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

To find the effective half-life \( T_{1/2} \) of a radioactive sample that disintegrates via two independent decay processes, we need to use the concept of effective half-life in parallel decay modes.

Given:

• First decay process with half-life: \( T_{1/2}^{(1)} \)
• Second decay process with half-life: \( T_{1/2}^{(2)} \)

The concept of multiple non-competing processes with parallel or simultaneous decay is given by:

\(\frac{1}{T_{1/2}} = \frac{1}{T_{1/2}^{(1)}} + \frac{1}{T_{1/2}^{(2)}}\)

Taking the reciprocal gives us the formula for the effective half-life:

\(T_{1/2} = \frac{T_{1/2}^{(1)} \cdot T_{1/2}^{(2)}}{T_{1/2}^{(1)} + T_{1/2}^{(2)}}\)

This means the effective half-life is derived from the reciprocal of the sum of the reciprocals of the individual half-lives.

Given the options, the correct answer is:

• T_{1/2} = \frac{T_{1/2}^{(1)} \cdot T_{1/2}^{(2)}}{T_{1/2}^{(1)} + T_{1/2}^{(2)}}

Thus, the effective half-life is calculated using the above formula.