The Gibbs Adsorption Isotherm

Drop a splash of dish soap on greasy water. The film races outward, thinning the grease in seconds. Something is piling up at the surface. That pileup is quietly lowering the surface tension. The Gibbs adsorption isotherm turns this everyday observation into a precise thermodynamic tool. It connects a measurable macroscopic quantity to a microscopic one. The macroscopic quantity is the slope of surface tension against solute activity. The microscopic one is how many excess solute molecules sit per unit area of interface. No molecular model, no fitted parameters — just the second law applied to a two-dimensional phase. This article builds the surface excess concept from the Gibbs dividing surface. It derives the isotherm rigorously and applies it to real surfactant data. It closes by pointing toward the Langmuir isotherm. That isotherm supplies the missing microscopic picture of how molecules actually pack at an interface.

Key takeaways

The Gibbs adsorption isotherm relates surface tension $\gamma$ , solute activity $a$ , and surface excess concentration $\Gamma$ .
$\Gamma$ is not a real physical density; it is defined relative to an imaginary "dividing surface" with zero excess solvent.
For a solute that lowers $\gamma$ as concentration rises, $\Gamma$ is positive: the solute accumulates at the interface.
The isotherm is derived from the Gibbs–Duhem equation applied separately to bulk and surface phases.
Plotting $\gamma$ against $\ln a$ for a surfactant gives a slope that yields the area occupied per adsorbed molecule.
Key idea: the rate at which surface tension falls with solute activity directly measures how much solute has accumulated at the interface, via $\Gamma = -(1/RT)(\partial\gamma/\partial \ln a)_T$ .

The core insight

Because the interface is itself a thermodynamic phase with its own Gibbs–Duhem relation, any change in the chemical potential of a bulk component must be balanced by a corresponding change in surface tension weighted by that component's surface excess — so measuring how $\gamma$ responds to bulk composition is equivalent to measuring surface composition directly.

The physical picture

A real interface between two bulk phases is not a mathematical plane; it is a thin region, a few molecular diameters thick, over which density and composition vary continuously from one bulk value to the other. Gibbs's insight was to replace this messy transition zone with an idealized sharp mathematical surface — the dividing surface — placed at some convenient location, and then to define surface excess quantities as the difference between what is actually present and what would be present if the bulk phases extended uniformly right up to that surface. For a solute $2$ dissolved in solvent $1$ , the surface excess concentration $\Gamma_2$ is defined after choosing the dividing surface so that the solvent's own surface excess is zero ( $\Gamma_1 = 0$ ). With that convention, $\Gamma_2$ has a clean physical meaning: it is the amount of solute per unit area that is "extra," over and above what a uniform bulk solution would contribute, sitting at the interface. If the solute prefers the surface to the bulk, $\Gamma_2 > 0$ ; if it is repelled from the surface, $\Gamma_2 < 0$ . Surfactant molecules — a polar head group attached to a nonpolar hydrocarbon tail — are strongly surface-active. The tail avoids water, the head is happy in water, so the molecule minimizes its free energy by sitting at the air–water interface, tail poking into air, head anchored in water. This accumulation does two things simultaneously: it increases $\Gamma_2$ , and it lowers $\gamma$ , because replacing some water–air contacts with surfactant–air contacts costs less energy. The Gibbs isotherm formalizes the link between these two effects.

Figure

The Gibbs dividing surface: real density profile versus the idealized step

The core relations

d\gamma = -\sum_i \Gamma_i\, d\mu_i \qquad (T \text{ constant})

where: $\gamma$ is surface tension, $\Gamma_i$ is the surface excess concentration (mol m⁻²) of component $i$ , and $\mu_i$ its chemical potential.

\Gamma_2 = -\frac{1}{RT}\left(\frac{\partial \gamma}{\partial \ln a_2}\right)_T

where: $\Gamma_2$ is the surface excess of solute 2 (solvent excess set to zero by choice of dividing surface), $a_2$ its activity, $R$ the gas constant, $T$ the temperature.

\Gamma_2 \approx -\frac{c_2}{RT}\left(\frac{\partial \gamma}{\partial c_2}\right)_T \quad (\text{dilute, } a_2 \approx c_2/c^\ominus)

where: $c_2$ is bulk molar concentration, valid when activity coefficients are near unity.

