Thermodynamics consistency

Thermodynamic consistency requites 5 1 = q 2y but this requirement can cause difficulties when attempts ate made to correlate data for sorbates of very different molecular size. For such systems it is common practice to ignore this requirement, thereby introducing an additional model parameter. This facihtates data fitting but it must be recognized that the equations ate then being used purely as a convenient empirical form with no theoretical foundation. 

This has the advantage that the expressions for the adsotbed-phase concentration ate simple and expHcit, and, as in the Langmuir expression, the effect of competition between sorbates is accounted for. However, the expression does not reduce to Henry s law in the low concentration limit and therefore violates the requirements of thermodynamic consistency. Whereas it may be useful as a basis for the correlation of experimental data, it should be treated with caution and should not be used as a basis for extrapolation beyond the experimental range. 

This rate equation must satisfy the boundary conditions imposed by the equiUbrium isotherm and it must be thermodynamically consistent so that the mass transfer rate falls to 2ero at equiUbrium. It maybe a linear driving force expression of the form... 

Not all of the isotherm models discussed in the following are rigorous in the sense of being thermodynamically consistent. For example, specific deficiencies in the Freundhch, Sips, Dubinin-Radushkevich, Toth, and vacancy solution models have been identified (14). 

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,... 

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. 

In some cases, reported data do not satisfy a consistency check, but these may be the only available data. In that case, it may be possible to smooth the data in order to obtain a set of partial molar quantities that is thermodynamically consistent. The procedure is simply to reconstmct the total molar property by a weighted mole fraction average of the n measured partial molar values and then recalculate normalised partial molar quantities. The new set should always be consistent. 

In order to ensure thermodynamic consistency, in almost all cases these properties are calculated from Tr. and the vapor pressure and liquid density correlation coefficients listed in those tables. This means that there will be slight differences between the values listed here and those in the DIPPR tables. Most of the differences are less than 1%, and almost all the rest are less than the estimated accuracy of the quantity in question. 

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. 

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. 

Evaluation of laboratoiy data. Location and confirmation of saddle ternaiy azeotropes and a check of thermodynamic consistency of data. 

The UCKRON AND VEKRON kinetics are not models for methanol synthesis. These test problems represent assumed four and six elementary step mechanisms, which are thermodynamically consistent and for which the rate expression could be expressed by rigorous analytical solution and without the assumption of rate limiting steps. The exact solution was more important for the test problems in engineering, than it was to match the presently preferred theory on mechanism. 

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. 

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. 

Vapor-liquid equilibria activity coefficients, 173-179 constants, 179 equation of state, 171-172 high pressure, 170-184 thermodynamic consistency, 179-184... 

This example provides the proof that we promised earlier that (dU/dV)T = 0 for the ideal gas. We also obtain the same equation for AS as we derived earlier using AS = qrev/T. It also shows that AH — 0 for the isothermal expansion of ideal gas and gives a thermodynamically consistent valuef for AG. That is, in the final state... 

Thermodynamic variables are related through a number of different thermodynamic equations. Experimentalists who measure thermodynamic properties by one method often check the results using other relationships. This test checks for thermodynamic consistency, which must follow if the results are to be trusted. 

As previously noted the constancy of catalyst potential UWr during the formation of the Pt-(12xl2)-Na adlayer, followed by a rapid decrease in catalyst potential and work function when more Na is forced to adsorb on the surface, (Fig. 5.54) is thermodynamically consistent with the formation of an ordered layer whose chemical potential is independent of coverage. 

Clayton, T. D., Byrne, R. H., Breland, J. A. et al. (1995). The role of pH measurements in modern oceanic C02-system characteristics precision and thermodynamic consistency. Deep-Sea Res. II42,411 30. 

The positive values obtained in practically all cases indicate that all these models may be plausible representations of the data and indeed, the correlation coefBcients, R, are greater than 0.9. Thus, statistical compliance is not a sufficient basis for model discrimination. Specifically, the thermodynamic consistency of the estimates, as proposed by Boudart et al. [3], is appropriate further scrutinizing criterion during kinetic modelling and has been gainfully employed in other reactions [4-6]. 

The following illustration indicates one application of the principle of thermodynamic consistency. 

ILLUSTRATION 5.2 APPLICATION OF THE PRINCIPLE OF THERMODYNAMIC CONSISTENCY TO AN ABSORPTION PROCESS... 

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. 

Thermodynamically consistent forms may be obtained by choosing different values of n. 

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