• # question_answer A radioactive element decays by$\beta -emission$. A detector records n beta particles in 2s and in next $2s$ it records $0.75n$ beta particle. Find mean life correct to nearest whole number-(Given $\ell n\,2=0.6931,\text{ }\ell n\text{ }3\,\,=\,\,0986$) A) $17s~$                                    B) $7\,s$C) $5\text{ }s$                             D) $15\text{ }s$

$Let\text{ }{{N}_{0}}\text{ }=\text{ }initial\text{ }number\text{ }of\text{ }nuclei$ $~n={{N}_{0}}-{{N}_{0}}{{e}^{-\lambda \,\times \,2}}$ $\,1.75\,n={{N}_{0}}-{{N}_{0}}{{e}^{-4\lambda }}$ $\frac{1}{1.75}=\frac{1-{{e}^{-2\lambda }}}{1-{{e}^{-4\lambda }}}$ $1+{{e}^{-2\lambda }}\,\,=\,\,1.75$ ${{e}^{2\lambda }}\,\,=\,\,\frac{1}{0.75}\,\,=\,\,1.33$ $\,2\lambda =\ell n\,(1.33)$ $\tau \,\,=\frac{1}{\lambda }\,\,=\,\,\frac{2}{\ell n(1.33)}\,\,=\,\,7s$