Fick's First Law of Diffusion

1. Introduction to Molecular Diffusion

In chemical processing, mass transfer occurs whenever there is a concentration gradient. If you drop a drop of ink into a glass of still water, the ink will slowly spread out until the water is a uniform color. This spontaneous movement of molecules from a region of high concentration to a region of low concentration is driven by random molecular motion (Brownian motion) and is known as molecular diffusion.

In 1855, physiologist Adolf Fick recognized that mass diffusion behaves mathematically exactly like heat conduction (Fourier's Law). He formulated what is now known as Fick's First Law of Diffusion.

2. Fick's First Law Equation

Fick's First Law states that the molar flux of a species A ($J_A$) relative to the mixture's average velocity is directly proportional to the concentration gradient of A.

Mathematically, it is expressed as: $$J_A = - D_{AB} \frac{dC_A}{dz}$$

Where: - $J_A$ = Molar diffusion flux of species A ($\text{mol} / (\text{m}^2 \cdot \text{s})$) - $D_{AB}$ = Binary diffusion coefficient (or Diffusivity) of A through medium B ($\text{m}^2/\text{s}$) - $C_A$ = Molar concentration of A ($\text{mol}/\text{m}^3$) - $z$ = Distance in the direction of diffusion ($\text{m}$) - $\frac{dC_A}{dz}$ = Concentration gradient

The Negative Sign

The negative sign is crucial: it indicates that diffusion mathematically occurs in the direction of decreasing concentration (from high to low). Without the negative sign, the flux would incorrectly point up the gradient.

3. The Diffusion Coefficient ($D_{AB}$)

The diffusivity ($D_{AB}$) dictates how easily molecules move through the medium. - In Gases: Molecules are far apart. Diffusivity is high (typically $10^{-5} \text{ m}^2/\text{s}$). It increases strongly with temperature and decreases with pressure. - In Liquids: Molecules are tightly packed. Diffusivity is low (typically $10^{-9} \text{ m}^2/\text{s}$). It depends on solvent viscosity. - In Solids: Diffusion relies on crystal lattice vacancies. Diffusivity is extremely low (typically $10^{-14} \text{ m}^2/\text{s}$).

4. Worked Example: Diffusion Across a Thin Film

A membrane 0.1 mm thick separates a gas mixture containing Helium. The concentration of Helium on the high-pressure side is $0.05 \text{ mol/m}^3$, and on the low-pressure side it is $0.01 \text{ mol/m}^3$. The diffusion coefficient of Helium through this specific polymeric membrane is $D = 1.5 \times 10^{-10} \text{ m}^2/\text{s}$.

Calculate the steady-state molar flux of Helium.

Step 1: Identify Parameters - $\Delta z = 0.1 \text{ mm} = 0.0001 \text{ m}$ - $\Delta C = C_{final} - C_{initial} = 0.01 - 0.05 = -0.04 \text{ mol/m}^3$

Step 2: Apply Fick's Law (Linearized for steady-state flat film) $$J_A = - D \frac{\Delta C}{\Delta z}$$ $$J_A = - (1.5 \times 10^{-10}) \frac{-0.04}{0.0001}$$ $$J_A = 1.5 \times 10^{-10} \times 400$$ $$J_A = \mathbf{6.0 \times 10^{-8} \text{ mol} / (\text{m}^2 \cdot \text{s})}$$

This flux dictates exactly how much Helium leaks through a given area of the membrane every second.