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# The mean lives of radioactive substance are 1620 year and 405 year for $\;\alpha\;$ emission and $\;\beta\;$ emission respectively . Find time during which $\;{\large\frac{3}{4}}^{th}\;$ of a sample will decay if it is decaying both by $\;\alpha\;$ emission and $\;\beta\;$ emission simultaneously .

$(a)\;469 years\qquad(b)\;512 years \qquad(c)\;449 years \qquad(d)\;412 years$

The decay constant $\;\lambda\;$ is reciprocal of mean life
$\Rightarrow \lambda_{\alpha}=\large\frac{1}{1620}\;$ per year and $\lambda_{\Beta }=\large\frac{1}{405}\;$ per year.
Total decay constant , $\; \lambda=\lambda_{\alpha}+\lambda_{\beta}\; =\large\frac{1}{1620}+\large\frac{1}{405}=\large\frac{1}{324}$ per year
When $\;(\large\frac{3}{4})$$^{th}\; part of sample has disintegrated \;N=\large\frac{N_{0}}{4} \Rightarrow \large\frac{N_{0}}{4}$$=N_{0}e^{-\lambda t}$$\quad or \quad e^{\lambda t}=4 \rightarrow \lambda t =ln 4 \Rightarrow t=\large\frac{1}{\lambda}$$\;\ln 2^2 = \large\frac{2}{\lambda}$$\;\ln 2=2 \times 324 \times 0.693 = 449$ years.
edited Aug 12, 2014