Fick's Law Calculators

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Fick's first law of diffusion describes the flux of a substance moving from high to low concentration: J = −D × (dC/dx), where J is the flux (mol/m²/s), D is the diffusion coefficient (m²/s), and dC/dx is the concentration gradient. The negative sign indicates flow down the gradient. Fick's second law describes how concentration changes with time: ∂C/∂t = D × ∂²C/∂x². In biology, Fick's law governs gas exchange in the lungs (O₂ and CO₂ across the alveolar membrane), drug absorption, and nutrient transport. The rate of diffusion increases with surface area, concentration gradient, and diffusion coefficient, and decreases with membrane thickness.

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Fick's First Law

J = −D × (dC/dx)

J = diffusion flux (mol m⁻² s⁻¹). D = diffusion coefficient (m² s⁻¹). dC/dx = concentration gradient (mol m⁻⁴). Example: O₂ diffuses from alveolus (PO₂ = 100 mmHg) to blood (PO₂ = 40 mmHg); diffusion rate ∝ ΔP × A / (D × T). A = surface area; T = membrane thickness.

Fick's Second Law

∂C/∂t = D × ∂²C/∂x²

Describes how the concentration profile changes with time during diffusion. At steady state: ∂C/∂t = 0 → linear concentration gradient. Time to diffuse distance x: t ≈ x²/(2D). Example: glucose (D ≈ 7×10⁻¹⁰ m²/s) diffusing 10 μm: t = (10⁻⁵)² / (2 × 7×10⁻¹⁰) = 7.1 × 10⁻³ s ≈ 7 ms.

Fick's Law in Lung Gas Exchange

Rate of gas transfer = D × A × ΔP / T. D = diffusion coefficient (CO₂ diffuses ~20× faster than O₂). A = alveolar surface area (70 m² in healthy lungs). ΔP = partial pressure gradient. T = alveolar membrane thickness (~0.5 μm). Doubling A (e.g., exercise → more alveolar capillary recruitment) doubles transfer rate.

Glossary

Fick's First Law
J = −D × (dC/dx); flux = diffusion coefficient × concentration gradient; flux flows down the gradient (negative sign); doubles with doubled gradient or halved membrane thickness.
Fick's Second Law
∂C/∂t = D × ∂²C/∂x²; describes time evolution of concentration during diffusion; t ≈ x²/2D; explains why diffusion is fast at μm scale but too slow for mm-scale transport.
Diffusion Coefficient (D)
Molecule-specific constant (m²/s) in Fick's laws; decreases with molecule size (D ∝ 1/√M); increases with temperature; decreases in viscous media; CO₂ diffuses ~20× faster than O₂ in tissue.

Frequently Asked Questions

Fick's first law: J = −D × (dC/dx). J = flux = moles crossing a unit area per unit time (mol m⁻² s⁻¹). D = diffusion coefficient (m² s⁻¹); depends on molecule size, temperature, and medium. dC/dx = concentration gradient across the diffusion distance. The negative sign: flux is in the direction of decreasing concentration (downhill). Practical form: for a membrane of thickness T: J = D × (C₁ − C₂) / T. Doubling the concentration gradient doubles the flux. Halving the membrane thickness doubles the flux.

Fick's second law: ∂C/∂t = D × ∂²C/∂x². Describes how the concentration profile evolves with time during non-steady-state diffusion. Used when: diffusion has not yet reached steady state (transient diffusion); modeling drug absorption over time; neuronal signaling (action potential propagation involves ion diffusion); heat conduction (mathematically identical: Fourier's law). Simplified estimate: time for a molecule to diffuse a distance x ≈ x²/(2D). Example: small ion (D = 10⁻⁹ m²/s) diffusing 1 mm: t = (10⁻³)²/(2×10⁻⁹) = 500 s ≈ 8 minutes — explains why diffusion is fast across cell membranes (nm–μm) but too slow for long-distance transport in organisms.

Gas exchange rate across the alveolar membrane: V̇_gas = D × A × (P₁ − P₂) / T. D = diffusion coefficient for the gas in tissue. A = alveolar surface area (adult ~70 m², ~500 million alveoli). P₁ − P₂ = partial pressure difference (driving force). T = membrane thickness (~0.5 μm average). O₂: P_alveolar = 100 mmHg; P_capillary = 40 mmHg → ΔP = 60 mmHg. CO₂: P_capillary = 46 mmHg; P_alveolar = 40 mmHg → ΔP = 6 mmHg. Yet CO₂ transfer rate ≈ O₂ because D_CO₂ ≈ 20 × D_O₂ in tissue (CO₂ is much more soluble in water). Factors reducing gas exchange: reduced A (emphysema → alveolar wall destruction); increased T (pulmonary edema, fibrosis).

Diffusion rate ∝ (D × A × ΔC) / T. D (diffusion coefficient): larger molecules diffuse more slowly (D ∝ 1/√M). D increases with temperature (molecules move faster). D decreases in more viscous media. Surface area (A): larger exchange surface = faster diffusion; maximized in lungs (alveoli), gut (villi, microvilli), and cells (large surface-to-volume ratio). Concentration gradient (ΔC): maintained by: ventilation (removes CO₂, replenishes O₂ in alveoli); blood flow (carries O₂ away from alveoli, brings CO₂). Membrane thickness (T): thinner = faster diffusion; alveolar membrane is ~0.5 μm; swelling in pulmonary edema increases T → impaired gas exchange.