Define the term half-life and illustrate it with an example.

Radioactive decay is a foundational concept in nuclear physics and chemistry, describing the process by which an unstable atomic nucleus loses energy by emitting radiation. This spontaneous transformation results in the nucleus changing to a different state or element. One aspect that often confuses students is differentiating between the types of radiation emitted, such as alpha, beta, and gamma radiation. Each type has different properties and implications for stability and safety.

For example, alpha particles can be stopped by a sheet of paper, whereas gamma rays require lead or several inches of concrete to be halted. Understanding the characteristics of these particles is crucial in fields ranging from medical treatments to energy generation. It's also important to recognize that while the term 'decay' might imply that something is 'rotting away', in physics, it merely means that the nucleus is transitioning to a more stable configuration. This subtle yet significant understanding helps clarify the nature of the process.

Exponential decay is a term used to describe a process where the decrease in a quantity is proportional to its current value, leading to a rapid decline initially, which slows over time. It's not just in radioactive materials; exponential decay can occur in populations, bank accounts, and even in learning retention rates over time.

Mathematically, exponential decay is modeled by the formula \( N(t) = N_0 e^{-\frac{t}{\tau}} \), where \( N(t) \) is the amount of substance at time \( t \), \( N_0 \) is the initial amount, \( \tau \) is the mean lifetime, and \( e \) is Euler's number, approximately equal to 2.71828. This equation provides a powerful tool to predict how quickly a substance will decline. For instance, with medical isotopes used in diagnostic imaging, knowing the rate of decay is essential for calculating correct dosages and ensuring patient safety.

Carbon-14 dating, also known as radiocarbon dating, is an invaluable tool in archaeology and geology for determining the age of organic materials. It relies on the principle of radioactive decay, specifically the half-life of Carbon-14, which is approximately 5730 years.

By measuring the ratio of Carbon-14 to Carbon-12 in a sample and comparing it to the ratio in the atmosphere, scientists can determine when the organism was last taking in carbon — essentially, when it was still alive. This technique has been pivotal in piecing together historical timelines and understanding changes in the environment. While Carbon-14 dating is extremely useful, it isn't without limitations. For example, it's less effective for samples over 50,000 years old, as the remaining Carbon-14 levels are so low that they're difficult to measure accurately. Additionally, factors like contamination can affect the results, which is why precise laboratory techniques are crucial for accurate age estimation.