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Q.

Two nuclei P, Q have equal no. of atoms at t = 0. Their half-life are 3 hours, 9 hours. Compare their rates of disintegration after 18 hrs from the start.

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a

3 : 1

b

16 : 3

c

3 : 16

d

1 : 3

answer is A.

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Detailed Solution

Form $T = \frac{0 .693}{λ} \Rightarrow \frac{λ_{1}}{λ_{2}} = \frac{T_{2}}{T_{1}} = \frac{9}{3} = \frac{3}{1}$
$\begin{array}{l} N = \frac{N_{0}}{2^{t / T}} \Rightarrow \frac{N_{1}}{N_{2}} = \frac{2^{t / T_{2}}}{2^{t / T_{1}}} = \frac{2^{18 / 9}}{2^{18 / 3}} = \frac{2^{2}}{2^{6}} \\ \frac{N_{1}}{N_{2}} = \frac{4}{64} = \frac{1}{16}; ∴ \frac{dN}{dt} = λN; A = λN \\ \Rightarrow \frac{A_{1}}{A_{2}} = \frac{N_{1}}{N_{2}} \times \frac{λ_{1}}{λ_{2}} = \frac{1}{16} \times \frac{3}{1} = \frac{3}{16} \end{array}$

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