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The half-life period of a radioactive element x is the same as the mean life time of another radioactive element y. Initially both of them have the same number of atoms. Then which radioactive element will decay faster?

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Hint: Use the formula of half life and mean life to find the relation between the decay constants.

 

Complete step-by-step answer:

Rate of decay = $\dfrac{dN}{dt} = -\lambda N$, N is the original number of atoms and $\lambda$ is the decay constant.

The negative sign indicates the decay/reduction in atoms with time.

Half life = $t_{\frac{1}{2}} = \dfrac{ln2}{\lambda} = \dfrac{0.693}{\lambda}$

Mean life = $\tau = \dfrac{1}{\lambda}$

It is given that the half life period of x is the same as the mean life of y.

$({t_{\frac{1}{2}}})_x = {\tau}_y$

$\Rightarrow \dfrac{0.693}{{\lambda}_x} = \dfrac{1}{{\lambda}_y}$

$\Rightarrow {\lambda}_x = 0.693 \times {\lambda}_y$

$\therefore {\lambda}_y > {\lambda}_x$

Initial number of atoms i.e. N is the same for both radioactive materials x and y.

Higher the decay constant, faster the decay.

$\therefore$ Radioactive element y will decay faster.

 

Note: To solve such problems, one must be thorough with his concepts on chemical kinetics and should either know the required formula or how to derive them. There is another way to solve the above problem, which is, we can find the half life of element y in terms of half life of element x. We will get that the half life of x is more, so naturally we can say that x will decay slower than y or y will decay faster. 

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