Question

A radioactive nucleus decays by two different processes. The half life for the first process is $10\text{s}$ and that for the second is $100\text{s}$. The effective half life of the nucleus is close to

• A. $6\text{s}$
• B. $9\text{s}$
• C. $12\text{s}$
• D. $15\text{s}$

Answer 1

The correct option is B $9\text{s}$

Given:

$T_1=10\text{s}$ ; $T_2=100\text{s}$

${\lambda }_1=\frac{\ln 2}{T_1};{\lambda }_2=\frac{\ln 2}{T_2};{\lambda }_{eq}=\frac{\ln 2}{T_{eq}}$

We know that, In case of decay occurring by different processes,

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

$\Rightarrow \frac{\ln 2}{T_{eq}}=\frac{\ln 2}{T_1}+\frac{\ln 2}{T_2}$

$\begin{array}{c}{T}_{eq}=\frac{T_1{T}_2}{T_1+T_2}\end{array}$

$\Rightarrow T_{eq}=\frac{10\times 100}{10+100}=\frac{1000}{110}=\frac{100}{11}$

$\begin{array}{c}{T}_{eq}=9\text{s}\end{array}$

Hence, option $\left(B\right)$ is correct.

Why this question? To know the different processes of radioactivity occuring simultaneously of the same nucleus.