Question

A radioactive sample disintegrates via two independent decay processes having half-lives$T_{1/2}^1$ and $T_{1/2}^2$ respectively. The effective half-life, $T_{1/2}$ of the nuclei is

• A. None of the above
• B. $T_{1/2}=T_{1/2}^1+T_{1/2}^2$
• C. $T_{1/2}=\frac{T_{1/2}^1+T_{1/2}^2}{T_{1/2}^1-T_{1/2}^2}$
• D. $T_{1/2}=\frac{T_{1/2}^1{T}_{1/2}^2}{T_{1/2}^1+T_{1/2}^2}$

Answer 1

The correct option is D

$T_{1/2}=\frac{T_{1/2}^1{T}_{1/2}^2}{T_{1/2}^1+T_{1/2}^2}$

Step 1: Given Data

Half-lives of first process$=T_{1/2}^1$

Half-lives of second process $=T_{1/2}^2$

Effective half-life $=T_{1/2}$

Step 2: Find effective half-life, $T_{1/2}$ of the nuclei

Let decay constant for first process $={\lambda }_1$

Let decay constant for second process $={\lambda }_2$

Let size of a population of radioactive atoms at a given time $t$ $=N$

Let the amount by which the population decreases in time $dt$ $=dN$

the rate of change $\frac{dN}{dt}$

The effective half-life, $T_{1/2}^1$ of the nuclei is

${\left(\frac{dN}{dt}\right)}_1=N{\lambda }_1$

The effective half-life, $T_{1/2}^2$ of the nuclei is

${\left(\frac{dN}{dt}\right)}_2=N{\lambda }_2$

The effective half-life, $T_{1/2}$ of the nuclei is

$$\begin{gathered} \left(\frac{dN}{dt}\right)={\left(\frac{dN}{dt}\right)}_1+{\left(\frac{dN}{dt}\right)}_2 \\ N{\lambda }_{eff}=N{\lambda }_1+N{\lambda }_2 \\ \frac{1}{T_{1/2}}=\frac{1}{T_{1/2}^1}+\frac{1}{T_{1/2}^2} \\ {T}_{1/2}=\frac{T_{1/2}^1{T}_{1/2}^2}{T_{1/2}^1+T_{1/2}^2} \end{gathered}$$

Hence, option D is correct.