1. Describe general features of the uranium series.
2. Use the first-order rate equation to determine the amount of a radioisotope remaining after a given time.
3. Calculate the age of a rock sample given information about the amount of a particular radioisotope present in the rock.

Natural Radioactivity. A number of isotopes exist in nature that have an n : p ratio that places them outside the belt of stability. Radioisotopes that occur naturally on Earth give rise to natural radioactivity. Uranium, thorium, radon, potassium-40, carbon-12, and tritium (H-3) are naturally occurring radioisotopes. Uranium, which is fairly abundant in Earth's crust, is the start of a radioactive decay series. The series is a sequence of decay reactions that change U-238 ultimately to a stable isotope of lead. The uranium series is shown in Table 23.3 of the textbook. The uranium decay series includes all the elements between lead and uranium.

All of the radioisotopes in this series decay by alpha or beta emission. In the first step U-238 decays by alpha emission.

The decay product thorium-234 is also radioactive and decays by beta emission.

The product is also radioactive and the series continues by a number of steps to end at Pb-206.

Figure 23.2. The uranium decay series involves 14 steps as uranium decays eventually into lead.

Rates of Decay. Radioactive decay rates obey first-order kinetics.

decay rate = number of atoms disintegrating per unit time = lN

where l is the first-order rate constant, called the decay constant, and N is the number of atoms of the particular radioisotope present in the sample being studied. Recall that the half-life is related to the rate constant by

t_1/2 = 0.693/l

The integrated first-order equation is

Where N is the number of atoms of the radioisotope present in the sample after time t has elapsed, and N₀ is the number of atoms of the radioisotope present initially. If N₀, N, and t are known, then we can calculate the rate constant, l.

The object of radioactive dating is to determine the age of geological and archaeological samples and specimens. The age (t in the calculation) of certain rocks, for instance, can be estimated from analysis of the number of atoms of a particular radioisotope present now (N), as compared to the number present originally when the rock was formed (N₀).

Rearranging the above equation, the age t is given by:

The value of the initial number of atoms N₀ is the sum, N + D, where D is the number of daughter nuclei resulting from the decay of atoms of the radioisotope. The original number of atoms of a radioisotope present in a rock sample is equal to the number N remaining at time t, plus the number of daughter atoms (D).