Anthropologists can estimate the age of a bone or other sample of organic matter by its carbon- 14 content. The carbon- 14 in a living organism is constant until the organism dies, after which carbon-14 decays with first- order kinetics and a half-life of 5730 years. Suppose a bone from an ancient human contains \(19.5 \%\) of the C-14 found in living organisms. How old is the bone?

At the heart of carbon-14 dating is the principle of radioactive isotope decay. Radioactive isotopes, such as carbon-14, are unstable forms of elements that release energy by emitting radiation, transforming into a different element over time. This decay occurs predictably and at a fixed rate, characterized by what we call a half-life. For instance, carbon-14, a key player in archeological dating, transforms into nitrogen-14 as it decays.

It's important to note that once an organism dies, it no longer exchanges carbon with its environment, making the carbon-14 it contains at the time of death a closed system. Over time, the carbon-14 decays while the carbon-12, a stable isotope, remains unchanged. This predictable decay rate of carbon-14 allows scientists to calculate the age of formerly living materials by measuring the remaining carbon-14.

The half-life of a radioactive isotope is the time it takes for half of the atoms in a sample to decay. It's a pivotal concept in understanding radioactive decay as it directly influences age determination of an archeological sample. The half-life remains constant regardless of the initial amount of the isotope or the size of the sample.

For carbon-14, the half-life is approximately 5730 years. When we know this value, we can estimate the time that has elapsed since the death of the organism by measuring how much carbon-14 remains in comparison to the stable carbon-12. With each passing half-life, the amount of carbon-14 is reduced by half, forming a time marker for scientists to use in their calculations.

First-order kinetics is a term that describes a scenario where the rate of reaction is directly proportional to the concentration of a single reactant. In the context of radioactive decay, and specifically carbon-14 dating, this means that the rate at which carbon-14 decays is proportional to the amount of carbon-14 present in the sample at any given time.

Using mathematical language, the first-order decay formula is written as \( N(t) = N_0 (1/2)^{t/T} \), where \( N(t) \) is the amount of the substance at time 't', \( N_0 \) is the original amount, 't' is the elapsed time, and 'T' is the half-life of the substance. This relationship allows us to predict how much of the radioisotope will remain after a certain period has passed, which is instrumental in calculating the age of archeological samples.

Carbon-14 dating is a powerful tool used in archaeology to determine the age of organic artifacts. It relies on the principles of radioactive isotope decay and first-order kinetics as discussed earlier. The process begins by measuring the ratio of carbon-14 to carbon-12 in the artifact. By comparing this ratio to the expected ratio in a living organism, scientists can determine how many half-lives have passed since the death of the organism.

In the provided exercise, with a residual carbon-14 content of 19.5%, we can apply the decay formula and calculate that \( t = 5730 \times \frac{\ln(0.195)}{\ln(1/2)} \) to find the time elapsed since the organism's death. Using these calculations, the age of organic materials, such as bone, can be estimated with a fair degree of accuracy. Carbon-14 dating thus serves as a window into the past, helping us to piece together the timeline of human history and prehistory.