Physical Chemistry Third Edition

1106 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics

vibrational partition functions≈1. Do your equilibrium

constants depend on temperature? Why or why not?

a. H 2 +D 2 
2HD

b. HF+D
DF+H

26.26Without doing any detailed calculations, estimate the
equilibrium constant for each of the gas-phase reactions.
State any assumptions. Do your equilibrium constants
depend on temperature? Why or why not?
a. HF+D 2
DF+HD
b. CH 4 +D 2
CH 3 D+HD

26.27Show that the equilibrium constant for a racemization
reaction equals unity.

26.28The ionization energy of the hydrogen atom is equal to
13.60 eV. Find the equilibrium constant for the
dissociation of a hydrogen atom into a proton and an
electron at 10000 K.

26.29Consider a hypotheticalcis–transisomerization in the gas
phase:
C
T
Assume that the only important differences between the
two isomers are that one of the moments of inertia of the
transisomer is 8.3% higher than that of thecisisomer

and that one of the vibrational degrees of freedom has a

higher frequency in thecisisomer:

ν(trans) 6. 45 × 1013 s−^1

ν(cis) 7. 54 × 1013 s−^1

Assume that the ground-state energy of thecisisomer is

8.15× 10 −^21 J higher than that of thetransisomer. Find

the value of the equilibrium constant for this reaction at

500.0 K.

26.30Consider a substance that can exist in two tautomeric

forms, A and B. Define a partition function for the

combined forms

zzA+zB

where the same zero of energy (say the ground-state

energy of form A) must be used for both forms. Let the

energy of the ground state of form B be denoted byε0,B.

a.In terms ofzAandzB, what fraction of the molecules

will be form B? Form A?

b.In terms ofzAandzBwhat is the equilibrium constant

for the reaction

A
B

How does this relate to the expression for the equilibrium

constant derived as in the chapter?

26.4 The Activated Complex Theory of

Bimolecular Chemical Reaction Rates in

Dilute Gases

We write the chemical equation for a hypothetical reaction as

aA+bB→dD+fF (26.4-1)

where the capital letters stand for chemical formulas and the lower-case letters represent

stoichiometric coefficients. We define the forward rate of the reaction, denoted byrf:

rf−

1

a

d[A]

dt

−

1

b

d[B]

dt

−

1

d

d[D]

dt



1

f

d[F]

dt

(definition ofrf) (26.4-2)

There is a large class of chemical reactions in which the forward reaction rate is

proportional to the concentration of each reactant raised to some power:

rf

1

a

d[A]

dt

kf[A]α[B]β (26.4-3)

Equation (26.4-3) is called arate law with definite orders. The proportionality con-

stantkfis called theforward rate constant. It is not a true constant. It depends on