Diffusion Coefficients

What sets D_AB, why gases diffuse fast, and how temperature shifts diffusivity.

propertiesscalingestimation
⏱️ About 17 min

Diffusion in air can be minutes; diffusion in a liquid can be days. The same law applies—so what changes?

💡
The big idea: The diffusion coefficient D_AB captures how easily species mix at the molecular scale; it depends strongly on phase and (for gases) on temperature and pressure.
🎯 By the end, you'll be able to
  • Describe how D_AB depends on species pair, phase, temperature, and pressure
  • Recall typical order-of-magnitude ranges for gas and liquid diffusivities
  • Estimate gas-phase D_AB at a new temperature using a T^1.5 scaling
📎 Helpful to know first

What D_AB represents

The binary diffusion coefficient D_AB is a transport property that measures how rapidly species A and B intermingle due to molecular motion. In Fick’s law it scales the flux produced by a given concentration gradient.

D_AB is not universal: it depends on the two species (size, interactions), the phase (gas/liquid/solid), and the state (temperature, pressure).

🔑 Orders of magnitude (very typical)

Gases: D_AB ~ 1×10⁻5 to 3×10⁻5 m²/s near 300 K and 1 atm.

Liquids: D_AB ~ 1×10⁻10 to 1×10⁻9 m²/s (much smaller because molecules are crowded).

Solids: often 1×10⁻14 m²/s or lower (diffusion can be extremely slow).

Temperature and pressure trends (gases)

In gases, diffusivity generally increases strongly with temperature and decreases with pressure. A common engineering approximation at constant pressure is:

D_AB ∝ T3/2.

This is not exact, but it is often good enough for quick estimates over moderate temperature ranges.

\[ \frac{D_{AB,2}}{D_{AB,1}} = \left(\frac{T_2}{T_1}\right)^{3/2}\quad(\text{gas, approx., constant }P) \]
Simple gas-phase temperature scaling used for engineering estimates.
⚠️ Don’t apply gas scaling to liquids

Liquid diffusivities follow different physics (often linked to viscosity and molecular size). Using D ∝ T^1.5 is generally inappropriate for liquids.

Species-pair dependence

Even at the same T and P, D_AB varies by species pair. For example, water vapor in air is commonly around 2.6×10⁻5 m²/s near room temperature, while heavier organic vapors in air can be closer to 1×10⁻5 m²/s.

📝 Worked example: The gas-phase diffusion coefficient of water vapor in air is D_AB,1 = 2.60×10⁻5 m²/s at T1 = 300 K and 1 atm. Estimate D_AB,2 at T2 = 330 K at the same pressure using D ∝ T^1.5.
  1. Use scaling: D2 = D1 (T2/T1)^(3/2).
  2. Compute ratio: T2/T1 = 330/300 = 1.10.
  3. Compute power: (1.10)^(3/2) = (1.10)^(1.5) = 1.10 × sqrt(1.10) = 1.10 × 1.04881 = 1.15369.
  4. Compute D2: 2.60×10⁻5 × 1.15369 = 3.00×10⁻5 m²/s (3.000×10⁻5 to 3.001×10⁻5 depending on rounding).
✓ D_AB,2 ≈ 3.00×10⁻5 m²/s
✏️ Practice: A binary gas diffusivity is D_AB,1 = 1.80×10⁻5 m²/s at T1 = 290 K (constant pressure). Estimate D_AB,2 at T2 = 350 K using D ∝ T^1.5.
m^2/s
Solution
  1. Scaling: D2 = D1 (T2/T1)^(3/2).
  2. Temperature ratio: T2/T1 = 350/290 = 1.2068965517.
  3. Compute (T2/T1)^(1.5): 1.2068965517 × sqrt(1.2068965517).
  4. sqrt(1.2068965517) = 1.0985884360, so factor = 1.2068965517 × 1.0985884360 = 1.3258825952.
  5. D2 = 1.80×10⁻5 × 1.3258825952 = 2.3865886713×10⁻5 m²/s.

Check your understanding

1. Which diffusivity range is most typical for a binary gas mixture near 300 K and 1 atm?
Gas-phase diffusivities are commonly on the order of 10⁻5 m²/s near ambient conditions.
2. At constant pressure, increasing temperature generally makes gas-phase D_AB:
Molecules move faster and collide differently at higher T; D_AB usually increases strongly with T.
3. The simplified scaling D_AB ∝ T^1.5 is primarily used for:
The T^1.5 scaling is a common engineering approximation for gas-phase diffusivity at constant pressure.
🔎 Related lessons
Next: Steady-State Diffusion Through a Stagnant Film · soon
✅ Key takeaways
  • D_AB depends on the species pair and the phase (gas >> liquid >> solid).
  • Typical gas diffusivities are ~10⁻5 m²/s; typical liquid diffusivities are ~10⁻10 to 10⁻9 m²/s.
  • For gases at constant pressure, a useful estimate is D_AB ∝ T^1.5.
➡️ With D_AB in hand, we can solve an important steady-state case where diffusion of A occurs through a stagnant film of B, producing a logarithmic flux expression.
Want to test yourself on this? Try the Chemical Aptitude test →