Half-life of $C^{14}$ is around _____ years.

5700

The correct answer is option 2. 5700.

The half-life of a radioactive isotope is the time it takes for half of the atoms in a sample of that isotope to undergo radioactive decay. In the case of carbon-14 (\(^{14}\text{C}\)), it is a radioactive isotope of carbon that is naturally present in the Earth's atmosphere. Here's how the half-life of carbon-14 works:

1. Carbon-14 is continually produced in the Earth's atmosphere by cosmic rays, which interact with nitrogen-14 (\(^{14}\text{N}\)) atoms to form carbon-14 (\(^{14}\text{C}\)) atoms.

2. Once produced, carbon-14 atoms react with oxygen in the atmosphere to form carbon dioxide (\(\text{CO}_2\)), which then gets incorporated into plants during photosynthesis.

3. Animals, including humans, consume plants (or other animals that have consumed plants), incorporating carbon-14 into their tissues.

4. Carbon-14 undergoes radioactive decay by emitting a beta particle (an electron) and transforming into a stable nitrogen-14 atom. This process occurs randomly for each individual carbon-14 atom in a sample.

5. The half-life of carbon-14 is approximately 5700 years. This means that if you start with a certain amount of carbon-14 atoms, after 5700 years, half of those atoms will have decayed into nitrogen-14 atoms.

6. After another 5700 years (a total of 11,400 years), half of the remaining carbon-14 atoms will have decayed, leaving only one-quarter of the original amount. This pattern continues for subsequent half-lives.

The significance of carbon-14's half-life lies in its application to radiocarbon dating. By measuring the ratio of carbon-14 to stable carbon-12 (\(^{12}\text{C}\)) in a sample, scientists can determine how long ago the organism from which the sample originated died. This technique is widely used in archaeology, anthropology, and other fields to date organic materials such as bones, wood, and textiles.