Equilibrium isotherms mixtures

The chromatographic resolution of bi-naphthol enantiomers was considered for simulation purposes [18]. The chiral stationary phase is 3,5-dinitrobenzoyl phenyl-glycine bonded to silica gel and a mixture of 72 28 (v/v) heptane/isopropanol was used as eluent. The adsorption equilibrium isotherms, measured at 25 °C, are of bi-Langmuir type and were proposed by the Separex group ... 

The separation of bi-naphthol enantiomers can be performed using a Pirkle-type stationary phase, the 3,5-dinitrobenzoyl phenylglycine covalently bonded to silica gel. Eight columns (105 mm length) were packed with particle diameter of 25 0 fiva. The solvent is a 72 28 (v/v) heptane isopropanol mixture. The feed concentration is 2.9 g for each enantiomer. The adsorption equilibrium isotherms were determined by the Separex group and already reported in Equation (28) [33]. 

Example 14 Calculation of Band Profiles in Displacement Chromatography An equimolar mixture of two components (concentrations cf = cf = 1 arbitrary unit) is separated with a displacer with concentration Ci = 2. The equilibrium isotherm is ... 

The adsorption equilibria of methane, ethane and their mixture into single-walled carbon nanotuhes (SWNTs) were studied by using a Grand Canonical Monte Carlo (GCMC) method. The equilibrium isotherms of methane and ethane and the selectivity from their equimolar mixture were reported. 

Up to now, numerous studies have been conducted on their synthesis [9,10], treatment [5,13] and physical properties [4], However only limited number of studies has been carried out on die adsorption of gas in CNTs, including experimental works [8,11] and molecular simulations [3,7,14-lS]. Adsorption behavior depends strongly on the microporous structure of the particular adsorbent. In this work the effect of pore size on the adsorption behavior is of interest. The adsorption equilibria of methane, ethane and their mixture into SWNTs were studied by using a Grand Canonical Monte Carlo (GCMC) method. We reported equilibrium isotherms of methane and ethane, and the selectivity from their equimolar mixture. 

The exact shape of a breakthrough curve is mainly determined by the functional form of the underlying equilibrium isotherms of the sample components, but secondary factors such as diffusion and mass-transfer kinetics also have influence. The capacity of the column is an important parameter in frontal chromatography, because it determines when the column is saturated with the sample components and, therefore, is no longer able to adsorb more sample. The mixture then flows through the column with its original composition. 

Fang, F., Szleifer, I. Competitive adsorption in model charged protein mixtures Equilibrium isotherms and kinetic behavior. J. Chem. Phys. 2003,119,1053-65. 

We have attempted to present here, in a rather condensed form, a vievc of the present status of the fxmdamentals of preparative and nonlinear chromatography. The fundamental problems and the various models used to model chromatography are discussed first (Chapter 2). As the thermodynamics of phase equilibrium is central to the separation process, whatever model is used, we devote two chapters to the discussion of equilibrium isotherms, for single components (Chapter 3) and mixtures (Chapter 4). A chapter on the problems of dispersion, mass transfer and flow rate in chromatography (Chapter 5) completes the fundamental bases needed for the thorough discussion of preparative chromatography. 

All cases of practical importance in liquid chromatography deal with the separation of multicomponent feed mixtures. As shown in Chapter 2, the combination of the mass balance equations for the components of the feed, their isotherm equations, and a chromatography model that accounts for the kinetics of mass transfer between the two phases of the system permits the calculation of the individual band profiles of these compounds. To address this problem, we need first to understand, measure, and model the equilibrium isotherms of multicomponent mixtures. These equilibria are more complex than single-component ones, due to the competition between the different components for interaction with the stationary phase, a phenomenon that is imderstood but not yet predictable. We observe that the adsorption isotherms of the different compounds that are simultaneously present in a solution are almost always neither linear nor independent. In a finite-concentration solution, the amount of a component adsorbed at equilib-... 

