Question

A certain radioactive material is known to decay at a rate proportional to the amount present. Initially there is $50kg$ of the material present and after two hours it is observed that the material has lost 10%of its original mass, then the

Answer 1

$\frac{dN}{dt}\propto N$
$\frac{dN}{dt}=-\lambda N$
$\frac{dN}{N}=-\lambda dt$
$\ell nN=-\lambda t+c$
$t=0,N=50$
$\ell nN=-\lambda t+\ell n50$
$N=50e^{-\lambda t}$ ......(1)
At $t=2,N=45$
$45=50e^{-\lambda .2}$
$\frac{9}{10}=e^{-2\lambda }$
$\sqrt{\frac{9}{10}}=e^{-\lambda }$
$\therefore N=50(\frac{9}{10}{)}^{t/2}$ ......(2)
At $t=4$
$N=50(0.9{)}^2$
When $N=25kg25=50(0.9{)}^{t/2}$
$\Rightarrow \frac{1}{2}=(0.9{)}^{t/2}t=\frac{2\ell n\frac{1}{2}}{\ell n0.9}$