Gibbs energy-composition sections

The tables in this section contain values of the enthalpy and Gibbs energy of formation, entropy, and heat capacity at 298.15 K (25°C). No values are given in these tables for metal alloys or other solid solutions, for fused salts, or for substances of undefined chemical composition. 

The mutual solubility of two liquids A and B depends, in general, on how much the molecules of each liquid tend to attract those of its own kind, relative to their tendency to attract those of the other. This tendency is measured by the excess Gibbs energy of mixing of the two liquids (see section 2.4), Am gL, which is related to the partial vapor pressures p/ and of the two liquids A and B in the mixture. If the composition of the system is given by and Wb moles of the respective components in a given phase, their mole fractions in this phase are... 

We wish to determine under isothermal and isobaric conditions the concentration of defects as a function of the solid solution composition (e.g NB in alloy (A, B)). Consider a vacancy, the formation Gibbs energy of which is now a function of NB. In ideal (A, B) solutions, we may safely assume that the local composition in the vicinity of the vacancy does not differ much from Ns and /VA in the undisturbed bulk. Therefore, we may write the vacancy formation Gibbs energy Gy(NE) (see Eqn. (2.50)) as a series expansion G%(NE) = Gv(0) + A Gv Ab+ higher order terms, so that AGv = Gv(Nb = l)-Gv(AfB = 0). It is still true (as was shown in Section 2.3) that the vacancy chemical potential /Uy in the homogeneous equilibrium alloy is zero. Thus, we have (see Eqn. (2.57))... 

In this section two prediction techniques are discussed, namely, the gas gravity method and the Kvsi method. While both techniques enable the user to determine the pressure and temperature of hydrate formation from a gas, only the KVSI method allows the hydrate composition calculation. Calculations via the statistical thermodynamics method combined with Gibbs energy minimization (Chapter 5) provide access to the hydrate composition and other hydrate properties, such as the fraction of each cavity filled by various molecule types and the phase amounts. 

Further transformed Gibbs energies of formation are especially useful in calculating equilibrium compositions by computer programs that accept conservation matrices and vectors of initial amounts, as discussed in the next section. 

Gibbs energy curves have been calculated for the hexagonal and cubic phases for various hypothetical (Ti,Al)N deposition temperatures (Figure 9). The point of intersection of these curves at each temperature defines the composition at which there is a transition from one structure to the other. It can be seen that this composition is nearly temperature independent and has a calculated value of around 0.7 mol fraction AIN, Experimental studies of the extent of the metastable cubic range in the section AlN-TiN have been carried out by Knotek and Leyendecker and more recently, with... 

The second comment concerns the choice of standard states. Clearly, in defining the process of solvation, one must specify the thermodynamic variables under which the process is carried out. Here we used the temperature T, the pressure P, and the composition N1 ..., Nc of the system into which we added the solvaton. In the traditional definitions of solvation, one needs to specify, in addition to these variables, a standard state for the solute in both the ideal gas phase and in the liquid phase. In our definition, there is no need to specify any standard state for the solvaton. This is quite clear from the definition of the solvation process yet there exists some confusion in the literature regarding the standard state involved in the definition of the solvation process. The confusion arises from the fact that Ap is determined experimentally in a similar way as one of the conventional standard Gibbs energy of solvation. The latter does involve a choice of standard state, but the solvation process as defined in this section does not. For more details, see the next two sections. 

Suppose that we have availshle an expression for g , the molar excess Gibbs energy, which holds for the entire composition ranga. We can Red activity coefficients, as discusred in Section 1.4, from the relationship... 

The temperature of a liquid mi.xture is reduced so that solids form. However, unlike the illustrations in Section 12.3, on solidification, a solid mixture (rather than pure solids) is formed. Also, the liquid phase is not ideal. Assuming that the nonideality of the liquid and solid mixtures can be described by the same one-constant Margules excess Gibbs energy expression, derive the equations for the compositions of the coexisting liquid and solid phases as a function of the freezing point of the mixlure and the pure-component propenies. 

The first two differential coefficients express the change in Gibbs energy with T and p at constant composition as reported in Sections 1.6.2 and 1.6.4. After... 

Figure V-30 Section of the calculated phase diagram for the composition range 0.30 < JCs < 0.55. Dashed lines have been extrapolated, indicating regions where phase equilibria cannot be calculated by using the Gibbs energy fimctions of Table V-15.

Summary. Section 2.4 illustrates the extension of rational thermodynamics methodology on mixtures with chemical reaction(s) using a very simple model of two-component uniform mixture. The composition variable(s) enters the constitutive equations, cf. (2.76)-(2.79). In a uniform mixture, the classical chemical thermodynamics was obtained, i.e., its validity also in nonequilibrium covered by this model was demonstrated, cf. e.g., (2.82), (2.83), (2.85), (2.87), (2.88). Traditional quantities known from the equilibrium chemical thermodynamics may be thus introduced and used out of equilibrium—affinity by (2.89), chemical potential by (2.93) or (2.100), or Gibbs energy by (2.97). Gibbs and Gibbs-Duhem equations also remain valid,... 

We have shown how models for volumetric equations of state can be used with stability criteria to predict vapor-liquid phase separations. However, not all phase equilibria are conveniently described by volumetric equations of state for example, liquid-liquid, solid-solid, and solid-fluid equilibria are usually correlated using models for the excess Gibbs energy g. When solid phases are present, one motivation for not using a PvT equation is to avoid the introduction of spurious fluid-solid critical points, as discussed in 8.2.5. A second motivation is that properties of liquids and solids are little affected by moderate changes in pressure, so PvT equations can be unnecessarily complicated when applied to condensed phases. In contrast, g -models often do not contain pressure or density instead, they attempt to account only for the effects of temperature and composition. Such models are thereby limited to descriptions of phase separations that are driven by diffusional instabilities, and the stability behavior must be of class I (see 8.4.2). In this section we show how a g -model can describe liquid-liquid and solid-solid equilibria. 

At this point we have a fundamental problem. Given the relationship between Gibbs energies and compositions for ideal solutions we have developed, how do we handle deviations from this behavior What mathematical form should our equations for nonideality take There is a variety of approaches for this. The most general is to develop an equation of state, and there is a variety of types of those (Chapter 13). Then there are different approaches for dilute and concentrated solutions, and for electrolytes and nonelectrolytes. In this section we look at some fairly general methods which have been applied to many solid and liquid solutions. 

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