# BET theory

Editor-In-Chief: C. Michael Gibson, M.S., M.D. 

## Overview

BET theory is a well-known rule for the physical adsorption of gas molecules on a solid surface, that is 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 an article about the BET theory in a journal for the first time; “BET” consists of the first initials of their family names.

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 expressed by (1):

${\frac {1}{v\left[\left({P_{0}}/{P}\right)-1\right]}}={\frac {c-1}{v_{m}c}}\left({\frac {P}{P_{0}}}\right)+{\frac {1}{v_{m}c}}\ \ \ \ \ \ \ (1)$ $P$ 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 $v_{m}$ is the monolayer adsorbed gas quantity. $c$ is the BET constant, which is expressed by (2):

$c=\exp \left({\frac {E_{1}-E_{L}}{RT}}\right)\ \ \ \ \ \ \ (2)$ $E_{1}$ is the heat of adsorption for the first layer, and $E_{L}$ is that for the second and higher layers and is equal to the heat of liquefaction.

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

$v_{m}={\frac {1}{A+I}}\ \ \ \ \ \ \ (3)$ $c=1+{\frac {A}{I}}\ \ \ \ \ \ \ (4)$ The BET method is widely used in surface science for the calculation of surface areas of solids by physical adsorption of gas molecules. A total surface area $S_{total}$ and a specific surface area $S$ are evaluated by the following equations:

$S_{BET,total}={\frac {\left(v_{m}Ns\right)}{V}}\ \ \ \ \ \ \ (5)$ $S_{BET}={\frac {S_{total}}{a}}\ \ \ \ \ \ \ (6)$ $N$ : Avogadro's number, $s$ : adsorption cross section, $V$ : molar volume of adsorbent gas $a$ : molar weight of adsorbed species

## Examples

### 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 vapour 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 $v_{m}$ 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

$v_{m}={\frac {1}{A+I}}=0.0408g/g$ $c=1+{\frac {A}{I}}=73.6$ From this the specific BET surface area $S_{BET}$ can be calculated by use of the above mentioned equation (one water molecule covers $s=0.114nm^{2}$ ). It follows thus $S_{BET}=156m^{2}/g$ which means that hardened cement paste has an inner surface of 156 square meters per g of cement.

### Activated Carbon

For example, activated carbon, which is a strong adsorbate and usually has an adsorption cross section $s$ of 0.16 nm2 for nitrogen adsorption at liquid nitrogen temperature, is revealed from experimental data to have a large surface area around 3000 m² g-1. 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 layer 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. 