Question

A radioactive nucleus can decay by two different processes. If the half lives of the first and second decay processes are $5\times {10}^3\text{years}$ and ${10}^5\text{years}$ respectively, then the effective half life of the nucleus is-(nearly)

• A. $476.2\text{years}$
• B. $47.62\text{years}$
• C. $4.762\text{years}$
• D. $4762\text{years}$

Answer 1

The correct option is D $4762\text{years}$

Since, the two decay processes are mutually exclusive, the effective half life of the nucleus is given by, $\lambda ={\lambda }_1+{\lambda }_2$

$\Rightarrow \frac{1}{T}={\frac{1}{T}}_1+{\frac{1}{T}}_2$

$\Rightarrow T=\frac{T_1{T}_2}{T_1+T_2}=\frac{5\times {10}^3\times {10}^5}{(5\times {10}^3+{10}^5)}$

$=\frac{5\times {10}^3\times {10}^5}{5\times {10}^3[1+20]}=\frac{100000}{21}$

$\approx 4762\text{years}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(D\right)$ is the correct answer.