BET theory
Encyclopedia
BET theory aims to explain the physical adsorption
of gas
molecule
s on a solid
surface
and serves as the basis for an important analysis technique for the measurement of the specific surface area of a material. In 1938, Stephen Brunauer, Paul Hugh Emmett
, and Edward Teller
published an article about the BET theory in a journal for the first time; “BET” consists of the first initials of their family names.
, which is a theory for monolayer
molecular adsorption, to multilayer adsorption with the following hypotheses: (a) gas molecules physically adsorb on a solid in layers infinitely; (b) there is no interaction between each adsorption layer; and (c) the Langmuir theory can be applied to each layer. The resulting BET equation is expressed by (1):
and 
are the equilibrium
and the saturation pressure of adsorbates at the temperature of adsorption,
is the adsorbed gas quantity (for example, in volume units), and 
is the monolayer
adsorbed gas quantity.
is the BET constant, which is expressed by (2):
is the heat of adsorption for the first layer, and 
is that for the second and higher layers and is equal to the heat of liquefaction
.
Equation (1) is an adsorption isotherm and can be plotted as a straight line with
on the y-axis and 
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 
. The value of the slope 
and the y-intercept 
of the line are used to calculate the monolayer adsorbed gas quantity 
and the BET constant 
. The following equations can be used:
The BET method is widely used in surface
science for the calculation of surface areas
of solid
s by physical adsorption of gas molecules. A total surface area
and a specific surface area
are evaluated by the following equations:
where
is in units of volume which are also the units of the molar volume of the adsorbate gas
, 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:
Let us 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 Rads,i-1 for molecules on a layer (i-1) (i.e. formation of a layer i) is proportional to both its fractional surface θi-1 and to the pressure P; and that the desorption rate Rdes,i on a layer i is also proportional to its fractional surface θi:
Where ki 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 constant are assumed similar whatever the surface.
Assuming a Arrhenius law for desorption, the related constants can be expressed as :
Where Ei is the heat of adsorption, equals to E1 at the sample surface and to EL otherwise.
paste. If the quantity of adsorbed water vapor is measured at different levels of relative humidity a BET plot is obtained.
From the slope
and y-intersection 
on the plot it is possible to calculate 
and the BET constant 
. In case of cement paste hardened in water (T=97°C), the slope of the line is 
and the y-intersection 
; from this follows
From this the specific BET surface area
can be calculated by use of the above mentioned equation (one water molecule covers 
). It follows thus 
which means that hardened cement paste has an inner surface of 156 square meters per g of cement.
, which is a strong adsorbate and usually has an adsorption cross section
of 0.16 nm2 for nitrogen
adsorption at liquid nitrogen
temperature, is revealed from experimental data to have a large surface area around 3000 m² g-1. Moreover, in the field of solid catalysis
, the surface area of catalysts is an important factor in catalytic activity
. Porous inorganic materials such as mesoporous silica
and layer clay minerals
have high surface areas of several hundred m² g-1 calculated by the BET method, indicating the possibility of application for efficient catalytic materials.
Adsorption
Adsorption is the adhesion of atoms, ions, biomolecules or molecules of gas, liquid, or dissolved solids to a surface. This process creates a film of the adsorbate on the surface of the adsorbent. It differs from absorption, in which a fluid permeates or is dissolved by a liquid or solid...
of gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...
molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s on a solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
and serves as the basis for an important analysis technique for the measurement of the specific surface area of a material. In 1938, Stephen Brunauer, Paul Hugh Emmett
Paul H. Emmett
-Biography:He was born in Portland, Oregon. After completing his baccalaureate at Oregon Agricultural College , Emmett went on to the California Institute of Technology, where he earned his Ph.D. He was also a classmate and close friend of Linus Pauling at both institutions...
, and Edward Teller
Edward Teller
Edward Teller was a Hungarian-American theoretical physicist, known colloquially as "the father of the hydrogen bomb," even though he did not care for the title. Teller made numerous contributions to nuclear and molecular physics, spectroscopy , and surface physics...
published an article about the BET theory in a journal for the first time; “BET” consists of the first initials of their family names.
Concept
The concept of the theory is an extension of the Langmuir theoryLangmuir equation
The Langmuir equation relates the coverage or adsorption of molecules on a solid surface to gas pressure or concentration of a medium above the solid surface at a fixed temperature. The equation was developed by Irving Langmuir in 1916...
