Fick's Law of Diffusion

How concentration gradients drive molecular transport (and why it mirrors heat and momentum transport).

introtransport-lawsworked-examples
⏱️ About 18 min

If you open a bottle of perfume across the room, you can smell it minutes later—no fan required. What “pushes” the molecules?

💡
The big idea: Diffusion is the molecular-level transport driven by concentration gradients, and its governing law has the same form as the core laws for heat and momentum transport.
🎯 By the end, you'll be able to
  • State Fick’s first law and define the molar diffusive flux
  • Use a linear gradient approximation to compute flux
  • Recognize the analogy between diffusion, conduction, and viscosity
  • Track signs and directions of flux correctly
📎 Helpful to know first
  • Shell-and-Tube Design Basics

Diffusion: transport driven by gradients

Diffusion is molecular motion that causes species to spread from regions of high concentration to low concentration. Even in a perfectly still fluid, random thermal motion produces a net transport whenever there is a concentration gradient.

In many engineering settings we model diffusion as one-dimensional through a slab, film, or boundary layer and compute the flux from the concentration gradient.

\[ J_A = -D_{AB}\,\frac{dC_A}{dx} \]
Fick’s first law (binary mixture): diffusive molar flux of A relative to the molar-average motion.
🔑 Sign convention matters

If concentration decreases in the +x direction (dC_A/dx < 0), then J_A is positive: A diffuses in the +x direction. The negative sign ensures diffusion is “downhill” in concentration.

Analogy: the three transport laws share the same form

Fick’s law is the mass-transfer analog of the other two core transport laws:

Fourier’s law (heat conduction): q" = −k dT/dx, and Newton’s law of viscosity: τ = μ du/dy.

Each says “flux = (property) × (driving-force gradient)” with a negative sign that enforces flow from high to low (temperature, velocity, concentration).

✨ What is J_A versus N_A?

J_A is the diffusive molar flux relative to a reference velocity (commonly the molar-average velocity). N_A is the total molar flux (diffusion + bulk convection). In this lesson, we focus on pure diffusion so the distinction is minimal.

Linear-gradient approximation

If C_A varies approximately linearly across a thickness L, then dC_A/dx ≈ (C_A2 − C_A1)/L. This is often accurate for steady diffusion in a homogeneous medium without reactions.

📝 Worked example: A gas mixture at 300 K has species A diffusing in the x-direction. The diffusion coefficient is D_AB = 2.6×10⁻⁵ m²/s. The concentration drops from C_A1 = 1.80 mol/m³ at x=0 to C_A2 = 0.60 mol/m³ at x = 2.0 mm. Estimate the diffusive molar flux J_A (positive in +x).
  1. Assume a linear profile: dC_A/dx ≈ (C_A2 − C_A1)/L.
  2. Compute gradient: (0.60 − 1.80) mol/m³ / (2.0×10⁻3 m) = −1.20 / 0.002 = −600 mol/m⁴.
  3. Apply Fick’s law: J_A = −D_AB(dC_A/dx) = −(2.6×10⁻5)(−600) mol/(m²·s).
  4. Multiply: 2.6×10⁻5 × 600 = 1.56×10⁻2 mol/(m²·s).
✓ J_A = 1.56×10⁻2 mol/(m²·s) in the +x direction
✏️ Practice: In a polymer film of thickness L = 0.50 mm, a dissolved solute has D_AB = 1.0×10⁻9 m²/s. Concentration drops from 20 mol/m³ at x=0 to 5 mol/m³ at x=L. Using a linear gradient, what is the magnitude of J_A?
mol/(m^2·s)
Solution
  1. Gradient: dC_A/dx ≈ (5 − 20) / (0.50×10⁻3) = −15 / 5.0×10⁻4 = −3.0×10⁴ mol/m⁴.
  2. Flux: J_A = −D_AB(dC_A/dx) = −(1.0×10⁻9)(−3.0×10⁴) = 3.0×10⁻5 mol/(m²·s).
  3. Magnitude: |J_A| = 3.0×10⁻5 mol/(m²·s).

Check your understanding

1. Fick’s first law states that the diffusive molar flux of A is proportional to the:
J_A depends on dC_A/dx; diffusion is driven by concentration gradients.
2. If C_A decreases with x (dC_A/dx < 0), then J_A is:
J_A = −D(dC_A/dx); a negative gradient gives a positive flux in +x.
3. Which law has the same mathematical form as Fick’s first law?
Fourier’s law q" = −k dT/dx mirrors J_A = −D dC_A/dx.
🔎 Related lessons
Next: Diffusion Coefficients
✅ Key takeaways
  • Diffusion is molecular transport caused by concentration gradients.
  • Fick’s first law: J_A = −D_AB dC_A/dx.
  • The negative sign enforces transport from high to low concentration.
  • Fick, Fourier, and Newton transport laws share the same “flux = property × gradient” structure.
➡️ Next we focus on the key material property in Fick’s law: the diffusion coefficient, and how it changes with phase and temperature.
Want to test yourself on this? Try the Chemical Aptitude test →