Question

A radioactive nucleus can decay by two different processes. The half-life for the first process is t₁ and that for the second process is t₂ then what will be the effective half-life t of the nucleus is ______.

Options

• t = t₁ + t₂

• `1/"t" = "t"_1 + "t"_2`

• t = `("t"_1"t"_2)/("t"_1 + "t"_2)`

• t = t₁ - t₂

Answer 1

A radioactive nucleus can decay by two different processes. The half-life for the first process is t₁ and that for the second process is t₂ then what will be the effective half-life t of the nucleus is `underline("t" = ("t"_1"t"_2)/("t"_1 + "t"_2))`.

Explanation:

Decay constant for the processes are

`lambda_1 = 0.693/"t"_1, lambda_2 = 0.693/"t"_2`

The probability that an active nucleus decays by the first process in small time dt is λ₁ dt.

Similarly for second decay. The probability that it decays either by first process or second process is (λ₂ dt + λ₂ dt).

If effective decay constant is λ this probability is also equal to λ dt

∴ λ dt = λ₁ dt + λ₂ dt

λ = λ₁ + λ₂

`therefore 0.693/"t" = 0.693/"t"_1 + 0.693/"t"_2`

`1/"t" = 1/"t"_1 + 1/"t"_2`