MathConnections
CH 3 CHCH 2 CH3CH2CH2CH2
5.6 Kinetic–Molecular Theory and Ideal Versus Real Gases
The ability to model the behavior of gases numerically was an important achieve- ment that has broad applicability in many areas of science and engineering. The similarity of the behavior of different gases commonly encountered in the laboratory allowed the construction of models to describe how atoms must behave. But in many important practical settings, gases do not always behave ideally. So the simple ideal gas law calculations that we have been doing may not always be an accurate model of reality. Nonideal behavior is frequently seen in gases at very high pressures, such as might be found in the cylinder of an engine or in an industrial processing plant.
Can an atomic perspective developed for ideal gases help explain these real world cases? Perhaps more importantly, can an improved model defi ne quantitative meth- ods for dealing with real gases such as those encountered in a running engine? In this section, we will develop a very useful model called the kinetic–molecular theory, which provides connections between the observed macroscopic properties of gases, the gas law equation, and the behavior of gas molecules on a microscopic scale. Once we understand this model, we can use it to see how gases at high pressure might de- viate from ideal behavior.
Postulates of the Model
Because observation of individual gas molecules is not generally feasible, we must begin with postulates that are accepted as reasonable when we construct a model.
The kinetic-molecular theory is also called the kinetic theory of gases or simply the kinetic theory.
The kinetic-molecular theory is also called the kinetic theory of gases or simply the kinetic theory.
For the kinetic–molecular theory of gases, the following postulates form the bedrock of the model.
1. A gas is made up of a vast number of particles, and these particles are in constant random motion.
2. Particles in a gas are infi nitely small; they occupy no volume.
3. Particles in a gas move in straight lines except when they collide with other mol- ecules or with the walls of the container. Collisions with each other and with the walls of the container are elastic, so that the total kinetic energy of the particles is conserved.
4. Particles in a gas interact with each other only when collisions occur.
5. The average kinetic energy of the particles in a gas is proportional to the absolute temperature of the gas and does not depend on the identity of the gas.
As we consider the implications of these postulates, we will also comment on their validity or the conditions under which they are not particularly accurate. Starting from these postulates, it is actually possible to derive the ideal gas law. Although that result is a great example of the synergy between scientifi c theory and experimental observa- tions, it is a bit beyond our reach here. So instead we will consider the connections between the postulates of the kinetic theory and our practical experience with gases.
How do these postulates help to explain the observed behavior of various gases?
The statement that a gas consists of a large number of particles is consistent with what we know about the size of individual molecules. And the postulate that the mol- ecules are in constant random motion is consistent with the observation that gases readily expand to fi ll their containers. The postulates are also consistent with what we said at the outset of this chapter about the origin of gas pressure. The molecules in a gas exert force when they collide with the wall of the container, and this accounts for the relationship between pressure and volume. As the volume of the container increases, the walls move farther apart and so each molecule will strike the wall less often. Thus, each molecule will exert less force on the wall in a given time interval, reducing the pressure. So the kinetic theory is consistent with the inverse relationship between P and V that we’ve seen in the gas law equation.
How can the kinetic theory account for the effect of changes in temperature? The situation here is a little more complicated. We need to consider postulate #5 carefully:
The kinetic energy of the particles in a gas is proportional to the absolute temperature of the gas and does not depend on the identity of the gas. We know from physics that kinetic energy is related to speed and to mass: KE = 1/2 mv2. So if the kinetic energy depends on the temperature, as stated in the postulate, then the speed of the molecules must be a function of temperature. More specifi cally, increasing the temperature must increase the speed of the molecules. When the molecules move faster, they will collide with the walls more often, and they will also impart a greater force on the wall during each collision. Both of these effects will lead to an increase in gas pressure with increasing temperature, as predicted by the gas law.
If you were paying close attention to the statement of postulate #5, though, you may be wondering about the inclusion of the word “average” in “average kinetic energy.” That word is included because some molecules in a collection of randomly moving gas molecules move very fast, whereas others are virtually standing still at any given mo- ment. A moment later the identity of fast moving and slow moving particles may have changed due to collisions, but there will still be both kinds. So it isn’t really correct to think about the molecules as having a particular speed, and we should actually say that the average molecular speed increases with increasing temperature. This situation is best approached using a distribution function to describe the range of speeds.
Distribution functions are common mathematical devices used in science and also in economics, social sciences, and education. Your percentile scores on the SAT and other standardized tests are predicated on a distribution function for the students who take the exam. If you scored in the 85th percentile, for example, you would be expected to score higher than 85% of those taking the exam. This ranking is expected to hold independent
of the number of students who have actually taken the test, so that your percentile is not revised every time another round of tests is administered to more students.
The distribution function that describes the speeds of a collection of gas particles is known as the Maxwell-Boltzmann distribution of speeds. This function, which was originally derived from a detailed consideration of the postulates of the kinetic theory, predicts the fraction of the molecules in a gas that travel at a particular speed.
It has since been verifi ed by a variety of clever experiments that allow measuring the speed distribution in the laboratory.
