Question

A radioactive sample $S_1$ having an activity of $5\mu Ci$ has twice the number of nuclei as another sample $S_2$ which has an activity of $10\mu Ci$. The half lives of $S_1$ and $S_2$ can be

• A. $5$ years each
• B. $10$ years each
• C. $20$ years and $10$ years, respectively
• D. $20$ years and $5$ years, respectively

Answer 1

The correct option is D $20$ years and $5$ years, respectively

Given: $R_1=5\mu Ci,R_2=10\mu Ci,N_1=2N_2$


By, $\lambda N=R\frac{0.693}{T}N=RT=\frac{0.693}{R}N$

For sample $S_1$​:
$T_1=\frac{0.693}{5}{N}_1$​​

$T_1=\frac{0.693}{5}2N_2$​​ .....................(i),

For sample $S_2$​:

$T_2=\frac{0.693}{10}{N}_2$​​ ....................eq2,

By dividing eq1 by eq2, we get:
$\frac{T_1}{T_2}=\frac{4}{1}$

It is clear that the ratio of half lives of the two samples is $4:1$, therefore half lives can be $20$ years and $5$ years respectively.