Calculating Half life using the integrated rate equation

The time required for the concentration of a reactant to diminish to one half of the original is known as the half life (t_1/2).

The mathematical relationships between the half life and the rate constant for both first and second order reactions is given below. A detailed derivation of these is at the bottom of the page.
 

First Order Reaction	Second Order Reaction
Integrated rate equation: ln[A] = -kt + ln[A]0	Integrated rate equation: 1  =   kt   +  1 [A] [A]0
At t = t1/2, [A] = [A]0/2  t1/2  =   ln 2   k	At t = t1/2, [A] = [A]0/2 t1/2 = 1 k[A]0

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Measurement of the dependence of half life on concentration can be used to determine reaction order.
 

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Derivation of half life equations from first and second order integrated rate equations:

First order reaction		Second Order Reaction
Integrated rate equation: ln[A] = -kt + ln[A]0		Integrated rate equation: 1 = kt + 1 [A] [A]0
At t = t1/2, [A] = [A]0/2. Substitute ln [A]0 = -kt1/2 + ln[A]0   2		At t = t1/2, [A] = [A]0/2. Substitute 2 =  kt1/2  + 1 [A]0 [A]0
Rearrange ln [A]0 - ln[A]0 = -kt1/2 2		Rearrange -kt1/2 = 1 - 2 [A]0 [A]0
Simplify ln [A]0/2  = ln 1  = -ln 2 [A]0 2		Simplify -kt1/2 = -1 [A]0
-ln 2 = -kt1/2
t1/2  =   ln 2 k		t1/2   =  1 k[A]0