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.