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en>Ohnoitsjamie: Reverted edits by 206.71.251.194 (talk) to last version by Stesmo

2014-12-16T16:00:41Z

Reverted edits by 206.71.251.194 (talk) to last version by Stesmo

← Older revision		Revision as of 18:00, 16 December 2014
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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}}~~

	'''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>

	~~==Concept==~~
	~~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

	~~:<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>~~

	~~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'',

	:~~<math> c = \exp\left(\frac{E_1 - E_\mathrm{L}}{RT}\right), \qquad (2)</math>~~

	~~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]]~~.

	~~[[Image:BET-1~~.~~jpg\|300px\|thumb\|BET plot]]~~

	~~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:~~
	~~:<math>v_m = \frac{1}{A+I}\qquad (3)</math>~~
	~~:<math>c = 1+\frac{A}{I}.\qquad (4)</math>~~

	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
	~~:<math>S_\mathrm{total} = \frac{\left ( v_\mathrm{m} N s \right )}{V}, \qquad (5)</math>~~
	~~:<math>S_\mathrm{BET} = \frac{S_\mathrm{total}}{a}, \qquad (6)</math>~~
	~~where <math>v_\mathrm{m}</math> is in units of volume which are also the units of the molar volume of the adsorbate gas,~~
	<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.

	~~== Derivation ==~~
	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:
	~~# 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 phenomenon are homogeneous, i.e. same heat of adsorption for a given molecule layer.~~
	~~#* it is E<sub>1</sub> 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 E<sub>L<~~/~~sub> 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)~~

	~~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>:
	~~:<math>R_{\mathrm{ads},i-1} = k_i P \Theta_{i-1}</math>~~
	~~:<math>R_{\mathrm{des},i} = k_{-i} \Theta_i,</math>~~
	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.
	~~Assuming an Arrhenius law for desorption, the related constants can be expressed as~~
	~~:<math>k_i = \exp(-E_i/RT),</math>~~
	~~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.~~

	~~== 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 <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~~
	~~:<math>v_\mathrm{m} = \frac{1}{A+I}=0.0408,</math>~~
	~~:<math>c = 1+\frac{A}{I}=73.6 .</math>~~
	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.
	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."

	~~=== Activated Carbon ===~~
	~~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~~.

	~~==See also==~~
	*[[Adsorption]]
	*[[Capillary condensation]]
	*[[Surface tension]]
	*[[Sorption isotherm]]

	~~==References==~~
	~~<references/>~~

	~~[[Category:Scientific techniques]]~~
	~~[[Category:Physical chemistry]]~~
	~~[[Category:Gas technologies]]~~

en>Trivialist: Undid revision 590313812 by David Vincenzo (talk) rv refspam

2014-01-12T03:59:42Z

Undid revision 590313812 by David Vincenzo (talk) rv refspam

New page

{{distinguish|Blade element theory}}
{{refimprove|date=January 2011}}

'''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>

==Concept==
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

:<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>

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'',

:<math> c = \exp\left(\frac{E_1 - E_\mathrm{L}}{RT}\right), \qquad (2)</math>

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]].

[[Image:BET-1.jpg|300px|thumb|BET plot]]

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:
:<math>v_m = \frac{1}{A+I}\qquad (3)</math>
:<math>c = 1+\frac{A}{I}.\qquad (4)</math>

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
:<math>S_\mathrm{total} = \frac{\left ( v_\mathrm{m} N s \right )}{V}, \qquad (5)</math>
:<math>S_\mathrm{BET} = \frac{S_\mathrm{total}}{a}, \qquad (6)</math>
where <math>v_\mathrm{m}</math> is in units of volume which are also the units of the molar volume of the adsorbate gas,
<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.

== Derivation ==
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:
# 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 phenomenon 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)

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 ''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'':
:<math>R_{\mathrm{ads},i-1} = k_i P \Theta_{i-1}</math>
:<math>R_{\mathrm{des},i} = k_{-i} \Theta_i,</math>
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 an Arrhenius law for desorption, the related constants can be expressed as
:<math>k_i = \exp(-E_i/RT),</math>
where ''E''''i'' is the heat of adsorption, equal 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 <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
:<math>v_\mathrm{m} = \frac{1}{A+I}=0.0408,</math>
:<math>c = 1+\frac{A}{I}=73.6 .</math>
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.
However, the article on [[Portland cement#Cement grinding|Portland cement]] states that "Typical values are 320–380 m2·kg−1 for general purpose cements, and 450–650 m2·kg−1 for "rapid hardening" cements."

=== Activated Carbon ===
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 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 [[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-1 calculated by the BET method, indicating the possibility of application for efficient catalytic materials.

==See also==
*[[Adsorption]]
*[[Capillary condensation]]
*[[Surface tension]]
*[[Sorption isotherm]]

==References==
<references/>

[[Category:Scientific techniques]]
[[Category:Physical chemistry]]
[[Category:Gas technologies]]