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 :

Solution

Physics sir · 3 steps

Two independent decays act together, so their decay constants add. Then turn λ back into half-life.

1. Rates add. Each process removes nuclei at its own rate, λ₁ N and λ₂ N. Together, dN/dt = -(λ₁ + λ₂)N, so the effective decay constant is λ = λ₁ + λ₂.
2. Switch to half-life. For any decay, λ = (ln 2)/T_(1/2). Put this in for all three decay constants and cancel ln 2 from every term.

   1/T_(1/2) = 1/T_(1/2)⁽¹⁾ + 1/T_(1/2)⁽²⁾ ⇒ T_(1/2) = (T_(1/2)⁽¹⁾ T_(1/2)⁽²⁾)/(T_(1/2)⁽¹⁾ + T_(1/2)⁽²⁾)

3. So. The half-lives combine like resistors in parallel, which is option (c). The effective half-life is shorter than either one, as it should be when two routes drain the same sample.

The idea to keep. For parallel decay routes, add the decay constants, not the half-lives. Half-lives then combine as reciprocals, like resistors in parallel.

Answer from the NTA final answer key. Solution by PrepWiser.