Physical Chemistry Third Edition

1064 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

25.16The value of the molecular partition function of an argon
atom confined in a volume of 0.0250 m^3 (25.0 L) at
298 .15 K is given in Example 25.3.
a. Find the value ofnx(assumed equal tonyandnz) that
will make the energy of an argon atom equal to
3 kBT/2 at this temperature, assuming that the volume
is a cube.
b. Find the probability of the state with this value of
nx,ny, andnz.

25.17Calculate the ratio of the population of thev 1
vibrational state of CO to thev0 vibrational state at
(a) 298.15 K, (b) 500 K, (c) 1000 K, and (d) 5000 K.
(e) What is the limit asT→∞?

25.18Calculate the ratio of the population of thev 2

vibrational state of I 2 to thev0 vibrational state at

(a) 298.15 K, (b) 500 K, (c) 1000 K, and (d) 5000 K.

(e) What is the limit asT→∞?

25.19Calculate the ratio of the population of theJ 1

rotational level of CO to theJ0 rotational level at

(a) 298.15 K, (b) 1000 K. (c) What is the limit as

T→∞?

25.20Calculate the ratio of the population of theJ 2

rotational level of I 2 to theJ0 rotational level at

(a) 298.15 K, (b) 1000 K. (c) What is the limit as

T→∞?

25.4 The Calculation of Molecular Partition

Functions

In the previous section we obtained a general formula for the translational partition

function. In this section we obtain formulas for the other factors in the molecular parti-

tion function for dilute gases and carry out example calculations of partition functions.

Monatomic Gases

We have already determined that the molecular partition function for a dilute monatomic

gas is the product of a translational partition function and an electronic partition func-

tion. We obtained a formula for the translational partition function in Eq. (25.3-21):

ztr

(

2 πmkBT

h^2

) 3 / 2

V (25.4-1)

With our replacement ofβby 1/(kBT), Eq. (25.3-11) for the electronic partition function

becomes

zelg 0 e−ε^0 /kBT+g 1 e−ε^1 /kBT+ ··· (25.4-2)

Except for hydrogen-like atoms, there is no simple formula for atomic electronic energy

eigenvalues, so the sum in Eq. (25.4-2) must be added up explicitly. However, for a

typical monatomic gas such as helium or argon, the ground electronic energy level is a

nondegenerate^1 Sstate, and the first excited state is at least 1 eV higher than the ground-

level. In this case, all terms past the first term are negligible at ordinary temperatures,

and we can write, to an excellent approximation:

zel≈g 0 e−ε^0 /kBT (25.4-3)

If the energy of the ground-level is chosen to equal zero and if the ground-level is

nondegenerate, we have

zel≈1(ε 0 0,g 0 1) (25.4-4)