# Fluid Statics

## Pressures in the Body

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the concept of pressure the in human body.
2. Explain systolic and diastolic blood pressures.
3. Describe pressures in the eye, lungs, spinal column, bladder, and skeletal system.

### Pressure in the Body

Next to taking a person’s temperature and weight, measuring blood pressure is the most common of all medical examinations. Control of high blood pressure is largely responsible for the significant decreases in heart attack and stroke fatalities achieved in the last three decades. The pressures in various parts of the body can be measured and often provide valuable medical indicators. In this section, we consider a few examples together with some of the physics that accompanies them.
 lists some of the measured pressures in mm Hg, the units most commonly quoted.

### Blood Pressure

Common arterial blood pressure measurements typically produce values of 120 mm Hg and 80 mm Hg, respectively, for systolic and diastolic pressures. Both pressures have health implications. When systolic pressure is chronically high, the risk of stroke and heart attack is increased. If, however, it is too low, fainting is a problem. Systolic pressure increases dramatically during exercise to increase blood flow and returns to normal afterward. This change produces no ill effects and, in fact, may be beneficial to the tone of the circulatory system. Diastolic pressure can be an indicator of fluid balance. When low, it may indicate that a person is hemorrhaging internally and needs a transfusion. Conversely, high diastolic pressure indicates a ballooning of the blood vessels, which may be due to the transfusion of too much fluid into the circulatory system. High diastolic pressure is also an indication that blood vessels are not dilating properly to pass blood through. This can seriously strain the heart in its attempt to pump blood.
Blood leaves the heart at about 120 mm Hg but its pressure continues to decrease (to almost 0) as it goes from the aorta to smaller arteries to small veins (see ). The pressure differences in the circulation system are caused by blood flow through the system as well as the position of the person. For a person standing up, the pressure in the feet will be larger than at the heart due to the weight of the blood . If we assume that the distance between the heart and the feet of a person in an upright position is 1.4 m, then the increase in pressure in the feet relative to that in the heart (for a static column of blood) is given by
Standing a long time can lead to an accumulation of blood in the legs and swelling. This is the reason why soldiers who are required to stand still for long periods of time have been known to faint. Elastic bandages around the calf can help prevent this accumulation and can also help provide increased pressure to enable the veins to send blood back up to the heart. For similar reasons, doctors recommend tight stockings for long-haul flights.
Blood pressure may also be measured in the major veins, the heart chambers, arteries to the brain, and the lungs. But these pressures are usually only monitored during surgery or for patients in intensive care since the measurements are invasive. To obtain these pressure measurements, qualified health care workers thread thin tubes, called catheters, into appropriate locations to transmit pressures to external measuring devices.
The heart consists of two pumps—the right side forcing blood through the lungs and the left causing blood to flow through the rest of the body (). Right-heart failure, for example, results in a rise in the pressure in the vena cavae and a drop in pressure in the arteries to the lungs. Left-heart failure results in a rise in the pressure entering the left side of the heart and a drop in aortal pressure. Implications of these and other pressures on flow in the circulatory system will be discussed in more detail in Fluid Dynamics and Its Biological and Medical Applications.

### Pressure in the Eye

The shape of the eye is maintained by fluid pressure, called intraocular pressure, which is normally in the range of 12.0 to 24.0 mm Hg. When the circulation of fluid in the eye is blocked, it can lead to a buildup in pressure, a condition called glaucoma. The net pressure can become as great as 85.0 mm Hg, an abnormally large pressure that can permanently damage the optic nerve. To get an idea of the force involved, suppose the back of the eye has an area of , and the net pressure is 85.0 mm Hg. Force is given by

. To get

 in newtons, we convert the area to

 (
). Then we calculate as follows:
This force is the weight of about a 680-g mass. A mass of 680 g resting on the eye (imagine 1.5 lb resting on your eye) would be sufficient to cause it damage. (A normal force here would be the weight of about 120 g, less than one-quarter of our initial value.)
People over 40 years of age are at greatest risk of developing glaucoma and should have their intraocular pressure tested routinely. Most measurements involve exerting a force on the (anesthetized) eye over some area (a pressure) and observing the eye’s response. A noncontact approach uses a puff of air and a measurement is made of the force needed to indent the eye (). If the intraocular pressure is high, the eye will deform less and rebound more vigorously than normal. Excessive intraocular pressures can be detected reliably and sometimes controlled effectively.

### Pressure Associated with the Lungs

The pressure inside the lungs increases and decreases with each breath. The pressure drops to below atmospheric pressure (negative gauge pressure) when you inhale, causing air to flow into the lungs. It increases above atmospheric pressure (positive gauge pressure) when you exhale, forcing air out.
Lung pressure is controlled by several mechanisms. Muscle action in the diaphragm and rib cage is necessary for inhalation; this muscle action increases the volume of the lungs thereby reducing the pressure within them . Surface tension in the alveoli creates a positive pressure opposing inhalation. (See Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action.) You can exhale without muscle action by letting surface tension in the alveoli create its own positive pressure. Muscle action can add to this positive pressure to produce forced exhalation, such as when you blow up a balloon, blow out a candle, or cough.
The lungs, in fact, would collapse due to the surface tension in the alveoli, if they were not attached to the inside of the chest wall by liquid adhesion. The gauge pressure in the liquid attaching the lungs to the inside of the chest wall is thus negative, ranging from  to  during exhalation and inhalation, respectively. If air is allowed to enter the chest cavity, it breaks the attachment, and one or both lungs may collapse. Suction is applied to the chest cavity of surgery patients and trauma victims to reestablish negative pressure and inflate the lungs.

### Other Pressures in the Body

### Spinal Column and Skull

Normally, there is a 5- to12-mm Hg pressure in the fluid surrounding the brain and filling the spinal column. This cerebrospinal fluid serves many purposes, one of which is to supply flotation to the brain. The buoyant force supplied by the fluid nearly equals the weight of the brain, since their densities are nearly equal. If there is a loss of fluid, the brain rests on the inside of the skull, causing severe headaches, constricted blood flow, and serious damage. Spinal fluid pressure is measured by means of a needle inserted between vertebrae that transmits the pressure to a suitable measuring device.

### Bladder Pressure

This bodily pressure is one of which we are often aware. In fact, there is a relationship between our awareness of this pressure and a subsequent increase in it. Bladder pressure climbs steadily from zero to about 25 mm Hg as the bladder fills to its normal capacity of . This pressure triggers the micturition reflex, which stimulates the feeling of needing to urinate. What is more, it also causes muscles around the bladder to contract, raising the pressure to over 100 mm Hg, accentuating the sensation. Coughing, straining, tensing in cold weather, wearing tight clothes, and experiencing simple nervous tension all can increase bladder pressure and trigger this reflex. So can the weight of a pregnant woman’s fetus, especially if it is kicking vigorously or pushing down with its head! Bladder pressure can be measured by a catheter or by inserting a needle through the bladder wall and transmitting the pressure to an appropriate measuring device. One hazard of high bladder pressure (sometimes created by an obstruction), is that such pressure can force urine back into the kidneys, causing potentially severe damage.

### Pressures in the Skeletal System

These pressures are the largest in the body, due both to the high values of initial force, and the small areas to which this force is applied, such as in the joints.. For example, when a person lifts an object improperly, a force of 5000 N may be created between vertebrae in the spine, and this may be applied to an area as small as . The pressure created is

 or about 50 atm! This pressure can damage both the spinal discs (the cartilage between vertebrae), as well as the bony vertebrae themselves. Even under normal circumstances, forces between vertebrae in the spine are large enough to create pressures of several atmospheres. Most causes of excessive pressure in the skeletal system can be avoided by lifting properly and avoiding extreme physical activity. (See Forces and Torques in Muscles and Joints.)
There are many other interesting and medically significant pressures in the body. For example, pressure caused by various muscle actions drives food and waste through the digestive system. Stomach pressure behaves much like bladder pressure and is tied to the sensation of hunger. Pressure in the relaxed esophagus is normally negative because pressure in the chest cavity is normally negative. Positive pressure in the stomach may thus force acid into the esophagus, causing “heartburn.” Pressure in the middle ear can result in significant force on the eardrum if it differs greatly from atmospheric pressure, such as while scuba diving. The decrease in external pressure is also noticeable during plane flights (due to a decrease in the weight of air above relative to that at the Earth’s surface). The Eustachian tubes connect the middle ear to the throat and allow us to equalize pressure in the middle ear to avoid an imbalance of force on the eardrum.
Many pressures in the human body are associated with the flow of fluids. Fluid flow will be discussed in detail in the Fluid Dynamics and Its Biological and Medical Applications.

### Section Summary

1. Measuring blood pressure is among the most common of all medical examinations.
2. The pressures in various parts of the body can be measured and often provide valuable medical indicators.
3. The shape of the eye is maintained by fluid pressure, called intraocular pressure.
4. When the circulation of fluid in the eye is blocked, it can lead to a buildup in pressure, a condition called glaucoma.
5. Some of the other pressures in the body are spinal and skull pressures, bladder pressure, pressures in the skeletal system.

### Problems & Exercises


# Fluid Dynamics and Its Biological and Medical Applications

## Introduction to Fluid Dynamics and Its Biological and Medical Applications

We have dealt with many situations in which fluids are static. But by their very definition, fluids flow. Examples come easily—a column of smoke rises from a camp fire, water streams from a fire hose, blood courses through your veins. Why does rising smoke curl and twist? How does a nozzle increase the speed of water emerging from a hose? How does the body regulate blood flow? The physics of fluids in motion—fluid dynamics—allows us to answer these and many other questions.


