Question

After a time interval equal to $3$ half lives; how many times would the activity of a radio active element be of its initial activity?

• A. $2^3$
• B. $3^2$
• C. $1_{/3^2}$
• D. $1_{/2^3}$

Answer 1

The correct option is D $1_{/2^3}$

Activity at time t: $=A_t=A_o{e}^{-\lambda t}$

$\lambda =\frac{ln2}{t_{\circ }}$

$A_t=A_o{e}^{-\frac{ln2}{t_{\circ }}t}$

where $t_{\circ }=$half life

$A_t=$ Activity at time $t$

$A_o=$ Activity at time $t=0$

$\lambda =$ decay constant

$t_{\circ }=$ half life

Given:$t=3t_o$

$A_t=A_o{e}^{-\frac{ln2}{t_{\circ }}3t_o}$

$A_t=A_o{e}^{-3\frac{ln2}{t_{\circ }}{t}_o}$

$A_t=A_o{e}^{-3ln2}$

$A_t=A_o{e}^{-ln{2}^3}$

$A_t=\frac{A_o}{8}$

Hence the activity after $3t_o$ will be $\frac{1}{8}$ times of initial activity.

Hence the correct option is (D).