# THE KINETIC THEORY OF GASES

## Molar Specific Heat of an Ideal Gas

The energy required to raise the temperature of n moles of gas from T_{i} to T_{f} depends on the path taken between the initial and final states. To understand this, let us consider an ideal gas undergoing several processes such that the change in temperature is ΔT = T_{f} – T_{i} for all processes. The temperature change can be achieved by taking a variety of paths from one isotherm to another, as shown in Figure 21.3. Because ΔT is the same for each path, the change in internal energy ΔE_{int} is the same for all paths. However, we know from the first law, Q = ΔE_{int} + W, that the heat Q is different for each path because W (the area under the curves) is different for each path. Thus, the heat associated with a given change in temperature does not have a unique value.

We can address this difficulty by defining specific heats for two processes that frequently occur: changes at constant volume and changes at constant pressure. Because the number of moles is a convenient measure of the amount of gas, we define the molar specific heats associated with these processes with the following equations:

Where C_{V} is the molar specific heat at constant volume and C_{P} is the molar specific heat at constant pressure. When we heat a gas at constant pressure, not only does the internal energy of the gas increase, but the gas also does work because of the change in volume. Therefore, the heat Q_{constant P} must account for both the increase in internal energy and the transfer of energy out of the system by work, and so Q_{constant P} is greater than Q_{constant V} . Thus, C_{P} is greater than C_{V} .

In the previous topic, we found that the temperature of a gas is a measure of the average translational kinetic energy of the gas molecules. This kinetic energy is associated with the motion of the center of mass of each molecule. It does not include the energy associated with the internal motion of the molecule - namely, vibrations and rotations about the center of mass. This should not be surprising because the simple kinetic theory model assumes a structureless molecule.

In view of this, let us first consider the simplest case of an ideal monatomic gas, that is, a gas containing one atom per molecule, such as helium, neon, or argon. When energy is added to a monatomic gas in a container of fixed volume (by heating, for example), all of the added energy goes into increasing the translational kinetic energy of the atoms. There is no other way to store the energy in a monatomic gas. Therefore, from Equation 21.6, we see that the total internal energy E_{int} of N molecules (or n mol) of an ideal monatomic gas is

Note that for a monatomic ideal gas, E_{int} is a function of T only, and the functional relationship is given by Equation 21.10. In general, the internal energy of an ideal gas is a function of T only, and the exact relationship depends on the type of gas, as we shall soon explore.

If energy is transferred by heat to a system at constant volume, then no work is done by the system. That is, W = ∫ PdV = 0 for a constant-volume process. Hence, from the first law of thermodynamics, we see that

In other words, all of the energy transferred by heat goes into increasing the internal energy (and temperature) of the system. A constant-volume process from i to f is described in Figure 21.4, where ΔT is the temperature difference between the two isotherms. Substituting the expression for Q given by Equation 21.8 into Equation 21.11, we obtain

If the molar specific heat is constant, we can express the internal energy of a gas as

This equation applies to all ideal gases - to gases having more than one atom per molecule, as well as to monatomic ideal gases.
In the limit of infinitesimal changes, we can use Equation 21.12 to express the molar specific heat at constant volume as

Let us now apply the results of this discussion to the monatomic gas that we have been studying. Substituting the internal energy from Equation 21.10 into Equation 21.13, we find that

This expression predicts a value of C_{V} = 3/2 R = 12.5 J/mole.K for all monatomic gases. This is in excellent agreement with measured values of molar specific heats for such gases as helium, neon, argon, and xenon over a wide range of temperatures ( Table 21.2).

Now suppose that the gas is taken along the constant-pressure path i → f ʹ shown in Figure 21.4. Along this path, the temperature again increases by ΔT. The energy that must be transferred by heat to the gas in this process is Q = nC_{P}ΔT. Because the volume increases in this process, the work done by the gas is W = PΔV, where P is the constant pressure at which the process occurs.

Applying the first law to this process, we have

In this case, the energy added to the gas by heat is channeled as follows: Part of it does external work (that is, it goes into moving a piston), and the remainder increases the internal energy of the gas. But the change in internal energy for the process i → f ʹ is equal to that for the process i → f because E_{int} depends only on temperature for an ideal gas and because ΔT is the same for both processes. In addition, because PV = nRT, we note that for a constant-pressure process, PΔV = nRΔT.
Substituting this value for PΔV into Equation 21.15 with ΔE_{int} = nCV ΔT (Eq. 21.12) gives

This expression applies to any ideal gas. It predicts that the molar specific heat of an ideal gas at constant pressure is greater than the molar specific heat at constant volume by an amount R, the universal gas constant (which has the value 8.31 J/mol . K). This expression is applicable to real gases, as the data in Table 21.2 show.

Because C_{V} = 3/2 R for a monatomic ideal gas, Equation 21.16 predicts a value C_{P} = 5/2 R = 20.8 J/mol.K for the molar specific heat of a monatomic gas at constant pressure. The ratio of these heat capacities is a dimensionless quantity γ (Greek letter gamma):

Theoretical values of C_{P} and γ are in excellent agreement with experimental values obtained for monatomic gases, but they are in serious disagreement with the values for the more complex gases (see Table 21.2). This is not surprising because the value C_{V} = 3/2 R was derived for a monatomic ideal gas, and we expect some additional contribution to the molar specific heat from the internal structure of the more complex molecules. In coming topics, we describe the effect of molecular structure on the molar specific heat of a gas. We shall find that the internal energy— and, hence, the molar specific heat—of a complex gas must include contributions from the rotational and the vibrational motions of the molecule.

We have seen that the molar specific heats of gases at constant pressure are greater than the molar specific heats at constant volume. This difference is a consequence of the fact that in a constant-volume process, no work is done and all of the energy transferred by heat goes into increasing the internal energy (and temperature) of the gas, whereas in a constant-pressure process, some of the energy transferred by heat is transferred out as work done by the gas as it expands. In the case of solids and liquids heated at constant pressure, very little work is done because the thermal expansion is small. Consequently, C_{P} and C_{V} are approximately equal for solids and liquids.

Next Topic:- Adiabatic Processes for an Ideal Gas