# Fluid Dynamics and Its Biological and Medical Applications

## Flow Rate and Its Relation to Velocity

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate flow rate.
2. Define units of volume.
3. Describe incompressible fluids.
4. Explain the consequences of the equation of continuity.
Flow rate  is defined to be the volume of fluid passing by some location through an area during a period of time, as seen in . In symbols, this can be written as
where  is the volume and  is the elapsed time.
The SI unit for flow rate is , but a number of other units for  are in common use. For example, the heart of a resting adult pumps blood at a rate of 5.00 liters per minute (L/min). Note that a liter (L) is 1/1000 of a cubic meter or 1000 cubic centimeters ( or ). In this text we shall use whatever metric units are most convenient for a given situation.
Flow rate and velocity are related, but quite different, physical quantities. To make the distinction clear, think about the flow rate of a river. The greater the velocity of the water, the greater the flow rate of the river. But flow rate also depends on the size of the river. A rapid mountain stream carries far less water than the Amazon River in Brazil, for example. The precise relationship between flow rate  and velocity  is
where  is the cross-sectional area and  is the average velocity. This equation seems logical enough. The relationship tells us that flow rate is directly proportional to both the magnitude of the average velocity (hereafter referred to as the speed) and the size of a river, pipe, or other conduit. The larger the conduit, the greater its cross-sectional area.  illustrates how this relationship is obtained. The shaded cylinder has a volume
which flows past the point  in a time . Dividing both sides of this relationship by  gives
We note that  and the average speed is

. Thus the equation becomes .
 shows an incompressible fluid flowing along a pipe of decreasing radius. Because the fluid is incompressible, the same amount of fluid must flow past any point in the tube in a given time to ensure continuity of flow. In this case, because the cross-sectional area of the pipe decreases, the velocity must necessarily increase. This logic can be extended to say that the flow rate must be the same at all points along the pipe. In particular, for points 1 and 2,
This is called the equation of continuity and is valid for any incompressible fluid. The consequences of the equation of continuity can be observed when water flows from a hose into a narrow spray nozzle: it emerges with a large speed—that is the purpose of the nozzle. Conversely, when a river empties into one end of a reservoir, the water slows considerably, perhaps picking up speed again when it leaves the other end of the reservoir. In other words, speed increases when cross-sectional area decreases, and speed decreases when cross-sectional area increases.
Since liquids are essentially incompressible, the equation of continuity is valid for all liquids. However, gases are compressible, and so the equation must be applied with caution to gases if they are subjected to compression or expansion.
The solution to the last part of the example shows that speed is inversely proportional to the square of the radius of the tube, making for large effects when radius varies. We can blow out a candle at quite a distance, for example, by pursing our lips, whereas blowing on a candle with our mouth wide open is quite ineffective.
In many situations, including in the cardiovascular system, branching of the flow occurs. The blood is pumped from the heart into arteries that subdivide into smaller arteries (arterioles) which branch into very fine vessels called capillaries. In this situation, continuity of flow is maintained but it is the sum of the flow rates in each of the branches in any portion along the tube that is maintained. The equation of continuity in a more general form becomes
where  and  are the number of branches in each of the sections along the tube.

### Test Prep for AP Courses

### Section Summary

1. Flow rate  is defined to be the volume  flowing past a point in time

, or

 where
 is volume and

 is time.
2. The SI unit of volume is
.
3. Another common unit is the liter (L), which is .
4. Flow rate and velocity are related by
 where  is the cross-sectional area of the flow and

 is its average velocity.
5. For incompressible fluids, flow rate at various points is constant. That is,

### Conceptual Questions

### Problems & Exercises


# Fluid Dynamics and Its Biological and Medical Applications

## Bernoulli’s Equation

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the terms in Bernoulli’s equation.
2. Explain how Bernoulli’s equation is related to conservation of energy.
3. Explain how to derive Bernoulli’s principle from Bernoulli’s equation.
4. Calculate with Bernoulli’s principle.
5. List some applications of Bernoulli’s principle.
When a fluid flows into a narrower channel, its speed increases. That means its kinetic energy also increases. Where does that change in kinetic energy come from? The increased kinetic energy comes from the net work done on the fluid to push it into the channel and the work done on the fluid by the gravitational force, if the fluid changes vertical position. Recall the work-energy theorem,
There is a pressure difference when the channel narrows. This pressure difference results in a net force on the fluid: recall that pressure times area equals force. The net work done increases the fluid’s kinetic energy. As a result, the pressure will drop in a rapidly-moving fluid, whether or not the fluid is confined to a tube.
There are a number of common examples of pressure dropping in rapidly-moving fluids. Shower curtains have a disagreeable habit of bulging into the shower stall when the shower is on. The high-velocity stream of water and air creates a region of lower pressure inside the shower, and standard atmospheric pressure on the other side. The pressure difference results in a net force inward pushing the curtain in. You may also have noticed that when passing a truck on the highway, your car tends to veer toward it. The reason is the same—the high velocity of the air between the car and the truck creates a region of lower pressure, and the vehicles are pushed together by greater pressure on the outside. (See .) This effect was observed as far back as the mid-1800s, when it was found that trains passing in opposite directions tipped precariously toward one another.

### Bernoulli’s Equation

The relationship between pressure and velocity in fluids is described quantitatively by Bernoulli’s equation, named after its discoverer, the Swiss scientist Daniel Bernoulli (1700–1782). Bernoulli’s equation states that for an incompressible, frictionless fluid, the following sum is constant:
where  is the absolute pressure,  is the fluid density,  is the velocity of the fluid,  is the height above some reference point, and  is the acceleration due to gravity. If we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant. Let the subscripts 1 and 2 refer to any two points along the path that the bit of fluid follows; Bernoulli’s equation becomes
Bernoulli’s equation is a form of the conservation of energy principle. Note that the second and third terms are the kinetic and potential energy with  replaced by . In fact, each term in the equation has units of energy per unit volume. We can prove this for the second term by substituting   into it and gathering terms:
So  is the kinetic energy per unit volume. Making the same substitution into the third term in the equation, we find
so  is the gravitational potential energy per unit volume. Note that pressure  has units of energy per unit volume, too. Since , its units are . If we multiply these by m/m, we obtain , or energy per unit volume. Bernoulli’s equation is, in fact, just a convenient statement of conservation of energy for an incompressible fluid in the absence of friction.
The general form of Bernoulli’s equation has three terms in it, and it is broadly applicable. To understand it better, we will look at a number of specific situations that simplify and illustrate its use and meaning.

### Bernoulli’s Equation for Static Fluids

Let us first consider the very simple situation where the fluid is static—that is, . Bernoulli’s equation in that case is
We can further simplify the equation by taking  (we can always choose some height to be zero, just as we often have done for other situations involving the gravitational force, and take all other heights to be relative to this). In that case, we get
This equation tells us that, in static fluids, pressure increases with depth. As we go from point 1 to point 2 in the fluid, the depth increases by , and consequently,  is greater than  by an amount . In the very simplest case,

 is zero at the top of the fluid, and we get the familiar relationship

. (Recall that

 and

) Bernoulli’s equation includes the fact that the pressure due to the weight of a fluid is

. Although we introduce Bernoulli’s equation for fluid flow, it includes much of what we studied for static fluids in the preceding chapter.

### Bernoulli’s Principle—Bernoulli’s Equation at Constant Depth

Another important situation is one in which the fluid moves but its depth is constant—that is, . Under that condition, Bernoulli’s equation becomes
Situations in which fluid flows at a constant depth are so important that this equation is often called Bernoulli’s principle. It is Bernoulli’s equation for fluids at constant depth. (Note again that this applies to a small volume of fluid as we follow it along its path.) As we have just discussed, pressure drops as speed increases in a moving fluid. We can see this from Bernoulli’s principle. For example, if  is greater than  in the equation, then  must be less than  for the equality to hold.

### Applications of Bernoulli’s Principle

There are a number of devices and situations in which fluid flows at a constant height and, thus, can be analyzed with Bernoulli’s principle.

### Entrainment

People have long put the Bernoulli principle to work by using reduced pressure in high-velocity fluids to move things about. With a higher pressure on the outside, the high-velocity fluid forces other fluids into the stream. This process is called entrainment. Entrainment devices have been in use since ancient times, particularly as pumps to raise water small heights, as in draining swamps, fields, or other low-lying areas. Some other devices that use the concept of entrainment are shown in .

### Wings and Sails

The airplane wing is a beautiful example of Bernoulli’s principle in action. (a) shows the characteristic shape of a wing. The wing is tilted upward at a small angle and the upper surface is longer, causing air to flow faster over it. The pressure on top of the wing is therefore reduced, creating a net upward force or lift. (Wings can also gain lift by pushing air downward, utilizing the conservation of momentum principle. The deflected air molecules result in an upward force on the wing — Newton’s third law.) Sails also have the characteristic shape of a wing. (See (b).) The pressure on the front side of the sail, , is lower than the pressure on the back of the sail, . This results in a forward force and even allows you to sail into the wind.

### Velocity measurement

 shows two devices that measure fluid velocity based on Bernoulli’s principle. The manometer in (a) is connected to two tubes that are small enough not to appreciably disturb the flow. The tube facing the oncoming fluid creates a dead spot having zero velocity () in front of it, while fluid passing the other tube has velocity . This means that Bernoulli’s principle as stated in

     becomes
Thus pressure  over the second opening is reduced by , and so the fluid in the manometer rises by

   on the side connected to the second opening, where
(Recall that the symbol  means “proportional to.”) Solving for , we see that
(b) shows a version of this device that is in common use for measuring various fluid velocities; such devices are frequently used as air speed indicators in aircraft.

