Question

Two species of radioactive atoms are mixed in equal numbers. The disintegration constant of the first species is $\lambda $ and of the second is $\lambda /3$. After a long time the mixture will behave as a species with mean life of approximately:
$\begin{gathered}
  {\text{A}}{\text{. }}\dfrac{{0.70}}{\lambda } \\
  {\text{B}}{\text{. }}\dfrac{{2.10}}{\lambda } \\
  {\text{C}}{\text{. }}\dfrac{{1.00}}{\lambda } \\
  {\text{D}}{\text{. }}\dfrac{{0.52}}{\lambda } \\
\end{gathered} $

Answer 1

Hint: When we combine two species of the radioactive samples having different decay constants then the resultant decay constant is equal to the sum of the decay constants of the two species. The mean life of a radioactive sample is equal to the reciprocal of the decay constant for the radioactive sample.
Formula used:
Mean life of a radioactive sample is given in terms of the decay constant of the sample by the following expression:
$\tau = \dfrac{1}{\lambda }$

Complete answer:
We are given two different species of radioactive samples. The decay constant for the first species is given as
${\lambda _1} = \lambda $
The decay constant for the second species is given as
${\lambda _2} = \dfrac{\lambda }{3}$
Now these two different species of radioactive material are mixed together. After a long time, the mixture will behave as a species which has the decay constant given as the sum of the initial decay constants of the individual species. Therefore, the resultant decay constant is given as
$\lambda ' = {\lambda _1} + {\lambda _2} = \lambda + \dfrac{\lambda }{3} = \dfrac{{4\lambda }}{3}$
Now we need to find out the mean life of this mixture. The mean life of a radioactive sample is given as the reciprocal of the decay constant of the mixture. Therefore, the mean life of the mixture is
$\tau = \dfrac{1}{{\lambda '}} = \dfrac{3}{{4\lambda }} = \dfrac{{0.75}}{\lambda }$
This value is approximately equal to $\dfrac{{0.7}}{\lambda }$.

So, the correct answer is “Option A”.

Additional Information:
The half-life of a radioactive sample can be defined as the time taken by the sample to reduce to half of its initial concentration. It is related to the decay constant of the radioactive material by the following expression.
${T_{1/2}} = \dfrac{{0.693}}{\lambda }$
The mean life of a sample is the average of the lifetimes of the nuclei in the given radioactive sample.

Note:
The decay constant of a radioactive material is a measure of the rate at which the nuclei in the sample of the material decay. The value of decay constant depends on the half-life of the nuclei in the sample.