Explain the following: Factors affecting rate of reaction.

Establish the following relation for a first-order reaction:

$\mathrm{k}=\frac{2.303}{\mathrm{t}}\log \frac{\mathrm{a}}{(\mathrm{a}-\mathrm{x})}$

Where the terms have their usual significance. Hence, show that the half-life $\left({\mathrm{t}}_{1/2}\right)$ of a first-order reaction is independent of initial concentration.

(a)Write down the rate equation for a first order reaction.

(b)Deduce the mathematical expression for half-life period for such a reaction.

(c) If the rate constant of a first order reaction at a certain temperature is $1.5\times {10}^{-1}{\mathrm{s}}^{-1}$ and ${\mathrm{t}}_1$ and ${\mathrm{t}}_2$ are the respective times for $50\%$ and $75\%$ completion of the reaction, determine the ratio of ${\mathrm{t}}_2$ to ${\mathrm{t}}_1.$