Two radioactive materials X₁ and X₂ have decay constants 6 and 3A respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of X₁ to that of X₂ will be $\frac{1}{e}$ after a time

$\frac{1}{6 λ}$

$\frac{1}{3 λ}$

$\frac{3}{6 λ}$

$\frac{6}{9 λ}$

Solution

$\frac{1}{3 λ}$

The ratio of number of nuclei of X₁ to that of X₂ after time t,

∴ $\frac{N_{1}}{N_{2}} = \frac{1}{e}$

∴ $\frac{N_{0} e^{- 6 λt}}{N_{0} e^{- 3 λt}} = \frac{1}{e}$ ( $∵$ N = N₀ e^-λt )

$\frac{1}{e^{3 λt}} = \frac{1}{e}$

⇒ 3λt = 1

∴ t = $\frac{1}{3 λ}$

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