, which is a theory for monolayer
Monolayer
- Chemistry :A Langmuir monolayer or insoluble monolayer is a one-molecule thick layer of an insoluble organic material spread onto an aqueous subphase. Traditional compounds used to prepare Langmuir monolayers are amphiphilic materials that possess a hydrophilic headgroup and a hydrophobic tail...
molecular adsorption, to multilayer adsorption with the following hypotheses: (a) gas molecules physically adsorb on a solid in layers infinitely; (b) there is no interaction between each adsorption layer; and (c) the Langmuir theory can be applied to each layer. The resulting BET equation is expressed by (1):
and are the equilibriumDynamic equilibrium
A dynamic equilibrium exists once a reversible reaction ceases to change its ratio of reactants/products, but substances move between the chemicals at an equal rate, meaning there is no net change. It is a particular example of a system in a steady state...
and the saturation pressure of adsorbates at the temperature of adsorption,
is the adsorbed gas quantity (for example, in volume units), and is the monolayerMonolayer
- Chemistry :A Langmuir monolayer or insoluble monolayer is a one-molecule thick layer of an insoluble organic material spread onto an aqueous subphase. Traditional compounds used to prepare Langmuir monolayers are amphiphilic materials that possess a hydrophilic headgroup and a hydrophobic tail...
adsorbed gas quantity.
is the BET constant, which is expressed by (2): is the heat of adsorption for the first layer, and is that for the second and higher layers and is equal to the heat of liquefactionLiquefaction
Liquefaction may refer to:* Liquefaction, the general process of becoming liquid* Soil liquefaction, the process by which sediments become suspended* Liquefaction of gases in physics, chemistry, and thermal engineering* Liquefactive necrosis in pathology...
.
Equation (1) is an adsorption isotherm and can be plotted as a straight line with
on the y-axis and 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 . The value of the slope and the y-intercept of the line are used to calculate the monolayer adsorbed gas quantity and the BET constant . The following equations can be used:The BET method is widely used in surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
science for the calculation of surface areas
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
of solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
s by physical adsorption of gas molecules. A total surface area
and a specific surface areaSpecific surface area
Specific surface area is a material property of solids which measures the total surface area per unit of mass, solid or bulk volume, or cross-sectional area...
are evaluated by the following equations:where
is in units of volume which are also the units of the molar volume of the adsorbate gas : Avogadro's numberAvogadro's number In chemistry and physics, the Avogadro constant is defined as the ratio of the number of constituent particles N in a sample to the amount of substance n through the relationship NA = N/n. Thus, it is the proportionality factor that relates the molar mass of an entity, i.e... , |
 : adsorption cross section of the adsorbing species, |
 : molar volume of adsorbate gas |
 : mass of adsorbent (in g) |
Derivation
Similar to the derivation of Langmuir theoryLangmuir equation
The Langmuir equation relates the coverage or adsorption of molecules on a solid surface to gas pressure or concentration of a medium above the solid surface at a fixed temperature. The equation was developed by Irving Langmuir in 1916...
, 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:
-
- 1. Adsorptions occur only on well-defined sites of the sample surface (one per molecule)
- 2. The only considered molecular interaction is the following one: a molecule can act as a single adsorption site for a molecule of the upper layer.
- 3. The uppermost molecule layer is in equilibrium with the gas phase, i.e. similar molecule adsorption and desorption rates.
- 4. The desorption is a kinetically-limited process, i.e. a heat of adsorption must be provided:
-
- 4.1. these phenomenon are homogeneous, i.e. same heat of adsorption for a given molecule layer.
- 4.2. it is E1 for the first layer, i.e. the heat of adsorption at the solid sample surface
- 4.3. 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.
-
- 5. At the saturation pressure, the molecule layer number tends to infinity (i.e. equivalent to the sample being surrounded by a liquid phase)
Let us 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 Rads,i-1 for molecules on a layer (i-1) (i.e. formation of a layer i) is proportional to both its fractional surface θi-1 and to the pressure P; and that the desorption rate Rdes,i on a layer i is also proportional to its fractional surface θi:
-
-
-
- Rads,i-1 = ki*P*θi-1 (1)
- Rdes,i = k-i*θi (2)
-
-
Where ki 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 constant are assumed similar whatever the surface.
