A radioactive sample disintegrates via two independent

Question:

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

  1. (1) None of the above

  2. (2) $\mathrm{T}_{1 / 2}=\mathrm{T}_{1 / 2}^{(1)}+\mathrm{T}_{1 / 2}^{(2)}$

  3. (3) $\mathrm{T}_{1 / 2}=\frac{\mathrm{T}_{1 / 2}^{(1)} \mathrm{T}_{1 / 2}^{(2)}}{\mathrm{T}_{1 / 2}^{(1)}+\mathrm{T}_{1 / 2}^{(2)}}$

  4. (4) $\mathrm{T}_{1 / 2}=\frac{\mathrm{T}_{1 / 2}^{(t)}+\mathrm{T}_{1 / 2}^{(2)}}{\mathrm{T}_{1 / 2}^{(1)}-\mathrm{T}_{1 / 2}^{(2)}}$


Correct Option: , 3

Solution:

(3)

$\lambda_{\text {eq }}=\lambda_{1}+\lambda_{2}$

$\frac{1}{\mathrm{~T}_{1 / 2}}=\frac{1}{\mathrm{~T}_{1 / 2}^{(1)}}+\frac{1}{\mathrm{~T}_{1 / 2}^{(2)}}$

$\mathrm{T}_{1 / 2}=\frac{\mathrm{T}_{1 / 2}^{(1)} \mathrm{T}_{1 / 2}^{(2)}}{\mathrm{T}_{1 / 2}^{(1)}+\mathrm{T}_{1 / 2}^{(2)}}$

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