Langmuir isotherm deviations from

The Hybrid Method of Mass Balance (HMMB) This method is a modification of the MMB method in which, instead of measuring a series of values of C which is cumbersome, these concentrations are estimated through Eqs. 4.85 and 4.86 by using the isotherm parameters, , and determined by the MCV method. This hybrid approach employs MCV only to estimate the mezzanine concentrations, so it is more acciurate than MCV when the actual isotherm deviates from the Langmuir competitive isotherm. 

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. 

The deviations from the Szyszkowski-Langmuir adsorption theory have led to the proposal of a munber of models for the equihbrium adsorption of surfactants at the gas-Uquid interface. The aim of this paper is to critically analyze the theories and assess their applicabihty to the adsorption of both ionic and nonionic surfactants at the gas-hquid interface. The thermodynamic approach of Butler [14] and the Lucassen-Reynders dividing surface [15] will be used to describe the adsorption layer state and adsorption isotherm as a function of partial molecular area for adsorbed nonionic surfactants. The traditional approach with the Gibbs dividing surface and Gibbs adsorption isotherm, and the Gouy-Chapman electrical double layer electrostatics will be used to describe the adsorption of ionic surfactants and ionic-nonionic surfactant mixtures. The fimdamental modeling of the adsorption processes and the molecular interactions in the adsorption layers will be developed to predict the parameters of the proposed models and improve the adsorption models for ionic surfactants. Finally, experimental data for surface tension will be used to validate the proposed adsorption models. 

We should not be too surprised that the Langmuir equation often yields only an empirical isotherm. There are several reasons why real systems are likely to deviate from the theoretical model ... 

Other soft metals, such as silver, showed very similar behavior (Figure 5a). However, as the softness decreases, the adsorption isotherms deviate significantly from the ideal Langmuir curves (Figure 5b). 

The diffusion of the electroactive ions is both physical and due to electron transfer reactions.45 The occurrence of either or both mechanisms is a function of the electroactive species present. It has been observed that the detailed electrochemical behaviour of the electroactive species often deviates from the ideal thin film behaviour. For example, for an ideal nemstian reaction under Langmuir isotherm conditions there should be no splitting between the anodic and cathodic peaks in the cyclic voltammogram further, for a one-electron charge at 25 °C the width at half peak height should be 90.6 mV.4 In practice a difference between anodic and cathodic potentials may be finite even at slow scan rates. This arises from kinetic effects of phase formation and of interconversion between different forms of the polymer-confined electroactive molecules with different standard potentials.46... 

The concept of unrelaxed volume in glassy polymers is used to interpret sorption and transport data for pure and mixed penetrants A review of recent sorption and permeation data for mixed penetrants indicates that competition for sorption sites associated with unrelaxed gaps between chain segments is a general feature of gas/glassy polymer systems This observation provides convincing support for the use of the Langmuir isotherm to describe deviations from simple Henry s law sorption behavior. 

In systems that operate mainly by adsorption, it is also common to find isotherms that deviate from linearity. The resulting isotherm, known as a Langmuir isotherm (Figure 3.1 b), arises when the concentration of analyte in the mobile phase exceeds the linear level that is, the stationary phase does not have the capacity to sorb all the analyte that would be... 

Figure 6.28 compares measured and simulated profiles for the batch separation of EMD53986. Very good agreement between theory (solid lines) and experiment (symbols) is achieved using the multi-component modified-Langmuir isotherm (Fig. 6.21). Also shown are the simulation results neglecting component interaction by using only the single-component isotherms (dashed line), which deviate strongly from the observed mixture behavior. Typical for competitive adsorption is the displacement of the weaker retained R-enantiomer and the peak expansion of the stronger adsorbed S-enantiomer.

The Langmuir isotherm has been readily extended to Hquid-soHd equilibria, first on an empirical basis, then on a more fundamental one. This problem is discussed in the next section (Section 3.2.1.1). The Freimdlich isotherm (Section 3.2.2.4), first used for gas-solid isotherms, has also been extended to liquid-solid equilibria. These isotherms have permitted a correct description of experimental results in a variety of experimental studies involving dilute solutions of a strongly adsorbed component in a pure solvent. The pressure is replaced by the concentration in the equation of the isotherm. As expected from the derivation already discussed, the Langmuir isotherm appears to accoxmt fairly well for adsorption data acquired at low or moderate concentrations. At high concentrations, on the other hand, the activity coefficients of the species in solution are concentration dependent and systematic deviations from Langmuir adsorption behavior are observed. 

