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| {{distinguish|Blade element theory}}
| | Sportspersons Bykowski from Port Hawkesbury, really likes leathercrafting, property developers [http://www.ampaulz.com/1054/slew-of-new-rental-launches-in-2013/ new condos in singapore] singapore and hot rods. Of late had a family voyage to Wet Tropics of Queensland. |
| {{refimprove|date=January 2011}}
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| '''Brunauer–Emmett–Teller''' ('''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 H. Emmett|Paul Hugh Emmett]], and [[Edward Teller]] published the first article about the BET theory in the [[Journal of the American Chemical Society]].<ref>S. Brunauer, P. H. Emmett and E. Teller, ''J. Am. Chem. Soc.'', 1938, '''60''', 309. {{doi|10.1021/ja01269a023}}</ref>
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| ==Concept==
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| The concept of the theory is an extension of the [[Langmuir equation|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
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| :<math> \frac{1}{v \left [ \left ( {p_0}/{p} \right ) -1 \right ]} = \frac{c-1}{v_\mathrm{m} c} \left ( \frac{p}{p_0} \right ) + \frac{1}{v_m c}, \qquad (1)</math>
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| where <math>p</math> and <math>p_0</math> are the [[Dynamic equilibrium|equilibrium]] and the [[saturation pressure]] of adsorbates at the temperature of adsorption, <math>v</math> is the adsorbed gas quantity (for example, in volume units), and <math>v_\mathrm{m}</math> is the [[monolayer]] adsorbed gas quantity. <math>c</math> is the ''BET constant'',
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| :<math> c = \exp\left(\frac{E_1 - E_\mathrm{L}}{RT}\right), \qquad (2)</math> | |
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| where <math>E_1</math> is the heat of adsorption for the first layer, and <math>E_\mathrm{L}</math> is that for the second and higher layers and is equal to the heat of [[liquefaction]].
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| [[Image:BET-1.jpg|300px|thumb|BET plot]]
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| Equation (1) is an [[adsorption isotherm]] and can be plotted as a straight line with <math> {1}/{v [ ({p_0}/{p}) -1 ]}</math> on the y-axis and <math> \phi={p}/{p_0} </math> 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 <math>0.05 < {p}/{p_0} < 0.35</math>. The value of the slope <math>A</math> and the y-intercept <math>I</math> of the line are used to calculate the monolayer adsorbed gas quantity <math>v_\mathrm{m}</math> and the BET constant <math>c</math>. The following equations can be used:
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| :<math>v_m = \frac{1}{A+I}\qquad (3)</math>
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| :<math>c = 1+\frac{A}{I}.\qquad (4)</math>
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| The BET method is widely used in [[surface]] science for the calculation of [[area|surface areas]] of [[solid]]s by physical adsorption of gas molecules. The total surface area <math>S_\mathrm{total}</math> and the [[specific surface area]] <math>S_\mathrm{BET}</math> are given by
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| :<math>S_\mathrm{total} = \frac{\left ( v_\mathrm{m} N s \right )}{V}, \qquad (5)</math>
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| :<math>S_\mathrm{BET} = \frac{S_\mathrm{total}}{a}, \qquad (6)</math>
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| where <math>v_\mathrm{m}</math> is in units of volume which are also the units of the molar volume of the adsorbate gas,
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| <math>N</math> is [[Avogadro's number]], <math>s</math> the adsorption cross section of the adsorbing species, <math>V</math> the molar volume of the adsorbate gas, and <math>a</math> the mass of the adsorbent.
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| == Derivation ==
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| The BET theory can be derived similar to the [[Langmuir equation|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:
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| # Adsorptions occur only on well-defined sites of the sample surface (one per molecule)
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| # 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.
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| # The uppermost molecule layer is in equilibrium with the gas phase, i.e. similar molecule adsorption and desorption rates.
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| # The desorption is a kinetically-limited process, i.e. a heat of adsorption must be provided:
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| #* these phenomenon are homogeneous, i.e. same heat of adsorption for a given molecule layer.
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| #* it is E<sub>1</sub> for the first layer, i.e. the heat of adsorption at the solid sample surface
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| #* the other layers are assumed similar and can be represented as condensed species, i.e. liquid state. Hence, the heat of adsorption is E<sub>L</sub> is equal to the heat of liquefaction.
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| # At the saturation pressure, the molecule layer number tends to infinity (i.e. equivalent to the sample being surrounded by a liquid phase)
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| Let us consider a given amount of solid sample in a controlled atmosphere. Let ''θ<sub>i</sub>'' 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''<sub>ads,''i''-1</sub> for molecules on a layer (''i''-1) (i.e. formation of a layer ''i'') is proportional to both its fractional surface ''θ''<sub>''i''-1</sub> and to the pressure ''P'', and that the desorption rate ''R''<sub>des,''i''</sub> on a layer ''i'' is also proportional to its fractional surface ''θ''<sub>''i''</sub>:
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| :<math>R_{\mathrm{ads},i-1} = k_i P \Theta_{i-1}</math>
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| :<math>R_{\mathrm{des},i} = k_{-i} \Theta_i,</math>
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| where ''k''<sub>''i''</sub> and ''k''<sub>-''i''</sub> 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.
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| Assuming an Arrhenius law for desorption, the related constants can be expressed as
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| :<math>k_i = \exp(-E_i/RT),</math>
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| where ''E''<sub>''i''</sub> is the heat of adsorption, equal to ''E''<sub>1</sub> at the sample surface and to ''E''<sub>L</sub> otherwise.
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| == Example ==
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| === Cement paste ===
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| 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.
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| From the slope <math>A</math> and y-intersection <math>I</math> on the plot it is possible to calculate <math>v_\mathrm{m}</math> and the BET constant <math>c</math>. In case of cement paste hardened in water (''T'' = 97°C), the slope of the line is <math>A=24.20</math> and the y-intersection <math>I=0.33</math>; from this follows
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| :<math>v_\mathrm{m} = \frac{1}{A+I}=0.0408,</math>
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| :<math>c = 1+\frac{A}{I}=73.6 .</math>
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| From this the specific BET surface area <math>S_\mathrm{BET}</math> can be calculated by use of the above mentioned equation (one water molecule covers <math>s=0.114 \mathrm{nm}^2</math>). It follows thus <math>S_\mathrm{BET} = 156 \mathrm{m}^2/\mathrm{g}</math> which means that hardened cement paste has an inner surface of 156 square meters per g of cement.
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| However, the article on [[Portland cement#Cement grinding|Portland cement]] states that "Typical values are 320–380 m<sup>2</sup>·kg<sup>−1</sup> for general purpose cements, and 450–650 m<sup>2</sup>·kg<sup>−1</sup> for "rapid hardening" cements."
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| === Activated Carbon ===
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| For example, [[activated carbon]], which is a strong adsorbate and usually has an adsorption [[cross section (physics)|cross section]] <math>s</math> of 0.16 nm<sup>2</sup> for [[nitrogen]] adsorption at [[liquid nitrogen]] temperature, is revealed from experimental data to have a large surface area around 3000 m² g<sup>-1</sup>. Moreover, in the field of solid [[catalysis]], the surface area of [[catalyst]]s is an important factor in [[catalysis|catalytic activity]]. Porous inorganic materials such as [[mesoporous material|mesoporous silica]] and layer [[clay|clay minerals]] have high surface areas of several hundred m² g<sup>-1</sup> calculated by the BET method, indicating the possibility of application for efficient catalytic materials.
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| ==See also==
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| *[[Adsorption]]
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| *[[Capillary condensation]]
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| *[[Surface tension]]
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| *[[Sorption isotherm]]
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| ==References==
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| <references/>
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| [[Category:Scientific techniques]]
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| [[Category:Physical chemistry]]
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| [[Category:Gas technologies]]
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Sportspersons Bykowski from Port Hawkesbury, really likes leathercrafting, property developers new condos in singapore singapore and hot rods. Of late had a family voyage to Wet Tropics of Queensland.