Half life formula

What is the half life Period?

The time taken for half of reactions to complete or the time at which the concentration of the reactant is reduced to half of its original value is called the half life period of the reaction.

Half -life is generally represented by the symbol t_1/2. The term is commonly used in nuclear physics to describe the radioactive decay or how long stable atoms survive. The term half-life period was discovered by Ernest Rutherford to study the age determination of rocks. The value of the half-life depends on the order of the reaction.

Table of Contents

• Half life period formula for Zero order reaction
• Half life period formula for First order reaction
• Half life period formula for Second order reaction
• Half life period formula for nth order reaction
• Problem practice
• Frequently Asked Questions

Half life formula for Zero order reaction

A zero order reaction implies that the rate of the reaction does not depend on the concentration of the reactant. For a general reaction;

A  → Product 

The rate law of zero order kinetics is:  [A]₀ = [A] + kt

Where [A]₀ = Initial concentration of reactant at time(t)  = 0

            [A] = concentration of reactant at time (t) = t 
When  time(t) = t_1/2 , the concentration of the reaction([A]) = [A]₀/2
From the  rate law of zero order kinetics is:  [A]₀ = [A]₀/2  + k  t_1/2

         t_1/2 = {[A]₀–  [A]₀/2 }/ k

\(\begin{array}{l}t_{1/2} =\frac{ [A]_{0}}{2k}\end{array} \)

Half life formula for First order reaction

A zero order reaction implies that the rate of the reaction does not depend on the concentration of the reactant. For a general reaction;

A  → Product 

The rate law of zero order kinetics is:  ln [A]₀ = ln [A] + k t

Where [A]₀ = Initial concentration of reactant at time(t)  = 0

            [A] = concentration of reactant at time (t) = t 

When  time(t) = t_1/2 , the concentration of the reaction([A]) = [A]₀/2

From the  rate law of zero order kinetics is:  ln [A]₀ =  ln{ [A]₀/2}  + k t_1/2

 ln [A]₀ = ln[A]₀ – ln 2 + k t_1/2

t_1/2 = ln 2/ k

\(\begin{array}{l}t_{1/2} = \frac{0.693}{k}\end{array} \)

Half life formula for Second order reaction

A zero order reaction implies that the rate of the reaction does not depend on the concentration of the reactant. For a general reaction;

2A  → Product 

The rate law of zero order kinetics is:  1/[A] =1/ [A]₀+ k t

Where [A]₀ = Initial concentration of reactant at time(t)  = 0

            [A] = concentration of reactant at time (t) = t 

When  time(t) = t_1/2 , the concentration of the reaction([A]) = [A]₀/2

From the  rate law of zero order kinetics is:  1/{ [A]₀/2} =1/ [A]₀+ k t

\(\begin{array}{l}t_{1/2} =\frac{1}{k [A]_{0}}\end{array} \)

Half life formula for nth order reaction

A zero order reaction implies that the rate of the reaction does not depend on the concentration of the reactant. For a general reaction;

nA  → Product 

The rate law of zero order kinetics is:  1/[A]^n-1 =1/ [A]₀ ^n-1 + (n-1) kt

Where [A]₀ = Initial concentration of reactant at time(t)  = 0

            [A] = concentration of reactant at time (t) = t 

When  time(t) = t_1/2 , the concentration of the reaction([A]) = [A]₀/2

Half life formula of nth order reaction is

\(\begin{array}{l} t_{1/2} = \frac{2^{(n-1)} -1}{[A]_{0}^{(n-1)} (n-1) k }\end{array} \)

Problem practice

If the half life period for a first order reaction in A is 2 minutes , How long will it take [A]  to reach 25% of its initial concentration? 

Solution:  For 1st order reaction  k = 0.693/t_1/2  = 0.693/ 2min = 0.3465 min-1

      [A]₀ = 100 %   ;  [A] = 25% ,  

 t= (2.303/k ) x log { [A]₀/[A] }  =  (2.303/0.3465) x log (100/25)= 4 min