Adsorbents pure component isotherm equations

Equation (1) is the central equation of LAST, specifying the equality of chemical potential in the bulk gas and the adsorbed phase (which is assumed to be ideal in the sense of Raoult s law). Equation (2) calculates the spreading pressure from the pure-component isotherm. The total amount adsorbed and the selectivity are given by equations (3) and (4), respectively. 

The selectivity of 2 ( 2,1) at these conditions is given by Eq.(3). The quantity ni P) in the above equation is the pure component amount adsorbed for gas 1 at total column pressure P. Experimental measurements are required for 1 (obtained from the infinite dilution system) and data for pure component isotherm (obtained independently using a volumetric technique) to calculate selectivity (LHS of Eq.3). A similar equation can be written for the infinite dilution of gas 1. 

The approach of IAS of Myers and Prausnitz presented in Sections 5.3 and 5.4 is widely used to calculate the multicomponent adsorption isotherm for systems not deviated too far from ideality. For binary systems, the treatment of LeVan and Vermeulen presented below provides a useful solution for the adsorbed phase compositions when the pure component isotherms follow either Langmuir equation or Freundlich equation. These expressions are in the form of series, which converges rapidly. These arise as a result of the analytical expression of the spreading pressure in terms of the gaseous partial pressures and the application of the Gibbs isotherm equation. 

Consider a binary adsorbed mixture for which each pure component obeys the Langmuir equation, Eq. (16-13). Let n = 4 mol/kg, nl =. 3 mol/kg, Kipi = K2P2 = 1. Use the ideal adsorbed-solution theory to determine ni and n. Substituting the pure component Langmuir isotherm... 

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 pure component adsorption equilibrium of ethane and propane are measured on Norit AC at three temperatures (30, 60 and 90 °C). All experimental data of two species at three temperatures are employed simultaneously to fit the isotherm equation to extract the isothermal parameters. Since an extended Langmuir equation is used to describe the local multicomponent isotherm, the maximum adsorbed capacity is forced to be the same for ethane and propane in order to satisfy the thermodynamic consistency. The saturation capacity was assumed to be temperature dependent while the other parameters, bo and u], are temperature independent but species dependent. The derived isotherm parameters for ethane and propane are tabulated in Table 1. The experimental data (symbols) and the model fittings (solid lines)... 

The DA isotherm contains parameters related separately to the solid and the adsorbates. The advantage of using the potential theory for predicting gaseous mixture adsorption equilibria is that the pure component characteristic curves are generally independent of temperature. Thus having obtained isotherm information of a pure component at one temperature, the mixture equilibria at other temperatures and pressures can be predicted. The DA equation and its... 

Kapoor et al. [79] proposed a heterogeneous extended Langmuir (HEL) model for the description of multicomponent equilibria on heterogeneous adsorbents. With the integral equation approach of Eq. (16), the general isotherm for a pure component system can be simplified as... 

These equations form a system of partial differential equations of the second order. Examples of two complete systems are given in Table 2.1 (a binary mixture and a pure mobile phase or a mobile phase containing only weakly adsorbed additives, a two-component system) and Table 2.2 (a binary mixture and a binary mobile phase with a strongly adsorbed additive, a three-component system). For the sake of simplicity, the equilibrium-dispersive model (see Section 2.2.2) has been used in both cases. The problem of the choice of the isotherm model will be discussed in the next two chapters. 

The Langmuir isotherm has been readily extended to Hquid-soHd equilibria, first on an empirical basis, then on a more fundamental one. This problem is discussed in the next section (Section 3.2.1.1). The Freimdlich isotherm (Section 3.2.2.4), first used for gas-solid isotherms, has also been extended to liquid-solid equilibria. These isotherms have permitted a correct description of experimental results in a variety of experimental studies involving dilute solutions of a strongly adsorbed component in a pure solvent. The pressure is replaced by the concentration in the equation of the isotherm. As expected from the derivation already discussed, the Langmuir isotherm appears to accoxmt fairly well for adsorption data acquired at low or moderate concentrations. At high concentrations, on the other hand, the activity coefficients of the species in solution are concentration dependent and systematic deviations from Langmuir adsorption behavior are observed. 

For true adsorption, several equations for adsorption isotherms have been derived, based on various theories. Such equations often are applied to water sorption isotherms of foods as well. However, one cannot speak of adsorption in the case of most foods, as mentioned above, because there is no (or a very limited) phase surface onto which water can adsorb. Moreover, most foods contain numerous components even if phase surfaces were present they must be very inhomogeneous. In the author s opinion, it therefore makes little sense to use such equations. Only for relatively simple and homogeneous systems, like pure starch granules, can some theories be more or less applicable, but not for real foods. Mathematical fitting of experimental data may be useful for practical purposes, and since the equations generally have three or four adjustable parameters, a reasonable fit can often be obtained. But one cannot attribute physical significance to the parameters derived in this way, such as a monolayer water content. ... 

When solid particles are immersed in liquid medium, solid-liquid interfacial interactions will cause the formation of an adsorption layer on their surface. The material content of the adsorption layer is the adsorption capacity of the solid adsorbent, which may be determined in binary hquid mixtures if the so-called adsorption excess isotherm is known. Due to adsorption, the initial composition of the hquid mixture, x , changes to the equilibrium concentration Xj, where n = n +rf2, the mass content of the interfacial phase (e.g., mmol/g). This change, Xj — Xj = Axj, can be determined by simple analytical methods. The relationship between the reduced adsorption excess amount calculated from the change in concentration, = n°(x — xj), and the material content of the interfacial layer is given by the Ostwald-de Izaguirre equation [1-5]. In the case of purely physical adsorption of binary mixtures, the material content of the adsorption layer ( = — n) for component 1 is... 

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