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.

The following is a first-order reaction:
$N_2O_5 \text{ (solution)} \rightarrow 2 NO_2 \text{ (solution)} + \frac{1}{2} O_2 \text{ (g)}$
In this reaction,$CCl_4$ is the solvent. The rate constant is $k = 5.0 \times 10^{-4} \ s^{-1}$. The initial concentration of $N_2O_5$ is $0.25 \ mol \ L^{-1}$.
$(i)$ What will be the initial rate of reaction?
$(ii)$ Calculate the half-life $(t_{1/2})$.
$(iii)$ How much time is required to complete $75\%$ of the reaction?
$(iv)$ Calculate the concentration of $N_2O_5$ and $NO_2$ after $30 \ min$.

The inactivation rate of a viral preparation is proportional to the amount of virus. In the first minute after preparation,$10 \%$ of the virus is inactivated. The rate constant for viral inactivation is $..... \times 10^{-3} \ min^{-1}$. (Nearest integer)
[Use : $\ln 10 = 2.303; \log_{10} 3 = 0.477; \text{property of logarithm} : \log x^y = y \log x$]

The first order reaction $A_{(g)} \rightarrow B_{(g)} + 2C_{(g)}$ occurs at $25^{\circ} C$. After $24 \ min$,the ratio of the concentration of products to the concentration of the reactant is $1:3$. What is the half-life of the reaction (in $min$)? $(\log 1.11 = 0.046)$

The rate constant for a first order reaction is $20 \, min^{-1}$. The time required for the initial concentration of the reactant to reduce to its $\frac{1}{32}$ level is $........ \times 10^{-2} \, min$. (Nearest integer) (Given: $\ln 10 = 2.303, \log 2 = 0.3010$)

If a $75\%$ first-order reaction is completed in $32 \text{ minutes}$,how many minutes will it take for the same reaction to be $50\%$ completed?