Question

A radioactive sample can decay by two different processes. The half-life for the first process is $T_1$ and that for the second process is $T_2$. The effective half-life $T$ of the radioactive sample is,

• A. $T=\frac{T_1-T_2}{T_1{T}_2}$
  
• B. $T=\frac{T_1+T_2}{T_1{T}_2}$
  
• C. $T=\frac{T_1{T}_2}{T_1+T_2}$
  
• D. $T=T_1+T_2$
  

Answer 1

The correct option is C $T=\frac{T_1{T}_2}{T_1+T_2}$


We know that, $\lambda =\frac{0.693}{T}$
$\Rightarrow \lambda \propto \frac{1}{T}$

$\Rightarrow {\lambda }_1=\frac{\ln 2}{T_1}$, ${\lambda }_2=\frac{\ln 2}{T_2}$, And $\lambda =\frac{\ln 2}{T}$

Now, $\frac{dN}{dt}=-\lambda N$

Total rate of decay is,

$\frac{dN}{dt}=-({\lambda }_1+{\lambda }_2)N=-{\lambda }_{eff}N$

$\therefore {\lambda }_{eff}={\lambda }_1+{\lambda }_2$

$\frac{\ln 2}{T}=\frac{\ln 2}{T_1}+\frac{\ln 2}{T_2}$

$\frac{1}{T}=\frac{1}{T_1}+\frac{1}{T_2}$

$T=\frac{T_1{T}_2}{T_1+T_2}$

Hence, $\left(C\right)$ is the correct answer.