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Adsorption is accumulation of a substance at an interface. Binding a solute to a packing surface can delay its passage through a chromatography column. Ideal size-exclusion chromatography separates through SEC pore access without such binding; extra retention from adsorption requires a different interpretation.

For a gas contacting a solid, adsorption concentrates molecules at the boundary; absorption takes material into the bulk. A porous solid can have a large internal surface area: binding on pore walls is still adsorption, even though those walls lie inside the particle. Both processes can occur together. Desorption returns adsorbed material to the surrounding phase. The useful question is therefore not just how much material disappeared from the gas, but where it went and whether it returns when the pressure falls.

An adsorption site model turns this spatial distinction into counting. With twelve identical adsorption sites and three bound molecules, the surface coverage is $\theta =3/12=1/4$. Filling six gives $\theta =1/2$; filling twelve gives a complete monolayer. These are configurations illustrating the count, rather than measured arrangements or claims that all real surfaces have twelve sites.

The chemical potential of an ideal single-occupancy lattice gas with occupied-site internal partition function $z$ and surface coverage $\theta$. The ratio $\theta /(1-\theta )$ accounts for the entropy of distributing particles among occupied and vacant adsorption sites.

An adsorption site is a position or local environment at which a chemical species can be retained at a boundary. Treating all adsorption sites as identical is an assumption about their binding properties, not a definition of adsorption.

Colin D. Bain coauthored research on monolayer.^[ref]

A monolayer is a single layer of chemical atoms or molecules. In the Langmuir adsorption model, a complete monolayer occupies every adsorption site; partial surface coverage leaves vacancies within that first layer. One layer need not mean one molecule per surface chemical atom.

Competitive adsorption occurs when different chemical species seek the same adsorption sites. For intact molecules $A$ and $B$, retain identical adsorption sites and no interactions, but replace the vacant fraction by $1-{\theta }_A-{\theta }_B$. The same equilibrium balance givesHere $P_A$ and $P_B$ are partial pressures, not total pressure. Increasing $P_B$ can reduce ${\theta }_A$ at unchanged $P_A$: replacing $P_A$ by total pressure in a single-species Langmuir isotherm loses this competition.

In dissociative adsorption, an arriving molecule splits into bound fragments. For the idealized process $A_2+2\ast \rightleftharpoons 2A\ast$, let $\ast$ denote a vacant adsorption site and let $\theta$ count sites occupied by $A$. A mean-field approximation with independent vacancies takes the forward rate proportional to $P(1-\theta {)}^2$ and the reverse rate proportional to ${\theta }^2$. Equating them yieldswith a positive coefficient $K$ of inverse-pressure units that includes the chosen pair-counting convention. This square-root dependence illustrates why the intact-molecule Langmuir isotherm cannot simply be assumed for splitting molecules. Requirements for neighboring vacancies or interactions can further change the mean-field approximation.

Consider an ideal gas in chemical equilibrium with homogeneous, identical adsorption sites at fixed temperature. Each intact molecule occupies one site; only one monolayer is allowed. Occupation of distinct sites is independent: neighboring bound molecules do not change binding or release rates. The simplest Langmuir adsorption model has one adsorbing chemical species, no splitting of molecules, and no competitors. Irving Langmuir's 1918 paper treats simple adsorption alongside more complicated cases; its scope exceeds this particular model.^[ref]

The derivation is a balance between arrivals finding vacancies and departures from occupied adsorption sites. Let $k_{\mathrm{a}}P$ be the arrival-and-binding rate per vacant site and $k_{\mathrm{d}}$ the departure rate per occupied site. Thus $k_{\mathrm{a}}$ has units of inverse pressure per unit time, whereas $k_{\mathrm{d}}$ has units of inverse time. The mean surface coverage obeysAt chemical equilibrium, setting the two terms equal givesThe product $bP$ is dimensionless. Here $P$ is the adsorbing gas's partial pressure, expressed in modern bar, with $1\mathrm{bar}=10^5\mathrm{Pa}$, and $b$ is expressed in ${\mathrm{bar}}^{-1}$. The pascal is the SI unit of pressure. This dimensional $b$ is not itself a dimensionless thermodynamic equilibrium constant. A larger $b$ gives greater surface coverage at the same $P$ and temperature, but does not specify how quickly chemical equilibrium is reached: multiplying both rate coefficients by the same factor leaves $b$ unchanged.

Choose $b=0.5$, $2$, and $8{\mathrm{bar}}^{-1}$ as illustrative parameters at one fixed temperature, without assigning a particular material. At $P=0.5\mathrm{bar}$, substitution gives, respectively,Solving $\theta =1/2$ gives $P_{1/2}=1/b$: the three half-occupation pressures are $2$, $0.5$, and $0.125\mathrm{bar}$. For $b=2{\mathrm{bar}}^{-1}$, reaching $\theta =0.9$ instead requires $bP=9$, or $P=4.5\mathrm{bar}$. If the complete-monolayer capacity is stipulated to be $q_{\mathrm{m}}=10\mathrm{mmol}{\mathrm{g}}^{-1}$, then $q=q_{\mathrm{m}}\theta$ predicts $q=5\mathrm{mmol}{\mathrm{g}}^{-1}$ at $P=0.5\mathrm{bar}$. This last number is an original hypothetical example, not a measured uptake.

