Thermodynamic consistency relation

For multicomponent systems, the expression for y here employed may be shown equivalent to that involved in the cluster diagram technique (6), which is currently being employed in a variety of problems. The present derivation shows that the starting expressions satisfy the thermodynamic consistency relation embodied by the adsorption isotherm. It is, however, important to observe that any direct application of these alternative rigorous approaches, which is of necessity of an approximate nature, leads to some violation of the complete internal equilibrium conditions. Similarly, calculations of surface tension which employ the adsorption equation as a starting point invariably violate mechanical equilibrium in some order of approximation. 

B. Conditions of Thermodynamic Consistence Relating to isotherm Equations... 

The function v /(9) and its thermodynamic definition [Eq. (43)] make it possible to determine the conditions of thermodynamic consistence relating to isotherm equations. Evidently, this determination is vahd under conditions which must also be fulfilled by Eq. (43). These conditions are as follows ... 

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. 

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. 

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,... 

The approximate kinetic formats discussed above face inherent difficulties to account for fundamental physicochemical properties of biochemical reactions, such as the Haldane relation discussed in Section III.C.4 a major drawback when aiming to formulate thermodynamically consistent models. 

O. Penrose and P.C. Fife. On the relation between the standard phase-field model and a thermodynamically consistent phase-field model. Physica D, 69(1-2) 107-113, 1993. 

Shown in Figure 18.9 is a comparison of ArH° obtained from calorimetric measurements (solid line) and ATH° obtained from the Marshall-Frank equation,15 which is an expression relating K to T that gives ArH° from (dinK/dT)p. The agreement between the two methods is another example of thermodynamic consistency. 

Any hnear combination of steps in the reaction scheme that leads to an overall stoichiometric reaction that converts reactants and products gives rise to a relationship of thermodynamic consistency. Specifically, if stoichiometric coefficients of the linear combination of steps i that lead to an overall stoichiometric reaction, then the values of AH and AS- for these steps are related by the following equations ... 

In order to introduce basic equations and quantities, a preliminary survey is made in Section II of the statistical mechanics foundations of the structural theories of fluids. In particular, the definitions of the structural functions and their relationships with thermodynamic quantities, as the internal energy, the pressure, and the isothermal compressibility, are briefly recalled together with the exact equations that relate them to the interparticular potential. We take advantage of the survey of these quantities to introduce what is a natural constraint, namely, the thermodynamic consistency. 

This relationship corresponds to what is called a thermodynamic consistency condition, whose fulfilment is one of the criterions of accuracy of a self-consistent 1ET since S(q = 0) is calculated from the structure (9) and %T is obtained from the pressure (8), the left-hand side (l.h.s.) and right-hand side (r.h.s.) of the last relation have to coincide with a precision of 1%. This important aspect is developed below. 

The advantage of this choice of the X dependence for the correlation functions and the bridge function relies on the fact that the excess chemical potential, and the one-particle bridge function as well, can be determined unambiguously in terms of B(r) as soon as n and m are known. To address this problem, the authors proposed to determine the couple of parameters (n m) in using the Gibbs-Duhem relation. This amounts to obtaining values of n and m from Eq. (87), which is considered as supplementary thermodynamic consistency condition that have to be fulfiled. 

Select the criterion to be used for thermodynamic consistency. Deviations from thermodynamic consistency arise as a result of experimental errors. Impurities in the samples used for vapor-liquid equilibrium measurements are often the source of error. A complete set of vapor-liquid equilibrium data includes temperature T. pressure P. liquid composition x, and vapor composition y,. Usual practice is to convert these data into activity coefficients by the following equation, which is a rearranged form of the equation that rigorously defines K values (i.e., defines the ratio y, /x, under Related Calculations in Example 3.1) ... 

One way to estimate infinite-dilution activity coefficients is related to the preceding example on thermodynamic consistency in addition, it takes advantage of the fact that the plot of In Yi/x2 versus X is considerably more linear than is In Y versus X. ... 

This identification now makes for a completely consistent relation between the statistical properties of the gas and the thermodynamic varia-ables. The total energy of the gas, which is simply its kinetic energy for ideal monatomic gases, becomes [Eq. (VI.2.2)] for 1 mole... 

Consider the simple case where the radial distribution function in the fluid is zero for radii less than a cut-off value determined by the size of the hard core of the solute, and one beyond that value. Calculate the value of the parameter a appearing in the equation of state Eq. (4.1) for a potential of the form cr , where c is a constant and n is an integer. An example is the Lennard-Jones potential where = 6 for the long-ranged attractive interaction. What happens if n <37 Explain what happens physically to resolve this problem. See Widom (1963) for a discussion of the issue of thermodynamic consistency when constructing van der Waals and related approximations. 

Dependences dG vs. n of N2O and CO2 for NaLSX, as referred to bofli the boiling tempoature of each gas and the gas pressure, p° 760 toir, as standard state are shown in Fig. 5, together with related data for the N20-CaLSX system. Values dC7 fcff all systems change from negative ones to zero as n increases, approaches and exceeds the micropore>saturation capacities. This proves thermodynamic consistency and correctoess of the SIM data. 

The Gibbs-Duhem equation for ternary mixtures is used to analyze the quality of experimental data pertaining to the solubility of drugs and other poorly soluble solids in a binary mixed solvent. In order to test the quality of the data, a thermodynamic consistency test is suggested. This test is based on the thermodynamic relation between the solubilities of a solid in a binary mixed solvent at two different compositions and the activity coefficients of the constituents of the solute-free mixed solvent. The suggested test is applicable to all kinds of systems with the following limitations (1) the solubility of the solid should be low, (2) the above two compositions of the mixed solvent should be close enough to each other. 

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... 

The results obtained from the analytic solution are similar to those obtained previously using purely numerical methods. One of the advantages of the analytic solution is that it is possible to make changes to the closure relation which systematically improve the accuracy of the results. One method of doing this is by adding a Yukawa tail to the site-site direct correlation function. The two parameters in the Yukawa tail (that is, its amplitude and decay rate) can be chosen to give thermodynamic consistency and improve the accuracy of the site-site correlation function g y. This approach has been pursued Cummings and Morriss. " ... 

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. 

A thermodynamic model that satisfies all of the necessary consistency relations can be devised by writing an expression for the excess Gibbs energy and applying appropriate partial differentiation with respect to the number of moles of the components (see Wolery, in prep.). If the excess Gibbs energy is defined with reference to mole fraction ideality, the differential equations are... 

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