2.0171e-7
mol/m²
1.2147e+17
molecules/m²
8.2324
nm²
823.24
Ų
2.0171e-7
mol/m²
1.2147e+17
molecules/m²
8.2324
nm²
823.24
Ų
The Gibbs Adsorption Calculator computes the surface excess concentration (Γ) using the Gibbs adsorption equation, which relates changes in surface tension to the accumulation of solute molecules at an interface. This fundamental thermodynamic equation is a cornerstone of surface chemistry and colloid science, enabling the determination of how many molecules adsorb at a surface from measurable quantities. When surfactants or other surface-active solutes are added to a solution, the surface tension decreases, and the Gibbs equation quantifies the resulting surface excess. This calculator is essential for understanding surfactant adsorption, Langmuir monolayers, detergent performance, emulsion stability, and biological membrane interfaces. The output includes surface excess in mol/m² and the molecular area, which reveals the packing density of molecules at the interface.
The Gibbs adsorption equation for a single non-ionic solute is:
$$\Gamma = -\frac{1}{RT} \cdot \frac{d\gamma}{d\ln C} = -\frac{C}{RT} \cdot \frac{d\gamma}{dC}$$
where Γ is the surface excess concentration (mol/m²), R = 8.314 J/(mol·K) is the gas constant, T is the absolute temperature, γ is the surface tension, and C is the bulk concentration. For ionic surfactants, an additional factor of 1/2 (or 1/(n+1) for multivalent ions) may be needed depending on the counterion behavior.
The molecular area at the interface is calculated from:
$$A = \frac{1}{\Gamma \cdot N_A}$$
where N_A is Avogadro's number. Typical molecular areas for surfactants at maximum packing range from 0.2 to 0.6 nm² (20–60 Ų) per molecule.
The quantity dγ/dC is the slope of the surface tension vs. concentration curve, which must be determined experimentally (e.g., from tensiometry data).
A positive surface excess (Γ > 0) means the solute accumulates at the surface — this is the case for surfactants and other surface-active molecules. A negative surface excess (Γ < 0) means the solute is depleted at the surface — inorganic salts like NaCl slightly increase water's surface tension and have negative surface excess. The molecular area provides insight into molecular packing and orientation. Small areas (~0.2 nm²) indicate tightly packed, vertically oriented molecules. Larger areas (~0.5–1.0 nm²) suggest tilted or loosely packed configurations. Comparing the molecular area with the cross-sectional area of the hydrophobic chain reveals whether packing is dominated by headgroup or chain interactions.
Inputs
Results
Sodium dodecyl sulfate at 5 mM with dγ/dC = -0.05 N·L/(m·mol) gives a surface excess of ~1.01 × 10⁻⁷ mol/m², corresponding to ~165 Ų per molecule — still below maximum packing.
Inputs
Results
Near the critical micelle concentration, very steep dγ/dC slopes give high surface excess values approaching the limiting molecular area of ~40–55 Ų for typical surfactants. Values below ~20 Ų suggest multilayer or more complex adsorption behavior.
Surface excess (Γ) is the amount of solute per unit area at the interface in excess of what would be present if the bulk concentration extended uniformly up to the surface. It is defined relative to a chosen dividing surface (the Gibbs dividing surface). Positive Γ means more solute at the surface than in the bulk.
Measure surface tension (γ) at several concentrations below the CMC using a tensiometer (Wilhelmy plate, Du Noüy ring, or pendant drop). Plot γ vs. C (or γ vs. ln C). The slope of γ vs. C at the concentration of interest gives dγ/dC. For γ vs. ln C plots, the slope directly gives -RTΓ.
Above the CMC, the surface is saturated and surface tension is approximately constant (dγ/dC ≈ 0). The Gibbs equation still applies formally, but it becomes uninformative because the surface excess reaches a constant maximum value. The equation is most useful below and approaching the CMC.
For a 1:1 ionic surfactant (e.g., SDS → Na⁺ + DS⁻) without added salt, the Gibbs equation becomes Γ = -(1/2RT)(dγ/d ln C) because both ions contribute to the chemical potential. With excess salt (swamping electrolyte), the 1/2 factor is dropped because the counterion concentration is essentially constant.
For typical surfactants at the air-water interface near CMC: Γ_max ≈ 3–5 × 10⁻⁶ mol/m², corresponding to molecular areas of 35–55 Ų. For phospholipids at equilibrium: Γ ≈ 2–3 × 10⁻⁶ mol/m² (~55–80 Ų). Long-chain alcohols: Γ ≈ 5–8 × 10⁻⁶ mol/m² (~20–35 Ų).
The Gibbs dividing surface is an imaginary plane that defines where the interface is located. It is chosen so that the surface excess of the solvent is zero (Γ_solvent = 0). This convention means all calculated surface excess values are relative to this choice. It simplifies the mathematics but makes the surface excess a relative, not absolute, quantity.
The Gibbs equation is general and applies to any interface, but for solid surfaces, it is difficult to measure surface tension changes directly. Instead, techniques like contact angle measurements, adsorption isotherms, or calorimetry are used to characterize solid surface adsorption. The Gibbs equation is most directly applied to fluid interfaces.
The molecular area (A = 1/(ΓNA)) reveals how molecules pack at the interface. If A matches the cross-sectional area of the hydrocarbon chain (~20 Ų), molecules are close-packed and vertically oriented. If A matches the headgroup area (~40–60 Ų), headgroup repulsion limits packing. The transition reveals the dominant interaction at the interface.
Temperature affects surface excess through two mechanisms: (1) the 1/RT factor in the Gibbs equation directly — higher T reduces Γ for the same dγ/dC, and (2) temperature changes dγ/dC itself by altering adsorption thermodynamics and bulk solution behavior. Generally, surface excess decreases with increasing temperature.
Surface pressure (π = γ₀ - γ) is the reduction in surface tension caused by the adsorbed film. The Gibbs equation can be rewritten as: Γ = (C/RT)(dπ/dC). At low surface coverage, π is related to Γ by the 2D ideal gas law: πA = kT, where A = 1/(ΓNA). This connects Gibbs adsorption to Langmuir monolayer thermodynamics.
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