Physical Chemistry Third Edition

1118 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics

From Eq. (25.2-17), we can write

∂ln(Wmp)

∂Ni

−α+βεi (26.5-12)

so that

dSstk

∑

i

(−α+βεi)dNikBβ

∑

i

εidNikBβdqrev (26.5-13)

where we have used the fact that since the number of particles is fixed, thedNivalues

must sum to zero, and have used Eq. (26.5-10). If the statistical entropy is identi-

fied with the thermodynamic entropy, we have an alternative proof thatβ 1 /(kBT).

Ifβ 1 /(kBT) is assumed, then we have a demonstration that the statistical entropy

is the same as the thermodynamic entropy except for an additive constant.

Summary of the Chapter

The thermodynamic functions of a dilute gas can be calculated from the molecular

partition function of the gas. The necessary formulas are based on the postulates of

statistical mechanics and on the definition of the statistical entropy

SstkBln(Ω)

whereΩis the total number of system mechanical states that might be occupied, given

the information that we possess about the state of the system. Working formulas were

presented for all of the common equilibrium thermodynamic variables of a dilute gas.

A theoretical expression for the equilibrium constant for a chemical reaction in a

dilute gas mixture was obtained:

Ke−∆ε^0 /kBT

∏s

j 1

(

z′◦j

NAv

)vj

wherez′jis the molecular partition function of substance numberj, given that the

volume of the system is equal to the molar volume of a dilute gas at pressureP◦and

at the temperature of the system.

The statistical mechanical theory of rate constants known as the activated complex

theory or the transition state theory is based on the assumption that an activated complex

of high potential energy forms during the progress of a reaction, and that this activated

complex can be assumed to be at equilibrium with the reactants. Use of the expression

for the equilibrium constant in terms of partition functions and an approximate theory

for the rate of passage through the transition state gives an expression for the rate

constant. For a bimolecular reaction of a diatomic molecule CD and an atom F the rate

constant is given by

k

kBT

h

e−∆ε

‡

0 /kBT

(z‡CDF/NAvV)

(z′CD/NAvV)(z′F/NAvV)