy of the particles in that gas. Even in a very hot gas there will be some particles moving very slowly. However, the average will be high. Pressure and Collisions[ edit ] Pressure exists because the gas molecules are in continuous random motion, and they will constantly strike the walls of their container. Pressure will increase as the speed of the molecules increases, due to greater forces of collision. Pressure will also increase as the mass of the molecules increase. A small, slow molecule has less momentum than a large, fast molecule, which explains their difference in pressure. Question for the reader There are two jars of ideal gas, for example. In Jar A there is nitrogen gas (N2). In Jar B there is methane gas (CH4). Both jars are at the same temperature. Which will have greater pressure?

Question for the reader Now, Jars A and B both have propane gas (C3H8). Jar A is at 300 K and Jar B is at 500 K. Which will have greater pressure? Gas Laws Gas Laws One of the most amazing things about gases is that, despite wide differences in chemical properties, all the gases more or less obey the gas laws. The gas laws deal with how gases behave with respect to pressure, volume, temperature, and amount. Pressure Gases are the only state of matter that can be compressed very tightly or expanded to fill a very large space. Pressure is force per unit area, calculated by dividing the force by the area on which the force acts. The earth's gravity acts on air molecules to create a force, that of the air pushing on the earth. This is called atmospheric pressure. The units of pressure that are used are pascal (Pa), standard atmosphere (atm), and torr. 1 atm is the average pressure at sea level. It is normally used as a standard unit of pressure. The SI unit though, is the pascal. 101,325 pascals equals 1 atm. For laboratory work the atmosphere is very large.

A more convient unit is the torr. 760 torr equals 1 atm. A torr is the same unit as the mmHg (millimeter of mercury). It is the pressure that is needed to raise a tube of mercury 1 millimeter. The Gas Laws: Pressure Volume Temperature Relationships Boyle's Law: The Pressure-Volume Law Boyle's law or the pressure-volume law states that the volume of a given amount of gas held at constant temperature varies inversely with the applied pressure when the temperature and mass are constant. Another way to describing it is saying that their products are constant. PV = C When pressure goes up, volume goes down. When volume goes up, pressure goes down. From the equation above, this can be derived: P1V1 = P2V2 = P3V3 etc. This equation states that the product of the initial volume and pressure is equal to the product of the volume and pressure after a change in one of them under constant temperature. For example, if the initial volume was 500 mL at a pressure of 760 torr, when the volume is compressed to 450 mL, what is the pressure?

Plug in the values: (760 torr)(500 mL) = P2(450 mL) 760 torr x 500 mL/450 mL = P2 844 torr = P2 The pressure is 844 torr after compression. Charles' Law: The Temperature-Volume Law This law states that the volume of a given amount of gas held at constant pressure is directly proportional to the Kelvin temperature. V Same as before, a constant can be put in: V / T = C As the volume goes up, the temperature also goes up, and vice-versa. Also same as before, initial and final volumes and temperatures under constant pressure can be calculated. V1 / T1 = V2 / T2 = V3 / T3 etc. Gay-Lussac's Law: The Pressure Temperature Law This law states that the pressure of a given amount of gas held at constant volume is directly proportional to the Kelvin temperature. P Same as before, a constant can be put in: P / T = C As the pressure goes up, the temperature also goes up, and vice-versa.

Also same as before, initial and final volumes and temperatures under constant pressure can be calculated. P1 / T1 = P2 / T2 = P3 / T3 etc. Avogadro's Law: The Volume Amount Law Amedeo Avogadro (1776-1856) Gives the relationship between volume and amount when pressure and temperature are held constant. Remember amount is measured in moles. Also, since volume is one of the variables, that means the container holding the gas is flexible in some way and can expand or contract. If the amount of gas in a container is increased, the volume increases. If the amount of gas in a container is decreased, the volume decreases.

