Modern Physics 3 Question 16
16. The half-life period of a radioactive element $x$ is same as the mean life time of another radioactive element $y$. Initially both of them have the same number of atoms. Then
(1999, 3M)
(a) $x$ and $y$ have the same decay rate initially
(b) $x$ and $y$ decay at the same rate always
(c) $y$ will decay at a faster rate than $x$
(d) $x$ will decay at a faster rate than $y$
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Solution:
- $\left(t _{1 / 2}\right) _x=\left(t _{\text {mean }}\right) _y$
$$ \begin{array}{rlrl} \text { or } & \frac{0.693}{\lambda _x} & =\frac{1}{\lambda _y} \\ \therefore & \lambda _x & =0.693 \lambda _y \\ \lambda _x & <\lambda _y \end{array} $$
or Rate of decay $=\lambda N$
Initially number of atoms $(N)$ of both are equal but since $\lambda _y>\lambda _x$, therefore, $y$ will decay at a faster rate than $x$.
