Phases coexist in equilibrium

The shaded region is that part of the phase diagram where liquid and vapor phases coexist in equilibrium, somewhat in analogy to the boiling line for a pure fluid. The ordinary liquid state exists on the high-pressure, low-temperature side of the two-phase region, and the ordinary gas state exists on the other side at low pressure and high temperature. As with our earlier example, we can transform any Type I mixture... 

A phase boundary for a single-component system shows the conditions at which two phases coexist in equilibrium. Recall the equilibrium condition for the phase equilibrium (eq. 2.2). Letp and Tchange infinitesimally but in a way that leaves the two phases a and /3 in equilibrium. The changes in chemical potential must be identical, and hence... 

A two-dimensional illustration of three phases a, ft and % in equilibrium is shown in Figure 6.9. Two phases coexist in equilibrium in planes perpendicular to the lines indicated in the two-dimensional figure and all three phases coexist along a common line also perpendicular to the plane of the drawing. Each of the three two-phase boundaries, which meet at the point of contact, has a characteristic interfacial tension, e.g. ca for the interface, which tends to reduce the area of the... 

A phase diagram describes how a system reacts to changing conditions of pressure and temperature and consists of a field in which only one phase is stable, separated by boundary curves along which a combination of phases coexist in equilibrium. 

When the liquid and solid-solution phases coexist in equilibrium, the chemical potential of component A must be the same in the liquid and solid phases and similarly for and C. Therefore the sum of the chemical potentials of A and C as well as that for and C is the same in both phases, i.e.,... 

At the temperature where a phase change occurs, the two phases coexist in equilibrium and AG, the free-energy difference between the phases, is zero AG = AH — TAS = 0. Rearranging this equation gives AS = AH/T, where both AH and T are known. Remember that T must be expressed in kelvins. 

Melting point The temperature at which solid and liquid phases coexist in equilibrium. 

The point O represents a set of conditions under which all three phases coexist in equilibrium and is called the triple point. 

This behaviour has a particular importance for the soil removal process in detergency. During the oil removal from stained fabrics or hard surfaces, ternary systems occur where three phases coexist in equilibrium. As already pointed out above, in this region the interfacial tension is particularly low. Because the interfacial tension is generally the restraining force,... 

Fig. 6. Critical particle concentration, p Ip0, above which a dilute disordered phase and a concentrated ordered phase coexist in equilibrium for a sterically stabilized dispersion, as a function of the particle radius, a. System polyisobutene-stabilized silica particles in cyclohexane, 6=5 nm, T = 308 K, xi = 0.47, x2 0.10, A 4.54kT and v = 0.10.

When a system consists of saturated-liquid and saturated-vapor phases coexisting in equilibrium, the total value of any extensive property of the two-phase system is the sum of the total properties of the phases. Written for the volume, this relation is... 

For a specific polymer, critical concentrations and temperatures depend on the solvent. In Fig. 15.42b the concentration condition has already been illustrated on the basis of solution viscosity. Much work has been reported on PpPTA in sulphuric acid and of PpPBA in dimethylacetamide/lithium chloride. Besides, Boerstoel (1998), Boerstoel et al. (2001) and Northolt et al. (2001) studied liquid crystalline solutions of cellulose in phosphoric acid. In Fig. 16.27 a simple example of the phase behaviour of PpPTA in sulphuric acid (see also Chap. 19) is shown (Dobb, 1985). In this figure it is indicated that a direct transition from mesophase to isotropic liquid may exist. This is not necessarily true, however, as it has been found that in some solutions the nematic mesophase and isotropic phase coexist in equilibrium (Collyer, 1996). Such behaviour was found by Aharoni (1980) for a 50/50 copolymer of //-hexyl and n-propylisocyanate in toluene and shown in Fig. 16.28. Clearing temperatures for PpPTA (Twaron or Kevlar , PIPD (or M5), PABI and cellulose in their respective solvents are illustrated in Fig. 16.29. The rigidity of the polymer chains increases in the order of cellulose, PpPTA, PIPD. The very rigid PIPD has a LC phase already at very low concentrations. Even cellulose, which, in principle, is able to freely rotate around the ether bond, forms a LC phase at relatively low concentrations. 

Two-dimensional phase diagrams are often displayed in the form of In [p] against 1/7 (at a constant specific amount adsorbed), which provides a convenient way of indicating the conditions for the coexistence of two phases (see Figure 4.3). Indeed, the application of the Phase Rule indicates that when two adsorbed phases coexist in equilibrium, the system has one degree of freedom therefore, at constant... 

S. A system composed of ethane hydrate, water, and ethane is classed aa a two-component system when Gibbs phase rule is applied since it could be formed from water and ethane. What is the variance of this system when a solid, a liquid, and a vapor phase coexist in equilibrium If the temperature of this three-phase system is specified, would it be possible to alter the pressure without the disappaaranoe of a phase ... 