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. 

This isotherm model was designed by Fowler and Guggenheim [12] to correct for the first-order deviations from the Langmuir isotherm. It assumes ideal adsorption on a set of localized sites on a homogeneous surface, with weak interactions between molecules adsorbed on neighboring sites. It assumes also that the interaction energy between two sorbate molecules is small enough that the random... 

The retention factor imder linear conditions is related to the initial slope of the isotherm k = Fdqf dC). It is easy to check that this relation applies when the isotherm coefficients are calculated. Deviations from this relation are not unusual, and can often be ascribed to the heterogeneity of the adsorbent smface that can be explained, for example, by a small concentration of underivatized sUanol groups on the surface of chemically bonded silica. It may be that an additional Langmuir isotherm term would be necessary to account for the contribution of these groups, term that could become saturated at very low concentrations [172]. 

Figure 3.47 Dependence of the isotherm determined by ECP (or FACP) on the column efficiency. The ECP method is based on the ideal model profile (cf Eq. 7.4). A Langmuir isotherm (solid line, b) is used to calculate the band profiles obtained with columns of different efficiencies ( L/ = 10%). The profiles (a) are used to derive the isotherm following the ECP method. The isotherms differ from the initial Langmuir isotherm. The best fit of the data to a Langmuir model generates significant model errors, with deviations of the order of 1% between the initial and the measured coefficients of the isotherm for N = 5000 plates, larger at lower efficiencies and loading factors.

The FA method gives isotherm data. To be useful in preparative chromatography, these data must be fitted to an isotherm model. There are presently no numerical procedures available to smooth the data from multidimensional plots, similar to the 2-D splines or French curves and obtain purely empirical isotherms. Therefore, the major difficulty is the selection of adequate models. The Langmuir isotherm is too simplistic in most cases, and the LeVan-Vermeulen isotherm is complicated and difficult to use as a fitting fimction. Several methods have been described to extract the "best" set of Langmuir parameters which could accormt for a set of competitive adsorption data [108]. These methods have been compared. The most suitable method seems to depend on the aim of the determination and on the deviation of the system from true Langmuir behavior [108]. 

The method has been used for the determination of competitive isotherms in cases in which the deviation from the Langmuir model is moderate [115]. An HPLC chromatograph configured so as to permit injecting into the column a wide rectangular pulse, e.g., by pumping into it either the pure mobile phase or, for a known time, a solution of the compound of interest, was used to make the measurements described [115]. Excellent agreement was observed with other experimental data and with the experimental band profiles recorded in overloaded elution for binary samples of various compositions [48]. 

As expected, the two methods give basically the same results [16]. Both theories are valid only in the case of competitive Langmuir isotherms. This restriction should be kept in mind, because deviations from Langmuir behavior are the most probable explanation for the not so inconsequential disagreements observed sometimes between theoretical predictions and experimental observations reported in the literature [14,34]. 

At higher values of 0 small deviations from the Langmuir isotherm are corrected in (9.42a) by introduction of an empirical constant k [86b] ... 

An example of where the Scatchard transformation shows significant deviation from a rectangular hyperbola is shown in Figure 4.4. The direct presentation of the data shows little deviation from the saturation binding curve as defined by the Langmuir adsorption isotherm. The data at 10 and 30 nM yield slightly underestimated... 

Consider now adsorbed molecular or ionic species that are, practically speaking, immobilized in the soil. Unless the soil is extremely acid, metals such as Cu, Cr, and Pb fall into this category. Also, certain anions such as phosphate bond so strongly on minerals that they too behave as immobile elements. The property that all of these ions have in common is that their sorption isotherms are not reversible within a time scale relevant to soil processes the adsorption (forward) isotherm is usually approximated closely by a Langmuir function of the strong-affinity type, but the desorptioii (backward) isotherm deviates markedly from the adsorption isotherm. This kind of nonequilibrium behavior, depicted in Figure 9.6, is sometimes referred to as hysteresis. Possible reasons for hysteresis in chemisorption are discussed in Chapter 4. 

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