Question

A radioactive isotope having a half-life of 3 days was received after 12 days. It was found that there were 3 gm of the isotope in the container. The initial weight of the isotope when packed was

• A. $12g$
• B. $24g$
• C. $36g$
• D. $48g$

Answer 1

Answer: (D)

$48g$

$\begin{array}{cc}T& =\text{Total time}\\ n& =\text{no. of half}\text{lives}\end{array}$

$\begin{array}{c}{T}_{1/2}=\text{Half life}=3\text{days}\\ A=\text{left after 12 days}=3gm\\ A_0=\text{Initially present}=?\\ T=12\text{days}\\ T=n\times T_{1/2}\\ 12=n\times 3\\ 4=n\text{(i.e. 4 half lifes)}\end{array}$

$\begin{array}{cc}A& =A_0{\left(\frac{1}{2}\right)}^n\\ \Rightarrow 3& =A\circ {\left(\frac{1}{2}\right)}^4\\ \Rightarrow & 3\times 2^4=A_0\\ \Rightarrow & A_0=48gm\end{array}$