Question

Calculate the half-life period of a radioactive substance if its activity drops to $\dfrac{1}{{16}}$th of its initial value in $30$ years.

Answer 1

By using the initial and the total activity of the substance we will find the total number of half lives by converting those values in the power of $\dfrac{1}{2}$. After that we will divide the total time with respect to the number of half lives and eventually find the half-life.

Complete step-by-step solution:
In nuclear physics, half-life is defined as the time required for a quantity of a substance to reduce to half of its initial quantity.
Activity of a substance is defined as the value which signifies the amount of any substance being decayed in unit time.
For the given set of problem we have to use the relation,
$\dfrac{N}{{{N_0}}} = {\left( {\dfrac{1}{2}} \right)^n} - - - - - (1)$
The given variables here are defined as-
$N =$ the final amount undecayed quantity.
${N_0} =$ the initial amount of substance.
$n =$ number of half-life.
According to the given question, $N = \dfrac{{{N_0}}}{{16}}$
Substituting the value in equation $\left( 1 \right)$ we get,
$\left( {\dfrac{1}{{16}}} \right) = {\left( {\dfrac{1}{2}} \right)^n}$
Converting the value into the power of $\dfrac{1}{2}$ we get,
${\left( {\dfrac{1}{2}} \right)^4} = {\left( {\dfrac{1}{2}} \right)^n}$
Comparing the equations we get, $n = 4 - - - - \left( 2 \right)$.
So, we have got the number of half-life as $4$ and thus using the total time as given in the question we will find the time period of half-life.
Total time period given in the question$= 30$ years.
The relation is given as,
${T_{\dfrac{1}{2}}} = \dfrac{t}{n} - - - - - - \left( 3 \right)$
The variables are defined as-
${T_{\dfrac{1}{2}}}$ $=$ half-life.
$t =$ total time.
$n =$ number of half-life.
Substituting the value of $t = 30$ is given in the question and $n$from $\left( 2 \right)$ into equation $\left( 3 \right)$ we get,
${T_{\dfrac{1}{2}}} = \dfrac{{30}}{4} = 7.5$
Therefore, the half-life of this radioactive substance is $7.5$ years.

Note: This method is only applicable when the ratio of initial activity to the final activity of a radioactive substance can be converted to the power of $\dfrac{1}{2}$. Either we have to use the Radioactive Decay Law to find the decay constant and hence find the half life.