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(B) For a first order reaction, the relationship between half-life $(t_{1/2})$ and rate constant $(k)$ is given by $t_{1/2} = \frac{0.693}{k}$.For rea

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Reaction $(1)$ is slower than reaction $(2)$.

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(B) For a first order reaction, the relationship between half-life $(t_{1/2})$ and rate constant $(k)$ is given by $t_{1/2} = \frac{0.693}{k}$.For reaction $(1)$: $A \rightarrow B$, $k = 0.693 \ min^{-1}$. Therefore, $t_{1/2} = \frac{0.693}{0.693} = 1.0 \ min$.For reaction $(2)$: $A \rightarrow C$, $t_{1/2} = 0.693 \ min$. Therefore, $k = \frac{0.693}{0.693} = 1.0 \ min^{-1}$.Comparing the rate constants, for reaction $(1)$, $k = 0.693 \ min^{-1}$ and for reaction $(2)$, $k = 1.0 \ min^{-1}$.Since the rate of a first order reaction is $Rate = k[A]$, and the initial concentration of $A$ is the same for both, the reaction with the higher rate constant is faster.Since $1.0 > 0.693$, reaction $(2)$ is faster than reaction $(1)$, meaning reaction $(1)$ is slower than reaction $(2)$.

Consider the following two first order reactions occurring at $298 \ K$ with same initial concentration of $A$: $(1)$ $A \rightarrow B$; rate constant, $k=0.693 \ min^{-1}$ $(2)$ $A \rightarrow C$; half-life, $t_{1/2}=0.693 \ min$. Choose the correct option.

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