Demystifying Half-Life Problems

The half-life of a substance is the amount of time it takes for that substance to deplete by 50 percent. A typical problem goes as follows:

The half-life of uranium is 4.5 billion years. Originally, you have 1 kg of uranium. How much uranium will you end up with in 1 million years?

Lots of these word problem can be set up with equations. The important thing is to find the right equation for the job.

Let \( f(t) \) denote the amount of uranium you have at time \( t \). Since at time \( t = 0 \) you originally have 1 kg of uranium, we can write: $$ f(0) = 1 $$

Note that the half-life of uranium is 4.5 billion years, so in 4.5 billion years, you will only have half a kilogram left. We can write this as: $$ f(4,500,000,000) = 0.5 $$

More generally: $$ f(t) = 0.5^{t/4,500,000,000} $$ Note that when \( t = 4,500,000,000 \),

\begin{equation} \begin{split} f(t) & =0.5^{4,500,000,000/4,500,000,000} \\ & = 0.5^1 \\ & = 0.5 \end{split} \end{equation}

as above.

So, to determine the amount of uranium you will end up having in 1 million years, we can use the above formula with \( t = 1,000,000 \). The answer is:

\begin{equation} \begin{split} f(1,000,000) & =0.5^{1,000,000/4,500,000,000} \\ & = 0.999845 \dots \end{split} \end{equation}

So the answer is that you will have around \( 0.9998 \) kg of uranium. Essentially, almost all of it is still there after a million years.

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