Fick's Law of Diffusion Equation, Derivation & Rearrangements

Core idea

Overview

This principle governs how substances like oxygen and carbon dioxide move across biological membranes in the lungs and capillaries. Maximizing surface area and minimizing membrane thickness are key physiological strategies to ensure efficient gas exchange. Understanding this law explains why respiratory distress occurs when the diffusion pathway is widened by fluid or scarring.

When to use: Apply this when calculating or evaluating the efficiency of gas exchange across a biological barrier, such as the alveolar-capillary membrane.

Why it matters: It explains the fundamental limitations of gas exchange in the human body, dictating the design of lungs, gills, and capillary networks.

Symbols

Variables

Rate = Rate of diffusion, A = Surface Area, \Delta C = Concentration Gradient, T = Membrane Thickness

R a t e

Rate of diffusion

V a r iab l e

A

Surface Area

V a r iab l e

Δ C

Concentration Gradient

V a r iab l e

T

Membrane Thickness

V a r iab l e

Walkthrough

Derivation

Derivation of Fick's Law of Diffusion

Fick's Law is derived from the principle of conservation of mass in a fluid medium, relating the flux of a solute to the spatial variation of its concentration. It describes how diffusion naturally moves particles from areas of high concentration to low concentration.

The medium is isotropic and homogeneous.
Diffusion occurs under steady-state conditions (concentration gradient remains constant over time).
The membrane is thin and acts as a passive barrier to solute transport.

1

Defining Net Flux

The net flux (J) is proportional to the concentration gradient (dc/dx) and the diffusion coefficient (D). The negative sign indicates movement from high to low concentration.

J = - D \frac{d c}{d x}

Note: This is known as Fick's First Law.

2

Integrating across the membrane

For a membrane of thickness Δx, we approximate the concentration gradient as the difference between concentrations C1 and C2 divided by the membrane thickness.

J = D \frac{C _{1} - C _{2}}{Δ x}

Note: C1 - C2 represents the concentration gradient.

3

Accounting for Surface Area

To find the total rate of diffusion, we multiply the flux by the cross-sectional surface area (A) available for exchange.

R a t e = J \times A = \frac{D \times A \times ( C _{1} - C _{2} )}{Δ x}

Note: Since D is constant for a specific solute, we express this as Rate ∝ (Area × Concentration Gradient) / Thickness.

Result

R a t e = J \times A = \frac{D \times A \times ( C _{1} - C _{2} )}{Δ x}

Source: AQA A-Level Biology Specification / Guyton and Hall Textbook of Medical Physiology

One free problem

Practice Problem

If the surface area for gas exchange is 50 m?, the concentration gradient is 2 mol/m?, and the membrane thickness is 0.05 m, what is the rate of diffusion assuming a constant of 1?

Surface Area50

Concentration Gradient2

Membrane Thickness0.05

Solve for: $R a t e$

Hint: Divide the product of Area and Gradient by the Thickness.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In pulmonary edema, fluid accumulates in the interstitial space of the lungs, increasing the thickness of the membrane and drastically reducing the rate of oxygen diffusion into the blood.

Study smarter

Tips

Remember that the rate is proportional, not equal, unless given a specific diffusion constant.
Always ensure units for area and thickness are consistent before calculating.
Think of this as a trade-off equation: to increase rate, you must increase area or gradient, or decrease thickness.

Avoid these traps

Common Mistakes

Confusing inversely proportional by putting thickness in the numerator.
Forgetting that the concentration gradient is the difference in concentration between the two sides of the membrane.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Fick's Law is derived from the principle of conservation of mass in a fluid medium, relating the flux of a solute to the spatial variation of its concentration. It describes how diffusion naturally moves particles from areas of high concentration to low concentration.

Apply this when calculating or evaluating the efficiency of gas exchange across a biological barrier, such as the alveolar-capillary membrane.

It explains the fundamental limitations of gas exchange in the human body, dictating the design of lungs, gills, and capillary networks.

Confusing inversely proportional by putting thickness in the numerator. Forgetting that the concentration gradient is the difference in concentration between the two sides of the membrane.