Half Life (t1/2)



Half Life of a reaction is time required to reduce the initial concentration to half. The half-life of a reaction depends on the order of reaction. The variation of half-life with order is given as 

t1/2  ∝ [A0]1-n

Where, [A0] = initial concentration

    n = Order of reaction

 for, Zeroth order reaction

t1/2  ∝ [A0]

for first order reaction,

t1/2   is independent of initial concentration

for second order reaction,

 

t1/2  ∝ [1/A0]

Integrated rate law expression 

For Zeroth order reaction

                        A            →        product

Initially            a                      o

at t = t1           (a -x)                x

 

The rate of reaction at time ‘t’

half life

Integrating on both sides, we get,

half life

when t = 0, x = 0

            c = 0

x = kt

k = x/t

When t = t1/2, x = a/2

k = a/2t1/2

t1/2 = a/2k

 

For first Order Reaction

A       →        product

Initially            a                      o

At, t = t           (a -x)                x1

The rate of reaction at time ‘t’

 half life

Integrating on both sides, we get,

 half life

 when t = 0, x = 0

-ln a = c

-ln (a-x)  = kt – ln a

half life

 

When t = t1/2,

x = a/2

i.e. (a-x) = a/2

half life

So, half-life of first order reaction is independent of initial concentration.

 

 

Numerical

The half-life of a first order reaction is 50 mins. Calculate the time required to complete 75% of the reaction.

Given,

Half Life (t1/2) = 50 mins

Then,

Rate constant (k)

half life

 

= 0.01386 min-1

Again,

initial concentration (a) = 100 (let) then

at time t,

half life

Concentration left (a-x) = 100-75 = 25

We have,

half life

 

Calculate the half period of first order reaction when rate constant is 5 year-1.

We have,

For 1st order reaction

t1/2 = 0.693/5

= 0.1386 year