Gibbs energy representation

Figure 4.12 (a) Gibbs energy representation of the phases in the system Zr02-Ca0 at 1900 K. McaO - MzrO = TSZ n°t deluded for clarity, (b) Calculated phase diagram of the system Zr02 Ca0. Thermodynamic data are taken from reference [9]. 

The last of Eqs. (51) defines a thermodynamic fundamental equation for G = G(N, P, 7) in the Gibbs energy representation. Note that passing from one ensemble to the other amounts to a Legendre transformation in macroscopic thermodynamics [39]. Vq is just an arbitrary volume used to keep the partition function dimensionless. Its choice is not important, as it just adds an arbitrary constant to the free energy. The NPT partition function can also be factorized into the ideal gas and excess contributions. The configurational integral in this case is ... 

Guffey and Wehe (1972) used excess Gibbs energy equations proposed by Renon (1968a, 1968b) and Blac)c (1959) to calculate multicomponent LLE. They concluded that prediction of ternary data from binary data is not reliable, but that quarternary LLE can be predicted from accurate ternary representations. Here, we carry these results a step further we outline a systematic procedure for determining binary parameters which are suitable for multicomponent LLE. 

This chapter will begin, very briefly, with the thermodynamic representation of Gibbs energy for stoichiometric compounds before concentrating on the situation when mixing occurs in a phase. 

The starting point for thermodynamic description, whether in the calculus-based or the geometry-based formalism, is the Gibbs fundamental equation for a given equilibrium state S. In the energy representation, this is expressed as... 

In equation 33, the superscript I refers to the use of method I, a T) is the activity of component i in the stoichiometric liquid (si) at the temperature of interest, AHj is the molar enthalpy of fusion of the compound ij, and ACp[ij] is the difference between the molar heat capacities of the stoichiometric liquid and the compound ij. This representation requires values of the Gibbs energy of mixing and heat capacity for the stoichiometric liquid mixture as a function of temperature in a range for which the mixture is not stable and thus generally not observable. When equation 33 is combined with equations 23 and 24 in the limit of the AC binary system, it is termed the fusion equation for the liquidus (107-111). 

Figure 5.9. Representation of the Gibbs energy at constant pressure for a...

Equation (12.31) relates the emf of a cell to the change of the Gibbs energy for the change of state, based on n faradays, that takes place within the cell when the leads are short-circuited. The change of state is spontaneous, the value of AG is negative, and therefore the emf is positive. We conclude that the emf of any actual cell is always positive. However, the representation of a cell may be written in two ways. For the particular cell that we have been discussing, we may depict it as either... 

Analytical representation of the excess Gibbs energy of a system impll knowledge of the standard-state fugacities ft and of the frv. -xt relationshi Since an equation expressing /, as a function of x, cannot recognize a solubili limit, it implies an extrapolation of the /i-vs.-X[ curve from the solubility I to X) = 1, at which point /, = This provides a fictitious or hypothetical va for the fugadty of pure species 1 that serves to establish a Lewis/ Randall 1 for this species, as shown by Fig. 12.21. ft is also the basis for calculation of activity coefficient of species 1 ... 

Figure 2.5. Schematic representation of electronic potential energy surfaces 1, consecutive conformational and solvatational equilibrium processes with the essential change in the nuclear coordinates Q and the standard Gibbs energy AG0 2, consecutive non-equilibrium processes with small changes in Q and AG0 3, 4, equilibrium (full line) and non-equilibrium (broken line) processes in the normal and inverted Marcus regions respectively. (Likhtenshtein, 1996) Reproduced in permission.

Figure 2.6. Schematic representation of the dependence of the ET constants logarithm on the equilibrium Gibbs energy AG0 1, non-equilibrium conformational and solvational processes 2, partial non-equilibrium processes, J.n and AGoneq are slightly dependent on AG0 3, equilibrium processes. Arrows a and b are conditions for the maximum X = AGo and A.1 1 = AGonK respectively. (Likhtenshtein, 1996). Reproduced in permission.

The question remains, however, of whether the solution is in fact infinitely dilute at a solute concentration of xi. Only if this is true is it valid to assume that yi = y - Literature values of solubility data for several compounds in water were used to obtain parameters for the UNIQUAC and NRTL excess Gibbs energy expressions, and y values for these compounds were calculated. The calculated values are compared with inverse solubility data in Table I. The inverse solubility predicts lower values of y in all cases. However, the difference becomes smaller as the solubility decreases, and for compounds with solubility less than 0,5% the difference is less than 10%. It has been shown that these excess Gibbs energy expressions, while very useful, are not the exact representation of the composition dependence of activity coefficient all expressions have difficulty in representing liquid-liquid equilibria (43-44). Thus, extrapolating these expressions to infinite dilution may be in error. It is therefore inconclusive as to the correctness of using the inverse solubility to calculate... 

The quantity of radical cations formed by dissociation of is dependent on the polarity of the solvent with less polar solvents such as SO2 or SO2CIF favoring the formation of radical (mono-)cations if compared to polar solvents Hke HE or oleum. This can be understood by quantum chemical calculations with inclusion of solvation energies that approximate solvents of different polarities. A measure for the polarity of a solvent is the dielectric constant (DC) and a graphic representation of the calculated Gibbs energies of selected dissociation equilibria of in solvents with dielectric constants between 1 and 30 is given in Fig. 6 [3]. For comparison the DC of SO2 at 298 K is 14 and that of HF is 83. 

Although generally used for pure fluids, Equation (4.307) is sometimes also used to find the fugacity of a mixture, (() , =fjp- The fugacity of a mixture is just an altered representation of the Gibbs energy of a mixture, RT nf = g- g. ... 

The next step is to perform a simultaneous regression of NaCl(aq) apparent molal volumes from 25-350 C. Over this wide range of temperature, however, and particularly above 300 C, standard-state properties based on the infinitely dilute reference state exhibit a very complex behavior (7,8), which is related to various peculiarities of the solvent. Thus in their representation of NaCl(aq) volumetric properties, Rogers and Pitzer (7) adopted a reference composition of a hydrated fused salt, NaCl IOH2O, to minimize the P and T dependence of the reference state volume and to adequately fit volumetric ta to 300°C and 1 kb. In this study the (supercooled) fused salt is used as the reference state. The equation for the apparent molal volume on this basis can be easily derived from that for the excess Gibbs energy of Pitzer and Simonson (, and is given by ... 

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