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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.

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.

Irving Langmuir investigated gas adsorption on solid surfaces. The Langmuir adsorption model treats a surface as a set of sites that can each be empty or occupied, balancing adsorption against desorption.

The finite number of sites produces saturation: increasing gas pressure cannot raise single-layer occupancy above one molecule per site in this model. Lateral interactions and multilayer formation require a different description.^[ref]

A chemist working on physical chemistry.

Biographical archive: tile.loc.gov/storage-services/service/gdc/gdcfindingaidpdfs/ms012160/ms012160.pdf

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.

For independent spin-one-half adsorbates whose two spin states are split only at the surface, the one-site partition function contains $2\cosh ({\mu }_{\mathrm{B}}B/k_{\mathrm{B}}T)$. Relative to the unsplit bulk spin degeneracy, this increases the Langmuir adsorption model affinity by a hyperbolic-cosine factor. At fixed temperature and gas pressure, increasing the magnitude of the magnetic field increases surface coverage in this model.

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.

Use the ideal Langmuir adsorption model: identical independent sites, at most one atom per site, no lateral interactions, and no additional thermally active adsorbed degrees of freedom beyond the two spin states. Treat the atoms as indistinguishable and localized after adsorption; only site occupancy and spin choices distinguish configurations.

There are $\left(\genfrac{}{}{0ex}{}{M}{N}\right)$ ways to select $N$ occupied sites. For any such selection, independent spin sums contribute $z^N$. ThusWith $q_{\mathrm{ads}}=Mz$ as defined in part (b), this isEquivalently, starting from labelled particles gives $M!/(M-N)!$ distinct assignments to different sites; division by $N!$ removes particle relabellings. The $M^{-N}$ removes the site multiplicity already included in each full one-particle function.

If $q_{\mathrm{ads}}$ denotes the one-site internal factor instead, $f(M,N)=\left(\genfrac{}{}{0ex}{}{M}{N}\right)$ and $Q_N=\left(\genfrac{}{}{0ex}{}{M}{N}\right)q_{\mathrm{ads}}^N$. These are identical physical results, not different counting models. In particular, the complete-function convention gives $f(M,1)=1$ and hence $Q_1=q_{\mathrm{ads}}$.

At chemical equilibrium, adsorption and desorption balance:Canceling $M$ and rearranging givesThus $K_{\mathrm{ads}}=b$ when the kinetic and thermodynamic descriptions refer to the same ideal surface and energy convention. In particular,The ratio fixes the adsorption isotherm, while the individual rate constants also determine relaxation toward equilibrium. Here $k_a$ has units of inverse pressure per time and $k_d$ of inverse time, so $K_{\mathrm{ads}}$ is dimensional. This equivalent-site arrival/departure argument is the simple case of the Langmuir adsorption model.^[ref]

For the uniform-binding multilayer adsorption model, ${\theta }_j$ describes sites of exactly height $j$, not the fraction whose height is at least $j$. At equilibrium the upward and downward fluxes on each adjacent pair of heights balance:For completeness, stationarity alone gives a height-independent probability current; the reflecting boundary at height zero sets that current to zero. This justifies pairwise balance for a normalizable equilibrium distribution.

Writing $x=K_{\mathrm{ads}}p$ gives the recurrence ${\theta }_j=x{\theta }_{j-1}$. Successive substitution then yieldsAn equilibrium geometric distribution of unbounded heights requires $0\le x<1$. The first-layer-only Langmuir adsorption model instead truncates the allowed heights at one, which changes the normalization and removes this convergence restriction.

The Langmuir adsorption model assumes equivalent independent sites, a maximum of one adsorbate per site, no lateral adsorbate interactions, reversible adsorption-desorption equilibrium and non-dissociative adsorption here. Surface coverage obeys $\theta =bC/(1+bC)$ for the solution analogue.

BET adsorption allows multiple layers: occupation of one layer creates sites for further adsorption above it, and higher-layer binding is treated differently from first-layer binding, commonly using a condensation-like energy. Langmuir saturation is monolayer saturation; the BET model is not simply Langmuir with a different numerical equilibrium constant.