Physical Chemistry Third Edition

1088 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics

EXAMPLE26.2

Show that ln(N−1) differs from ln(N) by a term that is proportional to 1/N.

Solution

ln(N−1)ln(N)+ln

(

1 −

1

N

)

The Taylor series that represents ln(1−x)is

ln(1−x)ln(1)+

1

1!

(

dln(1−x)

dx

)

x 0

x+O(x^2 )

 0 −x+O(x^2 )

whereO(x^2 ) stands for terms that have powers ofxat least as great asx^2.

ln(N−1)ln(N)−

1

N

Exercise 26.4

Equation (26.1-27) can also be obtained by pretending thatNcan take on any real value and

performing the differentiation in Eq. (26.1-23). Carry out this differentiation.

The Gibbs Energy of a Dilute Gas

The Gibbs energy is defined by

GA+PV (26.1-28)

so that

G−NkBTln

(z

N

)

−NkBT+NkBT

G−NkBTln

(z

N

)

(26.1-29)

Note that

GNμ (26.1-30)

This is a version of Euler’s theorem, which is discussed in Part I of this textbook.

PROBLEMS

Section 26.1: The Statistical Thermodynamics of a
Dilute Gas

26.1 Consider the two isotopic substances^17 O 2 and^16 O^18 O.
Round off the atomic masses in amu to the nearest integer.

Answer the following questions without detailed

calculation:

a.How will the translational partition functions of the two

molecules compare at equal volumes and equal

temperatures?