Nucleus A decays into B with decay constant \(\lambda_{\mathrm{A}}\) and B decays into a stable product \(C\) with decay constant \(\lambda_{B}\). A pure sample starts with \(N_{0}\) nuclei \(\mathrm{A}\) at \(t=0 .\) Find an expression for the total activity of the sample at time \(t.\)

Nuclear decay, also known as radioactive decay, is a process by which an unstable atomic nucleus loses energy by emitting radiation. This can occur in several forms, such as the emission of alpha particles, beta particles, or gamma rays. During this process, one element can transform into another—a concept central to nuclear chemistry and physics.

Nuclear decay is inherently a random process at the level of single atoms, but it follows specific statistical laws when looking at a large number of atoms. This enables us to predict the behavior of a radioactive sample over time, despite the random nature of individual atomic decay events.

Understanding nuclear decay is essential for a myriad of applications, from archaeological dating methods, such as carbon dating, to medical treatments using radiopharmaceuticals, and even for energy production in nuclear reactors.

The decay constant, represented by the symbol \(\lambda\), is a crucial parameter in nuclear physics that characterizes the rate at which a radioactive substance undergoes decay. It defines the probability per unit time that a given nucleus will decay. Essentially, this constant tells us how 'quickly' or 'slowly' an isotope will decay.

The value of the decay constant is unique for each radioactive isotope and enables the calculation of an important value known as the half-life—the time required for half of the radioactive nuclei in a sample to decay. The higher the decay constant, the shorter the half-life, meaning the substance will become non-radioactive more quickly.

By using the decay constant, we can create formulas to calculate not only the remaining number of nuclei over time but also the activity of a radioactive sample, which is directly proportional to the number of undecayed nuclei and the decay constant itself.

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. When dealing with radioactive decay, differential equations are used to model the rate at which certain processes happen—such as the rate of decay of a radioactive material.

In the context of our original exercise, we use the differential equation \(\frac{dN_B}{dt} = \lambda_A N_A - \lambda_B N_B\) to explain how the number of type-B nuclei changes over time. This rate depends on the decay of A to B (creating more B nuclei) and the decay of B into some other product C (reducing the amount of B nuclei).

Solving differential equations like this can be complex, but the solutions are powerful tools that allow us to predict how the system will evolve over time. They are foundational in fields beyond nuclear physics, including engineering, economics, and biology.

Exponential decay is a specific type of decrease that occurs at a rate proportional to the value at that instant. It is a common pattern observed not just in radioactive decay, but in other natural phenomena as well, such as population decline and cooling of hot objects.

In radioactive decay, the number of undecayed nuclei decreases exponentially over time, a relationship that can be described by the formula \(N(t) = N_0 \cdot e^{-\lambda t}\), where \(N(t)\) is the quantity at time \(t\), \(N_0\) is the initial quantity, \(\lambda\) is the decay constant, and \(e\) is the base of the natural logarithm.

This exponential nature stems from the fact that each nucleus has a fixed probability of decaying per unit time, independent of time, and irrespective of the presence or absence of other nuclei. Therefore, the rate of decay is always proportional to the number of nuclei that have not yet decayed.