V FW = gRT / PV ; V = (0.1000 g)(0.0821 L-atm / mole-K)(295.3 K) / (0.0928 atm)(0.256 L) = 102 g / mole FW of CHF2 = 51.0 g / mole ; 102 / 51.0 = 2 ; C2H2F4 If the equation above is rearranged further, g / V = P x FW / RT = density you get an expression of the density of the gas as a function of T and FW. EXAMPLE 3: Compare the density of He and air (average FW = 28 g/mole) at 25.0 oC and 1.00 atm.

dHe = (4.003 g / mole)(1.00 atm) / (0.0821 L-atm / mole-K)(298 K) = 0.164 g / L dair = (28.0 g / mole)(1.00 atm) / (0.0821 L-atm / mole-K)(298 K) = 1.14 g / L EXAMPLE 4: Compare the density of air at 25.0 oC and air at 1807 oC and 1.00 atm.

dHe = (28.0 g / mole)(1.00 atm) / (0.0821 L-atm / mole-K)(298 K) = 1.14 g / L dair = (28.0 g / mole)(1.00 atm) / (0.0821 L-atm / mole-K)(2080 K) = 0.164 g / L Partial Pressures Dalton's Law of Partial Pressures states that the total pressure of a mixture of nonreacting gases is the sum of their individual partial pressures. Ptotal = Pa + Pb + Pc + ... Ptotal = naRT / V + nbRT / V + ncRT / V + ... or Ptotal = (na+ nb+ nc+ ... )RT / V The pressure in a flask containing a mixture of 1 mole of 0.20 mole O2 and 0.80 mole N2 would be the same as the same flask holding 1 mole of O2. Partial pressures are useful when gases are collected by bubbling through water (displacement).

The gas collected is saturated in water vapor which contibutes to the total number of moles of gas in the container. EXAMPLE 5: A sample of H2 was prepared in the laboratory by the reaction: Mg(s) + 2 HCl(aq) MgCl2(aq) + H2(g) 456 mL of gas was collected at 22.0 oC. The total pressure in the flask was 742 torr. How many moles of H2 were collected? The vapor pressure of H2O at 22.0 oC is 19.8 torr. nH2 = PH2V / RT ; nH2 = (0.9503 atm)(0.456 L) / (0.0821 L-atm / mole-K)(295 K) = 0.0179 mole H2. Non-Ideal Gases Johannes Diderik van der Waals (1837-1923) The ideal gas equation (PV=nRT) provides a valuable model of the relations between volume, pressure, temperature and number of particles in a gas.

As an ideal model it serves as a reference for the behavior of real gases. The ideal gas equation makes some simplifying assumptions which are obviously not quite true. Real molecules do have volume and do attract each other. All gases depart from ideal behavior under conditions of low temperature (when liquefaction begins) and high pressure (molecules are more crowed so the volume of the molecule becomes important). Refinements to the ideal gas equation can be made to correct for these deviations. In 1873 J. D. van der Waals proposed his equation, known as the van der Waals equation. As there are attractive forces between molecules, the pressure is lower than the ideal value. To account for this the pressure term is augmented by an attractive force term a/V2. Likewise real molecules have a volume. The volume of the molecules is represented by the term b. The term b is a function of a spherical diameter d known as the van der Waals diameter.

The van der Waals equation for n moles of gas is: The values for a and b, below, are determined empirically: Avogadro's Law: Volume and Amount About Watch and Favorite Watch Watching this resources will notify you when proposed changes or new versions are created so you can keep track of improvements that have been made. Favorite Favoriting this resource allows you to save it in the “My Resources” tab of your account. There, you can easily access this resource later when you’re ready to customize it or assign it to your students. Avogadro's Law: Volume and Amount Avogadro's Law states that at the same temperature and pressure, equal volumes of different gases contain an equal number of particles. Learning Objective State Avogadro's Law and its underlying assumptions Key Points The number of molecules or atoms in a specific volume of ideal gas is independent of size or the gas' molar mass . Avogadro's Law is stated mathematically as follows: $\frac{V}{n} = k$ , where V is the volume of the gas, n is the number of moles of the gas, and k is a proportionality constant.

Volume ratios must be related to the relative numbers of molecules that react; this relationship was crucial in establishing the formulas of simple molecules at a time when the distinction between atoms and molecules was not clearly understood. Term Full Text Definition of Avogadro's Law Avogadro's Law (sometimes referred to as Avogadro's hypothesis or Avogadro's principle) is a gas law; it states that under the same pressure and temperature conditions, equal volumes of all gases contain the same number of molecules. The law is named after Amedeo Avogadro who, in 1811, hypothesized that two given samples of an ideal gas—of the same volume and at the same temperature and pressure—contain the same number of molecules; thus, the number of molecules or atoms in a specific volume of ideal gas is independent of their size or the molar mass of the gas. For example, 1.00 L of N2 gas and 1.00 L of Cl2 gas contain the same number of molecules at Standard Temperature and Pressure (STP).

