Question

Nuclei of a radioactive element are being produced at a constant rate $\alpha$. The element has a decay constant $\lambda$. At time t = 0, there are $N_0$ nuclei of the element.

The number N of nuclei at time t is

• A. $\frac{1}{\lambda }(\alpha -\lambda N_0)e^{-\lambda t}$
• B. $\frac{1}{\lambda }(\alpha +\lambda N_0)e^{-\lambda t}]$
• C. $\frac{1}{\lambda }[\alpha +(\alpha -\lambda N_0\left)e^{-\lambda t}\right]$
• D. $\frac{1}{\lambda }[\alpha -(\alpha -\lambda N_0\left)e^{-\lambda t}\right]$

Answer 1

The correct option is D

$\frac{1}{\lambda }[\alpha -(\alpha -\lambda N_0\left)e^{-\lambda t}\right]$

To calculate the number N of nuclei at time t (given that the number at t = 0 is $N_0$), we integrate the above expression from N = $N_0$ to N = N and from t = 0 to t = t, We have

${\int }_{N_0}^N\frac{dN}{\alpha -\lambda N}={\int }_0^{t}dt$

or $-\frac{1}{\lambda }|lo{g}_e(\alpha -\lambda N){|}_{N_0}^N=|t{|}_0^t$

or $-\frac{1}{\lambda }lo{g}_e\left(\frac{\alpha -\lambda N}{\alpha -\lambda N_0}\right)=t$

or $lo{g}_e\left(\frac{\alpha -\lambda N}{\alpha -\lambda N_0}\right)=-\lambda t\text{or}\frac{\alpha -\lambda N}{\alpha -\lambda N_0}=e^{-\lambda t}$

or $N=\frac{1}{\lambda }[\alpha -(\alpha -\lambda N_0\left)e^{-\lambda t}\right]$