Question

$90$% of a radioactive sample is left undecayed after time $t$ has elapsed. What percentage of the intial sample will decay in a total time $2t$:

• A. $20$%
• B. $19$%
• C. $40$%
• D. $38$%

Answer 1

The correct option is B $19$%

$\begin{array}{c}\text{Given,}\\ \text{after time (t) sample is undecayed 90 \%}\\ \text{means it decayed}10\%\text{of initial}\end{array}$

$\begin{array}{c}\text{Let initial amount}\left(A_0\right)=100\\ \text{after time (t)}\left(A_t\right)=90\text{(left)}\end{array}$

$\begin{array}{c}\text{we know}\\ \begin{array}{cc}{A}_t& =A_0{e}^{-\lambda t}\\ 90=& 100e^{-\lambda t}\\ e^{-\lambda t}& =\frac{9}{10}-\left(i\right)\end{array}\end{array}$

$\begin{array}{cc}\text{for time}& =2t\\ A_t& =A_0{e}^{-2\lambda t}\\ A_t& =100\times {\left(e^{-\lambda t}\right)}^2\\ & =100\times {\left(\frac{9}{10}\right)}^2\text{; from equation (i)}\\ A_t& =\frac{81}{100}\text{(left)}\end{array}$

$\begin{array}{c}\text{(\%) of amount left=}81\%\\ \text{So, decayed ammount}=19\%\end{array}$