# 5.5 Radioactive Half-Life

Using the general principles of radioactive decay, scientists have developed a simple equation to calculate a rock’s age. Let us examine how a scientist might use these principles to develop such an equation. Our goal is to give the age of a rock sample in terms of the number of atoms that have decayed. Suppose a radioactive isotope has a half-life of 1 million years. A certain number of these atoms are trapped in a crystal as molten lava cools to form a rock. After 1 million years, half of them would be left. After 2 million years, half of that amount would be left. This would be ½ × ½ = 1/4 of the original number. How many would be left after 3 half-lives, or 3 million years? It would half that number again, or ½ × ½ × ½ = 1/8. From the progression of half-lives, it is easy to see how to convert from this specific description to a formula for the general case. Let’s define the fraction of the atoms that are left as F. Suppose that we wait N half-lives, and ask how many atoms would be left. Based on the paragraph above, we see that the answer would be:

F = (1/2)N

To make the same equation even more useful if you have a calculator, we can take the logarithm of both sides (base 10 logarithm), which gives:

log F = N log (1/2) = -0.301 N

We can check this result. From the discussion above, we know that when N = 3 half lives, the fraction of atoms left is 1/8. Substituting N = 3, we get log F = -0.903. A calculator confirms F = 0.125, or 1/8.

Suppose we want to determine the age of a rock crystal and its potassium atoms. A particular radioactive form of potassium decays with a half-life of 1.25 billion years (known to 3 significant digits), yielding a certain form of argon atoms. Suppose we measure the argon and potassium in the rock crystal, and we find that 58% of the radioactive potassium has already decayed into argon, while 42% of the original radioactive potassium atoms are left in the crystal. How old is the rock? Our measurement has told us that F is 0.42, and so our equation gives -0.376 = -0.301 N. Thus, N = 1.25 half-lives. That would mean that the rock is 1.62 billion years old.

The half-life of the radioactive tracer used should be roughly the same as the expected age of the rock sample. Suppose we have a lava sample that is suspected to be about 100,000 years old. Potassium-40 has a much longer half-life. So N is given by 10^{5} / 1.25 × 10^{9} = 0.000077. Using the equation above to solve for F we get F = 0.99994. The problem in this case is that very few potassium atoms will have decayed into argon atoms — only 6 out of every 100,000. Since potassium-40 is rare to start with, the measurement becomes very difficult. We should look for a more appropriate tracer with a shorter half-life.

In practice, there are many complexities in measuring the original numbers of radioactive atoms and the numbers that have decayed, but a discussion of radioactive decay shows the way in which general principles can be converted into a simple equation that covers all cases. The goal in scientific work is often to formulate a general equation that describes a phenomenon. Others can then make careful measurements, plug in the numbers, and get answers easily.