Question

The half-life of a radioactive element is 10 years. What percentage of it will decay in 100 years?
A) $99.9{\scriptstyle{}^{0}/{}_{0}}$
B)$10{\scriptstyle{}^{0}/{}_{0}}$
C)$50{\scriptstyle{}^{0}/{}_{0}}$
D) $66.5{\scriptstyle{}^{0}/{}_{0}}$

Answer 1

The time of the decay of the substance at the time ‘t’ is related to the number of half-life residing in ‘t’, $\text{t=n}{{\text{t}}_{\text{1/2}}}$.The amount of substance not decaying is find out by the equation, $\left[ \text{A} \right]\text{=}\dfrac{\left[ {{\text{A}}_{\text{0}}} \right]}{{{\text{2}}^{\text{n}}}}$. The amount of substance decays at time ‘t’ is obtained by considering the total amount of substance as a $100{\scriptstyle{}^{0}/{}_{0}}$

Complete answer:
The half-life ${{\text{t}}_{\text{1/2}}}$ for a radioactive element is defined as the amount of the time required for a given amount of substance to reduce by the half of its initial amount because of the decay or emission of
We are given with the half-life of radioactive elements ${{\text{t}}_{\text{1/2}}}=10\text{ years}$
The time at which the total amount of substance undergoes the decay,$100\text{ years}$
Let the initial amount of the radioactive substance be equal to the $\left[ {{\text{A}}_{\text{0}}} \right]$ at the time $\text{t=0}$
Consider the number of substance decays at the $\text{t=100 years}$ is equal to the$\left[ \text{A} \right]$.
We know that the time required for the decay of radioactive material is equal to the ‘n’ number of times to the half-life of an element.
The formula is as:
$\text{t=n}{{\text{t}}_{\text{1/2}}}$
Let’s first find out the total number of half-lives of 10 years present in the time of 100 years.
$\text{100=n}\times \text{10}$
Rearrange the formula for n we get,
$\text{n=}\dfrac{100}{10}$
$\text{n=}10$
Thus, the total number of half-life residing in the time ‘t’ is 10.
We have the other formula which relates the initial amount of radioactive element $\left[ {{\text{A}}_{\text{0}}} \right]$ is related to the amount of substance at a time ‘t’.
$\left[ \text{A} \right]\text{=}\dfrac{\left[ {{\text{A}}_{\text{0}}} \right]}{{{\text{2}}^{\text{n}}}}$
Let us substitute values for ‘n’.
$\text{=}\dfrac{\left[ {{\text{A}}_{\text{0}}} \right]}{{{\text{2}}^{\text{10}}}}$
$\text{=}\dfrac{\left[ {{\text{A}}_{\text{0}}} \right]}{1024}$
Since,${{2}^{10}}=1024$
$\left[ \text{A} \right]\text{=}\left[ {{\text{A}}_{\text{0}}} \right]\times 9.765\times {{10}^{-4}}$
Let's find out the percentages of the radioactive substance undergoes the decay at the 100 years
${\scriptstyle{}^{\text{0}}/{}_{\text{0}}}\text{ radioactive substance decay=100-}\dfrac{\left[ \text{A} \right]}{\left[ {{\text{A}}_{\text{0}}} \right]}$
Put the values in the above equation.
$\text{=100-}\dfrac{\left[ {{\text{A}}_{\text{0}}} \right]\times 9.765\times {{10}^{-4}}}{\left[ {{\text{A}}_{\text{0}}} \right]}$
$\text{=100-}\dfrac{\left[ {{\text{A}}_{\text{0}}} \right]\times 9.765\times {{10}^{-4}}}{\left[ {{\text{A}}_{\text{0}}} \right]}$
$\text{=100-}9.765\times {{10}^{-4}}{\scriptstyle{}^{0}/{}_{0}}$
$\text{=}99.9{\scriptstyle{}^{0}/{}_{0}}$
The amount of radioactive substances decays at the 100 years is equal to the $99.9{\scriptstyle{}^{0}/{}_{0}}$.

Hence, (A) is the correct option.

Note:
Here we are asked to find out the amount of substance decays at the time t hence always subtract the number of substance decays from the 100 to get the desired answer.