Question

In how many months, ${\left( {\dfrac{3}{4}} \right)^{th}}$ of the substance will decay, in half-life of the radioactive substance is $2$ months?
A. $4$ months
B. $6$ months
C. $8$ months
D. $14$ months

Answer 1

This problem is based on radioactive decay. Radioactive decay is also known as nuclear decay because disintegration of nucleus will take place in radioactive decay by the process by which an unstable nucleus loses energy by radiation. The material containing the unstable nuclei is said to be radioactive material. In this problem we need to find how much time will radioactive substance will take to decay ${\left( {\dfrac{3}{4}} \right)^{th}}$ the given substance by considering half life as $2$ months.

Formulas used:
$$ $N = {N_0}{e^{ - \lambda t}}$
$\lambda = \dfrac{{\ln \left( 2 \right)}}{{t\dfrac{{_1}}{2}}}$

Complete step by step answer:
We know that,
$N = {N_0}{e^{ - \lambda t}}$ ……….. $\left( 1 \right)$
Where, $N =$ Number of atoms at time $\left( t \right)$, ${N_0} =$ Initial number of elements present and $\lambda =$ Decay constant.
And $\lambda = \dfrac{{\ln \left( 2 \right)}}{{t\dfrac{{_1}}{2}}}$
${t_{\dfrac{1}{2}}} =$ Half life.
Given,
$N = {N_0} - \dfrac{3}{4}{N_0}$
$\Rightarrow N = \dfrac{1}{4}{N_0}$ ………… $\left( 2 \right)$
$\Rightarrow {t_{\dfrac{1}{2}}} = 2$ months.
Substituting in equation $\left( 1 \right)$ we get
$\dfrac{1}{4}{N_0} = {N_0}{e^{ - \lambda t}}$

On simplifying the above equation can be written as
$\dfrac{1}{4} = {e^{ - \lambda t}}$
Taking $\ln$ on both sides,
$\ln \left( {\dfrac{1}{4}} \right) = \ln \left( {{e^{ - \lambda t}}} \right)$
$\Rightarrow \ln \left( 1 \right) - \ln \left( 4 \right) = - \lambda t\ln \left( e \right)$
On simplifying the above equation we get,
$- \ln {\left( 2 \right)^2} = - \lambda t\ln \left( e \right)$
$\Rightarrow - 2\ln \left( 2 \right) = \dfrac{{\ln \left( 2 \right)}}{{{t_{\dfrac{1}{2}}}}}t\left( 1 \right)$
Therefore , $t = 2{t_{\dfrac{1}{2}}}$ ……….. $\left( 3 \right)$
Substituting, given data in equation $\left( 3 \right)$ we get
$t = 2 \times 2$ months
Therefore, $t = 4$ months.
That is, in $4$ months, ${\left( {\dfrac{3}{4}} \right)^{th}}$ of substance will decay.

Hence, option A is correct.

Note: Half life is defined as the time that is required for the radioactive material to reduce to half of its initial value. Half life describes how much time a radioactive substance undergoes radioactive decay which is generally differentiated as two types : exponential or non-exponential decay. There are three types of decay namely alpha decay, beta decay and gamma decay. All decay will involve emitting one or more photons or particles.