Equation pure component isotherm

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. 

Binary Isotherm. The method to calculate the binary sorption isotherm from the pure component isotherms is outlined below. Equation (1) can be rearranged to obtain... 

The pure component isotherms were fitted to either the Henry s Law equation or to... 

Note that the total pressure is used as the upper limit of the integral. The RHS of the above equation can be evaluated because all variables (y, P, and the single component isotherm equations) are known. If the pure component isotherm can be approximated by a Langmuir equation, then the initial estimate of the spreading pressure can be taken as ... 

Consider a system containing N components and the pure component isotherm of each component can be described by the following Langmuir equation... 

The above conclusion for the Langmuir equation does not readily apply to other isotherms. For example, if the pure component isotherm is described by the Sips equation... 

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. 

If the pure component isotherm takes the form of Freundlich equation ... 

The extension of the potential theory was studied by Bering et al (1963), Doong and Yang (1988) and Mehta and Dannes (1985) to multicomponent systems. We shall present below a brief account of a potential theory put forward by Doong and Yang (1988). The approach is simple in concept, and it results in analytical solution for the multicomponent adsorption isotherm. The basic assumption of their model is that there is no lateral interaction between molecules of different types and pure component isotherm data are described by the DA equation. With this assumption, the parameters of the DA equation (Wq, Eq, n) of each species are not affected by the presence of the other species, but the volume available for each species is reduced. This means that the volume available for the species i is ... 

Thus, for a system consisting of N components wih specified partial pressure in the gas phase, there are 2N + 1 independent variables to be solved x(/), P and 7t. Equations (l)-(4) provide the 2A + 1 necessary equations. Since the equations are nonlinear and the integrals in Eq. (2) cannot be solved analytically for most of the pure component isotherm equations, the 2N + 1 equations have to be solved numerically. 

To avoid the numerical integration procedure and iterative scheme in the lAST, O Brien and Myers [17] introduced an algorithm that allows for fast computation of multicomponent adsorption equilibria. The pure component isotherm equation was chosen as... 

Since the I AST has a rigorous thermodynamics basis, the pure component isotherms used are also expected to meet the thermodynamics requirement. Myers and coworkers [21,22] pointed out that the thermodynamic consistency for the single-component isotherm equations means that they should not have a singular value and be exaet at the origin, i.e.. 

The application of the DR equation in lAST was studied by Richter et al. [15] and was used recently by Lavanchy et al. [62] to predict the binary adsorption equilibria of chlorobenzene-carbon tetrachloride vapor mixtures on activated carbon. This is simply a formal application of the lAST theory with the DR equation being used to describe the pure component isotherm. However, due to some special properties of the DR equation, this combination generates some unique features and deserves some elaboration here. 

The HEL model is a noniterative model and has a reasonable prediction capability. With the pure component isotherm parameters derived from the LUD equation, the model can be employed to predict the adsorption equilibria of gas mixtures at any other temperature. Since the LUD equation is flexible in correlating the pure component data, the HEL model prediction for a gas mixture is generally good. Kapoor et al. [79] demonstrated in their study that the HEL model gave satisfactory results for a number of systems, which were comparable in many cases to the results from the lAST or even HIAST model. 

Integration of the pure-component isotherm according to the Gibbs equation [Eq. (3.50)] yields, for each component, the relationship between spreading pressure and equilibrium pressure ... 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

Equations of state are also used for pure components. Given such an equation written in terms of the two-dimensional spreading pressure 7C, the corresponding isotherm is easily determined, as described later for mixtures [see Eq. (16-42)]. The two-dimensional equivalent of an ideal gas is an ideal surface gas, which is described by... 

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... 

Pure component experimental data for sorption of methane and krypton on 5A zeolite at 238, 255. and 271K, and in the pressure range of 0 to 97.36 kPa were also obtained during this work (shown in Figures 3 and U). Further sorption data for methane on 5A zeolite (10, 13, 1 0, and for krypton on 5A zeolite (10. 15) are also plotted for other temperatures, all of which appear to be consistent. These experimental data were used to derive the energy and entropy parameters in equation U for the isotherm model of Schirmer et al. by a minimization of a sum of squares optimization procedure. 

The IAST model of layers and Prausnitz (3) was also used to calculate the mixture equilibria for this system using the theoretical isotherm of Schirmer et al. (equation U) as the pure component model. The theoretical results for the two models were identical to three significant figures. This may indicate that the two models are equivalent, although this has not been proved. 

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 ... 

Scamehorn et. al. ( ) also developed a reduced adsorption equation to describe the adsorption of mixtures of anionic surfactants, which are members of homologous series. The equations were semi-empirical and were based on ideal solution theory and the theory of corresponding states. To apply these equations, a critical concentration for each pure component in the mixture is chosen, so that when the equilibrium concentrations of the pure component adsorption isotherms are divided by their critical concentrations, the adsorption isotherms would coincide. The advantage of... 

The adsorption of binary mixtures of anionic surfactants in the bilayer region has also been modeled by using just the pure component adsorption isotherms and ideal solution theory to describe the formation of mixed admicelles (3 ). Positive deviation from ideality in the mixed admicelle phase was reported, and the non-ideality was attributed to the planar shape of the admicelle. However, a computational error was made in comparison of the ideal solution theory equations to the experimental data, even though the theoretical equations presented were correct. Thus, the positive deviation from ideal mixed admicelle formation was in error. 

The method of predicting the mixture adsorption isotherms is to first select the feed mole fractions of interest and to pick an adsorption level within Region II. The pure component standard states are determined from the total equilibrium concentration that occurs at that set level of adsorption for the pure surfactant component adsorption isotherms. The total equilibrium mixture concentration corresponding to the selected adsorption level is then calculated from Equation 8. This procedure is repeated at different levels of adsorption until enough points are collected to completely descibe the mixture adsorption isotherm curve. 

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