Avogadro's Law is stated mathematically as: $\frac{V}{n} = k$ V is the volume of the gas, n is the number of moles of the gas, and k is a proportionality constant. As an example, equal volumes of molecular hydrogen and nitrogen contain the same number of molecules and observe ideal gas behavior when they are at the same temperature and pressure. In practice, real gases show small deviations from the ideal behavior and do not adhere to the law perfectly; the law is still a useful approximation for scientists, however. The model contains gas molecules under constant pressure. The barrier moves when the volume of gas expands or contracts. Run the model and select different numbers of molecules from the drop-down menu. What is the relationship between the number of molecules and the volume of a gas? (Note: Although the atoms in this model are in a flat plane, volume is calculated using 0.1 nm as the depth of the container.) Significance of Avogadro's Law Discovering that the volume of a gas was directly proportional to the number of particles it contained was crucial in establishing the formulas for simple molecules at a time (around 1811) when the distinction between atoms and molecules was not clearly understood.

In particular, the existence of diatomic molecules of elements such as H2, O2, and Cl2 was not recognized until the results of experiments involving gas volumes was interpreted. Early chemists calculated the molecular weight of oxygen using the incorrect formula HO for water. This lead to the molecular weight of oxygen being miscalculated as 8, rather than 16. However, when chemists found that an assumed reaction of H + Cl $\rightarrow$ HCl yielded twice the volume of HCl, they realized hydrogen and chlorine were diatomic molecules. The chemists revised their reaction equation to be H2 + Cl2 $\rightarrow$ When chemists revisited their water experiment and their hypothesis that $HO \rightarrow H + O$ , they discovered that the volume of hydrogen gas consumed was twice that of oxygen. By Avogadro's Law, this meant that hydrogen and oxygen were combining in a 2:1 ratio.

This discovery led to the correct molecular formula for water (H2O) and the correct reaction $2H_2O \rightarrow 2H_2 + O_2$ 9.2: Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law - Chemistry LibreTexts 9.2: Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law Last updated Skills to Develop Identify the mathematical relationships between the various properties of gases Use the ideal gas law, and related gas laws, to compute the values of various gas properties under specified conditions During the seventeenth and especially eighteenth centuries, driven both by a desire to understand nature and a quest to make balloons in which they could fly ( Figure ), a number of scientists established the relationships between the macroscopic physical properties of gases, that is, pressure, volume, temperature, and amount of gas. Although their measurements were not precise by today’s standards, they were able to determine the mathematical relationships between pairs of these variables (e.g., pressure and temperature, pressure and volume) that hold for an ideal gas—a hypothetical construct that real gases approximate under certain conditions.

Eventually, these individual laws were combined into a single equation—the ideal gas law—that relates gas quantities for gases and is quite accurate for low pressures and moderate temperatures. We will consider the key developments in individual relationships (for pedagogical reasons not quite in historical order), then put them together in the ideal gas law. In 1783, the first (a) hydrogen-filled balloon flight, (b) manned hot air balloon flight, and (c) manned hydrogen-filled balloon flight occurred. When the hydrogen-filled balloon depicted in (a) landed, the frightened villagers of Gonesse reportedly destroyed it with pitchforks and knives. The launch of the latter was reportedly viewed by 400,000 people in Paris. Pressure and Temperature: Amontons’s Law Imagine filling a rigid container attached to a pressure gauge with gas and then sealing the container so that no gas may escape. If the container is cooled, the gas inside likewise gets colder and its pressure is observed to decrease. Since the container is rigid and tightly sealed, both the volume and number of moles of gas remain constant. If we heat the sphere, the gas inside gets hotter ( Figure ) and the pressure increases.