### Test Prep for AP Courses

### Summary

1. Bernoulli’s equation states that the sum on each side of the following equation is constant, or the same at any two points in an incompressible frictionless fluid:

2. Bernoulli’s principle is Bernoulli’s equation applied to situations in which depth is constant. The terms involving depth (or height h ) subtract out, yielding

3. Bernoulli’s principle has many applications, including entrainment, wings and sails, and velocity measurement.

### Conceptual Questions

### Problems & Exercises


# Fluid Dynamics and Its Biological and Medical Applications

## The Most General Applications of Bernoulli’s Equation

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate using Torricelli’s theorem.
2. Calculate power in fluid flow.

### Torricelli’s Theorem

 shows water gushing from a large tube through a dam. What is its speed as it emerges? Interestingly, if resistance is negligible, the speed is just what it would be if the water fell a distance  from the surface of the reservoir; the water’s speed is independent of the size of the opening. Let us check this out. Bernoulli’s equation must be used since the depth is not constant. We consider water flowing from the surface (point 1) to the tube’s outlet (point 2). Bernoulli’s equation as stated in previously is
Both  and  equal atmospheric pressure ( is atmospheric pressure because it is the pressure at the top of the reservoir.  must be atmospheric pressure, since the emerging water is surrounded by the atmosphere and cannot have a pressure different from atmospheric pressure.) and subtract out of the equation, leaving
Solving this equation for , noting that the density

   cancels (because the fluid is incompressible), yields
We let ; the equation then becomes
where  is the height dropped by the water. This is simply a kinematic equation for any object falling a distance  with negligible resistance. In fluids, this last equation is called Torricelli’s theorem. Note that the result is independent of the velocity’s direction, just as we found when applying conservation of energy to falling objects.
All preceding applications of Bernoulli’s equation involved simplifying conditions, such as constant height or constant pressure. The next example is a more general application of Bernoulli’s equation in which pressure, velocity, and height all change. (See .)

### Power in Fluid Flow

Power is the rate at which work is done or energy in any form is used or supplied. To see the relationship of power to fluid flow, consider Bernoulli’s equation:
All three terms have units of energy per unit volume, as discussed in the previous section. Now, considering units, if we multiply energy per unit volume by flow rate (volume per unit time), we get units of power. That is, . This means that if we multiply Bernoulli’s equation by flow rate , we get power. In equation form, this is
Each term has a clear physical meaning. For example,  is the power supplied to a fluid, perhaps by a pump, to give it its pressure . Similarly,  is the power supplied to a fluid to give it its kinetic energy. And  is the power going to gravitational potential energy.

### Test Prep for AP Courses

### Summary

1. Power in fluid flow is given by the equation  where the first term is power associated with pressure, the second is power associated with velocity, and the third is power associated with height.

### Conceptual Questions

### Problems & Exercises


# Fluid Dynamics and Its Biological and Medical Applications

## Viscosity and Laminar Flow; Poiseuille’s Law

### Learning Objectives

By the end of this section, you will be able to:
1. Define laminar flow and turbulent flow.
2. Explain what viscosity is.
3. Calculate flow and resistance with Poiseuille’s law.
4. Explain how pressure drops due to resistance.

### Laminar Flow and Viscosity

When you pour yourself a glass of juice, the liquid flows freely and quickly. But when you pour syrup on your pancakes, that liquid flows slowly and sticks to the pitcher. The difference is fluid friction, both within the fluid itself and between the fluid and its surroundings. We call this property of fluids viscosity. Juice has low viscosity, whereas syrup has high viscosity. In the previous sections we have considered ideal fluids with little or no viscosity. In this section, we will investigate what factors, including viscosity, affect the rate of fluid flow.
The precise definition of viscosity is based on laminar, or nonturbulent, flow. Before we can define viscosity, then, we need to define laminar flow and turbulent flow.  shows both types of flow. Laminar flow is characterized by the smooth flow of the fluid in layers that do not mix. Turbulent flow, or turbulence, is characterized by eddies and swirls that mix layers of fluid together.
 shows schematically how laminar and turbulent flow differ. Layers flow without mixing when flow is laminar. When there is turbulence, the layers mix, and there are significant velocities in directions other than the overall direction of flow. The lines that are shown in many illustrations are the paths followed by small volumes of fluids. These are called streamlines. Streamlines are smooth and continuous when flow is laminar, but break up and mix when flow is turbulent. Turbulence has two main causes. First, any obstruction or sharp corner, such as in a faucet, creates turbulence by imparting velocities perpendicular to the flow. Second, high speeds cause turbulence. The drag both between adjacent layers of fluid and between the fluid and its surroundings forms swirls and eddies, if the speed is great enough. We shall concentrate on laminar flow for the remainder of this section, leaving certain aspects of turbulence for later sections.
 shows how viscosity is measured for a fluid. Two parallel plates have the specific fluid between them. The bottom plate is held fixed, while the top plate is moved to the right, dragging fluid with it. The layer (or lamina) of fluid in contact with either plate does not move relative to the plate, and so the top layer moves at  while the bottom layer remains at rest. Each successive layer from the top down exerts a force on the one below it, trying to drag it along, producing a continuous variation in speed from  to 0 as shown. Care is taken to insure that the flow is laminar; that is, the layers do not mix. The motion in  is like a continuous shearing motion. Fluids have zero shear strength, but the rate at which they are sheared is related to the same geometrical factors  and  as is shear deformation for solids.
A force  is required to keep the top plate in  moving at a constant velocity , and experiments have shown that this force depends on four factors. First,  is directly proportional to  (until the speed is so high that turbulence occurs—then a much larger force is needed, and it has a more complicated dependence on ). Second,  is proportional to the area  of the plate. This relationship seems reasonable, since  is directly proportional to the amount of fluid being moved. Third,  is inversely proportional to the distance between the plates . This relationship is also reasonable;  is like a lever arm, and the greater the lever arm, the less force that is needed. Fourth,  is directly proportional to the coefficient of viscosity, . The greater the viscosity, the greater the force required. These dependencies are combined into the equation
which gives us a working definition of fluid viscosity . Solving for  gives
which defines viscosity in terms of how it is measured. The SI unit of viscosity is .  lists the coefficients of viscosity for various fluids.
Viscosity varies from one fluid to another by several orders of magnitude. As you might expect, the viscosities of gases are much less than those of liquids, and these viscosities are often temperature dependent. The viscosity of blood can be reduced by aspirin consumption, allowing it to flow more easily around the body. (When used over the long term in low doses, aspirin can help prevent heart attacks, and reduce the risk of blood clotting.)

### Laminar Flow Confined to Tubes—Poiseuille’s Law

What causes flow? The answer, not surprisingly, is pressure difference. In fact, there is a very simple relationship between horizontal flow and pressure. Flow rate  is in the direction from high to low pressure. The greater the pressure differential between two points, the greater the flow rate. This relationship can be stated as
where  and  are the pressures at two points, such as at either end of a tube, and  is the resistance to flow. The resistance  includes everything, except pressure, that affects flow rate. For example,  is greater for a long tube than for a short one. The greater the viscosity of a fluid, the greater the value of . Turbulence greatly increases , whereas increasing the diameter of a tube decreases .
If viscosity is zero, the fluid is frictionless and the resistance to flow is also zero. Comparing frictionless flow in a tube to viscous flow, as in , we see that for a viscous fluid, speed is greatest at midstream because of drag at the boundaries. We can see the effect of viscosity in a Bunsen burner flame, even though the viscosity of natural gas is small.
The resistance  to laminar flow of an incompressible fluid having viscosity  through a horizontal tube of uniform radius  and length , such as the one in , is given by
This equation is called Poiseuille’s law for resistance after the French scientist J. L. Poiseuille (1799–1869), who derived it in an attempt to understand the flow of blood, an often turbulent fluid.
Let us examine Poiseuille’s expression for  to see if it makes good intuitive sense. We see that resistance is directly proportional to both fluid viscosity  and the length  of a tube. After all, both of these directly affect the amount of friction encountered—the greater either is, the greater the resistance and the smaller the flow. The radius  of a tube affects the resistance, which again makes sense, because the greater the radius, the greater the flow (all other factors remaining the same). But it is surprising that  is raised to the fourth power in Poiseuille’s law. This exponent means that any change in the radius of a tube has a very large effect on resistance. For example, doubling the radius of a tube decreases resistance by a factor of .
Taken together,  and  give the following expression for flow rate:

This equation describes laminar flow through a tube. It is sometimes called Poiseuille’s law for laminar flow, or simply Poiseuille’s law.
The circulatory system provides many examples of Poiseuille’s law in action—with blood flow regulated by changes in vessel size and blood pressure. Blood vessels are not rigid but elastic. Adjustments to blood flow are primarily made by varying the size of the vessels, since the resistance is so sensitive to the radius. During vigorous exercise, blood vessels are selectively dilated to important muscles and organs and blood pressure increases. This creates both greater overall blood flow and increased flow to specific areas. Conversely, decreases in vessel radii, perhaps from plaques in the arteries, can greatly reduce blood flow. If a vessel’s radius is reduced by only 5% (to 0.95 of its original value), the flow rate is reduced to about  of its original value. A 19% decrease in flow is caused by a 5% decrease in radius. The body may compensate by increasing blood pressure by 19%, but this presents hazards to the heart and any vessel that has weakened walls. Another example comes from automobile engine oil. If you have a car with an oil pressure gauge, you may notice that oil pressure is high when the engine is cold. Motor oil has greater viscosity when cold than when warm, and so pressure must be greater to pump the same amount of cold oil.

