Phase Gibbs function

The Clausius-Clapeyron equation provides a relationship between the thermodynamic properties for the relationship psat = psat(T) for a pure substance involving two-phase equilibrium. In its derivation it incorporates the Gibbs function (G), named after the nineteenth century scientist, Willard Gibbs. The Gibbs function per unit mass is defined... 

Let us apply Equation (6.8) to the two-phase liquid-vapor equilibrium requirement for a pure substance, namely p = p T) only. This applies to the mixed-phase region under the dome in Figure 6.5. In that region along a p-constant line, we must also have T constant. Then for all state changes along this horizontal line, under the p—v dome, dg = 0 from Equation (6.8b). The pure end states must then have equal Gibbs functions ... 

Figure 5.2 Graph of molar Gibbs function Gm as a function of temperature. Inset, at temperatures below r(meit) the phase transition from liquid to solid involves a negative change in Gibbs function, so it is spontaneous...

Earlier, on p. 181, we looked at the phase changes of a single-component system (our examples included the melting of an ice cube) in terms of changes in the molar Gibbs function AGm. In a similar manner, we now look at changes in the Gibbs function for each component within the mixture and because several components participate, we need to consider more variables, to describe both the host and the contaminant. 

We look once more at Figure 5.18, but this time we concentrate on the thinner lines. These lines are seen to be parallel to the bold lines, but have been displaced down the page. These thin lines represent the values of Gm of the host within the mixture (i.e. the once pure material following contamination). The line for the solid mixture has been displaced to a lesser extent than the line for the liquid, simply because the Gibbs function for liquid phases is more sensitive to contamination. 

Thermodynamic stability. Let us consider a liquid binary mixture of two species, a and b, crystallizing to a solid solution. Let G be the free enthalpy (Gibbs) function of each phase, either solid or liquid. Calculate the change AG in molar free enthalpy when a liquid of molar composition XUqb crystallizes to a solid of molar composition... 

APPLICATION OF THE GIBBS FUNCTION AND THE PLANCK FUNCTION TO SOME PHASE CHANGES... 

CALCULATION OF CHANGE IN THE GIBBS FUNCTION FOR SPONTANEOUS PHASE CHANGE... 

Thus far we have restricted our attention to phase changes in which equilibrium is maintained. It also is useful, however, to find procedures for calculating the change in the Gibbs function in transformations that are known to be spontaneous, for example, the freezing of supercooled water at —10°C ... 

The condition of Equation (13.7) can be met only if p,j = p,n, which is the condition of transfer equilibrium between phases. Or, to put the argument differently, if the chemical potentials (escaping tendencies) of a substance in two phases differ, spontaneous transfer will occur from the phase of higher chemical potential to the phase of lower chemical potential, with a decrease in the Gibbs function of the system, until the chemical potentials are equal (see Section 10.5). For each component present in aU p phases, (p 1) equations of the form of Equation (13.7) provide constraints at transfer equilibrium. Furthermore, an equation of the form of Equation (13.7) can be written for each one of the C components in the system in transfer equUibrium between any two phases. Thus, C(p — 1) independent relationships among the chemical potentials can be written. As chemical potentials are functions of the mole fractions at constant temperamre and pressure, C(p — 1) relationships exist among the mole fractions. If we sum the independent relationships for temperature. 

The preceding examples illustrate some methods that can be used to combine data for the Gibbs functions for pure phases with information on the Gibbs function for constituents of a solution to calculate changes in the Gibbs function for chemical... 

Equation (5.30) holds for the simple case of a phase with the formula A, B)i(C, D)i. But for more complex phases the function for the Gibbs reference energy surface may be generalised by arranging the site fractions in a (f + c) matrix if there are I sublattices and c components. 

To better understand the complexity of situation, it is useful to apply some thermodynamic considerations. The separation process is governed by the change of the Gibbs function due to transfer of solute molecules between the mobile and the stationary phase. One can write... 

