Isotherm single solute

A. Seidel-Morgenstem, Experimental determination of single solute and competitive adsorption isotherms. J. Chromatogr.A 1037 (2004) 255-272. 

The relationship between qf and Cf is given by the single solute adsorption isotherm ... 

A number of attempts have been made to modify the IAS model (Eqs. 22-29) to improve its accuracy and reduce computational efforts. Using the IAS model, DiGiano et al. [80] derived a Simplified Competitive Equilibrium Adsorption Model (SCAM). This model, which is based on the Freundlich isotherm, assumes the single-solute isotherms of all the components are equal and it utilizes average isotherm constants when this assumption is not valid. The IAS model equations have been reduced to a single expression ... 

In this section, the basic theory required for the analysis and interpretation of adsorption and ion-exchange kinetics in batch systems is presented. For this analysis, we consider the transient adsorption of a single solute from a dilute solution in a constant volume, well-mixed batch system, or equivalently, adsorption of a pure gas. Moreover, uniform spherical particles and isothermal conditions are assumed. Finally, diffusion coefficients are considered to be constant. Heat transfer has not been taken into account in the following analysis, since adsorption and ion exchange are not chemical reactions and occur principally with little evolution or uptake of heat. Furthermore, in environmental applications,... 

The equilibrium concentrations in the solid and liquid phases at the surface of the adsorbent particle are qsfc end Cs and afc, bk and 8k are isotherm constants for each solute corresponding to the single solute isotherm equation (4). 

The system of equations (1) to (10) provide the basis for predicting multicomponent rate profiles. The input parameters required are the mass transfer and diffusion coefficients for each solute, the single solute isotherm constants, and the mixture equilibria correlation coefficients. Estimation of these equilibrium and rate parameters are discussed in the following sections. 

The equations (20) to (30) provide the basis for predicting the adsorption rate profiles for the binary system. The input parameters required for the model are the single-solute film transfer and surface diffusion coefficients, the single-solute isotherm constants and the mixture equilibria coefficients. The rate parameters were obtained from single solute rate data (20), and the equilibrium parameters were obtained from single and multi-solute equilibrium data. 

Adsorption of albumin, y-globulin, and fibrinogen from single solutions onto several hydrophobic polymers was studied using internal reflection IR spectroscopy. The adsorption isotherms have a Langmuir-type form. The calculated rate and amount of protein adsorbed was dependent on the polymer substrate and the flow rate of the solution. Competitive adsorption experiments were also investigated to determine the specific adsorption of each I-labelled protein from a mixture of proteins. Platelet adhesion to these proteinated surfaces is discussed in relation to a model previously proposed. 

Theory The Freundlich isotherm developed in 1924 by Freundich (16) is widely used to study oil processing adsorption. It is applied as an empirical expression to describe the reversible adsorption of a single solute from aqueous solution at equilibrium at a fixed temperature, and it is expressed as... 

Quinones et al. measured the isotherm data of three aromatic alcohols on Symmetry Ci8 column with methanol/water (50 50, v/v) as the mobile phase [129]. Data were derived from both FA and FACP experiments, at 30°C (see Figure 3.46). There is a very good agreement between the data obtained by the two different methods. The coincidence between FA and FACP demonstrates the excellent ther-mod3mamic consistency of the measured single solute data (in part a consequence of the use of highly efficient columns). 

Myers and Prausnitz [49] developed the ideal adsorbed solution (IAS) model in order to predict thermodjmamically consistent multicomponent isotherms of gas mixtures, using only experimental data acquired for single solute adsorption. The initial equation of the IAS theory for gases is... 

