SECTION VI. The Classical Ideal Gas, Maxwell– Boltzmann Statistics, nonideal Systems

The Classical Ideal Gas

Introduction

Following our discussion of the quantum gas cases in Chapters 13 and 14, we are now in a position to consider the classical ideal gas in some detail. For simplicity, we shall initially consider systems of particles without internal degrees of freedom, such as monatomic gases, but this restriction will be listed later in this chapter. In Chapter 12, we considered the classical limit of the Bose-Einstein and Fermi-Dirac distributions. In the classical limit, the partition function for a system of N particles

-|Л'

in a gas is given by Equation 12.24, Z = (1 IN'.) ^V= (lIN) z^N. with z as the single-particle

r

partition function. The factor /N allows for the indistinguishability of particles, as discussed in Chapters 4 and 12. Because internal energy contributions are not considered, the states r, with energy eigenvalues e_n correspond to the particle in a box state considered in Chapter 4. Equation

4.5 gives the energy eigenvalues as e_r = (n²Ti²l2m)(IV^ll}}[ii_x² +n_v² +m²), where the quantum numbers n_x, n„ and n, take integral values.

As noted in Chapter 4, in the classical limit, a large number of eigenstates are populated, albeit in a sparse way, with the mean number of particles in a given state (n) <V_a, where Vq = A³/(3mk_BT)^m is the quantum volume and V_A = (VIN) is the volume per particle. The inequality may be rewritten in the following form:

which is convenient for the comparison that is made below. From Equation 4.5, the spacing between adjacent energy levels with quantum numbers (n„ n_y, »,,) and (n_x+1, n_y, n.) is to a good approximation

Comparison of Equations 16.1 and 16.2 suggests that к_вГ» Ae_r so that the thermal energy per degree of freedom is very much larger than the spacing of the energy levels, as depicted in Figure 16.1.

The energy states are so closely spaced that they form a quasi-continuum. Because the factor е~^' in Equation 12.24 varies slowly with e„ it is permissible to replace the sum by an integral in the evaluation of the partition function Z. This approach is made use of in Section 16.2. Once the partition function has been determined, we use the bridge relationship F = -k_BT InZ to obtain the Helmholtz potential and from F the entropy of the ideal monatomic gas. For polyatomic molecules, allowance must be made for internal degrees of freedom, and this topic is discussed in Section 16.5. This chapter concludes with a proof of the equipartition theorem and a brief discussion of the Maxwell velocity distribution for a classical gas.

FIGURE 16.1 Schematic representation of a range of energy levels for a particle in a box. For gases in the classical limit, the thermal energy per degree of freedom ~k_BT is much larger than the spacing between singleparticle levels.

The Partition Function for an Ideal Classical Gas

If the gas particles do not possess internal energies, the single-particle partition function for an ideal gas may be written as

To simplify the expression, introduce К = (я²й²/2/иУ^2Л) so that the partition function may be written as

Each of the identical summations may, to a good approximation, be replaced by an integral over n Уе-^~ - J c“^№r An, with the lower limit taken as zero, with negligible error, because so many

states contribute to the integral. The integral is given in Appendix A, e~' d.v = yJn/2, and,

J 0

with .v = (0K)^m n, it follows that f f e~^^K" d.v = —^jnJfJK 1. The single-particle partition function is therefore given by ^ '

The partition function for a gas of N molecules follows from Equation 12.24:

and hence,

where the use has been made of Stirling’s formula for In N

Exercise 16.1

Show that the single-particle partition function z may, to a good approximation, be written in terms of the quantum volume Vq = (h²/3mk_BT)^y2 as z~V/Vq. Obtain an expression for the partition function for N particles in terms of the quantum volume and the atomic volume.

From Equation 16.4, the single-particle partition function is given by

For N particles, we write the N particle partition function as

An advantage of the simple form for In Z in terms of V_A = V/N and V_Q is that it is memorable and easy to write down. Furthermore, the form emphasizes the importance of the classical limit condition V_A» V_Q.

Thermodynamics of an Ideal Gas

From Equations 10.49 and 16.6, an expression for the Helmholtz potential for an ideal gas is immediately obtained

The entropy is as follows:

This is the Sackur-Tetrode equation, given previously in the equivalent form in Equation 5.15. The entropy is obtained correctly as an extensive quality because the factor 1/7V!, which takes account of the indistinguishability of particles, is introduced in a natural way in taking the classical limit of the quantum distributions. For constant particle density (i.e., V/N constant) and Tconstant, Equation

16.8 shows that S is proportional to N. The entropy expression does not allow for possible spin states of the ideal gas particles. If the particles have angular momentum specified by quantum number 7, then an additional term Nk_B ln(2./+1) must be added to the entropy, corresponding to the (2J+ l)^,v possible spin states.

Exercise 16.2

Use the expression for the Helmholtz potential F to obtain expressions for thermodynamic quantities of interest, including pressure, specific heat, and chemical potential. (Several of these results have been discussed previously in Section 5.4.)

The pressure is given by P = ~(dF/dV)_T = Nk_BT/V, which is the ideal gas equation of state.

3 3

The mean energy is {E) = -d Z/[i = ^Nk_BT or, per particle, {£) = (£}/Д' = — k_BT, in agreement with the equipartition of energy theorem, with three translational degrees of freedom for each particle.

3 3

The heat capacity of the gas follows directly: CV = (d(E)/dT)_v = — Nk_B, or for 1 mol cy = — R,

which is the familiar result for the specific heat of a monatomic ideal gas.

Finally, the chemical potential is given by

As noted in Chapter 11,//is proportional to T and depends on the particle concentration N/V. We have seen that // <0 in the classical limit of the quantum distributions.//may again be written in a simple form by making use of the quantum volume V_q and the atomic volume V_A. To a good approxi-

mation, we obtain p-k_BT 1п(Уд/У_л)-^-[ln(2n/3) + l] , with Vq^V^ in the classical limit, as

emphasized previously. Representative values for a classical gas at standard temperature and pressure lead to Vq/V_a~ 10~⁴ so that // < 0. In general, for given T, large values for V_A, which correspond to a low particle density, lead to relatively large negative values for //. For systems not in equilibrium, where there is a gradient in //, particles will tend to move from regions of high concentration to regions of low concentration. When V_Q and V_A become comparable, the classical limit no longer applies and the appropriate quantum statistics expressions must be used.

Exercise 16.3

Use the expression for//derived above to obtain a simple classical limit form for the fugacity X = e**, which was defined in Chapter 11 in our discussion of the quantum distributions.

From the expression for//given above, we obtain A—0.2(Vq/VX). It is clear that 2