Fick's First Law Calculator
Solve the classic diffusion relation J = −D · (ΔC/Δx) for any of its four unknowns — diffusion flux, diffusion coefficient, concentration difference, or membrane thickness — with step-by-step math, a live membrane schematic, and companion diffusion metrics.
D is the material's transport fingerprint
The diffusion coefficient measures how quickly a species migrates through a medium under a unit concentration gradient. It is not a universal constant — it depends on both the diffusing species and the medium. Hydrogen moves through air at about 6×10⁻⁵ m²/s but through palladium at roughly 10⁻⁸ m²/s. The same ion can diffuse a billion times faster in water than in a polymer. D also depends strongly on temperature, usually following an Arrhenius form D = D₀ exp(−Eₐ/RT).
Flux is proportional to the gradient
The driving force for diffusion is not concentration itself but the concentration gradient, ΔC/Δx. Doubling the concentration difference doubles the flux; doubling the membrane thickness halves it. This is why thinner membranes and steeper gradients are the two levers engineers use to increase permeation rates. It is also why diffusion across very thick materials is negligibly slow — the same ΔC spread over a larger distance gives a proportionally smaller gradient.
Temperature changes D exponentially
Unlike the linear dependence of flux on ΔC and Δx, the dependence on temperature is exponential. A typical activation energy of 40 kJ/mol for diffusion in a polymer gives a fourfold increase in D between 20 °C and 60 °C. This is why food spoilage accelerates in the heat, why drug release from a polymer matrix is strongly temperature-dependent, and why hydrogen embrittlement of steels is a high-temperature phenomenon. The linear Fick relation holds at constant temperature; D itself must be updated when temperature changes.
Steady state vs. transient diffusion
Fick's first law describes steady-state diffusion — the flux is constant in time and there is no accumulation anywhere in the medium. Real systems start transient: a concentration profile evolves over time until it reaches the linear steady-state profile that Fick's first law describes. The characteristic time to reach steady state is τ ≈ Δx²/D, which for a 1 mm polymer membrane with D = 10⁻¹² m²/s is about 17 minutes, but for a 1 cm water layer with D = 10⁻⁹ m²/s is under two minutes.
How to Use This Calculator
Fick's first law links four quantities: J, D, ΔC and Δx. Pick which one you want the tool to solve, then fill in the other three.
1. Choose what to solve for
Pick J, D, ΔC or Δx from the toggle. The calculator rearranges J = −D·(ΔC/Δx) automatically for whichever quantity you select, and greys out that field.
2. Use realistic presets
Material presets cover gas, water, NaCl solution, polymer, glass and metal. Concentration presets span 0.1 to 10,000 mol/m³ and thickness presets run from 1 µm to 100 mm.
3. Match units carefully
J is in mol/(m²·s), D in m²/s, ΔC in mol/m³, and Δx in metres. All sliders are logarithmic, so drag for order-of-magnitude control or type an exact value such as 1e-9 directly into the number field.
4. Read the result and verdict
The result card shows the solved value plus equivalents — mol/(m²·h) and mol/(cm²·s) for flux, cm²/s and mm²/s for D, mol/L for ΔC, and mm/µm for thickness. A colour-coded verdict classifies flux magnitude.
5. Check the schematic and charts
The schematic shows the feed side, membrane and permeate side with the flux arrows passing through. The chart plots flux against thickness for three values of D.
6. Switch to Advanced for depth
Advanced mode adds a chart of J against ΔC, a sensitivity check on a +1 % input change, and companion metrics including concentration gradient, permeance, characteristic diffusion time, and diffusion length.
What This Tool Offers
Built around Fick's first law, J = −D·(ΔC/Δx), the fundamental link between a material's diffusivity, the driving gradient, and the resulting mass flux.
Four-way solving
Solve for flux, diffusion coefficient, concentration difference, or membrane thickness. The calculator rearranges Fick's law for whichever variable you choose.
Live membrane schematic
A feed-side, membrane, permeate-side diagram with flux arrows whose thickness scales with the computed flux, a Δx dimension line, and a ΔC bracket.
Flux-class verdict
Each result is classified from very low (barrier film) through polymer membrane and liquid-phase diffusion, up to gas-phase and extreme flux, with a coloured badge.