The effect of temperature on gas pressure: When the hot plate is off, the pressure of the gas in the sphere is relatively low. As the gas is heated, the pressure of the gas in the sphere increases. This relationship between temperature and pressure is observed for any sample of gas confined to a constant volume. An example of experimental pressure-temperature data is shown for a sample of air under these conditions in Figure . We find that temperature and pressure are linearly related, and if the temperature is on the kelvin scale, then P and T are directly proportional (again, when volume and moles of gas are held constant); if the temperature on the kelvin scale increases by a certain factor, the gas pressure increases by the same factor. For a constant volume and amount of air, the pressure and temperature are directly proportional, provided the temperature is in kelvin. (Measurements cannot be made at lower temperatures because of the condensation of the gas.) When this line is extrapolated to lower pressures, it reaches a pressure of 0 at –273 °C, which is 0 on the kelvin scale and the lowest possible temperature, called absolute zero.

Guillaume Amontons was the first to empirically establish the relationship between the pressure and the temperature of a gas (~1700), and Joseph Louis Gay-Lussac determined the relationship more precisely (~1800). Because of this, the P-T relationship for gases is known as either Amontons’s law or Gay-Lussac’s law. Under either name, it states that the pressure of a given amount of gas is directly proportional to its temperature on the kelvin scale when the volume is held constant. Mathematically, this can be written: \(P∝T\ce{\:or\:}P=\ce{constant}×T\ce{\:or\:}P=k×T\) where ∝ means “is proportional to,” and k is a proportionality constant that depends on the identity, amount, and volume of the gas. For a confined, constant volume of gas, the ratio \(\dfrac{P}{T}\) is therefore constant (i.e., \(\dfrac{P}{T}=k\)).

If the gas is initially in “Condition 1” (with P = P1 and T = T1), and then changes to “Condition 2” (with P = P2 and T = T2), we have that \(\dfrac{P_1}{T_1}=k\) which reduces to \(\dfrac{P_1}{T_1}=\dfrac{P_2}{T_2}\) . This equation is useful for pressure-temperature calculations for a confined gas at constant volume. Note that temperatures must be on the kelvin scale for any gas law calculations (0 on the kelvin scale and the lowest possible temperature is called absolute zero). (Also note that there are at least three ways we can describe how the pressure of a gas changes as its temperature changes: We can use a table of values, a graph, or a mathematical equation.) Predicting Change in Pressure with Temperature A can of hair spray is used until it is empty except for the propellant, isobutane gas. (a) On the can is the warning “Store only at temperatures below 120 °F (48.8 °C). Do not incinerate.” Why?

(b) The gas in the can is initially at 24 °C and 360 kPa, and the can has a volume of 350 mL. If the can is left in a car that reaches 50 °C on a hot day, what is the new pressure in the can? Solution (a) The can contains an amount of isobutane gas at a constant volume, so if the temperature is increased by heating, the pressure will increase proportionately. High temperature could lead to high pressure, causing the can to burst. (Also, isobutane is combustible, so incineration could cause the can to explode.) (b) We are looking for a pressure change due to a temperature change at constant volume, so we will use Amontons’s/Gay-Lussac’s law.

Taking P1 and T1 as the initial values, T2 as the temperature where the pressure is unknown and P2 as the unknown pressure, and converting °C to K, we have: \(\dfrac{P_1}{T_1}=\dfrac{P_2}{T_2}\textrm{ which means that }\dfrac{360\:\ce{kPa}}{297\:\ce K}=\dfrac{P_2}{323\:\ce K}\) Rearranging and solving gives: \(P_2=\mathrm{\dfrac{360\:kPa×323\cancel{K}}{297\cancel{K}}=390\:kPa}\) Check Your Learning A sample of nitrogen, N2, occupies 45.0 mL at 27 °C and 600 torr. What pressure will it have if cooled to –73 °C while the volume remains constant? Answer: 400 torr Volume and Temperature: Charles’s Law If we fill a balloon with air and seal it, the balloon contains a specific amount of air at atmospheric pressure, let’s say 1 atm.

If we put the balloon in a refrigerator, the gas inside gets cold and the balloon shrinks (although both the amount of gas and its pressure remain constant). If we make the balloon very cold, it will shrink a great deal, and it expands again when it warms up. This video shows how cooling and heating a gas causes its volume to decrease or increase, respectively. These examples of the effect of temperature on the volume of a given amount of a confined gas at constant pressure are true in general: The volume increases as the temperature increases, and decreases as the temperature decreases. Volume-temperature data for a 1-mole sample of methane gas at 1 atm are listed and graphed in Figure . <figcaption>The volume and temperature are linearly related for 1 mole of methane gas at a constant pressure of 1 atm. If the temperature is in kelvin, volume and temperature are directly proportional. The line stops at 111 K because methane liquefies at this temperature; when extrapolated, it intersects the graph’s origin, representing a temperature of absolute zero.