The shape follows directly from the finite number of adsorption sites. For $bP\ll 1$, $\theta \approx bP$: nearly every arriving molecule finds a vacancy. For $bP\gg 1$, $1-\theta =1/(1+bP)\approx 1/(bP)$: vacancies become scarce. More formally,Thus surface coverage rises with diminishing slope and approaches one only as $P\to \infty$. This saturation is a site-capacity limit, not a claim that the gas has reached its condensation pressure. A second layer, different site binding strengths, interactions between bound molecules, or a changing surface invalidate parts of the setup. A good fit alone does not establish identical adsorption sites: a restricted pressure range can hide such differences. The Langmuir adsorption model is useful because it makes the capacity and the pressure scale of filling separate, testable quantities.

A monolayer is a single layer of chemical atoms or molecules. In the Langmuir adsorption model, a complete monolayer occupies every adsorption site; partial surface coverage leaves vacancies within that first layer. One layer need not mean one molecule per surface chemical atom.

Both in-plane orientations minimize the one-site energy. Since every adsorption site is occupied, the sites label independent choices even though the molecules are chemically identical. The ground state of the whole system therefore hasThe ground-state degeneracy is extensive: independently choosing either in-plane orientation at each site costs no energy. There is no extra vacancy or free-translational contribution in this model.

Write $\beta =(k_{\mathrm{B}}T{)}^{-1}$ and $x=\beta \delta$. The one-site partition function includes both degenerate lower states,The occupied adsorption sites are distinguishable positions, so the canonical partition function is a product, without an additional $N!$ divisor:Differentiating $\ln Z_N$ gives the internal energy,The final factor is the population of the perpendicular orientation. The shared binding offset $-\epsilon$ shifts all energy levels equally and does not determine their relative populations.

Take the energy of an isolated, field-free gas atom as zero, and let $E_b>0$ be the field-independent binding energy at one adsorption site. The two adsorbed spin energies areHere ${\mu }_{\mathrm{B}}$ is the Bohr magneton. The lower state is stabilized by increasing $|B|$; the assignment of the two signs to names for the spin projections is a convention.

Let ${\epsilon }_s$ be the energy of one adsorbed atom, using the same energy reference as the gas-phase value ${\epsilon }_0$. For equivalent, noninteracting adsorption sites, choosing the occupied positions givesEvery configuration has energy $N_1{\epsilon }_s$, so the canonical lattice-gas partition function isThere is no further $1/N_1!$ factor: the combinatorial choice already counts indistinguishable adsorbates correctly. The Helmholtz free energy, using Stirling's approximation for a macroscopic number of occupied and empty sites, isHence the adsorbed-phase chemical potential at fixed $T,M$ isThe logarithm comes from configurational entropy: adding an atom is less favorable when vacancies are scarce. In the gas, the stipulated expression can be writtenwhere $\Lambda$ is the thermal de Broglie wavelength. Equality of chemical potentials at chemical equilibrium givesThereforeThe binding energy $E_b={\epsilon }_0-{\epsilon }_s$ is positive for favorable binding. The units of $b$ are inverse pressure, making $bp$ dimensionless. Choosing the adsorbed energy as zero simply sets ${\epsilon }_s=0$ and leaves the same physics; it must not remove the relative binding energy. This is the Langmuir isotherm, with low-pressure linear uptake and high-pressure monolayer saturation.

Dividing the total adsorbed population by the number of underlying adsorption sites givesThis multilayer uptake can exceed one because it is a mean column height. The occupied-site fraction remains $1-{\theta }_0=bp<1$. If each layer has the same binding energy and microscopic states assumed in the single-layer calculation, the corresponding thermodynamic expression for $b$ is also the one obtained there; the kinetic ratio is the definition supplied by this multilayer model.

In a one-molecule-per-adsorption site description, surface coverage is $\theta =N_{\mathrm{occupied}}/N_{\mathrm{sites}}$, with $0\le \theta \le 1$. For a macroscopic sample at chemical equilibrium, $\theta$ usually denotes the mean occupied fraction. Inferring it from total uptake also requires the capacity of a complete monolayer.

An ideal gas whose particles move in an area $A$ with two independent translational momentum components. In its dilute classical limit the one-particle translational partition function is $q=A/{\Lambda }^2$, where $\Lambda$ is the thermal de Broglie wavelength, and $Z_N=q^N/N!$. The translational internal energy is $N{k}_{\mathrm{B}}T$, and the surface pressure conjugate to area is $N{k}_{\mathrm{B}}T/A$. A mobile dilute adsorbed gas is an example; a fully occupied array of fixed adsorption sites has different state counting.