Logarithmic sliders
Because every variable spans many orders of magnitude, all four sliders are logarithmic — a single drag sweeps from 10⁻²⁴ to 10⁻⁴ for D, for example.
Dual comparison charts
Flux against thickness for three values of D, plus an advanced flux against concentration difference chart for three thicknesses.
Reference tables
Diffusion coefficients by material pair, a dynamic flux lookup grid at the current D, and a full variable reference table with unit notes.
Reference Tables & Conversions
A diffusion-coefficient unit converter, a dynamic flux lookup grid, and typical diffusion coefficients by material pair.
Diffusion coefficient unit converter
Diffusion flux J (mol/(m²·s)) at D = 1.000 × 10⁻⁹ m²/s
Computed from J = D · ΔC / Δx. Rows are concentration difference ΔC, columns are thickness Δx. Values in mol/(m²·s).
| ΔC (mol/m³) | Δx 1 µm | Δx 10 µm | Δx 100 µm | Δx 1 mm | Δx 10 mm | Δx 100 mm |
|---|
Typical Diffusion Coefficients
Representative values near 25 °C. D depends strongly on temperature, concentration and the medium's microstructure.
| Diffusing species / medium | D (m²/s) | D (cm²/s) | Notes |
|---|---|---|---|
| H₂ in air (STP) | 6.1×10⁻⁵ | 6.1×10⁻¹ | Fastest common gas |
| Water vapour in air | 2.5×10⁻⁵ | 2.5×10⁻¹ | Evaporation, drying |
| O₂ in air | 2.0×10⁻⁵ | 2.0×10⁻¹ | Combustion, respiration |
| CO₂ in air | 1.6×10⁻⁵ | 1.6×10⁻¹ | Photosynthesis, GHG exchange |
| Ethanol in water | 1.2×10⁻⁹ | 1.2×10⁻⁵ | Liquid-phase |
| NaCl in water (25 °C) | 1.5×10⁻⁹ | 1.5×10⁻⁵ | Reference electrolyte |
| Sucrose in water | 5.2×10⁻¹⁰ | 5.2×10⁻⁶ | Larger molecule, lower D |
| O₂ in water (25 °C) | 2.1×10⁻⁹ | 2.1×10⁻⁵ | Dissolved gas |
| H₂ in palladium (300 °C) | 1×10⁻⁸ | 1×10⁻⁴ | Unusually fast for a metal |
| C in γ-iron (1000 °C) | 1×10⁻¹¹ | 1×10⁻⁷ | Carburising, steel heat treatment |
| Water in epoxy | 1×10⁻¹³ | 1×10⁻⁹ | Composite moisture uptake |
| CO₂ in PET | 1×10⁻¹³ | 1×10⁻⁹ | Beverage bottle barrier |
| O₂ in PET | 5×10⁻¹⁴ | 5×10⁻¹⁰ | Food packaging |
| Cu in Cu (self, 800 °C) | 1×10⁻²⁴ | 1×10⁻²⁰ | Solid-state metal diffusion is extremely slow |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| J | Diffusion flux (magnitude) | 10⁻⁸ – 10⁻¹ | mol/(m²·s) |
| D | Diffusion coefficient | 10⁻²⁴ – 10⁻⁴ | m²/s |
| ΔC | Concentration difference | 0.1 – 10⁴ | mol/m³ |
| Δx | Membrane thickness / diffusion distance | 10⁻⁷ – 1 | m |
| ∇C | Concentration gradient ΔC/Δx | 10⁻⁴ – 10⁹ | mol/m⁴ |
| P | Permeance D/Δx | 10⁻²⁴ – 10⁻² | m/s |
| τ | Characteristic diffusion time Δx²/D | 10⁻⁶ – 10¹² | s |
| Eₐ | Activation energy for diffusion | 5 – 200 | kJ/mol |
What Does J = −D · (ΔC/Δx) Measure?
Fick's first law connects four quantities that anyone working with diffusion, membranes, or transport processes deals with constantly: the flux of a species (J), the diffusion coefficient of that species in the medium (D), the concentration difference driving the flow (ΔC), and the distance over which that difference is imposed (Δx). Together they describe how much material crosses a unit area per unit time, and hence how fast a membrane permeates, how quickly a drug releases from a polymer, how fast a contaminant spreads through soil, or how rapidly a gas leaks through a packaging film.