Figure 4.2S Competitive isotherms of phenol and p-cresol on an octadecylsilica. In these two figures, the top isotherm is the single solute isotherm. Isotherms denoted A, B, and C correspond to solutions with 1 3,1 1, and 3 1 mole ratios of phenol and p-cresol, respectively, (a) Competitive isotherms of p-cresol. (b) Competitive isotherms of phenol. Reproduced with permission from J. Jacobson, J. Frenz and Cs. Horvdth, Ind. Eng. Chem. (Res.), 26 (1987) 43 (Fig. 4), 1987 American Chemical Society, (c) Competitive isotherms of N-benzoyl D- (o) and L-alanine v>) in the racemic mixture single component isotherms of N-benzoyl D- (+) and L-alanine (A). Reproduced with permission from S.C. Jacobson, A. Felinger and G. Guiockon, Biotechnol. Progr., 8 (1992) 533 (Fig. 2), 1992 American Chemical Society.

A common feature of solubility data for single solutes dissolved in a supercritical fluid is the existence of what has been termed crossover pressure (Chimowitz and Pennisi 1986). This is a pressure around which isotherms at various near-critical temperatures tend to converge. Figure 3 shows this behavior for phenanthrene solubility in carbon dioxide. Below the crossover pressure, an isobaric increase in temperature causes a solubility decrease, so that the solute is retrograde. 

We are concerned in this book primarily with a description of the motion of fluids under the action of some applied force and with convective heat transfer in moving fluids that are not isothermal. We also consider a few analogous mass transfer problems involving the convective transport of a single solute in a solvent. 

On continuing to alter the temperature in the same direction as before, the relative shifting of the solubility curves becomes more marked, as shown in Fig. 124. At the temperature of this isothermal, the solution saturated for the double salt now lies in a region of distinct unsaturation with respect to the single salts and the double salt can now exist as solid phase in contact with solu- tions containing both relatively more of A (curve ED), and relatively more of B (curve DF), than is contained in the double salt itself. 

The determination of adsorption capacities in multicomponent solutions requires an appropriate description of the single solute isotherm. Freundlich s model may be selected, with the division of isotherms into linear concentration ranges introduced 1 Tien [3],... 

The abilily of activated carbon to remove a broad spectrum of orj nic compounds from wastewater is well documented. The Freundlich single-solute isotherm constants, as tabulated in [32], are reproduced in Table 2. These data illustrate clearly the wide range of organic compounds of different structures, sizes, functionality, etc. that can be adsorbed by an activated carbon. It can also be observed that these compounds exhibit different adsorption characteristics. Indeed, the adsorption behavior is affected by various factors related to molecular structure [33], like the adsorbate solubility, the kind of substituent groups of aromatic compounds, the size of tire molecule, its polarity and its hydrophobicity. 

Figure 3.9 Sorption isotherms of parathion (squares) and lindane (circles) as single and binary solutes from water on Woodburn soil fom = 0.019) at 20°C. Open symbols are for single solute data and solid symbols for binary-solute data. [Reproduced from C. T. Chiou, T. D. Shoup, and P. E. Porter, Mechanistic roles of soil humus and minerals in the sorption of nonionic compounds from aqueous and organic solutions , Org. Geochem., 8, 9, Copyright 1985, with permission from Elsevier.]...

Bliimel, C., Hugo, P., and Seidel-Morgenstern, A. (1999) Quantification of single solute and competitive adsorption isotherms using a closed-loop perturbation method. /. Chromatogr. A, 865, 51-71. 

Zhang, W, Shan, Y, and Seidel-Morgenstem, A. (2006) Breakthrough curves and elution profiles of single solutes in case of adsorption isotherms with two inflection points. J. Chromatogr. A, 1107, 216-225. 

The equilibrium sorption isotherms are one of the most important data to understand the mechanism of the sorption. They describe the ratio between the quantity of sorbate retained by the sorbent and that remaining in the solution at the constant temperature at equilibrium and are important from both theoretical and practical points of view. The parameters obtained from the isotherm models provide important information not only about the sorption mechanisms but also about the surface properties and affinities of the sorbent. The best known adsorption models in the linearized form for the single-solute systems are ... 

Marczewski, A. W., and M. Jaroniec. 1983. A new isotherm equation for single-solute adsorption from dilute solutions on energetically heterogeneous solids. Monatshefte fiir Chemie-Chemical Monthly 114, no. 6-7 711-715. doi 10.1007/BF01134184. 

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