Fick's Law of Diffusion Calculator

Compute diffusion flux, diffusion coefficient, concentration gradient, or membrane thickness for 1D steady-state diffusion using Fick's first law.

1. Choose what to solve for

We use the 1D steady-state form: \( J = -D \, \dfrac{\Delta C}{\Delta x} \).

2. Enter known values

Typical values: gases ~10⁻⁵ m²/s, liquids ~10⁻⁹ m²/s, solids ~10⁻¹⁴ m²/s.

Use a positive ΔC (high side minus low side). The sign of J is handled by the formula.

3. Results

Fick's first law: definition and formula

Fick's first law describes steady-state diffusion: the net movement of particles from regions of high concentration to regions of low concentration when the concentration profile does not change with time.

\( J = -D \, \dfrac{dC}{dx} \)

For a uniform layer of thickness \( \Delta x \) with a linear concentration drop from \( C_1 \) to \( C_2 \):

\( J = -D \, \dfrac{C_2 - C_1}{\Delta x} = D \, \dfrac{C_1 - C_2}{\Delta x} = D \, \dfrac{\Delta C}{\Delta x} \)

  • J – diffusion flux, amount crossing unit area per unit time (e.g. mol/(m²·s) or g/(m²·s))
  • D – diffusion coefficient (m²/s)
  • C – concentration (mol/m³, mol/L, g/m³, g/L, etc.)
  • x – position (m)

How this calculator works

This tool assumes one-dimensional, steady-state diffusion through a flat layer (membrane, film, stagnant fluid, soil layer, etc.) with a linear concentration gradient between the two faces. Internally it:

  1. Converts all inputs to SI units (m, s, mol/m³ or g/m³).
  2. Uses the algebraic form of Fick's first law: \( J = D \, \dfrac{\Delta C}{\Delta x} \).
  3. Solves for the requested variable:
    • \( J = D \, \dfrac{\Delta C}{\Delta x} \)
    • \( D = J \, \dfrac{\Delta x}{\Delta C} \)
    • \( \dfrac{\Delta C}{\Delta x} = \dfrac{J}{D} \)
    • \( \Delta x = D \, \dfrac{\Delta C}{J} \)
  4. Converts the result back to your chosen output units.

Typical diffusion coefficient values

  • Small molecules in gases: \( D \sim 10^{-5} \,\text{m}^2/\text{s} \)
  • Small solutes in water: \( D \sim 10^{-9} \,\text{m}^2/\text{s} \)
  • Diffusion in solids: \( D \sim 10^{-14} \) to \( 10^{-20} \,\text{m}^2/\text{s} \)

Worked example: gas diffusion across a membrane

Suppose oxygen diffuses through a 0.5 mm thick membrane. The concentration on the high side is 0.25 mol/L and on the low side 0.05 mol/L. The diffusion coefficient is \( D = 2.0 \times 10^{-9} \,\text{m}^2/\text{s} \). What is the flux?

  1. Convert units:
    • Thickness: \( \Delta x = 0.5 \,\text{mm} = 5.0 \times 10^{-4} \,\text{m} \)
    • Concentration difference: \( \Delta C = 0.25 - 0.05 = 0.20 \,\text{mol/L} = 0.20 \times 1000 = 200 \,\text{mol/m}^3 \)
  2. Apply Fick's law:
    \( J = D \, \dfrac{\Delta C}{\Delta x} = 2.0 \times 10^{-9} \,\dfrac{\text{m}^2}{\text{s}} \cdot \dfrac{200 \,\text{mol/m}^3}{5.0 \times 10^{-4} \,\text{m}} \)
  3. Compute:
    • \( \dfrac{200}{5.0 \times 10^{-4}} = 4.0 \times 10^{5} \,\text{mol/m}^4 \)
    • \( J = 2.0 \times 10^{-9} \times 4.0 \times 10^{5} = 8.0 \times 10^{-4} \,\text{mol/(m}^2\text{·s)} \)

The calculator reproduces this result when you enter the same values and choose to solve for flux J.

Limitations and assumptions

  • One-dimensional diffusion (properties uniform in the plane of the layer).
  • Steady state: concentrations at each face and the flux do not change with time.
  • Constant diffusion coefficient D across the layer.
  • No convection or bulk flow; transport is purely diffusive.

For time-dependent diffusion (e.g. uptake into a slab, transient diffusion in soils), Fick's second law and appropriate boundary conditions are required. Those problems generally need numerical solution and are not covered by this simple calculator.

FAQ

Is the sign of J important?

In vector form, the negative sign in \( J = -D \, dC/dx \) indicates that diffusion is from high to low concentration. In many engineering and life-science applications we are interested in the magnitude of the flux, so this calculator reports a positive value assuming you enter ΔC as high minus low concentration.

Can I use mass concentration instead of molar concentration?

Yes. If you use g/L or g/m³ for ΔC and a flux unit in g/(area·time), the diffusion coefficient D will be in units of m²/s but implicitly refers to mass diffusion rather than molar diffusion. Be consistent: do not mix molar and mass units in the same calculation.

How does temperature affect diffusion?

Diffusion coefficients generally increase with temperature. For many systems, D follows an Arrhenius-type dependence \( D = D_0 \exp(-E_a / RT) \), where \( E_a \) is an activation energy. This calculator assumes D is already known at the temperature of interest.