Physical Chemistry , 1st ed.

For atomic and molecular systems, we actually have such expressions: they
come from the application of quantum mechanics to the translations, rota-
tions, vibrations, and electronic states of atoms and molecules. Admittedly,
Boltzmann didn’t have quantum mechanics, since he developed the rudiments
of statistical mechanics about 50 years before quantum mechanics was formu-
lated. In fact, some of his expressions are incorrect by not including Planck’s
constant (Boltzmann was unaware of its existence for most of his life). But in
the calculation of thermodynamic values, the Planck’s constants cancel. Their
omission was, ultimately, unnoticed. However, in the material to come, we will
use the quantum-mechanical basis of energy levels.
We start by assuming that our sample consists of a monatomic gas, like He
or Ne (or any other monatomic gas, like Hg vapor). Such a sample has only
three types of energy states: electronic, nuclear, and translational. Of these
three, electronic and nuclear states are states withinthe atoms. Only transla-
tional energy states relate the position of the atom as a whole, rather than re-
lating the relative positions of the subatomic particles of the atom.
The partition function of a monatomic gas is a product of three separate
partition functions defined by the translational energy levels, the electronic en-
ergy levels, and the nuclear energy levels:

qqtransqelectqnucl (17.48)

Further, we will presume at this point that the translational partition function,
qtrans, is the major contributor to the thermodynamic properties of a
monatomic gas. (We will justify this by using the kinetic theory of gases, which
is covered in Chapter 19. The relative contributions ofqelectand qnucwill be
considered in Chapter 18.) Therefore, for a monatomic ideal gas, we are as-
suming that

qqtrans
How shall we model the translational motions of monatomic gases? Well,
we can apply the particle-in-a-box approximation to the straight-line motions
of the atoms in three-dimensional space. From Chapter 10, we know that for
a particle in a three-dimensional box, the quantum-mechanically allowed en-
ergy levels are

    

8

h

m

2



n

a^2

x^2 n

b^2

y^2 n

c^2

z^2



The variable mis the mass of the particle in the box, and so in this case rep-
resents the mass of the individual gaseous atom or molecule,not the molar
mass. For simplicity’s sake, we will arbitrarily assume that we are working in a
cubic system, so that abc:

    

8 m

h^2

a^2

(nx^2 ny^2 nz^2 ) (17.49)

In terms of the volume of the system V, if the system is cubic, then Va^3.
Therefore,a^2 must equal V2/3. Equation 17.49 becomes

    

8 m

h

V

2

2/3(nx

(^2) ny (^2) nz (^2) ) (17.50)
Therefore, our expression for qis (assuming that the translational states are all
singly degenerate)
17.6 The Partition Function: Monatomic Gases 605