Question

At time $t=0$, a container has $N_0$ radioactive atoms with a decay constant $\lambda$. In addition, $C$ numbers of atoms of the same type are being added to the container per unit time. How many atoms of this type are there at $t=T$ ?

• A. $\frac{C}{\lambda }{e}^{-\lambda T}-N_0{e}^{-\lambda T}$
• B. $\frac{C}{\lambda }{e}^{-\lambda T}+N_0{e}^{-\lambda T}$
• C. $\frac{C}{\lambda }(1-e^{-\lambda T})+N_0{e}^{-\lambda T}$
• D. $\frac{C}{\lambda }(1+e^{-\lambda T})+N_0{e}^{-\lambda T}$

Answer 1

The correct option is C $\frac{C}{\lambda }(1-e^{-\lambda T})+N_0{e}^{-\lambda T}$

At $t=0$ initial nucleon $=N_0$
Additionally, at a constant rate $(\lambda N-C)$
$=-\frac{dN}{dt}$
$\underset{0}{\overset{t}{\int }}dt=\underset{N_0}{\overset{N}{\int }}(-\frac{dN}{\lambda N-C})$
$t=-\frac{1}{\lambda }\ln (\lambda N-C{)}_{N_0}^N$
$t=-\frac{1}{\lambda }\ln \left(\frac{\lambda N-C}{\lambda N_0-C}\right)$
$\Rightarrow e^{-\lambda t}=\frac{\lambda N-C}{\lambda N_0-C}$
$\lambda N-C=e^{-\lambda t}(\lambda N_0-C)$
$N=\frac{C}{\lambda }+\frac{e^{-\lambda t}}{\lambda }(\lambda N_0-C)$
$=\frac{C}{\lambda }+N_0{e}^{-\lambda t}-\frac{C}{\lambda }{e}^{-\lambda t}$
$=\frac{C}{\lambda }[1-e^{-\left(\lambda t\right)}]+N_0{e}^{-\lambda t}$
At $t=T\Rightarrow N=\frac{C}{\lambda }[1-e^{-\left(\lambda T\right)}]+N_0{e}^{-\lambda T}$