Question

Obtain a relation between half-life of a radioactive substance and decay constant ($\lambda$)

Answer 1

Take population as ${{\text{N}}_0}$ in t=0 and N at any given time t.

Complete step by step solution:
Given, we need to find out the relation between half life of a radioactive substance and decay constant.
Now, let N be the size of the population of radioactive atoms at a given time t. dN is the amount by which the population of the radioactive atoms decreases in time dT
So, the rate of change can be given by the equation:

$$\Rightarrow \dfrac{{{\text{dN}}}}{{{\text{dT}}}} = - \lambda {\text{N}} \\\ \Rightarrow \dfrac{{{\text{dN}}}}{{\text{N}}} = - \lambda {\text{dT}} \\\$$

Integrating both sides we get,

$$\Rightarrow \int {\dfrac{{{\text{dN}}}}{{\text{N}}}} = - \lambda \int {{\text{dT}}} \\\ \Rightarrow {\text{N = }}{{\text{N}}_0}{e^{ - \lambda {\text{T}}}} \\\$$

Where ${{\text{N}}_0}$ is the population of initial radioactive atoms at time T = 0.

Half life is the time required to decay half the original population of radioactive atoms.
${\text{N = }}\dfrac{{{{\text{N}}_0}}}{2}{\text{ at T = }}{{\text{T}}_{(\dfrac{1}{2})}}$
$\Rightarrow \dfrac{{{{\text{N}}_0}}}{2} = {{\text{N}}_0}{e^{ - \lambda {{\text{T}}_{(\dfrac{1}{2})}}}}$
On cancelling out ${{\text{N}}_0}$we get,
$\Rightarrow \dfrac{1}{2} = {e^{ - \lambda {{\text{T}}_{(\dfrac{1}{2})}}}}$

Now on cross-multiplying we get,
$\Rightarrow {e^{\lambda {{\text{T}}_{(\dfrac{1}{2})}}}} = 2$
Putting the logarithm function on both sides we get,
${\log _e}{e^{\lambda {{\text{T}}_{(\dfrac{1}{2})}}}} = {\log _e}2$
Using formula and cancelling out log in LHS we get,
$\Rightarrow \lambda {{\text{T}}_{(\dfrac{1}{2})}} = {\log _e}2$
Making T as the subject of the formula we get,
$\Rightarrow {{\text{T}}_{(\dfrac{1}{2})}} = \dfrac{{{{\log }_e}2}}{\lambda }$
On putting the value of log2 we get,
$\Rightarrow {{\text{T}}_{(\dfrac{1}{2})}} = \dfrac{{0.693}}{\lambda }$
Thus, the decay constant and half life of a radioactive substance is related by
${{\text{T}}_{(\dfrac{1}{2})}} = \dfrac{{0.693}}{\lambda }$

Additional information:
Half-life of a radioactive substance can be defined as the time taken for a given amount of the substance to become reduced by half as a consequence of decay, and therefore, the emission of radiation .

Note: Decay constant is proportionality constant between the size of a population of radioactive atoms and the rate at which the population decreases because of radioactive decay.