### Flow and Resistance as Causes of Pressure Drops

You may have noticed that water pressure in your home might be lower than normal on hot summer days when there is more use. This pressure drop occurs in the water main before it reaches your home. Let us consider flow through the water main as illustrated in . We can understand why the pressure  to the home drops during times of heavy use by rearranging
to
where, in this case,  is the pressure at the water works and  is the resistance of the water main. During times of heavy use, the flow rate  is large. This means that  must also be large. Thus  must decrease. It is correct to think of flow and resistance as causing the pressure to drop from  to .  is valid for both laminar and turbulent flows.
We can use  to analyze pressure drops occurring in more complex systems in which the tube radius is not the same everywhere. Resistance will be much greater in narrow places, such as an obstructed coronary artery. For a given flow rate , the pressure drop will be greatest where the tube is most narrow. This is how water faucets control flow. Additionally,  is greatly increased by turbulence, and a constriction that creates turbulence greatly reduces the pressure downstream. Plaque in an artery reduces pressure and hence flow, both by its resistance and by the turbulence it creates.
 is a schematic of the human circulatory system, showing average blood pressures in its major parts for an adult at rest. Pressure created by the heart’s two pumps, the right and left ventricles, is reduced by the resistance of the blood vessels as the blood flows through them. The left ventricle increases arterial blood pressure that drives the flow of blood through all parts of the body except the lungs. The right ventricle receives the lower pressure blood from two major veins and pumps it through the lungs for gas exchange with atmospheric gases – the disposal of carbon dioxide from the blood and the replenishment of oxygen. Only one major organ is shown schematically, with typical branching of arteries to ever smaller vessels, the smallest of which are the capillaries, and rejoining of small veins into larger ones. Similar branching takes place in a variety of organs in the body, and the circulatory system has considerable flexibility in flow regulation to these organs by the dilation and constriction of the arteries leading to them and the capillaries within them. The sensitivity of flow to tube radius makes this flexibility possible over a large range of flow rates.
Each branching of larger vessels into smaller vessels increases the total cross-sectional area of the tubes through which the blood flows. For example, an artery with a cross section of  may branch into 20 smaller arteries, each with cross sections of

, with a total of

. In that manner, the resistance of the branchings is reduced so that pressure is not entirely lost. Moreover, because

 and

   increases through branching, the average velocity of the blood in the smaller vessels is reduced. The blood velocity in the aorta () is about 25 cm/s, while in the capillaries (

 in diameter) the velocity is about 1 mm/s. This reduced velocity allows the blood to exchange substances with the cells in the capillaries and alveoli in particular.

### Section Summary

1. Laminar flow is characterized by smooth flow of the fluid in layers that do not mix.
2. Turbulence is characterized by eddies and swirls that mix layers of fluid together.
3. Fluid viscosity  is due to friction within a fluid. Representative values are given in . Viscosity has units of

      or

     .
4. Flow is proportional to pressure difference and inversely proportional to resistance:

5. For laminar flow in a tube, Poiseuille’s law for resistance states that

6. Poiseuille’s law for flow in a tube is

7. The pressure drop caused by flow and resistance is given by

### Conceptual Questions

### Problems & Exercises


# Fluid Dynamics and Its Biological and Medical Applications

## The Onset of Turbulence

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate Reynolds number.
2. Use the Reynolds number for a system to determine whether it is laminar or turbulent.
Sometimes we can predict if flow will be laminar or turbulent. We know that flow in a very smooth tube or around a smooth, streamlined object will be laminar at low velocity. We also know that at high velocity, even flow in a smooth tube or around a smooth object will experience turbulence. In between, it is more difficult to predict. In fact, at intermediate velocities, flow may oscillate back and forth indefinitely between laminar and turbulent.
An occlusion, or narrowing, of an artery, such as shown in , is likely to cause turbulence because of the irregularity of the blockage, as well as the complexity of blood as a fluid. Turbulence in the circulatory system is noisy and can sometimes be detected with a stethoscope, such as when measuring diastolic pressure in the upper arm’s partially collapsed brachial artery. These turbulent sounds, at the onset of blood flow when the cuff pressure becomes sufficiently small, are called Korotkoff sounds. Aneurysms, or ballooning of arteries, create significant turbulence and can sometimes be detected with a stethoscope. Heart murmurs, consistent with their name, are sounds produced by turbulent flow around damaged and insufficiently closed heart valves. Ultrasound can also be used to detect turbulence as a medical indicator in a process analogous to Doppler-shift radar used to detect storms.
An indicator called the Reynolds number  can reveal whether flow is laminar or turbulent. For flow in a tube of uniform diameter, the Reynolds number is defined as
where  is the fluid density,  its speed,  its viscosity, and  the tube radius. The Reynolds number is a unitless quantity. Experiments have revealed that  is related to the onset of turbulence. For  below about 2000, flow is laminar. For  above about 3000, flow is turbulent. For values of  between about 2000 and 3000, flow is unstable—that is, it can be laminar, but small obstructions and surface roughness can make it turbulent, and it may oscillate randomly between being laminar and turbulent. The blood flow through most of the body is a quiet, laminar flow. The exception is in the aorta, where the speed of the blood flow rises above a critical value of 35 m/s and becomes turbulent.
The topic of chaos has become quite popular over the last few decades. A system is defined to be chaotic when its behavior is so sensitive to some factor that it is extremely difficult to predict. The field of chaos is the study of chaotic behavior. A good example of chaotic behavior is the flow of a fluid with a Reynolds number between 2000 and 3000. Whether or not the flow is turbulent is difficult, but not impossible, to predict—the difficulty lies in the extremely sensitive dependence on factors like roughness and obstructions on the nature of the flow. A tiny variation in one factor has an exaggerated (or nonlinear) effect on the flow. Phenomena as disparate as turbulence, the orbit of Pluto, and the onset of irregular heartbeats are chaotic and can be analyzed with similar techniques.

### Section Summary

1. The Reynolds number  can reveal whether flow is laminar or turbulent. It is
2. For  below about 2000, flow is laminar. For  above about 3000, flow is turbulent. For values of  between 2000 and 3000, it may be either or both.

### Conceptual Questions

### Problems & Exercises


# Fluid Dynamics and Its Biological and Medical Applications

## Motion of an Object in a Viscous Fluid

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate the Reynolds number for an object moving through a fluid.
2. Explain whether the Reynolds number indicates laminar or turbulent flow.
3. Describe the conditions under which an object has a terminal speed.
A moving object in a viscous fluid is equivalent to a stationary object in a flowing fluid stream. (For example, when you ride a bicycle at 10 m/s in still air, you feel the air in your face exactly as if you were stationary in a 10-m/s wind.) Flow of the stationary fluid around a moving object may be laminar, turbulent, or a combination of the two. Just as with flow in tubes, it is possible to predict when a moving object creates turbulence. We use another form of the Reynolds number , defined for an object moving in a fluid to be
where  is a characteristic length of the object (a sphere’s diameter, for example),  the fluid density,

 its viscosity, and

 the object’s speed in the fluid. If

 is less than about 1, flow around the object can be laminar, particularly if the object has a smooth shape. The transition to turbulent flow occurs for  between 1 and about 10, depending on surface roughness and so on. Depending on the surface, there can be a turbulent wake behind the object with some laminar flow over its surface. For an

 between 10 and

, the flow may be either laminar or turbulent and may oscillate between the two. For

 greater than about

, the flow is entirely turbulent, even at the surface of the object. (See .) Laminar flow occurs mostly when the objects in the fluid are small, such as raindrops, pollen, and blood cells in plasma.
One of the consequences of viscosity is a resistance force called viscous drag

 that is exerted on a moving object. This force typically depends on the object’s speed (in contrast with simple friction). Experiments have shown that for laminar flow (

 less than about one) viscous drag is proportional to speed, whereas for  between about 10 and

, viscous drag is proportional to speed squared. (This relationship is a strong dependence and is pertinent to bicycle racing, where even a small headwind causes significantly increased drag on the racer. Cyclists take turns being the leader in the pack for this reason.) For

 greater than

, drag increases dramatically and behaves with greater complexity. For laminar flow around a sphere,

 is proportional to fluid viscosity

, the object’s characteristic size

, and its speed

. All of which makes sense—the more viscous the fluid and the larger the object, the more drag we expect. Recall Stoke’s law

. For the special case of a small sphere of radius

 moving slowly in a fluid of viscosity

, the drag force

 is given by
An interesting consequence of the increase in  with speed is that an object falling through a fluid will not continue to accelerate indefinitely (as it would if we neglect air resistance, for example). Instead, viscous drag increases, slowing acceleration, until a critical speed, called the terminal speed, is reached and the acceleration of the object becomes zero. Once this happens, the object continues to fall at constant speed (the terminal speed). This is the case for particles of sand falling in the ocean, cells falling in a centrifuge, and sky divers falling through the air.  shows some of the factors that affect terminal speed. There is a viscous drag on the object that depends on the viscosity of the fluid and the size of the object. But there is also a buoyant force that depends on the density of the object relative to the fluid. Terminal speed will be greatest for low-viscosity fluids and objects with high densities and small sizes. Thus a skydiver falls more slowly with outspread limbs than when they are in a pike position—head first with hands at their side and legs together.
Knowledge of terminal speed is useful for estimating sedimentation rates of small particles. We know from watching mud settle out of dirty water that sedimentation is usually a slow process. Centrifuges are used to speed sedimentation by creating accelerated frames in which gravitational acceleration is replaced by centripetal acceleration, which can be much greater, increasing the terminal speed.

