The rate of a first-order reaction is $1.5 \times 10^{-2} \, mol \, L^{-1} \, min^{-1}$ when the concentration of the reactant is $0.5 \, M$. Find the half-life $(t_{1/2})$ of the reaction in $min$.

Find the rate constant of a first-order reaction in $sec^{-1}$ having a half-life of $2.5 \ hours$.

Drug $X$ becomes ineffective after $50 \%$ decomposition. The original concentration of drug in a bottle was $16 \ mg/mL$ which becomes $4 \ mg/mL$ in $12 \ months$. The expiry time of the drug in months is $..........$ Assume that the decomposition of the drug follows first order kinetics.

For a first-order gas-phase reaction $A(g) \to B(g) + C(g)$, let $p_i$ be the initial pressure of gas $A$ and $p_t$ be the total pressure of the reaction mixture at time $t$. The expression for the rate constant $(k)$ is:

If the rate constant of a reaction is $x \, sec^{-1}$,by what factor does the rate increase if the initial concentration of $A$ is tripled?

Given below are two statements:
Statement $I$: The graph of $t_{1/2}$ versus initial concentration $[R]_0$ for a first-order reaction is a horizontal line.
Statement $II$: The graph of $\log \frac{[R]_0}{[R]}$ versus time $t$ for a first-order reaction is a straight line passing through the origin with a slope equal to $\frac{k}{2.303}$.
In the light of the above statements,choose the correct answer from the options given below: