Chromatography equilibrium model

In many applications, such as chromatography, equilibrium titrations or kinetics, where series of absorption spectra are recorded, the individual rows in Y, C and R correspond to a solution at a particular elution time, added volume or reaction time. Due to the evolutionary character of these experiments, the rows are ordered and this particular property will be exploited by important model-free analysis methods described in Chapter 5, Model-Free Analyses. 

The equilibrium models of nonlinear chromatography assume that there always is an instantaneous equilibrium between the mobile phase and the stationary phase. That model is widely applied for the separation of small molecules, when mass transfer or diffusion in the stagnant pores of the mobile phase does not have a significant impact on the band profile. 

The equilibrium models of chromatography are given by the mass balance equation given in Equation 10.8 and a proper isotherm equation, q = f(C), should be used to relate the mobile phase and stationary phase concentrations. 

Guiochon, G., Golshan-Shirazi, S. Solutions of the equilibrium and semi-equilibrium models of chromatography, J. Chromatogr., 1990a, 506,495. 

Ideal model of chromatography A model of chromatography assuming no axial dispersion and no mass transfer resistance, i.e., that the column efficiency is infinite (fi = 0). This model is accurate for high-efficiency, strongly overloaded columns. It permits an easy study of the influence of the thermodynamics of phase equilibrium (i.e., of the isotherm) on the band profiles and the separation. See Chapters 7 to 9. 

Linear chromatography A model of chromatography assuming that the equilibrium isotherm is linear. This model accounts for the various sources of band broadening in the column, and permits their study independently of the properties of the isotherm. Synonymous with analytical chromatography. 

The most likely causes of non-equilibrium behavior in groundwater (as in chromatography) are illustrated on Figure 1. Two approaches to modeling slow sorption-desorption were presented in fitting breakthrough curves 1) use of an equilibrium model with a dispersion coefficient that includes both... 

Figure 6 Partitioning of LiCl between water and 1-octanol at 25°C, as taken from ref. [2l6]. The lithium distribution ratios Du were determined at 1 1 initial phase ratio by use of ion chromatography (IC), inductively coupled plasma (ICP) atomic emission spectrometry, and Li NMR spectrometry. A correction was made for the slight volume changes due to the mutual solubility of 1-octanol and water. Error bars are indicated only for the ICP data, which were the least precise data obtained by the three techniques. The solid curved line represents the equilibrium model calculated by SXLSQl using the values of log/Cs= = —6.85 and logX, = — 2.74 (Table 12). The dashed curved line is an extrapolation of the model to indicate the approach to the calculated asymptotic value of the distribution ratio at infinite dilution (3.76 X 10...

A three-phase equilibrium model for partitioning of solute, E, between a bulk aqueous phase and dissolved cyclodextrin, and between the bulk aqueous phase and the stationary phase, L3, allows one to derive equations relating capacity factor, k, to the molar concentration of CD. The equation given below is similar to that derived for micellar chromatography which also assumes a three-phase model (12. 131,... 

Cyclohexane Equilibrium model Gas chromatography Multi component systems Universal quasi-chemical model... 

For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

The competitive adsorption isotherms were determined experimentally for the separation of chiral epoxide enantiomers at 25 °C by the adsorption-desorption method [37]. A mass balance allows the knowledge of the concentration of each component retained in the particle, q, in equilibrium with the feed concentration, < In fact includes both the adsorbed phase concentration and the concentration in the fluid inside pores. This overall retained concentration is used to be consistent with the models presented for the SMB simulations based on homogeneous particles. The bed porosity was taken as = 0.4 since the total porosity was measured as Ej = 0.67 and the particle porosity of microcrystalline cellulose triacetate is p = 0.45 [38]. This procedure provides one point of the adsorption isotherm for each component (Cp q. The determination of the complete isotherm will require a set of experiments using different feed concentrations. To support the measured isotherms, a dynamic method of frontal chromatography is implemented based on the analysis of the response curves to a step change in feed concentration (adsorption) followed by the desorption of the column with pure eluent. It is well known that often the selectivity factor decreases with the increase of the concentration of chiral species and therefore the linear -i- Langmuir competitive isotherm was used ... 

To simulate the empirical concentration profiles, an appropriate mass-transfer model has to be used. One of the simplest models is the model based on the equilibrium-dispersive model, frequently used in column chromatography [1]. It can be given by the following equation ... 

Figure 4.17 General phenonenaloglcal retention model for a solute that participates in a secondary chemical equilibrium in liquid chromatography. A - solute, X - equilibrant, AX analyte-equilibrant coeplex, Kjq - secondary chemical equilibrium constant, and and are the primary distribution constants for A and AX, respectively, between the mobile and stationary phases.

Based on high performance liquid chromatography (HPLC) studies regarding the equilibration of isomeric fractions of P-carotene isomers at 45°C, a model consisting of two reversible concurrent isomerization reactions was developed by Pesek et al. 1990. Under dark storage conditions at 45°C, a P-carotene solution reached an equilibrium after 4-6 days yielding approximately 66% aW-trans-, 8% of 9-cis-, and 25% of 13-d.s-P-carotene. The observed rate constant (k) for the formation of the 13-d.v- isomer was faster than that of the 9-d.s-p-carotene isomer, and the back rate constants toward the all -trans- isomer were intrinsically faster as compared to the formation of d,v-isomcrs of P-carotene (Chart 12.1). 

Liquid chromatography (LC) and, in particular, high performance liquid chromatography (HPLC), is at present the most popular and widely used separation procedure based on a quasi-equilibrium -type of molecular distribution between two phases. Officially, LC is defined as a physical method... in which the components to be separated are distributed between two phases, one of which is stationary (stationary phase) while the other (the mobile phase) moves in a definite direction [ 1 ]. In other words, all chromatographic methods have one thing in common and that is the dynamic separation of a substance mixture in a flow system. Since the interphase molecular distribution of the respective substances is the main condition of the separation layer functionality in this method, chromatography can be considered as an excellent model of other methods based on similar distributions and carried out at dynamic conditions. 

The collection of examples is extensive and includes relatively simple data analysis tasks such as polynomial fits they are used to develop the principles of data analysis. Some chemical processes will be discussed extensively they include kinetics, equilibrium investigations and chromatography. Kinetics and equilibrium investigations are often reasonably complex processes, delivering complicated data sets and thus require fairly complex modelling and fitting algorithms. These processes serve as examples for the advanced analysis methods. 

Figure 4.2 Protein transformations in reversed-phase chromatography for a two-state model. The native folded state can exist in either the mobile phase (Fm) or the stationary phase (Fs), as can the unfolded state (Um, Us). The equilibrium constants (k) for interconversions of the four species are indicated. (Reproduced from X.M. Lu, K. Benedek, and B.L. Karger, J. Chromatogr., 359 19 [1986]. With permission from Elsevier Science.)...

E Szoekoe, J Gyimesi, Z Szakacs, M Tarnai. Equilibrium binding model of bile salt—mediated chiral micellar electrokinetic capillary chromatography. Electrophoresis 20 2754-2760, 1999. 

The assumption of linear chromatography fails in most preparative applications. At high concentrations, the molecules of the various components of the feed and the mobile phase compete for the adsorption on an adsorbent surface with finite capacity. The problem of relating the stationary phase concentration of a component to the mobile phase concentration of the entire component in mobile phase is complex. In most cases, however, it suffices to take in consideration only a few other species to calculate the concentration of one of the components in the stationary phase at equilibrium. In order to model nonlinear chromatography, one needs physically realistic model isotherm equations for the adsorption from dilute solutions. 

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