Question

The decay constants of a radioactive substance for $\alpha$ and $\beta$ emisson are ${\lambda }_{\alpha }$ and ${\lambda }_{\beta }$ respectively. If the substance emits $\alpha$ and $\beta$ simultaneously, then the average half life of the material will be:

• A. $\frac{1}{2}({\lambda }_{\alpha }+{\lambda }_{\beta })$
• B. ${\lambda }_{\alpha }+{\lambda }_{\beta }$
• C. $\frac{2{\lambda }_{\alpha }{\lambda }_{\beta }}{{\lambda }_{\alpha }+{\lambda }_{\beta }}$
• D. $\frac{{\lambda }_{\alpha }{\lambda }_{\beta }}{{\lambda }_{\alpha }+{\lambda }_{\beta }}$

Answer 1

The correct option is D $\frac{{\lambda }_{\alpha }{\lambda }_{\beta }}{{\lambda }_{\alpha }+{\lambda }_{\beta }}$

$\begin{array}{c}\text{Given,}\\ \text{Substance decay constant}\end{array}$

$\text{when,}\alpha \text{decay.}\lambda \alpha$

$\text{when,}\beta \text{decay}\lambda \beta$

$\begin{array}{c}\text{If. Substance emit.}\alpha \text{and}\beta \text{simantaneously}\\ \alpha ,\beta \text{decay}{\lambda }_{\text{net}}={\lambda }_{\alpha }+{\lambda }_{\beta }\end{array}$

$\begin{array}{c}\text{we know}\\ T_{1/2}\left(\text{half life}\right)=\frac{\ln 2}{\lambda }\end{array}$

$\begin{array}{c}\text{For}\alpha \text{decay.}\\ T_{1/2}=\frac{\ln 2}{{\lambda }_{\alpha }}\end{array}$

$\begin{array}{c}\text{For.}\beta \text{decay}\\ T_{\frac{1}{2}}=\frac{\ln 2}{{\lambda }_{\beta }}\end{array}$

$\begin{array}{c}\text{For}\alpha ,\beta \text{both}\\ T_{1/2}=\frac{\ln 2}{{\lambda }_{\alpha }+{\lambda }_{\beta }}\\ \text{putting value of}{\lambda }_{\alpha }\text{and}{\lambda }_{\beta }\end{array}$

$T_{1/2}\left(\text{averge}\right)=\frac{T_{\alpha }{T}_B}{T_{\alpha }+T_B}$