Equilibrium thermodynamic consistency

Vapor-liquid equilibrium data are said to be thermodynamically consistent when they satisfy the Gibbs-Duhem equation. When the data satisfy this equation, it is likely, but by no means guaranteed, that they are correct however, if they do not satisfy this equation, it is certain that they are incorrect. 

The three areas are found by graphical integration. The thermodynamic consistency test consists of comparing the sum of the three areas [left-hand side of Eq. (81)] with the right-hand side of Eq. (81). The three areas depend upon equilibrium data for the composition range x2 = 0 to x2 = x2. However, the right-hand side of Eq. (81) depends only on equilibrium data at the upper limit x2 = x2. The comparison indicated by Eq. (81) should be made for several values of x2 up to and including the critical composition. 

It should be clear that the most likely or physical rate of first entropy production is neither minimal nor maximal these would correspond to values of the heat flux of oc. The conventional first entropy does not provide any variational principle for heat flow, or for nonequilibrium dynamics more generally. This is consistent with the introductory remarks about the second law of equilibrium thermodynamics, Eq. (1), namely, that this law and the first entropy that in invokes are independent of time. In the literature one finds claims for both extreme theorems some claim that the rate of entropy production is... 

If it is assumed that in more concentrated solutions the rate of the forward reaction continues to follow this rate expression, what forms of the reverse rate are thermodynamically consistent in concentrated acid solution Equilibrium is to be established with respect to equation A when written in the N204 form. It may be assumed that the dependence on N02 and N204 concentrations may be lumped together by equation C. 

Note that the parameters in Eq. (110) are interdependent. To facilitate a thermodynamically consistent analysis, a more suitable representation is given in terms of the equilibrium constant... 

The semi-empirical Pitzer equation for modeling equilibrium in aqueous electrolyte systems has been extended in a thermodynamically consistent manner to allow for molecular as well as ionic solutes. Under limiting conditions, the extended model reduces to the well-known Setschenow equation for the salting out effect of molecular solutes. To test the validity of the model, correlations of vapor-liquid equilibrium data were carried out for three systems the hydrochloric acid aqueous solution at 298.15°K and concentrations up to 18 molal the NH3-CO2 aqueous solution studied by van Krevelen, et al. 

The simple formula makes this method very attractive. Although not thermodynamically consistent, this expression (Eq. 21) has been shown to provide a reasonably good empirical correlation of binary equilibrium data for a number of simple gases on molecular sieve adsorbents [34,73 - 75]. However, because of the lack of a proper theoretical foundation this approach should be treated with caution. 

For a given feed (fixed C o, . ) and using conversion of key component as a measure of the composition and extent of reaction, the versus T plot has the general shape shown in Fig. 9.3. This plot can be prepared either from a thermodynamically consistent rate expression for the reaction (the rate must be zero at equilibrium) or by interpolating from a given set of kinetic data in conjunction with thermodynamic information on the equilibrium. Naturally, the reliability of all the calculations and predictions that follow are directly dependent on the accuracy of this chart. Hence, it is imperative to obtain good kinetic data to construct this chart. 

Equilibrium thermodynamics is one of the pillars supporting the safety analyses of radioactive waste repositories. Thermodynamic constants are used for modelling reference porewaters, calculating radionuclide solubility limits, deriving case-specific sorption coefficients, and analysing experimental results. It is essential to use the same data base in all instances of the modelling chain in order to ensure internally consistent results. 

A review is presented of techniques for the correlation and prediction of vapor-liquid equilibrium data in systems consisting of two volatile components and a salt dissolved in the liquid phase, and for the testing of such data for thermodynamic consistency. The complex interactions comprising salt effect in systems which in effect consist of a concentrated electrolyte in a mixed solvent composed of two liquid components, one or both of which may be polar, are discussed. The difficulties inherent in their characterization and quantitative treatment are described. Attempts to correlate, predict, and test data for thermodynamic consistency in such systems are reviewed under the following headings correlation at fixed liquid composition, extension to entire liquid composition range, prediction from pure-component properties, use of correlations based on the Gibbs-Duhem equation, and the recent special binary approach. 

This rate equation must satisfy the boundary conditions imposed by the equilibrium isotherm and it must be thermodynamically consistent so that the mass transfer rate falls to zero at equilibrium. 

The case of binary solid-liquid equilibrium permits one to focus on liquid-phase nonidealities because the activity coefficient of solid component ij, Yjj, equals unity. Aselage et al. (148) investigated the liquid-solution behavior in the well-characterized Ga-Sb and In-Sb systems. The availability of a thermodynamically consistent data base (measurements of liquidus, component activity, and enthalpy of mixing) provided the opportunity to examine a variety of solution models. Little difference was found among seven models in their ability to fit the combined data base, although asymmetric models are expected to perform better in some systems. 

Thus, when values of (A/if - A/if) are plotted as a function of xt, the area between the best smooth curve drawn through the points and the composition axis must be zero within the experimental accuracy. This test concerns the thermodynamic consistency of the data as a whole rather than that of each individual set of experimental values. It also applies strictly to the liquid solution at the arbitrary pressure P0 and only to the two-phase system at equilibrium through the calculation of A fi [ and A /if from the experimental data. 

The most fundamental experimental observation on which equilibrium thermodynamics is based on is the observation that all externally unforced macroscopic systems (with some exceptions, namely glasses, that we shall mention later in Section 4) can be prepared in such a way that their behavior shows some universal features. Subsequent investigation of these features leads then to the formulation of equilibrium thermodynamics. The preparation process consists of letting the macroscopic systems evolve sufficiently long time without external influences. The states reached when the preparation process is completed are called equilibrium states. The approach to equilibrium states is thus a primary experience the behavior at equilibrium states is the secondary experience. An investigation of the secondary experience leads to equilibrium thermodynamics. We may expect that an investigation of the primary experience (i.e., an... 

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. 

Furthermore, the CME framework has been shown to be consistent with the general theory of non-equilibrium thermodynamics [109], and the recently developed... 

THERMODYNAMIC CONSISTENCY OF EXPERIMENTAL VAPOR-LIQUID EQUILIBRIUM DATA 3.8... 

Vapor-liquid equilibrium data for the ethanol/water system (subscripts 1 and 2, respectively) at 70°C (158°F, 343 K) are given in the three left columns of Table 3.1. Check to see if the data are thermodynamically consistent. 

Select the criterion to be used for thermodynamic consistency. Deviations from thermodynamic consistency arise as a result of experimental errors. Impurities in the samples used for vapor-liquid equilibrium measurements are often the source of error. A complete set of vapor-liquid equilibrium data includes temperature T. pressure P. liquid composition x, and vapor composition y,. Usual practice is to convert these data into activity coefficients by the following equation, which is a rearranged form of the equation that rigorously defines K values (i.e., defines the ratio y, /x, under Related Calculations in Example 3.1) ... 

LLE data cannot be checked for thermodynamic consistency as can be done for vapor-liquid equilibria data (Sorensen and Arlt, 1979). In VLE systems, one of the set of equilibrium values (T, P, Xj, yj) is calculated from the other and checked against experimental data. For LLE systems, pressure has a small influence on the other quantities. Therefore, pressure cannot be included in such a consistency test. 

One method of evaluating the overall thermodynamic consistency of binary, isobaric L-V equilibrium data is based on the equation... 

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