text: The BET theory can be derived similarly to the 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 phenomena are homogeneous, i.e. same heat of adsorption for a given molecule layer.
  #* it is E 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 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)
  Consider a given amount of solid sample in a controlled atmosphere. Let θ 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 for molecules on a layer (i-1) (i.e. formation of a layer i) is proportional to both its fractional surface θ and to the pressure P, and that the desorption rate R on a layer i is also proportional to its fractional surface θ:
  where k and k 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 constants are assumed similar whatever the surface.
  Assuming an Arrhenius law for desorption, the related constants can be expressed as
  where E is the heat of adsorption, equal to E at the sample surface and to E otherwise.
  Consider some substance A. The adsorption of A onto an available surface site  produces a new site  on the first layer. In summary,
  Extending this to higher order layers one obtains
  and similarly
  Denoting the activity of the number of available sites of the th layer with   and the partial pressure of A with , the last equilibrium can be written
  It follows that the coverage of the first layer can be written
  and that the coverage of the second layer can be written
  Realising that the adsorption of A onto the second layer is equivalent to adsorption of A onto its own liquid phase, the rate constant for  should be the same, which results in the recursion
  In order to simplify some infinite summations, let  and let . Then the th layer coverage can written
  if . The coverage of any layer is defined as the relative number of available sites. An alternative definition, which leads to a set of coverage's that are numerically to those resulting from the original way of defining surface coverage, is that  denotes the relative number of sites covered by only  adsorbents. Doing so it is easy to see that the total volume of adsorbed molecules can be written as the sum
  where  is the molecular volume. Employing the fact that this sum is the first derivative of a geometric sum, the volume becomes
  Since the total coverage of a mono-layer must be unity, the full mono-layer coverage must be
  In order to properly make the substitution for , the restriction  forces us to take the zeroth contribution outside the summation, resulting in
  Lastly, defining the excess coverage as , the excess volume relative to the volume of an adsorbed mono-layer becomes
  where the last equality was obtained by making use of the series expansions presented above. The constant  must be interpreted as the relative binding affinity the substance A has towards a surface, relative to its own liquid. If  then the initial part of the isotherm will be reminiscent of the Langmuir isotherm which reaches a plateau at full mono-layer coverage, whereas  means the mono-layer will have a slow build-up. Another thing to note is that in order for the geometric substitutions to hold, . The isotherm above exhibits a singularity at . Since  one can write , implying that . This means that  must be true, ultimately resulting in .