Molecular Diffusion and Mass Transfer Fundamentals

Introduction

Mass transfer is the movement of a chemical species from one location to another because of a concentration, chemical-potential, or pressure difference. In mechanical engineering it is important in drying, combustion, evaporation, humidification, absorption, membrane separation, catalytic reactors, corrosion, fuel cells, and pollutant control. Molecular diffusion is the transport mechanism caused by the random thermal motion of molecules. Unlike convection, it does not require bulk fluid motion.

This chapter develops the differential and integral descriptions of diffusion in gases, liquids, and solids. It distinguishes species flux from total molar flux, establishes Fick’s laws, derives useful steady one-dimensional results, and introduces transient diffusion and boundary conditions.

Learning objectives

By the end of this chapter, the reader should be able to:

1. Define concentration, velocity, molar flux, mass flux, and diffusivity.
2. Apply Fick’s first and second laws to gases, liquids, and solids.
3. Distinguish equimolar counter-diffusion from diffusion through a stagnant species.
4. Select appropriate boundary and initial conditions for transient diffusion.
5. Estimate binary diffusivity using suitable models and correlations.
6. Analyze practical diffusion systems and identify the limitations of ideal models.

1. Molecular basis and transport quantities

Molecules in a fluid move randomly because of thermal energy. If concentration is uniform, molecular movements occur in all directions and produce no net transport. A concentration gradient makes the molecular movements unequal in opposite directions, producing a net flux from high concentration to low concentration.

For a mixture containing species A and B, the molar concentration of A is

where n_A is the amount of A in moles and V is volume. The total molar concentration is c = c_A + c_B + .... The mole fraction is

for a gas mixture, and the liquid-phase mole fraction is conventionally written x_A. The mass concentration is ρ_A = m_A/V, and the mass fraction is w_A = ρ_A/ρ.

A species flux is the amount crossing a unit area per unit time. The molar flux of A relative to fixed laboratory coordinates is denoted N_A and has units mol·m⁻²·s⁻¹. The molar-average velocity of a mixture is

where v_i is the velocity of species i. The diffusive flux relative to the molar-average motion is

Thus,

The first term represents molecular diffusion and the second represents convection of A by bulk mixture motion. Confusing J_A and N_A is a frequent source of error.

2. Fick’s first law

For an isotropic, binary system at constant temperature and pressure, the molar diffusive flux of A is described by Fick’s first law:

Here D_AB is the binary molecular diffusivity of A in B, with SI units m²·s⁻¹. The negative sign indicates transport down the concentration gradient. In one dimension,

The law is most directly applicable when diffusivity is known or approximately constant, the mixture is dilute or binary, and local thermodynamic equilibrium exists. For concentrated, nonideal liquids, the chemical-potential gradient is the more fundamental driving force and a concentration-gradient form may require a thermodynamic correction.

For a stationary medium, the bulk velocity is zero and N_A = J_A. For a moving mixture, the total flux must be calculated from N_A = J_A + c_A v.

Constant-property plane wall

Consider a slab of thickness L in which the concentration at z=0 is c_A1 and at z=L is c_A2. At steady state, with constant D_AB and no reaction, the flux is constant. Integration gives

for a stationary medium. The units are (m²·s⁻¹)(mol·m⁻³)/m = mol·m⁻²·s⁻¹. The corresponding transfer rate through area A_s is dot n_A = A_s N_A.

3. Species conservation and Fick’s second law

A differential species balance states that accumulation equals inflow minus outflow plus generation. For species A,

where R_A is the rate of generation of A per unit volume, in mol·m⁻³·s⁻¹. For a stationary medium, constant diffusivity, and no reaction, substituting Fick’s first law gives Fick’s second law:

In Cartesian coordinates,

In one-dimensional diffusion,

These equations assume constant temperature, constant diffusivity, no bulk convection, no chemical reaction, and a continuum description. With reaction, a first-order homogeneous reaction R_A = −k c_A gives

where k has units s⁻¹.

4. Binary diffusion in gases

For an ideal gas mixture,

where P is total pressure, T is absolute temperature, and R is the universal gas constant. Therefore c_A = y_A P/(RT). At constant P and T,

where N = N_A + N_B is the total molar flux. This expression combines diffusion with bulk molar motion.

Equimolar counter-diffusion

In equimolar counter-diffusion, A moves in one direction while B moves oppositely at equal molar rates:

Consequently N=0, and for constant P and T,

For a plane layer,

This model is appropriate when both species diffuse and the net molar flow is negligible.

Diffusion of A through stagnant B

If B is stagnant, N_B=0, so N=N_A. The total flux is not equal to the diffusive flux because A’s movement creates a bulk molar flow. At constant P and T,

Integration across a layer gives

The logarithmic correction is essential. Replacing it with a simple mole-fraction difference incorrectly assumes equimolar counter-diffusion. The result applies to steady, one-dimensional, constant-property diffusion of A through non-diffusing B.

