Adsorption theoretical treatment

The following several sections deal with various theories or models for adsorption. It turns out that not only is the adsorption isotherm the most convenient form in which to obtain and plot experimental data, but it is also the form in which theoretical treatments are most easily developed. One of the first demands of a theory for adsorption then, is that it give an experimentally correct adsorption isotherm. Later, it is shown that this test is insufficient and that a more sensitive test of the various models requires a consideration of how the energy and entropy of adsorption vary with the amount adsorbed. Nowadays, a further expectation is that the model not violate the molecular picture revealed by surface diffraction, microscopy, and spectroscopy data, see Chapter VIII and Section XVIII-2 Steele [8] discusses this picture with particular reference to physical adsorption. 

The quantitative solution of the problem, i.e. simultaneous determination of both the sequence of surface chemical steps and the ratios of the rate constants of adsorption-desorption processes to the rate constants of surface reactions from experimental kinetic data, is extraordinarily difficult. The attempt made by Smith and Prater 82) in a study of cyclohexane-cyclohexene-benzene interconversion, using elegant mathematic procedures based on the previous theoretical treatment 28), has met with only partial success. Nevertheless, their work is an example of how a sophisticated approach to the quantitative solution of a coupled heterogeneous catalytic system should be employed if the system is studied as a whole. 

Interestingly, the efficiency for producing hot electrons scales with the reaction exoergicity.64 A recent theoretical treatment of such devices has shown that the efficiency with which H-atom adsorption on Cu(lll)... 

Virtually all theoretical treatments of adsorption phenomena are based on or can be readily related to the analysis developed by Langmuir (5,6). The Langmuir isotherm corresponds to a highly idealized type 6f adsorption and the analysis is predicated on the following key assumptions. 

Many individuals have developed more elegant theoretical treatments of adsorption processes since Brunauer, Emmett, and Teller published their classic paper. Nonetheless, the BET and Langmuir isotherms are the most significant ones for chemical engineering applications. 

Hesselink attempted to calculate theoretical adsorption isotherms for flexible polyelectrolyte chains using one train and one tail conformation (1) and loop-train conformation (2) as functions of the surface charge, polyion charge density, ionic strength, as well as molecular weight. His theoretical treatment led to extensive conclusions, which can be compared with the relevant experimental data. 

Theoretical treatment of such structures is favored by the fact that the azimuthal component of the angular dependence of the adsorption potential is determined by the arrangement symmetry for the substrate atoms closest to the adsorbate, i.e., by the substrate crystal lattice. As a result, an isolated molecule can have several... 

A theoretical treatment of the effect caused by the competition between the sine-like angular-dependent component of the adsorption potential and dipole lateral interaction demonstrated that the values 6 are the same in the ground state and at the phase transition temperature.81 Study of the structure and dynamics for the CO monolayer adsorbed on the NaCl(lOO) surface using the molecular dynamics method has also led to the inference that angles 0j are practically equalized in a wide temperature range.82 That is why the following consideration of orientational structures and excitations in a system of adsorbed molecules will imply, for the sake of simplicity, the constant value of the inclination angle ty =0(see Fig. 2.14) which is due to the adsorption potential u pj,q>j). 

The transport of electro active species from the bulk of the solution to the electrode may be governed not only by diffusion but also by adsorption of the species on the electrode surface. When both the mechanisms are operative, the overall electrochemical process may give considerably complicated results. The theoretical treatment is complex and of limited interest to inorganic chemists, therefore, a qualitative approach will be adopted to identify the presence of adsorption phenomena. 

The extension of the ideas presented in Sections 5.8 and 5.10 to the theoretical treatment of isotope separation by gas chromatography is straightforward. The isotope effects observed in chromatography are governed by the isotopic ratio of Henry s Law constants (for gas-liquid separations), or adsorption constants (for... 

The description of bonding at transition metal surfaces presented here has been based on a combination of detailed experiments and quantitative theoretical treatments. Adsorption of simple molecules on transition metal surfaces has been extremely well characterized experimentally both in terms of geometrical structure, vibrational properties, electronic structure, kinetics, and thermo-chemistry [1-3]. The wealth of high-quality experimental data forms a unique basis for the testing of theoretical methods, and it has become clear that density functional theory calculations, using a semi-local description of exchange and correlation effects, can provide a semi-quantitative description of surface adsorption phenomena [4-6]. Given that the DFT calculations describe reality semi-quantitatively, we can use them as a basis for the analysis of catalytic processes at surfaces. 

Hesselink23) attempted to calculate adsorption isotherms for flexible polyelectrolytes. He assumed that, when adsorbed on a surface, a flexible polyelectrolyte takes a conformation consisting of one train and one tail. The theoretical treatment of Hoeve et al.4I) (cf. B.3.1) for non-ionic polymers was extended by taking into account the change in electrical free energy that occurs when the polyelectrolyte is brought from the solution onto the interface. The partition function Q for a system of N polyelectrolytes each consisting of n units, in which Na polyions are adsorbed on the surface of area S and Nf(Nf = N - N ) polyions remain in the bulk solution of volume V, is then represented by... 

Some interesting conclusions can be drawn from this theoretical treatment of the adsorption of polyelectrolyte. The calculated isotherms are of the high-affinity type as is... 

