Gibbs-Duhem consistency

I well remember my shock when I recognized that COSMO-RS was not thermodynamically consistent, just after having quit my safe position at Bayer AG and had started my own company. But within an hour I had located the origin of the inconsistency in the solvent-size correction, and within a day I found an expression for the size correction that ensures Gibbs-Duhem consistency, while leaving the limits of pure solvents unchanged compared with our original COSMO-RS expression ... 

The thermodynamic consistency of this expression follows directly from the fact that the chemical potential corrections are now calculated as composition derivatives of a thermodynamic potential. Since this expression left the limits of pure solvents unchanged, and since almost only these limits are of importance for our COSMO-RS parameterization data set, we could use the existing COSMO-RS parameterization in combination with the new Gibbs-Duhem-consistent solvent size correction. 

Mathematical Consistency. Consistency requirements based on the property of exact differentials can be apphed to smooth and extrapolate experimental data (2,3). An example is the use of the Gibbs-Duhem coexistence equation to estimate vapor mole fractions from total pressure versus Hquid mole fraction data for a binary mixture. 

The well-known Gibbs-Duhem equation (2,3,18) is a special mathematical redundance test which is expressed in terms of the chemical potential (3,18). The general Duhem test procedure can be appHed to any set of partial molar quantities. It is also possible to perform an overall consistency test over a composition range with the integrated form of the Duhem equation (2). 

Because experimental measurements are subject to systematic error, sets of values of In y and In yg determined by experiment may not satisfy, that is, may not be consistent with, the Gibbs/Duhem equation. Thus, Eq. (4-289) applied to sets of experimental values becomes a test of the thermodynamic consistency of the data, rather than a valid general relationship. 

The right-hand side of this equation is exactly the quantity that Eq. (4-289), the Gibbs/Duhem equation, requires to be zero for consistent data. The residual on the left is therefore a direct measure of deviations from the Gibbs/Diihem equation. The extent to which values of this residual fail to scatter about zero measures the departure of the data from consistency with respect to this equation. 

If the experimental values P and w are closely reproduced by the correlating equation for g, then these residues, evaluated at the experimental values of X, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P and y . Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-X data this is possible because the P-x -y data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters Ot, b, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals 6P. Known as Barkers method Austral. ]. Chem., 6, pp. 207-210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y values however, it provides a better fit than does the procedure that minimizes the sum of the squares of the 6g residuals. 

Worth noting is the fact that Barkers method does not require experimental yf values. Thus the correlating parameters Ot, b, and so on, can be ev uated from a P-X data subset. Common practice now is, in fact, to measure just such data. They are, of course, not subject to a test for consistency by the Gibbs/Duhem equation. The worlds store of X T.E data has been compiled by Gmehling et al. (Vapor-Liquid Lquilibiium Data Collection, Chemistiy Data Series, vol. I, parts 1-8, DECHEMA, Frankfurt am Main, 1979-1990). 

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. 

A consistency test described by Chueh and Muirbrook (C4) extends to isothermal high-pressure data the integral (area) test given by Redlich and Kister (Rl) and Herington (H2) for isothermal low-pressure data. [A similar extension has been given by Thompson and Edmister (T2)]. For a binary system at constant temperature, the Gibbs-Duhem equation is written... 

Equations 11 and 12 are not written for constant molality, and can not be easily used with the Gibbs-Duhem equation to obtain an analytical expression for the activity of water in the ternary solution. However, it is possible to propose a separate equation for the activity coefficient of water that is consistent with the proposed model of concentrated solutions. 

Using the data of Fig. 38 and data obtained by attempting to duplicate the run, Allen and her coworkers determined the activity coefficients presented in Fig. 39. The two sets of data arc in quite good agreement except at lower mole fractions of DBO, which correspond to the later phases of a run when contamination became significant. Since the activity coefficients for each of the two species were determined from the data, the consistency of the results can be tested by applying the Gibbs-Duhem equation,... 

In theory, once the activity of an electrolyte in solution is known, the activity of the solvent can be determined by the Gibbs-Duhem integration (see section 2.11). In practice, the calculation is prohibitive, because of the chemical complexity of most aqueous solutions of geochemical interest. Semiempirical approximations are therefore preferred, such as that proposed by Helgeson (1969), consisting of a simulation of the properties of the H20-NaCl system up to a solute... 