5. Diffusion in liquids

Diffusivities in liquids are usually much smaller than in gases because molecules are closely packed. Typical liquid diffusivities are approximately 10⁻¹⁰ to 10⁻⁹ m²·s⁻¹, whereas gas diffusivities are commonly about 10⁻⁵ m²·s⁻¹.

For a dilute solute in a stationary liquid, the simple form is

For concentrated or nonideal solutions, activity and chemical-potential effects become important. A practical expression may be written using an effective diffusivity or a thermodynamic factor, but the factor must be obtained from measured equilibrium data or a validated thermodynamic model.

Liquid diffusion is strongly affected by temperature, viscosity, solute size, and solvent structure. A commonly used estimate for dilute solutes is the Stokes–Einstein relation:

where k_B is Boltzmann’s constant, μ is dynamic viscosity, and r_A is an effective hydrodynamic radius. This relation assumes a spherical solute, continuum hydrodynamics, and dilute conditions; it can be inaccurate for molecular-scale solvents, electrolytes, and associating liquids.

6. Diffusion in solids

Diffusion in solids occurs through vacancies, interstitial sites, grain boundaries, or other defects. Fick’s laws remain useful at the continuum scale, although the microscopic mechanism differs from fluid diffusion.

For a planar solid with constant diffusivity, the steady rate is

In a solid, the concentration may be expressed as mol·m⁻³, mass fraction, or another concentration measure. The chosen form must be consistent with the diffusivity and boundary data.

Diffusivity in solids is often strongly temperature dependent. A widely used model is

where D₀ is the pre-exponential factor, E_D is activation energy in J·mol⁻¹, R is the gas constant, and T is kelvin. Taking logarithms gives

Thus a plot of ln D_A against 1/T is approximately linear when a single diffusion mechanism dominates.

7. Estimating diffusivity

Diffusivity should ideally be obtained from reliable experimental data at the relevant temperature, pressure, composition, and phase. Correlations are useful for preliminary engineering calculations.

For gases, kinetic-theory correlations generally predict that diffusivity increases approximately with T^3/2 to T^1.75 and decreases approximately inversely with pressure. For liquids, empirical correlations frequently involve viscosity and molecular volume. No correlation should be extrapolated far beyond its validated range.

8. Boundary and initial conditions

A diffusion equation requires both spatial boundary conditions and an initial condition for a transient problem.

Common boundary conditions include:

• Specified concentration: c_A(0,t)=c_As.
• Specified flux: −D_AB ∂c_A/∂z = N_A,s.
• Impermeable boundary: N_A=0, often written ∂c_A/∂z=0.
• Convective interfacial condition:

where k_c is a mass-transfer coefficient, c_A,b is bulk concentration, and c_A,i is interfacial concentration.

For a semi-infinite solid initially at uniform concentration c_A,0, suddenly exposed at z=0 to a fixed surface concentration c_As, the solution is

where erf is the error function, z is distance, and t is time. The characteristic diffusion length is approximately

This estimate is useful before solving the full equation. If L_d is much smaller than a body thickness, the semi-infinite approximation may be reasonable.

The Fourier-like diffusion time scale is

It is an order-of-magnitude estimate, not an exact time to uniformity. For example, if L=2×10^-3 m and D=1×10^-10 m²·s⁻¹, then

9. Dimensionless interpretation

The Fourier number for mass diffusion is

It measures the extent of diffusion penetration. Small values indicate limited penetration; values of order unity indicate substantial diffusion across the characteristic length.

When interfacial resistance is present, the mass-transfer Biot number is

A large Bi_m indicates that internal diffusion resistance is important relative to external film resistance. A small Bi_m indicates that the internal concentration is relatively uniform and the external film may control.

10. Worked example: stagnant-gas diffusion

A layer of stagnant nitrogen has thickness 5 mm. Ammonia diffuses through it at 298 K and 1 atm. At the two surfaces, the ammonia mole fractions are 0.20 and 0.05. Take D_NH3-N2=2.0×10^-5 m²·s⁻¹. Find the molar flux of ammonia.

For an ideal gas,

Because nitrogen is stagnant, use the logarithmic expression. Let y_A1=0.20, y_A2=0.05, and L=0.005 m. Then

The positive direction is from the surface with mole fraction 0.20 toward the surface with mole fraction 0.05. An equimolar calculation would give a different value and would violate the stated stagnant-nitrogen condition.

11. Applied case: moisture diffusion during polymer drying

Background. Consider a polymer film containing dissolved moisture. The film is dried in warm air. Product quality depends on reducing moisture without causing cracking or excessive thermal exposure.