The theoretical treatment of adsorption from solution, however, is,... 

Interaction of the noble gas atom with condensed matter is considerably more complicated and is usually approximate simply by summing or integrating potentials pairwise. Such treatments are necessarily crude nevertheless, they allow an appraisal of the general features of an interaction and often provide realistic numerical values as well. Young and Crowell (1962), for example, review theoretical treatments of noble gas adsorption along these lines predicted potentials for adsorption on various forms of carbon, to consider one example, range from a few hundred calories per mole for He to a few kilocalories per mole for Xe, in reasonable agreement with observed heats of adsorption. 

The Henry constant J Cis a function of T but not P. (In some theoretical treatments, the Henry constant is the ratio of fugacity to quantity adsorbed, i.e., the inverse of the sense used here.) It is generally expected that adsorption will be governed by Henry s law at sufficiently low pressures. It is possible to construct theoretical models for adsorption in which an isotherm does not reduce to Henry s law, Equation (2.3), even in the limit P —> 0, but it is not clear that such situations obtain in practice and doubtful that they are important in noble gas geochemistry. 

There is a wealth of both theoretical treatment and empirical data for the phenomenon of solution, including empirical data for noble gas solution. As with adsorption, however, data for solution in geochemically important materials are sparse. A prominent exception, the only one, is water, extensive data for which are presented in Chapter 4. Most of the available data for other materials of principal geochemical interest are summarized in Table 2.3. 

Polymer molecules are often employed to stabilize colloids [1]. In most theoretical treatments of the effect of polymer adsorption [2-5], only the steric force is taken into account, and the steric force and the traditional double-layer force for particles devoid of hairs are assumed to be additive. The steric force is a short-range interaction which acts only when the chains on the surfaces of the two particles interpenetrate [6-8]. However, in addition to this short-range interaction, a hairy surface can also generate another effect, because it can change the dielectric constant in the vicinity of the surface. 

Variation of heats of adsor-ption on different major crystal faces of copper. The effect of the crystal face of the adsorbent on physical adsorption has been treated theoretically by Barrer (153) for covalent surfaces and by Orr (151) and by Lenel (154) for dielectric surfaces. The vertical interactions between a nonpolar molecule and a polycrystalline metal surface have been independently treated by Lennard-Jones (155), by Bardeen (156), and by Margenau and Pollard (157). Considering the theoretical limitations involved in the last treatment, the observed agreement between the values calculated theoretically and the experimental values is acceptable. At present no explicit theoretical treatment of the physical adsorption of a nonpolar gas molecule on a single crystal metal surface in terms of the crystal parameter and geometry of the latter is available. 

If a sound theoretical adsorption wave expression could be applied to this problem, it would not only greatly reduce the number of experiments necessary to define completely the geometry of the bed, but could also be useful in the elucidation of the mechanism of adsorption of various gases on different types of adsorbents, from which information could be derived for their improvement or to indicate when the adsorbent has attained its maximum efficiency. Theoretical treatment of the problem has been made by various investigators (73,74,75). However, these workers did not have sufficient experimental data to support their views. The problem of adsorption by charcoal was treated by Wicke (76) and a number of useful differential equations derived. However, a real... 

Preliminary to a more detailed theoretical treatment of the problem, it may be of interest to express the above observations from another point of view. One characteristic of the surface behavior of a solid is the adsorption isotherm. If it were possible within a few minutes to obtain a sufficient number of points to construct an isotherm, it would be most interesting to follow the changes in the adsorption isotherm of the diamond surface as it decays to its normal surface behavior. Figure 7 indicates that the decay is almost independent of the presence of argon. One possible procedure is to introduce different amounts of argon to the evacuated sample and by making a cross plot of the amounts adsorbed at constant time of cooling construct the desired isotherms. 

An alternative approach is by the application of an approximate theory. At present, the most useful theoretical treatment for the estimation of the equilibrium properties is generally considered to be the density functional theory (DFT). This involves the derivation of the density profile, p(r), of the inhomogeneous fluid at a solid surface or within a given set of pores. Once p(r) is known, the adsorption isotherm and other thermodynamic properties, such as the energy of adsorption, can be calculated. The advantage of DFT is its speed and relative ease of calculation, but there is a risk of oversimplification through the introduction of approximate forms of the required functionals (Gubbins, 1997). 

The applicability of the Saam-Cole theory has been tested by Findenegg and his co-workers (1993,1994). Their adsorption measurements of certain organic vapours on carefully selected grades of controlled-pore glass provide semi-quantitative confirmation of the theoretical treatment adopted so far. However, it is evident that some refinement is required in the assessment of 0(rp) for materials of small pore size and that the experimental choice of the mesoporous adsorbent is important To make further progress it will be necessary to study adsorbents having narrow size and shape distributions of easily accessible mesopores. 

Everett and Powl (1976) applied both the 9-3 and the 10-4 expressions in their theoretical treatment of potential energy profiles for the adsorption of small molecules in slit-like and cylindrical micropores. As one would expect, the two corresponding potential energy curves were of a similar shape, but the differences between them became greater as the pore size was reduced. Strictly, the replacement of the summation by integration is dependent on the distance between the molecule and the surface plane, becoming more accurate as the distance is increased (Steele, 1974). 

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