In a first-order reaction,the concentration of the reactant decreases from $1.0 \, M$ to $0.25 \, M$ in $20 \, \text{minutes}$. What will be the rate constant of the reaction?

Calculate the rate constant for the first order reaction,$A \rightarrow B$ if the rate of reaction is $5.4 \times 10^{-6} \ mol \ dm^{-3} \ s^{-1}$ and $[A]=0.3 \ M$.

$N_{2}O_{5(g)} \rightarrow 2NO_{2(g)} + \frac{1}{2}O_{2(g)}$
In the above first order reaction,the initial concentration of $N_{2}O_{5}$ is $2.40 \times 10^{-2} \ mol \ L^{-1}$ at $318 \ K$. The concentration of $N_{2}O_{5}$ after $1 \ hour$ was $1.60 \times 10^{-2} \ mol \ L^{-1}$. The rate constant of the reaction at $318 \ K$ is $..... \times 10^{-3} \ min^{-1}$. (Nearest integer)
[Given: $\log 3 = 0.477, \log 5 = 0.699$]

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:

The integrated rate equation for a first-order reaction,$A \rightarrow \text{product}$,is

It takes $1 \, h$ for a first order reaction to go to $50 \, \%$ completion. The total time required for the same reaction to reach $87.5 \, \%$ completion will be $..... \, h$.