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For steady diffusion, Fick’s first law relates flux per area to the concentration gradient: particles move net from high toward low concentration.
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Formula
J in mol/(m2·s) · D in m2/s · C in mol/m3 · x in m; amount rate through area A: ṅ = J·A in mol/s
| Symbol | Meaning | Unit |
|---|---|---|
| Molar flux density in the positive x direction; amount per time and area | mol/(m2·s) | |
| Diffusion coefficient of the substance in the medium | m2/s | |
| Concentration of the diffusing substance in that medium | mol/m3 | |
| Position coordinate along the concentration gradient | m |
J = -D \cdot \frac{dC}{dx}For one-dimensional, steady diffusion in a homogeneous medium, J = −D·dC/dx. The minus sign means net flux runs opposite to the concentration gradient. J is flux density (mol per square metre per second), not concentration and not total amount per time. Through area A, the amount rate is ṅ = J·A. For a flat film of thickness L with constant D, J ≈ D(Chigh−Clow)/L when positive direction is chosen from high to low concentration. A time-varying profile requires Fick’s second law.
Applies to steady, one-dimensional diffusion in a homogeneous medium with constant D. J is amount per area and time; a changing profile requires Fick’s second law.
Diffusive flux is proportional to the concentration gradient and points toward lower concentration.
Diffusion coefficient from flux and gradient
Set the sign and positive x direction first.
Steady flux through a film
Applies for constant D and an approximately linear concentration profile across thickness L.
D=1×10−9 m2/s; C falls from 2.0 to 0.5 mol/m3 across 1 mm. What is J?
J≈D·ΔC/L = 1×10−9·1.5/1×10−3 = 1.5×10−6 mol/(m2·s), toward the lower concentration.
How do you convert flux density J through area A to amount rate?
For uniform J through a perpendicular area: ṅ=J·A.
J remains mol/(m2·s), while ṅ has units mol/s.
Confusing concentration units with flux units.
C has units mol/m3; J has units mol/(m2·s). Only the gradient dC/dx multiplied by D gives flux density.
Treating D·ΔC/L without direction as universally positive.
Define the x-axis. The differential form includes the minus sign; a positive magnitude form assumes direction from high to low concentration.
Formula cluster
Separates local flux from the total transport rate through an area.
Steady water film: D=1.0×10−9 m2/s, ΔC=1.5 mol/m3, L=1.0×10−3 m. J≈D·ΔC/L=1.5×10−6 mol/(m2·s). Across A=0.020 m2, ṅ=J·A=3.0×10−8 mol/s. Assumes a steady linear gradient.
Diffusion across thin tissue and membrane layers, gas exchange in the lungs, nutrient transport and estimating diffusion rates in biological media
Here is how to work through a typical Fick's First Law of Diffusion (J = −D·dC/dx) task step by step:
Task
D=1×10−9 m2/s; C falls from 2.0 to 0.5 mol/m3 across 1 mm. What is J?
Solution path
J≈D·ΔC/L = 1×10−9·1.5/1×10−3 = 1.5×10−6 mol/(m2·s), toward the lower concentration.
Task
How do you convert flux density J through area A to amount rate?
Solution path
For uniform J through a perpendicular area: ṅ=J·A.
J remains mol/(m2·s), while ṅ has units mol/s.
Fick’s first law describes steady diffusive flux: J = −D·dC/dx. J is the amount crossing a unit area per unit time; D is the diffusion coefficient and dC/dx the spatial concentration gradient. The minus sign means net flux points from high to low concentration. For a flat film of thickness L with a linear profile, use approximately J = D(Chigh−Clow)/L. The equation gives flux density, not concentration itself and not directly the total amount per second. It assumes D stays constant across the modeled region.
In SI units, C is measured in mol/m3, x in m and D in m2/s. The gradient dC/dx therefore has units mol/m4. Multiplying by D gives J in mol/(m2·s), an amount per area per time. This differs from an amount rate ṅ in mol/s. For uniform flux normal to an area A, calculate total transport with ṅ = J·A. A concentration given in mol/L must be multiplied by 1000 to convert to mol/m3 before an SI calculation. The sign of J also shows direction relative to the chosen x-axis.
The minus sign sets the physical direction. If x is positive to the right and concentration increases to the right, then dC/dx is positive but J is negative: particles flow net to the left, toward lower concentration. A gradient points toward the strongest increase, while diffusion runs downhill. In a magnitude calculation across a membrane, one often writes J = D(Chigh−Clow)/L and explicitly states the direction from high to low. Do not apply another minus sign in that convention, or you would reverse the direction twice.
For a uniform membrane of thickness L, constant diffusion coefficient D and an approximately linear, steady profile, use J ≈ D(Chigh−Clow)/L. Example: D=1×10−9 m2/s, L=1 mm and concentration difference 1.5 mol/m3 give J≈1.5×10−6 mol/(m2·s), from high to low concentration. Multiply by the effective area to get total amount per time: ṅ=J·A. This approximation ignores membrane binding, reactions, discontinuous D values and changes in the gradient; complex tissues often need a richer model. If boundary concentrations change over time, Fick’s second law describes how the profile evolves.
The first law relates flux to a spatial concentration profile that is already known and is often applied at steady state. If concentration changes over time, you must also describe how material enters and leaves a small volume element. For constant D, mass balance leads to Fick’s second law, ∂C/∂t = D·∂2C/∂x2. It is used, for example, to model a substance entering tissue after a sudden concentration change. The first law remains the local flux relation; combined with conservation, it yields the time-dependent diffusion equation.
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Curated exam set for "Fick's First Law of Diffusion":
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Which formula describes Fick's First Law of Diffusion?
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How do you rearrange Fick's First Law of Diffusion for Diffusion coefficient from flux and gradient?
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Which common mistake happens with Fick's First Law of Diffusion?
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