Question

Consider the two different first-order reactions given below: $\text{A + B} \rightarrow \text{C (Reaction 1)} \\\\\text{P} \rightarrow \text{Q (Reaction 2)}$The ratio of the half-life of Reaction 1 : Reaction 2 is $5 : 2$. If $t_1$ and $t_2$ represent the time taken to complete $\frac{2}{3}^\text{rd}$ and $\frac{4}{5}^\text{th}$ of Reaction 1 and Reaction 2, respectively, then the value of the ratio $t_1 : t_2$ is ____ $\times 10^{-1}$ (nearest integer).[Given: $\log_{10}(3) = 0.477$ and $\log_{10}(5) = 0.699$]

Answer 1

For first order reactions:

$K_1 t_1 = \ln\left(\frac{1}{1 - \frac{2}{3}}\right) = \ln 3$

$K_2 t_2 = \ln\left(\frac{1}{1 - \frac{4}{5}}\right) = \ln 5$

$\therefore K_1 t_1 = 5 \quad \text{and} \quad K_2 t_2 = 2$

$\frac{K_1}{K_2} = \frac{\ln 3}{\ln 5}$

$\frac{t_1}{t_2} = \frac{0.477}{0.699} \times 5 = 1.7 \times 10^{-1}$