Physical Chemistry Third Edition

1114 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics

where the equilibrium constantK

‡

cand the standard-state Gibbs energy change∆G‡

◦

are for the formation of the activated complex excluding the motion along the reaction

coordinate.

SvanteArrhenius,1859–1927,wasa
Swedish chemist who wonthe 1905
Nobel Prize inchemistry for his theory
ofdissociationandionizationof
electrolytes insolution.

There is an empirical formula that represents the temperature dependence of rate

constants, due to Arrhenius:

kAe−Ea/RT (26.4-17)

whereEais a parameter called the activation energy. The activation energyEais

sometimes defined as

EaRT^2

dln(k)

dT

(26.4-18)

Exercise 26.13

Show that Eq. (26.4-18) is compatible with Eq. (26.4-17) ifAis constant.

Differentiation of the natural logarithm ofkas represented in the first equality of

Eq. (26.4-16) gives

EaRT^2

(

dln(T)

T

+

dln(K‡c)

dT

)

(26.4-19)

From Eq. (7.6-4)

(

∂ln(K)

∂T

)

P

−

(

∂∆G◦/RT

∂T

)

P



∆G◦

RT^2

+

1

RT

(

∂∆G◦

∂T

)

P



∆G◦

RT^2

+

1

RT

∆S◦

∆H◦

RT^2

(26.4-20)

so that

EaRT+∆H◦‡ (26.4-21)

Using this relation and the relation∆G∆H−T∆S, which holds for isothermal

processes, Eq. (26.4-16) can be rewritten

k

kBT

h

1

c◦

e∆S

‡◦/R

e−∆H

‡◦/RT



kBT

h

1

c◦

e^1 e∆S

‡◦/R

e−Ea/RT (26.4-22)

The Arrhenius preexponential factor corresponds to the transition-state preexponential

factor

Ats

kBT

h

1

c◦e

e∆S

‡◦/R

(26.4-23)

The Arrhenius preexponential factor is temperature-independent, butAtsdepends on

the temperature. However, if we setAtsequal to the Arrhenius preexponential factor

we can relate∆S‡

◦

to the Arrhenius preexponential factor.