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.
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:
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.
Radioactivity is often used in determining how old something is; this is known as radioactive dating. When carbon-14 is used, as is often the case, the process is called radiocarbon dating, but radioactive dating can involve other radioactive nuclei. The trick is to use an appropriate half-life; for best results, the half-life should be on the order of, or somewhat smaller than, the age of the object.
Carbon-14 is used because all living things take up carbon from the atmosphere, so the proportion of carbon-14 in the carbon in a living organism is the same as the proportion in the carbon-14 in the carbon in the atmosphere. For many thousands of years this proportion has been about 1 atom of C-14 for every 8.3 x 10^11 atoms of carbon.
When an organism dies the carbon-14 slowly decays, so the proportion of C-14 is reduced over time. Carbon-14 has a half life of 5730 years, making it very useful for measuring ages of objects that are a few thousand to several tens of thousands of years old. To measure the age of something, then, you measure the activity of carbon-14, and compare it to the activity you'd expect it to have if it was brand new. Plugging these numbers into the decay equation along with the half-life, you can calculate the time period over which the nuclei decayed, which is the age of the object.
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