Thermodynamic consistency testing

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

Many additional consistency tests can be derived from phase equiUbrium constraints. From thermodynamics, the activity coefficient is known to be the fundamental basis of many properties and parameters of engineering interest. Therefore, data for such quantities as Henry s constant, octanol—water partition coefficient, aqueous solubiUty, and solubiUty of water in chemicals are related to solution activity coefficients and other properties through fundamental equiUbrium relationships (10,23,24). Accurate, consistent data should be expected to satisfy these and other thermodynamic requirements. Furthermore, equiUbrium models may permit a missing property value to be calculated from those values that are known (2). 

Thermodynamic consistency tests for binary vapor-liquid equilibria at low pressures have been described by many authors a good discussion is given in the monograph by Van Ness (VI). Extension of these methods to isothermal high-pressure equilibria presents two difficulties first, it is necessary to have experimental data for the density of the liquid mixture along the saturation line, and second, since the ideal gas law is not valid, it is necessary to calculate vapor-phase fugacity coefficients either from volumetric data for... 

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. 

To illustrate this thermodynamic consistency test, Figs. 15, 16, and 17 show plots of the appropriate functions needed to calculate Areas I, II, and III, respectively, for the nitrogen-carbon dioxide system at 0°C the data are taken from Muirbrook (M5). Fugacity coffiecients were calculated with the modified Redlich-Kwong equation (R4). 

Thermodynamic Consistency Test for Carbon Dioxide (I)-Nitrogen (2) at 0°C... 

The thermodynamic consistency test for binary systems described above can be extended to ternary (and higher) systems with techniques similar to those described by Herington (H3). The necessary calculations become quite tedious, and unless extensive multicomponent data are available, they are usually not worthwhile. 

Ammonia - Water System. Interaction parameter for the ammonia - water system was obtained using the data of Clifford and Hunter (1 4) and of Macriss et al. (15). A single - valued parameter was capable of representing the composition of the liquid phase reasonably well at all temperatures, however, the calculated amount of water in the vapor phase in the very high ammonia concentration region was somewhat lower than the data of Clifford and Hunter and Macriss et al. Edwards et al. (16) have applied a new thermodynamic consistency test to the data of Macriss et al and have concluded that the data appear to be inconsistent and that the reported water content of the vapor phase is too high. 

Correlating ln(Pc /Pc), with vapor pressure data For isotopomer pairs with the vapor pressure and VPIE established near Tc, a thermodynamic consistency test between ln(Tc7Tc) and ln(Pc /Pc), and calculation of ln(Pc7Pc) from ln(Tc7Tc) is possible. The critical pressure of the heavier isotopomer at its critical temperature, Pc(Tc), can be calculated from the lighter, Pc,(Tc7 provided the vapor pressure of the lighter between Tc and Tc, the VPIE, and Tc and Tc are known. For Tc < Tc ... 

In summary, to be thermodynamically rigorous, the salt presence must be recognized in calculating activity coefficients for use in correlating equations its degree of dissociation as a function of liquid composition, among other factors, must be considered also. Conversely, it may be possible to apply data that are believed to be consistent to a consistency test in order to calculate degree of dissociation as a function of liquid composition. 

The four procedures are thermodynamically equivalent and are therefore useful in performing consistency tests on available sets of data. As an ex-... 

The preceding test for thermodynamic data treats the data set as a whole. It does not determine whether individual data points are consistent. A point-consistency test has been proposed, but it is too cumbersome for manual calculations. 

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. 

There are no thermodynamic consistency tests that can be applied to the data. For each system in the infinite dilution VLE data base, the weight fraction activity coefficients were plotted as a function of temperature. In many cases considerable scatter was observed. Some data were found that were significantly outside the anticipated range or which showed contradictory behavior with temperature from that expected. These points were kept in the data base but are indicated by an "R", for Rejected, if they were judged to be clearly erroneous or by an "N", for Not recommended, if they appeared questionable but not obviously incorrect. Similarly, in the finite concentration VLE data base some points were judged to be significant outliers and are indicated accordingly. 

Marek (4). Here we try to obtain x-y-t data for the acetic acid-acetone system at 760 and 500 mm Hg and to show how considering acid dimerization improves the results for thermodynamic consistency tests. 

The vapor—liquid equilibrium data used in the calculations have been selected on the basis of the following two criteria (1) the thermodynamic consistency tests (the integral test and the point tesC ) should be fulfilled by the systems chosen, (2) the sets chosen should contain at least 10—15 experimental... 

Solubility of drugs in aqueous solutions. Part 5 Thermodynamic consistency test for the solubility data. 

