Question

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

Answer 1

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.

Additional information:
There are various types of radioactive decays available such as Alpha decay, Beta-Decay and Gamma Decay. These decays happen due to the instability in the nucleus of an atom. The more unstable the nucleus the higher would be the energy of radioactive decay. The lowest level of energy decay is in alpha decay, Beta decay has higher energy decay than alpha decay and in the Gamma decay it is the highest.

Note: Here the quantity of the substance remaining is one sixteenth of the original substance and we have given the time elapsed as 2 hours. Here apply the formula for Half-Life and calculate the known variable.