The relationship between the volume and temperature of a given amount of gas at constant pressure is known as Charles’s law in recognition of the French scientist and balloon flight pioneer Jacques Alexandre César Charles. Charles’s law states that the volume of a given amount of gas is directly proportional to its temperature on the kelvin scale when the pressure is held constant. Mathematically, this can be written as: \(VαT\ce{\:or\:}V=\ce{constant}·T\ce{\:or\:}V=k·T\ce{\:or\:}V_1/T_1=V_2/T_2\) with k being a proportionality constant that depends on the amount and pressure of the gas. For a confined, constant pressure gas sample, \(\dfrac{V}{T}\) is constant (i.e., the ratio = k), and as seen with the V-T relationship, this leads to another form of Charles’s law: \(\dfrac{V_1}{T_1}=\dfrac{V_2}{T_2}\).

Predicting Change in Volume with Temperature A sample of carbon dioxide, CO2, occupies 0.300 L at 10 °C and 750 torr. What volume will the gas have at 30 °C and 750 torr? Solution Because we are looking for the volume change caused by a temperature change at constant pressure, this is a job for Charles’s law.

Taking V1 and T1 as the initial values, T2 as the temperature at which the volume is unknown and V2 as the unknown volume, and converting °C into K we have: \(\dfrac{V_1}{T_1}=\dfrac{V_2}{T_2}\textrm{ which means that }\dfrac{0.300\:\ce L}{283\:\ce K}=\dfrac{V_2}{303\:\ce K}\) Rearranging and solving gives: \(V_2=\mathrm{\dfrac{0.300\:L×303\cancel{K}}{283\cancel{K}}=0.321\:L}\) This answer supports our expectation from Charles’s law, namely, that raising the gas temperature (from 283 K to 303 K) at a constant pressure will yield an increase in its volume (from 0.300 L to 0.321 L). Check Your Learning A sample of oxygen, O2, occupies 32.2 mL at 30 °C and 452 torr.

What volume will it occupy at –70 °C and the same pressure? Answer: 21.6 mL Measuring Temperature with a Volume Change Temperature is sometimes measured with a gas thermometer by observing the change in the volume of the gas as the temperature changes at constant pressure. The hydrogen in a particular hydrogen gas thermometer has a volume of 150.0 cm3 when immersed in a mixture of ice and water (0.00 °C). When immersed in boiling liquid ammonia, the volume of the hydrogen, at the same pressure, is 131.7 cm3. Find the temperature of boiling ammonia on the kelvin and Celsius scales. Solution A volume change caused by a temperature change at constant pressure means we should use Charles’s law.

Taking V1 and T1 as the initial values, T2 as the temperature at which the volume is unknown and V2 as the unknown volume, and converting °C into K we have: \(\dfrac{V_1}{T_1}=\dfrac{V_2}{T_2}\textrm{ which means that }\mathrm{\dfrac{150.0\:cm^3}{273.15\:K}}=\dfrac{131.7\:\ce{cm}^3}{T_2}\) Rearrangement gives \(T_2=\mathrm{\dfrac{131.7\cancel{cm}^3×273.15\:K}{150.0\:cm^3}=239.8\:K}\) Subtracting 273.15 from 239.8 K, we find that the temperature of the boiling ammonia on the Celsius scale is –33.4 °C.

Check Your Learning What is the volume of a sample of ethane at 467 K and 1.1 atm if it occupies 405 mL at 298 K and 1.1 atm? Answer: 635 mL Volume and Pressure: Boyle’s Law If we partially fill an airtight syringe with air, the syringe contains a specific amount of air at constant temperature, say 25 °C. If we slowly push in the plunger while keeping temperature constant, the gas in the syringe is compressed into a smaller volume and its pressure increases; if we pull out the plunger, the volume increases and the pressure decreases. This example of the effect of volume on the pressure of a given amount of a confined gas is true in general. Decreasing the volume of a contained gas will increase its pressure, and increasing its volume will decrease its pressure. In fact, if the volume increases by a certain factor, the pressure decreases by the same factor, and vice versa. Volume-pressure data for an air sample at room temperature are graphed in Figure .

