For steady, one-dimensional diffusion with constant diffusivity, Fick’s first law is:
J = −D · (dC/dx)
- J is the molar diffusion flux in mol/(m²·s).
- D is the diffusion coefficient in m²/s.
- dC/dx is the concentration gradient in mol/m⁴ (since C is mol/m³).
When area is provided, the molar flow rate is N = J·A. With molar mass, mass flux becomes jm = J·M.
- Select what you want to solve: J, D, or dC/dx.
- Choose a gradient method: two-point values or direct ΔC/Δx.
- Enter inputs with units. Add area to compute N, and molar mass for mass results.
- Press Calculate. Results appear above the form.
- Use Download CSV or Download PDF to export the summary.
Tip: If you only need directionless sizing, enable magnitude-only output.
| Case | D (m²/s) | C1 (mol/m³) | C2 (mol/m³) | x1 (m) | x2 (m) | Computed J (mol/m²·s) |
|---|---|---|---|---|---|---|
| Example | 1.0e−9 | 2.0 | 1.0 | 0 | 0.002 | 5.0e−7 |
1) What the flux number represents
Molar diffusion flux J expresses moles crossing a unit area per unit time, driven by a concentration gradient. Large magnitudes indicate strong molecular transport. The sign indicates direction relative to your x-axis, not whether diffusion is physically possible.
2) Governing relationship used
The calculator applies Fick's first law for steady, one-dimensional diffusion with constant diffusivity: J = -D(dC/dx). It is most appropriate when bulk convection is small, properties are uniform, and the region of interest can be approximated as a straight diffusion path.
3) Interpreting the minus sign
Diffusion moves from higher concentration toward lower concentration. If concentration decreases with increasing x, then dC/dx is negative and J becomes positive, meaning transport is in the +x direction. If you only need sizing, you can output the magnitude only.
4) Gradient from two points
When you enter C1 at x1 and C2 at x2, the tool computes a linear slope (C2 - C1)/(x2 - x1). This is a practical engineering approximation for thin layers, membranes, and short diffusion distances where the profile is near linear.
5) Direct DeltaC/DeltaX for film models
If you already know the concentration difference across a diffusion layer and its effective thickness, use the direct DeltaC/DeltaX option. This is common for stagnant films and boundary layers in mass transfer problems, where the layer thickness is estimated from correlations or measurements.
6) Selecting diffusivity realistically
Diffusivity D depends on temperature, phase, and species. Typical scales are about 1e-5 m^2/s in gases and 1e-9 m^2/s for many small solutes in liquids; solids can be much smaller. Use values measured for your medium and temperature whenever possible.
7) Units and molar-mass supported inputs
The calculator converts common units internally to a consistent basis: concentration to mol/m^3, distance to meters, and flux to mol/(m^2*s). If you enter concentration in g/L or kg/m^3, molar mass is used to convert to molar concentration for an accurate gradient.
8) From flux to total rate and quality checks
With area A, the tool reports the total molar transfer rate N = J*A, useful for membranes and planar interfaces. Confirm that x1 and x2 are distinct, units are correct, and the assumed one-dimensional path is reasonable before exporting results for documentation.
1) What is the difference between molar flux and mass flux?
Molar flux is in mol/(m^2*s). Mass flux is in kg/(m^2*s) and is found by multiplying molar flux by molar mass in kg/mol.
2) When should I use the direct DeltaC/DeltaX option?
Use it when you know the concentration change across a known diffusion thickness, such as a membrane, film, or boundary layer estimate.
3) Why can the calculated flux be negative?
Negative flux simply means transport is opposite your positive x direction. It reflects your coordinate choice, not a physical error.
4) Can I apply this to convection or turbulent mixing?
This model is molecular diffusion. For convection or turbulence, use an effective diffusivity or a mass transfer coefficient model instead.
5) What if diffusivity changes with temperature or concentration?
The calculator assumes constant D. Use a representative average, or compute flux with a model that accounts for D(C,T) variation.
6) How accurate is the two-point gradient method?
It works well when the profile is nearly linear between the points. For curved profiles, use smaller spacing or piecewise gradients.
7) What quick checks avoid unrealistic outputs?
Confirm unit scales, nonzero distance, correct point order, and a realistic D for your medium and temperature before exporting.