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 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) 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: :R_{\mathrm{ads},i-1} = k_i P \Theta_{i-1} :R_{\mathrm{des},i} = k_{-i} \Theta_i, 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 constants are assumed similar whatever the surface. Assuming an Arrhenius law for desorption, the related constants can be expressed as :k_i = \exp(-E_i/RT), where
Ei is the heat of adsorption, equal to
E1 at the sample surface and to
EL otherwise. Consider some substance A. The adsorption of A onto an available surface site (*)_0 produces a new site (*)_1 on the first layer. In summary, :A(g) + (*)_0 (*)_1 Extending this to higher order layers one obtains :A(g) + (*)_1 (*)_2 and similarly :A(g) + (*) n-1 (*)_n Denoting the activity of the number of available sites of the nth layer with \theta_n and the
partial pressure of A with P, the last equilibrium can be written : K_n = \frac{\theta_n}{P \theta_{n-1}} It follows that the coverage of the first layer can be written :\theta_1=K_1 P\theta_0 and that the coverage of the second layer can be written :\theta_2=K_2 P\theta_1=K_2 P K_1 P \theta_0 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 n>1 should be the same, which results in the recursion :\theta_n=(K_\ell P)^{n-1} P K_1 \theta_0 In order to simplify some infinite summations, let x=K_\ell P and let y=K_1 P. Then the nth layer coverage can written :\theta_n = c \theta_0 x^n, \quad n>0 if c=y/x. 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 \theta_n denotes the relative number of sites covered by only n adsorbents. Doing so it is easy to see that the total volume of adsorbed molecules can be written as the sum :V_\text{ads}=V_\text{m}\sum_{n=1}^{\infty} n \theta_n = V_\text{m} c\theta_0 x\sum_{n=1}^{\infty} nx^{n-1} where V_\text{m} is the molecular volume. Employing the fact that this sum is the first derivative of a geometric sum, the volume becomes :V_\text{ads}= V_\text{m} c\theta_0\frac{x}{(1-x)^2}, \quad |x| Since the total coverage of a mono-layer must be unity, the full mono-layer coverage must be :1=\sum_{n=1}^{\infty} \theta_n In order to properly make the substitution for \theta_n, the restriction n>1 forces us to take the zeroth contribution outside the summation, resulting in :\sum_{n=1}^{\infty} \theta_n=\theta_0 + c \theta_0 x\sum_{n=0}^{\infty} x^n = \theta_0 + c \theta_0 \frac{x}{1-x}, \quad |x| Lastly, defining the excess coverage as V_\text{excess}=V_\text{ads}/V_\text{mono}, the excess volume relative to the volume of an adsorbed mono-layer becomes :V_\text{excess}=\frac{\sum_{n=0}^{\infty} n \theta_n}{1} =\frac{\sum_{n=0}^{\infty} n \theta_n}{\sum_{n=0}^{\infty}\theta_n} =\frac{cx}{(1-x)(1+(c-1)x)} where the last equality was obtained by making use of the series expansions presented above. The constant c must be interpreted as the relative binding affinity the substance A has towards a surface, relative to its own liquid. If c>1 then the initial part of the isotherm will be reminiscent of the Langmuir isotherm which reaches a plateau at full mono-layer coverage, whereas c means the mono-layer will have a slow build-up. Another thing to note is that in order for the geometric substitutions to hold, x. The isotherm above exhibits a singularity at x^\ast=1. Since x=K_\ell P one can write x^\ast=K_\ell P^\ast , implying that K_\ell=1/P^\ast. This means that x=P/P^\ast must be true, ultimately resulting in x\in[0,1). == Finding the linear BET range ==