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# After 2 hours, only 1/16th of the original decaying nuclei were present. What is the half-life of the sample?

Last updated date: 13th Jun 2024
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Hint: Half-Life as the name suggests it is the required time interval for a radioactive sample to decay to its one-half. It can also be stated as the required time interval that is needed for a number of radioactive disintegration each second of a radioactive material to get to its one-half.

Complete step by step solution:
Find the Half-Life:
$N\left( t \right) = {N_o}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{{t_{1/2}}}}}}$;
Here:
$N\left( t \right)$= Quantity of the Substance remaining;
${N_o}$ = Quantity of the original substance;
t = Time elapsed;
${t_{1/2}}$= Half – Life.

Put in the given values:
$\left( {\dfrac{1}{{16}}} \right){N_o} = {N_o}{\left( {\dfrac{1}{2}} \right)^{\dfrac{2}{{{t_{1/2}}}}}}$;
Cancel out the common factors:
$\Rightarrow \left( {\dfrac{1}{{16}}} \right) = {\left( {\dfrac{1}{2}} \right)^{\dfrac{2}{{{t_{1/2}}}}}}$;
Make the base on the LHS to the RHS and compare their powers:
$\Rightarrow {\left( {\dfrac{1}{2}} \right)^4} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{2}{{{t_{1/2}}}}}}$;
Do the needed mathematical Calculations:
$\Rightarrow \dfrac{2}{{{t_{1/2}}}} = 4$;
So, the half-life would be:
$\Rightarrow 2 = 4 \times {t_{1/2}}$;
$\Rightarrow {t_{1/2}} = \dfrac{1}{2}$;
In terms of minutes;
${t_{1/2}} = 30\min$;

The half-life of the sample is 30min.