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# Ideal Monatomic Gas

Let us now practice calculating thermodynamic relations using the partition function by considering an example with which we are already quite familiar: namely, an ideal monatomic gas. Consider a gas consisting of identical monatomic molecules of mass , enclosed in a container of volume . Let us denote the position and momentum vectors of the th molecule by and , respectively. Because the gas is ideal, there are no interatomic forces, and the total energy is simply the sum of the individual kinetic energies of the molecules:

 (7.63)

where .

Let us treat the problem classically. In this approach, we divide up phase-space into cells of equal volume . Here, is the number of degrees of freedom, and is a small constant with dimensions of angular momentum that parameterizes the precision to which the positions and momenta of molecules are determined. (See Section 3.2.) Each cell in phase-space corresponds to a different state. The partition function is the sum of the Boltzmann factor taken over all possible states, where is the energy of state . Classically, we can approximate the summation over cells in phase-space as an integration over all phase-space. Thus,

 (7.64)

where is the number of degrees of freedom of a monatomic gas containing molecules. Making use of Equation (7.63), the previous expression reduces to

 (7.65)

Note that the integral over the coordinates of a given molecule simply yields the volume of the container, , because the energy, , is independent of the locations of the molecules in an ideal gas. There are such integrals, so we obtain the factor in the previous expression. Note, also, that each of the integrals over the molecular momenta in Equation (7.65) are identical: they differ only by irrelevant dummy variables of integration. It follows that the partition function of the gas is made up of the product of identical factors: that is,

 (7.66)

where

 (7.67)

is the partition function for a single molecule. Of course, this result is obvious, because we have already shown that the partition function for a system made up of a number of weakly interacting subsystems is just the product of the partition functions of the subsystems. (See Section 7.6.)

The integral in Equation (7.67) is easily evaluated:

 (7.68)

where use has been made of Equation (2.79). Thus,

 (7.69)

and

 (7.70)

The expression for the mean pressure, (7.50), yields

 (7.71)

which reduces to the ideal gas equation of state

 (7.72)

where use has been made of and . According to Equation (7.35), the mean energy of the gas is given by

 (7.73)

Note that the internal energy is a function of temperature alone, with no dependence on volume. The molar heat capacity at constant volume of the gas is given by

 (7.74)

so the mean energy can be written

 (7.75)

We have seen all of the previous results before. Let us now use the partition function to calculate a new result. The entropy of the gas can be calculated quite simply from the expression

 (7.76)

Thus,

 (7.77)

or

 (7.78)

where

 (7.79)

The previous expression for the entropy of an ideal gas is certainly a new result. Unfortunately, it is also quite obviously incorrect.

Next: Gibb's Paradox Up: Applications of Statistical Thermodynamics Previous: Partition Function
Richard Fitzpatrick 2016-01-25