BET Theory
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Brunauer–Emmett–Teller (BET) theory aims to explain the physical
adsorption Adsorption is the adhesion of atoms, ions or molecules from a gas, liquid or dissolved solid to a surface. This process creates a film of the ''adsorbate'' on the surface of the ''adsorbent''. This process differs from absorption, in which a ...
of gas
molecule A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
s on a
solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
and serves as the basis for an important analysis technique for the measurement of the
specific surface area Specific surface area (SSA) is a property of solids defined as the total surface area (SA) of a material per unit mass, (with units of m2/kg or m2/g). Alternatively, it may be defined as SA per solid or bulk volume (units of m2/m3 or m−1). I ...
of materials. The observations are very often referred to as physical adsorption or
physisorption Physisorption, also called physical adsorption, is a process in which the electronic structure of the atom or molecule is barely wikt:perturb, perturbed upon adsorption. Overview The fundamental interacting force of physisorption is Van der Waals ...
. In 1938, Stephen Brunauer, Paul Hugh Emmett, and
Edward Teller Edward Teller (; January 15, 1908 – September 9, 2003) was a Hungarian and American Theoretical physics, theoretical physicist and chemical engineer who is known colloquially as "the father of the hydrogen bomb" and one of the creators of ...
presented their theory in the ''
Journal of the American Chemical Society The ''Journal of the American Chemical Society'' (also known as JACS) is a weekly peer-reviewed scientific journal that was established in 1879 by the American Chemical Society. The journal has absorbed two other publications in its history, the ...
''. BET theory applies to systems of multilayer adsorption that usually utilizes a probing gas (called the adsorbate) that does not react chemically with the adsorptive (the material upon which the gas attaches to) to quantify specific surface area.
Nitrogen Nitrogen is a chemical element; it has Symbol (chemistry), symbol N and atomic number 7. Nitrogen is a Nonmetal (chemistry), nonmetal and the lightest member of pnictogen, group 15 of the periodic table, often called the Pnictogen, pnictogens. ...
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 N2 (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: # gas molecules physically adsorb on a solid in layers infinitely; # gas molecules only interact with adjacent layers; and # the Langmuir theory can be applied to each layer. # the enthalpy of adsorption for the first layer is constant and greater than the second (and higher). # the enthalpy of adsorption for the second (and higher) layers is the same as the enthalpy of liquefaction. The resulting BET equation is \theta=\frac where ''c'' is referred to as the BET C-constant, p_o is 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: \theta=n_/n_m. Here n_ is the amount of adsorbate and n_ is called the monolayer equivalent. The n_ is the entire amount that would be present as a monolayer (which is theoretically impossible for physical adsorption) that 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: : \frac = \frac \left ( \frac \right ) + \frac, \qquad (1) where p and p_0 are the equilibrium and the
saturation pressure Vapor pressure or equilibrium vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. The equilibrium vapor pressure is an indicati ...
of adsorbates at the temperature of adsorption, respectively; v is the adsorbed gas quantity (for example, in volume units) while v_\mathrm is the monolayer adsorbed gas quantity. c is the ''BET constant'', : c = \exp\left(\frac\right), \qquad (2) where E_1 is the heat of adsorption for the first layer, and E_\mathrm is that for the second and higher layers and is equal to the heat of
liquefaction In materials science, liquefaction is a process that generates a liquid from a solid or a gas or that generates a non-liquid phase which behaves in accordance with fluid dynamics. It occurs both naturally and artificially. As an example of t ...
or
heat of vaporization In thermodynamics, the enthalpy of vaporization (symbol ), also known as the (latent) heat of vaporization or heat of evaporation, is the amount of energy (enthalpy) that must be added to a liquid substance to Phase transition, transform a qua ...
. Equation (1) is an adsorption isotherm and can be plotted as a straight line with 1/ on the y-axis and \varphi=/ 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 0.05 < / < 0.35. The value of the slope A and the y-intercept I of the line are used to calculate the monolayer adsorbed gas quantity v_\mathrm and the BET constant c. The following equations can be used: :v_m = \frac\qquad (3) :c = 1+\frac.\qquad (4) The BET method is widely used in
materials science Materials science is an interdisciplinary field of researching and discovering materials. Materials engineering is an engineering field of finding uses for materials in other fields and industries. The intellectual origins of materials sci ...
for the calculation of surface areas of
solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
s by physical adsorption of gas molecules. The total surface area S_\mathrm and the
specific surface area Specific surface area (SSA) is a property of solids defined as the total surface area (SA) of a material per unit mass, (with units of m2/kg or m2/g). Alternatively, it may be defined as SA per solid or bulk volume (units of m2/m3 or m−1). I ...