When a bicontinuous cubic lipid-water phase is mechanically fragmented in the presence of a liposomal dispersion or of certain micellar solutions e.g. bile salt solution), a dispersion can be formed with high kinetic stability. In the polarising microscope it is sometimes possible to see an outer birefringent layer with radial symmetry (showing an extinction cross like that exhibited by a liposome). However, the core of these structures is isotropic. Such dispersions are formed in ternary systems, in a region where the cubic phase coexists in equilibrium with water and the L(x phase. The dispersion is due to a localisation of the La phase outside cubic particles. The structure has been confirmed by electron microscopy by Landh and Buchheim [15], and is shown in Fig. 5.4. It is natural to term these novel structures "cubosomes". They are an example of supra self-assembly. 

This situation is typical for systems exhibiting a miscibility gap and is associated with a sufficiently large positive deviation from the ideal behavior. In the system, two liquid phases coexist in equilibrium. These two liquids of composition corresponding... 

To be specific, let the two phases coexist in equilibrium at some temperature 2 and uniform hydrostatic pressure P. Then let the temperature be kept at throughout, let the magnitude of be specified at any moment as + 3p, and let the imposed cylindrical constriction rate be Cq. Then in the equilibrium state (7, Pj), both eo and dp are zero but we envisage a series of nonequi-Ubrium stationary-interface states where e and dp balance each other, neither being zero. What will such a state be like and how might it be reached ... 

The in situ formation of microemulsions can occur in washing processes depending on the oil type and conditions. During the oil removal from hard surfaces or fabrics ternary systems occur where two or three phases coexist in equilibrium. These systems are also referred to as Windsor I or Windsor III microemulsions. The effects were studied in detail for alkyl polyglycol ethers [77]. Depending on temperature different phases exist, having a three-phase region between the temperature T and Tu (see Fig. 1.3, Chapter 1). When... 

Equation (356) is identical to the Kelvin equation (Equation (347)) for saturation vapor pressures (Pv => P°) at r = rc. The P v parameter is the critical vapor pressure, which corresponds to the vapor pressure when the drop radius, r equals rc, which is also the radius of curvature for a spherical drop having the critical size. We should note that Equation (356) is valid when two phases coexist in equilibrium. 

EquiUbrium between Solid, Liquid and Vapour. The Triple Point.—From the Phase Rule, F = n + 2 — r, it follows that when oaie component is present in three coexisting phases, the system is invariant. Such a system can exist in stable equilibrium only at one definite temperature and one definite pressure. This definite temperature and pressure at which three phases coexist in equilibrium, as an invariant system, is called a triple point. Although the commonest triple point in a one-component system is the triple point, solid, liquid, vapour (S—L—V), other triple points are also possible when, as in the case of ice, sulphur, and other substances, polymorphic forms occur. Whether or not all the triple points can be experimentally realised will, of course, depend on circumstances. We shall, in the first place, consider the triple point S—L—Y. 

The transformation of liquid to solid is called freezing, and the reverse process is called melting, or fusion. The melting point of a solid or the freezing point of a liquid is the temperature at which solid and liquid phases coexist in equilibrium. The normal melting (or freezing) point of a substance is the temperature at which a substance melts (or... 

The non-physical van der Waals isotherm may be improved using the so-called Maxwell construction. It involves drawing the horizontal section AD, for which (8p/8V)T = 0, joining the two branches of the isotherm, EA and DF, corresponding to the liquid and gaseous phase of a system, respectively. It follows from the condition of equality of chemical potentials at a critical point that the section AD should be thus selected that the areas and S2 be equal. Between the points A and D the system is nonhomo-geneous, i.e. separated into two phases coexisting in equilibrium. The... 

Figure 11.3-3 shows the vapor-liquid and liquid-liquid equilibrium behavior computed for the system of methanol and n-hexane at various temperatures. Note that two liquid phases coexist in equilibrium to temperatures of about 43°C. Since liquids are relatively incompressible, the species liquid-phase fugacities are almost independent of pressure (see Illustrations 7.4-8 and 7.4-9), so that the liquid-liquid behavior is essentially independent of pressure, unless the pressure is very high, or low enough for the mixture to vaporize (this possibility will be considered shortly). The vapor-liquid equilibrium curves for this system at various pressures are also shown in the figure. Note that since the fugacity of a species in a vapor-phase mixture is directly proportional to pressure, the VLE curves are a function of pressure, even though the LLE curves are not. Also, since the methanol-hexane mixture is quite nonideal, and the pure component vapor pressures are similar in value, this system exhibits azeotropic behavior. 

Clearly there is some pressure at which the three curves intersect at the same temperature. This temperature and pressure define the triple point all three phases coexist in equilibrium at the triple point. 

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