# Derive the Relation Between Half Life and Rate Constant for a First Order Reaction - Chemistry

Derive the relation between half life and rate constant for a first order reaction

#### Solution

The integrated rate law for the first order reaction is given by the equation

k= 2.303/t log_10"[A]_o/[A]_t

Where [A]0 = initial concentration of the reactant at t = 0

The concentration falls to [A]t at time t from the start of the reaction.

The concentration of the reactant falls to [A]0/2 at time t1/2.
Therefore, t = t1/2
[A]t = [A]0/2
So, the equation can be written as

k= 2.303/t_(1/2) log_10"[A]_o /([A]_o/2)

=2.303/(t_1/2)log_10 2

=2.303/(t_1/2)*0.301=0.693/(t_1/2)

t_(1/2)=0.693/k

Concept: Integrated Rate Equations - Half-life of a Reaction
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