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

Hint: 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.