Isotherm multi-component Langmuir

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... 

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). 

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]

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

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. 

The corresponding equilibrium data for pure components and different mixtures are represented in Fig. 6.21 by filled rhombuses and triangles. Also shown are the results of different isotherm equations (solid and dashed lines) that have been fitted to the experimental data. In this case the modified multi-component Langmuir isotherm (Eq. 2.47)... 

Figure 6.29 shows a comparison between two different isotherm models (Fig. 6.21) used for validation agreement between theory and experiment is good using the modified multi-component Langmuir model (Fig. 6.21), while the symmetrical Langmuir isotherm leads to profiles shifted to earlier retention times. The structural... 

Fig. 6.30 Comparison of the simulated profiles for the modified multi-component Langmuir isotherm and the IAS equation (Fig. 6.26) (Cfeej = 4.4l1 V= lOmlmin-1, Vinj = 120ml, Vc = 54ml for additional data see Appendix B.l).

Determination of a Multi-component Langmuir Isotherm by Frontal Analysis... 

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

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

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.

Extension of the Series Solution of the LeVan-Vermeulen Isotherm to multi-component systems Frey and Rodrigues [56] have extended the binary LeVan-Vermeulen isotherm to the case of multicomponent systems. They showed that, if we assume that the adsorption isotherm of each single component follows Langmuir isotherm behavior, the multicomponent isotherm is given by [56]... 

Antia and Horvith [61] attempted to use these equations to obtain multi-component isotherms in cases in which each individual solute follows single-component Langmuir isotherm behavior, hence... 

Isotherms other than Langmuir s are not readily generalized for multi-component adsorption and are therefore rarely used in practice in kinetics of heterogeneous systems. An exception is the use of the Temkin isotherm in ammonia synthesis [31]. 

The adsorption window is demarcated by the adsorption equilibrium within the boundaries of the imposed pressure cycle and is determined by the multi-component adsorption isotherm and the pressure range. The Langmuir type isotherm is assumed. 

Fundamentals of sorption and sorption kinetics by zeohtes are described and analyzed in the first Chapter which was written by D. M. Ruthven. It includes the treatment of the sorption equilibrium in microporous sohds as described by basic laws as well as the discussion of appropriate models such as the Ideal Langmuir Model for mono- and multi-component systems, the Dual-Site Langmuir Model, the Unilan and Toth Model, and the Simphfied Statistical Model. Similarly, the Gibbs Adsorption Isotherm, the Dubinin-Polanyi Theory, and the Ideal Adsorbed Solution Theory are discussed. With respect to sorption kinetics, the cases of self-diffusion and transport diffusion are discriminated, their relationship is analyzed and, in this context, the Maxwell-Stefan Model discussed. Finally, basic aspects of measurements of micropore diffusion both under equilibrium and non-equilibrium conditions are elucidated. The important role of micropore diffusion in separation and catalytic processes is illustrated. 

The Langmuir model of adsorption easily can be extended to multi-component adsorption processes [7.2 Chap. 4.6, 7.3 Chap. 5, 7.29, 7.75, 7.76]. The resulting adsorption isotherm is... 

The Equilibrium Theory of chromatography is a very powerful tool to study and understand the dynamics of chromatographic columns for single component, binary and multi-component systems, whose retention behavior is described by any type of isotherm. The mathematical model equations are solved using the method of characteristics, and in the case of the Langmuir isotherm one finds out... 

In section IV the less retained component B has to be adsorbed and carried towards the raffinate port in order to regenerate the liquid phase. For the EMD53986 system with its multi-Langmuir isotherm, the corresponding constraint on the dimensionless flow rate ratio mIV is... 

We start this book with a chapter (Chapter 2) on the fundamentals of pure component equilibria. Results of this chapter are mainly applicable to ideal solids or surfaces, and rarely applied to real solids. Langmuir equation is the most celebrated equation, and therefore is the cornerstone of all theories of adsorption and is dealt with first. To generalise the fundamental theory for ideal solids, the Gibbs approach is introduced, and from which many fundamental isotherm equations, such as Volmer, Fowler-Guggenheim, Hill-de Boer, Jura-Harkins can be derived. A recent equation introduced by Nitta and co-workers is presented to allow for the multi-site adsorption. We finally close this chapter by presenting the vacancy solution theory of Danner and co-workers. The results of Chapter 2 are used as a basis for the... 

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