Question

What is the half-life of Uranium 234?

Answer 1

Hint: To solve this question, we first need to know what is half-life. The half-life of a substance is the time taken by it to decay or reduce to half of its original quantity. Half-life is used to describe exponential as well as non-exponential form of decay.

Complete answer:
When we talk about the decaying of a substance, it is usually the exponential decay of a substance. A substance is said to decay exponentially when it decays at a rate proportional to its current value.
The half-life of a substance that decays exponentially is constant throughout its lifetime.
Now, the relation between time and the amount of the substance can be given by the exponential decay equation.
\[N(t)={{N}_{0}}{{e}^{-\lambda t}}\]
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 the decay constant is given by $\lambda $.
The fraction of substance remaining when n half-lives have passed is given by $\dfrac{1}{{{2}^{n}}}$.
Now, we let us take the time taken for the substance to decay in half to be ${{t}_{\dfrac{1}{2}}}$.
So, when t = ${{t}_{\dfrac{1}{2}}}$, $N({{t}_{\dfrac{1}{2}}})=\dfrac{{{N}_{0}}}{2}$.
When we substitute these values in the exponential decay equation, we get
\[\begin{align}
  & \dfrac{{{N}_{0}}}{2}={{N}_{0}}{{e}^{-\lambda {{t}_{\dfrac{1}{2}}}}} \\
 & {{e}^{-\lambda {{t}_{\dfrac{1}{2}}}}}=\dfrac{1}{2} \\
\end{align}\]
Upon taking the log, we get
\[\begin{align}
  & {{\log }_{e}}{{e}^{-\lambda {{t}_{\dfrac{1}{2}}}}}={{\log }_{e}}\dfrac{1}{2} \\
 & -\lambda {{t}_{\dfrac{1}{2}}}=-\ln 2 \\
 & {{t}_{\dfrac{1}{2}}}=\dfrac{\ln 2}{\lambda } \\
 & {{t}_{\dfrac{1}{2}}}\cong \dfrac{0.693}{\lambda } \\
\end{align}\]
Now, the half-life of uranium-234 or $^{234}U$ has been calculated experimentally to be 246000 years.

Note:
It should be noted that the half-life of discrete entities like radioactive atoms describes the probability of the single unit of the entity decaying within its half-life time rather than the time taken to decay half of the single entity.