Surface Coverage & Gibbs Adsorption Calculator

Understanding the Gibbs adsorption and surface coverage calculator — concept and principle

Surfactants lower surface tension because they crowd into the air-water surface. The Gibbs adsorption isotherm lets you work out how many molecules sit there per square metre, and how much area each one occupies, from the slope of a surface-tension curve. This calculator is used with tensiometry data on detergents, emulsifiers and wetting agents, typically from the steep region just below the critical micelle concentration (CMC).

A steeper fall of surface tension with ln C means more molecules packed on each square metre.

How it is calculated

Γ = −(1 / nRT) · dγ/d(ln C), and area per molecule a = 1 / (Γ · N_A)

Slope dγ/d(ln C) in mN/m: divided by 1000 to give N/m. It is negative because γ falls as C rises.
T in °C, converted to kelvin; R = 8.314462618 J/(mol·K).
n = 1 for nonionic surfactants (or ionic ones in excess salt), 2 for a 1:1 ionic surfactant without added salt.
N_A = 6.02214076 × 10²³ /mol. Γ is shown in µmol/m² and a in Å² (1 Å² = 10⁻²⁰ m²).

The principle

Gibbs showed from thermodynamics that a solute which lowers surface tension must be enriched at the surface. At constant temperature, dγ = −Γ·RT·d(ln C) for a dilute ideal solution, so the steeper γ falls with ln C, the more is adsorbed. For an ionic surfactant without salt, the counter-ions adsorb along with the surfactant ions and both contribute, which gives the factor n = 2. Just below the CMC the surface is nearly saturated and the γ–ln C plot is close to straight, so one slope gives the maximum surface excess.

Worked example

Inputs: slope = −8 mN/m = −0.008 N/m, T = 25 °C, n = 1.
nRT = 1 × 8.314462618 × 298.15 = 2478.96 J/mol.
Γ = 0.008 / 2478.96 = 3.2272 × 10⁻⁶ mol/m² = 3.227 µmol/m².
Molecules per m²: 3.2272 × 10⁻⁶ × 6.02214 × 10²³ = 1.9434 × 10¹⁸.
a = 1 / 1.9434 × 10¹⁸ m² = 5.1455 × 10⁻¹⁹ m² = 51.455 Å².

Each point is the calculator re-run with only “n (1 nonionic, 2 ionic)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

n (1 nonionic, 2 ionic)	Surface excess Γ (µmol/m²)
0.25	12.90865457
0.5	6.454327287
0.75	4.302884858
1	3.227163644
1.25	2.581730915
1.5	2.151442429
2	1.613581822

How to read the result

Saturated monolayers of common surfactants usually give Γ of a few µmol/m² and areas of several tens of Å² per molecule. A small area means tightly packed, upright chains; a large area suggests a bulky head group, charge repulsion or incomplete coverage. Comparing a with the head-group cross-section helps judge how close the layer is to full packing.

Common mistakes

Using a slope in log₁₀ C; multiply by 1/2.303 (that is, divide by ln 10) to convert to ln C first.
Choosing n = 1 for an ionic surfactant in pure water, which doubles Γ and halves the area.
Taking the slope above the CMC, where γ is flat and the slope is near zero.

Keep in mind: The equation assumes dilute ideal behaviour and a pure surfactant. Trace surface-active impurities can create a minimum in the γ curve and distort the slope, so treat results as estimates.

Frequently asked questions

Does this calculator find the CMC itself?

No. The CMC is read from the break point where the γ–ln C curve levels off; this tool uses the slope just before that point.

When should I use n = 2?

For a 1:1 ionic surfactant such as SDS in water without added electrolyte. With a large excess of salt, use n = 1.

Why give the area in Å²?

Molecular head groups are a few ångströms across, so Å² gives readable numbers (1 nm² = 100 Å²).

Sources: J. W. Gibbs, "On the Equilibrium of Heterogeneous Substances", Transactions of the Connecticut Academy, 1876–1878 · M. J. Rosen and J. T. Kunjappu, Surfactants and Interfacial Phenomena, Wiley

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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