The Gibbs Energy and Phase Transitions

23 At —78.5°C, the vapor pressure of solid carbon dioxide is equal to 760 torr. The triple point is at 216.55 K and 5.112 atm. Find the average enthalpy change of sublimation. 

24 The normal boiling point of oxygen is 90.18 K. The vapor pressure at 100.0 K is equal to 2.509 atm. Find the enthalpy change of vaporization. 

25 The following data give the vapor pressure of liquid aluminum as a function of temperature  

Using a linear least-squares procedure, find the enthalpy change of vaporization of aluminum. 

26 Find the pressure necessary to lower the freezing temperature of wato- to — 10.00°C. 

---

Describe the general features of the Debye-Hiickel theory of 5 6 Review the concepts in Chapters 1 through 5 and prepare a electrolyte solutions. summary of the experimental and calculational methods that can be. , . .. used to measure or estimate the Gibbs energies of phase transitions 5.2 Describe the mechanism or proton conducbonm water., 0- r and chemical reactions. ... 

Phase transitions are classified according to the partial derivatives of the Gibbs energy. Ordinary phase transitions such as vaporizations, freezings, and so on, are called first-order phase transitions, which means that at least one of the first derivatives dG /dT) p or (dGta/dP)T is discontinuous at the phase transition. In most first-order transitions, both of these derivatives are discontinuous. From Chapter 4 we know that (9Gm/9r)/> is equal to -5m and that (9Gm/9F)r is equal to Fm- Figure 5.7 shows schematically the molar volume as a function of pressure as it would appear for a solid-liquid or a solid-solid transition. Figure 5.8 shows schematically the molar entropy as a function of temperature as it would appear for a liquid-vapor transition. 

The transitions between phases discussed in Section 10.1 are classed as first-order transitions. Ehrenfest [25] pointed out the possibility of higher-order transitions, so that second-order transitions would be those transitions for which both the Gibbs energy and its first partial derivatives would be continuous at a transition point, but the second partial derivatives would be discontinuous. Under such conditions the entropy and volume would be continuous. However, the heat capacity at constant pressure, the coefficient of expansion, and the coefficient of compressibility would be discontinuous. If we consider two systems, on either side of the transition point but infinitesimally close to it, then the molar entropies of the two systems must be equal. Also, the change of the molar entropies must be the same for a change of temperature or pressure. If we designate the two systems by a prime and a double prime, we have... 

First-order phase transitions, as is well known, are characterized by a continuous change in the Gibbs energy and by a stepwise change in its first derivatives, as... 

In order to examine the possible relationship between the bulk thermodynamics of binary transition metal-aluminum alloys and their tendency to form at underpotentials, the room-temperature free energies of several such alloys were calculated as a function of composition using the CALPHAD (CALculation of PHAse Diagrams) method [85]. The Gibbs energy of a particular phase, G, was calculated by using Eq. (14),... 

Figure 1. Chemical potentials of the three phases of matter (H, Q, and Q ), as defined by Eq. (2) as a function of the total pressure (left panel) and energy density of the H- and Q-phase as a function of the baryon number density (right panel). The hadronic phase is described with the GM3 model whereas for the Q and Q phases is employed the MIT-like bag model with ms = 150 MeV, B = 152.45 MeV/fm3 and as = 0. The vertical lines arrows on the right panel indicate the beginning and the end of the mixed hadron-quark phase defined according to the Gibbs criterion for phase equilibrium. On the left panel P0 denotes the static transition point.

As mentioned earlier, the Gibbs energy of adsorption can be analyzed using one of two independent electrical variables potential or charge density. The problem was discussed by Parsons and others, but it was not unequivocally solved because both variables are interconnected. Recent studies of the phase transition occurring at charged interfaces, performed at a controlled potential, show that if the potential is... 

From a thermodynamic viewpoint, we may imagine that, in an actinide metal, the model of the solid in which completely itinerant and bonding 5 f electrons exist and that in which the same electrons are localized, constitute the descriptions of two thermodynamic phases. The 5f-itinerant and the 5 f-localized phases may therefore have different crystal properties a different metallic volume, a different crystal structure. The system will choose that phase which, at a particular T and p (since we are dealing with metals, the system will have only one component) has the lower Gibbs free-energy. A phase transition will occur then the fugacity in the two possible phases is equal e.g. the pressure. To treat the transition, therefore, the free energies and the pressures of the two phases have to be compared. We recall that ... 

The common characteristics of phase transitions are that the Gibbs energy is continuous. Although the conditions of equilibrium and the continuity of the Gibbs energy demand that the chemical potential must be the same in the two phases at a transition point, the molar entropies and the molar volumes are not. If, then, we have two such phases in equilibrium, we have a set of two Gibbs-Duhem equations, the solution of which gives the Clapeyron equation (Eq. (5.73))... 

---