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Sections 30.7 - 30.11

Making a precise prediction of when an individual nucleus will decay is not possible; however, radioactive decay is governed by statistics, so it is very easy to predict the decay pattern of a large number of radioactive nuclei. The rate at which nuclei decay is proportional to N, the number of nuclei there are:

Whenever the rate at which something occurs is proportional to the number of objects, the number of objects will follow an exponential decay. In other words, the equation telling you how many objects there are at a particular time looks like this:

The decay constant is closely related to the half-life, which is the time it takes for half of the material to decay. Using the radioactive decay equation, it's easy to show that the half-life and the decay constant are related by:

The activity of a sample of radioactive material (i.e., a bunch of unstable nuclei) is measured in disintegrations per second, the SI unit for this being the becquerel.

Understanding half-life using M&M's

Please note that M&M's are perfectly safe, and are not radioactive. M&M's can be used as a model of a sample of radioactive nuclei, however, because when they lie on a flat surface they can be in one of just two states - they can lie with the M up or with the M down. Let one of those states (M down, say) represent decayed nuclei.

With a package of M&M's, you can model a sample of decaying nuclei like this:

• step 1 - count the number of M&M's you have.
• step 2 - throw them onto a flat surface, and count the number of M&M's with M up. Remove all the M down M&M's from the sample.
• step 3 - repeat step 2 until you have no M&M's left.

Every time you throw the M&M's, you've gone through one more half-life. Here's the data from one trial:

The above is just a single trial; you should try it yourself to see what you get. This trial shows something interesting, however. When you have a large number of particles, they follow the predicted behavior very closely. When you only have a small number, the inherent randomness of the decay process is a little more obvious.