<figcaption>When a gas occupies a smaller volume, it exerts a higher pressure; when it occupies a larger volume, it exerts a lower pressure (assuming the amount of gas and the temperature do not change). Since P and V are inversely proportional, a graph of \(\dfrac{1}{P}\) vs. V is linear. Unlike the P-T and V-T relationships, pressure and volume are not directly proportional to each other. Instead, P and V exhibit inverse proportionality: Increasing the pressure results in a decrease of the volume of the gas. Mathematically this can be written: \(Pα1/V\textrm{ or }P=k⋅1/V\textrm{ or }P⋅V=k\textrm{ or }P_1V_1=P_2V_2\) with k being a constant. Graphically, this relationship is shown by the straight line that results when plotting the inverse of the pressure \(\left(\dfrac{1}{P}\right)\) versus the volume (V), or the inverse of volume \(\left(\dfrac{1}{V}\right)\) versus the pressure (V).

Graphs with curved lines are difficult to read accurately at low or high values of the variables, and they are more difficult to use in fitting theoretical equations and parameters to experimental data. For those reasons, scientists often try to find a way to “linearize” their data. If we plot P versus V, we obtain a hyperbola (see Figure ). The relationship between pressure and volume is inversely proportional. (a) The graph of P vs. V is a parabola, whereas (b) the graph of \(\left(\dfrac{1}{P}\right)\) vs. V is linear. The relationship between the volume and pressure of a given amount of gas at constant temperature was first published by the English natural philosopher Robert Boyle over 300 years ago. It is summarized in the statement now known as Boyle’s law: The volume of a given amount of gas held at constant temperature is inversely proportional to the pressure under which it is measured. Volume of a Gas Sample The sample of gas in Figure has a volume of 15.0 mL at a pressure of 13.0 psi.

Determine the pressure of the gas at a volume of 7.5 mL, using: (a) the P-V graph in Figure (b) the vs. V graph in Figure (c) the Boyle’s law equation Comment on the likely accuracy of each method. Answer: (a) about 17–18 mL; (b) ~18 mL; (c) 17.7 mL; it was more difficult to estimate well from the P-V graph, so (a) is likely more inaccurate than (b); the calculation will be as accurate as the equation and measurements allow Breathing and Boyle’s Law What do you do about 20 times per minute for your whole life, without break, and often without even being aware of it? The answer, of course, is respiration, or breathing. How does it work? It turns out that the gas laws apply here. Your lungs take in gas that your body needs (oxygen) and get rid of waste gas (carbon dioxide). Lungs are made of spongy, stretchy tissue that expands and contracts while you breathe.

When you inhale, your diaphragm and intercostal muscles (the muscles between your ribs) contract, expanding your chest cavity and making your lung volume larger. The increase in volume leads to a decrease in pressure (Boyle’s law). This causes air to flow into the lungs (from high pressure to low pressure). When you exhale, the process reverses: Your diaphragm and rib muscles relax, your chest cavity contracts, and your lung volume decreases, causing the pressure to increase (Boyle’s law again), and air flows out of the lungs (from high pressure to low pressure). You then breathe in and out again, and again, repeating this Boyle’s law cycle for the rest of your life ( Figure ). Breathing occurs because expanding and contracting lung volume creates small pressure differences between your lungs and your surroundings, causing air to be drawn into and forced out of your lungs. Moles of Gas and Volume: Avogadro’s Law The Italian scientist Amedeo Avogadro advanced a hypothesis in 1811 to account for the behavior of gases, stating that equal volumes of all gases, measured under the same conditions of temperature and pressure, contain the same number of molecules.

Over time, this relationship was supported by many experimental observations as expressed by Avogadro’s law: For a confined gas, the volume (V) and number of moles (n) are directly proportional if the pressure and temperature both remain constant. In equation form, this is written as: \(V∝n\textrm{ or }V=k×n\textrm{ or }\dfrac{V_1}{n_1}=\dfrac{V_2}{n_2}\) Mathematical relationships can also be determined for the other variable pairs, such as P versus n, and n versus T. Visit this interactive PhET simulation to investigate the relationships between pressure, volume, temperature, and amount of gas. Use the simulation to examine the effect of changing one parameter on another while holding the other parameters constant (as described in the preceding sections on the various gas laws).