Problem. The surface moisture concentration falls rapidly when the film enters dry air, but moisture at the center must diffuse to the surface. The engineer must estimate drying time and determine whether external air-side resistance is important.

Analysis. Moisture movement inside the polymer can initially be approximated by one-dimensional Fickian diffusion. The relevant scale is t_d≈L²/D, where L is the half-thickness for a symmetric film. The diffusivity may vary strongly with moisture content and temperature, so a constant-D calculation is only a first estimate. The external condition is represented by N_A=k_c(c_A,b−c_A,s). Comparing Bi_m=k_cL/D indicates whether internal or external resistance dominates.

Approach. Measure or obtain moisture-dependent diffusivity, estimate Bi_m, and solve the transient diffusion equation with symmetry at the center, ∂c_A/∂z=0, and a convective boundary at the surface. Validate the predicted average moisture against gravimetric drying data. If shrinkage or cracking occurs, couple mass diffusion with heat transfer and mechanical stress analysis.

Outcome. A reliable design identifies the controlling resistance rather than simply increasing air velocity. If internal diffusion controls, increasing air velocity yields diminishing benefit; reducing film thickness or increasing temperature may be more effective, subject to polymer stability.

Learning points. Boundary conditions are physical statements, diffusivity is often composition-dependent, and model validation is essential before using a diffusion model for production decisions.

12. Limitations, safety, and professional practice

The ideal Fickian model may be inadequate when convection, reaction, phase change, swelling, porous-media effects, multicomponent coupling, or nonideal thermodynamics is significant. In porous solids, tortuosity and porosity modify the effective diffusivity. In gases at high pressure, Stefan–Maxwell equations are generally more appropriate than a simple binary Fick law. In electrolytes, electric fields can drive ionic transport.

Diffusion calculations also have safety implications. Flammable vapor diffusion determines ignition-zone composition; toxic-gas diffusion affects exposure assessment; and hydrogen diffusion contributes to leakage and material degradation. Engineers should use validated property data, document assumptions, apply appropriate occupational and process-safety standards, and avoid treating a preliminary estimate as a certified safety assessment.

Modern computational fluid dynamics and process simulators solve coupled momentum, heat, and species equations. Machine-learning models can assist property prediction and parameter estimation, but they require physically consistent training data and independent validation. Conservation laws, dimensional checks, and uncertainty bounds should remain central to engineering judgment.

Best-practice checklist

1. Define whether the required quantity is J_A or N_A.
2. State the reference frame and identify any bulk motion.
3. Check phase, temperature, pressure, composition, and units for diffusivity.
4. Select equimolar or stagnant-species equations only when their assumptions apply.
5. Write the species balance before simplifying it.
6. State all initial and boundary conditions.
7. Estimate L_d or t_d before undertaking detailed computation.
8. Check signs, limiting cases, dimensions, and conservation.
9. Validate uncertain diffusivity or mass-transfer coefficients with measurements or trusted data.
10. Report assumptions and uncertainty with the final result.

Chapter Summary

Molecular diffusion is species transport caused by random molecular motion and a thermodynamic driving force. Fick’s first law relates diffusive flux to a concentration gradient, while Fick’s second law follows from species conservation and describes transient concentration changes. Total species flux includes both diffusion and convection.

For gases, ideal-gas relations convert mole-fraction gradients into concentration gradients. Equimolar counter-diffusion has zero total molar flux, whereas diffusion through a stagnant species produces a logarithmic flux relation because the diffusing species carries a bulk molar flow. Liquids generally have much lower diffusivities than gases, and concentrated solutions may require thermodynamic corrections. Solid-state diffusivity is commonly temperature dependent and may follow an Arrhenius equation.

Transient problems require initial and boundary conditions. The diffusion length √(Dt), diffusion time scale L²/D, mass Fourier number, and mass Biot number provide valuable physical interpretation. Reliable engineering analysis depends on correct flux definitions, dimensional consistency, realistic properties, validated assumptions, and recognition of reaction, convection, nonideality, and porous-media limitations.

Key Terms

Molecular diffusion: Species transport caused by random molecular motion.

Molar flux: Molar transport rate per unit area.

Mass flux: Mass transport rate per unit area.

Diffusive flux: Species flux relative to a chosen mixture-average velocity.

Total species flux: Sum of diffusive and convective species flux.

Molecular diffusivity: Proportionality coefficient relating diffusive flux to a concentration gradient.

Fick’s first law: Constitutive law for diffusion flux.

Fick’s second law: Transient diffusion equation obtained from species conservation.

Equimolar counter-diffusion: Counter-transport in which two species have equal and opposite molar fluxes.

Stagnant species: Species with zero total molar flux in the chosen direction.

