Question

You have prepared a nice sample of $N_{\circ }$ radioactive nuclei of decay constant $\lambda$, and want to show it off to your friends. When they arrive units of time later, you open the lid and find there are only _______ nuclei left intact.

• A. $N_{\circ }{e}^{\lambda t}$
• B. $N_{\circ }{e}^{-\lambda t}$
• C. $N_{\circ }$
• D. $N_{\circ }\lambda t$

Answer 1

The correct option is B $N_{\circ }{e}^{-\lambda t}$

The decay constant given is $\lambda$. Also, at t=0, there were $N_{\circ }$ nuclei in the sample.
If you have N nuclei in the sample at a time t, the number of nuclei which will decay in the next time interval dt will be -
$dN=-\lambda Ndt$
$\Rightarrow$ $\frac{dN}{N}=-\lambda dt$
$\Rightarrow$ ${\int }_{N_{\circ }}^N\frac{dN}{N}=-\lambda {\int }_0^{t}dt$
$\Rightarrow$ $ln\frac{N}{N_{\circ }}=-\lambda t$
$\Rightarrow$ $\frac{N}{N_{\circ }}=e^{-\lambda t}$
$\Rightarrow$ $N=N_{\circ }{e}^{-\lambda t}$.
So if you had started with $N_{\circ }$ nuclei in the sample, when your friends arrive t units of time later, there will be only $N=N_{\circ }{e}^{-\lambda t}$.
P.S : If you ever get your hands on a nice sample of radioactive nuclei in a jar, do not 'open the lid'. Even better, never get your hands on a nice sample of radioactive nuclei.