Interestingly, the standard entropies (and in turn heat capacities) of both phases were found to be rather similar [69,70]. Considering the difference in standard entropy between F2(gas) and the mixture 02(gas) + H2(gas) taken in their standard states (which can be extracted from general thermodynamic tables), the difference between the entropy terms of the Gibbs function relative to HA and FA, around room temperature, is about 6.5 times lower than the difference between enthalpy terms (close to 125 kJ/mol as estimated from Tacker and Stormer [69]). This indicates that FA higher stability is mostly due to the lower enthalpy of formation of FA (more exothermic than for HA), and that it is not greatly affected by entropic factors. Jemal et al. [71] have studied some of the thermodynamic properties of FA and HA with varying cationic substitutions, and these authors linked the lower enthalpy of formation of FA compared to HA to the decrease in lattice volume in FA. 

Eq.(2.2-4) is the phase rule of Gibbs. According to this rule a state with II phases in a system with N components is frilly determined (all intensive thermodynamic properties can be calculated) if a number of F of the variables is chosen, provided that g of all phases as function of pressure, temperature and composition is known. 

Fig. 4 Schematic free energy diagram for the crystallization of polyethylene from the melt showing the specific Gibbs function (chemical potential) for melt (m), hexagonal (h), and orthorhombic (o), phases as function of temperature...

Employing standard states of a single solute in a physical state of infinite dilution in the liquid stationary phase at the temperature and pressure of the system and a single solute in the perfect gas state at unit pressure and the temperature of the system for the solute in the stationary and in the gaseous phase, respectively, we obtain for the standard molar Gibbs function of sorption of solute i, AG°p(/) [19] ... 

The heat capacity of thiazole was determined by adiabatic calorimetry from 5 to 340°K by Goursot and Westrum (295,296). A glass-type transition occurs between 145 and 175 K. Melting occurs at 239.53°K (-33.62°C) with an enthalpy increment of 2292 cal mole" and an entropy increment of 9.57 cal mole" -"K". Table 1-44 summarizes the variations as a function of temperature of the most important thermodynamic properties of thiazole molar heat capacity Cp, standard entropy S°, and Gibbs function -(G°-The variation of Cp for crystalline thiazole between 145 and 175°K reveals a marked inflection that has been attributed to a gain in molecular freedom within the crystal lattice. The heat capacity of the liquid phase varies nearly linearly with temperature to 310°K, at which temperature it rises more rapidly. This thermal behavior, which is not uncommon for nitrogen compounds, has been attributed to weak intermolecular association. The remarkable agreement of the third-law ideal-gas entropy at... 

Reproducibility in the system was good, but various uncertainties limit the precision of the data. For instance, the solid phase probably contained both a and p UH3. The enthalpy difference between these species is unknown, and the heat capacity data for UH3 does not extend to the relevant temperature range (316a). Recent analyses of all the data suggest best values of -126.99 and -72.61 kJ moF for the standard enthalpy and Gibbs function changes on formation at 298 K and an S° value of 63.67 J mol 1 for UH3 (3d, 316b). 

Figure 2.7. Gibbs function, system enthalpy, and system entropy variations with the extent of reaction for the dissolution of gaseous CO2 in water C02(g) = C02(a

Structure and orientation of a Me deposit on S in the initial stage of 3D Me bulk phase formation can be either independent of or influenced by the surface structure of S, which can be modified by 2D Meads overlayer formation and/or 2D Me-S surface alloy phase formation in the UPD range. Epitaxial behavior of 2D and 3D Me phases exists if some or all of their lattice parameters coincide with those of the top layer of S. The epitaxy is determined by a minimum of the Gibbs function at constant temperature and pressure. 

Based on the Gibbs functions, the temperatures for the invariant three-phase equilibria can be calculated ... 

It follows from the discussion in this paragraph that only standard differential thermodynamic functions can be calculated from any chromatographic distribution constant defined in whatever way. Also, it is necessary to always specify the choice of the standard states for the solute in both phases of the system. Without specifying the standard states the data on the thermodynamic functions calculated from chromatographic retention data lack any sense. When choosing certain standard states it may happen that the standard differential Gibbs function is identical with another form of the differential Gibbs function, or includes such a form situations described by equations 46 and 49 may serve as examples. The same also holds true for standard differential volumes, entropies and enthalpies (compare Section 1.8.3). However, every particular situation requires a special treatment. 

The molar excess Gibbs function G T,p, x) of a liquid mixture of A -f B containing mole fraction jc of B in the liquid phase at temperature T and pressure is defined by ... 

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