Question

Derive the relationship between half life and rate constant for first order reaction.

Answer 1

The integrated rate law for the first order reaction is
$t=\frac{2.303}{k}log\frac{a}{a-x}$......(1)
Here, t is time, k is the rate constant, a is the initial concentration and $a-x$ is the concentration at time t.
After one half life period, $t=t_{1/2}=$ half life period
and $a-x=\frac{a}{2}$
Substitute these in equation (1)
$t_{1/2}=\frac{2.303}{k}log\frac{a}{\frac{a}{2}}$
$t_{1/2}=\frac{2.303}{k}log2$
$t_{1/2}=\frac{0.693}{k}$
This is the relationship between half life and rate constant for first order reaction.