Question

A first order reaction takes \[69.3\]minutes for $50\% $ completion. How much time will be needed for $80\% $ completion?

Answer 1

Hint: Half-life, in radioactivity, the interval of time required for one half of the atomic nuclei of a radioactive sample to decay (change spontaneously into other nuclear species by emitting particles and energy), or, equivalently, the time interval required for the number of disintegrations per second of a radioactive material to decrease by one half.

Complete answer:
The half-life of a reaction is the time required for a reactant to reach one half its initial concentration or pressure. For a first-order reaction, the half-life is independent of concentration and constant over time.
The half-life of a reaction is the time required for the reactant concentration to decrease to one half its initial value. The half-life of a first-order reaction is a constant that is related to the rate constant for the reaction:${t_{\dfrac{1}{2}}} = \dfrac{{0.693}}{K}$. Radioactive decay reactions are first-order reactions.
$K = \dfrac{{\ln 2}}{{69.3}}{\min ^{ - 1}}$
${t_{\dfrac{1}{2}}} = 69.3\min = \dfrac{{\ln }}{k}$
For $80\% $ conversion, if we assume initial concentration to be ${a_ \circ }$,concentration left would be $\dfrac{{{a_ \circ }}}{5}$
$t \times \dfrac{{\ln 2}}{{69.3}} = \ln (\dfrac{{{a_ \circ }}}{{\dfrac{{{a_ \circ }}}{5}}})$
$t = \dfrac{{69.3\ln 5}}{{\ln 2}} = 161{\min ^{ - 1}}$

Note:
Half-lives are characteristic properties of the various unstable atomic nuclei and the particular way in which they decay. Alpha and beta decay are generally slower processes than gamma decay. Half-lives for beta decay range upward from one-hundredth of a second and, for alpha decay, upward from about one one-millionth of a second. Half-lives for gamma decay may be too short to measure (around ${10^{ - 14}}$second), though a wide range of half-lives for gamma emission has been reported.