What is the value of the rate constant for a first-order reaction,if it takes $15 \ min$ for the consumption of $20 \%$ of the reactants?

Show that in a first order reaction,the time required for completion of $99.9 \%$ is $10$ times the half-life $(t_{1/2})$ of the reaction.

Calculate the rate constant of the first order reaction if $80 \%$ of the reactant reacted in $15 \ minute$.

Hydrolysis of methyl acetate in aqueous solution has been studied by titrating the liberated acetic acid against sodium hydroxide. The concentration of the ester at different times is given below:

Time $(t)$ $\text{min}$	$0$	$30$	$60$	$90$
Conc. of ester $(C)$ $\text{mol L}^{-1}$	$0.850$	$0.800$	$0.754$	$0.710$

Show that it follows a pseudo first-order reaction as the concentration of $H_2O$ remains nearly constant $(54.2 \text{ mol L}^{-1})$ during the course of the reaction. What is the value of $k'$ in this reaction?

For the first-order gaseous reaction $A \rightarrow 2B + C$ at $373 \, K$,starting with pure $A$,the total pressure of the system after $10 \, min$ is $176 \, mm \, Hg$ and after a long time,it is $270 \, mm \, Hg$. Based on this information,calculate: $(1)$ Initial pressure of $A$,$(2)$ Pressure of $A$ after $10 \, min$,and $(3)$ Rate constant.

The half-life of a first order reaction is $30 \, min$. The time required for $75 \, \%$ completion of the same reaction will be $..... \, min$