Question

A radioactive nuclide X decays into nuclei T and Z by simultaneous disintegration as shown. Effective decay constant for the disintegration is

• A. ${\lambda }_1+{\lambda }_2$
• B. $\frac{{\lambda }_1{\lambda }_2}{{\lambda }_1+{\lambda }_2}$
• C. $\frac{{\lambda }_1+{\lambda }_2}{2}$
• D. $\frac{2{\lambda }_1{\lambda }_2}{{\lambda }_1+{\lambda }_2}$

Answer 1

The correct option is A ${\lambda }_1+{\lambda }_2$

$\begin{array}{c}\frac{dy}{dt}={\lambda }_{1}x---\text{for}x\to y\\ \frac{dz}{dt}={\lambda }_{2}x---\text{for}x\to z\\ \text{Now adding}\\ ({\lambda }_1+{\lambda }_2)x=\frac{dy}{dt}+\frac{dz}{dt}⟶\left(i\right)\\ \frac{dy}{dt}+\frac{dz}{dt}=-\frac{dx}{dt}⟶\text{(iii)}\end{array}$

$\begin{array}{c}\text{Rate of decrease of}x\\ \text{Would be equal to}\\ \text{net increase rale}\\ \text{of}y\text{and}z\text{.}\\ -\frac{dx}{dt}=({\lambda }_1+{\lambda }_2)x\\ \Rightarrow {\lambda }_{\text{effective}}={\lambda }_1+{\lambda }_2\end{array}$