Molecular Diffusivity & Binary Diffusion

The Core Parameter: $D_{AB}$

In mass transfer, the Diffusion Coefficient ($D_{AB}$) is the exact equivalent of thermal conductivity ($k$) in heat transfer. It represents the physical ease with which Component A can migrate through Component B.

Diffusivity values vary drastically depending on the physical state of the fluid:

• Gases: $D_{AB} \approx 10^{-5} \ m^2/s$. Gas molecules are far apart and move rapidly, so diffusion is fast.
• Liquids: $D_{AB} \approx 10^{-9} \ m^2/s$. Liquid molecules are tightly packed. Component A must physically push Component B molecules out of the way to move. Liquid diffusion is roughly 10,000 times slower than gas diffusion!
• Solids: $D_{AB} \approx 10^{-14} \ m^2/s$. Diffusion through a solid catalyst pore is excruciatingly slow.

Predicting Diffusivity in Gases: Chapman-Enskog

For non-polar gas mixtures at low to moderate pressures, engineers use the Chapman-Enskog kinetic theory to calculate $D_{AB}$. The equation proves that gas diffusivity is highly sensitive to temperature and pressure:

$$ D_{AB} \propto \frac{T^{1.5}}{P} $$

This relationship is critical for process design:

1. Temperature: If you raise the temperature in a gas scrubber, the molecules vibrate faster, and diffusivity increases exponentially. Mass transfer becomes much more efficient.
2. Pressure: If you increase the pressure, the gas molecules are squeezed together. The mean free path drops, and molecules constantly collide. Diffusivity is inversely proportional to pressure. Therefore, high-pressure absorbers have notoriously poor molecular diffusivity!

Predicting Diffusivity in Liquids: Wilke-Chang

Liquid diffusivity is much harder to predict because of complex intermolecular forces. The industry standard empirical correlation is the Wilke-Chang Equation (a modification of the Stokes-Einstein relation):

$$ D_{AB} = \frac{1.173 \times 10^{-16} \cdot (\phi M_B)^{0.5} \cdot T}{\mu_B \cdot V_A^{0.6}} $$

Where:

• $T$ = Absolute temperature (K)
• $\mu_B$ = Viscosity of the solvent B (Pa·s or kg/m·s)
• $V_A$ = Molar volume of the solute A at its normal boiling point ($m^3/kmol$)
• $M_B$ = Molecular weight of solvent B
• $\phi$ = Association factor of the solvent (2.6 for water, 1.0 for unassociated non-polar solvents like benzene)

The Role of Viscosity: Notice that $D_{AB}$ is inversely proportional to the solvent viscosity ($\mu_B$). If you try to absorb a gas into a highly viscous solvent (like heavy crude oil or cold TEG), the diffusivity plummets, and the absorption column must be built massively tall to compensate.