Brunauer–Emmett–Teller (BET) theory aims to explain the physical adsorption of gas molecules on a solid surface and serves as the basis for an important analysis technique for the measurement of the specific surface area of materials. The fundamental element of BET theory is associated with the adsorption of a gas on the material's surface. This phenomenon is caused by van der Waals forces that are created by a film of the adsorbate, which consists of atoms, ions, or molecules on the surface of a substance that adsorbs these particles.

Stephen Brunauer, Paul Emmet and Edward Teller published this theory in 1938 in the Journal of the American Chemical Society. It is a theory for multi-layer physisorption and is of profound significance in the development of this field.

The two most widely used equations are the BET and SSL (Single‐Site Langmuir Adsorption) models, the latter of which was first used by Langmuir himself in 1918. While neither model is strictly appropriate for the description of gas adsorption within narrow micropores, it has been shown that the BET model is often suitable for the estimation of the true surface area of microporous and mesoporous materials including MOFs and zeolites (despite that the number of multilayers is constrained) when proper consistency criteria are employed in determining the range of partial pressure over which tofitthedata.

The BET formalism prevalent in the literature is written as follows:

equation 1

where P₀ is the saturation pressure of the adsorbate, defined as:

equation 2

and ܿc is the BET constant, defined as:

equation 3

A common linearized rearrangement, where B is simply the “BET variable”, is:

 equation 4

Brunauer, Emmett, and Teller described three regions along the BET isotherm in Equation 4: a concave region at low pressure, a convex region at high pressure, and a linear region at intermediate pressure. They originally specified the relative pressure P/P0 range of 0.05‐0.3 as the linear region from which the number of surface sites, ߁, can be dependably extracted. In the case of high surface area, microporous materials such as zeolites and MOFs, however, this range has proven to be inadequate as a universal standard and a now widely accepted set of self‐consistency criteria have been proposed. Once the linear range is determined and ߁ is extracted, the BET surface area, SA_BET is calculated as:

equation 5

where SA_B is the surface area of a single binding site. The conventional cross‐sectional area of a N₂ molecule for BET surface area calculation is 0.162 nm², which allows researchers across different laboratories to have a standard for materials comparison. For example, in the case of IRMOF‐16, the BET surface area determined herein corresponds to 61.8 mmolg^-1 or 6030 m²g^-1. This is consistent with the geometrical surface area of the crystal (~6000 m²g^-1 as determined by a Monte Carlo integration technique). It should be noted that challenges still persist in using the BET model to accurately estimate the monolayer capacity of porous materials, especially those with a diversity of pore sizes.

The surface area is estimated using the Brunauer, Emmett & Teller (BET) equation, from a specific region of a gas adsorption isotherm. The gas adsorption isotherm is experimentally obtained as follows. Successive doses of an adsorptive gas probe, typically N2 at 77 K, are sent to the testing samples, preliminarily need to pretreat samples which aims to purify materails. The amount of gas molecules that can adsorb onto the surface of the material is derived from the evolution of the pressure in the system. The cumulative amount of adsorbate plotted with respect to the pressure is the adsorption isotherm.

Brunauer, Emmett & Teller developed a model for type II isotherms, which considers that gas molecules are adsorbed in monolayers, i.e. monomolecular layers. In the specific relative pressure range from 0.05 to 0.30 each monolayer evenly covers the previous one. By applying Langmuir theory to those monolayers they obtained the following BET equation。

The surface area is then estimated from the monolayer amount n_m and the cross-sectional area of a molecule of adsorbate.

The concept of the theory is an extension of the Langmuir theory, which is a theory for monolayer molecular adsorption, to multilayer adsorption with the following hypotheses:

1.         gas molecules physically adsorb on a solid in layers infinitely;

2.         gas molecules only interact with adjacent layers;

3.         the Langmuir theory can be applied to each layer;

4.         the enthalpy of adsorption for the first layer is constant and greater than the second (and higher);

5.         the enthalpy of adsorption for the second (and higher) layers is the same as the enthalpy of liquefaction.

