Question

When the density of a species \(A\) in a semi-infinite medium is known at the beginning and at the surface, explain how you would determine the concentration of the species \(A\) at a specified location and time.

Short answer

Answer: To determine the concentration of species A at a specified location and time, follow these steps: 1. Understand the diffusion equation (Fick's second law): \(\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}\). 2. Identify the given initial and boundary conditions for density. 3. Set up the diffusion equation with the given initial and boundary conditions. 4. Solve the diffusion equation using appropriate mathematical techniques, such as separation of variables, Fourier series, or Laplace transforms. 5. Evaluate the obtained solution, \(C(x, t)\), at the desired location and time to determine the concentration of species A at that point.

Step-by-step solution

Understand the Diffusion Equation

To determine the concentration of species A at a specified location and time, we need to employ the diffusion equation. The diffusion equation describes how the density of a substance changes over time and distance based on its diffusion constant. It is given by: Fick's second law of diffusion: \(\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}\) Where \(C\) is the concentration of the species A, \(t\) is time, \(x\) is the position, and \(D\) is the diffusion constant.

Identify Initial and Boundary Conditions

According to the problem, we are given initial and boundary conditions concerning the density of species A: 1. Initial condition: density is known at the beginning (i.e., at \(t=0\), density is \(C_0(x)\)). 2. Boundary condition: density is known at the surface (i.e., at \(x=0\), density is \(C_s(t)\)).

Set Up the Problem

To solve the diffusion equation, we need to set it up with the given initial and boundary conditions. For initial condition: At \(t=0\), we have \(C(x,0) = C_0(x)\) For boundary condition: At \(x=0\), we have \(C(0,t) = C_s(t)\)

Solve the Diffusion Equation

Next, we need to solve the diffusion equation to find the concentration \(C(x,t)\) that satisfies the initial and boundary conditions. Generally, this requires using separation of variables, Fourier series, and Laplace transforms. The exact method will depend on the specific functions for given initial and boundary conditions. For this exercise, a general solution method will not be provided as it requires specific functional forms for \(C_0(x)\) and \(C_s(t)\). However, once you have those functional forms, you can apply the appropriate mathematical techniques to solve the diffusion equation.

Determine the Concentration

After obtaining the specific solution, \(C(x,t)\), you can now determine the concentration of species A at any given location and time according to: \(C(x,t)\) - The concentration of species A at position x and time t. By evaluating this function at the desired location and time, you can determine the concentration of species A at the specified point.

Key concepts

Understanding Fick's Second Law of Diffusion

Fick's Second Law of Diffusion is fundamental to analyzing the spread of particles in a medium. It describes the rate at which the concentration of a substance changes with time due to diffusion. Formally stated, the law is represented by the partial differential equation:
\[\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}\]
Here, \(C\) stands for the concentration of the substance, \(t\) for time, \(D\) for the diffusion constant, and \(x\) for the spatial position. This law implies that diffusion causes substances to migrate from regions of higher concentration to areas of lower concentration, and the rate of change in concentration depends on the second spatial derivative, meaning it's related to how the concentration gradient changes in space.

Determining Concentration in Diffusion Problems

To figure out the concentration of a diffusing species at any given point and time, concentration determination is key. The process involves applying the diffusion equation according to the known initial and boundary conditions and solving for \(C(x,t)\).
For any specific location \(x\) and time \(t\), the concentration \(C\) can be calculated using the derived solution from Fick's second law. It's important to understand both the theory behind and practical applications of concentration determination to predict the behavior of diffusing particles in various materials and environments.

The Role of Boundary Conditions in Diffusion

Boundary conditions are the conditions specified at the boundaries of the domain over which the diffusion equation is solved. They are crucial in determining the solution to the diffusion problem. In our context, boundary condition refers to the known concentration at the surface over time, \(C_s(t)\).
For instance, if you consider a semi-infinite solid with a surface exposed to a constant concentration, the boundary condition at the surface would be mathematically stated as \(C(0,t) = C_s(t)\). Different types of boundary conditions can be applied, such as fixed concentration, flux of particles across a boundary, or even a combination of both.

Initial Conditions in Solving Diffusion Equations

Initial conditions specify the state of the system at the beginning of the observation, which, for diffusion problems, is the concentration distribution at time zero. The initial condition in a diffusion equation provides a starting point for solving the problem. It is often represented as \(C(x,0) = C_0(x)\), where \(C_0(x)\) is the known initial concentration profile over the spatial domain. Initial conditions are paired with boundary conditions to find a unique solution to the diffusion equation.

Diffusion Constant: A Key Parameter

The diffusion constant, denoted by \(D\), is a critical parameter in the diffusion equation, reflecting how quickly or slowly a species diffuses through a particular medium. It encompasses the physical properties of both the diffusing substance and the medium and can be influenced by temperature, pressure, and other environmental factors. The value of the diffusion constant must be known or estimated to solve Fick's second law and consequently determine the concentration at a certain location and time.

Mathematical Solving Techniques for Diffusion Equations

Solving the diffusion equation often involves advanced mathematical techniques due to the nature of partial differential equations. Methods such as separation of variables, Fourier series, and Laplace transforms are frequently employed, each contributing to the solution repertoire in different scenarios.

Separation of Variables

This technique involves breaking the equation into two separate ordinary differential equations, one in space and one in time, to simplify the problem and find a solution.

Fourier Series

It is helpful for problems with periodic boundary conditions or initial conditions, allowing for the representation of complex functions in terms of simpler sine and cosine functions.

Laplace Transforms

Laplace transforms convert the partial differential equation into an algebraic equation, which is often easier to solve. Once the solution is found in the Laplace domain, the inverse transform is used to revert to the time domain. Proper application of these techniques allows for the precise determination of concentration variations over time.