"Consider the following reaction, the rate expression of which is given below $$\begin{aligned} & \mathrm{A}+\mathrm{B} \rightarrow \mathrm{C} \\ & \text { rate }=\mathrm{k}[\mathrm{A}]^{1 / 2}[\mathrm{~B}]^{1 / 2} \end{aligned}$$ The reaction is initiated by taking $1 \mathrm{~M}$ concentration of $\mathrm{A}$ and $\mathrm{B}$ each. If the rate constant $(\mathrm{k})$ is $4.6 \times 10^{-2} \mathrm{~s}^{-1}$, then the time taken for $\mathrm{A}$ to become $0.1 \mathrm{~M}$ is _________ sec. (nearest integer)"

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$$\begin{aligned} & A+B \rightarrow C \\ & \frac{-d[A]}{d t}=k[A]^{1 / 2}[B]^{1 / 2} \end{aligned}$$

Since, $[A]=[B]$

$$\begin{aligned} & \Rightarrow \quad \frac{-d[A]}{d t}=k[A] \\ & \Rightarrow \quad k t=\ln \frac{[A]_0}{[A]} \\ & \Rightarrow \quad t=\frac{1}{4.6 \times 10^{-2}} \times \ln \left(\frac{1}{0.1}\right) \\ & =\frac{2.303}{4.6} \times 100 \approx 50 \end{aligned}$$

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