Question

What is the half-life period of a radioactive material if its activity drops to $1 / 16^{\text {th }}$ of its initial value of $30$ years ?

• A. $9.5$ years
• B. $8.5$ years
• C. $7.5$ years
• D. $10.5$ years

Answer 1

Answer: (C)

$ A = A _0 e ^{-\lambda t } $
$ \Rightarrow-\lambda t =\ln \left(\frac{ A }{ A _0}\right)$
$ \Rightarrow-\frac{\ln 2}{ t _{1 / 2}} \times 30=\ln \left(\frac{1}{16}\right) $
$\Rightarrow-\frac{\ln 2}{ t _{1 / 2}} \times 30=-4 \ln 2$
$ \Rightarrow t _{1 / 2}=\frac{30}{4}=7.5 \,yrs$