Question

Define half-life of a radioactive substance and write the relation of half life with the following:
(a) Radioactive decay constant (disintegration constant)
(b) Mean life of a radioactive substance.

Answer 1

Hint: A radioactive substance is a substance which is highly unstable and loses energy by the process of decaying. Time taken for the substance to reduce to half its initial amount is half- life of the substance. Use the formula $N={{N}_{0}}{{e}^{-\lambda t}}$ to find the relation between half-life and decay constant.

Formula used:
$N={{N}_{0}}{{e}^{-\lambda t}}$
${{t}_{mean}}=\dfrac{1}{\lambda }$

Complete step-by-step solution -
(a) Half-life of a radioactive substance is defined as the time required for a substance to reduce to half of the initial amount of the substance.
A radioactive substance or a particle is not a stable particle. Therefore, it loses all of its energy by decaying.
The rate of decay of a radioactive substance is directly proportional to the amount of substance present.
Suppose a radioactive substance is under the process of decaying. Let the initial amount of the substance be ${{N}_{0}}$. Let the amount of substance left at time t be N.
Then $-\dfrac{dN}{dt}\propto N$.
Therefore, we get $-\dfrac{dN}{dt}=\lambda N$.
Here, k is a proportional constant called decay constant.
When we integrate the above differential equation we get,
$N={{N}_{0}}{{e}^{-\lambda t}}$ ….. (i).
The above equation is an expression for the amount of the substance present (left) at time t.
Let us calculate the half-life of the substance, that is when $N=\dfrac{{{N}_{0}}}{2}$.
Let the half-life be denoted by ${{t}_{1/2}}$.
Substitute the value of N as $\dfrac{{{N}_{0}}}{2}$ and t as ${{t}_{1/2}}$.
This implies,
$\dfrac{{{N}_{0}}}{2}={{N}_{0}}{{e}^{-\lambda {{t}_{1/2}}}}$
$\Rightarrow \dfrac{1}{2}={{e}^{-\lambda {{t}_{1/2}}}}$
Take reciprocals on both the sides.
$\Rightarrow 2={{e}^{\lambda {{t}_{1/2}}}}$
This gives us that,
$\Rightarrow \ln 2=\lambda {{t}_{1/2}}$
$\Rightarrow {{t}_{1/2}}=\dfrac{\ln 2}{\lambda }$
This is the relation between the half-life of a substance and its decay constant.

(b) Mean life of a radioactive substance is the average life of an atom of that substance. That is the time required for that atom to decay completely.
It is denoted by ${{t}_{mean}}$ and found to be equal to $\dfrac{1}{\lambda }$.
We know the relation between half-life and decay constant.
i.e. ${{t}_{1/2}}=\dfrac{\ln 2}{\lambda }$
$\Rightarrow \lambda =\dfrac{\ln 2}{{{t}_{1/2}}}$.
Substitute this value of $\lambda $ in equation ${{t}_{mean}}=\dfrac{1}{\lambda }$.
Therefore,
$\Rightarrow {{t}_{mean}}=\dfrac{1}{\dfrac{\ln 2}{{{t}_{1/2}}}}$
$\Rightarrow {{t}_{mean}}=\dfrac{{{t}_{1/2}}}{\ln 2}$
$\Rightarrow {{t}_{1/2}}=\ln (2){{t}_{mean}}$
The above equation is the relation between the half –life and means (average) life of a radioactive substance.

Note: Do not confuse between half-life and mean life of a radioactive substance.
Half-life of a radioactive substance is the time taken for the amount of the substance to be reduced (decay) to half of the initial amount.
Whereas mean life is the average time taken for an atom of that substance to decay completely. In simple words, it is the life of all the atoms of the substance divided by the total number of atoms in the substance.