An adsorption isotherm is a curve or mathematical relationship describing the amount of a substance retained by adsorption as a function of its equilibrium pressure or concentration at a fixed temperature. It characterizes a particular adsorbate–adsorbent system under specified conditions. Adsorption isotherms are used to investigate surface interactions, quantify adsorption capacity, characterize pore structure, and represent equilibrium behavior in separation processes. The term can refer either to measured data or to an equation fitted to those data. (nvlpubs.nist.gov)
Physical meaning and representation
Adsorption is the enrichment of a substance at an interface, rather than its incorporation throughout the bulk of a material, as in absorption. The material providing the interface is the adsorbent, and the substance in its adsorbed state is the adsorbate. An isotherm records the equilibrium distribution between the interface and the surrounding gas or liquid. (old.goldbook.iupac.org)
For gas adsorption, a common representation is
[ q=q(p;T), ]
where (q) is the adsorbed amount per unit mass of adsorbent and (p) is the equilibrium gas pressure. For adsorption from solution, the corresponding expression is
[ q_e=q(C_e;T), ]
where (C_e) denotes the equilibrium concentration remaining in solution. The initial concentration is not interchangeable with (C_e), because adsorption changes the solution composition. Common loading units include mol kg(^{-1}), mmol g(^{-1}), and mg g(^{-1}). (nvlpubs.nist.gov)
Gas–solid isotherms often use relative pressure, (p/p_0), where (p_0) is the saturation vapor pressure at the measurement temperature. A measurement must also identify whether it reports an absolute adsorbed amount or a surface excess. Excess adsorption subtracts the amount that would occupy a defined adsorption-region volume at the bulk-fluid density; the distinction becomes particularly important at high pressure. (sol.rutgers.edu)
Principal mathematical models
Different equations embody different assumptions. Their parameters are therefore not automatically comparable, even when several equations fit the same observations.
Linear or Henry-type behavior
At sufficiently low loading, many systems approach a linear relationship:
[ q_e=K_H C_e ]
or, for a gas,
[ q=K_Hp. ]
The constant (K_H) expresses the initial slope, with units depending on the selected variables. This approximation does not describe saturation and is generally restricted to a low-concentration or low-pressure region. (nepis.epa.gov)
Langmuir isotherm
The Langmuir isotherm represents adsorption onto a finite population of equivalent sites, with one adsorbed molecule per site and no interactions between occupied sites:
[ q_e=\frac{q_{\max}bC_e}{1+bC_e}. ]
Here (q_{\max}) is the limiting capacity and (b) is an affinity parameter. The product (bC_e) must be dimensionless. For gases, pressure replaces concentration in the corresponding expression. (pmc.ncbi.nlm.nih.gov)
Writing the fractional coverage as (\theta=q/q_{\max}), the idealized balance between adsorption onto vacant sites and desorption from occupied sites is
[ k_a p(1-\theta)=k_d\theta. ]
Solving gives (\theta=bp/(1+bp)), with (b=k_a/k_d). The model is linear at low pressure and approaches saturation at high pressure. Irving Langmuir developed this surface-site description in early twentieth-century work, including his 1918 paper on adsorption by glass, mica, and platinum. A Langmuir-shaped curve alone does not establish that a real surface satisfies all the model assumptions. (diverdi.colostate.edu)
Freundlich isotherm
The Freundlich isotherm is an empirical power-law relationship:
[ q_e=K_F C_e^{1/n}. ]
It is frequently used to represent adsorption on heterogeneous surfaces over a limited concentration range. For (0<1/n<1), uptake increases with concentration but progressively less steeply. Unlike the Langmuir equation, it has no finite saturation capacity. The units and numerical value of (K_F) depend on the concentration units and exponent; (K_F) is not itself a maximum capacity. (ars.usda.gov)
BET isotherm
Brunauer–Emmett–Teller theory extends a monolayer description to multilayer adsorption. Its commonly used linear form is
[ \frac{x}{q(1-x)}
\frac{1}{q_m C} + \frac{C-1}{q_m C}x, \qquad x=\frac{p}{p_0}, ]
where (q_m) is the monolayer capacity and (C) is the BET constant. Stephen Brunauer, P. H. Emmett, and Edward Teller published the theory in 1938. (doi.org)
