Question

If a radioactive nucleus decays by two different processes, the half-life for first process is t₁ and for second process it is t₂. Then, the effective half life t of the nucleus is given by

(a) t = t₁ + t₂

(b) √t = √t₁ + √t₂

(c) t^-1 = t^-1₁ + t^-1₂

(d) None of these

Answer 1

Answer is (c) t^-1 = t^-1₁ + t^-1₂

Decay constant for first process, λ₁ = \(\frac{0.693}{t_1}\)

and decay constant for second process,  λ₂ = \(\frac{0.693}{t_2}\) 

The probability of an active nucleus decays by first process in small time is λ₁ dt, similarly for second process it is λ₂ dt. 

Total probability it decays by first process or by second process is (λ₁dt +λ₂dt)

λdt = λ₁ dt + λ₂ dt

λ = λ₂ + λ₂