# BET theory

## Concept

BET model of multilayer adsorption, that is, a random distribution of sites covered by one, two, three, etc., adsorbate molecules.

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:

1. gas molecules physically adsorb on a solid in layers infinitely;
2. gas molecules only interact with adjacent layers; and
3. the Langmuir theory can be applied to each layer.
4. the enthalpy of adsorption for the first layer is constant and greater than the second (and higher).
5. the enthalpy of adsorption for the second (and higher) layers is the same as the enthalpy of liquefaction.

The resulting BET equation is

${\displaystyle \theta ={\frac {cp}{(p_{o}-p){\bigl (}1+(c-1)(p/p_{o}{\bigr )})}}}$

where c is referred to as the BET C-constant, ${\displaystyle p_{o}}$is the vapor pressure of the adsorptive bulk liquid phase which would be at the temperature of the adsorbate and θ is the "surface coverage, defined as:

${\displaystyle \theta =n_{ads}/n_{m}}$.

Here ${\displaystyle n_{ads}}$is the amount of adsorbate and ${\displaystyle n_{m}}$ is called the monolayer equivalent. The ${\displaystyle n_{m}}$ is the amount that if it were all in a monolayer (which is theoretically impossible for physical adsorption) would cover the surface with exactly one layer of adsorbate. The above equation is usually rearranged to yield the following equation as a convenience for analysis:

${\displaystyle {\frac {1}{v\left[\left({p_{0}}/{p}\right)-1\right]}}={\frac {c-1}{v_{\mathrm {m} }c}}\left({\frac {p}{p_{0}}}\right)+{\frac {1}{v_{m}c}},\qquad (1)}$

where ${\displaystyle p}$ and ${\displaystyle p_{0}}$ are the equilibrium and the saturation pressure of adsorbates at the temperature of adsorption, ${\displaystyle v}$ is the adsorbed gas quantity (for example, in volume units), and ${\displaystyle v_{\mathrm {m} }}$ is the monolayer adsorbed gas quantity. ${\displaystyle c}$ is the BET constant,

${\displaystyle c=\exp \left({\frac {E_{1}-E_{\mathrm {L} }}{RT}}\right),\qquad (2)}$

where ${\displaystyle E_{1}}$ is the heat of adsorption for the first layer, and ${\displaystyle E_{\mathrm {L} }}$ is that for the second and higher layers and is equal to the heat of liquefaction or heat of vaporization.

BET plot

Equation (1) is an adsorption isotherm and can be plotted as a straight line with ${\displaystyle 1/{v[({p_{0}}/{p})-1]}}$ on the y-axis and ${\displaystyle \varphi ={p}/{p_{0}}}$ 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 ${\displaystyle 0.05<{p}/{p_{0}}<0.35}$. The value of the slope ${\displaystyle A}$ and the y-intercept ${\displaystyle I}$ of the line are used to calculate the monolayer adsorbed gas quantity ${\displaystyle v_{\mathrm {m} }}$ and the BET constant ${\displaystyle c}$. The following equations can be used:

${\displaystyle v_{m}={\frac {1}{A+I}}\qquad (3)}$
${\displaystyle c=1+{\frac {A}{I}}.\qquad (4)}$

The BET method is widely used in materials science for the calculation of surface areas of solids by physical adsorption of gas molecules. The total surface area ${\displaystyle S_{\mathrm {total} }}$ and the specific surface area ${\displaystyle S_{\mathrm {BET} }}$ are given by

${\displaystyle S_{\mathrm {total} }={\frac {\left(v_{\mathrm {m} }Ns\right)}{V}},\qquad (5)}$
${\displaystyle S_{\mathrm {BET} }={\frac {S_{\mathrm {total} }}{a}},\qquad (6)}$

where ${\displaystyle v_{\mathrm {m} }}$ is in units of volume which are also the units of the monolayer volume of the adsorbate gas, ${\displaystyle N}$ is the Avogadro number, ${\displaystyle s}$ the adsorption cross section of the adsorbate,[3] ${\displaystyle V}$ the molar volume of the adsorbate gas, and ${\displaystyle a}$ the mass of the solid sample or adsorbent.

