Question

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

Answer 1

Derivation:-

According to integrated law of rate:-

$k=\frac{2.303}{t}lo{g}_{10}\frac{[A{]}_o}{[A{]}_t}$

where $[A{]}_o$ is at $t=0$

the concentration of reactant falls to $\left[A_o\right]/2$ at $t_{1/2}$

$\therefore t=t_{1/2}$

$[A{]}_t=A_o/2$

$\therefore$ Equation for first order can be written as:-

$k=\frac{2.303}{t_{1/2}}lo{g}_{10}\frac{[A{]}_o}{\frac{[A{]}_o}{2}}$

$k=\frac{2.303}{t_{1/2}}lo{g}_{10}2=\frac{0.693}{t_{1/2}}$

$t_{1/2}=\frac{0.693}{k}$.