Question

A radioactive sample contains ${10}^{-3}kg$ each of two nuclear species $A$ and $B$ with half-life $4$ days and $8$ days, respectively. The ratio of the amounts of $A$ and $B$ after period of $16$ days is

• A. $1:2$
• B. $4:1$
• C. $1:4$
• D. $2:1$
• E. $1:1$

Answer 1

Answer: (C)

$1:4$

Given, half-life of species
$A=4days$
Half-life of species $B=8days$
For species $A,N=N_0{\left(\frac{1}{2}\right)}^n;n=\frac{16}{4}=4$
Now, $N_1=N_0{\left(\frac{1}{2}\right)}^4...\left(i\right)$
Similarly, for species $B$
$N_2=N_0{\left(\frac{1}{2}\right)}^2....\left(ii\right)$
On dividing Eq. (i) from Eq. (ii), we get
$\frac{N_1}{N_2}=\frac{(2{)}^2}{(2{)}^4}\Rightarrow \frac{N_1}{N_2}=\frac{1}{4}$
$\Rightarrow N_1:N_2=1:4$.