Adsorption on a Uniform Surface

The concept of a uniform surface is widely used in theoretical work on adsorption. A closer examination of this concept, and of the unexpected conclusions to which it may lead upon suitable selection of the various possibilities, may therefore be of some interest. 

Let us start with an analogy. An ideal crystal, in which all the atoms are exactly located at the nodes of a geometrically perfect space lattice, can be conceived only on classical grounds and at absolute zero. However, it is impossible to accept this somewhat naive concept because of the uncertainty principle and thermal agitation at T 0°K. This does not, however, mean that the idea of crystallinity loses all definiteness or that, for instance, a crystal can melt in a continuous process, as Frenkel [1] seems to suggest. 

Nevertheless, the definition of crystallinity must be somewhat modified. Landau [2] introduces the function p(x, y. z) as the probability of finding a particle (atom, electron, ion) in a particular position. In a crystal this function is periodic at any distance in a coordinate system x, y, z referred to some fixed group of particles in any disorderly state (gas, liquid), it becomes constant at sufficiently great distances. 

Analogously, one can imagine a uniform surface as a collection of identical potential holes only at 0°K. As a result of thermal agitation at T 0°, sectors with heats of adsorption differing from the mean will be continuously formed and destroyed on a surface which was originally ideally uniform. A uniform surface may be defined as one on which p(E) is the same for all surface elements, where p E) dE is the probability that the heat of adsorption on a given element lies between E and E + dE. 

In this way the uniformity of the surface, although violated instantaneously, is retained on the average, provided the time element is large compared with the time of relaxation of the thermal motion on the surface. We should immediately note that this time of relaxation is by no means necessarily of the order of 10-13 sec (the period of atomic oscillations in the lattice) since not only is simple displacement of atoms about their equilibrium positions possible, but also much more complicated and slower processes—for example, exchange of foreign dissolved atoms between the surface and the bulk of the crystal. From our point of view such a surface can be called uniform if each atom of the surface has the same probability of being replaced 

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Both extreme models of surface heterogeneity presented above can be readily used in computer simulation studies. Application of the patch wise model is amazingly simple, if one recalls that adsorption on each patch occurs independently of adsorption on any other patch and that boundary effects are neglected in this model. For simplicity let us assume here the so-called two-dimensional model of adsorption, which is based on the assumption that the adsorbed layer forms an individual thermodynamic phase, being in thermal equilibrium with the bulk uniform gas. In such a case, adsorption on a uniform surface (a single patch) can be represented as... 

Thus q0 is the heat of adsorption on a uniform surface with adsorption coefficient a0. According to (125), nonuniformity exponent may be regarded as the ratio to RT of the slope of the straight line in the plot of q as function of 8. For strongly nonuniform surfaces/is much greater than 1. 

Assuming that the second process is rapid, we obtain the following standard picture of adsorption on a uniform surface the equilibrium concentration q, which depends on the pressure of the gas, is determined by the Langmuir isotherm. The only difference from the standard picture is that the statistical sum for all states of the adsorbed molecule in a potential hole must be replaced by a combination of two statistical sums for all states of the adsorbed molecule and for all possible states of the surface element. This, of course, has no effect on the form of the Langmuir equation. Under very simple assumptions the kinetics of establishment of equilibrium will also not differ from those on a uniform surface. Thus, the initial velocity is proportional to the pressure and approaches equilibrium exponentially. 

An unusual dependence for adsorption on a uniform surface arises when it is assumed that the rate of change of the surface is considerably slower than the rate of adsorption (see figure). If adsorption and desorption occur rapidly, the state of the surface remains practically unaltered and we then get an adsorption isotherm corresponding to a non-uniform surface with a distribution p(E, 0) of the heat of adsorption (curve 1) [3]. However, when the time interval is considerable, slow adsorption accompanies changes in the properties of the surface, and the amount of gas adsorbed approaches that given by the Langmuir isotherm (curve 2, point B), which describes a state of complete equilibrium (see above). 

In the case of adsorption on a uniform surface, Henry s law is to be expected at low surface coverage [11, 22]. Unless the operational temperature is relatively high, the Henry s law region is normally restricted to a very small part of the isotherm (e.g., below p/p° l X 10 ) and the deviations may be in either direction. From a fundamental standpoint, the virial analysis of low-coverage adsorption data is important since it is analogous to the treatment of imperfect gases and nonideal solutions [22]. 

There are B equivalent sites for localized adsorption in the first layer, interactions between molecules in the first layer being neglected. This is just Fowler s model for Langmuir adsorption on a uniform surface without interactions. 

For physical adsorption there is no change in molecular state on adsorption (i.e., no association or dissociation). It follows that for adsorption on a uniform surface at sufficiently low concentrations such that all molecules are isolated from their nearest neighbors, the equilibrium relationship between fluid phase and adsorbed phase concentrations will be linear. This linear relationship is commonly referred to as Henry s law by analogy with the limiting behavior of solutions of gases in liquids and the constant of proportionality, which is simply the adsorption equilibrium constant and is referred to as the Henry constant. The Henry constant may be expressed in terms of either pressure or concentration ... 

