Question

An unstable nucleus A decays to an unstable nucleus \(\mathrm{B}\), which in turn decays to a stable nucleus. If at \(t=0\) s there are \(N_{A 0}\) and \(N_{B 0}\) nuclei present, derive an expression for \(N_{B}\), the number of \(B\) nuclei present, as a function of time

Short answer

Question: Derive an expression for the number of B nuclei, \(N_B\), as a function of time t, in a two-step radioactive decay process where nucleus A decays to B and then B decays to a stable nucleus. Answer: \(N_B(t) = \frac{\lambda_A N_{A0}}{\lambda_B-\lambda_A} \left(e^{-\lambda_A t} - e^{-\lambda_B t}\right) + N_{B0}e^{-\lambda_B t}\), where \(\lambda_A\) and \(\lambda_B\) are the decay constants for nuclei A and B, and \(N_{A0}\) and \(N_{B0}\) are the initial numbers of A and B nuclei, respectively.

Step-by-step solution

Write the rate equations for the decays

We will model the decay processes using the following rate equations: \[\frac{d N_A}{d t} = -\lambda_A N_A\] \[\frac{d N_B}{d t} = \lambda_A N_A - \lambda_B N_B\] The first equation models the decay of nucleus A, and the second equation models the formation of nucleus B as a product of A's decay as well as the decay of nucleus B.

Solve the equation for A

We'll solve the first rate equation to find the number of A nuclei as a function of time: \[N_A(t) = N_{A0}e^{-\lambda_A t}\] where \(N_{A0}\) is the initial number of A nuclei.

Rewrite the equation for B with A's solution

Now, we will substitute the expression for \(N_A(t)\) from the previous step into our second rate equation: \[\frac{d N_B}{d t} = \lambda_A N_{A0}e^{-\lambda_A t} - \lambda_B N_B\]

Solve B's differential equation

To find \(N_B(t)\), we need to solve the differential equation obtained in the previous step. We can rewrite the equation and integrate: \[\int\frac{d N_B}{\lambda_A N_{A0}e^{-\lambda_A t} - \lambda_B N_B} = \int dt\] Reorganize the differential equation and apply the integration factor method to solve it: \[N_B(t) = \frac{\lambda_A N_{A0}}{\lambda_B-\lambda_A} \left(e^{-\lambda_A t} - e^{-\lambda_B t}\right) + N_{B0}e^{-\lambda_B t}\] This expression gives us \(N_B(t)\), the number of B nuclei as a function of time t, with given initial conditions and decay constants.

Key concepts

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. This process transforms the original nucleus, known as the parent, into a different nucleus, referred to as the daughter.

Radioactive decay is essentially a random but statistically predictable process, governed by the laws of quantum mechanics. The rate of decay is characterized by the half-life, which is the time taken for half of a sample of radioactive material to decay.

The formula provided in the original exercise is crucial for understanding the behavior of radioactive substances. The equation \[\frac{d N_A}{d t} = -\lambda_A N_A\] represents the rate of decay of nuclei A with \(\lambda_A\) as the decay constant, which is dependent on the specific substance and type of decay.

To improve comprehension, it's important to recognize that this is not merely a mathematical equation, but it encapsulates the very nature of radioactive processes. By engaging with the mathematics, students can gain insights into the physical world, including phenomena such as radiometric dating, nuclear power generation, and even medical diagnostics.

Differential Equations

Differential equations are mathematical tools that describe the relationships involving rates of change. They are indispensable in modeling the behavior of dynamic systems in fields such as physics, engineering, biology, and economics.

The exercise presents a differential equation for nuclear decay by showing rate change expressions for the number of nuclei over time. These mathematical objects can be daunting, but with practice, they reveal the patterns of our universe. For example, when observing the equation \[\frac{d N_A}{d t} = -\lambda_A N_A\] it represents the continuous decline of nucleus A over time.

Solving Differential Equations

The solution method for a differential equation depends on its type. In our case, the equation exhibited exponential behavior, leading to the solution of \(N_A(t) = N_{A0}e^{-\lambda_A t}\), showcasing the exponential decay model.

To better understand the concept during study, it might help to progressively step through simpler differential equations before tackling complex ones like those in radioactive decay. Visualizing the decay process with graphs or simulations can also reinforce the connection between the abstract equations and the physical process they describe.

Exponential Decay

Exponential decay is a specific type of decrease that occurs at a rate proportional to the current value. It is often found in natural processes where the rate of change is directly linked to the current state, such as in radioactive decay or in the cooling of a hot object to room temperature.

In our example, the number of undecayed nuclei A decreases exponentially, as descriptive in the equation \[N_A(t) = N_{A0}e^{-\lambda_A t}\], where \(\lambda_A\) is the decay constant, \(N_{A0}\) is the initial count of nucleus A, and \(t\) is time.

Graphical Understanding

A graph of the exponential decay function reveals a curve that starts rapidly at first, but the rate of decrease slows down over time. This reflects the characteristic 'halving time', or half-life, of the substance.

To facilitate comprehension, one should focus on real-life applications such as luminescence from glow-in-the-dark objects or cooling coffee, which adhere to this mathematical pattern. Connecting such everyday experiences with the abstract concept of exponential decay can significantly enhance the understanding and appreciation of this fundamental principle in nature.