Isotherm multi-component

The type of treatment described here was originally introduced by Scott and Dullien [4], who confined attention to isothermal isobaric diffusion in binary mixtures. Similar equations were independently published shortly after by Rothfeld [5], and the method was later extended to multi-component mixtures by Silveston [6], Perhaps the most complete exposition is given by Mason and Evans [7],... 

Langmuir-Type Relations For systems composed of solutes that individually follow Langmuir isotherms, the traditional multi-component Langmuir equation, obtained via a kinetic derivation, is... 

The isotherm model of Schirmer et al. (T.) for sorption in molecular sieves is based on statistical thermodynamics in which the configuration integrals describing the sorption behaviour are extracted from the available data. The model does not presuppose any specific kind of sorption mechanism. The multi-component form of this isotherm derived by Loughlin and Roberts (8 ) is also not limited to any particular sorption mechanism,... 

Simulation data on mixture adsorption can be used to screen zeolites as adsorbents, but experimental data are necessary to validate the simulations and to accurately design the separation process. The first step of the process design is to obtain such data. However, the experimental assessment of multi-component adsorption equilibria and kinetics is not straightforward and is highly time-consuming. As a result, some theories have been developed that predict adsorption behaviour for a mixture based on the pure component equilibria [1,3]. The isotherm data have to be correlated before their use in a design model... 

Adsorption isotherm of Langmuir-Freundlich for multi-component is defined by... 

The P-V Diagram for a Multicomponent System. For a relatively volatile multicomponent system, a gasoline for example, an isotherm on the P-V diagram is similar to its counterpart for a binary system (Figure 23). However, it is commonly found that at the dew point the break in the P-V isotherm is not very pronounced in multi-component systems. Consequently, for systems of this type, it may be very difficult to fix the dew point in this manner. This experimental difficulty can be overcome by using a windowed cell and observing the pressure and volume when traces of liquid appear in the system. 

Adsorption isotherms of oxygen, nitrogen, carbon dioxide, and sulfur dioxide on hydrogen-mordenite were measured at several temperatures in the range of O —IOO C. The SO2 and CO2 had considerably greater affinity for the adsorbent than the O2 and Ng. Using the pure-component data, multi-component isotherms were predicted and compared with experimental results. The more strongly adsorbed species completely overwhelm the lesser adsorbed components (e.g., SO2 vs. N2). Wherever 2 species of approximately equal affinity are adsorbed (e.g., CO2 + SOg), the temperature sensitivity of the individual components influences the extent of the competition. 

Quantum chemistry approaches to zeolites are complemented by an active research community that uses classical force-field methods to study molecular adsorption and diffusion in zeolites and similar materials. This topic was comprehensively reviewed by Keil, Krishna, and Coppens in 2000.262 For more recent examples of activity in this area, see References 263-270. Examples of impressive agreement between adsorption isotherms and molecular dilfusivities predicted with calculations of this type and experimental data are available.271,272 There appear to be many future opportunities for linking the detailed understanding of multi-component adsorption and diffusion that is now emerging from this area with detailed quantum chemistry approaches to reactivity at active sites inside zeolites. 

VI. Validation of the accuracy of the perturbation peak method for determination of multi-component adsorption isotherm parameters in LC. 

The Langmuir isotherm model can be extended to multi-component systems [109], When several components are simultaneously present in a solution, the amount of each component adsorbed at equilibrium is smaller than if that component were alone [13] because the different components compete to be adsorbed on the stationary phase. The adsorption isotherm for the / th component in a multicomponent system is written ... 

The FACP and ECP methods cannot be used to determine adsorption isotherm parameters from multicomponent mixtures. By contrast FA can be used to determine multi-component adsorption data but it is a complex and time-consuming process [124, 125],... 

Single-component isotherm parameters cannot always predict elution profiles with satisfied accuracy [122, 123], Therefore, to be able to predict accurate overloaded multi-component elution profiles where competition occurs competitive adsorption isotherm parameters are often necessary. Measurement of isotherms from a mixture is also often necessary because the pure enantiomers are not always accessible in large quantities. However, there exist only a small number of reports on the determination of multi-component adsorption isotherm parameters. FA can be used to determine binary isotherm data but it is time-consuming. The PP method is an alternative method to determine isotherm parameters from binary mixtures. It has been reported that the PP method works well up to weakly non-linear conditions [118, 119],... 

If the PP method is used in a multi-component case it should also be noted that the determined isotherm parameters could not be assigned to specific components without additional information, e.g., by comparing computer simulations with an experimental chromatogram where the peaks can be identified. 

The purpose of this paper was twofold (i) to investigate how to visualize all four perturbation peaks on a quaternary concentration plateau and to (ii) validate the accuracy of the PP method for determination of isotherm parameters directly from quaternary mixtures of l/l/l/l compositions. For this purpose the technique developed and validated for the binary case in papers IV and V (the Lindholm-technique ) was extended to the multi-component case. The approach is to inject the same excess as deficiency for every second solute. Thus, in a quaternary system the excess of components 1 and 3 is the same as the deficiency of components 2 and 4, or the converse. The concentrations of the components in the sample can be calculated according to Csample i = Chateau i - (-1 ) n, for some number n chosen so that the injected concentrations always are equal to or greater than zero. This technique made all perturbation peaks clearly detectable, although the area were not the same for all of them, see Figure 22. 

