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**Hint:**To solve this question, you need to consider the decay of reactivity as a first-order reaction or process and thus use equations for first order reaction to find the answer to the asked question. The half-life of a first-order reaction is: ${t_{1/2}} = \dfrac{{{{\log }_e}2}}{\lambda }$

**Complete step by step answer:**

As explained in the hint section of the solution to the asked question, we need to consider the decay of reactivity as a first-order reaction or process and use its equation to first, find the value of $\lambda $ , which is the decay constant of the decaying quantity.

The equation of a first-order reaction is given as:

$N = {N_o}{e^{ - \lambda t}}$

Where, ${N_o}$ is the initial decay rate

$N$ is the decay rate at time $t$

And, $\lambda $ is the decay constant

The question has already told us that the initial decay rate is ${N_o}$

The decay rate after time $t = 5$ min is given as ${N_o}/e$

Substituting in the values, we get:

$\Rightarrow {N_o}/e = {N_o}{e^{ - 5\lambda }}$

After solving, we get:

$

\Rightarrow - 5\lambda = - 1 \\

\Rightarrow \lambda = \dfrac{1}{5} \\

$

Now, we have found the value of the decay constant as: $\lambda = \dfrac{1}{5}$

We can find the half-life for the reaction, which is basically what the question is asking since half-life is the time taken at which the activity of the quantity gets reduced to half.

We already know that for a first-order half-life can be found out using the equation:

${t_{1/2}} = \dfrac{{{{\log }_e}2}}{\lambda }$

Substituting in the value of the decay constant in the equation, we get:

$\Rightarrow {t_{1/2}} = \dfrac{{{{\log }_e}2}}{{\dfrac{1}{5}}}$

$\Rightarrow {t_{1/2}} = 5{\log _e}2$

**Hence, We can see that the option (D) is the correct option as the value matches what we found out by solving the question.**

**Note:**Many students do not take such reactions as first-order and stay confused about what and which formulae to use to solve the question. Another way of solving the question would have been to put $N = \dfrac{{{N_o}}}{2}$ and find the value of $t$ using the value of $\lambda = \dfrac{1}{5}$ .

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