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Statistical thermodynamics 257

Thermodynamic parameters and the partition function

It is possible to directly relate the partition function to the thermodynamic parameters of a
system. The thermally sourced internal energy, U, and the entropy, S, are given by:
2
N
U=Nk BT ∂(lnq)/∂T and S=k Blnq +U/T
Table 1. Partition functions for a diatomic molecule
Property Partition function Notes
2 1/2
Translational q trans =(2πmk B T/h ) V Based on the energy levels for a particle in a box
partition function (Topic G4)
2
2
Rotational partition q rot =8π lk B T/σh Assumes a rigid rotor. σ=1 for heteronuclear
function molecules, such as HF or HCl, and σ=2 for
homonuclear molecules such as H 2 , I 2 , etc.
Vibrational Assumes a harmonic oscillator in which only the
partition function lowest energy vibrational modes are thermally
accessible

For the special case of a monatomic gas, the only contribution to the
partitionfunction results from translational energy levels. This ultimately
yields theSackur-Tetrode equation for the entropy of a perfect
monatomic gas of mass, m,at a pressure, p:

Heat capacity

Partition functions allow calculation of the heat capacity of a system. The following
discussion of heat capacity applies to the constant volume heat capacity, from which
the constant pressure heat capacity may be easily calculated (Topic B1). For a gas,
substitution of q trans into the expression for U yields
E trans=3RT/2

Therefore the molar translational heat capacity is given by C trans=dE trans /dT= 3R/2, and
q rot and q vib can be likewise treated. It is found that, for a diatomic gas, both the molar
quantities C rot and C υib vary between 0 and R depending upon the ratio of kT to the
difference between energy levels, hv. For C rot or C υib, when , the heat capacity
is zero, rising to a molar value of R when . Generally, for a translation or a
rotation, the maximum heat capacity is equal to R/2 for each degree of freedom, that is,
each independent mode of motion. Thus, a gaseous diatomic molecule may have three
translational degrees of freedom (one for each orthogonal direction of motion), two