Gibbs line

From the defintion of P, we see that the diamond-3-Sn transition occurs at the volume where the first common tangent (Gibbs line) can be drawn between the diamond curve and the 3-Sn curve. In Fig. 5, the Gibbs line is given by the dashed line. Thus, the points 2 and 3 label the transition volumes between the structures and the slope of the Gibbs line provides the transition pressure. 

In most cases only a single tie line is required. When several are available, the choice of which one to use is somewhat arbitrary. However, our experience has shown that tie lines which are near the middle of the two-phase region are most useful for estimating the parameters. Tie lines close to the plait point are less useful, since no common models for the excess Gibbs energy can adequately describe the flat region near the... 

The type of behavior shown by the ethanol-water system reaches an extreme in the case of higher-molecular-weight solutes of the polar-nonpolar type, such as, soaps and detergents [91]. As illustrated in Fig. Ul-9e, the decrease in surface tension now takes place at very low concentrations sometimes showing a point of abrupt change in slope in a y/C plot [92]. The surface tension becomes essentially constant beyond a certain concentration identified with micelle formation (see Section XIII-5). The lines in Fig. III-9e are fits to Eq. III-57. The authors combined this analysis with the Gibbs equation (Section III-SB) to obtain the surface excess of surfactant and an alcohol cosurfactant. 

Fig. in-12. Verification of the Gibbs equation by the radioactive trace method. Observed (o) and calculated (line) values for for aqueous sodium dodecyl sulfate solutions. (From Ref. 108.)... 

Figure A2.5.2 shows schematically the behaviour of several thennodynamic fiinctions along a constant-pressure line (shown as a dotted line in Figure A2.5.1 )—the molar Gibbs free energy G(for a one-component system the same as...

Figure A2.5.2. Schematic representation of the behaviour of several thennodynamic fiinctions as a fiinction of temperature T at constant pressure for the one-component substance shown in figure A2.5.1. (The constant-pressure path is shown as a dotted line in figure A2.5.1.) (a) The molar Gibbs free energy Ci, (b) the molar enthalpy n, and (c) the molar heat capacity at constant pressure The fimctions shown are dimensionless...

Figure A2.5.15. The molar Gibbs free energy of mixing versus mole fraetionxfor a simple mixture at several temperatures. Beeause of the synuuetry of equation (A2.5.15) the tangent lines indieating two-phase equilibrium are horizontal. The dashed and dotted eiirves have the same signifieanee as in previous figures.

Ipatieff never received the honor he coveted the most, the Nobel Prize. However, he continued to publish voluminously and, ever the practical scientist, he obtained numerous patents. He continued to receive honors in the United States and internationally. He became a member of the National Academy of Sciences and received the prestigious Gibbs medal for his many achievements. Ipatieff lived long enough to see the petroleum industry transformed with process technologies that he created and that were rooted in his early scientific research. The first platforming plant, the cuhniiiation of his life s work in catalytic research, came on line just shortly before his death in 1952. 

Therefore, if we have information on the partial molar volumes and the excess Gibbs energy of the ternary system, we can use Eqs. (119)—(122) to find the ends of the tie lines which comprise the coexistence curve. 

FIGURE 7.26 For some substances and at certain pressures, the molar Gibbs free energy of the liquid phase might never lie lower than those of the other two phases. For such substances, the liquid is never the stable phase and, at constant pressure, the solid sublimes when the temperature is raised to the point of intersection of the solid and vapor lines. 

FIGURE 27.8 Specular reflectivity for a clean Au(lOO) surface in vacuum at 310 K ( ). The solid line is calculated for an ideally terminated lattice. The dashed line is a fit to the data with a reconstmcted surface with a 25% increase in the surface density combined with a surface relaxation that increases the space between the top and next layers by 19%. In addition, the data indicate that the top layer is buckled or cormgated with a buckling amplitude of 20%. (From Gibbs et al., 1988, with permission from the American Physical Society.)... 

SCHEME 2.3 Gibbs energy profiles for the benzylation of NH3 (a), H20 (b), and H2S (c) by o-QM in the gas phase (continuous line), water-catalyzed (S4-S6) and uncatalyzed (S1-S3), and in aqueous solution (dotted line, Slaq-S6aq) optimizing both reagents and TSs in aqueous solution [B3LYP-C-PCM/6-311 +G(d,p)]. Data are taken from Ref. [13]. 

The Gibbs Ensemble MC simulation methodology [17-19] enables direct simulations of phase equilibria in fluids. A schematic diagram of the technique is shown in Fig. 10.1. Let us consider a macroscopic system with two phases coexisting at equilibrium. Gibbs ensemble simulations are performed in two separate microscopic regions, each within periodic boundary conditions (denoted by the dashed lines in Fig. 10.1). The thermodynamic requirements for phase coexistence are that each... 

While the main driving force in [43, 44] was to avoid direct particle transfers, Escobedo and de Pablo [38] designed a pseudo-NPT method to avoid direct volume fluctuations which may be inefficient for polymeric systems, especially on lattices. Escobedo [45] extended the concept for bubble-point and dew-point calculations in a pseudo-Gibbs method and proposed extensions of the Gibbs-Duhem integration techniques for tracing coexistence lines in multicomponent systems [46]. 

If solvent is added to either of the solid eutectics represented by e or e in Fig. 25a or b, the undissolved solid retains this composition while the saturated solution maintains the composition E or E, respectively. Again, Gibbs phase rule [145,146] can provide further insight into these systems. If the solid enantiomers are solvated, the compositions of the equilibrium solids are displaced symmetrically along the DS or LS axes to an extent determined by the stoichiometry of the solvates. Similarly, if the racemic compound is solvated, the stoichiometry of the equilibrium solid is displaced from R along the line RS to an extent determined by the stoichiometry of the solvate. 

The general equation for a straight line is given below with the slightly modified Gibbs Free-Energy equation as a reference AG° = - AS°T + AH° (here AH° is assumed to be constant)... 

The calculation of the number of Frenkel defects in a crystal proceeds along lines parallel to those above. The introduction of Frenkel defects causes the Gibbs energy of the crystal to change by an amount AGp ... 

The nucleation rate kjv is the inverse of the average time between two pulses. By varying the overpotential r/, the dependence of the nucleation rate on r can be obtained. In Section 10.3 we showed that for three-dimensional nucleation the Gibbs energy of formation is proportional to r) 2. A similar analysis for the two-dimensional case gives a proportionality to if1 instead (see Problem 10.1). Hence a plot of In k/y versus ry"1 should result in a straight line, which is indeed observed (see Fig. 10.8). 

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 ... 

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