Introduction

Determining the surface area of a solid material is vitally important for many applications, particularly in surface sensitive ones such as catalysis. For a long time, measuring the surface are of solids, particularly those with porous or otherwise complicated surface geometries, was not readily possible. However, this changed in 1938 with the introduction of the BET (Brunnauer, Emmett and Teller) theory for physical adsorption, which allows estimating the surface area of solids.^[1]BET theory predicts the volume of adsorbed gas as a function of the gas partial pressure^[1]. The theory has several limitations, which cause it to only be valid in a small range of partial pressures^[1]. Still, it is very useful for getting quantitative measurements on the surface areas of solids and thus widely used. The first section of this page will give some necessary background information, explaining the concept of adsorption, the simple Langmuir adsorption theory and how adsorption can be used to estimate the surface area of solids. .The second section will present the BET theory, which is an extension of the Langmuir theory and the resulting BET equation. Finally, the last section will delve deeper into the limitations and applications of the BET equation.

Physisorption and the Langmuir isotherm

Adsorption is defined by IUPAC as an increase in the concentration of a substance at an interface.^[2]When discussing the BET theory, we need only concern ourselves with adsorption of a gas at a solid interface. This kind of adsorption can be divided into two types, chemisorption and physisorption. Physisorption is reversible adsorption which is caused by van der Waals forces, where there is no (signifcant) changes in the electronic structure of the adsorbed species^[2]. In simpler terms, there are no bonds formed or broken when the gas is adorped at the solid surface. In contrast, in chemisorption a chemical reaction happens between the adsorbed species and the solid, new bonds are formed^[2]. Physisorption is the physical phenomena on which BET surface analysis is based, thus it will be the focus of this page.

Physisorption (and adsorption in general) is usually visualized in a plot known as an isotherm. The data for an isotherm is collected by measuring the equilibrium adsorption of the gas on the solid at different gas partial pressures at a constant temperature. The plot has the gas partial pressure on the x-axis and the amount adsorbed on the y-axis. An example adsorption isotherm for the adsorption of nitrogen gas onto a La₂O₃-ZrO₂ catalyst support at 77K is given in Figure 1.

Figure 1. Adsorption isotherm for nitrogen onto La₂O₃-ZrO₂ at 77 K. The x-axis shows the relative gas partial pressure and the y-axis shows the volume of gas adsorbed per gram of solid (Figure: Joakim Kattelus, data provided by the Catalysis group at Aalto University).

The isotherm shows both the volume adsorbed at adsorption and at desorption. These were measured by first increasing the partial pressure of the gas, and then decreasing it to zero. The two volumes are the same at lower pressures but differ at higher pressures. This is called hysteresis, and caused here by the mesoporous structure of the solid^[1]. The gas entering the pores is condensed in them, due to the small pores changing the vapor pressure of the gas (capillary condensation). This pore condensation phenomena will be discussed in more detail later.

The Langmuir theory gives a simple method for estimating the amount of gas adsorbed at a constant temperature. It only concerns itself with monolayer adsorption, that it, the adsorption of only a single layer of adsorbed atoms or molecules on the surface of the solid^[1]. Langmuir theory also does several other assumptions:

• Adsorbed species do not interact with each other in any way^[1]
• Adsorption can only happen at certain specific surface sites, that can be assumed to be "flat"^[1]
• Adsorption is equally likely at all (free) adsorption sites^[1]
• Adsorption and desorption are elementary (one step) processes^[1]

Using these assumptions, a relatively simple expression for the amount adsorped can be obtained^[1]:

\[ \frac{V_{ads}}{V_m} = \theta = \frac{bP}{1+bP} \]

where \( V_{ads} \) is the volume of adsorbed gas, \( V_m \) is the volume of one monolayer of the adsorbed gas, \( \int_{-\infty}^\infty \mbox{e}^{-x^2} \mbox{d}x = \sqrt{\pi} \) is the monolayer coverage, \( b \) is a constant and \( P \) the partial pressure of the adsorbed gas. The monolayer coverage is defined as the fraction of the available sites on the catalyst surface that have attached an adsorbed species. The monolayer volume is the property of interest that has to be measured. When the monolayer volume is known, the surface area can be calculated easily from it if one knows the area occupied by one adsorbed molecule. The number of adsorbed molecules is simply

\[ N_{ads} = \frac{V_m}{V_{mol}}N_A \]

and the specific surface area is

\[ S = N_{ads} A_{ads}=\frac{V_m N_A A_{ads}}{V_{mol}} \]

where \( S \) is the specific surface area of the solid (m²/g), \( V_m \) is the volume of one monolayer of the adsorbed gas at STP, \( N_A \) is Avogadro constant (1/mol) \( A_{ads} \) the surface area of the adsorbed molecule, and \( V_{mol} \) is the molar volume at standard temperature and pressure (22.7 dm³/mol).

Unfortunately, real physisorption seldom forms a single, uniform monolayer. Using the Langmuir isotherm equation to estimate the monolayer volume thus does not work well for real solids. This was long an unsolved problem, until the breaktrough came in the form of the introduction of BET theory.

BET theory

BET theory can be viewed as an extension of Langmuir theory, which gets rid of the assumption of just one monolayer formed^[1]. In essence, BET theory makes the same assumptions as the Langmuir theory, but assumes several monolayers can form on top of each other^[1]. Each single monolayer is modeled using the Langmuir theory for a monolayer. BET theory thus does the following assumptions:

• The first monolayer is formed with a heat of adsorption^[1] \( \Delta H_1 \)

• Subsequent layers are formed on top of the previous layers with a heat of adsorption^[1]
  \( \Delta H_L \)

• The second and subsequent layers can start forming before the first layer has reached full coverage^[1]

Using these assumptions, the BET equation can be derived. It has the following form:^[1][3]

\[ V_{ads} = \frac{V_m c P}{(P_0 - P)(1+(c-1)P/P_0)} \]

where \( V_{ads} \) is the volume of adsorped gas, \( V_m \) is the volume of one monolayer of the adsorbed gas, \( c \) is a constant related to the adsorption enthalpies that will be considered later, \( P \) the partial pressure of the gas, and \( P_0 \) the vapor pressure of the gas at the specified temperature.

