EQ-FICK-DIFFUSION · Mass Transport
Fick's First Law of Diffusion (1D)
D*dC_dx + J = 0
Derivative form
D*Derivative(C(x), x) + J = 0
Variables
variable
D
diffusion coefficient of the solute in the solvent
- Object
- diffusing_medium
- Property
- DiffusionCoefficient
- Context
- dilute
variable
J
molar flux (amount per area per time) in the +x direction
- Object
- diffusing_medium
- Property
- MolarFlux
- Context
- dilute
variable
dC_dx
concentration gradient in the +x direction (atomic alias for ∂C/∂x)
- Object
- diffusing_medium
- Property
- ConcentrationGradient
- Context
- dilute
Axioms
classical constant_coefficients deterministic differential dilute_limit homogeneous isotropic linear
Assumptions
- Dilute solute: no solute–solute interactions distort D
- Isotropic solvent (D independent of direction)
- No bulk flow (pure diffusive transport; no advection)
- Fickian regime: Gaussian displacement statistics (not anomalous diffusion)
Derivation
- Empirical; Fick, Über Diffusion, Annalen der Physik, 1855
- Derivable from the random walk: for a Brownian particle with step variance σ² per time τ, D = σ²/(2τ) (Einstein 1905)
- Microscopically: the chemical-potential gradient ∇μ drives the flux, and for dilute ideal solutions μ = μ₀ + RT ln C, so J ∝ -∇μ ∝ -∇C
- STRUCTURAL IDENTITY: substitute (q,k,T) → (J,D,C) into Fourier's law and this equation emerges verbatim
References
- Fick, Ann. Phys. 170 (1855), 59-86
- Crank, The Mathematics of Diffusion, 2nd ed., §1.1
- Einstein, Ann. Phys. 17 (1905), 549 (Brownian motion)