Question

Concentration of the radioactive element after one average life is _________ of the original concentration.

• A. $\frac{1}{2}$
• B. $\frac{1}{e}$
• C. $\frac{1}{e^2}$
• D. none of these

Answer 1

The correct option is B $\frac{1}{e}$

Average life time is defined as the life time of a single isolated nucleus. Let us imagine, a single nucleus which decays in $1$ second. Assuming $1$ second time interval to be very small, the rate of change of nuclei would be $1/1$ (because -dN = 1 and dt = 1).

We can also see that since $\frac{-dN}{dt}=\lambda N$, for a single isolated nucleus $N=1$, $\frac{-dN}{dt}=\lambda$. Therefore, in this present case, $\lambda =1$.

Now, let us assume, the same nucleus decays in $2$ seconds, we can see that $\frac{-dN}{dt}$, i.e. $\lambda$ is equal to $\frac{1}{2}$. You will also notice that in the first case, the nucleus survived for $1$ second and in the second case, it survived for $2$ seconds. Therefore, the life time of a single isolated nucleus is $\frac{1}{\lambda }$.

As we know, average life is $t_{avg}=\frac{1}{\lambda }$

So, the concentration of the radioactive element after one average life is

$N=N_o{e}^{-\lambda \ast t=-1}$

$N=N_o{e}^{-1}$