### Section Summary

1. When an object moves in a fluid, there is a different form of the Reynolds number  which indicates whether flow is laminar or turbulent.
2. For  less than about one, flow is laminar.
3. For  greater than

, flow is entirely turbulent.

### Conceptual Questions


# Fluid Dynamics and Its Biological and Medical Applications

## Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

### Learning Objectives

By the end of this section, you will be able to:
1. Define diffusion, osmosis, dialysis, and active transport.
2. Calculate diffusion rates.

### Diffusion

There is something fishy about the ice cube from your freezer—how did it pick up those food odors? How does soaking a sprained ankle in Epsom salt reduce swelling? The answer to these questions are related to atomic and molecular transport phenomena—another mode of fluid motion. Atoms and molecules are in constant motion at any temperature. In fluids they move about randomly even in the absence of macroscopic flow. This motion is called a random walk and is illustrated in . Diffusion is the movement of substances due to random thermal molecular motion. Fluids, like fish fumes or odors entering ice cubes, can even diffuse through solids.
Diffusion is a slow process over macroscopic distances. The densities of common materials are great enough that molecules cannot travel very far before having a collision that can scatter them in any direction, including straight backward. It can be shown that the average distance  that a molecule travels is proportional to the square root of time:
where  stands for the root-mean-square distance and is the statistical average for the process. The quantity  is the diffusion constant for the particular molecule in a specific medium.  lists representative values of  for various substances, in units of .
Note that  gets progressively smaller for more massive molecules. This decrease is because the average molecular speed at a given temperature is inversely proportional to molecular mass. Thus the more massive molecules diffuse more slowly. Another interesting point is that  for oxygen in air is much greater than  for oxygen in water. In water, an oxygen molecule makes many more collisions in its random walk and is slowed considerably. In water, an oxygen molecule moves only about  in 1 s. (Each molecule actually collides about  times per second!). Finally, note that diffusion constants increase with temperature, because average molecular speed increases with temperature. This is because the average kinetic energy of molecules, , is proportional to absolute temperature.
Because diffusion is typically very slow, its most important effects occur over small distances. For example, the cornea of the eye gets most of its oxygen by diffusion through the thin tear layer covering it.

### The Rate and Direction of Diffusion

If you very carefully place a drop of food coloring in a still glass of water, it will slowly diffuse into the colorless surroundings until its concentration is the same everywhere. This type of diffusion is called free diffusion, because there are no barriers inhibiting it. Let us examine its direction and rate. Molecular motion is random in direction, and so simple chance dictates that more molecules will move out of a region of high concentration than into it. The net rate of diffusion is higher initially than after the process is partially completed. (See .)
The net rate of diffusion is proportional to the concentration difference. Many more molecules will leave a region of high concentration than will enter it from a region of low concentration. In fact, if the concentrations were the same, there would be  no net movement. The net rate of diffusion is also proportional to the diffusion constant , which is determined experimentally. The farther a molecule can diffuse in a given time, the more likely it is to leave the region of high concentration. Many of the factors that affect the rate are hidden in the diffusion constant . For example, temperature and cohesive and adhesive forces all affect values of .
Diffusion is the dominant mechanism by which the exchange of nutrients and waste products occur between the blood and tissue, and between air and blood in the lungs. In the evolutionary process, as organisms became larger, they needed quicker methods of transportation than net diffusion, because of the larger distances involved in the transport, leading to the development of circulatory systems. Less sophisticated, single-celled organisms still rely totally on diffusion for the removal of waste products and the uptake of nutrients.

### Osmosis and Dialysis—Diffusion across Membranes

Some of the most interesting examples of diffusion occur through barriers that affect the rates of diffusion. For example, when you soak a swollen ankle in Epsom salt, water diffuses through your skin. Many substances regularly move through cell membranes; oxygen moves in, carbon dioxide moves out, nutrients go in, and wastes go out, for example. Because membranes are thin structures (typically  to  m across) diffusion rates through them can be high. Diffusion through membranes is an important method of transport.
Membranes are generally selectively permeable, or semipermeable. (See .) One type of semipermeable membrane has small pores that allow only small molecules to pass through. In other types of membranes, the molecules may actually dissolve in the membrane or react with molecules in the membrane while moving across. Membrane function, in fact, is the subject of much current research, involving not only physiology but also chemistry and physics.
Osmosis is the transport of water through a semipermeable membrane from a region of high concentration to a region of low concentration. Osmosis is driven by the imbalance in water concentration. For example, water is more concentrated in your body than in Epsom salt. When you soak a swollen ankle in Epsom salt, the water moves out of your body into the lower-concentration region in the salt. Similarly, dialysis is the transport of any other molecule through a semipermeable membrane due to its concentration difference. Both osmosis and dialysis are used by the kidneys to cleanse the blood.
Osmosis can create a substantial pressure. Consider what happens if osmosis continues for some time, as illustrated in . Water moves by osmosis from the left into the region on the right, where it is less concentrated, causing the solution on the right to rise. This movement will continue until the pressure  created by the extra height of fluid on the right is large enough to stop further osmosis. This pressure is called a back pressure. The back pressure  that stops osmosis is also called the relative osmotic pressure if neither solution is pure water, and it is called the osmotic pressure if one solution is pure water. Osmotic pressure can be large, depending on the size of the concentration difference. For example, if pure water and sea water are separated by a semipermeable membrane that passes no salt, osmotic pressure will be 25.9 atm. This value means that water will diffuse through the membrane until the salt water surface rises 268 m above the pure-water surface! One example of pressure created by osmosis is turgor in plants (many wilt when too dry). Turgor describes the condition of a plant in which the fluid in a cell exerts a pressure against the cell wall. This pressure gives the plant support. Dialysis can similarly cause substantial pressures.
Reverse osmosis and reverse dialysis (also called filtration) are processes that occur when back pressure is sufficient to reverse the normal direction of substances through membranes. Back pressure can be created naturally as on the right side of . (A piston can also create this pressure.) Reverse osmosis can be used to desalinate water by simply forcing it through a membrane that will not pass salt. Similarly, reverse dialysis can be used to filter out any substance that a given membrane will not pass.
One further example of the movement of substances through membranes deserves mention. We sometimes find that substances pass in the direction opposite to what we expect. Cypress tree roots, for example, extract pure water from salt water, although osmosis would move it in the opposite direction. This is not reverse osmosis, because there is no back pressure to cause it. What is happening is called active transport, a process in which a living membrane expends energy to move substances across it. Many living membranes move water and other substances by active transport. The kidneys, for example, not only use osmosis and dialysis—they also employ significant active transport to move substances into and out of blood. In fact, it is estimated that at least 25% of the body’s energy is expended on active transport of substances at the cellular level. The study of active transport carries us into the realms of microbiology, biophysics, and biochemistry and it is a fascinating application of the laws of nature to living structures.

### Section Summary

1. Diffusion is the movement of substances due to random thermal molecular motion.
2. The average distance  a molecule travels by diffusion in a given amount of time is given bywhere
3. Osmosis is the transport of water through a semipermeable membrane from a region of high concentration to a region of low concentration.
4. Dialysis is the transport of any other molecule through a semipermeable membrane due to its concentration difference.
5. Both processes can be reversed by back pressure.
6. Active transport is a process in which a living membrane expends energy to move substances across it.

### Conceptual Questions

### Problem Exercises


# Temperature, Kinetic Theory, and the Gas Laws

## Introduction to Temperature, Kinetic Theory, and the Gas Laws

Heat is something familiar to each of us. We feel the warmth of the summer Sun, the chill of a clear summer night, the heat of coffee after a winter stroll, and the cooling effect of our sweat. Heat transfer is maintained by temperature differences. Manifestations of heat transfer—the movement of heat energy from one place or material to another—are apparent throughout the universe. Heat from beneath Earth’s surface is brought to the surface in flows of incandescent lava. The Sun warms Earth’s surface and is the source of much of the energy we find on it. Rising levels of atmospheric carbon dioxide threaten to trap more of the Sun’s energy, perhaps fundamentally altering the ecosphere. In space, supernovas explode, briefly radiating more heat than an entire galaxy does.
What is heat? How do we define it? How is it related to temperature? What are heat’s effects? How is it related to other forms of energy and to work? We will find that, in spite of the richness of the phenomena, there is a small set of underlying physical principles that unite the subjects and tie them to other fields.


# Temperature, Kinetic Theory, and the Gas Laws

## Temperature

### Learning Objectives

By the end of this section, you will be able to:
1. Define temperature.
2. Convert temperatures between the Celsius, Fahrenheit, and Kelvin scales.
3. Define thermal equilibrium.
4. State the zeroth law of thermodynamics.
The concept of temperature has evolved from the common concepts of hot and cold. Human perception of what feels hot or cold is a relative one. For example, if you place one hand in hot water and the other in cold water, and then place both hands in tepid water, the tepid water will feel cool to the hand that was in hot water, and warm to the one that was in cold water. The scientific definition of temperature is less ambiguous than your senses of hot and cold. Temperature is operationally defined to be what we measure with a thermometer. (Many physical quantities are defined solely in terms of how they are measured. We shall see later how temperature is related to the kinetic energies of atoms and molecules, a more physical explanation.) Two accurate thermometers, one placed in hot water and the other in cold water, will show the hot water to have a higher temperature. If they are then placed in the tepid water, both will give identical readings (within measurement uncertainties). In this section, we discuss temperature, its measurement by thermometers, and its relationship to thermal equilibrium. Again, temperature is the quantity measured by a thermometer.
Any physical property that depends on temperature, and whose response to temperature is reproducible, can be used as the basis of a thermometer. Because many physical properties depend on temperature, the variety of thermometers is remarkable. For example, volume increases with temperature for most substances. This property is the basis for the common alcohol thermometer, the old mercury thermometer, and the bimetallic strip (). Other properties used to measure temperature include electrical resistance and color, as shown in , and the emission of infrared radiation, as shown in .

