JEE Main 2025 Jan 23 Shift 1, Physics Q6: Radioactivity

JEE Main2025Jan 23, Shift 1Physics

A radioactive nucleus $n_{2}$ has 3 times the decay constant as compared to the decay constant of another radioactive nucleus $n_{1}$ . If initial number of both nuclei are the same, what is the ratio of number of nuclei of $n_{2}$ to the number of nuclei of $n_{1}$ , after one half-life of $n_{1}$ ?

A
$1/4$
B
$1/8$
C
$4$
D
$8$

Solution

Let $λ_{1} = λ$ and $λ_{2} = 3 λ$ . Both species start with $N_{0}$ nuclei. After time $t$ :

$N_{1} (t) = N_{0} e^{- λ t}, N_{2} (t) = N_{0} e^{- 3 λ t}$ .

So $\frac{N _{2}}{N _{1}} = e^{- 2 λ t}$ .

At one half-life of $n_{1}$ , $t = \frac{ln 2}{λ}$ , giving:

$\frac{N _{2}}{N _{1}} = e^{- 2 l n 2} = 2^{- 2} = \frac{1}{4}$ .

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JEE Main 2025 Jan 23 Shift 1, Physics Q6: Radioactivity

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