Question

Time for half change for a first order reaction is $25{\text{ min}}$. What will be the time required for $99\%$ completion of reaction.

Answer 1

To solve this question we must know the equation for the rate constant of first order reaction. A reaction in which the rate of the reaction is directly proportional to the concentration of one of the reactant species is known as the first order reaction. First calculate the rate constant using the equation of half-life of first order reaction. Then calculate the rate constant at $99\%$ completion.

Complete solution:
We know that the expression for half life of a first order reaction is as follows:
${t_{{\text{1/2}}}} = \dfrac{{0.693}}{k}$
Where ${t_{{\text{1/2}}}}$ is the half-life time for the reaction,
$k$ is the rate constant for the reaction.
We are given that the time for half change for a first order reaction is $25{\text{ min}}$. Thus,
$25{\text{ min}} = \dfrac{{0.693}}{k}$
$k = \dfrac{{0.693}}{{25{\text{ min}}}}$
$k = 0.02772{\text{ mi}}{{\text{n}}^{ - 1}}$
Thus, the rate constant for the reaction is $0.02772{\text{ mi}}{{\text{n}}^{ - 1}}$.
We know the equation for the rate constant of a first order reaction is,
$k = \dfrac{{2.303}}{t}\log \dfrac{{{{\left[ a \right]}^0}}}{{\left[ a \right]}}$
Where $k$ is the rate constant of a first order reaction,
$t$ is time,
${\left[ a \right]^0}$ is the initial concentration of the reactant,
$\left[ a \right]$ is the final concentration of the reactant.
Rearrange the equation for the time as follows:
$t = \dfrac{{2.303}}{k}\log \dfrac{{{{\left[ a \right]}^0}}}{{\left[ a \right]}}$
The expression when reaction is $90\%$ complete is as follows:
Let the initial concentration of the reactant be 100. When the reaction is $99\%$ complete the final concentration will be $100 - 99 = 1$. Thus,
${t_{99\% }} = \dfrac{{2.303}}{{0.02772{\text{ mi}}{{\text{n}}^{ - 1}}}}\log \dfrac{{\left( {100} \right)}}{{\left( 1 \right)}}$
${t_{99\% }} = \dfrac{{2.303}}{{0.02772{\text{ mi}}{{\text{n}}^{ - 1}}}} \times 2$
${t_{99\% }} = 166.16{\text{ min}}$

Thus, the time required for $99\%$ completion of reaction is $166.16{\text{ min}}$.

Note: The unit of rate constant for first order reaction is ${\text{mi}}{{\text{n}}^{ - 1}}$. The units do not contain concentration terms. Thus, we can say that the rate constant of a first order reaction is independent of the concentration of the reactant.