BET theory

Not to be confused with Blade element theory.

Brunauer–Emmett–Teller (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 the first article about the BET theory in the Journal of the American Chemical Society.[1] The BET theory refers to multi layer adsorption, and usually adopts non-corrosive gases (like nitrogen, Ar, CO2 etc.) as adsorbates to determine the surface area data.

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

$\frac{1}{v \left [ \left ( {p_0}/{p} \right ) -1 \right ]} = \frac{c-1}{v_\mathrm{m} c} \left ( \frac{p}{p_0} \right ) + \frac{1}{v_m c}, \qquad (1)$

where $p$ 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 $v_\mathrm{m}$ is the monolayer adsorbed gas quantity. $c$ is the BET constant,

$c = \exp\left(\frac{E_1 - E_\mathrm{L}}{RT}\right), \qquad (2)$

where $E_1$ is the heat of adsorption for the first layer, and $E_\mathrm{L}$ is that for the second and higher layers and is equal to the heat of liquefaction.

BET plot

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 $v_\mathrm{m}$ and the BET constant $c$. The following equations can be used:

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

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

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

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

Derivation

The BET theory can be derived similar 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:[2]

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 phenomenon 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)

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:

$R_{\mathrm{ads},i-1} = k_i P \Theta_{i-1}$
$R_{\mathrm{des},i} = k_{-i} \Theta_i,$

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 constant 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.

Applications

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 $v_\mathrm{m}$ 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

$v_\mathrm{m} = \frac{1}{A+I}=0.0408,$
$c = 1+\frac{A}{I}=73.6 .$

From this the specific BET surface area $S_\mathrm{BET}$ can be calculated by use of the above-mentioned equation (one water molecule covers $s=0.114 \mathrm{nm}^2$). It follows thus $S_\mathrm{BET} = 156 \mathrm{m}^2/\mathrm{g}$ which means that hardened cement paste has an inner surface of 156 square meters per g of cement. However, the article on Portland cement states that "Typical values are 320–380 m2·kg−1 for general purpose cements, and 450–650 m2·kg−1 for "rapid hardening" cements."

Activated Carbon

For example, activated carbon strongly adsorbs many gases and has an adsorption cross section $s$ of 0.16 nm2 for nitrogen adsorption at liquid nitrogen temperature. BET theory has been applied to measure the specific surface area of activated carbon from experimental data, demonstrating a large specific surface area of around 3000 m² g−1.[citation needed] 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 layered 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.