Chemistry, Third edition

252 14 · SPEED OF CHEMICAL REACTIONS

Examples of rate expressions found by

experiment

Decomposition of ethanal

Ethanal (CH 3 CHO) molecules may be broken up by heating:

CH 3 CHO(g)CH 4 (g)CO(g)

Experiments show the rate expression to be

rate of reaction k[CH 3 CHO]1.5

14.4

BOX 14.2

Dependence of the rate constant kupon temperature: the

Arrhenius equation

Experiments show that the way that the rate

constantfor a chemical reaction varies with

temperature can be described by the

Arrhenius equation:

kAeE/RT

wherekis the rate constant at temperature T

(K),Ais a constant for the reaction, Ris the

gas constant (8.3145 J mol^1 K^1 ) and E(in

J mol^1 ) is the so-called Arrhenius activation

energyfor the reaction. The Arrhenius

equation is useful because it is found to apply

even if the reaction under study does not

occur in one simple step.

Taking natural logs (ln) to both sides of the

Arrhenius equation gives an equation of the

form

ymxc

thus

lnklnA(E)^1

RT

If we have measured the rate constants for a

named reaction at various temperatures, we

can plot ln kagainst 1/T, giving a straight line

graph of slope E/R. This is the way that

Arrhenius activation energies are calculated

from experimental data.

The Arrhenius activation energy Eisnotthe

same as the reaction activation energy EA

(which is only meaningful for single-step

reactions). It is better to look upon the

Arrhenius activation energy simply as a useful

experimental index which shows how quickly

the rate constant of a reaction changes as

the temperature is changed:

●IfE0, the rate constant is independent

of temperature.

●IfEis small, the rate constant has only a

slight dependence upon temperature (Fig.

14.4(c)).

●IfEis large, the rate constant changes

dramatically with changes in temperature

(Fig. 14.4(a)).

If the rate constant of a reaction is k 1 at

temperatureT 1 , and k 2 at temperature T 2 , the

Arrhenius equations for both temperatures

may be combined to give the expression

k 1 E 1 1

ln(—)—(——)

k 2 R T 2 T 1

Provided that we know the Arrhenius

activation energy for the reaction, this

equation may be used to calculate the rate

constant at one temperature, knowing the

rate constant at another temperature. For

example, if for a particular chemical reaction,

k 1 1.0 10 ^3 mol^1 dm^3 s^1 at 300 K

and

E50 kJ mol^1 (i.e. 50 000 J mol^1 ),

then, at 320 K,

ln^10 ^3 50 000^11

(——k 2 )———8.3145 320(———— 300 )1.25

or

(

10 ^3

)e

1.250.287

k 2

so that k 2  10 ^3 /0.287 3.5

10 ^3 mol^1 dm^3 s^1. In other words, the rate

constant has increased by a factor of 3.5 for

an increase in temperature of only 20 K.