Fick's First Law of Diffusion formula
| Symbol | Quantity | SI unit | Meaning |
|---|---|---|---|
| J | Diffusion flux | mol/(m²·s) | Amount of substance crossing unit area per unit time |
| D | Diffusion coefficient | m²/s | How quickly the species diffuses in the medium |
| C₁ | Concentration at x₁ | mol/m³ | Concentration on the high side (position x₁) |
| C₂ | Concentration at x₂ | mol/m³ | Concentration on the low side (position x₂) |
| Δx | Distance (thickness) | m | Distance between the two positions, e.g. membrane or plate thickness |
What is Fick's First Law of Diffusion?
Fick's First Law of Diffusion: The diffusion flux is proportional to the negative concentration gradient: matter flows from high to low concentration at a rate set by the diffusion coefficient. — Adolf Fick, 1855
Fick's first law describes steady-state diffusion — the situation where the concentration profile no longer changes with time. It says the flux J of a diffusing species is directly proportional to the concentration gradient, and the constant of proportionality is the diffusion coefficient D.
The minus sign tells you that diffusion runs down the gradient, from high to low concentration. When the gradient is linear (as it is across a thin membrane or plate at steady state), the derivative becomes a simple difference, (C₁ − C₂)/Δx, and that is what this calculator uses.
Typical uses include gas permeation through a metal sheet, drug release across a membrane, oxygen transport across tissue, dopant flux in semiconductors and moisture transport through packaging films.
How to use this calculator
- Choose the quantity you want to find in the Calculate menu. It is highlighted as the result field.
- Enter the known values. Scientific notation such as
1.6e-11is accepted. - Select a unit for each value. Units are converted to SI automatically.
- The answer updates as you type. Open Show calculation steps to see the working, or use Copy result to share a link with your inputs.
Worked example
A small solute (D = 1.0 × 10⁻⁹ m²/s) diffuses across a 2 mm thick stagnant water layer. The concentration is 0.5 mol/L on one side and 0.1 mol/L on the other. What is the flux?
- Convert to SI: ΔC = 0.4 mol/L = 400 mol/m³, Δx = 0.002 m.
- Gradient: ΔC/Δx = 400 / 0.002 = 2 × 10⁵ mol/m⁴.
- Flux: J = 1.0 × 10⁻⁹ × 2 × 10⁵ = 2.0 × 10⁻⁴ mol/(m²·s).
Multiply the flux by the area to get the total transfer rate in mol/s.
Frequently asked questions
Why is there a minus sign in Fick's first law?
The concentration gradient dC/dx points toward increasing concentration, but particles move toward lower concentration. The minus sign makes the flux positive in the direction of decreasing concentration. In the difference form (C₁ − C₂)/Δx with C₁ > C₂, the flux comes out positive from 1 toward 2.
What units does the diffusion coefficient have?
D has units of area per time — m²/s in SI, and cm²/s is very common in the literature (1 cm²/s = 10⁻⁴ m²/s). Gases diffuse at about 10⁻⁵ m²/s, small molecules in liquids at about 10⁻⁹ m²/s, and atoms in solids at 10⁻¹⁰ to 10⁻²⁰ m²/s or lower.
When should I use Fick's second law instead?
Use the first law when the concentration profile is steady (not changing with time). If concentrations are still evolving — for example during carburizing, doping or the early stages of drug release — use Fick's second law.
Can I use mass concentration instead of molar concentration?
Yes. The law is linear, so if you enter concentrations in kg/m³ the flux will be in kg/(m²·s). Just keep the units consistent.