Fick's First Law Formula
The defining differential form of the law is:
For a membrane of finite thickness with steady-state linear concentration profile, this becomes the finite-difference form used by this calculator:
Rearranged to solve for the other three unknowns:
ΔC = −J · Δx / D
Δx = −D · ΔC / J
Working with magnitudes (which is standard engineering practice), the negative sign is dropped and the four equations become:
D = J · Δx / ΔC
ΔC = J · Δx / D
Δx = D · ΔC / J
Diffusion Variables
| Symbol | Meaning | Notes |
|---|---|---|
| J | Diffusion flux | Amount of substance crossing unit area per unit time |
| D | Diffusion coefficient | Material property, temperature-dependent; higher D means faster diffusion |
| ΔC | Concentration difference | C₁ − C₂ across the membrane; always positive in magnitude form |
| Δx | Membrane thickness | The distance over which the concentration drops |
| ∇C | Concentration gradient | ΔC / Δx; the actual driving force |
| P | Permeance | D / Δx; a membrane property independent of the driving concentration |
How Each Variable Affects the Flux
Because the relation is linear in D, ΔC and 1/Δx, each has a predictable effect on flux. Doubling the diffusion coefficient doubles the flux at the same gradient; doubling the concentration difference doubles it again; doubling the thickness halves it. In practice the three are coupled: a thinner membrane often also has a different D because its microstructure differs, and a higher concentration may change D if the species plasticises the polymer or alters the solvent viscosity.
The quantity that often matters more than J itself is the permeance, P = D / Δx. Two membranes of different thickness and different D can have the same permeance — the same flux for a given driving concentration — which is why membrane datasheets often quote permeance directly rather than D and Δx separately. Permeance has units of m/s and is the membrane equivalent of a mass-transfer coefficient. Its reciprocal, Δx / D, has units of s/m and is the mass-transfer resistance.
Fick's Law Calculation Examples
J = 1×10⁻⁹ × 100 / 0.001 = 1.000×10⁻⁴ mol/(m²·s)
D = 1×10⁻⁴ × 0.001 / 100 = 1.000×10⁻⁹ m²/s
ΔC = 1×10⁻⁴ × 0.001 / 1×10⁻⁹ = 100 mol/m³
Δx = 1×10⁻⁹ × 100 / 1×10⁻⁴ = 0.001 m = 1 mm
J = 5×10⁻¹⁴ × 10 / 25×10⁻⁶ = 2×10⁻⁸ mol/(m²·s)
Frequently Asked Questions
Fick's first law, J = −D·(dC/dx), describes steady-state diffusion — the flux is constant in time and the concentration profile is fixed. It says nothing about how the concentration evolves. Fick's second law, ∂C/∂t = D·∂²C/∂x², describes transient diffusion — how concentration changes with time and position as the profile evolves toward steady state. The second law is obtained by combining the first law with a mass balance on a differential element. In practice, engineers use the first law for membrane permeation and steady flux problems, and the second law for diffusion times, concentration profiles during transients, and drug-release kinetics.
The negative sign tells you that diffusion always carries material down the concentration gradient — from high concentration to low concentration — not up it. If C decreases in the +x direction, then dC/dx is negative and the minus sign makes J positive, i.e. flux in the +x direction. If C increases in the +x direction, dC/dx is positive and the minus sign makes J negative, i.e. flux in the −x direction. Either way, the flux always points from high to low concentration. The sign is a statement of the second law of thermodynamics: diffusion is an irreversible process that increases entropy.
D typically follows an Arrhenius relationship: D = D₀ · exp(−Eₐ / R·T), where Eₐ is the activation energy for diffusion, R is the gas constant and T is absolute temperature. Activation energies range from 5–20 kJ/mol for gases diffusing in gases, 15–40 kJ/mol for small molecules in liquids, and 40–200 kJ/mol for atoms in solids. A typical polymer with Eₐ = 40 kJ/mol shows D increasing roughly fourfold between 20 °C and 60 °C. This is why diffusion is dramatically faster at high temperature and why barrier properties of packaging degrade in hot storage.
