Question

How many milligrams of a $15.0mg$ sample of radium-226 remain after $6396\,years$ if the half life of radium-226 is $1599\,years$?

Answer 1

Hint: Radium-226 is a radioactive element and continuously undergoes nuclear reactions due to which it emits particles and species and hence its amount depletes over time. Half life is the time taken to reduce to half of its amount. We calculate the number of half lives completed in the given time. The remaining amount after n lives is the initial amount divided by two to the power n.

Formula used:
$A'=\dfrac{A}{{{2}^{n}}}$

Complete step-by-step answer:
A radioactive material is one which undergoes radioactive decay in which it emits alpha, beta and gamma particles. Radioactive materials are highly unstable and hence continuously undergo nuclear reactions to emit different species and undergo chemical changes.
The half life of a radioactive material is the time taken by it to reduce to half of its value. Given, the half life of radium-226 is $1599\,years$. Number of half lives that complete in $6396\,years$ will be-
$\dfrac{6396}{1599}=4$
Therefore, in $6396\,years$ 4 half lives of radium-226 are completed. The amount of a radioactive material remaining after n half lives will be-
$A'=\dfrac{A}{{{2}^{n}}}$
Here, $A$ is the initial amount of the material
$A'$ is the remaining amount
Given, $A=15.0mg,\,n=4$ given values are substituted in the above equation to get,
$\begin{align}
  & A'=\dfrac{15.0}{{{2}^{4}}} \\
 & \Rightarrow A'=\dfrac{15}{16} \\
 & \therefore A'=0.94mg \\
\end{align}$
The remaining amount is $0.94mg$.
Therefore, the amount of radium-226 remaining after $6396\,years$ is $0.94mg$.

Note: After a material undergoes radioactive reactions, its chemical properties change. In alpha decay helium like atoms called alpha particles are emitted, in beta decay electron-like species called beta particles are emitted and in gamma decay electromagnetic radiation is emitted. Mostly very heavy atoms undergo nuclear reactions.