BET theory
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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.
Contents
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.
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 \( \phi={p}/{p_0} \) 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
| \(N\): 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 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.
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 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.
See also
References
- ↑ S. Brunauer, P. H. Emmett and E. Teller, J. Am. Chem. Soc., 1938, 60, 309. doi:10.1021/ja01269a023
