Formula used
Fick’s first law relates diffusion flux to the concentration gradient: J = −D · (dC/dx). Here, J is molar flux, D is diffusivity, and dC/dx is the spatial concentration gradient.
Using two points, the gradient is approximated by: dC/dx ≈ (C₂ − C₁) / (x₂ − x₁). The negative sign means flux points from high concentration toward low concentration.
How to use this calculator
- Enter D and select its units.
- Pick concentration units. Add molar mass if needed.
- Select two‑point mode or direct dC/dx mode.
- Choose the length unit for x or gradient denominator.
- Select output units and press Calculate.
- Export your results using CSV or PDF buttons.
Example data table
| Case | D (m²/s) | C1 (mol/m³) | x1 (m) | C2 (mol/m³) | x2 (m) | dC/dx (mol/m⁴) | J (mol/(m²·s)) |
|---|---|---|---|---|---|---|---|
| 1 | 2.0×10⁻⁹ | 1200 | 0 | 800 | 0.002 | −200,000 | +4.0×10⁻⁴ |
| 2 | 1.1×10⁻¹⁰ | 50 | 0 | 20 | 0.005 | −6,000 | +6.6×10⁻⁷ |
Values above are illustrative. Real diffusivities depend on material and temperature.
Technical article
1) Why diffusion flux matters in design
Diffusion governs how species move through membranes, coatings, porous media, and biological tissues. Engineers use flux to size barriers, predict contamination spread, and estimate delivery rates in films or gels. A reliable flux estimate supports material selection, safety margins, and performance targets.
2) Interpreting Fick’s first law correctly
Fick’s first law states that flux is proportional to the concentration gradient and opposes it. The negative sign ensures transport occurs from higher concentration to lower concentration under steady conditions. When the gradient is negative along +x, the computed flux becomes positive, indicating flow toward +x.
3) Choosing diffusion coefficient values
The diffusion coefficient depends strongly on temperature, microstructure, and the diffusing species. In liquids, typical small‑molecule diffusivities range around 10−9 m²/s, while in polymers and dense solids they can drop to 10−12–10−16 m²/s. Use experimentally measured data whenever possible and document conditions.
4) Building a gradient from measurements
Two‑point gradients are common when you have concentration measurements at two locations. The calculator estimates dC/dx using (C2−C1)/(x2−x1). For best accuracy, choose points across a region where concentration changes approximately linearly and avoid extremely small spacing that amplifies noise.
5) Units, conversions, and molar mass
Flux computations are sensitive to unit consistency. This tool converts diffusivity to m²/s and concentration to mol/m³ internally, then returns flux in your selected units. If you enter concentration as g/m³ or mg/L, molar mass (g/mol) is required to convert mass concentration to molar concentration.
6) Practical boundary conditions in layers
Many real systems involve layers: air–film–substrate or liquid–membrane–liquid. Under steady diffusion with constant D in a layer, flux is constant through that layer. You can compute flux from the layer’s gradient and then relate it to interfacial concentrations using partitioning or continuity assumptions.
7) When the steady, one‑dimensional model breaks down
Fick’s first law applies best to steady diffusion with constant diffusivity. Time‑dependent processes, strong convection, or concentration‑dependent diffusivity require a more complete model. If gradients evolve with time or geometry is complex, consider solving the diffusion equation numerically with appropriate boundary conditions.
8) Reporting results for audits and collaboration
Professional workflows benefit from traceable calculations. Exporting a CSV captures inputs, units, gradients, and flux values for spreadsheets and lab notebooks. The PDF report helps document assumptions, supports reviews, and simplifies comparison across materials, temperatures, and thicknesses during design iterations.
FAQs
1) What does a negative flux mean?
A negative flux means the net diffusion is toward the −x direction. It occurs when the product −D·(dC/dx) is negative, based on your chosen coordinate axis.
2) Can I use this for gases and porous solids?
Yes, if you have an appropriate diffusion coefficient for the medium and conditions. For porous materials, use an effective diffusivity that already accounts for tortuosity and porosity.
3) Why is molar mass required for mg/L or g/m³?
Mass concentration must be converted to molar concentration to use Fick’s law in molar units. The conversion is C(mol/m³)=C(g/m³)/M(g/mol); mg/L is numerically g/m³.
4) Which gradient mode should I choose?
Use two‑point mode when you have concentrations at two positions. Use direct mode when you already know dC/dx from a fit, simulation, or analytical model.
5) Does this include convection or drift?
No. This calculator is for diffusion flux from Fick’s first law only. If convection is important, total flux must include an advective term, typically v·C.
6) How do I improve accuracy with experimental data?
Measure concentration over multiple positions, fit a line to obtain dC/dx, and use the fitted slope as the direct gradient input. Also confirm D matches your temperature and material state.
7) Why does the calculator show both J and |J|?
J includes direction based on your axis choice, which helps interpret transport direction. |J| is the magnitude, useful for comparing transport rates across cases regardless of sign.