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. 

Pitzer s solution to the problem was the development of a set of analytical equations that are thermodynamically consistent after transformations through the Gibbs-Duhem equation. These equations are known as the Pitzer equations, in recognition of the major role that he played in developing them and the major contributions he made in the understanding of electrolyte solutions through a lifetime of work. We will now summarize these equations and describe their usefulness. For details of the derivation we refer the reader to Pitzer s original paper.6... 

Three different uses of the Gibbs-Duhem equation associated with the integral method are discussed in this section (A) the calculation of the excess chemical potential of one component when that of the other component is known (B) the determination of the minimum number of intensive variables that must be measured in a study of isothermal vapor-liquid equilibria and the calculation of the values of other variables and (C) the study of the thermodynamic consistency of the data when the data are redundant. 

C) In many experimental studies, all of the intensive variables are determined, giving a redundancy of experimental data. However, Equations (10.70) and (10.73) afford a means of checking the thermodynamic consistency of the data at each experimental point for the separate cases. Thus, for Equation (10.70), the required slope of the curve of P versus ylt consistent with the thermodynamic requirements of the Gibbs-Duhem equations, can be calculated at each experimental point from the measured values of P, xt, and at the experimental temperature. This slope must agree within the experimental error with the slope, at the same composition, of the best curve... 

Thus, another approach consists in selecting some boundary conditions and properties. It is obvious that all exact correlation functions must satisfy and incorporate them in the closure expressions at the outset, so that the resulting correlations and properties are consistent with these criteria. These criteria have to include the class of Zero-Separation Theorems (ZSTs) [71,72] on the cavity function v(r), the indirect correlation function y(r) and the bridge function B(r) at zero separation (r = 0). As will be seen, this concept is necessary to treat various problems for open systems, such as phase equilibria. For example, the calculation of the excess chemical potential fi(iex is much more difficult to achieve than the calculation of usual thermodynamic properties since one of the constraints it has to satisfy is the Gibbs-Duhem relation... 

The advantage of this choice of the X dependence for the correlation functions and the bridge function relies on the fact that the excess chemical potential, and the one-particle bridge function as well, can be determined unambiguously in terms of B(r) as soon as n and m are known. To address this problem, the authors proposed to determine the couple of parameters (n m) in using the Gibbs-Duhem relation. This amounts to obtaining values of n and m from Eq. (87), which is considered as supplementary thermodynamic consistency condition that have to be fulfiled. 

The present relations differ from the KM approximation since the factor 3 is replaced by the bridge function at zero separation. This feature does not seem to be unreasonable because, from diagrammatic expansions, B (r) = B r)/3 is supposed to be accurate only at very low densities. Eq. (112) presents two advantages at high density i) it provides a closed-form expression for Bother fluids than the HS model and ii) it allows to ensure a consistent calculation of the excess chemical potential by requiring only the use of the pressure consistency condition (the Gibbs-Duhem constraint, no longer required, is nevertheless implicitly satisfied within 1%). 

The problem is that the data are not consistent with the Gibbs/ Duhem equation. That is, the experimental values of In yt and In y2 do not conform to Eq. (11.70). However, the values of In y and In y2 found from the correlation necessarily obey this equation the two sets of values therefore cannot possibly agree, and the resulting correlation cannot provide a precise representation of the complete set of P-x,- data. Although this is true regardless of the means of data reduction, the method just described produces a correlation that is unnecessarily divergent from the experimental values. 

The nonidealities of equilibrium mixtures result from various combinations of molecular interactions one such interaction that has been recently studied is molecular association. Molecules of fatty acids such as acetic acid typically form dimers and, to a lesser extent, trimers by hydrogen bonding in the vapor and liquid states. Failure of the equilibrium data of binary systems containing acetic acid to meet established criteria of consistency based upon the Gibbs-Duhem relation has been observed by Rius et al. (I), Campbell et al. (2), Herington (3), and... 

Tpo obtain vapor-liquid equilibrium data for binary systems, it is now well established that under certain circumstances it can be more accurate and less time consuming to measure the boiling point, the total pressure, and the liquid composition and then use the Gibbs-Duhem relationship to predict vapor composition (I) rather than to measure it. The disadvantage is that there is no way of checking the thermodynamic consistency of the experimental data. 

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