Question

The radioactive sources A and B with half lives of $t$ hr and $2t$ hr respectively, initially contain the same number of radioactive atoms. At the end of $t$ hours, their rates of disintegration are in the ratio:

• A. $2\sqrt{2}:1$
• B. 1 : 8
• C. $\sqrt{2}:1$
• D. $ln2:1$

Answer 1

The correct option is C $\sqrt{2}:1$

For a first order reaction, $\frac{0.693}{t_{1/2}}=\frac{2.303}{t}log\frac{A_0}{A-t}$

For radioactive source $A$, the equation becomes $\frac{0.693}{t}=\frac{2.303}{t}log\frac{A_0}{A_t}$......(1)

For radioactive source $B$, the equation becomes $\frac{0.693}{t}=\frac{2.303}{2t}log\frac{A_0}{B_t}$......(2)

Divide equation (1) with equation (2).
$2log\frac{A_0}{A-t}=log\frac{A_0}{B_t}$.
At the end of t hours, their rates of disintegration are in the ratio $\sqrt{2}:1$