AkCalculators

Diffusion: Fick's Laws, physical meaning and practical applications

Overview. Diffusion is the process by which particles spread from regions of high concentration to regions of low concentration due to random thermal motion. It underlies many phenomena in physics, chemistry, biology and engineering — from gas mixing to nutrient transport in tissues and mass transfer across membranes. This article explains the two foundational descriptions of diffusion (Fick's First and Second Laws), derives the commonly used formulas, discusses typical values of diffusion coefficients in different media, and presents practical examples and limitations.

Fick's First Law: steady-state flux

Fick's First Law gives the steady-state flux of a diffusing species in response to a concentration gradient. In one dimension it is written as:

J = -D (dC/dx)

where J (mol·m⁻²·s⁻¹) is the molar flux across a surface, D (m²·s⁻¹) is the diffusion coefficient, and dC/dx is the concentration gradient (mol·m⁻³·m⁻¹ = mol·m⁻⁴). The negative sign indicates flux from high to low concentration. For a linear gradient across a thin slab of thickness Δx with concentrations C₁ and C₂ at opposing faces, the average steady flux becomes:

J = -D (C2 - C1) / Δx = D (C1 - C2) / Δx

To obtain total moles per second crossing an area A, multiply: Ṅ = J × A (mol·s⁻¹). This simple relation is the basis of the First Law calculator. It assumes a constant gradient (steady state), no convection, and constant D.

Fick's Second Law: time-dependent diffusion

When the concentration profile changes with time, the appropriate equation is Fick's Second Law (one-dimensional form):

∂C/∂t = D ∂²C/∂x²

This partial differential equation describes how concentration evolves. Exact solutions depend on initial and boundary conditions. A commonly used case is a semi-infinite medium (x ≥ 0) with initial uniform concentration C₀ for x > 0 and a sudden, constant surface concentration C_s applied at x = 0 for t > 0. The solution is expressed using the complementary error function (erfc):

C(x,t) = C_s - (C_s - C₀) · erf c( x / (2 √(D t) ) )

At the surface (x = 0) C = C_s; as x → ∞, C → C₀. The characteristic diffusion depth scales as √(D t): after time t, significant diffusion reaches distances on the order of √(D t). The calculator implements this erfc solution to give concentration at a chosen depth and to generate profiles vs x.

Typical diffusion coefficients

Diffusion coefficients vary by many orders of magnitude depending on medium and species. Representative values at ~25 °C:

• Gases (small molecules in air): 10⁻⁵ – 10⁻⁴ m²·s⁻¹
• Liquids (small ions, small molecules in water): ~10⁻⁹ – 10⁻¹⁰ m²·s⁻¹
• Polymers / solids (segmental diffusion): ≪10⁻¹² m²·s⁻¹

Use the temperature and porosity options to estimate effective diffusion in porous media (D_eff ≈ D · porosity / tortuosity).

Worked examples

1. Oxygen flux through a membrane: A membrane with area 1 cm² (1e-4 m²), thickness 1 μm (1e-6 m), D (O₂ in membrane) ≈ 2×10⁻⁹ m²·s⁻¹, concentration difference between sides 0.2 mol·m⁻³. Using Fick's First Law, J = D ΔC / Δx = (2e-9 × 0.2 / 1e-6) = 4e-4 mol·m⁻²·s⁻¹. For A = 1e-4 m², Ṅ = 4e-8 mol·s⁻¹.

2. Transient penetration into a polymer: For D = 1e-12 m²·s⁻¹ and t = 1 day (86,400 s), √(D t) ≈ √(8.64e-8) ≈ 2.94e-4 m (≈ 0.3 mm). This indicates slow penetration; use the erfc solution to compute concentration at a specified depth.

Assumptions, limitations and best practices

The simple equations assume:

• Isothermal conditions and constant D — many systems show concentration and temperature dependence of D.
• No convection or bulk flow — if present, combine diffusion with advection (convection-diffusion equation).
• Linear geometry and well-defined boundary conditions — disks, spherical sources, and finite slabs require adapted solutions.

For membranes and porous media, measuring or estimating effective diffusion coefficients (including tortuosity and porosity) is important. For electrolytes, migration in an electric field and ionic strength effects also matter.

Practical tips

• Check units — this calculator expects SI units (D in m²/s, concentrations in mol/m³).
• Compare √(D t) to your geometry to judge whether steady-state assumption is reasonable.
• When in doubt, simulate a transient profile — the erfc solution provides quick insight for semi-infinite cases.

Summary

Fick's First Law gives a simple, powerful estimate of steady flux, while Fick's Second Law captures time-dependent behavior with characteristic √(D t) scaling. This calculator implements both formulas and supplies exportable profiles so you can integrate these estimates into lab planning or engineering design.