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:
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- 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:
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- 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.
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- 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:
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- Rads,i-1 = ki*P*θi-1 (1)
- Rdes,i = k-i*θi (2)
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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 :
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- k-i = exp(-Ei/RT)
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Where Ei is the heat of adsorption, equals to E1 at the sample surface and to EL otherwise.
Read more about this topic: BET Theory