PHYSICAL CHEMISTRY IN BRIEF

CHAP. 9: CHEMICAL KINETICS [CONTENTS] 277

9.2.3.6 Kinetic equation

−

dcA

dτ

=kcAcB =⇒

dx

dτ

=k(cA0−x)(cB0−x). (9.37)

9.2.3.7 Integrated forms of the kinetic equation

For time as a function of the reactants concentrations it holds

kτ=

1

(cA0−cB0)

ln

cB0cA

cA0cB

=

1

(cA0−cB0)

ln

cB0(cA0−x)

cA0(cB0−x)

. (9.38)

For concentration as a function of time we write

cA=cA0−x , cB=cB0−x , x=cA0cB0

1 −exp[k(cA0−cB0)τ]

cB0−cA0exp[k(cA0−cB0)τ]

. (9.39)

9.2.3.8 Reaction half-life

The half-life of reaction is defined here for the component with a lower initial concentration.
WhencB0< cA0, thenτ 1 / 2 defined with respect to B is

τ 1 / 2 =

1

k(cA0−cB0)

ln

2 cA0−cB0

cA0

. (9.40)

Note: When the initial concentrations of the components are identical,cA0=cB0, then

their concentrations are identical in any moment,cA=cB. The reaction

A + B→products, (9.41)

changes to the type

2 A→products (9.42)

and equations (9.38) and (9.40) simplify to (9.32) and (9.34).
Figure9.3illustrates the time dependence of the concentration of the reactants and prod-
ucts.