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 find the concentration of species A at a specific location and time, use the error function solution of Fick's second law: \(C(x, t) = C_i + (C_s - C_i)(1 - erf[\frac{x}{2\sqrt{Dt}}])\) where \(C(x, t)\) is the concentration at position \(x\) and time \(t\), \(C_i\) is the initial concentration, \(C_s\) is the surface concentration, \(D\) is the diffusion coefficient, and \(erf\) denotes the error function. Plug in the given values of initial concentration, surface concentration, diffusion coefficient, location, and time, then compute the concentration of species A at the specified location and time.

Step-by-step solution

Understand the diffusion process

In a semi-infinite medium, species A will diffuse from regions of higher concentration to regions of lower concentration over time. The diffusion process is dictated by Fick's second law, which describes the relationship between the rate of change of concentration and the spatial variation of concentration. We will use Fick's second law to determine the concentration of the species.

Write down Fick's second law

Fick's second law is given as: \(\frac{\partial{C}}{\partial{t}} = D \frac{\partial^2{C}}{\partial{x^2}}\) Where \(C\) is the concentration of species A, \(t\) is the time, \(x\) is the distance from the surface of the medium, and \(D\) is the diffusion coefficient for species A.

Find the appropriate diffusion formula

Due to the given information about the initial and surface densities and the semi-infinite nature of the medium, we will require a formula called the "Error function solution of Fick's second law" to determine the concentration of species A. This formula is given as: \(C(x, t) = C_i + (C_s - C_i)(1 - erf[\frac{x}{2\sqrt{Dt}}])\) Where \(C(x,t)\) is the concentration of species A at position \(x\) and time \(t\), \(C_i\) is the initial concentration, \(C_s\) is the surface concentration, and \(erf\) denotes the error function.

Determine the concentration of species A

Given the values of initial concentration \(C_i\), surface concentration \(C_s\), diffusion coefficient \(D\), location \(x\), and time \(t\), we can calculate the concentration of species A at position \(x\) and time \(t\). Plug the given values into the error function solution of Fick's second law as shown in Step 3 and compute the concentration \(C(x, t)\). This value represents the concentration of species A at the specified location and time.