S_\mathrm are given by :S_\mathrm = \frac, \qquad (5) :S_\mathrm = \frac, \qquad (6) where v_\mathrm is in units of volume which are also the units of the monolayer volume of the adsorbate gas, N is the
Avogadro number The Avogadro constant, commonly denoted or , is an SI defining constant with an exact value of when expressed in reciprocal moles. It defines the ratio of the number of constituent particles to the amount of substance in a sample, where th ...
, s the adsorption cross section of the adsorbate, V the
molar volume In chemistry and related fields, the molar volume, symbol ''V''m, or \tilde V of a substance is the ratio of the volume (''V'') occupied by a substance to the amount of substance (''n''), usually at a given temperature and pressure. It is also eq ...
of the adsorbate gas, and a 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: # Adsorptions occur only on well-defined sites of the sample surface (one per molecule) # The only molecular interaction considered is the following one: a molecule can act as a single adsorption site for a molecule of the upper layer. # The uppermost molecule layer is in equilibrium with the gas phase, i.e. similar molecule adsorption and desorption rates. # The desorption is a kinetically limited process, i.e. a heat of adsorption must be provided: #* these phenomena are homogeneous, i.e. same heat of adsorption for a given molecule layer. #* it is E1 for the first layer, i.e. the heat of adsorption at the solid sample surface #* the other layers are assumed similar and can be represented as condensed species, i.e. liquid state. Hence, the heat of adsorption is EL is equal to the heat of liquefaction. # At the saturation pressure, the molecule layer number tends to infinity (i.e. equivalent to the sample being surrounded by a liquid phase) Consider a given amount of solid sample in a controlled atmosphere. Let ''θi'' 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''ads,''i''-1 for molecules on a layer (''i''-1) (i.e. formation of a layer ''i'') is proportional to both its fractional surface ''θ''''i''-1 and to the pressure ''P'', and that the desorption rate ''R''des,''i'' on a layer ''i'' is also proportional to its fractional surface ''θ''''i'': :R_ = k_i P \Theta_ :R_ = k_ \Theta_i, where ''k''''i'' and ''k''−''i'' 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 constants are assumed similar whatever the surface. Assuming an Arrhenius law for desorption, the related constants can be expressed as :k_i = \exp(-E_i/RT), where ''E''''i'' is the heat of adsorption, equal to ''E''1 at the sample surface and to ''E''L otherwise. Consider some substance A. The adsorption of A onto an available surface site (*)_0 produces a new site (*)_1 on the first layer. In summary, :A(g) + (*)_0 <=> (*)_1 Extending this to higher order layers one obtains :A(g) + (*)_1 <=> (*)_2 and similarly :A(g) + (*) n-1 <=> (*)_n Denoting the activity of the number of available sites of the nth layer with \theta_n and the partial pressure of A with P, the last equilibrium can be written : K_n = \frac It follows that the coverage of the first layer can be written :\theta_1=K_1 P\theta_0 and that the coverage of the second layer can be written :\theta_2=K_2 P\theta_1=K_2 P K_1 P \theta_0 Realising that the adsorption of A onto the second layer is equivalent to adsorption of A onto its own liquid phase, the rate constant for n>1 should be the same, which results in the recursion :\theta_n=(K_\ell P)^ P K_1 \theta_0 In order to simplify some infinite summations, let x=K_\ell P and let y=K_1 P. Then the nth layer coverage can written :\theta_n = c \theta_0 x^n, \quad n>0 if c=y/x. The coverage of any layer is defined as the relative number of available sites. An alternative definition, which leads to a set of coverage's that are numerically to those resulting from the original way of defining surface coverage, is that \theta_n denotes the relative number of sites covered by only n adsorbents. Doing so it is easy to see that the total volume of adsorbed molecules can be written as the sum :V_\text=V_\text\sum_^ n \theta_n = V_\text c\theta_0 x\sum_^ nx^ where V_\text is the molecular volume. Employing the fact that this sum is the first derivative of a geometric sum, the volume becomes :V_\text= V_\text c\theta_0\frac, \quad , x, < 1 Since the total coverage of a mono-layer must be unity, the full mono-layer coverage must be :1=\sum_^ \theta_n In order to properly make the substitution for \theta_n, the restriction n>1 forces us to take the zeroth contribution outside the summation, resulting in :\sum_^ \theta_n=\theta_0 + c \theta_0 x\sum_^ x^n = \theta_0 + c \theta_0 \frac, \quad , x, < 1 Lastly, defining the excess coverage as V_\text=V_\text/V_\text, the excess volume relative to the volume of an adsorbed mono-layer becomes :V_\text=\frac =\frac =\frac where the last equality was obtained by making use of the series expansions presented above. The constant c must be interpreted as the relative binding affinity the substance A has towards a surface, relative to its own liquid. If c>1 then the initial part of the isotherm will be reminiscent of the Langmuir isotherm which reaches a plateau at full mono-layer coverage, whereas c < 1 means the mono-layer will have a slow build-up. Another thing to note is that in order for the geometric substitutions to hold, x<1. The isotherm above exhibits a singularity at x^\ast=1. Since x=K_\ell P one can write x^\ast=K_\ell P^\ast , implying that K_\ell=1/P^\ast. This means that x=P/P^\ast must be true, ultimately resulting in x\in[0,1).


