Question

A radioactive nuclide can decay simultaneously by two different processes which have decay constants ${\lambda }_1$ and ${\lambda }_2$. The effective decay constant of the nuclide is $\lambda$

• A. $\lambda ={\lambda }_1+{\lambda }_2$
• B. $\lambda =\frac{1}{2}({\lambda }_1+{\lambda }_1)$
• C. $\frac{1}{\lambda }=\frac{1}{{\lambda }_1}+\frac{1}{{\lambda }_2}$
• D. $\lambda =\sqrt{{\lambda }_1{\lambda }_2}$

Answer 1

Answer: (A)

$\lambda ={\lambda }_1+{\lambda }_2$

$\begin{array}{c}\text{Given.}\\ \text{a radioactive nuclie. Let say}A\text{decayed}\\ \text{simantaneously by two process,}\end{array}$

$\begin{array}{c}-\frac{\delta A}{\delta t}\text{decrease in A}={\lambda }_{1}A\\ -\frac{\delta A}{\delta t}\text{decrease in A}={\lambda }_{2}A\end{array}$

$\begin{array}{c}\text{So, Net decrease in}A\\ \begin{array}{c}\frac{dA}{dt}=({\lambda }_1+{\lambda }_2)A\\ \text{so,}{\lambda }_{\text{net}}={\lambda }_1+{\lambda }_2\end{array}\end{array}$