Chemistry · Lesson

Diffusion, conduction and coupled transport

Use fluxes, displacements and equilibrium correlations on explicit dimensional bases. Recognize the constitutive and scale assumptions behind simple diffusion, heat conduction and coupled linear response.

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By the end, you can

Convert a per-area flux into an amount or energy throughput.
Calculate dimension-dependent RMS net displacement without calling it a path length.
Distinguish component and vector correlation conventions and compatible coupled forces.

A flux needs an area and a direction

In a simple one-dimensional constant-coefficient diffusion model, J_c=−D dc/dx, with D in m² s⁻¹ and concentration c in mol m⁻³. Amount flux J_c has units mol m⁻² s⁻¹. Multiply by area for amount per time. In a separate Fourier model, J_q=−κ dT/dx in W m⁻², where this κ is thermal conductivity in W m⁻¹ K⁻¹. It differs from the transmission factor used earlier, and area converts it into heat power.

These laws add constitutive information. They assume appropriate continuum, local and linear regimes with the stated coefficients and maintained boundary values. Advection, reactions, nonideal chemical potentials and electric fields can require additional terms. Heat and matter examples with separate supplied coefficients need not describe one actual material.

Diffusion describes a distribution of net displacements

For zero-drift, unbounded homogeneous normal diffusion, ⟨Δx²⟩=2Dt in one dimension and ⟨|Δr|²⟩=6Dt in three dimensions. The RMS net displacements are their square roots. A zero mean displacement can coexist with a nonzero spread. Neither RMS value is the total travelled path or a guaranteed first-passage time to a target.

For the stipulated D=1.00 × 10⁻⁹ m² s⁻¹, at 10.0 s the one-dimensional RMS is approximately 141 μm and the three-dimensional RMS approximately 245 μm. Quadrupling time doubles either RMS distance. Confinement, drift, anomalous transport and microscopic inertial times need a different description or regime.

RMS net displacement depends on both time and dimension

RMS net displacement / μm

Elapsed time / s

One-dimensional RMS
Three-dimensional RMS
— One-dimensional √(2Dt)

Underlying data

Elapsed time / s	RMS net displacement / μm
1	44.721359549995796 · One-dimensional RMS
4	89.44271909999159 · One-dimensional RMS
10	141.4213562373095 · One-dimensional RMS
100	447.21359549995793 · One-dimensional RMS
1	77.45966692414835 · Three-dimensional RMS
4	154.9193338482967 · Three-dimensional RMS
10	244.94897427831785 · Three-dimensional RMS
100	774.5966692414834 · Three-dimensional RMS

Original zero-drift unbounded homogeneous normal-diffusion model, D=1.00 × 10⁻⁹ m² s⁻¹ and negligible initial spatial width. The shown times lie in the stipulated normal-diffusion regime. The dashed curve draws the one-dimensional RMS; three-dimensional values are selected markers. These are ensemble net-displacement spreads, not travelled path lengths, measured trajectories or first-passage times.

An equilibrium correlation connects microscopic motion to transport

For a stationary zero-drift equilibrium model, let C_x(t)=⟨v_x(0)v_x(t)⟩. A normal-diffusion coefficient is D=∫₀∞C_x(t)dt for a single Cartesian component. In an isotropic three-dimensional dot-product convention, D=(1/3)∫₀∞⟨v(0)·v(t)⟩dt. Mixing these conventions introduces a factor-of-three error.

An invented C_x(t)=C₀exp(−t/τ), with C₀=100 m² s⁻² and τ=1.00 × 10⁻¹¹ s, gives D=C₀τ=1.00 × 10⁻⁹ m² s⁻¹. Stopping the integral at t=τ gives only 1−exp(−1)≈ 63.2% of that value. A finite trajectory needs sampling, drift, boundary and tail checks; an arbitrary correlation is not automatically an electrical-conductivity or viscosity observable.

