Langmuir adsorption isotherm single component

Here n is the number of components in the system, coefficients a, and b, are the coefficients of the single-component Langmuir adsorption isotherm for component /. The coefficient bt is the ratio of the rate constants of adsorption and desorption, so it is a thermodynamic constant. The ratio ajb, is the column saturation capacity of component / [13],... 

The treatment here is restricted to the Langmuir or constant separation factor isotherm, single-component adsorption, dilute systems, isothermal behavior, and mass-transfer resistances acting alone. References to extensions are given below. Different isotherms have been considered, and the theory is well understood for general isotherms. 

For example, the Langmuir adsorption isotherm specifically describes adsorption of a single gas-phase component on an otherwise bare surface. When more than one species is present or when chemical reactions occur, the functional form of the Langmuir adsorption isotherm is no longer applicable. Thus, although such simple functional expressions are very useful, they are not generally extensible to describe arbitrarily complex surface reaction mechanisms. 

The Langmuir adsorption isotherm describes the equilibrium between a single-component gas A and adsorbed species A(s) at a surface [237]. The expression relates the fraction of the surface 6a covered by adsorbed species as a function of the partial pressure pa exposed to the surface. The usual form of the Langmuir adsorption isotherm is... 

At this point, it is feasible to correlate the liquid-phase adsorption equilibrium single component data, with the help of isotherm equations developed for gas-phase adsorption, since, in principle, it is feasible to extend these isotherms to liquid-phase adsorption by the simple replacement of adsorbate pressure by concentration [92], These equations are the Langmuir, Freundlich, Sips, Toth, and Dubinin-Radushkevich equations [91-93], Nevertheless, the Langmuir and Freudlich equations are the most extensively applied to correlate liquid-phase adsorption data. [2,87],... 

Adsorption of single components in zeolites can often be described by a simple Langmuir isotherm, Eq. (18), or the Langmuir-Freundlich isotherm, Eq. (26) [50-52] ... 

The isotherm that is most easily understood theoretically and widely applicable to experimental data is, known as the Langmuir isotherm. This is also the simplest model of a nonlinear isotherm. It accounts well for the adsorption of single components on homogenous surfaces at low to moderate concentrations, or for the adsorption to isolated type-I or type-II sites [103, 110, 111]. The Langmuir adsorption isotherm equation is written ... 

Figure 12.29 Comparison of theoretical and experimental displacement separations of resorcinol and catechol by phenol. Calculations using the equilibrium-dispersive model, the LeVan- Vermeulen isotherm model, and single-component adsorption data. Experimental results on a 4.6x250 CIS Nucleosil 5 fim column, F = 0.4 carrier, water, Fj, = 0.2 mL/min, T = 20°C 1 1 mixture, = 0.5 mL displacer, 80 g/L phenol in water = 30%, Lf = 16.5%. (a) Calculation with LeVan-Vermeulen isotherm, (b) Calculation with quadratic isotherm, three floating parameters, (c) Calculation with competitive Langmuir isotherm, single-component isotherm parameters, (d) Calculation with Langmuir isotherm, best adjusted parameters. Reproduced with permission from. C. Bellot and J.S. Condoret, J. Chromatogr., 657 (1994) (Figs. 3c, 4c, 6c, 8c) 305.

Binary mixtures of 1,3,5-tri-tert-butyl-benzene (TTB) and 1,3,5-tri-hydroxyben-zene (PFIL) were used as the feed. The separation was carried out on a 0.46 x 25 cm coluiim packed with microcrystaUine cellulose triacetate, with an average particle size of 5 im. Two different volumes (10 and 50 jiL) of mixtures of different compositions (1 1, 1 3, and 3 1) were injected into the column. All the experiments were performed imder isocratic conditions, with pure methanol as the mobile phase, and low concentration samples. Teoh et al. [65] ignored the competitive adsorption between the two feed components on the grounds that their colimm was efficient, the band resolution rather high, and the samples used rather dilute. So, they assumed that each compoimd follows single-component Langmuir adsorption isotherm behavior, with... 

Dividing both sides of Equation 5.134 by the total concentration of surface sites Cm leads to the fractional form of the single component, associative Langmuir adsorption isotherm. 

