Physical Chemistry Third Edition

25.3 The Probability Distribution and the Molecular Partition Function 1057

The Molecular Partition Function of a Dilute Monatomic Gas

The parametersαandβcan depend on thermodynamic variables. Sinceαis now

expressed in terms ofβ, finding out howβdepends on thermodynamic variables

will complete the task of finding the most probable distribution. The expression of

Eq. (25.3-7) is used to do this, and it is necessary to know how the thermodynamic

energyUdepends on thermodynamic variables for a specific system.

For this system we choose a monatomic dilute gas confined in a rectangular box of

dimensionsa×b×c. It is an experimental fact that the thermodynamic energy of a

monatomic gas such as helium, neon, or argon is given for ordinary temperatures by

the relation

U

3

2

nRT+U 0 

3

2

NkBT+U 0 (25.3-8)

whereRis the ideal gas constant andTis the absolute temperature, and whereU 0

is a constant. The constantkBis Boltzmann’s constant, equal toR/NAv 1. 3807 ×

10 −^23 JK−^1. We choose to setU 0 equal to zero.

We now apply the expression forUin Eq. (25.3-7) to obtain a formula to compare

with Eq. (25.3-8). Atoms have only translational and electronic energy. The degeneracy

of an energy level is the product of the translational degeneracy and the electronic

degeneracy, and the energy is the sum of the translational and the electronic energy.

The molecular partition function is

z

∑

jtr

∑

jel

gjtrgjele−β(εjtr+εjel) (25.3-9)

wherejtris an abbreviation for the translational quantum numbers andjelis an abbre-

viation for the electronic quantum numbers. The two energies and degeneracies are

independent of each other, so the partition function is a product of two independent

sums:

z

∑

jtr

gjtre−βεjtr

∑

jel

gjele−βεjelztrzel (25.3-10)

The factorztris called thetranslational partition functionand the factorzelis called

theelectronic partition function. Since there are no general formulas forεjel,zelmust

be summed explicitly:

zelg (^0) ele−βε^0 el+g (^1) ele−βε^1 el+ ··· (25.3-11)
It is an experimental fact for inert gases such as helium, neon, and argon that the
electronic energy can be ignored near room temperature, and we will see in the next
section that this leads to a constant value forzel. We now focus on the translational
partition function.
If the potential energy is chosen to equal zero inside the box, the translational energy
eigenvalues are given by Eq. (22.1-4):
εtrεnxnynz
h^2
8 m

[

n^2 x

a^2

+

n^2 y

b^2

+

n^2 z

c^2

]

(25.3-12)