### Temperature Scales

Thermometers are used to measure temperature according to well-defined scales of measurement, which use pre-defined reference points to help compare quantities. The three most common temperature scales are the Fahrenheit, Celsius, and Kelvin scales. A temperature scale can be created by identifying two easily reproducible temperatures. The freezing and boiling temperatures of water at standard atmospheric pressure are commonly used.
The Celsius scale (which replaced the slightly different   scale) has the freezing point of water at  and the boiling point at . Its unit is the degree Celsius. On the Fahrenheit scale (still the most frequently used in the United States), the freezing point of water is at  and the boiling point is at . The unit of temperature on this scale is the degree Fahrenheit. Note that a temperature difference of one degree Celsius is greater than a temperature difference of one degree Fahrenheit. Only 100 Celsius degrees span the same range as 180 Fahrenheit degrees, thus one degree on the Celsius scale is 1.8 times larger than one degree on the Fahrenheit scale
The Kelvin scale is the temperature scale that is commonly used in science. It is an absolute temperature scale defined to have 0 K at the lowest possible temperature, called absolute zero. The official temperature unit on this scale is the kelvin, which is abbreviated K, and is not accompanied by a degree sign. The freezing and boiling points of water are 273.15 K and 373.15 K, respectively. Thus, the magnitude of temperature differences is the same in units of kelvins and degrees Celsius. Unlike other temperature scales, the Kelvin scale is an absolute scale. It is used extensively in scientific work because a number of physical quantities, such as the volume of an ideal gas, are directly related to absolute temperature. The kelvin is the SI unit used in scientific work.
The relationships between the three common temperature scales is shown in . Temperatures on these scales can be converted using the equations in .
Notice that the conversions between Fahrenheit and Kelvin look quite complicated. In fact, they are simple combinations of the conversions between Fahrenheit and Celsius, and the conversions between Celsius and Kelvin.

### Temperature Ranges in the Universe

 shows the wide range of temperatures found in the universe. Human beings have been known to survive with body temperatures within a small range, from  to   to ). The average normal body temperature is usually given as  (), and variations in this temperature can indicate a medical condition: a fever, an infection, a tumor, or circulatory problems (see ).
The lowest temperatures ever recorded have been measured during laboratory experiments:  at the Massachusetts Institute of Technology (USA), and  at Helsinki University of Technology (Finland). In comparison, the coldest recorded place on Earth’s surface is Vostok, Antarctica at 183 K , and the coldest place (outside the lab) known in the universe is the Boomerang Nebula, with a temperature of 1 K.

### Thermal Equilibrium and the Zeroth Law of Thermodynamics

Thermometers actually take their own temperature, not the temperature of the object they are measuring. This raises the question of how we can be certain that a thermometer measures the temperature of the object with which it is in contact. It is based on the fact that any two systems placed in thermal contact (meaning heat transfer can occur between them) will reach the same temperature. That is, heat will flow from the hotter object to the cooler one until they have exactly the same temperature. The objects are then in thermal equilibrium, and no further changes will occur. The systems interact and change because their temperatures differ, and the changes stop once their temperatures are the same. Thus, if enough time is allowed for this transfer of heat to run its course, the temperature a thermometer registers does represent the system with which it is in thermal equilibrium. Thermal equilibrium is established when two bodies are in contact with each other and can freely exchange energy.
Furthermore, experimentation has shown that if two systems, A and B, are in thermal equilibrium with each another, and B is in thermal equilibrium with a third system C, then A is also in thermal equilibrium with C. This conclusion may seem obvious, because all three have the same temperature, but it is basic to thermodynamics. It is called the zeroth law of thermodynamics.
This law was postulated in the 1930s, after the first and second laws of thermodynamics had been developed and named. It is called the zeroth law because it comes logically before the first and second laws (discussed in Thermodynamics). Suppose, for example, a cold metal block and a hot metal block are both placed on a metal plate at room temperature. Eventually the cold block and the plate will be in thermal equilibrium. In addition, the hot block and the plate will be in thermal equilibrium. By the zeroth law, we can conclude that the cold block and the hot block are also in thermal equilibrium.

### Section Summary

1. Temperature is the quantity measured by a thermometer.
2. Temperature is related to the average kinetic energy of atoms and molecules in a system.
3. Absolute zero is the temperature at which there is no molecular motion.
4. There are three main temperature scales: Celsius, Fahrenheit, and Kelvin.
5. Temperatures on one scale can be converted to temperatures on another scale using the following equations:

6. Systems are in thermal equilibrium when they have the same temperature.
7. Thermal equilibrium occurs when two bodies are in contact with each other and can freely exchange energy.
8. The zeroth law of thermodynamics states that when two systems, A and B, are in thermal equilibrium with each other, and B is in thermal equilibrium with a third system, C, then A is also in thermal equilibrium with C.

### Conceptual Questions

### Problems & Exercises


# Temperature, Kinetic Theory, and the Gas Laws

## Thermal Expansion of Solids and Liquids

### Learning Objectives

By the end of this section, you will be able to:
1. Define and describe thermal expansion.
2. Calculate the linear expansion of an object given its initial length, change in temperature, and coefficient of linear expansion.
3. Calculate the volume expansion of an object given its initial volume, change in temperature, and coefficient of volume expansion.
4. Calculate thermal stress on an object given its original volume, temperature change, volume change, and bulk modulus.
The expansion of alcohol in a thermometer is one of many commonly encountered examples of thermal expansion, the change in size or volume of a given mass with temperature. Hot air rises because its volume increases, which causes the hot air’s density to be smaller than the density of surrounding air, causing a buoyant (upward) force on the hot air. The same happens in all liquids and gases, driving natural heat transfer upwards in homes, oceans, and weather systems. Solids also undergo thermal expansion. Railroad tracks and bridges, for example, have expansion joints to allow them to freely expand and contract with temperature changes.
What are the basic properties of thermal expansion? First, thermal expansion is clearly related to temperature change. The greater the temperature change, the more a bimetallic strip will bend. Second, it depends on the material. In a thermometer, for example, the expansion of alcohol is much greater than the expansion of the glass containing it.
What is the underlying cause of thermal expansion? As is discussed in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature, an increase in temperature implies an increase in the kinetic energy of the individual atoms. In a solid, unlike in a gas, the atoms or molecules are closely packed together, but their kinetic energy (in the form of small, rapid vibrations) pushes neighboring atoms or molecules apart from each other. This neighbor-to-neighbor pushing results in a slightly greater distance, on average, between neighbors, and adds up to a larger size for the whole body. For most substances under ordinary conditions, there is no preferred direction, and an increase in temperature will increase the solid’s size by a certain fraction in each dimension.
 lists representative values of the coefficient of linear expansion, which may have units of  or 1/K. Because the size of a kelvin and a degree Celsius are the same, both  and  can be expressed in units of kelvins or degrees Celsius. The equation  is accurate for small changes in temperature and can be used for large changes in temperature if an average value of  is used.

### Thermal Expansion in Two and Three Dimensions

Objects expand in all dimensions, as illustrated in . That is, their areas and volumes, as well as their lengths, increase with temperature. Holes also get larger with temperature. If you cut a hole in a metal plate, the remaining material will expand exactly as it would if the plug was still in place. The plug would get bigger, and so the hole must get bigger too. (Think of the ring of neighboring atoms or molecules on the wall of the hole as pushing each other farther apart as temperature increases. Obviously, the ring of neighbors must get slightly larger, so the hole gets slightly larger).
In general, objects will expand with increasing temperature. Water is the most important exception to this rule. Water expands with increasing temperature (its density decreases) when it is at temperatures greater than . However, it expands with decreasing temperature when it is between  and  to . Water is densest at . (See .) Perhaps the most striking effect of this phenomenon is the freezing of water in a pond. When water near the surface cools down to  it is denser than the remaining water and thus will sink to the bottom. This “turnover” results in a layer of warmer water near the surface, which is then cooled. Eventually the pond has a uniform temperature of . If the temperature in the surface layer drops below , the water is less dense than the water below, and thus stays near the top. As a result, the pond surface can completely freeze over. The ice on top of liquid water provides an insulating layer from winter’s harsh exterior air temperatures. Fish and other aquatic life can survive in  water beneath ice, due to this unusual characteristic of water. It also produces circulation of water in the pond that is necessary for a healthy ecosystem of the body of water.