Assuming a Arrhenius law for desorption, the related constants can be expressed as :
-
-
-
- k-i = exp(-Ei/RT)
-
-
Where Ei is the heat of adsorption, equals to E1 at the sample surface and to EL otherwise.
Cement paste
By application of the BET theory it is possible to determine the inner surface of hardened cementCement
In the most general sense of the word, a cement is a binder, a substance that sets and hardens independently, and can bind other materials together. The word "cement" traces to the Romans, who used the term opus caementicium to describe masonry resembling modern concrete that was made from crushed...
paste. If the quantity of adsorbed water vapor is measured at different levels of relative humidity a BET plot is obtained.
From the slope
and y-intersection on the plot it is possible to calculate and the BET constant . In case of cement paste hardened in water (T=97°C), the slope of the line is and the y-intersection ; from this followsFrom this the specific BET surface area
can be calculated by use of the above mentioned equation (one water molecule covers ). It follows thus which means that hardened cement paste has an inner surface of 156 square meters per g of cement.Activated Carbon
For example, activated carbonActivated carbon
Activated carbon, also called activated charcoal, activated coal or carbo activatus, is a form of carbon that has been processed to make it extremely porous and thus to have a very large surface area available for adsorption or chemical reactions.The word activated in the name is sometimes replaced...
, which is a strong adsorbate and usually has an adsorption cross section
Cross section (physics)
A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....
of 0.16 nm2 for nitrogenNitrogen
Nitrogen is a chemical element that has the symbol N, atomic number of 7 and atomic mass 14.00674 u. Elemental nitrogen is a colorless, odorless, tasteless, and mostly inert diatomic gas at standard conditions, constituting 78.08% by volume of Earth's atmosphere...
adsorption at liquid nitrogen
Liquid nitrogen
Liquid nitrogen is nitrogen in a liquid state at a very low temperature. It is produced industrially by fractional distillation of liquid air. Liquid nitrogen is a colourless clear liquid with density of 0.807 g/mL at its boiling point and a dielectric constant of 1.4...
temperature, is revealed from experimental data to have a large surface area around 3000 m² g-1. Moreover, in the field of solid catalysis
Catalysis
Catalysis is the change in rate of a chemical reaction due to the participation of a substance called a catalyst. Unlike other reagents that participate in the chemical reaction, a catalyst is not consumed by the reaction itself. A catalyst may participate in multiple chemical transformations....
, the surface area of catalysts is an important factor in catalytic activity
Catalysis
Catalysis is the change in rate of a chemical reaction due to the participation of a substance called a catalyst. Unlike other reagents that participate in the chemical reaction, a catalyst is not consumed by the reaction itself. A catalyst may participate in multiple chemical transformations....
. Porous inorganic materials such as mesoporous silica
Mesoporous material
A mesoporous material is a material containing pores with diameters between 2 and 50 nm.Porous materials are classified into several kinds by their size...
and layer clay minerals
Clay
Clay is a general term including many combinations of one or more clay minerals with traces of metal oxides and organic matter. Geologic clay deposits are mostly composed of phyllosilicate minerals containing variable amounts of water trapped in the mineral structure.- Formation :Clay minerals...
have high surface areas of several hundred m² g-1 calculated by the BET method, indicating the possibility of application for efficient catalytic materials.
See also
- AdsorptionAdsorptionAdsorption is the adhesion of atoms, ions, biomolecules or molecules of gas, liquid, or dissolved solids to a surface. This process creates a film of the adsorbate on the surface of the adsorbent. It differs from absorption, in which a fluid permeates or is dissolved by a liquid or solid...
- Capillary condensationCapillary condensationCapillary Condensation is the "process by which multilayer adsorption from the vapor [phase] into a porous medium proceeds to the point at which pore spaces become filled with condensed liquid from the vapor [phase]." The unique aspect of capillary condensation is that vapor condensation occurs...
- MicromeriticsMicromeriticsMicromeritics is the science and technology of small particles. The knowledge and control of the size of particles is of importance in pharmacy and materials science. The size, and hence the surface area of a particle, can be related to the physical, chemical and pharmacologic properties of drugs...
- Surface tensionSurface tensionSurface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...
- Sorption isothermSorption isothermA sorption isotherm describes the equilibrium of the sorption of a material at a surface at constant temperature. It represents the amount of material bound at the surface as a function of the material present in the gas phase and/or in the solution...