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Molar diffusion rate through a plane layer
Moles per second of a species diffusing through a layer of a given area, from the flux multiplied by the area.
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- CHE-08
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- v1
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- Chemical
- Result
- ṅA [µmol/s]
The molar rate of diffusion through a plane layer is the flux times the area: ṅ = AJ = AD(C₁ − C₂)/L. It gives the total amount transferred per second, which is what a process engineer needs to size a membrane or estimate a leakage. Here A is in cm², the thickness in mm, and the result in µmol/s.
Moles per second of a species diffusing through a layer of a given area, from the flux multiplied by the area. It assumes steady diffusion, a straight concentration profile and no reaction or bulk flow.
Calculator
- Flux, per unit area160 µmol/(m² s)
- Amount transferred per hour576 µmol
- Area needed for 1 mmol/s6.25 m²
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 10 cm² layer 1 mm thick passes 0.16 µmol/s, about 0.576 mmol per hour, with a concentration drop of 80 mol/m³.
- Because the rate is in direct proportion to area and inversely proportional to thickness, a thin, large-area membrane is the way to move material quickly.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (µmol/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Area | 0.1 to 1,000 cm² | 0.0016 | 16 | 16 | +10% | +10 % |
| 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 Area
| Area (cm²) | Molar diffusion rate (µmol/s) |
|---|---|
| 0.1 | 0.0016 |
| 0.3162 | 0.00506 |
| 1 | 0.016 |
| 3.162 | 0.0506 |
| 10 | 0.16 |
| 31.62 | 0.506 |
| 100 | 1.6 |
| 316.2 | 5.06 |
| 1,000 | 16 |
Sweep one variable
Variables and how each one contributes
Molar diffusion rate
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 rate in direct proportion. Raising it by 0.5 (to 2.5) moves the result by +25 %, from 0.16 to 0.2 µmol/s.
Concentration at face 1
The concentration of the species at face 1 (high side), in mol/m³. Raises the rate 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 µmol/s.
Concentration at face 2
The concentration at face 2 (low side), in mol/m³. Lowers the rate: the driving concentration difference becomes smaller. Raising it by 10 (to 30) moves the result by −12.5 %, from 0.16 to 0.14 µmol/s.
Layer thickness
The thickness of the layer, in mm. Lowers the rate in inverse proportion. Raising it by 0.25 mm (to 1.25 mm) moves the result by −20 %, from 0.16 to 0.128 µmol/s.
Area
The area through which diffusion occurs, in cm². Raises the rate in direct proportion: more area transfers more. Raising it by 2 cm² (to 12 cm²) moves the result by +20 %, from 0.16 to 0.192 µmol/s.
Worked example with the values above
- Flux: J = D(C₁ − C₂)/L = 2×10−9 × 80 / 0.001 = 1.6×10−4 mol/(m² s).
- Area: 10 cm² = 0.001 m².
- ṅ = J A = 0.16 µmol/s.
Assumptions and limits
- Steady, one-dimensional diffusion through a uniform layer of constant D and a linear concentration profile.
- No reaction in the layer, uniform conditions across the area, and negligible bulk flow.
Common mistakes
- Using the area in m² with a result in µmol/s when it is entered in cm².
- Neglecting the surface equilibrium at the layer faces.
Statistics and uncertainty
The rate carries the errors of D (±10 % or worse), the concentrations at the faces and the area. For a membrane, small leaks and defects add to the area and can dominate a very low true diffusion rate.
Where it is used
- Leak rate of a gas or vapour through a seal or packaging wall.
- Rate of transfer across a membrane in a separation or sensor.
Maths you need
- Fick’s first law: flux proportional to the concentration gradient
- Molar rate: moles per unit time