Isotherms, heterogeneous surfaces

Fig. XVII.l. Various isotherms for gas sorption. Fraction of surface covered B as function of sorbate concentration (A), measured in units of iv 2.1. Curve SL for simple Langmuir isotherm homogeneous surface Eq. (XVII.3.3). Curve CL for complex Langmuir isotherm heterogeneous surface Eq. (XVII.3.5) (Xm = 1 Xo = O.IXm).

A somewhat subtle point of difficulty is the following. Adsorption isotherms are quite often entirely reversible in that adsorption and desorption curves are identical. On the other hand, the solid will not generally be an equilibrium crystal and, in fact, will often have quite a heterogeneous surface. The quantities ys and ysv are therefore not very well defined as separate quantities. It seems preferable to regard t, which is well defined in the case of reversible adsorption, as simply the change in interfacial free energy and to leave its further identification to treatments accepted as modelistic. 

Most calculations of f(Q) for a heterogeneous surface, using an adsorption isotherm assume a patchwise distribution of sites. Explain for what kind of local isotherm functions,/((2,P, T) this assumption is not necessary, and for which it is necessary. Give examples. 

Discuss physical situations in which it might be possible to observe a vertical step in the adsorption isotherm of a gas on a heterogeneous surface. 

There are several reasons for deviations from the LHHW kinetics Surface heterogeneity, surface reconstruction, adsorbate island formation and, most important, lateral coadsorbate interactions.18,19 All these factors lead to significant deviations from the fundamental assumption of the Langmuir isotherm, i.e. constancy of AHa (and AHB) with varying coverage. 

A typical adsorption process in electrocatalysis is chemisorption, characteristic primarily for solid metal electrodes. The chemisorbed substance is often chemically modified during the adsorption process. Then either the substance itself or some fragment of it is bonded chemically to the electrode. As electrodes mostly have physically heterogeneous surfaces (see Sections 4.3.3 and 5.5.5), the Temkin adsorption isotherm (Eq. 4.3.46) is suitable for characterizing the adsorption. 

Nevertheless, surfactant sorption isotherms on natural surfaces (sediments and biota) are generally non-linear, even at very low concentrations. Their behaviour may be explained by a Freundlich isotherm, which is adequate for anionic [3,8,14,20,30], cationic [7] and non-ionic surfactants [2,4,15,17] sorbed onto solids with heterogeneous surfaces. Recently, the virial-electrostatic isotherm has been proposed to explain anionic surfactant sorption this is of special interest since it can be interpreted on a mechanistic basis [20]. The virial equation is similar to a linear isotherm with an exponential factor, i.e. with a correction for the deviation caused by the heterogeneity of the surface or the energy of sorption. 

Adsorption Isotherm Measurements and Site-Selective Thermodynamics. For heterogeneous surfaces like CSPs, the adsorption isotherms are usually composite isotherms and often a Bi-Langmuir model (Equation 1.15) describes reasonably well the adsorption behavior [54]. 

Freundlich Adsorbents that follow the Freundlich isotherm equation are assumed to have a heterogeneous surface consisting of sites with different adsorption potentials, and each type of site is assumed to adsorb molecules, as in the Langmuir equation (Perry and Green, 1999) ... 

Temkin (1941) approached the development of an adsorption isotherm by considering a heterogeneous surface (Section 6.8.3) where no molecular interactions exist. He divided the surface into different patches, and since there are no interactions between molecules, in each patch the Langmuir isotherm can be applied [see Eq. (6.196)] (Fig. 6.98). Thus, for the ith patch,... 

This partial isotherm was constructed by considering a heterogeneous surface built by patches of different energy Up and the equation is then applicable only in the/th patch. This partial isotherm can be rearranged by grouping on the left the terms that are related to the heterogeneity of the surface, i.e., 0y and Up... 

This is the equation, the isotherm, we were seeking. It is a generalized isotherm for the adsorption of ionic species on a heterogeneous surface. It considers the adsorption reaction as a substitution process, with the possibility of transfer of charge between the ion and the electrode and also lateral interactions among adsorbed species. 

As already discussed, DFT can be used to predict the capillary condensation and capillary evaporation pressures for pores with homogeneous surface and well-defined geometry. To generate model adsorption isotherms for heterogeneous pores, it is convenient to employ hybrid models based on both DFT data for homogeneous pores and experimental data for flat heterogeneous surfaces [6-9]. Such model adsorption isotherms can be used to calculate PSDs in mesopore [6-9] and micropore [9] ranges. This approach is particularly useful for pores of diameter below 2-3 nm (micropores and narrow mesopores), where an assumption about the common t-curve for pores of different sizes is less accurate, which in turn makes the methods based on such an assumption (even properly calibrated ones) less reliable [18],... 

The differences observed in the adsorption isotherm are also qualitatively and quantitatively significant for the entropy. It has been recently shown that the isotherm of adsorption of an ideal adsorbate on a heterogeneous surface can be appreciably improved by taking into account the exact form of S from Eq. (7) instead of the approximate one arising from F-H theory [22], Results for the coverage dependence of the chemical potential (adsorption isotherm) and entropy per site are shown in figs. 1-2 for various fc-mer s sizes and interaction energies [attractive (w<0) as well as repulsive (w>0)]. 

Chapters 6 and 7 dealt with solid state reactions in which the product separates the reactants spatially. For binary (or quasi-binary) systems, reactive growth is the only mode possible for an isothermal heterogeneous solid state reaction if local equilibrium prevails and phase transitions are disregarded. In ternary (and higher) systems, another reactive growth mode can occur. This is the internal reaction mode. The reaction product does not form at the contacting surfaces of the two reactants as discussed in Chapters 6 and 7, but instead forms within the interior of one of the reactants or within a solvent crystal. 

Two-dimensional condensation—on homogeneous surfaces—leads to sudden jumps in the adsorption isotherm. These jumps may already be found at very low pressures of the gas which is in equilibrium with the adsorbed layer (242). Heterogeneous surfaces do not give rise to sudden jumps but to gradual slopes (Sec. V,12). There is sometimes a tendency to consider such jumps as indications of multimoleeular adsorption this is not correct. It is of course true that stepwise adsorption can also occur together with multimoleeular adsorption. (See also Sec. V,12.)... 

This explains the success of the well-known B.E.T. method for this analysis. After the excellent discussion by Hill (244) of the B.E.T. and the Hiittig theories, in which he points out the weaknesses of the first and the fallacy of the latter, and after the analysis by Halsey (245), who indicates when a B.E.T. isotherm of satisfactory character is obtained on a heterogeneous surface, little need be said here. 

For an energetically heterogeneous surface, where centers of different adsorption intensity are scattered, a superposition of the Langmuir isotherm results in... 

Fig. 6. A Frumkin-Fowler s adsorption isotherm for uniform surfaces. B, C Adsorption isotherm for heterogeneous surfaces with condensation [after Ref. 89]...

Fig. 7. Adsorption isotherm of a surfactant with patches and phase transition on heterogeneous surfaces [after Ref. 96]...

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