This paper is devoted to the verification of the quality of experimental data regarding the solubility of sparingly soluble solids, such as drugs, environmentally important substances, etc. in mixed solvents. A thermodynamic consistency test based on the Gibbs-Duhem equation for ternary mixtures is suggested. This test has the form of an equation, which connects the solubilities of the solid, and the activity coefficients of the constituents of the solute-free mixed solvent in two mixed solvents of close compositions. 

Keywords Drug solubility Mixed solvent Thermodynamic consistency test... 

Thermodynamic consistency tests are well known, and have been frequently used for vapour-liquid equilibrium data in binary mixtures (for reviews one can see Gmehling and Onken, 1977 Acree, 1984 Prausnitz et al., 1986). These tests are based on theGibbs-Duhem equation and allow one to grade the experimental data for vapor-liquid equilibrium in binary mixtures. A more difficult problem is the consistency of data regarding vapor-liquid equilibrium in ternary or multicomponent mixtures. However, several thermodynamic consistency tests, also based on the Gibbs-Duhem equation, were suggested for vapor-liquid equilibrium in ternary or multicomponent mixtures (Li and Lu, 1959 McDermott and Ellis, 1965). 

The McDermott and Ellis consistency test means that if the vapor-liquid equilibrium data for points c and d are correct, then Eq. (3) should be satisfied. Eq. (3) will be used to derive a thermodynamic consistency test for verifying the experimental data regarding the solubility of drugs in aqueous mixed solvents. 

Thermodynamic consistency test regarding the solubility of drugs in binary aqueous mixed solvents... 

Eq. (11) provides a thermodynamic consistency test for the solubility of poorly soluble substances, such as drugs, environmentally important substances, etc. in mixed solvents in terms of the... 

The main difficulties in this selection were the following 1) the total number of experimental data regarding the HOP in aqueous mixed solvents is small, much smaller that the number of experimental data regarding the solubilities of dmgs in aqueous mixed solvents (7). 2) There is no thermodynamic consistency test, such as those for vapor-liquid equilibrium (29), for checking the self-consistency of the data regarding the solubility of a solid in a mixed solvent. Therefore, it is difficult to evaluate whether the solubility data are accurate or contain errors. 

Our Eqs. (19) and (23) are rigorous thermodynamic relations which provide a relation between the protein solubility and the preferential binding parameter F2 These thermodynamic equations provide a consistency test between the protein solubility ( 4 ) preferential binding parameter... 

Data Quality Normally, it is not possible to evaluate LLE data for thermodynamic consistency [Sorenson and Arlt, Liquid-Liquid Equilibrium Data Collection, Binary Systems, vol. V, pt. 1 (DECHEMA, 1979), p. 12]. The thermodynamic consistency test for VLE data involves calculating an independently measured variable from the others (usually the vapor composition from the temperature, pressure, and liquid composition) and comparing the measurement with the calculated value. Since LLE data are only very weakly affected by change in pressure, this method is not feasible for LLE. However, if the data were produced by equilibration and analysis of both phases, then at least the data can be checked to determine how well the material balance closes. This can be done by plotting the total... 

The consistency test is carried out by plotting In (Y1/Y2) versus Xj and graphically evaluating the integral in Equation 1.31. The curve consists of a positive part and a negative part above and below the line IntYi /Y2) = 0- The data points are considered thermodynamically consistent, and the assumption of negligible effect of variable pressure is deemed valid if the areas above and below the line are equal. 

In Chapter 7 we found that although thermodynamics could not be used to predict the equation of state of a real fluid, it did provide certain consistency tests (i.e., Eqs. 7.2-12 and 7.2-13) that had to be satisfied by any equation of state. The situation is much the same here, in that starting from... 

This equation provides a thermodynamic consistency test for experimental activity coefficient data. As an illustration of its use, consider its application to the Weissman-Wood measurements, which were carried out at constant temperature but varying total pressure. In this case Eq. 10.2-13 reduces to... 

Figure 10.2-7 Thermodynamic consistency test for the activity coefficients of the benzene-2,2,4-trimethyl pentane system.

It is possible, as a result of cancellation, to satisfy the integral test of Eq. 10.2-13 while violating the differential form of the Gibbs-Duhem equation, Eq. 9.3-15, on which Eq. 10.2-13 is based, at some or all data points. In this case the experimental data should be rejected as thermodynamically inconsistent. Thus, the integral consistency test is a necessar). but not sufficient, condition for accepting experimental data. 

The ideality of vapour phase is taken as a working hypothesis. Experimental data together with calculated y, and / -values are presented in Table 6.6. The regression of the interaction parameter has been done with Aspen Plus. Note that the program performs automatically a thermodynamic consistency test. 

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