The BET equation is generally not valid at higher partial pressures, but it is valid at a high enough pressure range to be useful for determination of \( V_m \) for real solids. ^[1] Fortunately, finding the valid range of the BET equation is quite simple, as the equation can be written in the linear form:

\[ \frac{P}{V_{ads}(P_0 -P)} = \frac{1}{V_m c}+\frac{c-1}{V_m c}\frac{P}{P_0} \]

Thus, plotting \( \frac{P}{V_{ads}(P_0 -P)} \) as a function of \( \frac{P}{P_0} \) will give a linear plot with a slope of \( \frac{c-1}{V_m c} \) and intercept \( \frac{1}{V_m c} \) in the region in which the approximations for BET theory are valid. From these, \( V_m \) and \( c \) can be determined. An example of the BET equation fitted to the same experimental data as in Figure 1 is given in Figure 2 below:

Figure 2. The BET equation fitted to the experimental data shown in Figure 1. The equation is valid up to a relative pressure of approximately 0.4, but starts deviating from the experimental data at higher relative pressures. (Figure: Joakim Kattelus, data provided by the Catalysis group at Aalto University.)

From the figure, it can be seen that the BET equation works well at relatively low partial pressures, but deviates at higher pressures. The reason for the deviation is that the assumptions of BET theory no longer hold as partial pressure and thus the amount adsorbed becomes larger. Fortunately, the fact that the BET equation works well in the lower pressure region allows for effective and repeatable estimation of the surface area of the solid^[1]. It still is important to keep in mind that the surface area given by applying BET theory is only an estimate, for this reason this value is often reported as \( S_{BET} \) rather than \( S. \)

The parameter \( c \) in the BET equation is related to the enthalpy of adsorption and given by^[1]:

\[ c = e^{(\Delta H_1 - \Delta H_L)/RT} \]

Thus, fitting the BET equation also gives information on the difference between the adsorption enhalpies of the first and subsequent layers. Still, the big advantage of the BET isotherm is that these enthalpies do not have to be known to model the adsorption and determine the surface area^[1], since this allows applying the equation to a wide variety of solids.

Limitations of the model and discussion

The BET isotherm tends to overestimate the adsorption at high pressures and underestimate it at low pressures^[3]. One reason the BET isotherm may fail at low pressures or low surface areas is related to the accuracy of the measured pressure or volume change of the gas. The amount adsorbed is determined by measuring the change in the gas pressure or volume, due to part of the gas remaining adsorbed to the solid^[1]. If there is little adsorption, the measured value is small, and it may be comparable to the error of measurement^[1].  On the other hand, the BET isotherm may also fail at higher gas partial pressures due to several reasons, such as the assumptions made not holding up as the thickness of the adsorbed layer increases^[1]. For instance, \( \Delta H_L \) may cease to be constant. However, somewhat surprisingly a larger error is caused by the pore condensation phenomena mentioned in section 1^[1]. Any highly porous solid with a high surface area is made up of nanometer scale particles, which have empty space (pores) between them^[1]. The pores may be a variety of shapes, but cylindrical pores are often assumed for calculations. Within such a pore, the vapor pressure of the gas decreases in accordance to the Kelvin equation^[1]:

\[ \ln{P_0/P'} = \frac{2\gamma V_m \cos{\theta}}{rRT} \]

where \( P_0 \) is the vapor pressure of the gas, \( P' \) is the vapor pressure of the gas inside the pore, \( \gamma \) is the surface tension of the liquid, \( V \) is its molar volume, \( \cos{\theta} \) is the contact angle between liquid and gas, \( r \) is the radius of a cylindrical pore, \( R \) is the gas constant and \( T \) is the temperature^[1].

Thus, the vapor pressure of the gas is lower inside a pore, leading to the gas condensing inside the pores^[1], which in practice means more gas is retained in the solid than would be by only physisorption. As BET theory does not account for this, this pore condensation leads to large deviations from the linear BET isotherm as the partial pressure of the gas increases. Fitting the BET equation to the data in this region therefore overestimates the surface area.

However, pore condensation is not wholly a detrimental effect, since using the Kelvin equation above allows for the pore size distribution to be estimated^[1]. In essense, a uniform pore size distribution causes a sudden large increase in the retained gas as the value for \( P' \) is reached. Similarly, a less uniform pore size distribution with several different pores leads to a gradual change in the retained amount of gas, which still differs from that predicted by BET theory. This data can thus allow calculating the pore sizes, by assuming a certain shape for the pores and applying the Kelvin equation.

References

1.	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 J. R. Ross, “Chapter 2 - surfaces and adsorption” in Heterogeneous Catalysis, J. R. Ross, Ed., Elsevier, Amsterdam, 2012, pages 7–45, ISBN:978-0-444-53363-0. https://doi.org/10.1016/B978-0-444-53363-0.10002-7.
2.	1 2 3 IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). Online version (2019-) created by S. J. Chalk. ISBN 0-9678550-9-8. https://doi.org/10.1351/goldbook.
3.	1 2 BET isotherm. In Law,  A Dictionary of Chemistry, J., & Rennie, R. (Eds.),  Oxford University Press, 2020, eisbn: 9780191876783, Retrieved 21 Mar. 2021, https://www.oxfordreference.com/view/10.1093/acref/9780198841227.001.0001/acref-9780198841227-e-506.