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 t1/2. 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 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) = t1/2 , the concentration of the reaction([A]) = [A]0/2
From the  rate law of zero order kinetics is:  [A]0 = [A]0/2  + k  t1/2

         t1/2 = {[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) = t1/2 , 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 t1/2

 ln [A]0 = ln[A]0 – ln 2 + k t1/2

t1/2 = 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) = t1/2 , 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) = t1/2 , 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/t1/2  = 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

Frequently Asked Questions -FAQs

Q1

1. What is the half-life period of first order reaction?

The half-life of a reaction is the time required for the reactant concentration to decrease to one-half its initial value. The half-life of a first-order reaction does not depend upon the concentration of the reactant. It is a constant and related to the rate constant for the reaction: t1/2 = 0.693/k.

Q2

2. What is meant by half-life?

The time taken for half of the 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 period in radioactivity is defined as the interval of time required for one-half of the atomic nuclei of a radioactive sample to decay.

Q3

3. What is half-life used for?

The half-life of an isotope of Carbon-14 to determine the approximate age of organic objects. Half- life period is also used to describe the rate at which the isotope will decay and give off radiation. Using the half-life, it is possible to predict the amount of radioactive material that will remain after a given amount of time. Radioactive isotope cobalt-60 is used for radiotherapy.

Q4

4. What happens to the half-life period of a first order reaction if the initial concentration of the reactants is doubled?

If the initial concentration of the reactants is doubled, in the first order reaction, half life period of the reaction remains constant. Because the half-life of a first-order reaction does not depend upon the initial concentration of the reactant.

Q5

5. Write the half life formula for a first-order reaction.

Half life formula for a first order reaction is
t1/2 = 0.693/K

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