Fick's Law Diffusion Rate Calculator
Calculate diffusion flux from concentration gradient, diffusion coefficient, and membrane thickness.
For cell membranes, drug delivery, and gas exchange.
Fick’s first law of diffusion governs how molecules move through a medium when there is a difference in concentration between two regions. Particles move from where they are concentrated to where they are dilute, and the rate at which they do that scales with both the steepness of the gradient and a material-specific diffusion coefficient.
The formula:
J = −D × (dC / dx)
The negative sign captures the direction: flux flows from high to low concentration. For practical use, you usually drop the sign and work with magnitudes, treating the absolute value of the gradient as a positive number.
The variables:
- J = diffusion flux in mol/m²·s, the amount of substance crossing a unit area per second
- D = diffusion coefficient in m²/s, a property of the diffusing species and the medium
- dC/dx = concentration gradient in mol/m⁴, the change in concentration per unit distance through the medium
Adolf Fick, a German physiologist, derived this in 1855 by analogy with Fourier’s law of heat conduction and Ohm’s law of current. He was studying salt diffusion in water but the law turned out to govern every passive transport process in chemistry, biology, and materials science.
Worked example, oxygen across a cell membrane: Oxygen diffuses through a 10 μm cell membrane. Concentration is 8 mol/m³ outside, 2 mol/m³ inside, and D for oxygen in tissue is 2.1 × 10⁻⁹ m²/s.
|dC/dx| = (8 − 2) / (10 × 10⁻⁶) = 600,000 mol/m⁴ J = D × |dC/dx| = 2.1 × 10⁻⁹ × 600,000 = 1.26 × 10⁻³ mol/m²·s
For a 1 cm² patch of membrane, that is 1.26 × 10⁻⁷ mol/s = 76 picomoles/s. Multiply by Avogadro’s number and you get about 4.5 × 10¹³ molecules/s crossing the membrane. A single alveolar cell processes a steady torrent of oxygen molecules.
Worked example, transdermal drug patch: A nicotine patch delivers a drug through 1 mm of skin. Surface concentration is 500 mol/m³ (in the patch reservoir), interior is essentially 0, and D through skin is 1.0 × 10⁻¹⁰ m²/s.
|dC/dx| = 500 / 0.001 = 500,000 mol/m⁴ J = 1.0 × 10⁻¹⁰ × 500,000 = 5.0 × 10⁻⁵ mol/m²·s
Over a 10 cm² patch (10⁻³ m²): total flux = 5.0 × 10⁻⁸ mol/s = 4.3 × 10⁻³ mol/day. With nicotine’s molar mass of 162 g/mol that is about 0.7 g/day, which is the right order of magnitude (commercial patches deliver 7–21 mg over 24 hours, depending on dose).
Diffusion coefficients you will see in practice:
| Species in medium | D (m²/s) |
|---|---|
| Oxygen in water at 25°C | 2.0 × 10⁻⁹ |
| Glucose in water | 6.7 × 10⁻¹⁰ |
| Hemoglobin in water | 7.0 × 10⁻¹¹ |
| Small drug molecules through skin | 10⁻¹⁰ to 10⁻¹² |
| Water vapor in air | 2.4 × 10⁻⁵ |
| Carbon in iron (austenite, 900°C) | 1.5 × 10⁻¹¹ |
| Salt in water | 1.6 × 10⁻⁹ |
The 8-orders-of-magnitude span explains why some processes (gas exchange) happen in milliseconds and others (industrial carburizing) need hours at high temperature.
Why it is linear in the gradient: Fick’s law is a constitutive equation, which means it is empirical: it just describes what happens. The deeper reason it works is that random thermal motion is uncorrelated between molecules, so the net drift across any plane is proportional to the difference in particle counts on either side. In dense or strongly interacting systems (very concentrated solutions, near phase transitions) Fick’s law starts to fail, and you need more elaborate transport theory (Maxwell-Stefan equations).
Fick’s second law and time-dependent diffusion: For situations where the concentration profile evolves over time, you need Fick’s second law: ∂C/∂t = D × ∂²C/∂x². The first law gives the instantaneous flux at any point; the second law tracks how the whole concentration field changes. For steady-state problems (where flux is constant and concentrations have reached their equilibrium distribution) the first law alone is enough.
Practical applications:
- Pulmonary gas exchange (oxygen in, CO₂ out across alveolar membranes)
- Drug delivery through skin patches, mucosal surfaces, blood-brain barrier
- Dialysis machine design — clearance is essentially a Fick’s law calculation
- Pollutant dispersion in groundwater, atmosphere, oceans
- Industrial heat treatment (carburizing, nitriding steel surfaces)
- CO₂ uptake in plant photosynthesis through stomatal pores