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

Answer 1

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

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

By, $\lambda N=R$

$\frac{0.693}{T}N=R$

$T=\frac{0.693N}{R}$

For sample $S_1$:

$T_1=\frac{0.693N_1}{5}$

$T_1=\frac{0.693\times 2N_2}{5}$ ......................eq1,

For sample $S_2$:

$T_2=\frac{0.693N_2}{10}$ ....................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 $20$ ($4k$) years and $5$ ($1k$) years respectively.