Requirements for thermodynamic consistency

The Sips and related LRC (loading ratio correlation) models fail to propedy predict Henry s law behavior (as required for thermodynamic consistency) at the zero pressure limit (8). Thermodynamic inconsistency of the LRC model had also been noted by the original authors (17) nevertheless, the model has been found useful in predicting multicomponent performance from single component data and correlating multicomponent data (18). However, users of models lacking thermodynamic consistency must take due care, particularly in extrapolation beyond the range of actual experimental data. 

The mathematics required for thermodynamics consists for the most part of nothing more complex than differential and integral calculus. However, several aspects of the subject can be presented in various ways that are either more or less mathematically based, and the best way for various individuals depends on their mathematical background. The more mathematical treatments are elegant, concise, and satisfying to some people, and too abstract and divorced from reality for others. 

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. 

Rate equations and their coefficients in networks are not entirely independent. They are subject to two constraints thermodynamic consistency and so-called microscopic reversibility. For reversible reactions, the algebraic form of the rate equation of the forward reaction imposes a constraint on that of the rate equation of the reverse reaction. In addition, the requirements of thermodynamic consistency and microscopic reversibility can be used to verify that the postulated values of the coefficients constitute a self-consistent set, or to obtain a still missing coefficient value from those of the others. 

The general formula 6.4 to 6.6 for simple pathways is so important that a closer examination of its properties is warranted. First, we can see that it meets the requirement of thermodynamic consistency (Section 2.5.1) Setting rP = 0 for equilibrium one finds... 

Ruckenstein and Chi presented a thermodynamic treatment of a microemulsion consisting of fixed amounts of oil, water, and surfactant. The analysis yielded an optimum droplet diameter, demonstrated that the entropy of dispersion made an important contribution to microemulsion free energy, and confirmed that a very low interfacial tension of the droplets was required for thermodynamic stability. Indeed, to a first approximation, we can often estimate droplet size by taking the area per surfactant molecule as that for which interfacial tension would be zero. 

The PTEF is a dimensionless quantity, as required by thermodynamic consistency. The PTEF definition can be broadly applied, covering various kinetic models and being appropriate for various photochemical reactors, either homogenous or heterogeneous. 

Each of the above three methods employs a different data base. Most of the property values required for the evaluation of in Equations 7-9 have been experimentally determined for III—V systems and these three relationships can be used as a test for thermodynamic consistency. The first method, Equation 7, is most reliable at or near the binary compound melting temperature. As the temperature is lowered below the melting one, uncertainties in the extrapolated stoichiometric liquid heat capacity and component activity coefficients become important. The second method, Equation 8, is limited to the temperature range in which an experimental determination of AG. is feasible (e.g., high temperature galvanic cell). Method II is also valuable for "pinning down" the low temperature values of 0yp. Method III is the preferred procedure when estimating solution model parameters from liquidus data. Since the activity coefficients of the stochiometric liquid... 

The transformed Gibbs free energy clearly decreases in the presence of electric fields. This property is required for a consistent thermodynamic treatment of electric-chemical field effects. 

MSL models present the correct Henry s law limit and obey the requirement of thermodynamic consistency. This model is often used to account for the effect of the size difference in the study of multicomponent adsorption equilibria [29,30]. The MSL model can be extended to include effects such as lateral interactions in the adsorbed phase as well as surface heterogeneity [80]. 

In Eq. 3.4, the monolayer amount for component i in the mixed adsorbed phase remains the same as that in pure component adsorption. It is assumed that each species maintains its own molecular area (the area covered by one molecule that is not influenced by the presence of other species on the surface). This does not meet the requirement of thermodynamic consistency (which requires all monolayer amounts be equal). An empirical mixing rule is available for dealing with this problem (Yang, 1987). Nonetheless, Eq. 3.4 remains useful for practical design purposes. 

Here /if is the electrochemical potential of i in a secondary reference state (typically specified by choosing a reference electrode of a given kind [29]), ji is an activity coefficient on a mole-fraction basis, y, is the mole fraction of i, F is Faraday s constant, and O is the electric potential. (Note that y, is a sensible basis for composition because it leads the electrochemical potentials to satisfy the Gibbs-Duhem equation in the limit of infinite dilution, where y, goes to unity for all species.) For thermodynamic consistency it should be borne in mind that the balance of net charge in equilibrated phases requires... 

DIPPR Data Compilation of Pure Compound Properties http //dippr.byu.edu/ (subscription required). The DIPPR Chemical Database consists of both experimental data and temperature-dependent properties for over 2,000 pure chemicals. Data have been evalnated, correlated, and checked for thermodynamic consistency. Datasets, DIPPR-approved property constants, and regressed correlation coefficients for temperature-dependent properties are included. Calculation of temperature-dependent properties is possible in a variety of units (Standard, cgs, or British). Compounds are searched with either chemical names (or name fragments) or Chemical Abstracts Registry Nnmbers. Individual compound records also include both CAS and lUPAC names as well as trivial names. 

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. 

If a surface precipitate of metal hydroxy-polymer has formed on an adsorbent, the -pH relationship for the coated adsorbent should resemble closely that observed for particles consisting purely of the hydroxy-polymer or the hydrous oxide of the metal (15). This kind of evidence for Co(ll), La(lII), and Th(lV) precipitation on silica colloids was cited by James and Healy (15). It should be noted, however, that the increase in C toward a maximum value often occurs at pH values well below that required thermodynamically to induce bulk-solution homogeneous precipitation of a metal hydrous oxide (15, 16). If surface precipitation is in the incipient stage under these conditions, it must be a nucleation phenomenon. James and Healy (15) argue that the microscopic electric field at the surface of a charged adsorbent is sufficiently strong to lower the vicinal water activity and induce precipitation at pH values below that required for bulk-solution precipitation of a metal hydrous oxide. 

Diblock copolymers, especially those containing a block chemically identical to one of the blend components, are more effective than triblocks or graft copolymers. Thermodynamic calculations indicate that efficient compat-ibilisation can be achieved with multiblock copolymers [47], potentially for heterogeneous mixed blends. Miscibility of particular segments of the copolymer in one of the phases of the bend is required. Compatibilisers for blends consisting of mixtures of polyolefins are of major interest for recyclates. Random poly(ethylene-co-propylene) is an effective compatibiliser for LDPE-PP, HDPE-PP or LLDPE-PP blends. The impact performance of PE-PP was improved by the addition of very low density PE or elastomeric poly(styrene-block-(ethylene-co-butylene-l)-block styrene) triblock copolymers (SEBS) [52]. 

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... 

We examine, as an example, the exergy vector diagram for methanol synthesis to estimate the minimum exergy loss thermodynamically required for the synthesis reaction of methanol from methane [Ref. 16.]. First, we consider a direct (single step) synthesis of methanol from methane through a coupled-and-coupling reaction consisting of the oxidation of methane (objective reaction) and the dissociation of water molecule (coupled reaction) shown, respectively, as follows ... 

Equation 5.31 provides the basis for a test of both experimental precision and thermodynamic self-consistency when the ArG° values are calculated using values of In Kex determined with the help of Eq. 5.25c. Table 5.2 shows some typical examples of this test applied to cation exchange reactions on three montmoril-lonites.12 The values of b ArG°(3) expected according to Eq. 5.31b are given in the eighth column of the table. In each case the measured value of b ArG°(3) in column 7 is in fair agreement with the theoretical value based on the requirement of self-consistency, the mean deviation between measured and theoretical values being +0.62 kj mol ]. 

---