The accuracy of the adsorption isotherm parameters were validated in two step (1) the experimental perturbation retention data were compared with the calculated retention data using the adsorption isotherm parameters, and (2) by comparing simulated multi-component elution profiles with experimental ones. Figure 23 (a)-(d) shows the experimental (dashed lines) elution... 

In contrast to the well-developed thermodynamic methods for determining gas/ liquid equilibriums the theoretical determination of adsorption isotherms is not yet feasible. Only approaches to determining multi-component isotherms from experimentally determined single-component isotherms are known. Such approaches are explained in more detail in Section 2.5.2.3. Careful experimental determination of the adsorption isotherm is therefore absolutely necessary. The different approaches for isotherm determination are discussed in Chapter 6.5.7. 

Isotherm models have to be divided into single-component and multi-component models. Because most of multi-component models are directly derived from singlecomponent models the latter will be presented first. 

If mixtures of solutes are injected into a chromatographic system, not only interferences between the amount of each component and the adsorbent but also between the molecules of different solutes occur. The resulting displacement effects cannot be appropriately described with independent single-component isotherms. Therefore, an extension of single-component isotherms that also takes into account the interference is necessary. For the modified Langmuir isotherm the extension term is shown in Eq. 2.43, representing the multi-component Langmuir isotherm. 

The coupled isotherm equation takes into account the displacement of one component by the other with the term (b, c,j = i). Equation 2.43 is the general form of the multi-component Langmuir isotherm and is also called asymmetric as for each component i specific parameters bt j have to be determined. With symmetric coefficients only a set of bj parameters is taken into account and Eq. 2.43 reduces to Eq. 2.44,... 

Equations 2.45 and 2.46 show that simple multi-component Langmuir isotherms obviously can not explain the decrease in selectivity observed under increased loading factors. The extended form of the modified Langmuir isotherm, however, can represent these phenomena. The modified multi-component Langmuir isotherm is shown in Eq. 2.47 (Charton and Nicoud, 1995). 

The Bi-Langmuir isotherm can be extended in the same way to give the multi-component Bi-Langmuir isotherm. (Eq. 2.48) (Guiochon, 1994). 

Modified multi-component Langmuir and multi-component Bi-Langmuir isotherms offer a maximum flexibility for adjustment to measured data if all coefficients are chosen individually. But in the same way as for multi-component Langmuir isotherms (Eq. 2.43) it is possible to use, for Eqs. 2.47 and 2.48, constant Langmuir terms (by = by, bjy = b]i , b2y = b2u) as well as constant adjustment terms (Xj = X) or equal saturation capacities (qsaUii = [Pg.37]

As shown in Fig. 2.17, the point wise determination of the multi-component isotherm at given mixture compositions is performed iteratively to find the hypothetical concentrations (cfs) of all components that fulfill all necessary equations. Afterwards, the respective hypothetical loadings (efts) are calculated with the singlecomponent isotherm model. The real loadings qtot, q2 and q2 are then calculated with Eqs. 2.50 and 2.51. 

For practical applications the resulting points of the multi-component isotherm may be fitted to a suitable model equation. 

Point wise calculation of multi-component isotherm qA(cA,CB), qB(cA,cB) for the given mixture composition (cA,cB) by means of a parameter estimation problem (iterative procedure)... 

Fitting data to multi-component isotherm model... 

Fig. 2.17 Procedure to determine the multi-component isotherms from pure component data with the IAS theory for binary mixtures.

Fig. 2.19 Asymmetric multi-component Langmuir isotherm for different mass ratios (Hi =...

Figure 2.19a-c presents the multi-component isotherms of a two-component mixture for different concentrations (Eq. 2.43). In Fig. 2.19a the mass ratio of the first to the second eluting component is 1 9. In this case the isotherm for component B changes drastically while the equilibrium for component A is nearly unchanged. This relation changes for different mass ratios of 1 1 (Fig. 2.19b) and 9 1 (Fig. 2.19c). 

The terms qS3i,i represent the maximum loadings for each component. Interaction between the different components is considered by the sum of all Ncom P components. Equation 6.33 is the non-equilibrium form of the multi-component Langmuir isotherm (Eq. 2.44). 

The ideal model should be applied to get information about the thermodynamic behavior of a chromatographic column. Through work by Lapidus and Amundson (1952) and van Deemter et al. (1956) in the case of linear isotherms and by Glueckauf (1947, 1949) for nonlinear isotherms, considerable progress was made in understanding the influences of the isotherm shape on the elution profile. This work was later expanded to a comprehensive theory due to improved mathematics. Major contributions come from the application of nonlinear wave theory and the method of characteristics by Helfferich et al. (1970, 1996) and Rhee et al. (1970, 1986, 1989), who made analytical solutions available for Eqs. 6.41 and 6.42 for multi-component Langmuir isotherms. 

For multi-component mixtures and isotherms depending on more than one concentration, the partial differentials are replaced by total differentials. For two components they are ... 

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