Question

Two radioactive substances X and Y initially contain equal number of atoms. Their half-lives are $1$ hour and $2$ hours respectively. Then the ratio of their rates of disintegration after four hours is:

• A. $1:1$
• B. $2:1$
• C. $1:2$
• D. $2:3$

Answer 1

The correct option is C $1:2$

Given that their half lives are 1 hour and 2 hours.

Also we know that the half life is related to disintegration constant by:

$\lambda =\frac{k}{t_{half}}$

${\lambda }_x=\frac{k}{1}$

${\lambda }_y=\frac{k}{2}$

Hence, after 4 hours.

X - 4 half lives

Y - 2 half lives

If initially they have $N_0$ molecules each.

Then after time of 4 hours.

$\frac{N_0}{16}$ molecules of x

$\frac{N_0}{4}$ molecules of y

Rate of disintegration is equal to $\lambda \times N$

$R_x=\frac{k}{1}\times \frac{N_0}{16}$

$R_y=\frac{k}{2}\frac{N_0}{4}$

Hence we see that $R_y=2R_x$

Hence the ratio is $1:2$