Binary mixture: Mixture containing two chemical species.

Species conservation: Balance of accumulation, transport, and generation of a species.

Boundary condition: Physical or mathematical specification at a system boundary.

Initial condition: Concentration distribution at the start of a transient process.

Diffusion length: Characteristic penetration distance approximately equal to √(Dt).

Fourier number for mass diffusion: Dimensionless group Dt/L².

Mass Biot number: Ratio of external mass-transfer conductance to internal diffusive conductance.

Arrhenius diffusivity: Temperature-dependent diffusivity represented by an exponential activation-energy model.

Effective diffusivity: Macroscopic diffusivity adjusted for porous structure, tortuosity, or other restrictions.

Review Questions

5 Short Answer Questions

1. Distinguish diffusive flux J_A from total molar flux N_A.
2. State the assumptions behind the one-dimensional constant-property form of Fick’s first law.
3. Why does diffusion through stagnant B produce a logarithmic mole-fraction term?
4. Define the mass Fourier number and explain its physical meaning.
5. Name two factors that influence diffusivity in liquids or solids.

5 Long Answer Questions

1. Derive the one-dimensional steady diffusion equation for a stationary medium with constant diffusivity.
2. Explain equimolar counter-diffusion and derive its plane-layer flux expression for an ideal gas.
3. Starting with species conservation, derive Fick’s second law for a stationary medium with constant diffusivity and no reaction.
4. Compare diffusion in gases, liquids, and solids in terms of molecular mechanism, typical diffusivity, and engineering limitations.
5. Explain the physical meaning and engineering use of specified-concentration, specified-flux, impermeable, and convective boundary conditions.

3 Analytical / Application Questions

1. A gas species diffuses through a 3 mm stagnant layer at fixed temperature and pressure. Explain which data and equation are required to calculate flux, and identify two checks that should be made before accepting the result.
2. A polymer slab has thickness 4 mm and moisture diffusivity 2×10^-10 m²·s⁻¹. Estimate its diffusion time scale and discuss whether increasing air velocity will necessarily halve the drying time.
3. A catalytic porous pellet consumes species A by a first-order reaction. Formulate the governing differential equation and state physically appropriate centerline and external-surface boundary conditions.

Multiple Choice Questions with Answers

Question 1

Which statement correctly defines Fick’s first law for one-dimensional diffusion in a stationary medium?

A. J_A = D_AB dc_A/dz

B. J_A = −D_AB dc_A/dz

C. J_A = c_A v

D. J_A = k c_A

Correct Answer: B. J_A = −D_AB dc_A/dz

Question 2

In equimolar counter-diffusion, which condition applies?

A. N_A = N_B

B. N_A = 0

C. N_A = −N_B

D. N_B = 0

Correct Answer: C. N_A = −N_B

Question 3

For diffusion of A through stagnant B, what is the defining condition?

A. N_A = 0

B. N_B = 0

C. N_A = N_B

D. N_A + N_B = 0

Correct Answer: B. N_B = 0

Question 4

What is the SI unit of molecular diffusivity?

A. m·s⁻¹

B. mol·m⁻³

C. m²·s⁻¹

D. mol·m⁻²·s⁻¹

Correct Answer: C. m²·s⁻¹

Question 5

For constant diffusivity, no reaction, and no bulk convection, which equation describes transient diffusion?

A. ∂c_A/∂t = D_AB∇²c_A

B. ∂c_A/∂t = D_AB∇c_A

C. ∂c_A/∂t = −D_AB c_A

D. ∂c_A/∂t = 0

Correct Answer: A. ∂c_A/∂t = D_AB∇²c_A

Question 6

A process has D=1×10^-9 m²·s⁻¹ and L=1×10^-3 m. Its diffusion time scale L²/D is:

A. 1 s

B. 10 s

C. 1,000 s

D. 1,000,000 s

Correct Answer: C. 1,000 s

Question 7

For an ideal gas mixture, total molar concentration is given by:

A. c=RT/P

B. c=P/(RT)

C. c=P R T

D. c=T/(P R)

Correct Answer: B. c=P/(RT)

Question 8

Which dimensionless group is the mass Fourier number?

A. k_c L/D

B. ρ c_p L/k

C. D t/L²

D. μ/(ρD)

Correct Answer: C. D t/L²

Question 9

The Arrhenius model predicts that diffusivity generally does what when temperature increases?

A. Decreases exponentially

B. Increases exponentially

C. Remains exactly constant

D. Becomes zero

Correct Answer: B. Increases exponentially

Question 10

If the mass Biot number is very large, which resistance is generally important?

A. Internal diffusion resistance

B. Only radiation resistance

C. No mass-transfer resistance

D. Only electrical resistance

Correct Answer: A. Internal diffusion resistance