• # question_answer The time during which three-fourth of a sample will decay if decaying both by $\alpha$ and $\beta$ -emission simultaneously is 312 year. The mean life of this sample is 900 years for $\alpha$ - emission. Find the mean life of this sample for $\beta$ - emission. A)  550 years            B)  300 years C)  615 years                            D)  655 years

$N={{N}_{0}}{{e}^{-\lambda t}}$             $\frac{{{N}_{0}}}{4}\,={{N}_{0}}{{e}^{-\lambda t}}$             $\lambda t=\,\ln 4$             $t=\frac{1}{\lambda }\,\ln \,4=\frac{1}{\lambda }\ln \,{{2}^{2}}=\frac{2}{\lambda }In\,2=312$             $\lambda =\frac{2\ln 2}{312}\,\times 4.443\,\times {{10}^{-3}}$             $=\frac{1}{225}$             $\lambda ={{\lambda }_{\alpha }}\,+{{\lambda }_{B}}$             $\frac{1}{225}\,=\frac{1}{900}\,+{{\lambda }_{\beta }}$             ${{\lambda }_{\beta }}\,=\frac{1}{225}\,-\frac{1}{900}=\frac{1}{300}$ Decay constant is reciprocal of mean life. So, mean life for $\beta$ - emission = 300 years.