Let’s examine a plot of the Maxwell-Boltzmann distribution for a sample of CO2 gas at various temperatures, as shown in Figure 5.6. As with any unfamiliar graph, you should start by looking at what is plotted on each axis. Here the x axis shows a fairly familiar quantity: speed in meters per second. The y axis is a little less obvious. It shows the fraction of all the molecules in the sample that would be traveling at a par- ticular speed under the given conditions. (The actual mathematical form of the distri- bution function is discussed in the MathConnections box.) We’ve already established that the distribution of speeds must be a function of temperature, and the fi gure bears this out. At high temperatures, we see that the peak in the distribution shifts toward higher speeds. This is entirely consistent with postulate #5, which states that the ki- netic energy of the molecules must increase with temperature.
Furthermore, postulate #5 also says that the kinetic energy does not depend on the identity of the gas. But the mass of the gas molecules clearly depends on their identity. For the kinetic energy to remain fi xed as the mass changes, the speed dis- tribution must also be a function of mass. Figure 5.7 illustrates this by showing the speed distribution for various gases at the same temperature. You can readily see that on average lighter molecules travel faster than heavier molecules.
Because we do not care in which direction the particles are going, our discussion here is based on speed rather than velocity. Speed does not depend on direction and is always positive.
Because we do not care in which direction the particles are going, our discussion here is based on speed rather than velocity. Speed does not depend on direction and is always positive.
0
Molecular speed (m/s)
500 1000 1500 2000 2500
N(v)/N(total)
300 K 500 K
1000 K
Figure 5.6 ❚ The distributions of molecular speeds for CO2 molecules at three different temperatures are shown. The quantity on the y axis is the fraction of gas molecules moving at a particular speed. Notice that as the temperature increases, the fraction of molecules moving at higher speed increases.
0
Molecular speed (m/s)
500 1000 1500 2000 2500
N(v)/N(total)
O2 N2
H2O
He
Figure 5.7 ❚ Distributions of molecular speeds for four different gases at the same temperature (300 K). As in Figure 5.6, the quantity on the y axis is the fraction of gas molecules moving at a particular speed. Notice that as the molecular mass decreases, the fraction of the molecules moving at higher speed increases.
MathConnections
The actual equation for the Maxwell-Boltzmann distribution of speeds is some- what complicated, but it consists entirely of terms in forms that you should recognize from your math classes.
N(v)
— Ntotal
= 4p(M/2pRT )3/2v 2e−Mv2/2RT (5.11) N(v) is the number of molecules moving with speeds between v and v + Δv, where Δv is some small increment of speed. Ntotal is the total number of molecules.
So the left-hand side is the fraction of the molecules having speeds between v and v + Δv, which is also the quantity plotted on the y axis in Figures 5.6 and 5.7.
M is the molar mass of the gas, R is the gas constant, and T is the temperature. (As in the gas law, this must be on an absolute scale.) Units for all of these quantities must be chosen consistently, with particular care taken to ensure that the exponent (–Mv 2/2RT ) has no units. Now look at Figure 5.6 or 5.7 again and notice that the independent variable on the x axis is the speed. So we want to think of the right- hand side of the distribution function as an elaborate function of the speed, v. We see two terms containing v. The fi rst is just v 2, which will obviously increase as v increases. The exponential term, though, varies as exp(–v 2), which will decrease as v increases. So the overall shape of the distribution function refl ects a competi- tion between these two terms. At small values of v, the v 2 term dominates, and the distribution function increases. At larger values of v, the exponential term takes over, and the overall function decreases. So, between those two limits, the distri- bution function goes through a maximum.
The speed at which the peak in the plot occurs is known as the most probable speed, for fairly obvious reasons: more molecules travel at this speed than at any other. Although we will not do the derivation here, the most probable speed is given by Equation 5.12:
vmp =
√
‾‾‾ 2RT — M (5.12) If you know some calculus you can do the derivation. Remember that the most probable speed is the maximum in the distribution function. To fi nd this maxi- mum, we would take the derivative of Equation 5.11 and set it equal to zero.We might also want to describe our distribution by talking about the average speed. Because the distribution function is not symmetric, the most probable speed and the average speed will not be the same. In particular, the existence of the “tail” on the distribution curve at high speeds will pull the average to a speed higher than the most probable value. The average speed actually turns out to be 1.128 times the most probable speed.
Finally, sometimes we characterize the distribution by talking about some- thing called the root-mean-square speed. This value is actually the square root of the average value of v 2. At fi rst glance, it might sound like that should be the same as the average speed, but for the Boltzmann distribution, the root- mean- square speed is actually 1.085 times the average speed. (If this sounds wrong to you, try a simple test. Pick a few numbers at random and calculate their average.
Then square all of your numbers and take the average of the squares. Finally, take the square root of that result and you’ll see that it is larger than the average of the original numbers.) The root-mean-square speed is useful if we are interested in kinetic energy because the average kinetic energy is given by
KEavg= — 1
2 mvrms2 (5.13)
Real Gases and Limitations of the Kinetic Theory
In all of our calculations so far, we have begun by assuming that any gas or gas mixture behaves ideally. But the postulates of the kinetic theory—and the gas law itself—are strictly true only for purely hypothetical ideal gases. How good is the assumption that gases behave ideally, and under what kinds of conditions might we want to question this assumption? The answers to these questions can help us to develop further our understanding of gases at the molecular level. By learning how real gases deviate from ideal behavior, we will understand the nature of the kinetic–molecular theory better. We’ll start by taking a more critical look at some of the postulates of the theory.
