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Fick’s first law relates diffusive flux to a concentration gradient; for constant isotropic diffusivity, the second law gives transient concentration evolution.
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Formula
J in mol/(m2·s); D in m2/s; c in mol/m3; ∂c/∂t in mol/(m3·s)
| Symbol | Meaning | Unit |
|---|---|---|
| J⃗A | Molar diffusive flux of A relative to mean motion | mol/(m2·s) |
| Diffusion coefficient | m2/s | |
| Molar concentration | mol/m3 | |
| Time | s | |
| Spatial gradient or Laplacian | mol/m5 |
\vec J_A=-D_{AB}\nabla c_A,\quad \frac{\partial c_A}{\partial t}=D_{AB}\nabla^2c_AThe first-law form assumes a binary system with approximately constant isotropic D and no extra drift terms. The second law ∂c/∂t=D∇2c follows by combining flux conservation with Fick I for constant D, with no convection or reaction. Initial and boundary conditions are required. Nonideal concentrated mixtures may need chemical-potential gradients; multicomponent coupling and spatially varying D change the equation.
Fick’s second law ∂c/∂t=D∇2c assumes constant isotropic D and no convection or reaction; initial and boundary conditions are required.
Combine species conservation ∂c/∂t=−∇·J with Fick’s first law J=−D∇c.
Decay time of a sinusoidal mode
k is the spatial wavenumber and D is constant.
A 1D concentration is c(x,0)=c̄+C cos(kx). Find its amplitude after time t.
c(x,t)=c̄+C exp(−Dk2t)cos(kx); for D=10−9 m2/s and λ=1 mm, τ=1/(D(2π/λ)2)≈25.3 s.
Dropping the negative sign in Fick I or pulling D outside ∇ when D varies.
Use J=−D∇c; constant D gives D∇2c, while variable D requires ∇·(D∇c).
Formula cluster
The first law gives flux from a gradient; the second predicts concentration change from conservation and the flux law.
For a one-dimensional sinusoidal concentration c(x,0)=c̄+C cos(kx), constant D gives c(x,t)=c̄+C exp(−Dk2t)cos(kx). With D=1×10−9 m2/s and wavelength 1 mm, k=2π/0.001 m; the amplitude e-folding time is 1/(Dk2)≈25.3 s. A shorter wavelength decays faster.
Model both steady diffusive flux and transient smoothing of concentration fields.
Here is how to work through a typical Fick’s laws of diffusion flux and transient concentration (J⃗A = −DAB ∇cA) task step by step:
Task
A 1D concentration is c(x,0)=c̄+C cos(kx). Find its amplitude after time t.
Solution path
c(x,t)=c̄+C exp(−Dk2t)cos(kx); for D=10−9 m2/s and λ=1 mm, τ=1/(D(2π/λ)2)≈25.3 s.
The gradient points toward increasing concentration. In the simple model, diffusion transports particles from high to low concentration, opposite that direction. If ∂c/∂x is negative, J is positive along x. The minus sign carries direction and must remain when flux signs matter. Diffusive flux points against the concentration gradient; this form is a local linear model. Application: Model both steady diffusive flux and transient smoothing of concentration fields. The numerical example For a one-dimensional sinusoidal concentration c(x,0)=c̄+C cos(kx), constant D gives c(x,t)=c̄+C exp(−Dk2t)cos(kx). With D=1×10−9 m2/s and wavelength 1 mm, k=2π/0.001 m; the amplitude e-folding time is 1/(Dk2)≈25.3 s. A shorter wavelength decays faster anchors the idea in a concrete measurement case.
D has units of m2/s. Multiplying by a concentration gradient in mol/m4 gives mol/(m2·s), amount per area per time. This dimensional check helps distinguish diffusivity from a mass-transfer coefficient, which may use different units. Diffusive flux points against the concentration gradient; this form is a local linear model. The corresponding units are J in mol/(m2·s); D in m2/s; c in mol/m3; ∂c/∂t in mol/(m3·s). For a one-dimensional sinusoidal concentration c(x,0)=c̄+C cos(kx), constant D gives c(x,t)=c̄+C exp(−Dk2t)cos(kx). With D=1×10−9 m2/s and wavelength 1 mm, k=2π/0.001 m; the amplitude e-folding time is 1/(Dk2)≈25.3 s. A shorter wavelength decays faster. This checks that the reference quantities are compatible.
In concentrated or nonideal mixtures, concentration may not be the appropriate thermodynamic driving force. Temperature gradients, electric fields, reactions and multicomponent coupling can create additional fluxes. An extended transport model is then needed, with the reference mass- or molar-average velocity stated explicitly. Diffusive flux points against the concentration gradient; this form is a local linear model. Scope: The first-law form assumes a binary system with approximately constant isotropic D and no extra drift terms. The second law ∂c/∂t=D∇2c follows by combining flux conservation with Fick I for constant D, with no convection or reaction. Initial and boundary conditions are required. Nonideal concentrated mixtures may need chemical-potential gradients; multicomponent coupling and spatially varying D change the equation. A practical application is Model both steady diffusive flux and transient smoothing of concentration fields.
Combine the local species balance without convection or reaction, ∂c/∂t=−∇·J, with Fick’s first law J=−D∇c. For spatially constant isotropic D this gives ∂c/∂t=D∇2c. If D varies, retain the divergence form, ∂c/∂t=∇·(D∇c); D cannot simply be pulled outside the Laplacian. Diffusive flux points against the concentration gradient; this form is a local linear model. Check the calculation with this case: For a one-dimensional sinusoidal concentration c(x,0)=c̄+C cos(kx), constant D gives c(x,t)=c̄+C exp(−Dk2t)cos(kx). With D=1×10−9 m2/s and wavelength 1 mm, k=2π/0.001 m; the amplitude e-folding time is 1/(Dk2)≈25.3 s. A shorter wavelength decays faster. Keep the model limits and units in view.
A Fourier mode cos(kx) remains sinusoidal while its amplitude decays as exp(−Dk2t). With D=10−9 m2/s and wavelength 1 mm, k=2π/0.001 m and time constant 1/(Dk2) is about 25.3 s. Fine-scale concentration variations have larger k and smooth faster. This is for the ideal linear diffusion equation with suitable boundary conditions. Diffusive flux points against the concentration gradient; this form is a local linear model. The relation describes Fick’s first law relates diffusive flux to a concentration gradient; for constant isotropic diffusivity, the second law gives transient concentration evolution. This helps distinguish its target quantity from neighbouring quantities.
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Which formula describes Fick’s laws of diffusion flux and transient concentration?
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How do you rearrange Fick’s laws of diffusion flux and transient concentration for Decay time of a sinusoidal mode?
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Which common mistake happens with Fick’s laws of diffusion flux and transient concentration?
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