Gibbs space

In this section, we wish to derive the Gibbs-Duhem equation, the fundamental relationship between the allowed variations dRt of the intensive properties of a homogeneous (singlephase) system. Paradoxically, this relationship (which underlies the entire theory of phase equilibria to be developed in Chapter 7) is discovered by considering the fundamental nature of extensive properties Xu as well as the intrinsic scaling property of the fundamental equation U = U(S, V, n, n2,. .., nc) that derives from the extensive nature of U and its Gibbs-space arguments. 

Figure 9.1 Schematic Gibbs-space USV model in orthogonal (a) and nonorthogonal (b) axis systems, showing distortion of surface metric properties (e.g., distances, angles in marked triangle) with arbitrary change of axes.

For the entropy and internal energy, the canonical variables consist of extensive parameters. For a simple system, the extensive properties are S, U, and V. and the fundamental equations define a fundamental surface of entropy S = S(U,V) in the Gibbs space of S, U, and V. 

This relationship fully describes all of the stable equilibrium states of a simple system with n components. However, there is no single fundamental equation governing the properties of all materials. The fundamental equation is represented by a surface in (3 + n) dimensional space. Quasi-static processes can be represented by a curve on this surface. The points on this surface represent stable equilibrium states of this simple system. For the entropy and internal energy, the canonical variables consist of extensive parameters. For a simple system, the extensive properties are S, U, and V, and the fundamental equations define a fundamental surface of entropy S = S U,V) in the Gibbs space of S, U, and V. 

Let us cortsider a collection of N iderrtical qrrantum objects each with three degrees of freedom (e.g. one punctual gas molectrle). To define the movemerrt of these objects, we must first define the three spatial coordinates and the three qrrantities of movement (or three velocity componerrls), i.e. a total of six coordinates in a space with six dimensions. In this space, called the quarrtum space, an object is defined by a poirrt. For N objects, we rrse an hyperspace with 6N dimensions, called the Gibbs space of phase. In this hyperspace, it is a system state which is represented by a poirrt Each ensemble defined constitutes a complexion of the system. 

Another method of simulating chemical reactions is to separate the reaction and particle displacement steps. This kind of algorithm has been considered in Refs. 90, 153-156. In particular. Smith and Triska [153] have initiated a new route to simulate chemical equilibria in bulk systems. Their method, being in fact a generalization of the Gibbs ensemble Monte Carlo technique [157], has also been used to study chemical reactions at solid surfaces [90]. However, due to space limitations of the chapter, we have decided not to present these results. 

The direction of change of pressure occurring in the distillation of a mixture of changing composition is fixed by a very general rule, deduced by Gibbs (1876), and used by Konowalow as a consequence of some experiments of Pliicker (1854), who found that the vapour pressure over a mixture of alcohol and water is all the less the larger the space which the vapour has to saturate. The rule may be stated as follows —... 

If the property evaluated, for instance, the critical micelle concentration, can be approximated by a suitable plot, it is depicted in the ternary system as a concave area (e.g., cM area) located in the space above the Gibbs triangle as the basis for the distinct concentrations. The property axis describing the cM data stands vertically on the base triangle. 

It is beyond our control how the cross-links are spaced along the polymer chains during the vulcanization process. This extraordinary important fact demands a generalization of the Gibbs formula in statistical mechanics for amorphous materials that have fixed constraints of which the exact topology is unknown. Details of a modified Gibbs formula of polymer networks can be found in the pioneering paper of Deam and Edwards [13]. 

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.)... 

One consequence of the positivity of a is that A A < (AU)0. If we repeat the same reasoning for the backwards transformation, in (2.9), we obtain A A > (AU)V These inequalities, known as the Gibbs-Bogoliubov bounds on free energy, hold not only for Gaussian distributions, but for any arbitrary probability distribution function. To derive these bounds, we consider two spatial probability distribution functions, F and G, on a space defined by N particles. First, we show that... 

We have now derived the phase boundary between the two liquids. By analogy with our earlier examples, the two phases may exist as metastable states in a certain part of the p,T potential space. However, at some specific conditions the phases become mechanically unstable. These conditions correspond to the spinodal lines for the system. An analytical expression for the spinodals of the regular solution-type two-state model can be obtained by using the fact that the second derivative of the Gibbs energy with regards to xsi)B is zero at spinodal points. Hence,... 

Figure 6.1 Search for the minimum of the Gibbs function in a two-component space (nn and ni2 are mole numbers) with the mass conservation constraints Bn = q. The search direction is the projection of the gradient onto the constraint subspace. Minimum is attained when the gradient is orthogonal to the constraint direction, which is the geometrical expression of the Lagrange multiplier methods.

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