For the reaction $R \rightarrow P$,the half-life is independent of the initial concentration of the reactant,$R$. Which one of the following graphs is not correct for this reaction?

For a first-order reaction,the ratio of the time required for $99.9\%$ completion to the time required for $50\%$ completion is:

Calculate the half-life period of a first-order reaction if the rate constant of the reaction is $0.02 \ min^{-1}$. (in $min$)

Calculate the half-life of the first-order reaction $C_2H_4O_{(g)} \to CH_{4(g)} + CO_{(g)}$. If the initial pressure of $C_2H_4O_{(g)}$ is $80 \ mm$ and the total pressure at the end of $20 \ minutes$ is $120 \ mm$,find the half-life in $\text{min}$.

State whether the following sentences are true $(T)$ or false $(F)$.
$(a)$ The reaction of an ester with water produces an alcohol and an acid.
$(b)$ The concentration of water remains constant during the hydrolysis of an ester.
$(c)$ In the hydrolysis of an ester,$[H_2O]$ remains constant.

Calculate the half-life of a first-order reaction in minutes if the rate constant is $1 \times 10^{-3} \ sec^{-1}$.