A gram of radioactive material may contain more than 1021 atoms. Imagine that during the time period equivalent to its half-life, each atom flips a coin exactly once to decide whether or not it decays. Because we are dealing with so many atoms, there are equally as many coin flips, and probability theory says that half of the time, the coin will turn up "heads", and half "tails." Imagine that if the coin turns up "heads," the atom does not decay, but it does decay if it's "tails." Then after one half-life, exactly half of the atoms, having found their coins to end up "tails," will decay. Similarly after two half-lives, only 1/4 of the original atoms are left; after three half-lives, only 1/8 are left. In general, after n half-lives, (½)n survive.
Let's say we are going to measure the radioactivity of a certain substance (which you will do in this lab). First we measure the number of atoms of that substance. Let's define time so that it begins when we make the first measurement. Then at the beginning of our experiment, t = 0, we measure N(0) to be the number of atoms. The law of radioactive decay, formulated by Rutherford, states that the number of radioactive atoms N(t) observed at time t will decline in comparison with the number observed at the beginning of the experiment N(0) as a decaying exponential in time.
where t½ = te • ln(2) = 0.693 te
and where te is the time corresponding to a 1/e decrease in the number of atoms (1/te = l is called the decay constant)
We call t½ the half-life of the radioactive species. The half-life is the time period in which probability theory predicts that half of the original atoms of that substance will decay. The half-life of a single atom is the time period in which the probability that it will decay is one-half. Each radioactive substance has a characteristic half-life. Typical half-lives range from billions of years for geologically-important uranium, thorium and potassium, to small fractions of a second.
Atoms that have unstable nuclei have very specific patterns and rates of decay. Suppose we could conduct an experiment with a sample of Uranium-238 (238U) containing 8 billion atoms; we would have to let the experiment run for a very long time, but if we could observe this sample over a 4.5 billion year period and then count the number of 238U atoms, we would find that only 4 billion remain. The other 4 billion would have decayed by ejecting some nuclear constituents into some other atoms. If we waited another 4.5 billion years, only 2 billion uranium-238 atoms would be left.
Fill in the table 3.1 (decay of 238U ) on your lab sheet now.
The resulting situation in which we are left with 1/2, 1/4, 1/8, etc. of the original number of atoms after equal intervals of time is known as a geometric decrease. When the points for intermediate time intervals are filled in (if we were to make measurements at extremely short time intervals), we would find an exponential decay. This is characteristic of the situation in which the rate of decrease of some quantity is proportional to that quantity-a common occurrence in science. The opposite condition is called exponential growth (when the rate of increase of a quantity is proportional to that quantity) is also familiar-as in the growth of bacteria or of a bank account. When scientists wish to test data for agreement with an exponential law, they make use of a semi-log plot; graphing, in this case, the logarithm of the number of atoms (essentially our exponent n) with respect to time t. This type of plot converts an exponential curve into a straight line.
The JAVA applet running below should help you become more familiar with the idea of radioactive decay and the use of semi-log plots for graphic display. A test sample is visible in teh upper right. You can adjust the number of atoms in the sample and the half-life of the sample by moving the slides. The plot on the left is a linear, and the one on the right is a semi-log plot. Start the decay by pushing Decay.
Sketch the resulting plots on your lab sheet (3.2).
The key to the use of radioactive decay measurements for the aging of rocks is that each radioactive isotope has its own characteristic decay products. By observing the relative amounts of parent atoms and those produced by decay we can establish an estimate for when the decay began: when the rock was formed.
In our experiment, we will deal with much smaller numbers of radioactive atoms and we will have to take into account real experimental deviations from this idealized case. Be sure that you understand the concepts of half-life, exponential decay, and probability before you attempt the second part of the lab.
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