Mass Transfer: Diffusion Basics

Concept

Mass transfer is the net movement of mass from one location to another, usually meaning a specific chemical species moving through a mixture. The most fundamental mechanism of mass transfer is molecular diffusion, which occurs due to random molecular motion. When there is a difference in the concentration of a species (a concentration gradient) in a fluid or solid mixture, the species will naturally diffuse from the region of high concentration to the region of low concentration. The principle governing this phenomenon is Fick's First Law, which is highly analogous to Fourier's Law for heat conduction and Ohm's Law for electrical conduction.

Formula & Method

Fick's First Law for the steady-state, one-dimensional molecular diffusion of species A through a stationary medium B is:

$J_A = -D_{AB} \frac{dC_A}{dz}$

Where $J_A$ is the molar flux of species A, which tells us how many moles of A pass through a unit area per unit time. For a linear concentration gradient over a distance $\Delta z = z_2 - z_1$, the formula can be simplified to:

$J_A = D_{AB} \frac{C_{A1} - C_{A2}}{z_2 - z_1}$

The total molar rate of transfer of species A ($W_A$) across a cross-sectional area $S$ is:

$W_A = J_A S = S D_{AB} \frac{C_{A1} - C_{A2}}{\Delta z}$

Variables & Units

• $J_A$ = Molar flux of species A, in mol/(m$^2$·s) or kmol/(m$^2$·s).
• $W_A$ = Molar transfer rate of species A, in mol/s or kmol/s.
• $D_{AB}$ = Mass diffusivity (or diffusion coefficient) of species A in medium B, in m$^2$/s.
• $C_A$ = Molar concentration of species A, in mol/m$^3$ or kmol/m$^3$.
• $C_{A1}, C_{A2}$ = Concentrations at position 1 and 2, respectively.
• $z$ = Distance in the direction of diffusion, in meters (m).
• $\Delta z$ = Distance across which diffusion occurs, in meters (m).
• $S$ = Cross-sectional area, in square meters (m$^2$).

Worked Example

Problem: Ammonia gas (Species A) is diffusing through a straight tube containing stationary nitrogen gas (Species B) at a steady state. The tube has a length of $0.15 \text{ m}$ and a cross-sectional area of $0.005 \text{ m}^2$. The concentration of ammonia at the beginning of the tube is $0.20 \text{ mol/m}^3$ and at the end of the tube is $0.05 \text{ mol/m}^3$. The diffusion coefficient of ammonia in nitrogen under these conditions is $D_{AB} = 2.30 \times 10^{-5} \text{ m}^2\text{/s}$. Calculate the molar flux of ammonia and the total molar transfer rate.

Calculation:

1. Identify the parameters: $C_{A1} = 0.20 \text{ mol/m}^3$, $C_{A2} = 0.05 \text{ mol/m}^3$, $\Delta z = 0.15 \text{ m}$, $D_{AB} = 2.30 \times 10^{-5} \text{ m}^2\text{/s}$, $S = 0.005 \text{ m}^2$.
2. Calculate the concentration difference: $\Delta C_A = C_{A1} - C_{A2} = 0.20 - 0.05 = 0.15 \text{ mol/m}^3$
3. Calculate the molar flux ($J_A$) using Fick's First Law: $J_A = D_{AB} \frac{\Delta C_A}{\Delta z} = (2.30 \times 10^{-5}) \times \frac{0.15}{0.15}$ $J_A = (2.30 \times 10^{-5}) \times 1 = 2.30 \times 10^{-5} \text{ mol/(m}^2\text{·s)}$
4. Calculate the total molar transfer rate ($W_A$): $W_A = J_A \times S = (2.30 \times 10^{-5}) \times 0.005$ $W_A = 1.15 \times 10^{-7} \text{ mol/s}$

Engineering Meaning

Fick's law highlights that diffusion is entirely driven by the concentration gradient ($dC_A/dz$); molecules move to erase the difference in concentration. The diffusion coefficient $D_{AB}$ represents how easily species A can maneuver through species B. In gases, $D_{AB}$ is relatively large (rapid diffusion), whereas in liquids it is much smaller, and in solids it is exceedingly small (extremely slow diffusion). This principle forms the basis for designing separation processes like gas absorption, distillation, and membrane filtration.

Engineering Check

When applying Fick's First Law in this simplified form, ensure that the system is operating at a steady state (concentrations are not changing with time) and that the bulk fluid is stationary (no convective flow). If the fluid mixture is moving, convective mass transfer dominates, and the total flux must include the bulk motion term in addition to the diffusion term.

Explicit Exclusions

This foundational article explicitly excludes transient diffusion (Fick's Second Law), equimolar counterdiffusion, diffusion through a stagnant gas layer where bulk flow corrections (Stefan diffusion) are necessary, and convective mass transfer coefficients.