Once we establish the complete partition function of a molecular gaseous
species, we will consider one additional application of the partition function:
the chemical change. In the last chapter, a few exercises asked for a determina-
tion of the (something) of a physical process, like the expansion of a
monatomic gas. However, in chemistry we are often concerned with the change
in the chemicalidentity of a species—a chemical reaction. It may surprise you
to learn that the partition functions of each chemical species in a balanced
chemical reaction can be used to determine a characteristic property of that
reaction: its equilibrium constant.
18.1 Synopsis
First, we will define electronic and nuclear parts of the partition function,
which were ignored in the previous chapter. It will be demonstrated, though,
that for most systems they can be neglected. We also present counterexamples
in which the electronic or nuclear partition functions can’t be neglected—not
to be confusing, but as a lesson that they shouldn’t be ignored automatically.
In both of these cases, we will find that the partition functions can actually be
expressed in terms of the original definition ofq. But for the rotational and vi-
brational partition functions, this is not the case. We will be able to rewrite the
infinite summation over the energy levels to get a new expression for q. These
expressions will be determined with the help of quantum mechanics and the
equations for the quantized energies of an object rotating in three-dimensional
space (the 3-D rigid rotor) or vibrating in a Hooke’s-law type of oscillation
(the harmonic oscillator).
Recognizing that molecules are an important part of chemistry, we will de-
fine a molecular partition function,Q, that is the product of partition func-
tions from various energies of a molecule: translational, vibrational, rotational,
electronic, and nuclear.
A chemical reaction is a process that we can apply statistical thermody-
namics to. The Hor Svalue of a process is determined by the Hor Svalue
of the products minus the Hor Svalue of the reactants. In being able to cal-
culate Hor S(or any other state function) of product or reactant species, we
should be able to calculate the Hor Svalue of the process from a statisti-
cal thermodynamic perspective. We will ultimately find that the very concept
of an equilibrium constant—that is, that a constantdefines the long-term ex-
tent of any reaction—comes directly from statistical thermodynamics. This, if
anything, should establish the impact that a statisticalapproach to atoms and
molecules has on our understanding of chemistry.
There was some attempt to extend these ideas to phases other than gases.
Historically, some of the most useful extensions were to crystals. We will fin-
ish our (nowhere near exhaustive) treatment of “stat thermo” with a discus-
sion of this application. The discussion has its human aspects, because it is an
interesting example of a case in which Einstein was wrong. (Or at least, not as
“right” as others were.)
18.2 Separating q: The Nuclear and
Electronic Partition Functions
In the previous chapter, we suggested that the overall partition function qfor
a monatomic gas is
qqtransqelectqnuc (18.1)18.2 Separating q: The Nuclear and Electronic Partition Functions 617