A radioactive sample disintegrates via two independent decay processes having half lives $\mathrm{T}_{1 / 2}^{(1)}$ and $\mathrm{T}_{1 / 2}^{(2)}$ respectively. The effective half-life, $\mathrm{T}_{1 / 2}$ of the nuclei is :

(D) $T_{1 / 2}=\frac{T_{1 / 2}^{(1)} T_{1 / 2}^{(2)}}{T_{1 / 2}^{(1)}+T_{1 / 2}^{(2)}}$

(B) $\mathrm{T}_{1 / 2}=\mathrm{T}_{1 / 2}^{(1)}+\mathrm{T}_{1 / 2}^{(2)}$

(C) $\mathrm{T}_{1 / 2}=\frac{\mathrm{T}_{1 / 2}^{(1)}+\mathrm{T}_{1 / 2}^{(2)}}{\mathrm{T}_{1 / 2}^{(1)}-\mathrm{T}_{1 / 2}^{(2)}}$

JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Modern Physics

A radioactive sample disintegrates via two independent decay processes having half lives

T_{1/2}^{(1)}

and

T_{1/2}^{(2)}

respectively. The effective half-life,

T_{1/2}

of the nuclei is :

(A)None of the above
(B) $T_{1/2} = T_{1/2}^{(1)} + T_{1/2}^{(2)}$
(C) $T_{1/2} = \frac{T _{1/2}^{(1)} + T _{1/2}^{(2)}}{T _{1/2}^{(1)} - T _{1/2}^{(2)}}$
(D) $T_{1/2} = \frac{T _{1/2}^{(1)} T _{1/2}^{(2)}}{T _{1/2}^{(1)} + T _{1/2}^{(2)}}$ ✓ Correct

Answer: (D)

Two independent decay routes act in parallel, so the probability of decay per unit time is the sum of the two,

λ = λ_{1} + λ_{2}

. Writing each decay constant as

λ = \frac{ln 2}{T _{1/2}}

and cancelling

ln 2

\frac{1}{T _{1/2}} = \frac{1}{T _{1/2}^{(1)}} + \frac{1}{T _{1/2}^{(2)}}

so the effective half-life is

T_{1/2} = \frac{T _{1/2}^{(1)} T _{1/2}^{(2)}}{T _{1/2}^{(1)} + T _{1/2}^{(2)}}

JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Single correct option