Question

What is the time taken by the radioactive element to reduce to $\dfrac{1}{e}$ times?
A. Half-life
B. Mean life
C. (Half-life)/2
D. Twice the mean life.

Answer 1

Hint: A radioactive element tends to emit radioactive energy in the surrounding. The law of radioactivity gives the relation between the initial amount of element, final amount of element, decay constant and time taken to decay. By using that law of radioactivity, the time taken by the radioactive element to reduce to $\dfrac{1}{e}$ times is calculated.

Useful formula:
The law of radioactivity,
$N = {N_0}{e^{ - \lambda t}}$
Where, $t$ is the time taken by the element to decay, $N$ is the amount of element after decay for time $t$, ${N_0}$ is the amount of radioactive element initial and $\lambda $ is the decay constant.

Step by step solution:
(i) The radioactive element tends to decay, then the element reduces to $\dfrac{1}{e}$ times of the initial amount.
Thus,
$N = \dfrac{{{N_0}}}{e}$
By the law of radioactivity,
$N = {N_0}{e^{ - \lambda t}}\;.....................................\left( 1 \right)$
Substitute $N$ in equation (1), we get
 $
  \dfrac{{{N_0}}}{e} = {N_0}{e^{ - \lambda t}} \\
  \dfrac{1}{e} = {e^{ - \lambda t}} \\
  {e^{ - 1}} = {e^{ - \lambda t}} \\
 $
By taking log on both sides, we get
$
   - 1 = - \lambda t \\
  \lambda t = 1 \\
  t = \dfrac{1}{\lambda } \\
 $
Since, the time taken by the radioactive element to reduce $\dfrac{1}{e}$ times is $t = \dfrac{1}{\lambda }\;...........................\left( 2 \right)$
(ii) Mean life of radioactive element:
     The mean life of the radioactive element is defined as the average life of the radioactive element tends to decay. It is equal to the inverse of the decay constant.
${t_{avg}} = \dfrac{1}{\lambda }\;.................................\left( 3 \right)$
Where, ${t_{avg}}$ is the average lifetime of the radioactive element and $\lambda $ is the decay constant.

From the equations (2) and (3), we get
$t = {t_{avg}}$
Hence, the option (B) is correct.

Note: The half-life of the radioactive element is the time taken by the element to decay to its half of the initial amount. So, it is not equal to the inverse of the decay constant$\left( {\dfrac{1}{\lambda }} \right)$ of the radioactive element. Half of half-life defined as the time taken to decay to is $\dfrac{1}{4}$ th of initial amount. Hence, that is also not equal to $\left( {\dfrac{1}{\lambda }} \right)$ of the radioactive element.