The inferred monolayer capacity, together with an assumed molecular cross-sectional area, is used to estimate specific surface area. BET analysis applies only over an appropriate portion of an isotherm; it is not a universal description of pore filling or adsorption near saturation. (publications.iupac.org)
Classification of gas physisorption curves
The IUPAC classification distinguishes six main shapes of gas physisorption isotherms. It concerns observed curve morphology and adsorption processes, rather than a list of interchangeable fitting equations. (publications.iupac.org)
| Type | Characteristic interpretation |
|---|---|
| I | Rapid initial uptake followed by a limiting region, characteristic of micropore filling. |
| II | Monolayer followed by multilayer adsorption, typically on nonporous or macroporous solids. |
| III | Weak adsorbent–adsorbate interactions, without a clearly identifiable monolayer-completion point. |
| IV | Adsorption in mesoporous solids, including pore condensation; hysteresis may occur. |
| V | Weak initial interactions followed by pore filling, often with hysteresis. |
| VI | Stepwise, layer-by-layer adsorption on highly uniform surfaces. |
Micropores have widths below approximately 2 nm, mesopores span approximately 2–50 nm, and macropores are wider than approximately 50 nm. The limiting uptake of a Type I curve should not automatically be interpreted as a Langmuir monolayer: filling a confined pore is physically distinct from covering an open surface. (publications.iupac.org)
The 2015 IUPAC treatment further distinguishes Types I(a) and I(b), associated with different micropore-width distributions, and Types IV(a) and IV(b), distinguished by the presence or absence of hysteresis. (sol.rutgers.edu)
Measurement and hysteresis
Gas adsorption isotherms are commonly measured by volumetric methods, which infer uptake from gas balances, or gravimetric methods, which measure mass changes. Sample pretreatment, temperature control, accessible void volume, and sufficient equilibration time influence the resulting curve. Slow diffusion into narrow pores can make equilibrium difficult to establish. (sol.rutgers.edu)
Adsorption and desorption branches need not coincide. This difference is called adsorption hysteresis. In mesoporous materials it can reflect capillary condensation, metastability, pore blocking, and network effects. A hysteresis loop therefore contains structural information, but its shape is not a uniquely identifying image of pore geometry. (publications.iupac.org)
Applications and limits of interpretation
Isotherms provide the experimental basis for comparing adsorbents such as activated carbon, zeolites, and metal–organic frameworks. They also support surface-area and porosity analysis in porous materials. Adsorption databases combine such measurements with tools for fitting, mixture prediction, and breakthrough modeling. (sol.rutgers.edu)
Reliable interpretation requires attention to both physical assumptions and statistical behavior. Linearizing a nonlinear equation can alter error weighting and produce different parameter estimates from fitting the original relationship. Nonlinear fitting avoids that particular transformation, although uncertainty, parameter identifiability, and the experimental range remain important. A high coefficient of determination does not, by itself, demonstrate a particular microscopic mechanism. (pmc.ncbi.nlm.nih.gov)
An isotherm is an equilibrium description, not a measurement of adsorption rate. A single-component curve also does not directly establish behavior in a competing mixture. Consequently, equilibrium capacity, transport behavior, and multicomponent adsorption constitute distinct information requirements when interpreting or modeling an adsorption process. (sol.rutgers.edu)
References
- NIST recommended practice guide: the use of nomenclature in dispersion science and technologynvlpubs.nist.gov
- The Adsorption of Gases on Plane Surfaces of Glass, Mica and Platinumdiverdi.colostate.edu
- Adsorption Equations and Modelsars.usda.gov
- Understanding Variation in Partition Coefficient, Kd, Values, Volume 1: The Kd Model of Measurement, and Application of Chemical Reaction Codesnepis.epa.gov
- Adsorption of Gases in Multimolecular Layerstau.ac.il
- Nonlinear regression for treating adsorption isotherm data to characterize new sorbents: Advantages over linearization demonstrated with simulated and experimental datapmc.ncbi.nlm.nih.gov
- NIST Data Resources for Adsorptionnist.gov