Thermodynamic consistency, requirements

Thermodynamic consistency requires that qs i be equal to qs2, but it is common practice to ignore this requirement, thereby introducing an additional parameter. This is legitimate if the equations are to be used purely as an empirical correlation, but it should be recognized that since thermodynamic consistency is violated such expressions are not valid over the entire composition range. 

Thermodynamic consistency requires qsl = qs2, but this requirement can cause difficulties when attempts are 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 facilitates data fitting but it must be recognized that the equations are then being used purely as a convenient empirical form with no theoretical foundation. 

Thermodynamic consistency requires that the saturation limit (qs) must be the same for all components [22,23], so the model is clearly inappropriate for mixtures of molecules of very different sizes, even if the pure component isotherms conform to the Langmuir expression. 

Cotter has examined the postulates underlying the mean field approximation in the light of Widom s analysis of this general problem and has concluded that thermodynamic consistency requires that u should be proportional to V regardless of the nature of the intermolecular pair potential. However, in what follows we have assumed a dependence as in the original formulation of the theory by Maier and Saupe. 

Kemball, Ridcal, and Guggenheim and independently Broughton have shown that thermodynamic consistency requires = q 2- physical adsorption of molecules of widely different size such an assumption is unrealistic. If the extended Langmuir equation is regarded as an analytical description rather than a physical model, the use of different values of q for each component becomes permis.sible although the equations cannot then be expected to apply over the entire concentration range and extrapolation on such... 

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. 

Mathematical Consistency Requirements. Theoretical equations provide a method by which a data set s internal consistency can be tested or missing data can be derived from known values of related properties. The abiUty of data to fit a proven model may also provide insight into whether that data behaves correctiy and follows expected trends. For example, poor fit of vapor pressure versus temperature data to a generally accepted correlating equation could indicate systematic data error or bias. A simple sermlogarithmic form, (eg, the Antoine equation, eq. 8), has been shown to apply to most organic Hquids, so substantial deviation from this model might indicate a problem. Many other simple thermodynamics relations can provide useful data tests (1—5,18,21). 

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,... 

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... 

The fulfilment of this condition is termed as one of the thermodynamic consistency conditions. But, doing so requires the use of external information to the theory that cannot be used in a predictive manner. Of course, obtaining a full thermodynamic consistency without resorting to any external data is the practical purpose of self-consistent IETs. 

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. 

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. 

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... 

It has been said that only termination, but not dissociation, involves a collision partner M and that the ratio klm, ikcB, in the rate equation does not equal the dissociation equilibrium constant because the two coefficients are "not linked by detailed balancing" [16], However, this argument is without merit. In the absence of H2 (or any other species with which Br- can react), thermodynamic consistency and microscopic reversibility clearly require M to participate in dissociation if it does so in recombination. The addition of any species such as H2 that takes no part in the dissociation step may cause the system to deviate from thermodynamic dissociation equilibrium, but can obviously not alter the mechanism of dissociation. 

Prediction of the second normal stress difference in shear and thermodynamic consistency obviously requires the use of a different strain measure including of the Cauchy strain tensor in the form of the K-BKZ model. With the ratio of second to first normal stress difference as a new parameter, Wagner and Demarmels [32] have shown that this is also necessary for accurate prediction of other flow situations such as equibiaxial extension, for example. 

A basic problem in thermodynamics consists in determining the final equilibrium state that an isolated system reaches after starting out from a given set of initial conditions and constraints. In this matter we are guided by two corollaries of the First and Second Laws namely, that in an isolated system subjected to any change the entropy cannot decrease, and that its energy must remain constant. These requirements may not be sufficient to determine the final equilibrium state, in which case other experimental data or additional constraints must be inserted to provide a unique solution to the problem. 

Nowadays, for a thermodynainicist, /pVT-calorimetry (further referred to as scanning transitiometry, its patented and commercial name ) is the most accomplished experimental concept. It allows direct determinations of the most important thermodynamic derivatives it shows how, in practice, the Maxwell relations can be used to fully satisfy the thermodynamic consistency of those derivatives. Of particular interest is the use of pressure as an independent variable this is typically illustrated by the relatively newly established pressure-controlled scanning calorimeters (PCSC). - Basically, the isobaric expansibility Op(p,T) =il/v)(dv/dT)p can be considered as the key quantity from which the molar volume, v, can be obtained and therefore all subsequent molar thermodynamic derivatives with respect to pressure. Knowing the molar volume as a function of p at the reference temperature, Tg, the determination of the foregoing pressure derivatives only requires the measurement of the isobaric expansibilities as... 

Beside the condition (3.216) for the calculation of the molar excess Gibbs energy of mixing, the minimum necessary Gj coefficients for attaining thermodynamically consistent phase diagram and a reasonable standard deviation of approximation are required. The calculation is mostly performed assuming AfusH / T) and AG j / T). 

As we mentioned in Section II.C, one of the most important requirements, which is applicable to the overall structure of reaction schemes, is its thermodynamic consistency. In other words, the scheme, as written, must allow the process to proceed asymptotically to its equilibrium state at infinite time. This can be reached only if any elementary step is included into the overall scheme together with its reverse reaction. Let us consider the consequences for the description of the process kinetics. For the sake of simplicity, we assume that the reaction proceeds in the ideal gas mixture. The value of rate constants k( for forward and k( ) for reverse reactions must satisfy the connecting equation... 

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 ... 

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