Thermodynamic phase-equilibrium system

Alternatively, thermodynamic phase equilibrium in a model system can be evaluated by beginning the simulation with two (or more) phases in the same simulation volume, in direct physical contact (i.e., with a solid-fluid interface). This approach has succeeded [79], but its application can be problematic. Some of the issues have been reviewed by Frenkel and McTague [80]. Certainly the system must be large (recent studies [79,81,82] have employed from 1000 up to 65,000 particles) to permit the bulk nature of both phases to be represented. This is not as difficult for solid-liquid equilibrium as it is for vapor-liquid, because the solid and liquid densities are much more alike (it is a weaker first-order transition) and the interfacial free energy is smaller. However, the weakness of the transition also implies that a system out of equilibrium experiences a smaller driving force to the equilibrium condition. Consequently, equilibration of the system, particularly at the interface, may be slow. 

In the case of true thermodynamic phase equilibrium, in which the absolute minimum is attained for the system Gibbs free energy at given T and p, the solubility calculation is performed following the classical thermodynamic result which imposes the equality between the equilibrium chemical potential of the penetrant in the polymeric mixture and in the external phase Th equilibrium solute content, and... 

The PDLC system performance depends strongly on the final morphology of the liquid crystal domains dispersed inside the polymer matrix. The size, shape and distribution of liquid crystal domains are generally dictated not only by thermodynamic phase equilibrium, but also by the type of material used and by interfacial interactions [58-62]. 

Why must a chapter on macroemulsions consider the phase behavior and thermodynamics of oil-water-surfactant systems at all Indeed, macroemulsions are nonequilibrium systems, whereas the phase behavior concerns thermodynamically stable equilibrium systems — microemulsions and liquid crystals. The reason for it is that microemulsions and macroemulsions normally co-exist in the same system and their properties are interrelated. The surfactant monolayers covering micelles and macroemulsion droplets are in thermodynamic equilibrium. This equilibrium leads to many peculiar effects. 

More than one phase can coexist within the system at equilibrium. When this phenomenon occurs, a phase boundary separates the phases from each other. One of the major topics in chemical thermodynamics, phase equilibrium, is used to determine the chemical compositions of the different phases that coexist in a given mixture at a specified temperature and pressure. 

Vapor/liquid equilibrium (XT E) relationships (as well as other interphase equihbrium relationships) are needed in the solution of many engineering problems. The required data can be found by experiment, but such measurements are seldom easy, even for binaiy systems, and they become rapidly more difficult as the number of constituent species increases. This is the incentive for application of thermodynamics to the calculation of phase-equilibrium relationships. 

The N equations represented by Eq. (4-282) in conjunction with Eq. (4-284) may be used to solve for N unspecified phase-equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. The types of problems encountered for nonelectrolyte systems at low to moderate pressures (well below the critical pressure) are discussed by Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996). 

The distribution coefficient is an equilibrium constant and, therefore, is subject to the usual thermodynamic treatment of equilibrium systems. By expressing the distribution coefficient in terms of the standard free energy of solute exchange between the phases, the nature of the distribution can be understood and the influence of temperature on the coefficient revealed. However, the distribution of a solute between two phases can also be considered at the molecular level. It is clear that if a solute is distributed more extensively in one phase than the other, then the interactive forces that occur between the solute molecules and the molecules of that phase will be greater than the complementary forces between the solute molecules and those of the other phase. Thus, distribution can be considered to be as a result of differential molecular forces and the magnitude and nature of those intermolecular forces will determine the magnitude of the respective distribution coefficients. Both these explanations of solute distribution will be considered in this chapter, but the classical thermodynamic explanation of distribution will be treated first. 

The generic case is a subsystem with phase function x(T) that can be exchanged with a reservoir that imposes a thermodynamic force Xr. (The circumflex denoting a function of phase space will usually be dropped, since the argument T distinguishes the function from the macrostate label x.) This case includes the standard equilibrium systems as well as nonequilibrium systems in steady flux. The probability of a state T is the exponential of the associated entropy, which is the total entropy. However, as usual it is assumed (it can be shown) [9] that the... 

