Question

The half-life period of a radioactive element X is the same as the mean life of another radioactive element Y. Initially, both of them have the same numbers of atoms then
A. X and Y have the same decay rate initially
B. X and Y decays at the same rate always
C. Y will decay at a faster rate than X
D. X will decay at a faster rate than Y

Answer 1

Use the formulae for half-life period of a radioactive element and mean life of a radioactive element. These formulae give the relation between the half-life period of a radioactive element, mean life of a radioactive element and decay rates of the radioactive elements. The ratio of these decay rates gives information about which element decays faster.

Formula used:
The half-life period ${t_{1/2}}$ of a radioactive element is given by
${t_{1/2}} = \dfrac{{0.693}}{\lambda }$ …… (1)
Here, $\lambda$ is the decay constant for the radioactive decay.
The mean life period $T$ of a radioactive element is given by
$T = \dfrac{1}{\lambda }$ …… (2)
Here, $\lambda$ is the decay constant for the radioactive decay.

Complete step by step answer:
We have given that the half-life period of a radioactive element X is the same as the mean life of another radioactive element Y.
Also, the initial population of both the radioactive elements X and Y is the same.
Let ${t_{1/2}}$ be the half-life period of the radioactive element X and $T$ be the mean life period of the radioactive element Y.
Rewrite equation (1) for the half-life period of the radioactive element X.
${t_{1/2}} = \dfrac{{0.693}}{{{\lambda _X}}}$
Here, ${\lambda _X}$ is the decay rate constant for the radioactive element X.
Rewrite equation (2) for the mean life period of the radioactive element Y.
$T = \dfrac{1}{{{\lambda _Y}}}$
Here, ${\lambda _Y}$ is the decay rate constant for the radioactive element Y.
From the given information,
${t_{1/2}} = T$
Substitute $\dfrac{{0.693}}{{{\lambda _X}}}$ for ${t_{1/2}}$ and $\dfrac{1}{{{\lambda _Y}}}$ for $T$
in the above equation.
$\dfrac{{0.693}}{{{\lambda _X}}} = \dfrac{1}{{{\lambda _Y}}}$
$\Rightarrow \dfrac{{{\lambda _X}}}{{{\lambda _Y}}} = 0.693$
From the above equation, we can conclude that the decay rate of the element X is less than the decay rate of element Y.
$\Rightarrow {\lambda _Y} > {\lambda _X}$

Therefore, the element Y will decay faster than the element X.

So, the correct answer is “Option C”.

Note:
The same question can be solved in another way. One can determine the relation for the half-life period of the radioactive element Y in terms of the half-life period of the radioactive element X. This relation shows that the half-life of element X is more than the half-life of element Y. Hence, once can prove element X decays slower than element Y.