Isotherm anti-Langmuir

FIGURE 2.3 The anti-Langmuir isotherm of adsorption and the corresponding nonsymmetrical distribution of the analyte s concentration in the chromatographic band. 

Also, in this case, with the adsorbate concentration low enongh, the anti-Langmuir isotherm transforms into the linear equation and becomes the simplest isotherm of adsorption, as described by Henry s law. 

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). 

The exemplary peak profiles, simulated with use of Equation 2.20 for the linear, Langmuir, and the anti-Langmuir isotherms of adsorption are presented in Figure 2.21. 

The tendency of carboxylic acid analytes to form associative multimers can also be viewed as multilayer adsorption. Analysis of the concentration profiles presented in Fig. 2 reveals that for low concentrations of the analyte, peaks a and b are similar to the band profiles simulated by use of the Langmuir isotherm, whereas peaks c-f resemble profiles obtained by use of the anti-Langmuir isotherm (tailing toward the front of the chromatogram is more pronounced than tailing toward the start of the chromatogram.). 

Figure 14.20 Comparison between an experimental overloaded elution profile (symbols) and the profiles calculated with the anti-Langmuir isotherm and the FOR and the ED models. Butyl-benzene on a monolithic Cig silica column. Sample concentration, 6.0 g/L injection volume,...

Anti-Langmuir isotherm An isotherm which is convex downward or, by extension, an adsorption behavior which is opposite to the one observed with a Langmuir isotherm. With the Langmuir isotherm (see Chapter 3), the solute concentration at equilibrium in the adsorbent, q, increases less rapidly than the solution concentration, C. With an anti-Langmuir isotherm, q increases more rapidly... 

Examples of linear, Langmuir and anti-Langmuir isotherms and their effect on peak shape (no other interactions present). 

There are several isotherm models for which the isotherm shapes and peak prohles are very similar to that for the anti-Langmuir case. One of these models was devised by Fowler and Guggenheim [2], and it assumes ideal adsorption on a set of localized active sites with weak interactions among the molecules adsorbed on the neighboring active sites. It also assumes that the energy of interactions between the two adsorbed molecules is so small that the principle of random distribution of the adsorbed molecules on the adsorbent surface is not significandy affected. For the liquid-solid equilibria, the Fowler-Guggenheim isotherm has been empirically extended, and it is written as ... 

Also, the mititilayer isotherms have the anti-Langmuir shape. The mititilayer isotherm models can easily be derived, assuming an infinitely fast adsorption of the adsorbate on the adsorbent active sites, followed by a subsequent adsorption of the molecules on the first, the second, and consecutive adsorbed layers [7,8]. 

These isotherms are sometimes referred to as anti-Langmuir models because their initial curvature is convex down. The fact that the curvature of these isotherms at the origin and at low concentrations... 

Figure 3.1. A comparison of four types of isotherm (a) linear (b) Langmuir (c) anti-Langmuir (cl) chemisorption.

The opposite type of isotherm (anti-Langmuir) is shown in Figure 3.1c and its resulting peak shape in Figure 3.2c. In this situation, the analyte molecules, which are the first to adsorb, facilitate the sorption of additional molecules. For example, in LC when the planar molecule phenol adsorbs on alumina, the phenyl rings extend out from the surface, as... 

Figure 14 The ion-exchange isotherms for uni-univalent ion-exchange reactions (left) and the In dependence (right) of various types of reaction on X, (a) Langmuir type, (b) selectivity reversal, (c) anti-Langmuir type, (d) incomplete exchange, (e) hysteresis loop. (From M. Abe, in Ion Exchange Processes Advances and plications (A. Dyer, M. J. Hudson, and P. A. Williams, eds.). The Royal Society of Chemistry, Science Park, Cambridge, UK, p. 199, 1993. With permission.)...

Equation 6.49 is strictly valid only for the disperse part of the peak (Chapter 2.2.3). Depending on the shape of the isotherm, this is the rear part ( Langmuir ) or the front part ( anti-Langmuir ) of the peak (Fig 2.6). The sharp fronts ( Langmuir ) or tails ( anti-Langmuir ) of the peaks are called concentration discontinuities or shocks. To describe the movement of these shocks, the differential in Eq. 6.49 has to be replaced by discrete differences A, the secant of the isotherm, which describe the amplitudes of the concentration shocks in the mobile and stationary phases ... 

For anti-Langmuir-type (concave) isotherms Eq. 6.49 without i]n, prevails because in this case the disperse front originates from the front of the injection pulse. 

Other examples are isotherms for the isomers fructose and glucose. Figure 6.22 shows that the resulting isotherms exhibit an upward curvature and the slope of the isotherm is increased in the case of mixtures. This anti-Langmuir behavior is explained from the specific interaction between the hydrated solute molecules and the eluent (water) (Saska et al. 1991 and 1992). These isotherms are expressed by the following empirical correlations ... 

Besides the heterogeneity of the adsorbent surface, the second major reason for the adsorption of a compound to deviate from Langmuir isotherm behavior is that the adsorbed molecules interact. In this category, we find the Fowler isotherm, the anti-Langmuirian isotherm, and several S-shaped isotherm models, including the quadratic isotherm, the extended BET isotherm models, and the Moreau model. 

Anti-Langmuir type isotherms are more common in partition systems where solute-stationary phase interactions are relatively weak compared with solute-solute interactions or where column overload occurs as a result of large sample sizes. In this case, analyte molecules already sorbed to the stationary phase facilitate sorption of additional analyte. Thus, at increasing analyte concentration the distribution constant for the sorption of the analyte by the stationary phase increases due to increased sorption of analyte molecules by those analyte molecules already sorbed by the stationary phase. The resulting peak has a diffuse front and a sharp tail, and is described as a fronting peak. 

Alternatively, there may be cooperative absorption of the analyte so that, as the amount of analyte increases, the concentration in the stationary phase increases more rapidly than would otherwise be predicted. The plot of C ob against Cjiat consequently curves upward in a shape known as an anti-Langmuir sorption isotherm (Figure 2.13). This is less common than saturation of the stationary phase, but as some solutes (e.g., some acids) are absorbed in the stationary phase, they alter its nature so that they have... 

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