### Thermal Stress

Thermal stress is created by thermal expansion or contraction (see Elasticity: Stress and Strain for a discussion of stress and strain). Thermal stress can be destructive, such as when expanding gasoline ruptures a tank. It can also be useful: for example, when two parts are joined together by heating one in manufacturing, then slipping it over the other and allowing the combination to cool. Thermal stress can explain many phenomena, such as the weathering of rocks and pavement by the expansion of ice when it freezes.
Forces and pressures created by thermal stress are typically as great as that in the example above. Railroad tracks and roadways can buckle on hot days if they lack sufficient expansion joints. (See .) Power lines sag more in the summer than in the winter, and will snap in cold weather if there is insufficient slack. Cracks open and close in plaster walls as a house warms and cools. Glass cooking pans will crack if cooled rapidly or unevenly, because of differential contraction and the stresses it creates. (Pyrex® is less susceptible because of its small coefficient of thermal expansion.) Nuclear reactor pressure vessels are threatened by overly rapid cooling, and although none have failed, several have been cooled faster than considered desirable. Biological cells are ruptured when foods are frozen, detracting from their taste. Repeated thawing and freezing accentuate the damage. Even the oceans can be affected. A significant portion of the rise in sea level that is resulting from global warming is due to the thermal expansion of sea water.
Metal is regularly used in the human body for hip and knee implants. Most implants need to be replaced over time because, among other things, metal does not bond with bone. Researchers are trying to find better metal coatings that would allow metal-to-bone bonding. One challenge is to find a coating that has an expansion coefficient similar to that of metal. If the expansion coefficients are too different, the thermal stresses during the manufacturing process lead to cracks at the coating-metal interface.
Another example of thermal stress is found in the mouth. Dental fillings can expand differently from tooth enamel. It can give pain when eating ice cream or having a hot drink. Cracks might occur in the filling. Metal fillings (gold, silver, etc.) are being replaced by composite fillings (porcelain), which have smaller coefficients of expansion closer to those of teeth.

### Section Summary

1. Thermal expansion is the increase, or decrease, of the size (length, area, or volume) of a body due to a change in temperature.
2. Thermal expansion is large for gases, and relatively small, but not negligible, for liquids and solids.
3. Linear thermal expansion is

where  is the change in length ,  is the change in temperature, and  is the coefficient of linear expansion, which varies slightly with temperature.

4. The change in area due to thermal expansion is

where  is the change in area.

5. The change in volume due to thermal expansion is

where  is the coefficient of volume expansion and . Thermal stress is created when thermal expansion is constrained.

### Conceptual Questions

### Problems & Exercises


# Temperature, Kinetic Theory, and the Gas Laws

## The Ideal Gas Law

### Learning Objectives

By the end of this section, you will be able to:
1. State the ideal gas law in terms of molecules and in terms of moles.
2. Use the ideal gas law to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume.
3. Use Avogadro’s number to convert between number of molecules and number of moles.
In this section, we continue to explore the thermal behavior of gases. In particular, we examine the characteristics of atoms and molecules that compose gases. (Most gases, for example nitrogen, , and oxygen, , are composed of two or more atoms. We will primarily use the term “molecule” in discussing a gas because the term can also be applied to monatomic gases, such as helium.)
Gases are easily compressed. We can see evidence of this in , where you will note that gases have the largest coefficients of volume expansion. The large coefficients mean that gases expand and contract very rapidly with temperature changes. In addition, you will note that most gases expand at the same rate, or have the same . This raises the question as to why gases should all act in nearly the same way, when liquids and solids have widely varying expansion rates.
The answer lies in the large separation of atoms and molecules in gases, compared to their sizes, as illustrated in . Because atoms and molecules have large separations, forces between them can be ignored, except when they collide with each other during collisions. The motion of atoms and molecules (at temperatures well above the boiling temperature) is fast, such that the gas occupies all of the accessible volume and the expansion of gases is rapid. In contrast, in liquids and solids, atoms and molecules are closer together and are quite sensitive to the forces between them.
To get some idea of how pressure, temperature, and volume of a gas are related to one another, consider what happens when you pump air into an initially deflated tire. The tire’s volume first increases in direct proportion to the amount of air injected, without much increase in the tire pressure. Once the tire has expanded to nearly its full size, the walls limit volume expansion. If we continue to pump air into it, the pressure increases. The pressure will further increase when the car is driven and the tires move. Most manufacturers specify optimal tire pressure for cold tires. (See .)
At room temperatures, collisions between atoms and molecules can be ignored. In this case, the gas is called an ideal gas, in which case the relationship between the pressure, volume, and temperature is given by the equation of state called the ideal gas law.
The ideal gas law can be derived from basic principles, but was originally deduced from experimental measurements of Charles’ law (that volume occupied by a gas is proportional to temperature at a fixed pressure) and from Boyle’s law (that for a fixed temperature, the product  is a constant). In the ideal gas model, the volume occupied by its atoms and molecules is a negligible fraction of . The ideal gas law describes the behavior of real gases under most conditions. (Note, for example, that  is the total number of atoms and molecules, independent of the type of gas.)
Let us see how the ideal gas law is consistent with the behavior of filling the tire when it is pumped slowly and the temperature is constant. At first, the pressure  is essentially equal to atmospheric pressure, and the volume  increases in direct proportion to the number of atoms and molecules  put into the tire. Once the volume of the tire is constant, the equation  predicts that the pressure should increase in proportion to the number N of atoms and molecules.

### Moles and Avogadro’s Number

It is sometimes convenient to work with a unit other than molecules when measuring the amount of substance. A mole (abbreviated mol) is defined to be the amount of a substance that contains as many atoms or molecules as there are atoms in exactly 12 grams (0.012 kg) of carbon-12. The actual number of atoms or molecules in one mole is called Avogadro’s number, in recognition of Italian scientist Amedeo Avogadro (1776–1856). He developed the concept of the mole, based on the hypothesis that equal volumes of gas, at the same pressure and temperature, contain equal numbers of molecules. That is, the number is independent of the type of gas. This hypothesis has been confirmed, and the value of Avogadro’s number is

### The Ideal Gas Law Restated Using Moles

A very common expression of the ideal gas law uses the number of moles, , rather than the number of atoms and molecules, . We start from the ideal gas law,
and multiply and divide the equation by Avogadro’s number . This gives
Note that  is the number of moles. We define the universal gas constant , and obtain the ideal gas law in terms of moles.
The ideal gas law can be considered to be another manifestation of the law of conservation of energy (see Conservation of Energy). Work done on a gas results in an increase in its energy, increasing pressure and/or temperature, or decreasing volume. This increased energy can also be viewed as increased internal kinetic energy, given the gas’s atoms and molecules.

### The Ideal Gas Law and Energy

Let us now examine the role of energy in the behavior of gases. When you inflate a bike tire by hand, you do work by repeatedly exerting a force through a distance. This energy goes into increasing the pressure of air inside the tire and increasing the temperature of the pump and the air.
The ideal gas law is closely related to energy: the units on both sides are joules. The right-hand side of the ideal gas law in  is . This term is roughly the amount of translational kinetic energy of  atoms or molecules at an absolute temperature , as we shall see formally in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature. The left-hand side of the ideal gas law is , which also has the units of joules. We know from our study of fluids that pressure is one type of potential energy per unit volume, so pressure multiplied by volume is energy. The important point is that there is energy in a gas related to both its pressure and its volume. The energy can be changed when the gas is doing work as it expands—something we explore in Heat and Heat Transfer Methods—similar to what occurs in gasoline or steam engines and turbines.

### Test Prep for AP Courses

### Section Summary

1. The ideal gas law relates the pressure and volume of a gas to the number of gas molecules and the temperature of the gas.
2. The ideal gas law can be written in terms of the number of molecules of gas:

where  is pressure,  is volume,  is temperature,  is number of molecules, and  is the Boltzmann constant

3. A mole is the number of atoms in a 12-g sample of carbon-12.
4. The number of molecules in a mole is called Avogadro’s number ,

5. A mole of any substance has a mass in grams equal to its molecular weight, which can be determined from the periodic table of elements.
6. The ideal gas law can also be written and solved in terms of the number of moles of gas:

where  is number of moles and  is the universal gas constant,

7. The ideal gas law is generally valid at temperatures well above the boiling temperature.

### Conceptual Questions

### Problems & Exercises


# Temperature, Kinetic Theory, and the Gas Laws

## Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature

### Learning Objectives

By the end of this section, you will be able to:
1. Express the ideal gas law in terms of molecular mass and velocity.
2. Define thermal energy.
3. Calculate the kinetic energy of a gas molecule, given its temperature.
4. Describe the relationship between the temperature of a gas and the kinetic energy of atoms and molecules.
5. Describe the distribution of speeds of molecules in a gas.
We have developed macroscopic definitions of pressure and temperature. Pressure is the force divided by the area on which the force is exerted, and temperature is measured with a thermometer. We gain a better understanding of pressure and temperature from the kinetic theory of gases, which assumes that atoms and molecules are in continuous random motion.
 shows an elastic collision of a gas molecule with the wall of a container, so that it exerts a force on the wall (by Newton’s third law). Because a huge number of molecules will collide with the wall in a short time, we observe an average force per unit area. These collisions are the source of pressure in a gas. As the number of molecules increases, the number of collisions and thus the pressure increase. Similarly, the gas pressure is higher if the average velocity of molecules is higher. The actual relationship is derived in the Things Great and Small feature below. The following relationship is found:
where  is the pressure (average force per unit area),  is the volume of gas in the container,  is the number of molecules in the container,  is the mass of a molecule, and  is the average of the molecular speed squared.
What can we learn from this atomic and molecular version of the ideal gas law? We can derive a relationship between temperature and the average translational kinetic energy of molecules in a gas. Recall the previous expression of the ideal gas law:
Equating the right-hand side of this equation with the right-hand side of  gives
We can get the average kinetic energy of a molecule, , from the right-hand side of the equation by canceling  and multiplying by 3/2. This calculation produces the result that the average kinetic energy of a molecule is directly related to absolute temperature.
The average translational kinetic energy of a molecule, , is called thermal energy. The equation  is a molecular interpretation of temperature, and it has been found to be valid for gases and reasonably accurate in liquids and solids. It is another definition of temperature based on an expression of the molecular energy.
It is sometimes useful to rearrange , and solve for the average speed of molecules in a gas in terms of temperature,
where  stands for root-mean-square (rms) speed.

