Isotherms lateral interaction

Fig. XVII-4. Langmuir plus lateral interaction isotherms.

It must be remembered that, in general, the constants a and b of the van der Waals equation depend on volume and on temperature. Thus a number of variants are possible, and some of these and the corresponding adsorption isotherms are given in Table XVII-2. All of them lead to rather complex adsorption equations, but the general appearance of the family of isotherms from any one of them is as illustrated in Fig. XVII-11. The dotted line in the figure represents the presumed actual course of that particular isotherm and corresponds to a two-dimensional condensation from gas to liquid. Notice the general similarity to the plots of the Langmuir plus the lateral interaction equation shown in Fig. XVII-4. 

This type of isotherm is more realistic for describing chemisorption at intermediate 0a values but quickly leads to mathematically cumbersome or intractable expressions with many unknown parameters when one considers coadsorption of two gases. One needs to know how -AHa is affected both by 0A and by the coverages of all other adsorbates. Thus for all practical purposes the LHHW kinetics represent even today the only viable approach for formulating mathematically tractable, albeit usually highly inaccurate, rate expressions for catalytic kinetics. In Chapter 6 we will see a new, medium field type, approach which generalizes the LHHW kinetics by accounting also for lateral interactions. 

It should be noted that within the context of the Langmuir isotherm (energetically equivalent adsorption sites, no lateral interactions) Eq. (6.28), which relates two surface properties, i.e. aj and 0j, remains valid even when the surface activity of Sj, aj, is different from the gaseous activity, pJ5 i.e. when Pj(g) Pj(ad). 

Contrary to the last two isotherms, which take into the account interactions between the neighboring molecities ortiy, the Kiselev model assumes the singlecomponent localized adsorption, with the specific lateral interactions among all the adsorbed molecules in the monolayer [4—6]. The equation of the Kiselev isotherm is given below ... 

From the asymmetrical concentration profile with front tailing (see Figure 2.4b), it can correctly be deduced that (1) the adsorbent layer is already overloaded by the analyte (i.e., the analysis is being run in the nonlinear range of the adsorption isotherm) and (2) the lateral interactions (i.e., those of the self-associative type) among the analyte molecules take place. The easiest way to approximate this type of concentration profile is by using the anti-Langmuir isotherm (which has no physicochemical explanation yet models the cases with lateral interactions in a fairly accurate manner). 

Needless to add, lateral interactions among the enantiomeric antipodes can very strongly (and, of course, negatively) affect their preparative thin-layer (and also technological column) separation, carried out in the range of the nonlinear isotherm of adsorption. 

In this chapter, we are going to show that using the one- and the two-component multilayer adsorption isotherm models or the models taking into the account lateral interactions among the molecules in the monolayer (discussed in Section 2.1), the overload peak profiles presented in Section 2.4 can be qualitatively modeled. 

The experiments showing the influence of lateral interaction on coelution of the two species were discussed in Subsection 2.4.2. Figure 2.18 and Figure 2.19 give a comparison of single profiles of acid and ketone or of alcohol and ketone with those attained for the binary mixture. Very similar peak profiles can be obtained upon solving Equation 2.21 separately for the alcohol, acid, and ketone with isotherms (Equation 2.4 and Equation 2.7a), and for the binary mixture with the isotherms (Equation 2.9 and Equation 2.10). 

Temkin s isotherm can describe the effects of surface heterogeneity or of surface modification on adsorption but we should also take into account the lateral interactions between adsorbed molecules. For the adsorption of simple... 

The model based on formal kinetics was used to model the TPD curves of adsorbed CO molecules, based on the model previously reported [4], The desorption is strongly affected by the fast readsorption of CO on unoccupied Cu+ ions, thus, a quasiequilibrium state is a suitable approximation for the description of adsorption. A Langmuir type of adsorption isotherm was assumed for the CO adsorption on the Cu+ sites in zeolite, without considering lateral interactions among adsorbed molecules. 