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. A crowd-sourced study involving 61 research groups has shown that reproducibility of BET area determination from identical isotherms is, in some cases, problematic. Rouquerol et al. suggested a procedure that is based on two criteria: * C must be positive implying that any negative intercept on the BET plot indicates that one is outside the valid range of the BET equation. * Application of the BET equation must be limited to the range where the term V(1-P/P0) continuously increases with P/P0. These corrections are an attempt to salvage the BET theory, which is restricted to type II isotherms. Even while using this type, use of the data itself is restricted to 0.05 to 0.35 of P/P_0, routinely discarding 70% of the data. This restriction must be modified depending upon conditions.


Limitations of BET

Terrell L. Hill described BET as a theory that is "... extremely useful as a qualitative guide; but it is not quantitatively correct". Although BET adsorption isotherm is still extensively used for different applications and is used for specific surface area determinations of powders whose calculation is not sensitive to the simplifications of the BET theory. Both Hackerman's and Sing's group have highlighted the limitations of the BET method. Hackerman et al. noted the potential for 10% uncertainty in the method's values, with Sing's group attributed the significant variation in reported values of molecular area to the BET method's possible inaccurate assessment of monolayer capacity. In subsequent studies using the BET interpretation of nitrogen and water vapor adsorption isotherms, the reported area occupied by an adsorbed water molecule on fully hydroxylated silica ranged from 0.25 to 0.44 nm². Other issues with the BET include the fact that in certain cases, BET leads to anomalies such as reaching an infinite amount adsorbed at P/P_0 reaching unity, and in some cases, the constant C (surface binding energy) can be determined to be negative.