The Ideal Gas Law To this point, four separate laws have been discussed that relate pressure, volume, temperature, and the number of moles of the gas: Boyle’s law: PV = constant at constant T and n Amontons’s law: \(\dfrac{V}{n}\) = constant at constant P and T Combining these four laws yields the ideal gas law, a relation between the pressure, volume, temperature, and number of moles of a gas: \(PV=nRT\) where P is the pressure of a gas, V is its volume, n is the number of moles of the gas, T is its temperature on the kelvin scale, and R is a constant called the ideal gas constant or the universal gas constant. The units used to express pressure, volume, and temperature will determine the proper form of the gas constant as required by dimensional analysis, the most commonly encountered values being 0.08206 L atm mol–1 K–1 and 8.314 kPa L mol–1 K–1.

Gases whose properties of P, V, and T are accurately described by the ideal gas law (or the other gas laws) are said to exhibit ideal behavior or to approximate the traits of an ideal gas. An ideal gas is a hypothetical construct that may be used along with kinetic molecular theory to effectively explain the gas laws as will be described in a later module of this chapter. Although all the calculations presented in this module assume ideal behavior, this assumption is only reasonable for gases under conditions of relatively low pressure and high temperature. In the final module of this chapter, a modified gas law will be introduced that accounts for the non-ideal behavior observed for many gases at relatively high pressures and low temperatures. The ideal gas equation contains five terms, the gas constant R and the variable properties P, V, n, and T. Specifying any four of these terms will permit use of the ideal gas law to calculate the fifth term as demonstrated in the following example exercises. Using the Ideal Gas Law Methane, CH4, is being considered for use as an alternative automotive fuel to replace gasoline. One gallon of gasoline could be replaced by 655 g of CH4.

What is the volume of this much methane at 25 °C and 745 torr? Solution We must rearrange PV = nRT to solve for V: \(V=\dfrac{nRT}{P}\) If we choose to use R = 0.08206 L atm mol–1 K–1, then the amount must be in moles, temperature must be in kelvin, and pressure must be in atm.

Converting into the “right” units: \(n=\mathrm{655\cancel{g\:CH_4}×\dfrac{1\:mol}{16.043\cancel{g\: CH_4}}=40.8\:mol}\) \(T=\mathrm{25°C+273=298\:K}\) \(P=\mathrm{745\cancel{torr}×\dfrac{1atm}{760\cancel{torr}}=0.980\:atm}\) \(V=\dfrac{nRT}{P}=\mathrm{\dfrac{(40.8\cancel{mol})(0.08206\:L\cancel{atm mol^{–1}K^{–1}})(298\:K)}{0.980\cancel{atm}}=1.02×10^3\:L}\) It would require 1020 L (269 gal) of gaseous methane at about 1 atm of pressure to replace 1 gal of gasoline.

It requires a large container to hold enough methane at 1 atm to replace several gallons of gasoline. Check Your Learning Calculate the pressure in bar of 2520 moles of hydrogen gas stored at 27 °C in the 180-L storage tank of a modern hydrogen-powered car. Answer: 350 bar If the number of moles of an ideal gas are kept constant under two different sets of conditions, a useful mathematical relationship called the combined gas law is obtained: \(\dfrac{P_1V_1}{T_1}=\dfrac{P_2V_2}{T_2}\) using units of atm, L, and K. Both sets of conditions are equal to the product of n × R (where n = the number of moles of the gas and R is the ideal gas law constant). Example Using the Combined Gas Law When filled with air, a typical scuba tank with a volume of 13.2 L has a pressure of 153 atm ( Figure ).

If the water temperature is 27 °C, how many liters of air will such a tank provide to a diver’s lungs at a depth of approximately 70 feet in the ocean where the pressure is 3.13 atm? Scuba divers use compressed air to breathe while underwater.

(credit: modification of work by Mark Goodchild) Letting 1 represent the air in the scuba tank and 2 represent the air in the lungs, and noting that body temperature (the temperature the air will be in the lungs) is 37 °C, we have: \(\dfrac{P_1V_1}{T_1}=\dfrac{P_2V_2}{T_2}⟶\mathrm{\dfrac{(153\:atm)(13.2\:L)}{(300\:K)}=\dfrac{(3.13\:atm)(\mathit{V}_2)}{(310\:K)}}\) Solving for V2: \(V_2=\mathrm{\dfrac{(153\cancel{atm})(13.2\:L)(310\cancel{K})}{(300\cancel{K})(3.13\cancel{atm})}=667\:L}\) (Note: Be advised that this particular example is one in which the assumption of ideal gas behavior is not very reasonable, since it involves gases at relatively high pressures and low temperatures.

