Question

Protactinium-234 has a half-life of 1.17 minutes. How long does it take for a 10mg sample to decay to 2mg?

Answer 1

Hint: First we need to understand what is half-life. The time taken by any substance to reduce to half of its original quantity is known as the half-life of that substance. It is usually used to describe any form of decay be it exponentially or non-exponentially.

Complete step by step answer: When we talk about the decaying of a substance, it is usually the exponential decay of a substance. When a substance decays or decreases at a rate that is in proportion to its current value, it is known as the exponential decay of a substance.
For a substance that decays exponentially, its half-life is constant throughout its lifetime. This helps us to determine the exponential decay equation and the decrease in the quantity of a substance when several half-lives have passed.
Now, the exponential decay of a substance can be described as
     \[N(t)={{N}_{0}}{{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{{{t}_{\dfrac{1}{2}}}}}}\]
Where the initial quantity of a substance is given by ${{N}_{0}}$,
the final quantity of the undecayed substance after time t is given by N(t),
and ${{t}_{\dfrac{1}{2}}}$ is the half-life of the substance.
Now, it is given to us that
${{N}_{0}}$= 10mg
N(t)=2mg
And, ${{t}_{\dfrac{1}{2}}}$=1.17minutes.
By substituting these values in the exponential decay equation, we get
     \[\begin{align}
  & 2=10{{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{1.17}}} \\
 & \dfrac{1}{5}={{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{1.17}}} \\
\end{align}\]
Upon taking log on both sides,
     \[\begin{align}
  & \log \left( \dfrac{1}{5} \right)=\left( \dfrac{t}{1.17} \right)\log \left( \dfrac{1}{2} \right) \\
 & t=1.17\left( \dfrac{\log \left( \dfrac{1}{5} \right)}{\log \left( \dfrac{1}{2} \right)} \right) \\
 & t=1.17\left( \dfrac{-\log 5}{-\log 2} \right) \\
 & t=1.17\left( \dfrac{0.69897}{0.30103} \right) \\
 & t\cong 2.7166\min \\
\end{align}\]
So, it would take approximately 2.7166 minutes or about 2 minutes and 43 seconds for the Protactinium-234 sample which has a half-life of 1.17 minutes to decay from 10mg to 2mg.

Note: It is important to note that when the half-life is given for discrete entities like radioactive atoms, it describes the probability of a single unit of the entity decaying within its half-life time and not the time taken to decay the single entity in half as that is not possible.