A sample of \(\mathrm{F}-18\) has an initial decay rate of \(1.5 \times 10^{5} / \mathrm{s}\). How long will it take for the decay rate to fall to \(2.5 \times 10^{3} / \mathrm{s}\) ? (F-18 has a half-life of 1.83 hours.)

Radioactive decay is a fundamental process by which an unstable atomic nucleus loses energy by emitting particles or radiation. This process is spontaneous and occurs in unstable isotopes, which are variants of elements with the same number of protons but a different number of neutrons.

These isotopes, often referred to as radioisotopes, undergo decay to become more stable, which can result in the creation of a different element altogether. During this process, there are several types of decay that can occur, such as alpha decay, beta decay, and gamma decay, each named after the type of radiation emitted.

The rate of decay is unique for each radioisotope and is expressed in terms of half-life, which is the time required for half of the radioactive atoms to decay. Understanding the concept of half-life is crucial, as it not only gives insight into the stability of the nucleus but also has practical applications in fields like medicine, archaeology, and nuclear power.

The decay constant, represented by the symbol \( \lambda \), is a probability that defines the rate at which a particle will decay per unit time. It is a key factor in characterizing the radioactive properties of an isotope and is inherently related to its half-life.

The decay constant can be mathematically determined using the half-life \( t_{1/2} \) of the substance by the formula \( \lambda = \frac{\ln(2)}{t_{1/2}} \). This expression makes use of the natural logarithm of two, as the half-life is the time required for the quantity of the substance to reduce to half its initial value.

Knowing the decay constant allows us to compute the rate of decay at any given moment, which is vital for applications like calculating the required dosage of a radioactive tracer in medical imaging or determining the age of archaeological artifacts through radiocarbon dating.

Exponential decay describes the process by which the quantity of a radioactive substance decreases at a rate proportional to its current value. This behavior is governed by the exponential decay law, which frames how substances like radioisotopes lose their matter over time.

The mathematical formula for exponential decay is \( N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \), where \( N(t) \) represents the remaining quantity of the radioactive substance after time \( t \), \( N_0 \) is the initial quantity, and \( t_{1/2} \) is the half-life of the substance. The nature of exponential decay means that the substance diminishes rapidly at first, but the rate of loss slows down as time progresses.

This formula is crucial in predicting the future behavior of a radioactive sample and is used extensively in fields ranging from nuclear physics to environmental science. When solving problems related to exponential decay, students should carefully work through each step, ensuring to correctly apply the decay constant and the half-life to find the time it takes for a substance to decay to a specific level, as demonstrated in the original exercise.