Physics - Modern Physics Question with Solution | TestHub

Let at some instant of time t, number of atoms of the radioactive substance are N. It may decay either by α - emission on by β- emission. So, we can write,

${\left(\frac{-dN}{dt}\right)}_{\text{net}}={\left(\frac{-dN}{dt}\right)}_{\alpha }+{\left(\frac{-dN}{dt}\right)}_{\beta }$

If the effective decay constant is λ, then

$\lambda N={\lambda }_{\alpha }N+{\lambda }_{\beta }N$

∴$\lambda ={\lambda }_{\alpha }+{\lambda }_{\beta }=\frac{1}{1620}+\frac{1}{405}$

$=\frac{1}{324}{\text{year}}^{-1}$

Now,$\frac{N_0}{4}=N_0{e}^{-\lambda t}$

∴$-\lambda t=\text{ln}\left(\frac{1}{4}\right)=-1\text{.}386$

or$\left(\frac{1}{324}\right)t=1\text{.}386$

∴                  t = 449 yrs