The assertion that gas molecules occupy no volume may seem at odds with real- ity because we know that all matter occupies space. What is actually implied by this postulate? The essential idea is that, compared to the volume of empty space between particles, the volume of the particles themselves is not signifi cant. One way to assess the validity of this assertion is to defi ne the mean free path of the particles. The mean free path is the average distance a particle travels between collisions with other particles. In air at room temperature and atmospheric pressure, the mean free path is about 70 nm, a value 200 times larger than the typical radius of a small molecule like N2 or O2. (For comparison, the molecules in a liquid typically have a mean free path roughly the same as a molecular radius.) When we consider the cubic relationship of volume to distance, the difference in volume is on the order of 2003 or 8 × 106. So the volume of empty space in a gas at room temperature and pressure is on the order of 1 million times greater than the volume of the individual molecules; the assumption in the kinetic theory that the volume of the molecules is negligible seems reasonable under these conditions.
But under what circumstances would this assumption break down? We cannot generally increase the size of molecules substantially by changing conditions, so the only way the volumes become more comparable is to squeeze the molecules closer together by decreasing the volume. Thus under conditions of high pressure, when gases are highly compressed, the volume of individual molecules may become sig- nificant and the assumption that the gas will behave ideally can break down (see Figure 5.8a). This makes sense if we realize that one way to condense a gas into a liquid is to compress it, and of course we don’t expect the ideal gas law to hold for a liquid.
The postulates also assert that gas molecules always move in straight lines and interact with each other only through perfectly elastic collisions. Another way to put this is to say that the molecules neither attract nor repel one another. If this were strictly true, then molecules could never stick together, and it would be impossible to condense a gas into a liquid or a solid. Clearly, there must be some attractive forces between molecules, a topic we will explore further in Chapter 8. The ideal gas model works because under many conditions these attractive forces are negligibly small com- pared to the kinetic energy of the molecules themselves. The fact that these forces
(b) Low temperature (a) High pressure
Figure 5.8 ❚ The ideal gas model breaks down at high pressures and low temperatures. (a) At high pressures, the average distance between molecules decreases, and the assumption that the volume of the molecules themselves is negligible becomes less valid.
(b) At low temperatures, molecules move more slowly and the attractive forces between molecules can cause
“sticky” collisions, as shown here by the paired molecules.
lead to the formation of liquids and solids points us to the conditions under which the forces become important. We know that gases will condense if we lower the tempera- ture suffi ciently. The reason for this is that at lower temperatures the kinetic energy of the molecules decreases, and eventually the strength of the attractive forces between molecules will become comparable to the kinetic energy. At that point, molecules will begin to experience sticky collisions, as shown in Figure 5.8b. This means that pairs or small clusters of molecules might stay in contact for a short time, rather than un- dergoing simple billiard ball collisions. This tendency for molecules to stick together, however fl eetingly, will reduce the number of times that the molecules collide with the walls of the container, so the pressure of the gas will be lower than that predicted by the ideal gas equation.
Correcting the Ideal Gas Equation
When the postulates of the kinetic theory are not valid, the observed gas will not obey the ideal gas equation. In many cases, including a variety of important engineering ap- plications, gases need to be treated as nonideal, and empirical mathematical descriptions must be devised. There are many equations that may be used to describe the behavior of a real gas; the most commonly used is probably the van der Waals equation:
qP + an— 2
V 2 r (V − nb) = nRT (5.14) Here, a and b are called van der Waals constants. Unlike the universal gas con- stant (R), the values of these van der Waals constants must be set for each specifi c gas. If a gas behaves ideally, both a and b are zero, and the van der Waals equation reverts to the ideal gas equation. Investigating the connection of this equation to the ideal gas equation further, we see that the term involving a accounts for the at- traction between particles in the gas. The term involving b adjusts for the volume occupied by the gas particles. Table 5.2 provides the values of van der Waals con- stants for some common gases. Notice that larger molecules have larger values for b, An empirical equation contains one or
more adjustable parameters that are found from a best fi t to observed data rather than from a theoretical model.
An empirical equation contains one or more adjustable parameters that are found from a best fi t to observed data rather than from a theoretical model.
Table
❚
5.2Van der Waals constants for several common gases
Gas
a (atm L2 mol−2)
b (L mol−1)
Ammonia, NH3 4.170 0.03707
Argon, Ar 1.345 0.03219
Carbon dioxide, CO2 3.592 0.04267
Helium, He 0.034 0.0237
Hydrogen, H2 0.2444 0.02661
Hydrogen fl uoride, HF 9.433 0.0739
Methane, CH4 2.253 0.04278
Nitrogen, N2 1.390 0.03913
Oxygen, O2 1.360 0.03183
Sulfur dioxide, SO2 6.714 0.05636
Water, H2O 5.464 0.03049