Data on the adsorption of caprylic acid on a hydrophobic (mercury) surface in terms of a double logarithmic plot of Eq. (4.13) Panel a) compares the experimental values with a theoretical Langmuir isotherm, using the same values for the adsorption constant B for both curves. Panel b) shows that the adsorption process can be described by introducing the parameter a, which accounts for lateral interaction in the adsorption layer. Eq. (4.13) postulates a linear relation between the ordinate [= log [0/ 1 - 0)] - 2a 0 / (In 10)] and the abscissa (log c). If the correct value for a is inserted, a straight line results. For caprylic acid at pH 4, a value of a = 1.5 gives the best fit. 

When the amount of the sample is comparable to the adsorption capacity of the zone of the column the migrating molecules occupy, the analyte molecules compete for adsorption on the surface of the stationary phase. The molecules disturb the adsorption of other molecules, and that phenomenon is normally taken into account by nonlinear adsorption isotherms. The nonlinear adsorption isotherm arises from the fact that the equilibrium concentrations of the solute molecules in the stationary and the mobile phases are not directly proportional. The stationary phase has a finite adsorption capacity lateral interactions may arise between molecules in the adsorbed layer, and those lead to nonlinear isotherms. If we work in the concentration range where the isotherms are nonlinear, we arrive to the field of nonlinear chromatography where thermodynamics controls the peak shapes. The retention time, selectivity, plate number, peak width, and peak shape are no longer constant but depend on the sample size and several other factors. 

The various regions on the isotherm are determined by the lateral interaction between the surfactant molecules within the surface phase. In the dilute, gaseous state, the molecules can be considered to be negligible in size and non-interacting. Under these conditions the isotherms obey an ideal, two-dimensional gas equation of the form nA = kT. As the pressure is increased, a point is reached (at about 8 nm for myris-... 

Instead of changing the temperature it is also possible to determine lateral interactions from isothermal experiments/ or from multi-isotherm experiments in which the temperature is increased in steps. " ... 

Lateral interaction work of water adsoiption, 907 Lateral interactions of ionic adsoiption, 924, 944 Lateral interactions and Frumkin s isotherm, 938 Lattice gas models of adsorption, 965 Lattice spacing, 1276 Laue pattern, 793... 

The Frumkin isotherm is one of the earliest isotherms (1925) that deals with lateral interactions among adsorbed species (Fig 6.98). The isotherm can be written as59... 

In Section 6.8.10 we saw that the Temkin isotherm is based on the Langmuir isotherm. One advantage of the Temkin isotherm is that it considers the heterogeneity of the surface. However, like the Langmuir isotherm, it does not take into account lateral interactions between the adsorbates. 

Finally, the Flory-Huggins isotherm has the advantage of considering the size of the molecules as well as the replacement of adsorbed solvent molecules by the adsorbing molecule. However, its applicability to ionic systems depends on the parameters included in the term p—lateral interactions, surface heterogeneity, etc. 

However, when adsorption of ionic species takes place on solid electrodes, it is difficult to decide what particular characteristic—surface heterogeneity, transfer of charge, lateral interactions, displacement of adsorbed solvent, size of the ions, etc.—is dominant in the process or which one can be neglected. Nonetheless, would it not be possible to include all these effects in a single isotherm It is possible, although not easy. In the following sections we will introduce the development of one isotherm for ionic adsorption where many of these distinctive characteristics of ionic adsorption are considered. 

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. 

What does Eq. (6.246) mean This equation represents the adsorption process of ions on metallic surfaces. It includes several conditions that are characteristic of the adsorption process of ionic species, namely, surface heterogeneity, solvent displacement, charge transfer, lateral interactions, and ion size. However, is this equation capable of describing the adsorption process of ions In other words, what is the success of the isotherm described in Eq. (6.246) Figure 6.104 shows a comparison of data obtained experimentally for the adsorption of two ions—chloride and bisulfate—on polycrystalline platinum, with that obtained applying Eq. (6.246). The plots indicate that the theory is able to reproduce the experimental results quite satisfactorily. The isotherm may be considered a success in the theory of ionic adsorption. 

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