BET theory
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 observations are very often referred to as physical adsorption or physisorption. In 1938, Stephen Brunauer, Paul Hugh Emmett, and Edward Teller presented their theory in the "Journal of the American Chemical Society". BET theory applies to systems of multilayer adsorption that usually utilizes a probing gas (called the adsorbate) that do not react chemically with the adsorptive (the material  upon which the gas attaches to and the gas phase is called the adsorptive) to quantify specific surface area. Nitrogen is the most commonly employed gaseous adsorbate for probing surface(s). For this reason, standard BET analysis is most often conducted at the boiling temperature of N (77 K). Other probing adsorbates are also utilized, albeit less often, allowing the measurement of surface area at different temperatures and measurement scales. These include argon, carbon dioxide, and water. Specific surface area is a scale-dependent property, with no single true value of specific surface area definable, and thus quantities of specific surface  area determined through BET theory may depend on the adsorbate molecule utilized and its adsorption cross section. Concept. 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: The resulting BET equation is formula_1 where "c" is referred to as the BET C-constant, formula_2is the vapor pressure of the adsorptive bulk liquid phase which would be at the temperature of the adsorbate and θ is the "surface coverage, defined as: formula_3. Here formula_4is the amount of adsorbate and formula_5 is called the  monolayer equivalent. The formula_5 is the entire amount that would be present as a monolayer (which is theoretically impossible for physical adsorption) would cover the surface with exactly one layer of adsorbate. The above equation is usually rearranged to yield the following equation for the ease of analysis: where formula_8 and formula_9 are the equilibrium and the saturation pressure of adsorbates at the temperature of adsorption, respectively; formula_10 is the adsorbed gas quantity (for example, in volume units) while formula_11 is the monolayer adsorbed gas quantity. formula_12 is the "BET constant", where formula_14 is the heat of adsorption for the  first layer, and formula_15 is that for the second and higher layers and is equal to the heat of liquefaction or heat of vaporization. Equation (1) is an adsorption isotherm and can be plotted as a straight line with formula_16 on the y-axis and formula_17 on the x-axis according to experimental results. This plot is called a "BET plot". The linear relationship of this equation is maintained only in the range of formula_18. The value of the slope formula_19 and the y-intercept formula_20 of the line are used to calculate the monolayer adsorbed gas quantity formula_11 and  the BET constant formula_12. The following equations can be used: The BET method is widely used in materials science for the calculation of surface areas of solids by physical adsorption of gas molecules. The total surface area formula_25 and the specific surface area formula_26 are given by where formula_11 is in units of volume which are also the units of the monolayer volume of the adsorbate gas, formula_30 is the Avogadro number, formula_31 the adsorption cross section of the adsorbate, formula_32 the molar volume of the adsorbate gas, and formula_33 the mass of the solid sample or adsorbent. Derivation. The  BET theory can be derived similarly to the Langmuir theory, but by considering multilayered gas molecule adsorption, where it is not required for a layer to be completed before an upper layer formation starts. Furthermore, the authors made five assumptions: Consider a given amount of solid sample in a controlled atmosphere. Let "θ" be the fractional coverage of the sample surface covered by a number "i" of successive molecule layers. Let us assume that the adsorption rate "R" for molecules on a layer ("i"-1) (i.e. formation of a layer "i") is proportional to both its fractional surface "θ" and to  the pressure "P", and that the desorption rate "R" on a layer "i" is also proportional to its fractional surface "θ": where "k" and "k" are the kinetic constants (depending on the temperature) for the adsorption on the layer ("i"−1) and desorption on layer "i", respectively. For the adsorptions, these constant are assumed similar whatever the surface. Assuming an Arrhenius law for desorption, the related constants can be expressed as where "E" is the heat of adsorption, equal to "E" at the sample surface and to "E" otherwise. Finding the linear BET range. It is still not clear on how  to find the linear range of the BET plot for microporous materials in a way that reduces any subjectivity in the assessment of the monolayer capacity. Rouquerol et al. suggested a procedure that is based on two criteria: These corrections are an attempt to salvage the BET theory which is restricted to type II isotherm. Even with this type, use of the data is restricted to 0.05 to 0.35 of formula_37, routinely discarding 70% of the data. Even this restriction has to be modified depending upon conditions. The problems with the BET theory are multiple and reviewed by Sing. A  serious problem is that there is no relationship between the BET and the calorimetric measurements in experiments. It violates the Gibbs' phase rules. It is extremely unlikely that it measure correctly the surface area, formerly great advantage of the theory. It is based upon chemical equilibrium, which assumes localized chemical bond (this approach has been abandoned by the modern theories. See chapter 4, χ/ESW and Chapter 7, DFT or better NLDFT) in total contradiction to what is known about physical adsorption, which is based upon non-local intermolecular attractions. Two extreme problems is that in certain cases BET leads  to anomalies and the C constant can be negative, implying an imaginary energy. Applications. Cement and concrete. The rate of curing of concrete depends on the fineness of the cement and of the components used in its manufacture, which may include fly ash, silica fume and other materials, in addition to the calcinated limestone which causes it to harden. Although the Blaine air permeability method is often preferred, due to its simplicity and low cost, the nitrogen BET method is also used. When hydrated cement hardens, the calcium silicate hydrate (or C-S-H), which is responsible for the hardening  reaction, has a large specific surface area because of its high porosity. This porosity is related to a number of important properties of the material, including the strength and permeability, which in turn affect the properties of the resulting concrete. Measurement of the specific surface area using the BET method is useful for comparing different cements. This may be performed using adsorption isotherms measured in different ways, including the adsorption of water vapour at temperatures near ambient, and adsorption of nitrogen at 77 K (the boiling point of liquid nitrogen). Different methods of measuring cement paste surface areas often give  very different values, but for a single method the results are still useful for comparing different cements. Activated carbon. Activated carbon has strong affinity for many gases and has an adsorption cross section formula_31 of 0.162 nm for nitrogen adsorption at liquid-nitrogen temperature (77 K). BET theory can be applied to estimate the specific surface area of activated carbon from experimental data, demonstrating a large specific surface area, even around 3000 m/g. However, this surface area is largely overestimated due to enhanced adsorption in micropores, and more realistic methods should be used for its estimation, such as the  subtracting pore effect (SPE) method. Catalysis. In the field of solid catalysis, the surface area of catalysts is an important factor in catalytic activity. Inorganic materials such as mesoporous silica and layered clay minerals have high surface areas of several hundred m/g calculated by the BET method, indicating the possibility of application for efficient catalytic materials. Specific surface area calculation. The ISO 9277 standard for calculating the specific surface area of solids is based on the BET method.