A radioactive sample consists of two distinct species having equal number of atoms initially. The mean lifetime of one species is and that of the other is . The decay products in both cases are stable. A plot is made of the total number of radioactive nu

Question

A radioactive sample consists of two distinct species having equal number of atoms initially. The mean lifetime of one species is $τ$ and that of the other is $5 τ$ . The decay products in both cases are stable. A plot is made of the total number of radioactive nuclei as a function of time. Which of the following figures best represent the form of this plot? [2001]

Smart Achievers · Accepted Answer

(4)

$N_{1} = N_{0} e^{- λ_{1} t} = N_{0} e^{- t / τ}$ $\dots (i)$

Mean life time $τ = \frac{1}{λ_{1}}$

Similarly, $N_{2} = N_{0} e^{- λ_{2} t} = N_{0} e^{- t / 5 τ}$ $\dots (i i)$

as $5 τ = \frac{1}{λ_{2}}$

Adding equations (i) and (ii), we get

$N = N_{1} + N_{2} = N_{0} (e^{- t / τ} + e^{- t / 5 τ})$

The total number of radioactive nuclei $(N)$ as a function of time only decreases exponentially. Hence graph (4) correctly depicts this behavior.

Solution

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