A_m = \frac{1}{N_A \Gamma_2}

where: $A_m$ is the area occupied per adsorbed molecule at surface saturation, $N_A$ Avogadro's number.

Derivation AFrom the Gibbs–Duhem equation to the adsorption isotherm

Goal. Show that the slope of surface tension against solute activity measures the surface excess concentration of that solute.

Step 1. Define the surface as a thermodynamic system. Treat the interfacial region as a two-dimensional phase with its own extensive internal energy $U^\sigma$ , entropy $S^\sigma$ , area $A$ , and mole numbers $n_i^\sigma$ . Its fundamental equation, by analogy with a bulk phase but with $\gamma\,dA$ replacing $-p\,dV$ , is

dU^\sigma = T\,dS^\sigma + \gamma\,dA + \sum_i \mu_i\, dn_i^\sigma

Here $\gamma$ plays the role of a (negative) two-dimensional pressure: stretching the surface costs energy $\gamma\,dA$ , unlike expanding a bulk gas which releases energy.

Step 2. Integrate at constant intensive variables. Because $U^\sigma$ is a first-order homogeneous function of the extensive variables $S^\sigma, A, n_i^\sigma$ at fixed $T$ , $\gamma$ , $\mu_i$ , Euler's theorem gives

U^\sigma = TS^\sigma + \gamma A + \sum_i \mu_i n_i^\sigma

Step 3. Take the total differential and compare. Differentiating Step 2 fully,

dU^\sigma = T\,dS^\sigma + S^\sigma dT + \gamma\,dA + A\,d\gamma + \sum_i \mu_i\,dn_i^\sigma + \sum_i n_i^\sigma\,d\mu_i

Subtracting the Step 1 expression for $dU^\sigma$ term by term leaves the surface Gibbs–Duhem relation:

0 = S^\sigma dT + A\,d\gamma + \sum_i n_i^\sigma\, d\mu_i

Step 4. Hold temperature fixed and divide by area. At constant $T$ , $dT = 0$ , so

A\,d\gamma = -\sum_i n_i^\sigma\, d\mu_i

Dividing through by $A$ and using the definition of surface excess concentration, $\Gamma_i \equiv n_i^\sigma / A$ (the mole number assigned to the surface phase per unit area, relative to the dividing surface), gives

d\gamma = -\sum_i \Gamma_i\, d\mu_i

This is the general Gibbs adsorption equation, valid for any number of components.

Step 5. Specialize to solvent + one solute. For solvent (1) and solute (2), $d\gamma = -\Gamma_1 d\mu_1 - \Gamma_2 d\mu_2$ . Gibbs's convenient choice of dividing surface sets $\Gamma_1 = 0$ — the surface is drawn at exactly the position where the solvent has no excess. This is always geometrically possible for a two-component system. Then

d\gamma = -\Gamma_2\, d\mu_2

Step 6. Express $\mu_2$ through measurable activity. Since $\mu_2 = \mu_2^\ominus + RT\ln a_2$ at constant $T$ , we have $d\mu_2 = RT\, d\ln a_2$ . Substituting,

d\gamma = -\Gamma_2 RT\, d\ln a_2

Step 7. Solve for the surface excess.

\boxed{\Gamma_2 = -\frac{1}{RT}\left(\frac{\partial \gamma}{\partial \ln a_2}\right)_T}

Every quantity on the right is measurable: $\gamma$ from a tensiometer, $a_2$ from bulk thermodynamic data. No assumption about molecular packing, orientation, or a specific isotherm form was needed — the result rests only on the first and second laws applied to the surface phase.

Worked examples

Example 1 — Foundational: sodium dodecyl sulfate (SDS) below its critical micelle concentration. Surface tension measurements for aqueous SDS near room temperature ( $T = 298\ \text{K}$ ) show that, over the concentration range $2$ – $6\ \text{mmol dm}^{-3}$ , $\gamma$ decreases linearly with $\ln c$ with slope

\left(\frac{\partial \gamma}{\partial \ln c}\right)_T = -19.1\ \text{mN m}^{-1}

(a typical literature value for this system; concentrations here are low enough that $a_2 \approx c_2 / c^\ominus$ and activity coefficients are treated as unity). Step 1 — surface excess.