Despite the extensive use of the BET method, many authors have discussed the limitations that are inherently related when it is applied for the surface area determination of microporous materials. Since the method is based on gas adsorption, limitations are often related to monolayers. For instance,

1) the validity of n_m (the BET monolayer capacity) is problematic;

2) the monolayer structure is not the same on all surfaces, particularly when N2 isotherms are used since the molecule is quadrupolar;

3) at very low pressure ranges (P/P0) strong adsorption can involve localized monolayer coverage and/or primary micropore filling in the pores of molecular dimensions.

When characterizing materials with micropores below 20 Å, the biggest problem is usually related to micropore filling, which takes place rather than mono or multilayer coverage. This can lead to obtaining higher or overestimated surface areas; however, high rates of micropore filling can potentially be recognized.

The BET (Brunauer-Emmett-Teller) theory is critically important in materials science and engineering because it provides a quantitative measure of a material's surface area and porosity, which are key factors influencing its performance in various applications. Here’s why BET theory is essential for materials study:

1. Determines Specific Surface Area

The BET method calculates the specific surface area (m²/g) of a material by measuring gas (usually N₂) adsorption.

Why it matters: A higher surface area often means more active sites for reactions, crucial for catalysts, adsorbents, and energy storage materials.

2. Essential for Porous Materials Characterization

BET analysis helps classify materials based on porosity:

• Microporous (pores < 2 nm, e.g., zeolites, activated carbon).

• Mesoporous (2–50 nm, e.g., MCM-41, SBA-15).

• Macroporous (>50 nm, e.g., some ceramics).

Why it matters: Pore size and distribution affect diffusion, adsorption capacity, and selectivity in separation processes.

3. Critical for Catalyst Design & Performance

Catalysts rely on high surface area to maximize active sites for reactions.

BET analysis helps optimize:

• Supported catalysts (e.g., Pt/Al₂O₃ for fuel cells).

• Zeolites in petrochemical refining.

• Nanoparticle catalysts (e.g., Au/TiO₂).

Why it matters: Higher surface area = more reaction sites = better catalytic efficiency.

4. Impacts Adsorption & Separation Technologies

Used in designing:

• Activated carbon for water purification.

• MOFs (Metal-Organic Frameworks) for CO₂ capture.

• Molecular sieves for gas separation.

Why it matters: BET data helps predict how well a material can adsorb pollutants or separate gas mixtures.

5. Key for Battery & Energy Storage Materials

Electrode materials (e.g., graphene, porous carbons) need high surface area for:

• Supercapacitors (charge storage at the surface).

• Lithium-ion batteries (Li⁺ diffusion).

Why it matters: Larger surface area improves charge/discharge rates and capacity.

6. Pharmaceutical & Drug Delivery Applications

Porous carriers (e.g., silica nanoparticles) use BET analysis to optimize:

• Drug loading capacity.

• Controlled release kinetics.

Why it matters: Surface area affects how much drug can be adsorbed and released over time.

7. Quality Control in Industrial Processes

Industries use BET to ensure consistency in:

• Nanoparticle synthesis.

• Ceramic and polymer production.

• Coatings and thin films.

Why it matters: Reproducible surface properties ensure reliable product performance.

Limitations & Complementary Techniques

While BET is powerful, it has limitations:

• Less accurate for microporous materials (where pore filling ≠ multilayer adsorption).

• Often combined with:

• DFT (Density Functional Theory) for micropores.

• Mercury Porosimetry for larger pores.

• SEM/TEM for direct imaging.

BET theory is indispensable because surface area and porosity dictate how materials interact with gases, liquids, and other phases. Whether in catalysis, energy storage, environmental cleanup, or medicine, understanding these properties through BET analysis enables smarter material design and optimization.

The Langmuir adsorption theory (developed by Irving Langmuir in 1918) is a fundamental model describing monolayer adsorption of gases or solutes onto solid surfaces. Unlike the BET theory (which covers multilayer adsorption), Langmuir focuses on chemisorption and localized monolayer formation. Below are its key applications, purposes, and reasons for its widespread use.

