The time required for the concentration of a reactant to diminish to one half of the original is known as the half life (t1/2). The mathematical relationships between the half life and the rate constant for both first and zero order reactions are shown below as well as a derivation of these.
If the reaction is zero order in A, each successive t1/2 is smaller (because [A]0 is smaller as the reaction proceeds).
If the reaction is first order in A, successive t1/2 are equal
First half life: 100%-50%
Second half life: 50%-25%....and so on
Show/hide derivation of half life equations
Half life equations from integrated rate equations
| Zero order reaction: Integrated rate equation: [A] = –kt + [A]0 At t = t1/2, [A] = [A]0/2
|
 | First order reaction: Integrated rate equation: ln[A] = –kt + ln[A]0 At t = t1/2, [A] = [A]0/2
|
Experiments determining the dependence of half life on concentration can be used to determine reaction order.
If the reaction is zero order in A, each successive t1/2 is smaller (because [A]0 is smaller as the reaction proceeds).
If the reaction is first order in A, successive t1/2 are equal
First half life: 100%-50%
Second half life: 50%-25%....and so on
Show/hide derivation of half life equations
Half life equations from integrated rate equations
| Zero Order reaction: [A] = –kt + [A]0 At t = t1/2, [A] = [A]0/2 Substitute:
Rearrange:
Therefore:
|
First Order reaction: ln[A] = -kt + ln[A]0 At t = t1/2, [A] = [A]0/2. Substitute:
Rearrange:
Simplify:
Therefore:
|