The majority of physisorption isotherms (Fig. 1.14 Type I-VI) and hysteresis loops (Fig. 1.14 H1-H4) are classified by lUPAC [21]. Reversible Type 1 isotherms are given by microporous (see below) solids having relatively small external surface areas (e.g. activated carbon or zeolites). The sharp and steep initial rise is associated with capillary condensation in micropores which follow a different mechanism compared with mesopores. Reversible Type II isotherms are typical for non-porous or macroporous (see below) materials and represent unrestricted monolayer-multilayer adsorption. Point B indicates the stage at which multilayer adsorption starts and lies at the beginning of the almost linear middle section. Reversible Type III isotherms are not very common. They have an indistinct point B, since the adsorbent-adsorbate interactions are weak. An example for such a system is nitrogen on polyethylene. Type IV isotherms are very common and show characteristic hysteresis loops which arise from different adsorption and desorption mechanisms in mesopores (see below). Type V and Type VI isotherms are uncommon, and their interpretation is difficult. A Type VI isotherm can arise with stepwise multilayer adsorption on a uniform nonporous surface. 

Equations for desorption rate, r, could be obtained by calculations similar to those used for adsorption rate, r+, but there is a simpler method based on the following considerations that are analogous to those developed in Section VIII. On a uniform surface at gas pressure P and fugacity of the adsorbed layer p, adsorption rate is... 

We mentioned above the general agreement on the linearity of the initial sector of the adsorption isotherm on a uniform surface. The isotherms derived by Chakravarti and Dhar, g = Cp1 /( + p1/ ), and by Zeise, q = [ap/(p + )]1/2, are interesting in that, together with the nonlinearity of the isotherm at the beginning (q p1/" g y/p), they also reflect the tendency of adsorption towards saturation for p — oo. 

In conclusion, I would like to emphasize that the main object of this part of the paper has not been a quantitative theory of adsorption on non-uniform surfaces, but a struggle for the correct qualitative interpretation of a specific set of experimental data. 

The Type VI isotherm, in which the sharpness of the steps depends on the system and the temperature, represents stepwise multilayer adsorption on a uniform nonporous surface. The step height now represents the monolayer capacity for each adsorbed layer and, in the simplest case, remains nearly constant for two or three adsorbed layers. Amongst the best examples of Type VI isotherms are those obtained with argon or krypton on graphitised carbon blacks at liquid nitrogen temperature. 

The further development of the theory of nonuniform surfaces in the U.S.S.R. was helped by the mathematical methods of Zel dovich and Roginskil (200,201,331). A. V. Frost analyzed some work on the subject (mostly Russian) in a recent review (10) and concluded that an equation derived by him on the assumption that the reactants are adsorbed on a uniform surface and that no significant interactions take place between the adsorbed molecules, satisfactorily described many reactions on non-uniform surfaces including cracking of individual hydrocarbons and petroleum fractions, hydrogen disproportionation, and dehydration of alcohols. From the experimental results it was concluded that the catalytic centers on the surface were not identical with the adsorption centers. The catalysts used consisted of different samples of silica-alumina and pure alumina. 

Various attempts have been made to modify the Langmuir model. One of the best known is that of Fowler and Guggenheim (1939), which allowed for adsorbate-adsorbate interactions in a localized monolayer on a uniform surface. However, on an empirical basis the Fowler-Guggenheim equation turns out to be no more successful than the original Langmuir isotherm. The highly complex problem of localized adsorption on heterogeneous surfaces has been discussed by Rudzinski and Everett (1992). 

It is clear that on a uniform surface one would expect stepwise condensation. That this is not the case leads one to discard the assumption of uniformity, and assume that there is a distribution function of adsorption energies over the surface. If the surface is characterized by a distribution function. Vae, and the sites of equal energy are collected in patches, each patch will fill at a pressure p/po = erAE/RT. The total coverage will be... 

The equation describing the kinetics of adsorption on a nonuniform surface with a linear increase of activation energy of adsorption with coverage was first deduced by Brunauer et al. (82). The surface is divided into a series of equal areas ds each area constitutes a uniform element of surface having an activation energy for adsorption given by = Eq (x s, where s is the reference number of the uniform patch and a is a constant. The rate of adsorption is then given by... 

For induced heterogeneity on a uniform surface, the increase of activation energy is solely a function of the coverage q, and the adsorbate is always distributed uniformly over a constant adsorption potential surface, although this potential changes due to induced effects of coverage on the heat of adsorption. The rate equation, therefore, has the form... 

Although the obedience of the rate of adsorption to the Elovich equation is maintained, both for mobile and immobile layers, and for any functional relationship between — AH and E, when the heterogeneity is induced on a uniform surface, this is not so for the rate of desorption. 

One can describe adsorption versus time by various means. In the case of a first-order adsorption, which would be expected to occur on a uniform surface, the equation can be derived (Appendix B) from the Langmuir equation (43) and put in the form where... 

Very often the fall in heat of adsorption is more nearly linear than logarithmic, and it is this type of behavior that led to the derivation of the Temkin adsorption isotherm. The isotherm is, in fact, derived from the Langmuir adsorption isotherm by inserting the condition that the heat of adsorption decreases linearly with surface coverage. Such an effect can arise from repulsive forces on a uniform surface or from surface heterogeneity of the surface. 

Adsorption takes place on a uniform surface and the energies of adsorption of all molecules in the first layer are identical... 

As with adsorption, the rate constant and Edes can vary with coverage on a uniform surface, and the desorption rate can then be expressed as ... 

The differential activation energies Eg of the adsorption process for methanol, ethylene glycol, and formic acid on platinum are satisfied by the equation for the kinetics of activated adsorption on a uniformly inhomogeneous surface ... 

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