The Potential Theory

Polanyi [12] took a somewhat different approach to multilayer adsorption by assuming that dispersion forces play the determining role in adsorption, resulting in the existence of a potential field in the vicinity of the adsorbent surface. The adsorbed layer has the highest density at the solid surface and its density decreases as the distance from the surface increases. Thus, it is possible to draw equipotential surfaces as shown in Fig. 3.3. The space between each adjacent potential surface represents a defixute adsorption volume which is a function of the potential field. Mathematically, it can be represented as 

One way of plotting the characteristic curve can be obtained by considering the work required to transport molecules from the solid surface to the gas phase. Since at the adsorption equilibrium, the change of free energy of the system must be zero. The removal of gas molecules from the surface must, therefore, be compensated by the compression work on the surface. Thus 

Based on the theory of dispersion interaction, the ratio of the forces of attraction of different molecules is equal to that of polarizability of the molecules of the vapours. This ratio is called an affinity coefficient P and is introduced into the potential function [13]. Furthermore, a parameter k, reflecting the function of the size distribution of volume of the pores, is also included in 

---

Fig. XVII-13. Variation of the density with the adsorbed phase according to the potential theory.

Unfortunately none of the various proposed forms of the potential theory satisfy this criterion Equation XVII-78 clearly does not Eq. XVII-79 would, except that / includes the constant A, which contains the dispersion energy Uo, which, in turn, depends on the nature of the adsorbent. Equation XVII-82 fares no better if, according to its derivation, Uo reflects the surface polarity of the adsorbent (note Eq. VI-40). It would seem that after one or at most two layers of coverage, the adsorbate film is effectively insulated from the adsorbent. 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

Surface Area and Permeability or Porosity. Gas or solute adsorption is typicaUy used to evaluate surface area (74,75), and mercury porosimetry is used, ia coajuactioa with at least oae other particle-size analysis, eg, electron microscopy, to assess permeabUity (76). Experimental techniques and theoretical models have been developed to elucidate the nature and quantity of pores (74,77). These iaclude the kinetic approach to gas adsorptioa of Bmaauer, Emmett, and TeUer (78), known as the BET method and which is based on Langmuir s adsorption model (79), the potential theory of Polanyi (25,80) for gas adsorption, the experimental aspects of solute adsorption (25,81), and the principles of mercury porosimetry, based on the Young-Duprn expression (24,25). 

The primary current distribution is uniform on the hemisphere. Numerical calculations using the potential theory have shown that the current distribution is essentially uniform on the RHSE if the current density is less than 68% of the average limiting current density [47]. 

The potential theory postulates a unique relationship between the adsorption potential ep and the volume of adsorbed phase contained between that equipotential surface and the solid. It is convenient to express the adsorbed volume as the corresponding volume in the gas phase. 

According to the potential theory the volume V, defined by the adsorbent s surface and the equipotential plane , can contain adsorbate in three different conditions depending upon the temperature. Above the critical temperature the adsorbate can not be liquified and the gas in the adsorption volume V simply becomes more dense near the surface. At temperatures near, but less than the critical temperature, the adsorbate is viewed as a liquid near the surface and a vapor of decreasing density away from the surface. Substantially below the critical temperature... 

The potential theory asserts that the adsorption potential, when the adsorbate is in the liquid state, is given by... 

Which theory is suitable for a certain application The adsorption theory of Henry is applicable at low pressure. This, however, is natural since it can be viewed as the first term in a series of the adsorption function. A widely used adsorption isotherm equation is the BET equation. It usually fits experimental results for 0.05 < P/P0 < 0.35. For very small pressures the fit is not perfect due to the heterogeneity. For higher pressures the potential theory is more suitable at least for flat, homogeneous adsorbents. It often applies to P/Po values from 0.1 to 0.8. Practically for P/Po > 0.35 adsorption is often dominated by the porosity of the material. A more detailed description of adsorption is obtained by computer simulations [382],... 

In developing his later views on the social practice of science, Polanyi explicitly drew upon his earlier career experiences in the 1920s and 1930s. Two series of investigations at the boundaries of chemistry and physics were prominent among his examples of scientific practice and the distribution of merit within the scientific community. I turn now to these two cases the potential theory of adsorption on solid surfaces and X-ray studies of the solid state. 

Polanyi, The potential theory of adsorption," 1013, n. 2. Polanyi noted that Arnold T. Eucken had introduced the term Ad-sorptionspotential in 1914, a few months before Polanyi s first paper on the subject. 

Polanyi, The potential theory of adsorption, 1010. The chemist Hermann F. 

A major development in understanding the adsorption of gases and vapors on microporous carbons was provided by the potential theory of adsorption by Polanyi. This theory assumes that, on the adsorbent surface, the gas molecules are compressed by attractive forces acting between the surface and the molecules, and these forces decrease with the increasing distance from the surface. The force of attraction at any given point near the surface is measured by the adsorption potential (A), which can be defined as the work done to transfer a molecule from the gas phase to a given point above the surface. 

Dubinin MM. The potential theory of adsorption of gases and vapors for adsorbents with energetically nonuniform surfaces. Chem. Rev., 1960 60 235-241. 

Polanyi, M., Discussion of the Potential Theory of Adsorption in Adsorption of Gases and Vapors, by Stephen Brunauer, Princeton University Press, Princeton, N. J., 1947. 

Figure 1 displays the experimental characteristic curve obtained and the p values employed to generate it. The existence of very little scatter in the data demonstrates that the isotherms of the various adsorbates were successfully correlated as a single temperature-independent characteristic curve. This fact corroborates the applicability of the Potential theory to the carbon under study. 

This paper aims at presenting the results of the potential theory applied to high pressure and high temperature adsorption data for both sub and supercritical fluids. We used two different procedures for the calculation of the reference characteristic curve The first one is based on the works by Ozawa, Agarwal and Dubinin The second one is based on the works by Dhima and Neimark. The second method leads to satisfactory results as it is possible to obtain a unique characteristic curve but it requires the revision of the classical laws relating the characteristic curve to the structural properties of the adsorbent Using this procedure, it has been possible to point out the influence of the buoyancy effect on the adsorbed phase for the high pressure data and to propose a method to correct it... 

As to the theoretical treatments, methods derived from the potential theory have been used These methods are the easiest way to perform porosity analysis from experimental data related to different adsorbates... 

The potential theory is based on the concept of the characteristic curve [13,14] It is defined as the function of the adsorbed volume V (m kg ) versus the adsorption potential e (J mol ). V and b are calculated with the following equations ... 

The potential theory has been widely used in the last decades Such a success is mainly due to the predictive character of this method Once v f, f, and (3 are known, it is possible to calculate any isotherm if the reference characteristic curve of the adsorbent is known. A lot of research teams have tried to improve the initial theory by proposing different correlations for the determination of v> [15,16,17] The calculation of Ps and thus f, for supercritical fluids is another problem which has been much debated [18,19,20] Besides, it is possible to relate the reference characteristic curve to the porous structure of the adsorbent so that the theoretical treatments are very ofen used as characterization methods The way the mathematical expression of the reference characteristic curve can be related to the micropore size distribution function of the adsorbent has also been widely discussed in the literature [4,5,6] In this paper we consider two different methods for the calculation of v, and f ... 

---