Question

The half-life period of N¹³ is 10.1 min. Its life time is:
(A) 5.05 minute
(B) 20.2 minute ${N^{13}}$
(C) Infinity
(D) $\dfrac{{10.1}}{{0.6931}}\;\;{\rm{minute}}$

Answer 1

Hint: From the concept of radioactive decay, we can use the information that the total number of ${N^{13}}$ atoms represented by N will vanished after time t, so take the values $N = 0$ in the expression which is used for the determination of complete life time of ${N^{13}}$.

Complete step by step solution:
Given:
The half life period of ${N^{13}}$ is 10.1 minute.
By using decay constant, the total number of atoms present at time $t$ can be obtained as,
\[N = {N_o}{e^{ - \lambda t}}\]
Here, ${N_o}$ is the total number of atoms present originally when $t = 0$, here $t$ is the time, $N$ is the total number of atoms left after time $t$ and $\lambda $ is the decay constant.
For complete life, take $N = 0$, so the above equation becomes
\[0 = {N_o}{e^{ - \lambda t}}\]
Take log on the both side in the above equation
$\begin{array}{l}
\log 0 = {N_o}\left( { - \lambda t} \right)\\
t = \infty
\end{array}$

Therefore, the option (C) is the correct answer that is Infinity.

Note: To eliminate the exponential sign apply the concept of logarithm in the expression which relate the half life and decay constant. The value of the log 10 is undefined so put $\infty $ for log 10 and calculate the lifetime.