Krypton-87 has a rate constant of \(1.5 \times 10^{-4} \mathrm{~s}^{-1}\). What is the activity of a \(2.00-\mathrm{mg}\) sample?

Radioactive decay is a fundamental concept in nuclear physics and chemistry where unstable nuclei release energy by emitting radiation in the form of alpha particles, beta particles, or gamma rays. This fascinating process can transform one element into another and occurs spontaneously and at a constant rate characterized by the half-life of the radioactive isotope.

The activity of a radioactive sample, which is the measure of how many decays occur in a certain period of time, is directly proportional to the number of undecayed nuclei present in the sample. Hence, more atoms would generally mean a higher activity since there are more nuclei available to decay at any given moment.

Avogadro's number, a cornerstone of chemistry named after the scientist Amedeo Avogadro, is the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. Its value is approximately \(6.022 \times 10^{23} \) particles per mole.

This immense number helps us bridge the gap between the microscopic world of atoms and the macroscopic world we can measure, making it possible to work out the number of atoms in any given mass of a substance. Knowing Avogadro's number allows us to understand the relationship between mass at the macro-scale and the number of atoms or molecules, which is essential when investigating radioactive materials.

The decay constant, represented by the Greek letter \(\lambda\), is specific to each radioactive isotope and reflects the probability of decay of a nucleus per unit time. It's an essential part of the natural logarithmic law of radioactive decay, where the rate of decay is proportional to the number of nuclei which have not decayed.

The decay constant is inversely related to the half-life of the isotope, meaning that a large decay constant indicates a fast-decaying, highly radioactive substance, and vice versa. Understanding the decay constant is vital when predicting how quickly a sample will lose its radioactivity.

While 'decay constant' and 'rate constant' are terms that are often used interchangeably in the context of radioactive decay, the rate constant is generally applicable to many processes in physical sciences, such as chemical reactions. In the case of radioactive decay, the rate constant refers to how quickly the radioactive substance decays.

The higher the rate constant, the faster the substance undergoes decay, often resulting in a shorter half-life. In practical applications, the rate constant enables us to calculate the activity (the number of disintegrations per second) of a radioactive sample, which is crucial for understanding the sample's potential implications for health, dating archaeological artifacts, or managing nuclear waste.