Applications


Cement and concrete

The rate of curing of concrete depends on the fineness of the
cement A cement is a binder, a chemical substance used for construction that sets, hardens, and adheres to other materials to bind them together. Cement is seldom used on its own, but rather to bind sand and gravel ( aggregate) together. Cement mi ...
and of the components used in its manufacture, which may include fly ash, silica fume and other materials, in addition to the calcinated
limestone Limestone is a type of carbonate rock, carbonate sedimentary rock which is the main source of the material Lime (material), lime. It is composed mostly of the minerals calcite and aragonite, which are different Polymorphism (materials science) ...
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 A cement is a binder, a chemical substance used for construction that sets, hardens, and adheres to other materials to bind them together. Cement is seldom used on its own, but rather to bind sand and gravel ( aggregate) together. Cement mi ...
hardens, the
calcium silicate hydrate Calcium silicate hydrates (CSH or C-S-H) are the main products of the hydration of Portland cement and are primarily responsible for the strength of cement-based materials. They are the main binding phase (the "glue") in most concrete. Only well de ...
(or C-S-H), which is responsible for the hardening reaction, has a large
specific surface area Specific surface area (SSA) is a property of solids defined as the total surface area (SA) of a material per unit mass, (with units of m2/kg or m2/g). Alternatively, it may be defined as SA per solid or bulk volume (units of m2/m3 or m−1). I ...
because of its high
porosity Porosity or void fraction is a measure of the void (i.e. "empty") spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0% and 100%. Strictly speaking, some tests measure ...
. 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 Concrete is a composite material composed of aggregate bound together with a fluid cement that cures to a solid over time. It is the second-most-used substance (after water), the most–widely used building material, and the most-manufactur ...
. Measurement of the
specific surface area Specific surface area (SSA) is a property of solids defined as the total surface area (SA) of a material per unit mass, (with units of m2/kg or m2/g). Alternatively, it may be defined as SA per solid or bulk volume (units of m2/m3 or m−1). I ...
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 Water is an inorganic compound with the chemical formula . It is a transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance. It is the main constituent of Earth's hydrosphere and the fluids of all known liv ...
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 Activated carbon, also called activated charcoal, is a form of carbon commonly used to filter contaminants from water and air, among many other uses. It is processed (activated) to have small, low-volume pores that greatly increase the surface ar ...
has strong affinity for many gases and has an adsorption cross section s of 0.162 nm2 for
nitrogen Nitrogen is a chemical element; it has Symbol (chemistry), symbol N and atomic number 7. Nitrogen is a Nonmetal (chemistry), nonmetal and the lightest member of pnictogen, group 15 of the periodic table, often called the Pnictogen, pnictogens. ...
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 m2/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 Catalysis () is the increase in rate of a chemical reaction due to an added substance known as a catalyst (). Catalysts are not consumed by the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recycles quick ...
, the surface area of
catalyst Catalysis () is the increase in rate of a chemical reaction due to an added substance known as a catalyst (). Catalysts are not consumed by the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recycles quick ...
s is an important factor in catalytic activity. Inorganic materials such as
mesoporous silica Mesoporous silica is a form of silica that is characterised by its mesoporous structure, that is, having pores that range from 2 nm to 50 nm in diameter. According to IUPAC's terminology, mesoporosity sits between microporous (50  ...
and layered
clay minerals Clay minerals are hydrous aluminium phyllosilicates (e.g. kaolin, Al2 Si2 O5( OH)4), sometimes with variable amounts of iron, magnesium, alkali metals, alkaline earths, and other cations found on or near some planetary surfaces. Clay mineral ...
have high surface areas of several hundred m2/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. The method has also been adapted for determination of specific area of ceramics and non-ferrous metal powders.


Thermal desorption

In 2023, researchers in the United States developed a method to determine BET surface areas using a thermogravimetric analyzer (TGA).{{Cite journal, doi = 10.1021/acs.langmuir.3c00854, issn = 0743-7463, volume = 39, issue = 25, pages = 8814–8823, last1 = Stahlfeld, first1 = K., last2 = Belmont, first2 = E., title = BET and Kelvin Analyses by Thermogravimetric Desorption, journal = Langmuir, date = 2023 This method uses a TGA to heat a porous sample loaded with an adsorbate, the produced plot of sample mass vs. temperature is then mapped into a standard isotherm to which BET theory is applied as normal. Common fluids, e.g. water or toluene, can be used as adsorbates for the TGA method allowing the specific interactions of different adsorbates to be determined, as these frequently differ from the commonly used nitrogen.


See also

*
Adsorption Adsorption is the adhesion of atoms, ions or molecules from a gas, liquid or dissolved solid to a surface. This process creates a film of the ''adsorbate'' on the surface of the ''adsorbent''. This process differs from absorption, in which a ...
* Capillary condensation * Langmuir adsorption model * Mercury intrusion porosimetry *
Physisorption Physisorption, also called physical adsorption, is a process in which the electronic structure of the atom or molecule is barely wikt:perturb, perturbed upon adsorption. Overview The fundamental interacting force of physisorption is Van der Waals ...
*
Surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...


References

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