Physical Chemistry Third Edition

26.1 The Statistical Thermodynamics of a Dilute Gas 1087

The Helmholtz Energy of a Dilute Gas

Using the formula for the entropy, we can obtain a formula for the Helmholtz energy:

AU−TSU−T

[

U

T

+NkB+NkBln

(z

N

)]

A−NkBTln

(z

N

)

−NkBT (26.1-22)

The Chemical Potential of a Dilute Gas

In statistical mechanics, it is customary to define the chemical potential as a derivative

with respect to the number of molecules, not with respect to the amount in moles:

μ

(

∂A

∂N

)

T,V

(26.1-23)

This chemical potential is equal to the thermodynamic chemical potential divided by

Avogadro’s constant. The number of molecules is an integer, so we use a quotient of

finite differences to approximate the derivative in Eq. (26.1-23):

μ

AN−AN− 1

1

AN−AN− 1 (26.1-24)

where the subscripts indicate the number of molecules in the system.

μ−NkBTln(z/N)−NkBT−

[

−(N−1)kBTln

(

z

N− 1

)

−(N−1)kBT

]

−NkBTln(z)+NkBTln(N)−NkBT+(N−1)kBTln(z)

−(N−1)kBTln(N−1)+NkBT−kBT

−kBTln(z)+NkBTln

(

N

N− 1

)

+kBTln(N−1)−kBT (26.1-25)

SinceNis a large number, the second term of the right-hand side of the final version

of this equation can be approximated:

NkBTln

(

N

N− 1

)

−NkBTln

(

N− 1

N

)

−NkBTln

(

1 −

1

N

)

≈−NkBT

(

−

1

N

)

kBT (26.1-26)

Two terms cancel. SinceNis a large number (near 10^24 in most systems), ln(N−1)

can be replaced by ln(N) without serious error, so that

μ−kBTln

(z

N

)

(26.1-27)