Question

X and Y are two radioactive substances having ${\mathrm{N}}_1\mathrm{and}{\mathrm{N}}_2$nuclei. The half-life of X is the half-life of Y.After three half lives of Y , The ratio of ${\mathrm{N}}_1\mathrm{and}{\mathrm{N}}_2$will be equal to?

• A. $\frac{8}{1}$
• B. $\frac{1}{8}$
• C. $\frac{3}{1}$
• D. $\frac{1}{3}$

Answer 1

The correct option is A

$\frac{8}{1}$

Explanation for correct options:

A. $8/1$

Step 1:

• Half-life of X=t
• Half-life of Y=$2\mathrm{t}$

Step2:

• After three half-lives Y it will be:$3\mathrm{t}=3\times 2\mathrm{t}$
• $3\mathrm{t}=6\mathrm{t}$
• So for X, it is $6\mathrm{t}$

Step 3:

• After n half, it will be $\frac{{\mathrm{N}}_1}{2\mathrm{n}}$
• As X is six the half-lives will be=$\frac{{\mathrm{N}}_1}{{\left(2\right)}^6}=\frac{{\mathrm{N}}_1}{64}$
• As for Y is three half-lives it will be=$\frac{{\mathrm{N}}_2}{2^3}=\frac{{\mathrm{N}}_2}{8}$
• So the ratio of $\frac{{\mathrm{N}}_1}{{\mathrm{N}}_2}$will be= $\frac{{\mathrm{N}}_1}{64}=\frac{N_2}{8}$
• $\frac{{\mathrm{N}}_1}{{\mathrm{N}}_2}=\frac{64}{8}=\frac{8}{1}$

Hence option (A) is correct.,the ratio of ${\mathrm{N}}_1\mathrm{and}{\mathrm{N}}_2$will be equal to$\frac{8}{1}$