Question

Two nuclei $P,Q$ have equal no. of atoms at $t=0$ . Their half-lifes are $3$ hours , $9$ hours. Compare their rates of disintegration after $18$ hrs from the start.

• A. $3:16$
• B. $16:3$
• C. $1:3$
• D. $3:1$

Answer 1

The correct option is A $3:16$

The number of half-lives of P in $18hours$ is given as,

$n_1=\frac{t}{T_1}$

$=\frac{18}{3}$

$=6$

The number of nuclei P left after $6$ half-lives is given as,

$N_1=N{\left(\frac{1}{2}\right)}^6$

The number of half-lives of Q in $18hours$ is given as,

$n_2=\frac{t}{T_2}$

$=\frac{18}{9}$

$=2$

The number of nuclei Q left after $2$ half-lives is given as,

$N_2=N{\left(\frac{1}{2}\right)}^2$

The ratio of their decay rate is given as,

$\frac{R_1}{R_2}=\frac{T_2{N}_1}{T_1{N}_2}$

$\frac{R_1}{R_2}=\frac{9\times N{\left(\frac{1}{2}\right)}^6}{3\times N{\left(\frac{1}{2}\right)}^2}$

$\frac{R_1}{R_2}=\frac{3}{16}$

Thus, the rates of disintegration are in the ratio $3:16$.