Physical Chemistry Third Edition

1116 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics

calculated with the London–Eyring–Polanyi–Sato

(LEPS) method. For the reaction

H+HF→H+F 2

Their value of∆ε‡ 0 is 9. 0 × 10 −^21 J, from a barrier height

of 9. 3 × 10 −^21 J. They give the following values for the

activated complex at 298 K:z′rot‡ 111 .25,

z′vib‡ (1.00)(1.199)^2 (for the symmetric stretch and the

two degenerate bends). Find the value of the rate constant

at 298 K. Compare your value with that from the

experimental Arrhenius activation energy,9kJmol−^1 ,

and the Arrhenius preexponential factor,

6. 3 × 107 m^3 mol−^1 s−^1. What is an explanation for the

difference between the theoretical and experimental

values?

26.33Calculate the rate constant for the reaction of Example

26.12 at 550.0 K. Determine the Arrhenius activation

energy and compare it with the value of∆ε‡ 0.

26.34Calculate the forward rate constant at 300 K and at

1000 K for the reaction

H+H 2 →H 2 +H

using the activated complex theory. Assume a linear

activated complex with all vibrational partition functions

equal to unity. The intermolecular distance of hydrogen is

0. 741 × 10 −^10 m. For the activated complex take each

bond as 0. 942 × 10 −^10 m. Take the value of∆ε‡ 0 as

6. 8 × 10 −^20 J. What is the value of the equilibrium

constant for this reaction? What is the value of the rate

constant for the reverse reaction?

26.5 Miscellaneous Topics in Statistical

Thermodynamics

We close this chapter with three topics that help in interpreting the statistical mechanics

of dilute gases.

A Different Expression for the Probability Distribution

Now that we have the expression for the chemical potential in terms of the molecular

partition function, we can write Eq. (26.1-27) in the form

N

z

eμ/kBT (26.5-1)

This equation gives an expression for the Lagrange multiplier,α, in Eq. (25.2-23):

αμ/kBT (26.5-2)

The equation for the molecular probability distribution for a dilute gas can now be

written in the form

pj

Nj

N

gjeμ/kBTe−εj/kBTgje(μ−εj)/kBT (26.5-3)

The parametersαandβhave the same meaning in all cases. If the dilute occupa-

tion approximation cannot be used, Eqs. (25.2-26) and (25.2-29) can be rewritten for

noninteracting fermions and bosons:

Nj

gje(μ−εj)/kBT

1 +e(μ−εj)/kBT

(fermions) (26.5-4)

Nj

gje(μ−εj)/kBT

1 −e(μ−εj)/kBT

(bosons) (26.5-5)