Suppose you a have a 100,000-atom sample of a radioactive nuclide that decays with a half-life of 2.0 days. How many radioactive atoms are left after 10 days?

Understanding the half-life of a radioactive substance is crucial for grasping the fundamentals of radioactive decay. It answers the question: How long will it take for half of the radioactive nuclides in a sample to transform into something else? The mathematical formula that describes this process is:
\( N = N_0 \times (\frac{1}{2})^{\frac{t}{T}} \)
where:

• \( N \) is the number of remaining radioactive atoms.
• \( N_0 \) is the original number of radioactive atoms.
• \( t \) is the time that has elapsed.
• \( T \) is the half-life of the substance.

This formula is derived from the concept that, after each half-life period, half of the remaining radioactive atoms will have decayed. As a result, with each successive half-life, the number of remaining atoms is halved.

Radioactive nuclide decay, or simply radioactive decay, refers to the process by which an unstable atomic nucleus loses energy by emitting radiation. There are several types of radioactive decay, including alpha decay, beta decay, and gamma decay, each involving the release of different particles or energy waves.

The decay process follows an exponential decay pattern, where the rate of decay is proportional to the number of undecayed atoms. Therefore, rather than decaying at a constant number per unit time, the substance decays at a rate proportional to its current quantity, leading to the half-life phenomenon where the quantity of radioactive material decreases by half over a set period of time.

This is important in real-world applications such as carbon dating, medical diagnostics and treatments, and nuclear power generation, which all rely on an understanding of half-life and decay patterns.

Radioactivity calculations allow us to predict and understand the behavior of a sample of radioactive material over time. These calculations often involve using the half-life formula to determine either the remaining amount of a substance after a given period, the elapsed time given an initial and final quantity, or the initial quantity of a substance from a known half-life and elapsed time.

In the given problem, to find out how many radioactive atoms are left after 10 days, we calculated the number of elapsed half-lives and applied the half-life formula. Essential steps in these calculations include:

• Identifying the half-life (\( T \)) of the radioactive material.
• Measuring or stating the initial quantity of material (\( N_0 \)).
• Determining the time elapsed (\( t \)) for which we want to calculate the remaining quantity.

By mastering radioactivity calculations, one gains the ability to accurately discuss and analyze situations involving radioactive materials, which is valuable in fields such as physics, chemistry, environmental science, and particularly in healthcare, where radioisotopes play a role in diagnosis and treatment.