\Gamma_2 = -\frac{1}{RT}\left(\frac{\partial \gamma}{\partial \ln c}\right)_T = -\frac{(-19.1\times10^{-3}\ \text{N m}^{-1})}{(8.314\ \text{J K}^{-1}\text{mol}^{-1})(298\ \text{K})}

\Gamma_2 = \frac{19.1\times10^{-3}}{2478} \ \text{mol m}^{-2} = 7.71\times10^{-6}\ \text{mol m}^{-2}

Step 2 — area per molecule.

A_m = \frac{1}{N_A \Gamma_2} = \frac{1}{(6.022\times10^{23}\ \text{mol}^{-1})(7.71\times10^{-6}\ \text{mol m}^{-2})}

A_m = \frac{1}{4.64\times10^{18}\ \text{m}^{-2}} = 2.15\times10^{-19}\ \text{m}^2 = 21.5\ \text{Å}^2

Sanity check. A close-packed SDS monolayer is known experimentally to occupy roughly $30$ – $50\ \text{Å}^2$ per headgroup depending on ionic strength; our value of $21.5\ \text{Å}^2$ is somewhat below that because this slope was taken close to the critical micelle concentration where packing is near maximal but the ionic-strength correction (SDS is ionic, so the Gibbs equation technically needs a factor of 2 for the counter-ion, giving $A_m \approx 43\ \text{Å}^2$ ) has been omitted for pedagogical simplicity. The order of magnitude — a few tens of Å² per surfactant, consistent with close-packed hydrocarbon chains — is exactly right.

Example 2 — Applied: estimating maximum surface excess and cross-sectional area for a nonionic surfactant, then checking consistency with $\gamma$ at two concentrations. A nonionic surfactant in water at $T = 293\ \text{K}$ gives the following tensiometer data:

$c$ / mmol dm⁻³	$\gamma$ / mN m⁻¹
0.10	62.0
1.00	45.5

Assume dilute-solution behavior so $a_2 \approx c_2/c^\ominus$ , and estimate $\Gamma_2$ from the finite-difference slope of $\gamma$ against $\ln c$ . Step 1 — compute $\Delta \ln c$ .

\ln(1.00) - \ln(0.10) = \ln(10) = 2.303

Step 2 — compute $\Delta \gamma$ .

\Delta \gamma = (45.5 - 62.0)\ \text{mN m}^{-1} = -16.5\ \text{mN m}^{-1} = -16.5\times10^{-3}\ \text{N m}^{-1}

Step 3 — slope.

\left(\frac{\partial \gamma}{\partial \ln c}\right)_T \approx \frac{-16.5\times10^{-3}}{2.303}\ \text{N m}^{-1} = -7.17\times10^{-3}\ \text{N m}^{-1}

Step 4 — surface excess.

\Gamma_2 = -\frac{-7.17\times10^{-3}}{(8.314)(293)} \ \text{mol m}^{-2} = \frac{7.17\times10^{-3}}{2436}\ \text{mol m}^{-2} = 2.94\times10^{-6}\ \text{mol m}^{-2}

Step 5 — area per molecule.

A_m = \frac{1}{(6.022\times10^{23})(2.94\times10^{-6})}\ \text{m}^2 = \frac{1}{1.77\times10^{18}}\ \text{m}^2 = 5.65\times10^{-19}\ \text{m}^2 = 56.5\ \text{Å}^2

Sanity check. This is comparable to headgroup areas reported for nonionic ethoxylate surfactants ( $40$ – $70\ \text{Å}^2$ ), and it is larger than the SDS value above — reasonable, since nonionic surfactants often have bulkier, hydrated headgroups and no electrostatic repulsion forcing tighter or looser packing in a predictable direction; the number is physically sensible and not an order of magnitude off.

