Membrane Permeation (Solution–Diffusion Model) formula
| Symbol | Quantity | SI unit | Meaning |
|---|---|---|---|
| J | Flux through membrane | mol/(m²·s) | Steady-state flux across the membrane |
| D | Diffusion coefficient in membrane | m²/s | Diffusion coefficient of the solute inside the membrane |
| K | Partition coefficient | — | Partition coefficient (membrane concentration / solution concentration) |
| ΔC | Concentration difference | mol/m³ | Concentration difference between the two bulk solutions |
| L | Membrane thickness | m | Membrane thickness |
What is Membrane Permeation (Solution–Diffusion Model)?
Membrane Permeation (Solution–Diffusion Model): Steady-state flux through a membrane equals the permeability coefficient times the concentration difference across it, where permeability = D·K/L.
When a solute crosses a membrane, it first dissolves (partitions) into the membrane material, diffuses across it, and then leaves on the other side. This solution–diffusion model is Fick's first law with an extra factor K, the partition coefficient, which converts bulk-solution concentrations into concentrations just inside the membrane faces.
The group P = DK/L is the permeability coefficient (units of velocity, m/s or cm/s). It is the most widely reported quantity for skin permeation, drug absorption (e.g. Caco-2 and PAMPA assays), dialysis, packaging films and gas separation membranes.
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 drug has D = 1 × 10⁻⁶ cm²/s in a 100 µm membrane, with K = 0.8, and a 10 mmol/L concentration difference.
- SI: D = 1 × 10⁻¹⁰ m²/s, L = 1 × 10⁻⁴ m, ΔC = 10 mol/m³.
- P = DK/L = 1 × 10⁻¹⁰ × 0.8 / 10⁻⁴ = 8 × 10⁻⁷ m/s.
- J = P·ΔC = 8 × 10⁻⁶ mol/(m²·s).
Frequently asked questions
What does a partition coefficient greater than 1 mean?
The solute prefers the membrane phase over the surrounding solution. Its concentration just inside the membrane is higher than in the bulk, which increases the flux. Lipophilic drugs often have K > 1 in lipid membranes.
How is this related to Fick's first law?
It is Fick's first law applied inside the membrane. The concentrations at the two faces are K·C₁ and K·C₂, so the gradient is K·ΔC/L.