At a given instant, say t = 0, two radioactive substances A and B have equal activities. The ratio \frac{R_{B}}{R_{A}} of their activities after time t itself decays with time t as e–3t. If the half-life of A is ln2, the half-life of B is

Engineering

Physics

Radioactivity New

Question

At a given instant, say t = 0, two radioactive substances A and B have equal activities. The ratio $\frac{R_{B}}{R_{A}}$ of their activities after time t itself decays with time t as e^–3t. If the half-life of A is ln2, the half-life of B is

$\frac{ℓn 2}{4}$

4ln2

$\frac{ℓn 2}{4}$

2ln2

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R = R_Oe^–kt

^{$∴ \frac{R_{B}}{R_{A}} = \frac{R_{o} e^{- λ_{B} t}}{R_{o} e^{- λ_{B} t}} = e^{- (λ_{B} - λ_{A}) t} = e^{- 3 t}$}

⇒ λ_B – λ_A = 3

$\Rightarrow \frac{{ℓn}^{2}}{T_{B}} - \frac{ℓn 2}{ℓn 2} = 3 .$

$\Rightarrow T_{B} = \frac{ℓn 2}{4}$