Question

In a simple case of chain radioactive decay, a parent radioactive nucleus, A, decays with a decay constant \(\lambda_{1}\) into a daughter radioactive nucleus, \(\mathrm{B}\), which then decays with a decay constant \(\lambda_{2}\) to a stable nucleus, \(\mathrm{C}\). a) Write the equations describing the number of nuclei of each of the three types as a function of time, and derive expressions 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

Question: Write the equations for the number of each type of nuclei as a function of time and derive expressions for the number of daughter nuclei, N2, and the activity of the daughter nuclei, A2, as a function of time. Discuss the results for two cases: λ2 > λ1 (λ2 ≈ 10 λ1) and λ2 >> λ1 (λ2 ≈ 100 λ1). Answer: The number of each type of nuclei as a function of time is given by: N1(t) = N1(0) * e^{-λ1 * t} N2(t) = (λ1 * N1(0) * e^{-λ1 * t} * e^{λ2 * t}) / (λ2 - λ1) The activity of the daughter nuclei, A2(t), is given by: A2(t) = (λ1 * λ2 * N1(0) * e^{-λ1 * t}) / (λ2 - λ1) - (λ1 * λ2 * N1(0) * e^{-λ2 * t}) / (λ2 - λ1) For the two cases: 1. When λ2 > λ1 (λ2 ≈ 10 λ1): The number of daughter nuclei N2(t) will grow faster initially and slowly approach a steady state before decaying to nucleus C. 2. When λ2 >> λ1 (λ2 ≈ 100 λ1): The number of daughter nuclei N2(t) will grow quickly initially and then rapidly decay to the stable nucleus C. The transient growth and decay of nucleus B would be much faster compared to the first case.

Step-by-step solution

Write equations for the number of each type of nuclei as a function of time.

We start by writing the differential equations for the number of each type of nucleus. Let N1(t), N2(t), and N3(t) be the number of nuclei A, B, and C, respectively, as a function of time. Then we have the following differential equations: dN1/dt = -λ1 * N1(t) dN2/dt = λ1 * N1(t) - λ2 * N2(t) Now, let's solve these differential equations to get the number of each type of nuclei as a function of time.

Solve differential equations for N1(t) and N2(t).

To solve the first differential equation, we can integrate and obtain: N1(t) = N1(0) * e^{-λ1 * t} To solve the second differential equation, we use an integrating factor, μ(t) = e^{λ2 * t}. Then we rewrite the equation as: d(N2(t) * e^{λ2 * t})/dt = λ1 * N1(t) * e^{λ2 * t} Now, we can integrate both sides to get the solution: N2(t) = (λ1 * N1(0) * e^{-λ1 * t} * e^{λ2 * t}) / (λ2 - λ1)

Derive expressions for the number of daughter nuclei N2(t) and the activity of the daughter nuclei A2(t).

We have derived N2(t) in the previous step. Now we need to find A2(t), the activity of the daughter nuclei, which is the derivative of N2(t) concerning time: A2(t) = dN2/dt = (λ1 * λ2 * N1(0) * e^{-λ1 * t}) / (λ2 - λ1) - (λ1 * λ2 * N1(0) * e^{-λ2 * t}) / (λ2 - λ1)

Discuss the results for the two cases.

Now, we analyze the results for the two given cases: λ2 > λ1 (λ2 ≈ 10 λ1) and λ2 >> λ1 (λ2 ≈ 100 λ1). Case 1: λ2 > λ1 (λ2 ≈ 10 λ1) In this case, the ratio λ2/λ1 is not very large, so the daughter nuclei B decay relatively slower than the parent nuclei A. The number of daughter nuclei N2(t) will grow faster initially, due to the greater decay of nucleus A, and then it will slowly approach a steady state before decaying to nucleus C. Case 2: λ2 >> λ1 (λ2 ≈ 100 λ1) In this case, the ratio λ2/λ1 is much larger, meaning the daughter nuclei B decay much faster relative to the parent nuclei A. The number of daughter nuclei N2(t) will grow quickly initially and then rapidly decay to the stable nucleus C. The transient growth and decay of nucleus B would be much faster compared to the first case.

Key concepts

Decay Constant

The decay constant plays a crucial role in understanding the behavior of a radioactive substance. It is denoted by the symbol \( \lambda \) and essentially tells us how likely a given radioactive nucleus is to decay in a certain time interval. The probability that a nucleus will decay is proportional to \( \lambda \) and the time interval considered.

Mathematically, if we start with a certain number of unstable nuclei, \( N_0 \), the number remaining, \( N(t) \), after time \( t \) is given by the exponentiation of the negative decay constant times the time, \( N(t) = N_0 e^{-\lambda t} \). In the context of chain radioactive decay, the parents' and daughters' decay constants, \( \lambda_1 \) and \( \lambda_2 \), respectively, allow us to predict how these populations change over time and how swiftly one type of nucleus transitions to another.

Differential Equations in Physics

Differential equations are fundamental to physics as they describe how physical quantities evolve over time or space. These equations involve derivatives, which represent rates of change. In our case, differential equations model the rate at which radioactive nuclei decay.

To solve these equations, we use methods like separation of variables or integrating factors, allowing us to find the function that describes our system's behavior over time. Specifically, for radioactive decay, the solution to the differential equation involves calculating the number of nuclei remaining as a function of time, which then helps determine other related quantities like the activity of the substance.

Radioactive Nucleus

A radioactive nucleus is an unstable atom that can release energy by emitting radiation as it transforms into a more stable atom. The stability of a nucleus is determined by the balance between the protons and neutrons it contains. Too many or too few of either can lead to instability, resulting in radioactive decay.

Understanding the nature of a radioactive nucleus is vital in solving physics problems concerning nuclear reactions. Knowing the type of decay and the decay constant \( \lambda \) helps us predict how long a sample of radioactive nuclei will remain active or how quickly it will produce daughter nuclei through decay.

Activity of Radioactive Substances

The activity of a radioactive substance is a measure of its decay rate, specifically the number of disintegrations per second. It is denoted by the symbol \( A \) and its units are becquerel (Bq) in the International System of Units (SI). Activity is directly proportional to both the decay constant and the number of undecayed nuclei present at any given time.

Mathematically, activity can be described as \( A(t) = \lambda N(t) \), where \( N(t) \) is the number of nuclei and \( \lambda \) is the decay constant. For chain decay scenarios, where a parent nucleus decays into a daughter nucleus that also decays, calculating the activity involves solving the system of differential equations that describes the number of each type of nucleus over time.