258 14 · SPEED OF CHEMICAL REACTIONS
wherekis the first-order rate constant. Taking natural logs on both sides of this
equation, followed by rearrangement, givesln(
[A] 0
[A])kt (14.1)
tIf the time at which exactly half of the concentration of A has disappeared is
symbolised as t (^1) ⁄ 2 , then
[A]t (^1) ⁄ 2 [A] 0 /2 (14.2)
Substituting equation (14.2) into equation (14.1) gives
[A] 0
ln(
[A] 0 /2)
ln 2 kt (^1) ⁄ 2
and, since ln 2 0.693,
kt (^1) ⁄ 2 0.693
Note that:
●These equations apply to any first-order reaction of the type AB.
●The formal definition for the half-life(orreactant half-life) of a reaction, symbol-
isedt (^1) ⁄ 2 , is that it is the time taken for the concentration of a reactant to fall by half.
●The half-life of a first-order reaction does not depend upon the initial concentra-
tion of the reactant.
First-order reaction
The rearrangement ofN-bromoacetanilide to 4-bromoacetanilide
in chlorobenzene solvent is first order withk6.5 10 ^6 s^1 at 288 K. If the initial
concentration of N-bromoacetanilide was 0.010 mol dm^3 , calculate the concentration of N-
bromoacetanilide after 10 h.
Br COCH 3
N
H COCH 3
Br
N
Exercise 14L
Example of half-life – the decomposition of azomethane
The half-life for the decomposition of azomethane,
CH 3 N 2 CH 3 (g)CH 3 CH 3 (g) N 2 (g)
where
rate of reaction k[CH 3 N 2 CH 3 (g)]
is 2000 s at 180oC. This means that if the initial concentration of azomethane was
0.1 mol dm^3 , after 2000 s the concentration would have fallen to 0.05 mol dm^3 ,
after another 2000 s to 0.025 mol dm^3 , after a further 2000 s to 0.0125 mol dm^3 ,
and so on.