One of the interesting uses for half-life calculations involves radiocarbon dating, where the content of carbon-14 in organic (formally living matter) is used to calculate the age of a sample. The process begins in the upper atmosphere, where nitrogen is bombarded constantly by high-energy neutrons from the sun. Occasionally, one of these neutrons collides with a nitrogen nucleus and the isotope that is formed undergoes the following nuclear equation: $$ { }_{0}^{1} n+{ }_{7}^{14} N \rightarrow{ }_{1}^{1} \rho+{ }_{6}^{14} C $$ Plants take up atmospheric carbon dioxide by photosynthesis, and are ingested by animals, so every living thing is constantly exchanging carbon-14 with its environment as long as it lives. Once it dies, however, this exchange stops, and the amount of carbon-14 gradually decreases through radioactive decay with a half-life of about 5,730 years, following the nuclear equation shown below: $$ { }_{6}^{14} C \rightarrow_{-1}^{0} \beta+{ }_{7}^{14} N $$ Thus, by measuring the carbon-14/carbon-12 ratio in a sample and comparing it to the ratio observed in living things, the number of half-lives that have passed since new carbon-14 was absorbed by the object can be calculated.

Step by step solution

01

Understanding the problem

The problem describes the process of radiocarbon dating, which relies on understanding the decay of carbon-14 to nitrogen-14. The decay occurs with a half-life of 5,730 years. The ratio of carbon-14 to carbon-12 in a sample compared to that in living matter indicates the number of half-lives since the organism has died.

02

Defining Half-Life

The half-life of a radioactive isotope is the time required for half of the radioactive atoms in a sample to decay. For carbon-14, this time is approximately 5,730 years.

03

Using Half-Life in Radiocarbon Dating

The amount of carbon-14 in a dead organism decreases over time as it decays to nitrogen-14. By comparing the remaining carbon-14 in a sample with that of a living organism, the time since the organism's death can be estimated by calculating the number of half-lives that have elapsed.

04

Calculating the Age of a Sample

To calculate the age of a sample, the following steps are used: 1. Measure the current carbon-14/carbon-12 ratio in the sample. 2. Divide that by the ratio in a living organism to find the fraction remaining. 3. Calculate the number of half-lives that corresponds to this fraction. 4. Finally, multiply the number of half-lives by the half-life duration (5,730 years) to find the age of the sample.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-Life Calculations

When studying radioactive materials, one of the key concepts to grasp is the half-life calculation. The half-life refers to the amount of time it takes for half of the radioactive isotopes in a given sample to decay. This consistent rate of decay allows scientists to make predictions about the age of substances. For radiocarbon dating, carbon-14, a radioactive isotope of carbon, is the focus. Its half-life is approximately 5,730 years.

In practical examples, if we start with 1 gram of carbon-14, after 5,730 years, only 0.5 grams would remain unchanged; the rest would have decayed to nitrogen-14. After another 5,730 years, half of the remaining 0.5 grams would decay, leaving 0.25 grams, and so on. By measuring the amount of carbon-14 currently in a sample and applying half-life calculations, we can estimate how many half-lives have passed, and consequently, determine the age of an archaeological find or geological sample.

Carbon-14 Decay

Carbon-14 decay is a specific case of nuclear decay that informs the radiocarbon dating technique. It starts at a predictable rate once an organism dies and stops exchanging carbon with its environment. Over time, the carbon-14 isotopes within the organism undergo beta decay, transforming into stable nitrogen-14 isotopes. The equation for this process is:
\[ {}_{6}^{14} C \rightarrow{}_{-1}^{0} \beta + {}_{7}^{14} N \]
This equation illustrates a neutron in the carbon nucleus decaying into a proton, emitting an electron (beta particle) in the process. The atomic number increases by one, changing the element from carbon to nitrogen, while the mass number remains constant. Understanding this decay process allows archaeologists and geologists to trace the age of carbon-based remains by calculating how much carbon-14 has been lost over time. This calculation is based on comparing the amount of carbon-14 to carbon-12—a stable isotope of carbon that remains constant over time.

Nuclear Equations

Nuclear equations are symbolic representations of nuclear reactions, showing the transformation from reactants to products while conserving nucleons and charges. To this end, the sum of the mass numbers (top numbers) and the sum of the atomic numbers (bottom numbers) on both sides of the equation must be equal.

For instance, the formation of carbon-14 in the atmosphere is represented by the equation:
\[ {}_{0}^{1} n + {}_{7}^{14} N \rightarrow {}_{1}^{1} p + {}_{6}^{14} C \]
This equation demonstrates a neutron () striking a nitrogen-14 atom and causing it to lose a proton (p), thus transforming it into carbon-14. Having a balanced nuclear equation is crucial to understanding the details and implications of the reaction, such as the formation of carbon-14, which is the foundation for radiocarbon dating. In nuclear equations for decay processes, like carbon-14 decay, it's also key to acknowledge the release of other particles like beta particles or gamma rays, integral components of radioactive decay.