The adsorption properties of coal are normally represented by the adsorption isotherm which shows the amount of gas adsorbed at a certain temperature and partial pressure. For single component gas, the Langmuir adsorption isotherm (Langmuir, 191S) is usually used for coal and is given by... 

Langmuir isotherm or model Simple mathematical representation of a favorable (type I) isotherm defined by Eq. (2) for a single component and Eq. (4) for a binary mixture. The separation factor for a Langmuir system is independent of concentration. This makes the expression particularly useful for modeling adsorption column dynamics in multicomponent systems. 

The problem of predicting multicomponent adsorption equilibria from single-component isotherm data has attracted considerable attention, and several more sophisticated approaches have been developed, including the ideal adsorbed solution theory and the vacancy solution theory. These theories provide useful quantitative correlations for a number of binary and ternary systems, although available experimental data are somewhat limited. A simpler but purely empirical approach is to use a modified form of isotherm expression based on Langmuir-Freundlich or loading ratio correlation equations ... 

The Langmuir isotherm equation for the correlation of the liquid-phase adsorption equilibrium of a single component, can, in principle, as was previously stated, be extended to liquid-phase adsorption by the simple replacement of adsorbate pressure by concentration [2,87] ... 

As we have seen previously, the separation mechanism in pervaporation is explained by an adsorption-diffusion process. In this way, the selective adsorption of the components in the zeolite will be responsible for the selectivity in the separation. Adsorption is an exothermic nonactivated process. In general, the isotherm of adsorption on zeohtes follows a single site Langmuir-type isotherm [74]. 

Experimental data are often fitted more precisely to the adsorption isotherm when an additional term is introduced that covers the non-specific adsorption of the solute to the adsorbent. This modifies the equation to the single-component, modified Langmuir isotherm (Eq. 2.40) ... 

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.

With such a surface, covered with two different kinds of sites that behave independently (and on each of which the two basic principles of the Langmuir model, local adsorption and lack of adsorbate-adsorbate interactions, apply), the equilibrium isotherm results from the addition of the two independent contributions of the two types of sites [35]. Since in most cases the Langmuir isotherm is appropriate to account for single-component adsorption on a homogeneous surface (in practice if not in principle, see earlier), we have the following isotherm (see example in Figure 3.12.)... 

As a consequence, the competitive Langmuir isotherm model offers no possibility to account for a reversal in the elution order of two components with increasing concentration. On the contrary, experimental results show that such an inversion is possible, and that it is not even unusual when the column saturation capacities of the two single-component isotherms are different. For example, experimental adsorption data and chromatograms of mixtures of tmns- and a s-androsterone show an inversion of the elution order when the sample size increases (see later. Figure 4.8 and the related discussion below) [9-11]. 

This isotherm model has been used successfully to accoimt for the adsorption behavior of numerous compounds, particularly (but not only) pairs of enantiomers on different chiral stationary phases. For example, Zhou et ah [28] foimd that the competitive isotherms of two homologous peptides, kallidin and bradyki-nine are well described by the bi-Langmuir model (see Figure 4.3). However, most examples of applications of the bi-Langmuir isotherm are found with enantiomers. lire N-benzoyl derivatives of several amino acids were separated on bovine serum albumin immobilized on silica [26]. Figure 4.25c compares the competitive isotherms measured by frontal analysis with the racemic (1 1) mixture of N-benzoyl-D and L-alanine, and with the single-component isotherms of these compounds determined by ECP [29]. Charton et al. foimd that the competitive adsorption isotherms of the enantiomers of ketoprofen on cellulose tris-(4-methyl benzoate) are well accounted for by a bi-Langmuir isotherm [30]. Fornstedt et al. obtained the same results for several jS-blockers (amino-alcohols) on immobilized Cel-7A, a protein [31,32]. 

Similar results were obtained with the enantiomers of methyl mandelate separated on 4-methylcellulose tribenzoate immobilized on silica [30]. Figure 4.4a shows the experimental adsorption data for the two pure enantiomers (symbols), the best bi-Langmuir isotherms (solid lines) and the best LeVan-Vermeulen isotherms [33]. The data (symbols) were obtained by ECP. Figures 4.4b-d compare the competitive isotherm data measured with three mixtures of different composition and the isotherms calculated from the single component isotherms (Figure 4.4a) using the competitive bi-Langmuir model (Eq. 4.10). Results obtained... 

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