Conditions for phase equilibrium

Two copolymers are named B and C (to keep A for the solvents. Sect. 3). The phase equilibrium conditions for two coexisting phases (multiphase equilibria are again neglected) in continuous thermodynamics [35,40] read... 

The hquid-liquid equilibrium of blends containing more than two copolymers characterized by multivariate distribution functions can be treated by a straightforward extension of the methods discussed in the preceding part of this chapter and also in Sect. 3.2. Therefore, only a brief account will be presented here. The phase equilibrium conditions for two coexisting phases read... 

The application of continuous thermodynamics to block copolymer systems appears to be a difficult task. This is mainly due to the sophisticated thermodynamic models which have to be used, i.e. to the calculation of f and fg. Since there have been only first attempts [94], we will restrict ourselves to the case in which one diblock copolymer ensemble B(aP) characterized by a divariate distribution function is dissolved in one solvent A. The phase equilibrium conditions for this case are analogous to these for random copolymers. Thus, we have... 

With the phase equilibrium condition for even surfaces P (P. T) = m (P. T) and the relationships... 

Here, the phase equilibrium condition for the copolymer holds for all polymer species within the total segment number and chemical composition range of the system. This equation is valid for the total interval of the values of the identification variables r and y found in the system. Replacing the segment-molar chemical potentials for the solvents in (7) according to (2) and rearranging results in ... 

We then note that the equilibrium condition for reaction (3.7) now taking place not only at the tpb (three phase boundaries), but over the entire gas exposed Pt electrode surface is very similar to Eq. (3.28), i.e. 

Equilibrium conditions for the synthesis of intermetallic phases and compounds are summarized as a function of temperature and composition in the form of phase diagrams. Consequently, in the following subsections, phase relationships for group-IIA-group-IB metal systems are reviewed. Phase diagrams in ref. 1 are used as a baseline work published before this compilation is not specifically referred to, but that reported subsequently is used, as appropriate, to modify or replace these phase diagrams. 

Henry and Fauske (1971) developed a model for critical flow in nozzles and short tubes, which allows for nonequilibrium effects and considers a two-phase mixture upstream of the break by using an empirical correlation to relate actual dXIdp to the value (flXJdp) under equilibrium conditions. For a dispersed flow, they assumed that... 

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... 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

The equilibrium conditions for systems with curved interfaces [3] are in part identical to those defined earlier for heterogeneous phase equilibria where surface effects where negligible ... 

To establish the equilibrium conditions for pressure we will consider a movement of the dividing surface between the two phases a and [i. The dividing surface moves a distance d/ along its normal while the entropy, the total volume and the number of moles n, are kept constant. An infinitesimal change in the internal energy is now given by... 

Equation (6.27) is the Laplace equation, or Young-Laplace equation, which defines the equilibrium condition for the pressure difference over a curved surface. In Section 6.2 we will examine the consequences of surface or interface curvature for some important heterogeneous phase equilibria. 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

The equilibrium condition for the distribution of one solute between two liquid phases is conveniently considered in terms of the distribution law. Thus, at equilibrium, the ratio of the concentrations of the solute in the two phases is given by CE/CR = K, where K1 is the distribution constant. This relation will apply accurately only if both solvents are immiscible, and if there is no association or dissociation of the solute. If the solute forms molecules of different molecular weights, then the distribution law holds for each molecular species. Where the concentrations are small, the distribution law usually holds provided no chemical reaction occurs. 

In chapter 2 we defined equilibrium conditions for coexisting phases ... 

Why can the equilibrium condition for the pure phase equilibria of Model 1 be written as... 

For very many liquids, the entropy of vaporization at the normal boiling point is approximately 21 cal/mole °C water is not typical. The units for changes in entropy are the same as those for molar heat capacity, and care must be used to avoid confusion. When referring to an entropy change, a cal/mole °C is often called an entropy unit, abbreviated e.u. In order to avoid later misunderstanding, note now that this method of calculating AS from A HIT is valid only under equilibrium conditions. For transitions, for example, this method can be used only at temperatures where the two phases in question can coexist in equilibrium with each other. 

The equilibrium conditions for phase equilibria can be derived in the simplest way using the Gibbs energy G. According to the second law of thermodynamics, the total Gibbs energy of a closed system at constant temperature and pressure is minimum at equilibrium. If this condition is combined with the condition that the total number of moles of component i is constant in a closed system... 

Consider the reaction C2H4(e)+ HzO(w) - C2HsOH(a) at 200 °C and 34.5 bar. At these conditions the liquid phase is mixture of alcohol and water in which almost no ethene is dissolved and the vapour phase is a mixture of the three reactants. The equilibrium composition can be calculated from the equilibrium condition for the reaction in the gas phase together with the vapour liquid equilibrium conditions ... 

During vaporization of non-stoichiometric refractory carbides each element vaporizes at a different rate which is dependent on surface composition or relevant activities at the surface. When the initial bulk composition is near C/M = 1, the vaporization of C is much greater than that of M. As a result, the surface C content decreases and eventually approaches a constant value, which we will call the steady-state CVC (ssCVC). At the ssCVC, the vapor composition is nearly equal to the initial bulk composition. As C diffusion to the vaporizing surface reduces the C content of the bulk material, the surface composition asymptotically approaches the equilibrium CVC (eCVC). The rate at which eCVC is approached depends on the relative magnitudes of C vaporization and diffusion. When the eCVC has been reached, the surface and bulk C/M ratios are equal to the vapor composition. The intersection of the solid eCVC map with the solidus boundary of the monocarbide phase determines where melting occurs under equilibrium conditions for a particular atmosphere. 

We turn our attention in this chapter to systems in which chemical reactions occur. We are concerned not only with the equilibrium conditions for the reactions themselves, but also the effect of such reactions on phase equilibria and, conversely, the possible determination of chemical equilibria from known thermodynamic properties of solutions. Various expressions for the equilibrium constants are first developed from the basic condition of equilibrium. We then discuss successively the experimental determination of the values of the equilibrium constants, the dependence of the equilibrium constants on the temperature and on the pressure, and the standard changes of the Gibbs energy of formation. Equilibria involving the ionization of weak electrolytes and the determination of equilibrium constants for association and complex formation in solutions are also discussed. 

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