Gibbs energy phase transition

These criteria can be used to get information on the coefficients of eq. (2.48). In the high-symmetry phase, stable above the transition temperature, the order parameter r= 0 and the equilibrium conditions imply that the two first constants in the polynomial expansion are restricted to a = 0 and b > 0. If we assume that b < 0, the low-symmetry phase is stable since r > 0 then implies that AtrsG < 0. The transitional Gibbs energy is thus reduced to... 

The Landau theory predicts the symmetry conditions necessary for a transition to be thermodynamically of second order. The order parameter must in this case vary continuously from 0 to 1. The presence of odd-order coefficients in the expansion gives rise to two values of the transitional Gibbs energy that satisfy the equilibrium conditions. This is not consistent with a continuous change in r and thus corresponds to first-order phase transitions. For this reason all odd-order coefficients must be zero. Furthermore, the sign of b must change from positive to negative at the transition temperature. It is customary to express the temperature dependence of b as a linear function of temperature ... 

It is easily shown that a first-order phase transition is obtained for cases were d < 0, whereas behaviour at the borderline between first- and second-order transitions, tricritical behaviour, is obtained for d = 0. In the latter case the transitional Gibbs energy is... 

Figure 4.3 Behavior of thermodynamic variables at T for an idealized phase transition (a) Gibbs free energy and (b) entropy and volume.

Fig. 16. Gibbs energy-temperature diagram if FCC and ECC are present in the system. Ai-isotropic (undeformed) melt, A2-deformed melt (nematic phase) points 1 and 4 - melting temperatures of FCC and ECC under unconstrained conditions (transition into isotropic melt) points V and 2 -melting temperatures of FCC and ECC under isometric conditions (transition into nematic phase), point 3 - melting temperature of nematic phase (transition into isotropic melt but not completely randomized)...

Smit et al. [19] used the partition function given by (10.4) and a free energy minimization procedure to show that, for a system with a first-order phase transition, the two regions in a Gibbs ensemble simulation are expected to reach the correct equilibrium densities. 

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),... 

Second-order phase transitions are those for which the second derivatives of the chemical potential and of Gibbs free energy exhibit discontinuous changes at the transition temperature. During second-order transitions (at constant pressure), there is no latent heat of the phase change, but there is a discontinuity in heat capacity (i.e., heat capacity is different in the two... 

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... 

The X transition in liquid helium shown in Figures 11.5 and 11.6 is a second-order transition. Most phase transitions that follow the Clapeyron equation exhibit a nonzero value of A5m and AYmi that is, they show a discontinuity in 5 and Fm. the first derivatives of the Gibbs free energy Gm- Thus, they are caHA A first-order transitions. In contrast, the X transition shows a zero value of A5m and AVm and exhibits discontinuities in the second derivatives of Gm, such as the heat capacity Cpm-... 

The excess thermodynamic properties correlated with phase transitions are conveniently described in terms of a macroscopic order parameter Q. Formal relations between Q and the excess thermodynamic properties associated with a transition are conveniently derived by expanding the Gibbs free energy of transition in terms of a Landau potential ... 

Salje (1985) interpreted overlapping (displacive plus Al-Si substitutional) phase transitions in albite in the light of Landau theory (see section 2.8.1), assigning two distinct order parameters Q n and to displacive and substitutional disorder and expanding the excess Gibbs free energy of transition in the appropriate Landau form ... 

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 ... 

Paul Ehrenfest suggested a widely used classification of thermodynamic transition phenomena according to the lowest derivative of Gibbs free energy that exhibits a mathematical discontinuity at the phase transition. 

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