Figure

Surface tension versus ln(concentration): slope gives surface excess

Limiting cases & common traps

Situation	What happens	Why
$\gamma$ decreases with $c_2$	$\Gamma_2 > 0$ : positive adsorption	Solute lowers surface free energy by accumulating there
$\gamma$ increases with $c_2$ (e.g. simple electrolytes in water)	$\Gamma_2 < 0$ : negative adsorption	Ions are pulled into bulk away from the low-dielectric interface
Very dilute solute	$\Gamma_2 \propto$ slope of $\gamma$ vs. $c$ (linear regime)	$a_2 \approx c_2$ , Henry's-law-like behavior
Above the critical micelle concentration	$\gamma$ plateaus, isotherm slope $\to 0$	Surface is saturated; added surfactant forms micelles, not more surface excess
Ionic surfactant, no swamping electrolyte	Naive isotherm underestimates $\Gamma_2$ by up to a factor of 2	Counter-ion also has a surface excess and its $d\mu$ term cannot be dropped

The chemist behind it

Josiah Willard Gibbs published the entire framework of surface thermodynamics — dividing surfaces, surface excess quantities, and the adsorption isotherm — inside his monumental 1878 memoir On the Equilibrium of Heterogeneous Substances. Working largely in isolation at Yale, with no laboratory of his own, Gibbs derived results that experimentalists would not fully exploit for decades, until researchers such as Irving Langmuir connected the thermodynamic surface excess to explicit molecular models of monolayer packing.

Origins · 1 minJosiah Willard Gibbs and how this was discovered

Josiah Willard Gibbs

1839–1903 · American mathematical physicist who founded chemical thermodynamics and statistical mechanics, deriving the surface adsorption isotherm from pure thermodynamic reasoning decades before it was experimentally tested

Ask the tutor: How would the Gibbs isotherm change if I needed to account for both the surfactant and its counter-ion explicitly, and how does that connect to the Langmuir model of monolayer saturation?

Why is the Gibbs isotherm considered "model-free" compared to the Langmuir isotherm?

The Gibbs isotherm is derived purely from the first and second laws applied to a surface phase treated thermodynamically, using the Gibbs–Duhem relation. It makes no assumption about adsorption sites, monolayer saturation, or molecular geometry — it simply relates two measurable quantities, $\gamma$ and $a_2$ . The Langmuir isotherm, by contrast, assumes a specific microscopic mechanism: a fixed number of equivalent sites, each holding at most one molecule, with dynamic equilibrium between adsorption and desorption. Gibbs gives you $\Gamma_2$ ; Langmuir tells you why $\Gamma_2$ has the concentration dependence it does.

Can the Gibbs isotherm be applied to solid–gas or solid–liquid interfaces, not just liquid–vapor?

Yes in principle — the underlying thermodynamics does not require a liquid surface. In practice it is far harder to apply, because $\gamma$ for a solid surface is not directly measurable with a tensiometer or capillary-rise method the way liquid surface tension is. Solid surface energies are usually inferred indirectly, from contact angles (via the Young equation) or from calorimetric and computational methods, so the Gibbs isotherm is used far more often for liquid interfaces where $\gamma(c)$ can be measured directly and accurately.

What happens to the Gibbs isotherm above the critical micelle concentration?

Above the critical micelle concentration, surfactant monomers self-assemble into micelles in the bulk rather than continuing to adsorb at the interface, because the surface is already saturated. The chemical potential of free monomer becomes nearly constant with added surfactant, since extra surfactant simply forms more micelles at essentially fixed monomer activity. Consequently $\partial \gamma/\partial \ln c \to 0$ and $\gamma$ plateaus. The Gibbs isotherm still holds, but it correctly predicts no further increase in $\Gamma_2$ , since the monomer activity driving adsorption has stopped changing.

Does the sign of Γ₂ tell us anything about where the solute sits relative to the interface?

Yes, qualitatively. Positive $\Gamma_2$ (surface tension falls with concentration) means the solute is thermodynamically favored at the interface relative to the bulk — typical of amphiphiles and many organic solutes in water. Negative $\Gamma_2$ (surface tension rises with concentration, as for many simple inorganic salts in water) means the solute is depleted near the interface, pushed into the bulk by the unfavorable electrostatic environment of the low-dielectric air–water boundary. This underlies the classical Jones–Ray effect and Hofmeister-series behavior at electrolyte interfaces.