Adsorption Isotherms: Langmuir, Freundlich and BET Compared

Molecules sticking to a surface, not dissolving into it

Adsorption is molecules from a gas or liquid accumulating on a solid surface, distinct from absorption where they diffuse into the bulk of a material. As pressure (for a gas) or concentration (for a solution) rises, more of the surface's available sites get occupied — an adsorption isotherm is simply the curve of surface coverage plotted against pressure at fixed temperature, and its shape reveals what kind of bonding is holding the molecules on.

Langmuir: the simplest honest model

Irving Langmuir's 1918 model assumes a fixed number of identical, independent sites, each holding at most one molecule, with no interaction between neighbouring adsorbed molecules. Treating adsorption and desorption as a dynamic equilibrium — adsorption rate proportional to pressure times the fraction of empty sites, desorption rate proportional to the fraction of occupied sites — and setting the two equal gives:

θ = K·P / (1 + K·P)

θ = fractional surface coverage (0 to 1)
P = pressure (or concentration)
K = equilibrium (binding) constant — larger K means stronger adsorption

At low pressure θ grows almost linearly with P (plenty of empty sites, adsorption rate-limited by how many molecules arrive); at high pressure θ saturates toward 1 as the surface runs out of empty sites — a hyperbolic curve that flattens into a clean plateau at monolayer coverage, exactly one molecule thick.

Freundlich: when sites are not all equal

Real surfaces are rarely uniform — different crystal faces, edges, and defects bind adsorbate with different strengths. The empirical Freundlich isotherm (1909, predating Langmuir's derivation) captures that heterogeneity with a simple power law and fits many real systems, especially at moderate coverage, better than Langmuir does:

θ = K · P^(1/n)

n > 1 typically (n=1 recovers a linear isotherm)
no saturation plateau built into the equation itself

Because it is empirical rather than derived from a physical binding mechanism, Freundlich has no built-in saturation limit — it keeps rising indefinitely as P grows, which makes it a good local fit over a limited pressure range but a poor description of the full curve out to true monolayer saturation, where Langmuir's plateau behaviour is physically more realistic.

BET: when adsorption does not stop at one layer

Langmuir assumes at most one adsorbed layer, which fails badly for physisorption by weak van der Waals forces, where a second, third and further molecular layers can stack on top of the first well before it saturates. Brunauer, Emmett and Teller extended Langmuir's kinetic argument in 1938 to allow multilayer adsorption, treating each layer beyond the first as behaving like the adsorbate condensing on itself (its own liquid), while only the first layer feels the true solid-surface binding energy:

V / Vm  =  C·x / [ (1 - x)·(1 - x + C·x) ]

x = P / P0    (relative pressure, P0 = saturation vapour pressure)
Vm = volume adsorbed at monolayer coverage
C  = constant related to the heat of adsorption of the first layer

The resulting curve is S-shaped rather than a simple plateau: a Langmuir-like rise toward monolayer coverage, a shoulder near x ≈ 0.3, and then an upward sweep as multilayer condensation takes over approaching x = 1. Plotting the BET equation in its linearised form against experimental gas-adsorption data is, to this day, the standard laboratory method for measuring a solid's total surface area — porous catalysts, activated carbon, and battery electrode materials are routinely characterised this way.

Physisorption versus chemisorption

The underlying bonding sets which model actually fits. Physisorption — weak van der Waals attraction, no electron sharing, low heat of adsorption (a few kJ/mol) — is reversible, occurs at any temperature, and readily forms multiple layers, which is exactly the regime BET was built for. Chemisorption — genuine chemical bond formation, an order of magnitude larger heat of adsorption (tens to hundreds of kJ/mol) — is far more selective, typically limited to a single layer since the surface's chemical bonding capacity is what is being used up, and Langmuir's single-layer assumption fits this regime far more naturally.

Frequently asked questions

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