Half life dating equation
The most common method for measuring the age of ancient objects is carbon-14 dating.
The carbon-14 isotope, created continuously in the upper regions of Earth’s atmosphere, reacts with atmospheric oxygen or ozone to form .
For a given number of atoms, isotopes with shorter half-lives decay more rapidly, undergoing a greater number of radioactive decays per unit time than do isotopes with longer half-lives.
The half-lives of several isotopes are listed in In our earlier discussion, we used the half-life of a first-order reaction to calculate how long the reaction had been occurring.
But the way we think about half-life is, people have studied carbon and they said, look, if I start off with 10 grams-- if I have just a block of carbon that's 10 grams. Those five grams of carbon-14, every one of those atoms still has, over the next-- whatever that number was, 5,740 years-- after 5,740 years, all of those once again have a 50% chance. Well, after one billion years I'll say, well you know, it'll probably have turned into nitrogen-14 at that point, but I'm not sure. You don't know how well it calibrates against time. Any animal that eats a plant ingests a mixture of organic compounds that contains approximately the same proportions of carbon isotopes as those in the atmosphere.When the animal or plant dies, the carbon-14 nuclei in its tissues decay to nitrogen-14 nuclei by a radioactive process known as beta decay, which releases low-energy electrons (β particles) that can be detected and measured: \[ \ce \label\] The half-life for this reaction is 5700 ± 30 yr. Comparing the disintegrations per minute per gram of carbon from an archaeological sample with those from a recently living sample enables scientists to estimate the age of the artifact, as illustrated in Example 11.In fact, radioactive decay is a first-order process and can be described in terms of either the differential rate law () or the integrated rate law: \[N = N_0e^ \] \[\ln \dfrac=-kt \label\] Because radioactive decay is a first-order process, the time required for half of the nuclei in any sample of a radioactive isotope to decay is a constant, called the half-life of the isotope.
The half-life tells us how radioactive an isotope is (the number of decays per unit time); thus it is the most commonly cited property of any radioisotope.
This becomes evident when we rearrange the integrated rate law for a first-order reaction (Equation 14.21) to produce the following equation: Figure \(\Page Index\): The Half-Life of a First-Order Reaction.