Requirements of thermodynamic consistency

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. 

The predictive power of the approach becomes obvious by noting that the dependence of the coupling on T and / /, is completely governed by the requirement of thermodynamic consistency [11] Maxwell s relation,... 

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... 

The analysis of particular kinetic schemes and the results of simulations indicate that models which meet the requirement of thermodynamic consistency possess additional benefits, such as possibilities to describe multiple processes within one kinetic scheme ... 

Compiling a kinetic scheme that meets the requirement of thermodynamic consistency ... 

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. 

It is evident that function v(Al (9) does not meet both requirements of thermodynamic consistence [only Eq. (48) is fulfilled]. However, function /l ip) fulfils both requirements it can, therefore, be substituted into Eq. (55) and it is certain that after solving Eq. (55) we obtain a thermodynamically consistent isotherm equation ... 

Equation (74) fulfils both requirement of thermodynamic consistence. It is expected that the MFG equation (74), which considers explicitly the lateral interactions between molecules adsorbed, represents also the two-dimensional condensation. The formation of a liquid adsorbed layer can be expressed mathematically by the fact that the isotherm equationp = 0(9) has a local maximum and minimum, i.e., two values of 9 exist where... 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

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 ]. 

Self-consistency of postulated forward and reverse rate equations can be tested with the principles of thermodynamic consistency and so-called microscopic reversibility. The former invokes the fact that forward and reverse rates must be equal at equilibrium the latter is for loops in networks and can be stated as requiring that the products of the clockwise and counter-clockwise rate coefficients of the loop must be equal, or, for catalytic cycles, that the product of the forward coefficients must equal that of the reverse coefficients multiplied with the equilibrium constant of the catalyzed reaction. 

Self-consistency of postulated forward and reverse rate equations and their coefficients can be tested with the principles of thermodynamic consistency and so-called microscopic reversibility. The former invokes the fact that forward and reverse rates must be equal at equilibrium. The latter is for loops of parallel pathways and for catalytic cycles. Thermodynamic consistency allows the reverse rate equation to be constructed from the forward one if at least one of its reaction orders is known, and requires the ratio of the products of the forward and reverse rate coefficients to be equal to the thermodynamic equilibrium constant. Microscopic reversibility leads to several useful conclusions The products of the clockwise and counter-clockwise rate coefficients of a loop must be equal the product of the forward rate coefficients of a catalytic cycle must be equal that of the reverse rate coefficients multiplied with the equilibrium constant of the catalyzed reaction forward and reverse reaction must occur along the same pathway and the ratio of the products of forward and reverse rate coefficients must be the same along all parallel pathways from same reactants to same products. The latter two rules apply regardless of whether or not any of the reactions are catalytic. 

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. 

Not in all cases, the quality of the published data is suflScient. The quality of complete data (P, T, x,. y,) can be checked with the help of thermodynamic consistency tests. A large number of consistency tests have been developed. Most of ten the so-called area test is applied. The derivation of the required equations for the area test is started from the following equation (see Section 4.3) ... 

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. 

The subject of thermodynamics embodies the principles of energy conversion in macroscopic systems. The general restrictions which experience has shown to apply to all such conversions are known as the laws of thermodynamics. These laws are fundamental in the sense that they cannot be derived from anything more basic. They do not require to be proven, but their acceptance is based on the fact that the conclusions derived from them are consistent with the results of all known experiments. 

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