Fick Law Membrane Diffusion Calculator
Find the steady-state diffusion flux, permeability coefficient and transport rate across a biological membrane from Fick's first law.
🧪 What is Fick Law Membrane Diffusion?
Fick law membrane diffusion is the passive, downhill movement of a dissolved substance across a thin barrier, driven only by the difference in its concentration on the two sides. Adolf Fick published the relationship in 1855, and in its differential form it reads J = -D(dC/dx): the diffusive flux J is proportional to the concentration gradient, with the minus sign encoding the fact that matter always drifts from where it is abundant toward where it is scarce. For a membrane thin enough that the gradient inside it is essentially linear, the law collapses to the practical working form J = P(C₁ - C₂), where P = DK/Δx is the permeability coefficient.
Physiologists use this relationship constantly. It sets the oxygen flux across the alveolar wall in the lung, where a barrier only about half a micrometre thick lets enough oxygen through to sustain a human at rest. It predicts how quickly an anaesthetic or a lipophilic drug reaches the cytoplasm of a target cell. Pharmaceutical scientists apply the same equation to transdermal patches, controlled-release capsules and dialysis membranes, and tissue engineers use it to check whether nutrients can reach the centre of a scaffold before the cells there starve.
A common misconception is that a large diffusion coefficient automatically means fast membrane transport. It does not. Permeability is the product of three separate terms, and for a lipid bilayer the partition coefficient K usually dominates: a polar molecule such as glucose may diffuse perfectly well in water yet barely enter the hydrophobic core of the membrane, so its K is tiny and its permeability is negligible without a dedicated transporter. Another frequent error is applying Fick's first law to a system that has not reached steady state. The first law holds only once the concentration profile inside the barrier has stopped changing with time.
This calculator does the unit bookkeeping that trips people up. You supply the diffusion coefficient in cm²/s, the partition coefficient, the membrane thickness in micrometres, the two concentrations in millimolar, and the exposed area in cm². It converts everything to consistent CGS units, returns the permeability coefficient, the flux, the total transport rate and the raw concentration gradient, and plots how the flux changes as the barrier gets thicker or thinner.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Lipophilic Drug Across a Cell Membrane
D = 1 × 10⁻⁶ cm²/s, K = 0.01, Δx = 0.01 µm, C₁ = 5 mM, C₂ = 0 mM, A = 1 cm²
Example 2 — Urea Crossing a Plasma Membrane
D = 1.5 × 10⁻⁶ cm²/s, K = 0.005, Δx = 0.008 µm, C₁ = 10 mM, C₂ = 1 mM, A = 2 cm²
Example 3 — Glucose Through a 100 µm Hydrogel Slab
D = 6.7 × 10⁻⁷ cm²/s, K = 1, Δx = 100 µm, C₁ = 20 mM, C₂ = 2 mM, A = 10 cm²
❓ Frequently Asked Questions
🔗 Related Calculators
What is Fick's first law of diffusion?
Fick's first law states that the diffusive flux is proportional to the negative concentration gradient, J = -D(dC/dx). For a thin membrane of thickness dx separating two well-stirred compartments it simplifies to J = (D x K / dx)(C1 - C2), which is the form this calculator uses.
How do I calculate diffusion flux across a membrane?
Compute the permeability coefficient P = D x K / dx, then multiply it by the concentration difference: J = P x (C1 - C2). With D in cm2/s, dx in cm and concentrations in mol/cm3, the flux comes out in mol per cm2 per second. Multiply by membrane area to get the total transport rate.
What is the permeability coefficient P?
P is the lumped constant D x K / dx with units of cm/s. It combines how fast a solute diffuses inside the membrane (D), how strongly the membrane takes the solute up from water (K), and how far it has to travel (dx). Because P has velocity units, it is sometimes read as the effective speed at which solute crosses the barrier.
Why does the partition coefficient appear in the formula?
Because the concentration just inside the membrane surface is not the same as the concentration in the adjacent water. The partition coefficient K is the ratio of those two concentrations at equilibrium, so the true driving gradient inside the membrane is K times the aqueous concentration difference divided by the thickness.
What units should I use for this calculator?
Enter the diffusion coefficient in cm2/s, membrane thickness in micrometres, concentrations in millimolar, and membrane area in cm2. Results are reported as permeability in cm/s, flux in nanomoles per cm2 per second, and total transport rate in nanomoles per second.
How thick is a real cell membrane?
A lipid bilayer is roughly 4 to 5 nanometres thick, which is 0.004 to 0.005 micrometres. Engineered barriers are far thicker: a dialysis membrane is tens of micrometres and a hydrogel slab can be hundreds of micrometres, which is why their fluxes are orders of magnitude lower.
Does Fick's law apply to facilitated or active transport?
No. Fick's law describes passive diffusion driven purely by a concentration gradient. Carrier-mediated transport saturates at high substrate concentration and follows Michaelis-Menten kinetics, and active transport can move solute against its gradient using ATP, so neither is linear in the concentration difference.
What happens to flux if I double the membrane thickness?
The flux halves. P = D x K / dx is inversely proportional to thickness, so doubling dx halves P and therefore halves J at the same concentration difference. The chart on this page traces that inverse relationship across a range of thicknesses.
Why is the flux negative sometimes?
A negative flux simply means the net movement runs from side 2 to side 1 because C2 is larger than C1. The magnitude is what matters physically. Diffusion always moves solute down the gradient, so the direction flips whenever the concentration difference changes sign.
How is Fick's first law different from Fick's second law?
Fick's first law gives the flux at steady state, when concentrations no longer change with time. Fick's second law, dC/dt = D d2C/dx2, describes how the concentration profile evolves before steady state is reached. Use the first law for a thin membrane between two large reservoirs and the second law for transient problems.