# Radioactive-Decay Model

Throw one hundred coins, remove all those that come up tails, place them in a pile, repeat—you've got yourself a hands-on model for radioactive decay. The piles graphically show the meaning of the term “half-life.”

**COVID-19 Learning Note:** Half-life is also the way that scientists describe how long the coronavirus can last on different surfaces. Since the virus cannot reproduce outside of the body, it will slowly inactivate over time, with the exact amount of time depending on the surface it’s on. Scientists recently found that the half-life of the novel coronavirus on cardboard is around four hours. That means around half of the viruses present are inactivated after four hours, and another half four hours after that. After 24 hours, over 98% of the virus should be inactivated.

None needed.

Toss all the pennies onto a table surface. Remove the pennies that land tail side up and put them flat on the left side of the table, arranged in a tall column.

Gather up the remaining pennies and toss them again. Remove the pennies that land tail side up, and arrange them in a second column, right beside the first column. Repeat this experiment until all of the pennies have been removed. If no pennies come up tails on a toss, leave an empty column.

You can do the same thing with the wooden cubes, if you have them, removing the cubes that land red side up.

The chance that any penny will come up tails on any toss is always the same, 50 percent. However, once a penny has come up tails, it is removed. Thus, about half the pennies are left after the first toss.

Even though half of the remaining pennies come up tails on the second toss, there are fewer pennies to start with. After the first toss, about 1/2 of the original pennies are left; after the second, about 1/4; then 1/8, 1/16, and so on. These numbers can be written in terms of powers, or exponents, of 1/2: (1/2)^{1}, (1/2)^{2}, (1/2)^{3}, and (1/2)^{4}. This type of pattern—in which a quantity repeatedly decreases by a fixed fraction (in this case, 1/2)—is known as exponential decay (click to enlarge photo below).

Each time you toss the remaining pennies, about half of them are removed. The time it takes for half of the remaining pennies to be removed is called the half-life. The half-life of the pennies in this model is about one toss.

If you’re using painted wooden cubes, the probability that a cube will land red side up is 1/6. (Each cube has six sides, and only one of those sides is painted red.) It takes three tosses for about half the cubes to be removed, so the half-life of the cubes is about three tosses. After one toss, 5/6 remain; after two tosses, 5/6 of 5/6, or 25/36, remain; and after three tosses, (5/6)^{3} = 125/216 of the cubes are left.

Tossing the coins or cubes is an unpredictable, random process. Rarely will exactly 1/2 of the coins or 1/6 of the cubes decay on the first toss. However, if you repeat the first toss many, many times, the average number of coins that decay will approach 1/2 (or cubes that decay will approach 1/6).

In this model, the removal of a penny or a cube corresponds to the decay of a radioactive nucleus. The chance that a particular radioactive nucleus in a sample of identical nuclei will decay in each second is the same for each second that passes, just as the chance that a penny would come up tails was the same for each toss (1/2) or the chance that a cube would come up red was the same for each toss (1/6).

The smaller the chance of decay, the longer the half-life (time for half of the sample to decay) of the particular radioactive isotope. The cubes, for instance, have a longer half-life than the pennies. For uranium 238, the chance of decay is small: Its half-life is 4.5 billion years. For radon 217, the chance of decay is large: Its half-life is one thousandth of a second.

Some radioactive nuclei, called *mothers*, decay into other radioactive nuclei, called *daughters*. To simulate this process, start with 100 nickels. Toss them and replace each nickel that lands tail side up with a penny. Toss the pennies and the rest of the nickels together. Make a column with all the pennies that land tail side up, and replace all the nickels that land tail side up with more pennies. The nickels represent the mother nuclei; the pennies, the daughter nuclei. Repeat the experiment until all the coins are gone, if a toss results in no "tails" then leave the column empty." Notice that the columns of decayed pennies grow at first and then decay.