## Derivation

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:[4]

1. Adsorptions occur only on well-defined sites of the sample surface (one per molecule)
2. 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.
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:
• 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.
5. 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:

${\displaystyle R_{\mathrm {ads} ,i-1}=k_{i}P\Theta _{i-1}}$
${\displaystyle R_{\mathrm {des} ,i}=k_{-i}\Theta _{i},}$

where ki and ki 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

${\displaystyle k_{i}=\exp(-E_{i}/RT),}$

where Ei is the heat of adsorption, equal to E1 at the sample surface and to EL otherwise.

## Finding the linear BET range

It is still not clear on how to find the linear range of the BET plot for microporous materials in a way that reduces any subjectivity in the assessment of the monolayer capacity. Rouquerol et al.[5] suggested a procedure that is based on two criteria:

• C must be positive implying that any negative intercept on the BET plot indicates that one is outside the valid range of the BET equation.
• Application of the BET equation must be limited to the range where the term V(1-P/P0) continuously increases with P/P0.

These corrections are an attempt to salvage the BET theory which is restricted to type II isotherm. Even with this type, use of the data is restricted to 0.5 to 3.5 of ${\displaystyle P/P_{0}}$, routinely discarding 70% of the data. Even this restriction has to be modified depending upon conditions. The problems with the BET theory are multiple and reviewed by Sing.[6] A serious problem is that there is no relationship between the BET and the calorimetric measurements in experiments. It violates the Gibbs' phase rules. It is extremely unlikely that it measure correctly the surface area, formerly great advantage of the theory. It is based upon chemical equilibrium, which assumes localized chemical bond (this approach has been abandoned by the modern theories. See [7] chapter 4, χ/ESW and Chapter 7, DFT or better NLDFT) in total contradiction to what is known about physical adsorption, which is based upon non-local intermolecular attractions. Two extreme problems is that in certain cases BET leads to anomalies and the C constant can be negative, implying an imaginary energy.

## Applications

### Cement and concrete

The rate of curing of concrete depends on the fineness of the cement and of the components used in its manufacture, which may include fly ash, silica fume and other materials, in addition to the calcinated limestone which causes it to harden. Although the Blaine air permeability method is often preferred, due to its simplicity and low cost, the nitrogen BET method is also used.

When hydrated cement hardens, the calcium silicate hydrate (or C-S-H), which is responsible for the hardening reaction, has a large specific surface area because of its high porosity. This porosity is related to a number of important properties of the material, including the strength and permeability, which in turn affect the properties of the resulting concrete. Measurement of the specific surface area using the BET method is useful for comparing different cements. This may be performed using adsorption isotherms measured in different ways, including the adsorption of water vapour at temperatures near ambient, and adsorption of nitrogen at 77 K (the boiling point of liquid nitrogen). Different methods of measuring cement paste surface areas often give very different values, but for a single method the results are still useful for comparing different cements.

### Activated carbon

Activated carbon strongly adsorbs many gases and has an adsorption cross section ${\displaystyle s}$ of 0.162 nm2 for nitrogen adsorption at liquid-nitrogen temperature (77 K). BET theory can be applied to estimate the specific surface area of activated carbon from experimental data, demonstrating a large specific surface area, even around 3000 m2/g.[8] However, this surface area is largely overestimated due to enhanced adsorption in micropores,[5] and more realistic methods should be used for its estimation, such as the subtracting pore effect (SPE) method.[9]

### Catalysis

In the field of solid catalysis, the surface area of catalysts is an important factor in catalytic activity. Inorganic materials such as mesoporous silica and layered clay minerals have high surface areas of several hundred m2/g calculated by the BET method, indicating the possibility of application for efficient catalytic materials.

### Specific surface area calculation

The ISO 9277 standard for calculating the specific surface area of solids is based on the BET method.