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BET Theory

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  BET theory 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 a material. In 1938, Stephen Brunauer , Paul Hugh Emmett, and Edward Teller   published an article about the BET theory in a  journal [1]   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 theory, 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):  P   and  P  0  are the equilibrium and the saturation pressure of adsorbates at the temperature of adsorption, v  is the adsorbed gas quantity (for example, in volume units), and v m  is the monolayer  adsorbed gas quantity. c  is the BET constant, which is expressed by (2):  E  1  is the heat of adsorption for the first layer, and  E   L  is that for the second and higher layers and is equal to the heat of  liquefaction.  BET plot Equation (1) is an adsorption isotherm and can be plotted as a straight line with 1 / v [(  P  0  /  P  ) − 1] on the y -axis and ϕ  =  P   /  P  0  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 <  P   /  P  0  < 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 m  and the BET constant c . The following equations can be used: The BET method is widely used in surface science for the calculation of surface areas of  solids  by physical adsorption of gas molecules. A total surface area S  total   and a specific surface area  S   are evaluated by the following equations: where v m  is in units of volume which are also the units of the molar volume of the adsorbate gas : Avogadro's number ,   s : adsorption cross section of the adsorbing species, V  : molar volume of adsorbate gas a : mass of adsorbent (in g) Derivation Similar to the derivation of  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: 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 E 1  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 E L  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 θ i  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  ads, i -1  = k  i *P*θ i -1  (1)  R  des, i  = k  - i *θ i  (2) 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 constant are assumed similar whatever the surface. Assuming a 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, equals to E 1  at the sample surface and to E L  otherwise. Example Cement paste By application of the BET theory it is possible to determine the inner surface of hardened cement  paste. If the quantity of adsorbed water vapor is measured at different levels of relative humidity a BET plot is obtained. From the slope  A  and y-intersection  I   on the plot it is possible to calculate v m  and the BET constant c . In case of cement paste hardened in water (T=97°C), the slope of the line is  A  = 24.20 and the y-intersection  I   = 0.33; from this follows From this the specific BET surface area S   BET   can be calculated by use of the above mentioned equation (one water molecule covers  s  = 0.114 nm 2 ). It follows thus S   BET   = 156 m 2  /  g   which means that hardened cement paste has an inner surface of 156 square meters per g of cement. Activated Carbon For example, activated carbon, which is a strong adsorbate and usually has an adsorption cross section   s  of 0.16 nm 2  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. BET Main article: BET theory  Often molecules do form multilayers, that is, some are adsorbed on already adsorbed molecules and the Langmuir isotherm is not valid. In 1938 Stephen Brunauer , Paul Emmett, and Edward Teller  developed a model isotherm that takes that possibility into account. Their theory is called BET theory, after the initials in their last names. They modified Langmuir's mechanism as follows: A (g)  + S ⇌  AS A (g)  + AS ⇌  A 2 S A (g)  + A 2 S ⇌  A 3 S and so on   Langmuir isotherm (red) and BET isotherm (green) The derivation of the formula is more complicated than Langmuir's (see links for complete derivation). We obtain:  x  is the pressure divided by the vapor pressure for the adsorbate at that temperature (usually denoted  P   /  P  0 ), v  is the STP volume of adsorbed adsorbate, v mon  is the STP volume of the amount of adsorbate required to form a monolayer and c  is the equilibrium constant  K   we used in Langmuir isotherm multiplied by the vapor pressure of the adsorbate. The key assumption used in deriving the BET equation that the successive heats of adsorption for all layers except the first are equal to the heat of condensation of the adsorbate. The Langmuir isotherm is usually better for chemisorption and the BET isotherm works better for physisorption for non-microporous surfaces.
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