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 ${\displaystyle \frac{1}{2^n}}$.
Now, we let us take the time taken for the substance to decay in half to be $t_{{\displaystyle \frac{1}{2}}}$.
So, when t = $t_{{\displaystyle \frac{1}{2}}}$, $N(t_{{\displaystyle \frac{1}{2}}})={\displaystyle \frac{N_0}{2}}$.
When we substitute these values in the exponential decay equation, we get
$$\begin{array}{rl}& {\displaystyle \frac{N_0}{2}}=N_0{e}^{-\lambda t_{{\displaystyle \frac{1}{2}}}}\\ & e^{-\lambda t_{{\displaystyle \frac{1}{2}}}}={\displaystyle \frac{1}{2}}\end{array}$$
Upon taking the log, we get
$$\begin{array}{rl}& {\log }_e{e}^{-\lambda t_{{\displaystyle \frac{1}{2}}}}={\log }_e{\displaystyle \frac{1}{2}}\\ & -\lambda t_{{\displaystyle \frac{1}{2}}}=-\ln 2\\ & t_{{\displaystyle \frac{1}{2}}}={\displaystyle \frac{\ln 2}{\lambda }}\\ & t_{{\displaystyle \frac{1}{2}}}\cong {\displaystyle \frac{0.693}{\lambda }}\end{array}$$
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.