The agreement that was observed between the experimental results and the prediction of a competitive Langmuir model based on the use of single-component Langmuir isotherms in the case of the adsorption of enantiomeric derivatives of amino acids on immobilized serum albumin [26] is imusual. It demonstrates the validity of the competitive Langmuir model based on the use of the parameters of the single-component Langmuir model. However, as explained before, the experimental conditions are exceptionally favorable since the column saturation capacities for the two enantiomers are equal. Nevertheless, Zhou et ah have shown that it is possible, in certain favorable cases, to derive the equilibrium isotherms of the pure enantiomers and to calculate isotherm equilibrium data for any mixture of... 

The hypothesis of linear behavior of the equilibrium isotherm in analytical chromatography has three important consequences. First, the different components contained in a sample of a mixture behave independently of each other. They do not compete for interaction with the stationary phase because the sample size is small and the solutions are dilute. Therefore, the elution profiles and the retention times of the various components of a mixture are independent of the presence of other solutes and of their relative concentrations. Each band profile is the same as if the corresponding solute were alone, pure. As a consequence and in contrast with nonlinear chromatography, there is only one problem to solve in linear chromatography, the determination of the peak profile of a single component. 

The theory of the hodograph transform and the relationship derived between the equations of the two lines given by this transform in the case of a binary mixture and those of the competitive equilibrium isotherms were briefly presented in Section 8.1.2. The theory is easily extended to multicomponent mixtures, although in this case we must represent the hodograph transform in an n-dimensional coordinate system, Ci, C2, , C , or in its planar projections. If the solution presents a constant state (Figure 8.1), it is a simple wave solution, and there is a relationship between the concentrations of the different components in the eluent at the column exit (Figure 8.2). This result is valid for any convex-upward isotherm. In the particular case in which the competitive Langmuir isotherm apphes, these relationships are linear. 

Their analysis is based on the separation of a mixture of two components, A and B, whose single-component adsorption equilibrium isotherms are represented by the relationship = cia(Ca,Cb) and qe = b(C/i,Cb). The mixture is separated with a displacer D, of concentration Co and isotherm qo — qo Co), having an affinity for the sorbent that is greater than that of either component of the feed mixture considered. According to Antia and Horv4th [32], a necessary condition for the establishment of fully resolved bands in this process is expressed... 

Although for the sake of clarity the previous discussion was limited to the case of a binary mixture, these results are easily generalized to the study of an n-component mixture. Because of the coupling between the mobile phase components, the velocity eigenvalues are related to the slopes of the tangents to the n-dimensional isotherm surface, in the n composition path directions. These slopes can be calculated when the isotherm surface is known. Conversely, systematic measurement of the retention times of very small vacancy pulses for various compositions of the mobile phase may permit the determination of competitive equilibrium isotherms, but only if a proper isotherm model is available. Least-squares fitting of the set of slope data to the isotherm equations allows the calculation of the isotherm parameters. If an isotherm model, i.e., a set of competitive isotherm equations, is not available, the experimental data cannot be used to derive an empirical isotherm (see Chapter 4). 

For experimental verification of these models, Foplewska et al. [33] used binary mixtures of methanol-water and acetonitrile-water as the mobile phases and measured the adsorption equilibrium isotherms of cyclopentanone on two similar adsorbents having different degrees of sruface heterogeneity, a Cis non-endcapped and a Cig endcapped silica. Ehie to its structure, cyclopentanone exhibits affinity for adsorption on the bonded alkyl chains and for the polar, im-covered silica sruface of the adsorbent. Overloaded elution bands of cyclopentanone in piue water were recorded (Figrue 15.3) and the isotherms were derived using an inverse method (see Chapter 3). Five independent parameters (the excess coefficients and the eqiulibrirun constants for partition-adsorption and for... 

