Physical Chemistry Third Edition

25.3 The Probability Distribution and the Molecular Partition Function 1055

negligible. The potential energy of two molecules can be

represented by the Lennard–Jones formula exhibited in the

previous problem. For carbon dioxide,σ 4. 5 × 10 −^10 m

andε 2. 61 × 10 −^21 J. Estimate the energy of attraction

of 1.00 mol of carbon dioxide at molar volume 0.50 L

mol−^1 and temperature 298.15 K as follows: Calculate the

volume per molecule, and assume that each molecule

occupies a cubical volume. Estimate the average

nearest-neighbor distance as the distance from the center

of one cube to the center of the next cube. Evaluate the

potential energy of a pair of nearest-neighbor molecules.

Assume that each molecule is surrounded by twelve

nearest-neighbor molecules and neglect interactions with

more distant molecules. Remember that each

intermolecular potential energy is shared by two

molecules. Compare the potential energy with the kinetic

energy, which is given by gas kinetic theory as 3nRT /2,

wherenis the amount of gas in moles.

25.11 a. Use Lagrange’s method of undetermined multipliers to

find the constrained maximum of the function

fe−(x

(^2) +y (^2) )
subject to the constraintx+y2.
b. Find the maximum in an alternate way by replacingy
by 2−xin the original function and then finding the
maximum by setting the derivative of the function with
respect toxequal to zero.
25.12 a. Use Lagrange’s method of undetermined multipliers to
find the constrained maximum in the first octant (xand
yboth positive) of the functionfe−(x
(^2) +y (^2) )
subject
to the constraintx^2 +y^2 1.
b. Find the maximum in an alternate way by replacing
x^2 by 1−y^2 in the original function and then finding
the maximum by setting the derivative of the function
with respect toyequal to zero.
25.13Find the maximum of the functionfsin(x)e−y
2
subject
to the constraintx−y0.

25.3 The Probability Distribution and the

Molecular Partition Function

In this section, we evaluate the parametersαandβand introduce the molecular

partition function, a function that can be used to calculate values of thermodynamic

variables for a dilute gas.

The Molecular Partition Function and the Parameterα

To remove the parameterαfrom the distribution in Eq. (25.2-23) we substitute this

distribution into the constraint of Eq. (25.2-14a):

∑

j

gjeαe−βεjN (25.3-1)

Theeαfactor in Eq. (25.3-1) can be factored out of the sum and the resulting equation

can be solved foreα:

eαN

⎡

⎣

∑

j

gje−βεj

⎤

⎦

− 1



N

z

(25.3-2)

where we define themolecular partition functionz:

z

∑

j

gje−βεj (definition of the molecular partition function) (25.3-3)

This sum is over the molecular energy levels.