Question

A nuclear power plant supplying electrical power to a village uses a radioactive material of half life T years as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is 12.5% of the electrical power available from the plant at that time. If the plant is able to meet the total power needs of the village for a maximum period of nT years, then the value of n is

Answer 1

Hint: There is a nuclear power plant which supplies power to a village and we are given the half life of the radioactive element used as fuel in the plant. It is said that the total power requirement of the village is 12.5% of the power available at the plant in the beginning. It is also said that to meet the total power requirement the maximum time period is ‘n’ times the half life of the fuel. We know that after a half life, the number of nuclei at the beginning will halve. Using this concept we can solve the question.

Complete step-by-step answer:
In the question it is said that there is a power plant which supplies power to a village. The half life of the radioactive material used in the village as fuel is given as T years.
It is said that at the beginning the amount of power required by the village is 12.5% of the power available in the nuclear plant.
Let ‘${{N}_{0}}$’ be the number of nuclei when the available power is 100%.
Then after a half life we know that the number of nuclei and power gets decreased by half, i.e. the number of nuclei becomes $\dfrac{{{N}_{0}}}{2}$ and the power becomes 50%.
${{N}_{0}}\to \dfrac{{{N}_{0}}}{2}$ and $100%\to 50%$
After the second half life, this also gets decreased by half, i.e.
$\dfrac{{{N}_{0}}}{2}\to \dfrac{{{N}_{0}}}{4}$ and $50%\to 25%$
Now when it completes one more half life, i.e. the third half life, the number of nuclei and power gets halved, i.e.
$\dfrac{{{N}_{0}}}{4}\to \dfrac{{{N}_{0}}}{8}$ and $25%\to 12.5%$
From this we can see that the power becomes 12.5% after 3 half lives.
Therefore the value of n is 3.

Note: Radioactive decay is a process where the unstable nuclei of a radioactive element emits charged particles and energy and become stable.
In radioactivity, half life is the time period for the atomic nuclei of the radioactive element to get halved, i.e. if the initial concentration of a radioactive nucleus is ‘A’ then after a half life this concentration will become half, i.e. $\left( \dfrac{\text{A}}{\text{2}} \right)$ .