Ohmura, R. Uchida, T. Takeya, S. Nagao, J. Minagawa, H. Ebinuma, T. Narita, H. (2003 a). Clathrate hydrate formation in (methane + water + methylcyclohexanone) systems the first phase equilibrium data. J. Chem. Thermodynamics, 35, 2045-2054. 

The mixture CMC is plotted as a function of monomer composition in Figure 1 for an ideal system. Equation 1 can be seen to provide an excellent description of the mixture CMC (equal to Cm for this case). Ideal solution theory as described here has been widely used for ideal surfactant systems (4.6—18). Equation 2 can be used to predict the micellar surfactant composition at any monomer surfactant composition, as illustrated in Figure 2. This relation has been experimentally confirmed (ISIS) As seen in Figure 2, for an ideal system, if the ratio XA/yA < 1 at any composition, it will be so over the entire composition range. In classical phase equilibrium thermodynamic terms, the distribution coefficient between the micellar and monomer phases is independent of composition. 

Three-component systems, or ternary systems, are fundamentally no different from two-component systems in terms of their thermodynamics. Phases in eqnilibrium must still meet the equilibrium criteria [Eqs. (2.14)-(2.16)], except that there may now be as many as five coexisting phases in eqnilibrinm with each other. The phase rule still... 

VOCs), and to a decrease in production yields. Quantitation of these phenomena and determination of material balances and conversion yields remain the bases for process analysis and optimisation. Two kinds of parameters are required. The first is of thermodynamic nature, i.e. phase equilibrium, which requires the vapour pressure of each pure compound involved in the system, and its activity. The second is mass-transfer coefficients related to exchanges between all phases (gas and liquids) existing in the reaction process. 

The following criterion of phase equilibrium can be developed from the first and second laws of thermodynamics the equilibrium state for a closed multiphase system of constant, uniform temperature and pressure is the state for which the total Gibbs energy is a minimum, whence... 

For a PVT system of uniform T and P containing N species and 71 phases at thermodynamic equilibrium, the intensive state of the system is hilly determined by the values of T, P, and the (N — 1) independent mole fractions for each of the equilibrium phases. The total number of these variables is then 2 + tt(N — 1). The independent equations defining or constraining the equilibrium state are of three types equations 218 or 219 of phase-equilibrium, N(77 — 1) in number equation 245 of chemical reaction equilibrium, r in number and equations of special constraint, s in number. The total number of these equations is Ar(7r — 1) + r + s. The number of equations of reaction equilibrium r is the number of independent chemical reactions, and may be determined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

In summary, whether a reaction equilibrium or a phase equilibrium approach is adopted depends on the size of the micelles formed. In aqueous systems the phase equilibrium model is generally used. In Section 8.5 we see that thermodynamic analyses based on either model merge as n increases. Since a degree of approximation is introduced by using the phase equilibrium model to describe micellization, micelles are sometimes called pseudophases. 

It is apparent that CMC values can be expressed in a variety of different concentration units. The measured value of cCMC and hence of AG c for a particular system depends on the units chosen, so some uniformity must be established. The issue is ultimately a question of defining the standard state to which the superscript on AG C refers. When mole fractions are used for concentrations, AG c directly measures the free energy difference per mole between surfactant molecules in micelles and in water. To see how this comes about, it is instructive to examine Reaction (A) —this focuses attention on the surfactant and ignores bound counterions — from the point of view of a phase equilibrium. The thermodynamic criterion for a phase equilibrium is that the chemical potential of the surfactant (subscript 5) be the same in the micelle (superscript mic) and in water (superscript W) n = n. In general, pt, = + RTIn ah in which... 

The thermodynamic excess functions for the 2-propanol-water mixture and the effects of lithium chloride, lithium bromide, and calcium chloride on the phase equilibrium for this binary system have been studied in previous papers (2, 3). In this paper, the effects of lithium perchlorate on the vapor-liquid equilibrium at 75°, 50°, and 25°C for the 2-propanol-water system have been obtained by using a dynamic method with a modified Othmer still. This system was selected because lithium perchlorate may be more soluble in alcohol than in water (4). 

The concept of phase (cpaaLs appearance ) underlies some of the most remarkable phenomena of thermodynamics, and the complete elucidation of phase equilibrium phenomena represents the most famous achievement of Gibbsian thermodynamics. This chapter describes how the Gibbs principles extend almost effortlessly to such complex multiphase systems. 

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