Question

A radioactive substance decays to $\left(\frac{1}{16}\right)^{\text {th }}$ of its initial activity in $40$ days. The half life of the radioactive substance expressed in days is

• A. 10
• B. 20
• C. 5
• D. 2.5

Answer 1

Answer: (A)

Activity of radioactive substance, $A=A_{0} e^{-\lambda t}$
$\Rightarrow \frac{A}{A_{0}}=e^{-\lambda t}=e^{-\frac{0.693}{T_{1 / 2}} t} $
$\left[\because T_{1 / 2}=\frac{0.693}{\lambda}\right]$
$\Rightarrow \frac{1}{16}=e^{-\frac{0.693 \times 40}{T_{1 / 2}}} $
$\Rightarrow \frac{0.693 \times 40}{T_{1 / 2}}=\ln 16$
$\Rightarrow T_{1 / 2}=\frac{0.693 \times 40}{\ln 16}$
$=9.997 \approx 10$ days.