Question

A block having $12g$ of element is placed in a room. This element is a radioactive element with half-life of $15$ years. After how many years will there be just $1.5g$ of the element in the box?

• A. $40$ years
• B. $45$ years
• C. $20$ years
• D. $15$ years

Answer 1

The correct option is B $45$ years

Given, $T_{1/2}=15$ years
$N_0=12g$
$N=1.5g$
$\therefore N=N_0{e}^{-\lambda t}$
$\Rightarrow e^{-\lambda t}=\frac{N}{N_0}$
$\Rightarrow -\lambda t=\ln \left(\frac{N}{N_0}\right)$
$\Rightarrow t=\frac{1}{\lambda }\ln \left(\frac{N_0}{N}\right)=\frac{T_{1/2}}{0.693}\ln \left(\frac{N_0}{N}\right)$ $[\because \lambda =\frac{0.693}{T_{1/2}}]$
$=\frac{15}{0.693}\times \ln \left(\frac{12}{1.5}\right)$
$=45$ years.