Phase equilibria convexity

Inequalities (6) and (7) show that the surface is, at every point which corresponds with a homogeneous phase in stable equilibrium, convex downwards in every direction. 

However if, over some range of compositions, the mixture splits into two phases, then the single-phase equilibrium curve for g "(xi) will not be convex over all Xj. Similarly, the monotonicity of f x ) will be disrupted either by oscillations or by branching. These possibilities appear in Figure 8.13 oscillations occur in the f xi) curves for 30 and 60, while at 10 bar, the /i(xi) curve has divided into two distinct branches. These phenomena are caused by bifurcations in either the equation of state or the fugacity equation or both. Here we use those possibilities to identify four classes of instabilities that can lead to vapor-liquid phase separations in binary mixtures. 

All points on the two tangents HRi, HR2, to the curve of solutions represent heterogeneous systems composed of solid hydrate in contact with solutions. If the curve between Ri and R2 is convex the heterogeneous systems are stable, and inversely. At a given temperature and pressure the hydrate can be in equilibrium with two liquid phases of different composition, one containing relatively more, the other relatively less, salt than the hydrate. With rise of temperature the form of the curve and the altitude of H change ... 

The first one is that this particular form of H can also be used to prove the approach to equilibrium in the case of Boltzmann s kinetic equation for dilute gases. The Boltzmann equation is nonlinear and a different technique is needed to prove that all solutions tend to equilibrium. This technique is based on (5.6) other convex functions cannot be used. Incidentally, the Boltzmann equation is not a master equation for a probability density, but an evolution equation for the particle density in the six-dimensional one-particle phase space ( /i-space ). The linearized Boltzmann equation, however, has the same structure as a master equation (compare XIV.5). 

Solution. The outer interface is concave and the inner interface is convex with respect to the 0 phase. The concentrations of B maintained in equilibrium in the a phase at the outer interface, c (ft, and at the inner interface, c f, are given by Eq. 15.4. The concentration difference across the shell is therefore... 

A simple derivation of the Kelvin equation is presented by Broekhoff and van Dongen [16]. Imagine a gas B in physical adsorption equilibrium above a flat, a convex, and a concave surface, respectively (see Fig. 12.8). Considering a transfer of dN moles of vapour to the adsorbed phase at constant pressure and temperature, equilibrium requires that there will be no change in the free enthalpy of the system. 

It is well known (Defay and Prigogine, 1951) that a spherical interface of radius of curvature r and surface tension y can maintain mechanical equilibrium between two fluids at different pressures p" and p. The phase on the concave side of the interface experiences a pressure p" which is greater than that on the convex side. The mechanical equilibrium condition is given by the Laplace equation ... 

The local equilibrium surface alloy configuration and structure may be found by minimization of the surface free energy, or if several different phases may exist, by finding a convex hull of the lowest free energies of different phases at different alloy compositions (at T=0), or more generally by a common-tangent construction which is completely analogous to the usual treatment of the bulk systems. The procedure is illustrated in Fig. 4. 

Figure 18.6 shows two phases, 1 and 2, separated by an interface that is plane for the most part but has a portion in which phase 2 is convex the levels of the interface are different under the plane and curved portions. The densities of the two phases are and P2. Let Pi be the pressure in phase 1 at the plane surface separating the two phases this position is taken as the origin (z = 0) of the z-axis, which is directed downward. The pressures at the other positions are as indicated in the figure and p 2 are the pressures just inside phases 1 and 2 at the curved interface Pi and p 2 are related by Eq. (18.9). The condition of equilibrium is that the pressure at the depth z, which lies below both the plane and curved parts of the interface, must have the same value everywhere. Otherwise, at depth z, a flow of material would occur from one region to another. Equality of the pressures at the depth z requires that... 

AH estimations of the pore size of a material from its gas adsorption isotherm measurements are based on the well-known Kelvin equation suggested more than 100 years ago. It considers an equilibrium between the vapor phase and the bulk liquid at a constant temperature and relates the relative vapor pressure p/p to the radius r of the convex (plus) or concave (minus) spherical meniscus of the Hquid placed in a capillary ... 

In Figure 5.6, Equation (5.24) is plotted for a = 0.4, P = 0.35, T = 50°C. This has the UCST convex upward and LCST concave downward. The phase diagram is circular in nature from the standpoint of equilibrium. Eor stability. 

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