Question

In a simple case of chain radioactive decay, a parent radioactive species of nuclei, A, decays with a decay constant \(\lambda_{1}\) into a daughter radioactive species of nuclei, B, which then decays with a decay constant \(\lambda_{2}\) to a stable element C. a) Write the equations describing the number of nuclei in each of the three species as a function of time, and derive an expression for the number of daughter nuclei, \(N_{2}\), as a function of time, and for the activity of the daughter nuclei, \(A_{2},\) as a function of time. b) Discuss the results in the case when \(\lambda_{2}>\lambda_{1}\left(\lambda_{2} \approx 10 \lambda_{1}\right)\) and when \(\lambda_{2}>>\lambda_{1}\left(\lambda_{2} \approx 100 \lambda_{1}\right)\).

Short answer

Short Answer: The radioactive decay of parent nuclei A to daughter nuclei B and then to stable element C can be expressed with three equations. For A: \(N_{1}(t) = N_{1}(0)e^{-\lambda_{1}t}\), for B: \(N_{2}(t) = N_{1}(0)(\frac{\lambda_{1}}{\lambda_{2}-\lambda_{1}})(e^{(\lambda_{2}-\lambda_{1})t}-e^{-\lambda_{2}t})+N_{2}(0)e^{-\lambda_{2}t}\), and the activity of daughter nuclei B: \(A_{2}(t) = N_{1}(0)(\lambda_{1}e^{-\lambda_{1}t}-\lambda_{2}e^{-\lambda_{2}t})\). When \(\lambda_{2}>\lambda_{1}\), the decay of species B is faster than A, which affects the number and activity of daughter nuclei. As \(\lambda_{2}\) becomes much larger than \(\lambda_{1}\), the decay of species B will be extremely fast compared to A, leading to a rapid decrease in daughter nuclei's activity.

Step-by-step solution

- Write the equations of species A, B, and C as a function of time

We can start by writing the rate equations for the species A, B, and C as a function of time. These equations include the change in the number of nuclei as time passes. 1) For A: \(\frac{dN_{1}}{dt} = -\lambda_{1}N_{1}\) 2) For B: \(\frac{dN_{2}}{dt} = \lambda_{1}N_{1} - \lambda_{2}N_{2}\) 3) For C: \(\frac{dN_{3}}{dt} = \lambda_{2}N_{2}\) Now we'll move on to solve these differential equations.

- Solve the differential equation for species A

Separate the variables, integrate and rearrange to solve for \(N_{1}(t)\): \(\int_{N_{1}(0)}^{N_{1}(t)}\frac{dN_{1}}{N_{1}} = -\int_{0}^{t}\lambda_{1} dt\) \(\ln\frac{N_{1}(t)}{N_{1}(0)} = -\lambda_{1}t\) \(N_{1}(t) = N_{1}(0) e^{-\lambda_{1}t}\)

- Solve the differential equation for species B

First, we can find \(N_{1}(t)\) from the above equation and substitute it into the second differential equation: \(\frac{dN_{2}}{dt} = \lambda_{1}N_{1}(0)e^{-\lambda_{1}t} - \lambda_{2}N_{2}\) Now, we can solve this equation using the integrating factor method. Define \(y(t) = N_{2}(t)e^{\lambda_{2}t}\). Taking its derivative with respect to time, we have: \(\frac{dy}{dt} = e^{\lambda_{2}t}(-\lambda_{1}\lambda_{2}N_{1}(0)e^{-\lambda_{1}t} + \lambda_{1}N_{1}(0)e^{-\lambda_{1}t})\) Now we integrate with respect to time: \(\int_{y(0)}^{y(t)} dy = N_{1}(0)\int_{0}^{t}\lambda_{1}e^{(\lambda_{2}-\lambda_{1})t}dt\) \(y(t)-y(0) = N_{1}(0)(\frac{\lambda_{1}}{\lambda_{2}-\lambda_{1}})(e^{(\lambda_{2}-\lambda_{1})t}-1)\) Substituting back for \(y(t)\), we find: \(N_{2}(t)e^{\lambda_{2}t}-N_{2}(0) = N_{1}(0)(\frac{\lambda_{1}}{\lambda_{2}-\lambda_{1}})(e^{(\lambda_{2}-\lambda_{1})t}-1)\) Finally, we solve for \(N_{2}(t)\): \(N_{2}(t) = N_{1}(0)(\frac{\lambda_{1}}{\lambda_{2}-\lambda_{1}})(e^{(\lambda_{2}-\lambda_{1})t}-e^{-\lambda_{2}t})+N_{2}(0)e^{-\lambda_{2}t}\)

- Calculate the activity of daughter nuclei B

Now we can find the activity \(A_{2}\) by taking the derivative of \(N_{2}(t)\) with respect to time: \(A_{2}(t) = \frac{dN_{2}(t)}{dt}= N_{1}(0)(\lambda_{1}e^{-\lambda_{1}t}-\lambda_{2}e^{-\lambda_{2}t})\)

- Analyze the results for different values of \(\lambda_{2}\) and \(\lambda_{1}\)

a) When \(\lambda_{2} \approx 10\lambda_{1}\), the decay of species B is much faster than the decay of species A. In this case, the maximum number of daughter nuclei will be produced quickly, and their decay will affect the number of daughter nuclei more than the decay of parent nuclei A. The activity of daughter nuclei will also be high initially and then decrease as the number of parent nuclei decreases. b) When \(\lambda_{2} \approx 100\lambda_{1}\), the decay of species B will be extremely fast compared to species A. The daughter nuclei will quickly decay to the stable element C after being produced. The activity of daughter nuclei will be high initially but will decrease rapidly as the number of parent nuclei and daughter nuclei decreases.

Key concepts

Nuclear Physics

Nuclear physics is the field of physics that studies the properties and behavior of atomic nuclei, the central part of an atom where protons and neutrons are contained. The most important aspects of nuclear physics include the study of radioactive decay, nuclear reactions, and the energy that binds protons and neutrons together in the nucleus, known as binding energy.

Radioactive decay is one of the critical phenomena explained by nuclear physics and is the process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. This process is important in numerous applications, from medical imaging and treatment to energy generation in nuclear power plants.

Radioactive Decay Equations

Radioactive decay is a stochastic (random) process at the level of single atoms, yet it is deterministic for large numbers of atoms due to the law of large numbers. Because of this, we can describe the behavior of a radioactive sample statistically with decay equations.

These decay equations reflect how the number of particular types of atoms changes over time. The basic equation used to describe the decay of a species is known as the radioactive decay law: \( N(t) = N_0 e^{-\text{\(\backslash\)lambda} t} \), where \(N(t)\) is the number of undecayed nuclei at time t, \(N_0\) is the initial number of nuclei, and \(\text{\(\backslash\)lambda}\) is the decay constant, indicative of the probability of decay per time unit.

Daughter Nuclei Activity

In a chain radioactive decay, a parent nucleus decays into a daughter nucleus, which may itself be unstable and subject to further decay. The activity of the daughter nuclei, denoted \(A_{2}\), is defined as the rate at which they decay. It is given by the derivative of the number of daughter nuclei with respect to time, as shown in the solution provided: \( A_2(t) = \text{\(\backslash\)frac\text{\(\backslash\)slash}dN_2(t)\text{\(\backslash\)slash}dt} \).

Understanding the changes in activity over time is crucial for several fields, including radiation safety, where it's important to know how quickly a radioactive material becomes safe, and radiometric dating, where activity measurements are used to estimate the age of ancient objects.