Question

The half-life of plutonium-239 is 24,100 years. Of an original mass of 100g, how much plutonium-239 remains after 96,440 years?

Answer 1

Hint:In the above question, half-life of plutonium-239 is given as 24,100 years. It is given that initially the mass is 100g and is asked how much will remain after 96,440 years. We have to check how many half-life will result in 96,400 years and then we can compare it with the mass.

Complete step-by-step answer:The half-life of plutonium is 28 years. When the concentration of the substance is reduced to half then it is called half-life.
Let the concentration of strontium is ${{\text{A}}_{\text{o}}}$
Hence, after 1st half cycle, concentration will be $\dfrac{{{{\text{A}}_{\text{o}}}}}{{\text{2}}}$
After 2nd half cycle, concentration will be $\dfrac{{{{\text{A}}_{\text{o}}}{\text{/2}}}}{{\text{2}}}{\text{ = }}\dfrac{{{{\text{A}}_{\text{o}}}}}{{{{\text{2}}^{\text{2}}}}}$
After 3rd half cycle, concentration will be $\dfrac{{{{\text{A}}_{\text{o}}}{\text{/4}}}}{{\text{2}}}{\text{ = }}\dfrac{{{{\text{A}}_{\text{o}}}}}{{{{\text{2}}^{\text{3}}}}}$
and so on.
Hence after n half-cycle , concentration will be $\dfrac{{{{\text{A}}_{\text{o}}}}}{{{{\text{2}}^{\text{n}}}}}$
It is given in the question that the initial concentration of plutonium is 100mg and final concentration is x mg.
So, we have A=x mg and ${{\text{A}}_{\text{o}}}$= 100 mg.
Let us use the general equation, that is, A = $\dfrac{{{{\text{A}}_{\text{o}}}}}{{{{\text{2}}^{\text{n}}}}}$
Substituting the values, we get-
\[{\text{x = }}\dfrac{{{\text{100}}}}{{{{\text{2}}^{\text{n}}}}}\] (1)
So, we have to first find the value of n, that is, the number of half life cycles.
the 1st half cycle occurs after 24,100 years.
a half cycle occurs after 24,100n years.
According to the question, 24,100n =96,400.
Rearranging the above equation:
${\text{n = }}\dfrac{{{\text{96,400}}}}{{{\text{24,100}}}}{\text{ = 4}}$
Substituting the value of n in equation 1, we get:
\[{\text{x = }}\dfrac{{{\text{100}}}}{{{{\text{2}}^{\text{4}}}}}{\text{ = }}\dfrac{{{\text{100}}}}{{{\text{16}}}}{\text{ = 6}}{\text{.25}}\]g
Hence, \[{\text{6}}{\text{.25}}\] g of plutonium-239 remains after 96,440 years.

Note:The term half life is commonly used in nuclear physics to describe how quickly an unstable atom undergoes radioactive decay or how long stable atoms survive.