Question

A radioactive nucleus can decay by two differnt modes. The half life for the first mode is T₁ and that for the second mode is T₂ . The effective half life T of the nucleus would be

(1) T = T₁ + T₂

(2) \(\frac{1}{T}=\frac{1}{T_1}+\frac{1}{T_2}\)

(3) \(ln(\frac{1}{T})=ln(\frac{1}{T_1})+ln(\frac{1}{T_2})\)

(4) \(\frac{1}{T}=\frac{1}{(T_1+T_2)}\)

Answer 1

 (2) \(\frac{1}{T}=\frac{1}{T_1}+\frac{1}{T_2}\)

Denoting by λ₁ and λ₂ the decay constants of the two modes by which nucleus can decay we have

The probability that an active nucleus decay by first mode in a small time dt is λ₁ dt and that for the second mode is λ₂ dt. Then the probability that it decays either by first process or second process is (λ₁dt + λ₂ dt). Now we can say that λ is the effective decay constant. Hence the probability that the nucleus decay by effective value of λ in small time dt is λdt. 

Hence we must have