PHYSICAL CHEMISTRY IN BRIEF

CHAP. 9: CHEMICAL KINETICS [CONTENTS] 316

increases, it is highly probable that it is a reaction mediated by living organisms decaying at
higher temperatures. Fermentation is a typical example of such a reaction.

9.6.2 Arrhenius equation

For the dependence of the rate constant on temperature, Arrhenius suggested the relation

k=Ae−E

∗/(RT)

, (9.159)

whereAandE∗are constants independent of temperature. The constantAis called the pre-
exponential factor, constantE∗is the activation energy. The constantAis always positive, the
activation energy is positive in simple reactions (the rate constant increases with temperature).
In radioactive decays, E∗ = 0 because in this case the rate constant does not depend on
temperature [see9.6.1].
If we know the values of the rate constant at two temperatures, we can determineAand
E∗from the equations

E∗=R

T 2 T 1

T 2 −T 1

ln

k(T 2 )

k(T 1 )

, A=k(T 1 )eE

∗/(RT 1 )

. (9.160)

Note:If we know the values of the rate constant at more temperatures, we determineA

andE∗using the least squares method. This procedure is more reliable than the use of

equations (9.160).

In reversible reactions the difference between the activation energy of the direct reaction

→

E∗

and that of the reverse reaction

←

E∗equals the internal energy of reaction

∆rU=

→

E∗−

←

E∗. (9.161)

Example

For the decomposition of acetone

CH 3 COCH 3 →CH 2 =CO+CH 4

we know the values of the pre-exponential factorA= 1.5× 1015 s−^1 and the activation energy

E∗= 286.6 kJ mol−^1. Calculate the rate constant of this reaction at temperatureT= 850 K.