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Steady diffusion flux through a plane layer
Molar flux of a species through a flat layer in steady one-dimensional diffusion with a straight concentration profile.
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- CHE-07
- Rev
- v1
- Field
- Chemical
- Result
- JA [mmol/(m² s)]
Fick’s first law says a species diffuses from high to low concentration at a rate proportional to the gradient. For steady one-dimensional diffusion through a plane layer of thickness L with concentrations C₁ and C₂ at the two faces, the profile is a straight line and the molar flux is J = D(C₁ − C₂)/L. The diffusion coefficient D is about 10⁻⁹ m²/s for a solute in liquid and about 10⁻⁵ m²/s for a gas in a gas. The flux is shown in mmol/(m² s).
Molar flux of a species through a flat layer in steady one-dimensional diffusion with a straight concentration profile. It points from the high-concentration face to the low-concentration face.
Calculator
- Flux in mol/(m² s)1.6×10−4 mol/(m² s)
- Concentration gradient, (C₁ − C₂)/L80,000 mol/m⁴
- Characteristic diffusion time, L²/D500 s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A concentration drop of 80 mol/m³ across 1 mm with D = 2 × 10⁻⁹ m²/s drives a flux of 0.16 mmol/(m² s).
- The characteristic diffusion time L²/D is 500 s: halving the thickness cuts it by a factor of four, which is why thin membranes respond quickly while thick ones take a long time to reach steady state.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mmol/(m² s)) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Layer thickness | 0.01 to 50 mm | 0.0032 | 16 | 16 | +10% | −9.09 % |
| DAB Diffusion coefficient | 0.01 to 100 10⁻⁹ m²/s | 8e-4 | 8 | 7.999 | +10% | +10 % |
| CA,1 Concentration at face 1 | 0 to 1,000 mol/m³ | −0.04 | 1.96 | 2 | +10% | +12.5 % |
| CA,2 Concentration at face 2 | 0 to 1,000 mol/m³ | −1.8 | 0.2 | 2 | +10% | −2.5 % |
Sweep of Layer thickness
| Layer thickness (mm) | Molar flux (mmol/(m² s)) |
|---|---|
| 0.01 | 16 |
| 0.029 | 5.518 |
| 0.08409 | 1.903 |
| 0.2438 | 0.6562 |
| 0.7071 | 0.2263 |
| 2.05 | 0.07803 |
| 5.946 | 0.02691 |
| 17.24 | 0.009279 |
| 50 | 0.0032 |
Sweep one variable
Variables and how each one contributes
Molar flux
What the equation gives you. Everything else feeds it.
Diffusion coefficient
The diffusion coefficient of the species in the layer, in units of 10⁻⁹ m²/s. Raises the flux in direct proportion. Raising it by 0.5 (to 2.5) moves the result by +25 %, from 0.16 to 0.2 mmol/(m² s).
Concentration at face 1
The concentration of the species at face 1 (high side), in mol/m³. Raises the flux in direct proportion to the concentration drop. Raising it by 10 (to 110) moves the result by +12.5 %, from 0.16 to 0.18 mmol/(m² s).
Concentration at face 2
The concentration at face 2 (low side), in mol/m³. Lowers the flux: the driving concentration difference becomes smaller. Raising it by 10 (to 30) moves the result by −12.5 %, from 0.16 to 0.14 mmol/(m² s).
Layer thickness
The thickness of the layer, in mm. Lowers the flux in inverse proportion: a thicker layer means a gentler gradient. Raising it by 0.25 mm (to 1.25 mm) moves the result by −20 %, from 0.16 to 0.128 mmol/(m² s).
Worked example with the values above
- Concentration drop: C₁ − C₂ = 100 − 20 = 80 mol/m³; thickness L = 0.001 m.
- Gradient: 80 / 0.001 = 80,000 mol/m⁴; D = 2×10−9 m²/s.
- J = 2×10−9 × 80,000 = 1.6×10−4 mol/(m² s) = 0.16 mmol/(m² s).
Assumptions and limits
- Steady, one-dimensional diffusion through a uniform layer with constant D, and a linear concentration profile.
- No reaction in the layer and negligible bulk flow.
Common mistakes
- Using the bulk concentrations when the layer interface has a different (equilibrium) concentration.
- Mixing mm and m, or mol/dm³ and mol/m³.
Statistics and uncertainty
Diffusion coefficients depend on temperature (they roughly double for every 20 to 30 K in liquids) and are known to ±10 % at best, so the flux is no better. For a membrane, the surface concentrations are set by solubility rather than the bulk concentrations, which can change the result greatly.
Where it is used
- Gas or moisture permeation through a film, seal or wall, and the transfer rate in a membrane.
- Estimating the rate of diffusion-limited processes in chemical engineering.
Maths you need
- Concentration gradient: change of concentration over distance
- Molar flux: moles per unit area per unit time