D spans an enormous range. Gases diffusing in gases: 10⁻⁵ to 10⁻⁴ m²/s. Small molecules in water: 10⁻⁹ to 10⁻¹⁰ m²/s. Large molecules in water: 10⁻¹⁰ to 10⁻¹¹ m²/s. Small molecules in polymers: 10⁻¹¹ to 10⁻¹³ m²/s. Atoms in solid metals: 10⁻¹⁴ to 10⁻²⁴ m²/s, depending on temperature and whether the mechanism is interstitial or substitutional. The ratio between the fastest gas diffusion and the slowest solid-state metal diffusion is about twenty orders of magnitude, which is why the sliders in this calculator are logarithmic.
Steady state means that the concentration at every point in the medium is constant in time, even though material is flowing through. At steady state the flux is the same at every cross-section — no material accumulates anywhere — and the concentration profile across a uniform membrane is linear. Real systems reach this state only after a transient period. The characteristic time to reach steady state is τ ≈ Δx²/D. For a 100 µm polymer film with D = 10⁻¹² m²/s, τ ≈ 10 seconds. For a 1 cm water layer with D = 10⁻⁹ m²/s, τ ≈ 10⁵ seconds, about a day. The first law is valid only after steady state is established.
Flux is inversely proportional to thickness: halve Δx and you double J. This is why membranes used in gas separation, reverse osmosis, and dialysis are made as thin as mechanically possible — often 50–200 nm for the active layer. However, there is a limit: making the membrane thinner makes it more fragile and harder to manufacture without defects. Real membranes use an asymmetric structure with a very thin active layer on a thick porous support. The support contributes negligible resistance to diffusion, while the active layer controls the flux, so the effective Δx in the Fick calculation is the thickness of the active layer alone.
The SI unit of molar flux is mol/(m²·s). Other common forms are kg/(m²·s) for mass flux, and particles/(m²·s) for number flux. In CGS the molar flux is mol/(cm²·s), which is 10⁴ times smaller than mol/(m²·s). In membrane science, flux is sometimes quoted per unit pressure difference as a permeance in mol/(m²·s·Pa) or in GPU (gas permeation units) where 1 GPU = 10⁻⁶ cm³(STP)/(cm²·s·cmHg). The unit converter card above converts between the SI and CGS forms. The calculator uses mol/(m²·s) throughout.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| J | Diffusive flux | atoms/m2.s |
| D | Diffusion coefficient | m2/s |
| dC/dx | Concentration gradient | atoms/m3 per m |
What it means
Fick’s first law describes steady‑state diffusion: the flux J (amount crossing area per time) is proportional to the concentration gradient dC/dx, with D as the diffusion coefficient. The negative sign indicates flow from high to low concentration. This law is the foundation for modelling diffusion in solids, liquids, and gases. In materials science, it is applied to understand diffusion during heat treatment, oxidation, and doping of semiconductors. Engineers use it to calculate the rate at which atoms move, enabling predictions of case depth in carburising, decarburisation, and other diffusion‑controlled phenomena. The law assumes steady state; for transient diffusion, Fick’s second law is used. Understanding Fick’s first law is fundamental for materials processing, corrosion control, and thin‑film technology.
Worked example
Fick's First Law – Two Examples
Real‑World| Parameter | Value |
|---|---|
| D | 1×10⁻¹⁴ m²/s |
| dC/dx | 1×10²⁴ atoms/m⁴ |
| Parameter | Value |
|---|---|
| D | 5×10⁻¹³ m²/s |
| dC/dx | 5×10²³ atoms/m⁴ |
Common mistakes
- Same as ID 30 – see that entry (materials context).
- Note: In materials, D is often in m²/s, concentration gradient in atoms/m⁴ or kg/m⁴ – ensure consistent units.
Applications
Fick's first law (J = −D·dC/dx) governs steady‑state diffusion, relating the diffusive flux to the concentration gradient. It is essential for understanding mass transport in materials processing, from alloying to corrosion. Engineers use it to design diffusion barriers, to model the growth of oxide layers, and to optimise coating processes. In semiconductor manufacturing, it determines the doping profiles. In food packaging, it models the diffusion of moisture and gases. By applying Fick's law, materials engineers can predict concentration profiles, design diffusion couples, and control the rate of diffusion to achieve desired material properties.
- Design of diffusion barriers and protective coatings
- Oxidation and corrosion modelling
- Semiconductor doping and diffusion process design
- Hydrogen embrittlement and hydrogen diffusion analysis
- Food packaging and pharmaceutical controlled‑release systems
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