Question

A radioactive nucleus can decay in two different processes with half life $0.7\, hr$ and $0.3 \,hr$. The effective average life of the nucleus in minutes approximately is (value of $\ln 2=0.7)$

• A. 14
• B. 18
• C. 24
• D. 26

Answer 1

Answer: (B)

If a radioactive sample has more than one radioactive material then the effective half life.
$\frac{1}{T_{\text {eff } T / 2}}=\frac{1}{T_{1 \frac{1}{2}}}+\frac{1}{T_{2 \frac{1}{2}}}+\frac{1}{T_{3 \frac{1}{3}}}+\ldots$
Where, $T_{n \frac{1}{2}}=$ half life of $n^{\text {th }}$ material.
Here, effective half life of the net material,
$T_{\text {eff } \frac{1}{2}}=\frac{0.7 \times 0.3}{0.7+0.3}=0.21 \,hr$
$\left(\because T_{1 \frac{1}{2}}=0.7 hr , T _{2 \frac{1}{2}}=0.3 hr \right)$
As we know, $T_{1 / 2}=0.693 \tau_{\text {aveg. mean life }}$
$\Rightarrow \, \tau_{\text {aveg. mean life }} =\frac{T_{\text {eff } \frac{1}{2}}}{0.693} $ $=\frac{0.21}{0.693}=0.3 hr $
$\Rightarrow \, \tau_{\text {aveg. mean life }} =0.3 \times 60=18\, min $