In Chapter 14, we discussed the case of a single-component band. In practice, there are almost always several components present simultaneously, and they have different mass transfer properties. As seen in Chapter 4, the equilibrium isotherms of the different components of a mixture depend on the concentrations of all the components. Thus, as seen in Chapters 11 to 13, the mass balances of the different components are coupled, which makes more complex the solution of the multicomponent kinetic models. Because of the complexity of these models, approximate analytical solutions can be obtained only under the assumption of constant pattern conditions. In all other cases, only numerical solutions are possible. The problem is further complicated because the diffusion coefficients and the rate constants depend on the concentrations of the corresponding components and of all the other feed components. However, there are still relatively few papers that discuss this second form of coupling between component band profiles in great detail. In most cases, the investigations of mass transfer kinetics and the use of the kinetic models of chromatography in the literature assume that the rate constants and the diffusion coefficients are concentration independent. This seems to be an acceptable first-order approximation in many cases, albeit separation problems in which more sophisticated theoretical approaches are needed begin to appear as the accuracy of measru ments improve and more interest is paid to complex... 

The competitive equilibrium isotherm model best fitting the FA experimental data for the R and S enantiomers of 1-phenyl-l-propanol on cellulose tiibenzoate was the Toth model. This model was used to calculate the elution profiles of samples of mixtures of the two enantiomers [29]. The General Rate model combined with the Generalized Maxwell-Stefan equation (GR-GMS) was used to model and describe surface diffusion (see Chapter 5). The mass transfer kinetics is slow and this model fits the experimental data well over a wide concentration range with one single set of numerical parameters to account for the band profiles in a wide range of concentrations, as shown in Figure 16.24. 

Stefanie et al. [68] studied a closed loop SMB unit in which two solvent mixtures of different compositions are used as the feed solvent and as the desorbent for a binary separation. For such SMB systems, these authors derived the region of separation and showed how the optimum operating conditions can be found, using the equilibrium theory, i.e., neglecting axial dispersion and the mass transfer resistances, and assmning linear equilibrium isotherms. They also assumed in their calculations that the separation performance of the SG-TMB unit is the same as that of the SG-SMB. They used the following relationship to accoimt for the dependence of the affinity of the solutes for the solid phase in the presence of a fluid phase of variable composition i.e., for the variation of the initial slope of the isotherm of the solute or its a parameter with the solvent composition)... 

In the former case [32], the production rate of 99% pme enantiomers from the racemic mixture of R- and S-2-phenylbutyric acid was maximized as a function of the sample size and the mobile phase composition. The calculations were based on the column performance and the equilibrium isotherms of the two components (bi-Langmuir isotherms. Chapter 3). The separation was performed on immobilized bovine serum albumin, a chiral stationary phase, using water-methanol solution as the mobile phase. The retention times decrease with increasing methanol content, but so does the separation factor. For this reason, the optimum retention factor is around 3. Calculated production rates agree well with those measured (Table 18.4). The recovery yield is lower than predicted. 

Figure 7 shows an example of type IV isotherms for adsorption of trace water from toluene and p-xylene mixtures on Alcoa H-152 alumina at 22° C [22], The abcissa of the plot represents relative saturation of water (xi/x ), where x is the molar fraction of water at the solubility limit in the hydrocarbon liquid. Equilibrium isotherm models analogous to those used for pure water vapour adsorption can be derived for describing trace water adsorption from liquid mixtures [20-22],... 

This chapter considers the vapor-liquid equilibrium of mixtures, conditions for bubble and dew points of gaseous mixtures, isothermal equilibrium flash calculations, the design of distillation towers with valve trays, packed tower design. Smoker s equation for estimating the number of plates in a binary mixture, and finally, the computation of multi-component recovery and minimum trays in distillation columns. 

FIGURE 9.4 Pressure versus liquid mole fraction for isotherms of methane with m-cresol. Symbols for data from Simnick et al. (From J. J. Simnick, H. M. Sebastian, H. M. Lin, and K. C. Chao, 1979a, Gas-Liquid Equilibrium in Mixtures of Methane+m-Xylene, and Methane-Meta-Cresol, Fluid Phase Equilibria, 3, 145, With permission from Elsevier.) (-) Regression... 

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