Despite this limitation, the calculated volume can be viewed as a good “ballpark” estimate.) Exercise A sample of ammonia is found to occupy 0.250 L under laboratory conditions of 27 °C and 0.850 atm. Find the volume of this sample at 0 °C and 1.00 atm. Answer: The Interdependence between Ocean Depth and Pressure in Scuba Diving Whether scuba diving at the Great Barrier Reef in Australia (shown in Figure ) or in the Caribbean, divers must understand how pressure affects a number of issues related to their comfort and safety. <fi Scuba divers, whether at the Great Barrier Reef or in the Caribbean, must be aware of buoyancy, pressure equalization, and the amount of time they spend underwater, to avoid the risks associated with pressurized gases in the body. (credit: Kyle Taylor) Pressure increases with ocean depth, and the pressure changes most rapidly as divers reach the surface. The pressure a diver experiences is the sum of all pressures above the diver (from the water and the air).

Most pressure measurements are given in units of atmospheres, expressed as “atmospheres absolute” or ATA in the diving community: Every 33 feet of salt water represents 1 ATA of pressure in addition to 1 ATA of pressure from the atmosphere at sea level. As a diver descends, the increase in pressure causes the body’s air pockets in the ears and lungs to compress; on the ascent, the decrease in pressure causes these air pockets to expand, potentially rupturing eardrums or bursting the lungs. Divers must therefore undergo equalization by adding air to body airspaces on the descent by breathing normally and adding air to the mask by breathing out of the nose or adding air to the ears and sinuses by equalization techniques; the corollary is also true on ascent, divers must release air from the body to maintain equalization. Buoyancy, or the ability to control whether a diver sinks or floats, is controlled by the buoyancy compensator (BCD). If a diver is ascending, the air in his BCD expands because of lower pressure according to Boyle’s law (decreasing the pressure of gases increases the volume). The expanding air increases the buoyancy of the diver, and she or he begins to ascend.

The diver must vent air from the BCD or risk an uncontrolled ascent that could rupture the lungs. In descending, the increased pressure causes the air in the BCD to compress and the diver sinks much more quickly; the diver must add air to the BCD or risk an uncontrolled descent, facing much higher pressures near the ocean floor. The pressure also impacts how long a diver can stay underwater before ascending. The deeper a diver dives, the more compressed the air that is breathed because of increased pressure: If a diver dives 33 feet, the pressure is 2 ATA and the air would be compressed to one-half of its original volume. The diver uses up available air twice as fast as at the surface. Standard Conditions of Temperature and Pressure We have seen that the volume of a given quantity of gas and the number of molecules (moles) in a given volume of gas vary with changes in pressure and temperature. Chemists sometimes make comparisons against a standard temperature and pressure (STP) for reporting properties of gases: 273.15 K and 1 atm (101.325 kPa).

At STP, an ideal gas has a volume of about 22.4 L—this is referred to as the standard molar volume ( Figure ). Since the number of moles in a given volume of gas varies with pressure and temperature changes, chemists use standard temperature and pressure (273.15 K and 1 atm or 101.325 kPa) to report properties of gases. Key Concepts and Summary The behavior of gases can be described by several laws based on experimental observations of their properties. The pressure of a given amount of gas is directly proportional to its absolute temperature, provided that the volume does not change (Amontons’s law). The volume of a given gas sample is directly proportional to its absolute temperature at constant pressure (Charles’s law). The volume of a given amount of gas is inversely proportional to its pressure when temperature is held constant (Boyle’s law). Under the same conditions of temperature and pressure, equal volumes of all gases contain the same number of molecules (Avogadro’s law).

The equations describing these laws are special cases of the ideal gas law, PV = nRT, where P is the pressure of the gas, V is its volume, n is the number of moles of the gas, T is its kelvin temperature, and R is the ideal (universal) gas constant. Key Equations

Question: Which physicist's law states that equal volumes of all gases, measured at the same temperature and pressure, contain the same number of molecules?
Answer: Amadeo Avogadro