Question

A radioactive sample disintegrates via two independent decay processes having half lives T₁/₂\^{(1)} and T₁/₂\^{(2)} respectively. The effective half-life, T₁/₂ of the nuclei is :

• A. T₁/₂ = T₁/₂^{(1)} + T₁/₂^{(2)}
• B. T₁/₂ = T₁/₂^{(1)} - T₁/₂^{(2)}
• C. T₁/₂ = T₁/₂^{(1)} T₁/₂^{(2)} / (T₁/₂^{(1)} + T₁/₂^{(2)})
• D. None of the above

Hint:

Effective half-life for parallel decay follows the same "product over sum" rule as resistors in parallel.

Answer 1

The Correct Option is C

Step 1: For independent decay processes, the effective decay constant $\lambda$ is the sum of individual decay constants: $\lambda_{eff} = \lambda_1 + \lambda_2$.
Step 2: Since $\lambda = \frac{\ln 2}{T_{1/2}}$, we have $\frac{\ln 2}{T_{1/2}} = \frac{\ln 2}{T_{1/2}^{(1)}} + \frac{\ln 2}{T_{1/2}^{(2)}}$.
Step 3: $\frac{1}{T_{1/2}} = \frac{1}{T_{1/2}^{(1)}} + \frac{1}{T_{1/2}^{(2)}} \implies T_{1/2} = \frac{T_{1/2}^{(1)} T_{1/2}^{(2)}}{T_{1/2}^{(1)} + T_{1/2}^{(2)}}$.