Adsorption isotherms Thermodynamically consistent

Thermodynamically Consistent Isotherm Models. These models include both the statistical thermodynamic models and the models that can be derived from an assumed equation of state for the adsorbed phase plus the thermodynamics of the adsorbed phase, ie, the Gibbs adsorption isotherm,... 

For multicomponent systems, the expression for y here employed may be shown equivalent to that involved in the cluster diagram technique (6), which is currently being employed in a variety of problems. The present derivation shows that the starting expressions satisfy the thermodynamic consistency relation embodied by the adsorption isotherm. It is, however, important to observe that any direct application of these alternative rigorous approaches, which is of necessity of an approximate nature, leads to some violation of the complete internal equilibrium conditions. Similarly, calculations of surface tension which employ the adsorption equation as a starting point invariably violate mechanical equilibrium in some order of approximation. 

The relationships of Equations 5 and 2 are unquestionably valid for unlimited surface coverage on ideal external open (flat, planar, accessible) surfaces ranging from nil at E to infinity at E=0. All of the inherent assumptions (tabulated above) are equally valid as models for physical adsorption in internal constricted regions. These are classically denoted as ultramicropores ( 2 nm), micropores(<2 nm), mesopores (2<1000nm) and macropores (very large and difficult to define with adsorption isotherm). In these instances there are finite concentration limits corresponding to the volume (space, void) size domain(s). Although caution is needed to deduce models from thermodynamic data, we can expect to observe linear relationships over the respective domains. The results will be consistent with, albeit not absolute proof of the models. 

Because of the importance of these characterization methods, an experimental test of the thermodynamic consistency of the models was felt to be desirable. One such test is to determine how well the models and inversion methods used predict the temperature dependence of the isotherms as represented by the isosteric heat of adsorption. 

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 simplest equilibrium concept is that the extent of adsorption is proportional to the fluid-phase concentration, i.e., Henry s law [a in Table 14.3]. A principle of thermodynamic consistency is that isotherms should reduce to Henry s law at the limit of zero loading. If not, the equation (or data) is thermodynamically inconsistent and therefore fundamentally flawed. Conversely, there is pragmatic appeal to the premise that if an isotherm equation fits, one may use it, even though some scientific principle may be violated. The hazard of that approach arises if the data are flawed, causing an erroneous fit, which could lead to a defective design or mistaken predicted performance. 

Ilic, M., Flockerzi, D., and Seidel-Morgenstern, A. (2010) A thermodynamically consistent explicit competitive adsorption isotherm model based on second-order single component behaviour. /. Chromatogr. A, 1217, 2132-2137. 

We have shown in the last few sections the LAS theory as well as its computation implementation to obtain multicomponent adsorption isotherm. Since this theory is based on solution thermodynamics it can be applied to prove the thermodynamic consistency of the extended Langmuir equation. 

The thermodynamics consistency for binary adsorption equilibria isotherm equations was studied by Myers and Sircar [26]. Franses et al. [27] used similar thermodynamics arguments and proposed the following differential criterion for testing the thermodynamics consistency of adsorption isotherm equations of binary systems ... 

Franses, E.I., Siddiqui, FA., Ahn, D.J., Chang, C.-H., and Wang, N.-H.L., Thermodynamically consistent equilibrium adsorption isotherms for mixtures of different-sized molecules, Langmuir, 11, 3177, 1995. 

Hence, the interdependency of the adsorption isotherms F/X or, for that matter, r Pi) should satisfy Equation 3.89 to be thermodynamically consistent. As the Gibbs... 

An appendix to this paper discusses thermodynamic consistency tests for binary solute adsorption. Provided that the data are within the Henry s law region of the isotherm, it is shown that application of the Gibbs adsorption equation leads, for any closed path, to... 

The left-hand side is obtained by integration of the single solute data up to a concentration c, and the right-hand side from bisolute isotherms at constant (C1 + C2). The first of these criteria, using the data in the form of the empirical equation for bisolute adsorption previously proposed by Fritz and Schliinder," indicated that the data reported in this paper are thermodynamically consistent within the accuracy of experiment. No mention is made of the application of the second criterion. 

Davydov, Kiselev, and Sapozhnikov have presented adsorption isotherms for acetone + water, dioxan+water, and dioxan + acetone mixtures on KSK silica. All the isotherms exhibited azeotropic points at weight fractions of 0.45 acetone, =0.6 dioxan, and =0.9 dioxan, respectively, the first-named component of each pair being preferentially adsorbed up to this point. No check on the thermodynamic consistency of the three sets of results is... 

In Sect. 4 we present several adsorption isotherms which are solutions of the Maxwell relations of the Gibbs fundamental equation of the multicomponent adsorbate [7.15]. These isotherms are thermodynamically consistent generalizations of several of the empirical isotherms presented in Sect. 3 to (energetically) heterogenous sorbent materials with surfaces of fractal dimension. In Sect. 5 some general recommendations for use ofAIs in industrial adsorption processes are given. 

A thermodynamic consistent extension of the BET-isotherm to multicomponent systems (N > 1) and to real gas adsorptives will be presented in Sect. 4 of this Chapter. 

In the extraction systems the surfactant is distributed between the phases. Boguslavsky et al. systematically studied the adsorption of tetra-alkylammonium salts in such systems at the water-nitrobenzene interface [11, 76, 90, 91]. The values of the Volta potential at this interface are consistent with the thermodynamic distribution theory. The adsorption isotherms are formally described by the Frumkin equation with the increasing size of tetra-alkylammonium cation, the repulsion between the adsorbate particles increases. It testifies to the fact that the cations of alkylammonium... 

These considerations demonstrate the benefit of correlating spectroscopic data on the microscopic interactions between polymer and electrolyte molecules with thermodynamic data and models describing the adsorption isotherm to improve the knowledge on systems consisting of a basic polymer as PBI and protic (acidic) electrolytes. 

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