This is consistent with the assumption that each decay event is independent and its chance does not vary over time.where is the half-life of the element, is the time expired since the sample contained the initial number atoms of the nuclide, and is the remaining amount of the nuclide.We can measure directly, for example by using a radiation detector, and obtain a good estimate of by analyzing the chemical composition of the sample.

Although the time at which any individual atom will decay cannot be forecast, the time in which any given percentage of a sample will decay can be calculated to varying degrees of accuracy.

The time that it takes for half of a sample to decay is known as the half life of the isotope.

Some isotopes have half lives longer than the present age of the universe, but they are still subject to the same laws of quantum physics and will eventually decay, even if doing so at a time when all remaining atoms in the universe are separated by astronomical distances.

Various elements are used for dating different time periods; ones with relatively short half-lives like carbon-14 (or C) are useful for dating once-living objects (since they include atmospheric carbon from when they were alive) from about ten to fifty thousand years old. Longer-lived isotopes provide dating information for much older times.

The key is to measure an isotope that has had time to decay a measurable amount, but not so much as to only leave a trace remaining.

Given isotopes are useful for dating over a range from a fraction of their half life to about four or five times their half life.Symbolically, the process of radioactive decay can be expressed by the following differential equation, where N is the quantity of decaying nuclei and k is a positive number called the exponential decay constant.The meaning of this equation is that the rate of change of the number of nuclei over time is proportional only to the number of nuclei.You find a bone fragment and through analysis you determine that it contains 13% of its original carbon-14.The half-life of carbon-14 is approximately 5,730 years. Start with the equation for continuous growth and decay: Thus the bone is approximately 17,000 years old.(Our input data had two significant figures, so reporting a more accurate result would be meaningless.) A important limitation of radiometric dating often overlooked by layman (and not always made clear in scholarly works as well) is that any date is actually a range, following the 68–95–99.7 rule.