1. Key Applications of Langmuir Theory

A. Heterogeneous Catalysis

• Used to model catalytic surface reactions, where gas molecules adsorb onto active sites.

• Example: Ammonia synthesis (Haber process) relies on N₂ adsorption on Fe catalysts.

B. Gas Sensors & Adsorbents

• Helps design gas sensors (e.g., CO detectors) by predicting adsorption behavior.

• Used in activated carbon filters for pollutant removal.

C. Electrochemistry & Batteries

• Models electrode surface reactions, such as hydrogen adsorption in fuel cells.

D. Pharmaceutical & Drug Delivery

• Predicts drug adsorption onto carrier materials (e.g., nanoparticles).

E. Environmental Science

• Studies contaminant adsorption in soil/water treatment (e.g., heavy metals on clays).

2. Main Purposes of Langmuir Theory

A. Quantifying Surface Coverage

• Predicts how much gas/solute adsorbs at equilibrium.

• Helps optimize adsorbent efficiency (e.g., how much CO₂ a material can capture).

B. Determining Adsorption Capacity

• Calculates maximum monolayer coverage (θ = 1).

• Used to compare different adsorbents (e.g., activated carbon vs. zeolites).

C. Studying Binding Affinity

• The Langmuir constant (K) indicates adsorption strength:

• High K = Strong chemisorption (e.g., CO on Pt).

• Low K = Weak physisorption (e.g., N₂ on silica).

D. Modeling Reaction Kinetics

• Helps derive rate equations for surface-catalyzed reactions.

3. Why Langmuir Theory is Widely Used

A. Simplicity & Fundamental Basis

• Assumes:

• Monolayer adsorption only (no multilayers).

• Homogeneous surface (all sites are equivalent).

• No interactions between adsorbed molecules.

• Provides a baseline model for more complex theories (e.g., BET, Freundlich).

B. Works Well for Chemisorption

• Best for strong, localized adsorption (e.g., H₂ on metals, O₂ on catalysts).

C. Useful for Low to Moderate Pressures

• Accurate when gas pressure (or solute concentration) is not too high.

D. Basis for Advanced Models

• Extended by:

• BET Theory (for multilayer adsorption).

• Freundlich Isotherm (for heterogeneous surfaces).

• Temkin & Sips Isotherms (for varying adsorption energies)

4. Limitations of Langmuir Theory

Fails for multilayer adsorption (use BET instead).
Assumes uniform surface sites (real surfaces are often heterogeneous).
Neglects intermolecular interactions (e.g., lateral repulsion/attraction).

The Langmuir theory is essential for understanding monolayer adsorption in catalysis, gas sensing, environmental cleanup, and drug delivery. While simplified, it provides a foundation for more advanced adsorption models and remains widely used due to its predictive power and applicability to chemisorption-dominated systems.

Reference list:

• Determination of the specific surface area of solids by gas adsorption — BET method, ISO 9277:2010(E)

• Sing K.S.W., Everett D.H., Haul R.A.W., Moscou L., Pierotti R.A., Rouquérol J. and Siemieniewska T., IUPAC Recommendations 1984: Reporting Physisorption Data for Gas Solid Systems with Special Reference to the Determination of Surface Area and Porosity, Pure & Applied Chemistry 57, 1985, pp. 603-319

• P. C. Hiemenz, R. Rajagopalan, Principles of Colloid and Surface Chemistry, 3rd ed., CRC Press, 2016

•  J. Lyklema, Fundamentals of Interface and Colloid Science: Solid-Liquid Interfaces, Elsevier, 1995

• Filip Ambroz, Thomas J. Macdonald, Vladimir Martis, Ivan P. Parkin, Evaluation of the BET Theory for the Characterization of Meso and Microporous MOFs: small method, wiley, 2018

• Langmuir,I.,The Adsorption of Gases on Plane Surfaces of Glass, Mica and Platinum.J. Am. Chem. Soc.1918, 40,1361‐1403

• Walton,K.S.; Snurr,R.Q., Applicability of the BET Method for Determining Surface Areas of Microporous Metal‐Organic Frameworks.J.Am.Chem. Soc. 2007