Introduction & Context

Fick's Second Law of Diffusion describes the unsteady-state mass transfer process where the concentration of a solute changes over time at a specific location. In process engineering, this is critical for modeling systems where equilibrium has not yet been reached, such as the migration of contaminants in groundwater, the leaching of chemicals from solid matrices, or the transient diffusion of species across membranes.

The model assumes a semi-infinite medium, which is a standard approximation for systems where the diffusion front has not yet interacted with the physical boundaries of the domain. Understanding these transient profiles is essential for designing safe containment systems and predicting the temporal evolution of concentration gradients in chemical reactors and environmental systems.

Methodology & Formulas

The concentration profile C(z, t) is determined by solving the one-dimensional diffusion equation under the boundary conditions of a constant surface concentration and an initial uniform concentration. The solution is expressed using the Gaussian error function, erf(η).

The dimensionless argument η is defined as:

\[ \eta = \frac{z}{2\sqrt{D_{AB} \cdot t}} \]

The concentration at a specific depth z and time t is calculated as:

\[ \frac{C(z, t) - C_s}{C_i - C_s} = \text{erf}(\eta) \]

Rearranging for the local concentration C(z, t):

\[ C(z, t) = C_s + \text{erf}(\eta) \cdot (C_i - C_s) \]

The error function is approximated via Taylor series expansion for small values of η:

\[ \text{erf}(\eta) \approx \frac{2}{\sqrt{\pi}} \left( \eta - \frac{\eta^3}{3} + \frac{\eta^5}{10} - \frac{\eta^7}{42} + \frac{\eta^9}{216} \right) \]

Regime / Condition	Criteria
Semi-Infinite Validity	z_bottom > 4 \sqrt{D_{AB} \cdot t}
Error Function Saturation	η > 2.0 (where erf(η) ≈ 1.0)
Taylor Series Applicability	η ≤ 2.0

Fick's First Law describes steady-state diffusion where the concentration profile does not change over time. In contrast, Fick's Second Law is essential for process engineers because it accounts for unsteady-state conditions. Key differences include:

It models systems where concentration varies with both position and time.
It is represented by a partial differential equation, allowing for the prediction of concentration gradients during transient operations like startup, shutdown, or batch processing.
It is critical for calculating the time required to reach a specific concentration threshold in non-equilibrium systems.

To utilize the standard form of Fick's Second Law, process engineers typically assume the following conditions:

The diffusion coefficient is constant and independent of concentration.
The medium is homogeneous and isotropic.
There are no chemical reactions occurring within the diffusion path.
The system is isothermal, meaning temperature gradients do not influence the mass flux.

The error function (erf) solution is the analytical result for Fick's Second Law applied to a semi-infinite medium. You should apply this solution when:

The diffusion process occurs in a material where the depth of penetration is small relative to the total thickness of the medium.
You are modeling transient mass transfer into a solid or stagnant fluid from a constant surface concentration source.
The boundary conditions involve a fixed surface concentration and an initial uniform concentration throughout the bulk material.

Worked Example: Unsteady State Diffusion in a Stagnant Column

A process engineer is evaluating the contamination risk in a deep-well storage tank containing stagnant water. A surface spill has resulted in a constant concentration of a solute at the surface. We must determine if the solute will reach a sensor located at a depth of 0.05 meters after one hour of diffusion.

The system is modeled using Fick's Second Law for a semi-infinite medium:

\[ \frac{C(z,t) - C_s}{C_i - C_s} = \text{erf}\left( \frac{z}{2\sqrt{D_{AB}t}} \right) \]

Knowns:

Surface concentration (C_s): 0.100 kg/m³
Initial concentration (C_i): 0.000 kg/m³
Diffusion coefficient (D_AB): 1.000e-09 m²/s
Time elapsed (t): 3600.000 s
Depth of interest (z): 0.050 m

Step-by-Step Calculation:

Calculate the denominator of the error function argument: \( 2\sqrt{D_{AB}t} = 2\sqrt{1.000 \times 10^{-9} \times 3600} \approx 0.004 \) m.
Determine the dimensionless variable (η): \( \eta = \frac{z}{2\sqrt{D_{AB}t}} = \frac{0.050}{0.00379} \approx 13.176 \).
Evaluate the error function: Since η is significantly greater than 3, the value of \(\text{erf}(13.176)\) is effectively 1.000.
Solve for concentration at depth z: \( C(z,t) = C_s + (C_i - C_s) \times \text{erf}(\eta) \).
Substitute the values: \( C(0.05, 3600) = 0.100 + (0.000 - 0.100) \times 1.000 = 0.000 \) kg/m³.

Final Answer:

The concentration of the solute at a depth of 0.050 meters after 3600 seconds is 0.000 kg/m³. The diffusion front has only penetrated approximately 0.008 meters, indicating the sensor remains unaffected at this time.

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