Question

You have prepared a nice sample of $N_0$ 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. $N_0{e}^{\lambda t}$
• B. $N_0{e}^{-\lambda t}$
• C. $N_0$
• D. $N_0\lambda t$

Answer 1

The correct option is B $N_0{e}^{-\lambda t}$

The decay constant given is $\lambda$ . Also, at t = 0 , there were $N_0$ nuclei in the sample.
If you have N nuclei in the sample at a time , the number of nuclei which will decay in the next time interval dt will be -
$dN=-\lambda Ndt\Rightarrow \frac{dN}{N}=-\lambda dt\Rightarrow {\int }_{N_0}^N\frac{dN}{N}=-\lambda {\int }_0^{t}dt\Rightarrow In\frac{N}{N_0}=-\lambda t\Rightarrow \frac{N}{N_0}=e^{-\lambda t}\Rightarrow N=N_0{e}^{-\lambda t}$
So if you had started with $N_0$ nuclei in the sample, when your friends arrive units of time later, there will be only $N=N_0{e}^{-\lambda t}$
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.