Phase equilibrium defined

The activity coefficient y can be defined as the escaping tendency of a component relative to Raonlt s law in vapor-liqnid eqnihbrinm (see Sec. 4 in this handbook or Null, Phase Equilibrium in Process Design, Wiley-Interscience, 1970). 

Note that this equation holds for any component in a multi-component mixture. The integral on the right-hand side can only be evaluated if the vapor mole fraction y is known as a function of the mole fraction Xr in the still. Assuming phase equilibrium between liquid and vapor in the still, the vapor mole fraction y x ) is defined by the equilibrium curve. Agitation of the liquid in tire still and low boilup rates tend to improve the validity of this assumption. 

The result of the establishment of phase equilibrium is the formation of differences of inner and outer potentials of both phases. These differences are defined by the equations... 

The most broadly recognized theorem of chemical thermodynamics is probably the phase rule derived by Gibbs in 1875 (see Guggenheim, 1967 Denbigh, 1971). Gibbs phase rule defines the number of pieces of information needed to determine the state, but not the extent, of a chemical system at equilibrium. The result is the number of degrees of freedom Np possessed by the system. 

Phase equilibrium requires that A2 = Al and hence that the integral vanish. All conditions are satisfied if the points 1 and 2 are located such that the areas A = B. This geometry defines the Maxwell construction. It shows that stable liquid and vapour states correspond to minima in free energy and that AL = Ay when the external pressure line cuts off equal areas in the loops of the Van der Waals isotherm. At this pressure that corresponds to the saturated vapour pressure, a first-order phase transition occurs. 

The Clausius-Clapeyron equation provides a relationship between the thermodynamic properties for the relationship psat = psat(T) for a pure substance involving two-phase equilibrium. In its derivation it incorporates the Gibbs function (G), named after the nineteenth century scientist, Willard Gibbs. The Gibbs function per unit mass is defined... 

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 ... 

In bulk matter the quark-hadron mixed phase begins at the static transition point defined according to the Gibbs criterion for phase equilibrium... 

Figure 1. Chemical potentials of the three phases of matter (H, Q, and Q ), as defined by Eq. (2) as a function of the total pressure (left panel) and energy density of the H- and Q-phase as a function of the baryon number density (right panel). The hadronic phase is described with the GM3 model whereas for the Q and Q phases is employed the MIT-like bag model with ms = 150 MeV, B = 152.45 MeV/fm3 and as = 0. The vertical lines arrows on the right panel indicate the beginning and the end of the mixed hadron-quark phase defined according to the Gibbs criterion for phase equilibrium. On the left panel P0 denotes the static transition point.

Mixture property Define the model to be used for liquid activity coefficient calculation, specify the binary mixture (composition, temperature, pressure), select the solute to be extracted, the type of phase equilibrium calculation (VLE or LLE) and finally, specify desired solvent performance related properties (solvent power, selectivity, etc.)... 

Figure 2.12a. Building blocks of binary phase diagrams examples of single-phase (two-variant) and two-phase (mono-variant) fields. In the figure the indication is given of the phases existing in the various fields and respectively of their number. The phase equilibrium composition in the two-phase fields is defined by the boundary (saturation) lines of the single-phase regions. (Pt), (Ag),...

Sorption/desorption is one of the most important processes influencing movemement of organic pollutants in natural systems. Sorption with reference to a pollutant is its transfer from the aqueous phase to the solid phase on the other hand, desorption is its transfer from the solid phase to the aqueous phase. Similar to all interphase mass-transfers, the sorption/ desorption process can be defined by the final-phase equilibrium of the pollutant at the aqueous-solid phase interface and the time required to approach final equilibrium. 

Now consider the phase equilibrium of a mixture of isotopes (see Fig. 7.18c for example) with isotopic analyses for each phase carried out under equilibrium conditions. Labeling the isotopomers in the two component case as primed and unprimed, as before, the fractionation factor, a and a" = 1/a, are defined... 

It is also worthwhile to make some distinctions between methods of calculating phase equilibrium. For many years, equilibrium constants have been used to express the abundance of certain species in terms of the amounts of other arbitrarily chosen species, see for example Brinkley (1946,1947), