 # 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’ Integrating on both sides, we get, 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’

Integrating on both sides, we get,

when t = 0, x = 0
-ln a = c
-ln (a-x)  = kt – ln a When t = t1/2,
x = a/2
i.e. (a-x) = a/2 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) = 0.01386 min-1
Again,
initial concentration (a) = 100 (let) then
at time t, Concentration left (a-x) = 100-75 = 25
We have, 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