Question

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

Answer 1

Hint: 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.