Question

Half-life of a radioactive substance is 1 hour. The time taken for the disintegration of 87.5% substance is
A. 2 hrs
B. 3 hrs
C. 3.5 hrs
D. 4 hrs

Answer 1

Use the formula for the decay constant. Using this formula determines the half-life of the radioactive substance. Then use the formula for the population of the radioactive substance at any time t and determine the time required for the disintegration of the substance.

Formula used:
The decay constant $\lambda$ is given by
$\lambda = \dfrac{{0.693}}{T}$ …… (1)
Here, $T$ is the half-life of a radioactive element.
The equation for the population of the radioactive element present at any time is
$N = {N_0}{e^{ - \lambda t}}$ …… (2)
Here, ${N_0}$ is the initial population of the radioactive element, $N$ is the radioactive element at time $t$ and $\lambda$ is the decay constant of the decay.

Complete step by step solution:
We have given that the half-life of the radioactive substance is one hour.
$T = 1\,{\text{h}}$
We can determine the decay constant using equation (1).
Substitute for in equation (1).
$\lambda = \dfrac{{0.693}}{{1\,{\text{h}}}}$
$\Rightarrow \lambda = 0.693\,{{\text{h}}^{{\text{ - 1}}}}$
Hence, the decay constant is $0.693\,{{\text{h}}^{{\text{ - 1}}}}$.
The radioactive substance disintegrated 87.5%. Hence, the radioactive substance available at time $t$ is 12.5%.
Hence, the fraction of the radioactive substance as that of the initial population of the radioactive substance at time $t$ is 0.125.
$\dfrac{N}{{{N_0}}} = 0.125$
Rearrange equation (2) for $\dfrac{N}{{{N_0}}}$.
$\dfrac{N}{{{N_0}}} = {e^{ - \lambda t}}$
Substitute $0.125$ for $\dfrac{N}{{{N_0}}}$ and $0.693\,{{\text{h}}^{{\text{ - 1}}}}$ for $\lambda$ in the above equation.
$0.125 = {e^{ - \left( {0.693\,{{\text{h}}^{{\text{ - 1}}}}} \right)t}}$
Take natural log on both sides of the above equation.
${\log _e}\left( {0.125} \right) = {\log _e}{e^{ - \left( {0.693\,{{\text{h}}^{{\text{ - 1}}}}} \right)t}}$
$\Rightarrow {\log _e}\left( {0.125} \right) = - \left( {0.693\,{{\text{h}}^{{\text{ - 1}}}}} \right)t$
$\Rightarrow - 2.0794 = - \left( {0.693\,{{\text{h}}^{{\text{ - 1}}}}} \right)t$
$\Rightarrow t = \dfrac{{2.0794}}{{0.693\,{{\text{h}}^{{\text{ - 1}}}}}}$
$\Rightarrow t = 3\,{\text{h}}$
Therefore, the time required for the disintegration is $3\,{\text{h}}$.

So, the correct answer is “Option B”.

Note:
One can also solve the same question by another method. One can directly use the formula for the population of the radioactive substance at any time t in terms of the half-life period of the radioactive substance instead of determining the decay constant of the decay of the radioactive substance.