### Distribution of Molecular Speeds

The motion of molecules in a gas is random in magnitude and direction for individual molecules, but a gas of many molecules has a predictable distribution of molecular speeds. This distribution is called the Maxwell-Boltzmann distribution, after its originators, who calculated it based on kinetic theory, and has since been confirmed experimentally. (See .) The distribution has a long tail, because a few molecules may go several times the rms speed. The most probable speed  is less than the rms speed .  shows that the curve is shifted to higher speeds at higher temperatures, with a broader range of speeds.
The distribution of thermal speeds depends strongly on temperature. As temperature increases, the speeds are shifted to higher values and the distribution is broadened.
What is the implication of the change in distribution with temperature shown in  for humans? All other things being equal, if a person has a fever, they are likely to lose more water molecules, particularly from linings along moist cavities such as the lungs and mouth, creating a dry sensation in the mouth.

### Test Prep for AP Courses

### Section Summary

1. Kinetic theory is the atomistic description of gases as well as liquids and solids.
2. Kinetic theory models the properties of matter in terms of continuous random motion of atoms and molecules.
3. The ideal gas law can also be expressed as

where  is the pressure (average force per unit area),  is the volume of gas in the container,  is the number of molecules in the container,  is the mass of a molecule, and  is the average of the molecular speed squared.

4. Thermal energy is defined to be the average translational kinetic energy  of an atom or molecule.
5. The temperature of gases is proportional to the average translational kinetic energy of atoms and molecules.

    or

6. The motion of individual molecules in a gas is random in magnitude and direction. However, a gas of many molecules has a predictable distribution of molecular speeds, known as the Maxwell-Boltzmann distribution.

### Conceptual Questions

### Problems & Exercises


# Temperature, Kinetic Theory, and the Gas Laws

## Phase Changes

### Learning Objectives

By the end of this section, you will be able to:
1. Interpret a phase diagram.
2. State Dalton’s law.
3. Identify and describe the triple point of a gas from its phase diagram.
4. Describe the state of equilibrium between a liquid and a gas, a liquid and a solid, and a gas and a solid.
Up to now, we have considered the behavior of ideal gases. Real gases are like ideal gases at high temperatures. At lower temperatures, however, the interactions between the molecules and their volumes cannot be ignored. The molecules are very close (condensation occurs) and there is a dramatic decrease in volume, as seen in . The substance changes from a gas to a liquid. When a liquid is cooled to even lower temperatures, it becomes a solid. The volume never reaches zero because of the finite volume of the molecules.
High pressure may also cause a gas to change phase to a liquid. Carbon dioxide, for example, is a gas at room temperature and atmospheric pressure, but becomes a liquid under sufficiently high pressure. If the pressure is reduced, the temperature drops and the liquid carbon dioxide solidifies into a snow-like substance at the temperature . Solid  is called “dry ice.” Another example of a gas that can be in a liquid phase is liquid nitrogen .  is made by liquefaction of atmospheric air (through compression and cooling). It boils at 77 K  at atmospheric pressure.  is useful as a refrigerant and allows for the preservation of blood, sperm, and other biological materials. It is also used to reduce noise in electronic sensors and equipment, and to help cool down their current-carrying wires. In dermatology,  is used to freeze and painlessly remove warts and other growths from the skin.

### PV Diagrams

We can examine aspects of the behavior of a substance by plotting a graph of pressure versus volume, called a . When the substance behaves like an ideal gas, the ideal gas law describes the relationship between its pressure and volume. That is,
Now, assuming the number of molecules and the temperature are fixed,
For example, the volume of the gas will decrease as the pressure increases. If you plot the relationship  on a  diagram, you find a hyperbola.  shows a graph of pressure versus volume. The hyperbolas represent ideal-gas behavior at various fixed temperatures, and are called isotherms. At lower temperatures, the curves begin to look less like hyperbolas—the gas is not behaving ideally and may even contain liquid. There is a critical point—that is, a critical temperature—above which liquid cannot exist. At sufficiently high pressure above the critical point, the gas will have the density of a liquid but will not condense. Carbon dioxide, for example, cannot be liquefied at a temperature above . Critical pressure is the minimum pressure needed for liquid to exist at the critical temperature.  lists representative critical temperatures and pressures.

### Phase Diagrams

The plots of pressure versus temperatures provide considerable insight into thermal properties of substances. There are well-defined regions on these graphs that correspond to various phases of matter, so  graphs are called phase diagrams.  shows the phase diagram for water. Using the graph, if you know the pressure and temperature you can determine the phase of water. The solid lines—boundaries between phases—indicate temperatures and pressures at which the phases coexist (that is, they exist together in ratios, depending on pressure and temperature). For example, the boiling point of water is  at 1.00 atm. As the pressure increases, the boiling temperature rises steadily to  at a pressure of 218 atm. A pressure cooker (or even a covered pot) will cook food faster because the water can exist as a liquid at temperatures greater than  without all boiling away. The curve ends at a point called the critical point, because at higher temperatures the liquid phase does not exist at any pressure. The critical point occurs at the critical temperature, as you can see for water from . The critical temperature for oxygen is , so oxygen cannot be liquefied above this temperature.
Similarly, the curve between the solid and liquid regions in  gives the melting temperature at various pressures. For example, the melting point is  at 1.00 atm, as expected. Note that, at a fixed temperature, you can change the phase from solid (ice) to liquid (water) by increasing the pressure. Ice melts from pressure in the hands of a snowball maker. From the phase diagram, we can also say that the melting temperature of ice falls with increased pressure. When a car is driven over snow, the increased pressure from the tires melts the snowflakes; afterwards the water refreezes and forms an ice layer.
At sufficiently low pressures there is no liquid phase, but the substance can exist as either gas or solid. For water, there is no liquid phase at pressures below 0.00600 atm. The phase change from solid to gas is called sublimation. It accounts for large losses of snow pack that never make it into a river, the routine automatic defrosting of a freezer, and the freeze-drying process applied to many foods. Carbon dioxide, on the other hand, sublimates at standard atmospheric pressure of 1 atm. (The solid form of  is known as dry ice because it does not melt. Instead, it moves directly from the solid to the gas state.)
All three curves on the phase diagram meet at a single point, the triple point, where all three phases exist in equilibrium. For water, the triple point occurs at 273.16 K , and is a more accurate calibration temperature than the melting point of water at 1.00 atm, or 273.15 K . See  for the triple point values of other substances.

### Equilibrium

Liquid and gas phases are in equilibrium at the boiling temperature. (See .) If a substance is in a closed container at the boiling point, then the liquid is boiling and the gas is condensing at the same rate without net change in their relative amount. Molecules in the liquid escape as a gas at the same rate at which gas molecules stick to the liquid, or form droplets and become part of the liquid phase. The combination of temperature and pressure has to be “just right”; if the temperature and pressure are increased, equilibrium is maintained by the same increase of boiling and condensation rates.
One example of equilibrium between liquid and gas is that of water and steam at  and 1.00 atm. This temperature is the boiling point at that pressure, so they should exist in equilibrium. Why does an open pot of water at  boil completely away? The gas surrounding an open pot is not pure water: it is mixed with air. If pure water and steam are in a closed container at  and 1.00 atm, they would coexist—but with air over the pot, there are fewer water molecules to condense, and water boils. What about water at  and 1.00 atm? This temperature and pressure correspond to the liquid region, yet an open glass of water at this temperature will completely evaporate. Again, the gas around it is air and not pure water vapor, so that the reduced evaporation rate is greater than the condensation rate of water from dry air. If the glass is sealed, then the liquid phase remains. We call the gas phase a vapor when it exists, as it does for water at , at a temperature below the boiling temperature.

### Vapor Pressure, Partial Pressure, and Dalton’s Law

Vapor pressure is defined as the pressure at which a gas coexists with its solid or liquid phase. Vapor pressure is created by faster molecules that break away from the liquid or solid and enter the gas phase. The vapor pressure of a substance depends on both the substance and its temperature—an increase in temperature increases the vapor pressure.
Partial pressure is defined as the pressure a gas would create if it occupied the total volume available. In a mixture of gases, the total pressure is the sum of partial pressures of the component gases, assuming ideal gas behavior and no chemical reactions between the components. This law is known as Dalton’s law of partial pressures, after the English scientist John Dalton (1766–1844), who proposed it. Dalton’s law is based on kinetic theory, where each gas creates its pressure by molecular collisions, independent of other gases present. It is consistent with the fact that pressures add according to Pascal’s Principle. Thus water evaporates and ice sublimates when their vapor pressures exceed the partial pressure of water vapor in the surrounding mixture of gases. If their vapor pressures are less than the partial pressure of water vapor in the surrounding gas, liquid droplets or ice crystals (frost) form.

### Section Summary

1. Most substances have three distinct phases: gas, liquid, and solid.
2. Phase changes among the various phases of matter depend on temperature and pressure.
3. The existence of the three phases with respect to pressure and temperature can be described in a phase diagram.
4. Two phases coexist (i.e., they are in thermal equilibrium) at a set of pressures and temperatures. These are described as a line on a phase diagram.
5. The three phases coexist at a single pressure and temperature. This is known as the triple point and is described by a single point on a phase diagram.
6. A gas at a temperature below its boiling point is called a vapor.
7. Vapor pressure is the pressure at which a gas coexists with its solid or liquid phase.
8. Partial pressure is the pressure a gas would create if it existed alone.
9. Dalton’s law states that the total pressure is the sum of the partial pressures of all of the gases present.

### Conceptual Questions