Compatible forces can produce cross-fluxes

Near equilibrium, a coupled model can write J_i=Σ L_ij X_j using thermodynamically conjugate forces and fluxes. Their product sum is the entropy-production density on the chosen consistent scale. Reciprocal coefficient relations require the appropriate time-reversal conditions; magnetic fields, rotation or differing parity require care. Raw temperature and concentration gradients cannot be combined into a meaningful symmetric matrix merely by ignoring their units.

For a stipulated reduced-unit example, use L=[[2,0.5],[0.5,1]] and matched dimensionless conjugate scales with σ/σ_ref=X·J and σ_ref>0. For X=(0.0100,−0.0200), J=(0.0100,−0.0150) and scaled production is 0.000400. This matrix is symmetric and positive definite, with determinant 1.75. Symmetry alone would not ensure positive production for an arbitrary matrix.

Worked examples

Separate amount flux from amount throughput

Use an invented one-dimensional diffusion model with D=1.00 × 10⁻⁹ m² s⁻¹, maintained c_left=2.00 and c_right=0 mol m⁻³ across ℓ=1.00 × 10⁻³ m, and area 1.00 × 10⁻⁴ m². Find the uniform model flux and throughput.

dc/dx=(0−2.00)/(1.00 × 10⁻³)=−2000 mol m⁻⁴.
J_c=−D dc/dx=+2.00 × 10⁻⁶ mol m⁻² s⁻¹.
Multiply by the stated area.
The sign is toward decreasing concentration in this model.

Flux +2.00 × 10⁻⁶ mol m⁻² s⁻¹; amount throughput 2.00 × 10⁻¹⁰ mol s⁻¹ under the stated maintained-boundary assumptions.

State the dimension before estimating a spread

For D=1.00 × 10⁻⁹ m² s⁻¹ and t=10.0 s, calculate one- and three-dimensional RMS net displacement.

One dimension: √(2Dt)=√(2.00 × 10⁻⁸ m²).
This gives 1.41421 × 10⁻⁴ m, or approximately 141 μm.
Three dimensions: √(6Dt)=√3 times the one-dimensional value.
This gives 2.44949 × 10⁻⁴ m, or approximately 245 μm.

Approximately 141 μm in one dimension and 245 μm in three. Neither result predicts a deterministic travel distance or arrival time.

Keep the missing correlation tail visible

For the stipulated component correlation C_x=C₀exp(−t/τ), C₀=100 m² s⁻² and τ=1.00 × 10⁻¹¹ s, find the full integral and the fraction captured up to τ.

The full integral is C₀τ=1.00 × 10⁻⁹ m² s⁻¹.
Up to τ, the integral is C₀τ[1−exp(−1)].
The captured fraction is approximately 0.632121.
About 36.8% remains in the tail of this exact exponential model.

D=1.00 × 10⁻⁹ m² s⁻¹; stopping at τ captures only 63.2%. No finite-data sampling claim follows from this invented analytic example.

Try it yourself

How must time change to double RMS displacement in the same normal-diffusion model?
Show the answer and reasoning
Increase it by a factor of 4.
RMS displacement scales as√t.
A flux is 2.00 × 10⁻⁶ mol m⁻² s⁻¹ over area 1.00 × 10⁻⁴ m². What is the throughput?
Show the answer and reasoning
2.00 × 10⁻¹⁰ mol s⁻¹.
Flux and throughput differ by the area.
Does a single-component velocity-correlation integral require the isotropic vector formula’s extra 1/3?
Show the answer and reasoning
No.
The 1/3 belongs to the three-component dot-product convention.
Does a symmetric numerical response matrix alone prove nonnegative entropy production?
Show the answer and reasoning
No.
The compatible symmetric part must also satisfy the required positivity conditions.

What to carry forward

Multiply a flux by area before calling it throughput.
Name dimensionality and the meaning of displacement.
Keep correlation observable, drift and integration tails explicit.
Coupled coefficients need compatible conjugate scales and positivity.

References

Teaching resources and reference material supporting this lesson’s explanations and examples.