The rate of a first-order reaction is $$0.04\,mol\,{L^{ - 1}}{s^{ - 1}}$$   at $$10$$ $$sec$$  and $$0.03\,mol\,{L^{ - 1}}{s^{ - 1}}$$   at $$20$$ $$sec$$  after initiation of the reaction. The half-life period of the reaction is

Solution :
Given, order of reaction = 1
Rate of reaction at $$10\,s = 0.04\,mol\,{L^{ - 1}}{s^{ - 1}}$$
Rate of reaction at $$20\,s = 0.03\,mol\,{L^{ - 1}}{s^{ - 1}}$$
∴ Half-life period $$\left( {{t_{\frac{1}{2}}}} \right) = ?$$
We have the equation for rate-constant $$'k'$$  in first order reaction.
$$\eqalign{ & k = \frac{{2.303}}{t}{\text{log}}\frac{{{A_t}}}{{{A_0}}} = \frac{{2.303}}{{105}}{\text{log}}\frac{{0.04}}{{0.03}} \cr & \,\,\,\,\, = \frac{{2.303}}{{105}} \times 0.124 \cr & k = 0.028\,{s^{ - 1}} \cr & {\text{We know that,}} \cr & {t_{\frac{1}{2}}} = \frac{{0.693}}{k} = \frac{{0.693}}{{0.028773391\,{s^{ - 1}}}} \cr & \,\,\,\,\,\,\, = 24.14\,s \approx 24.1\,s \cr} $$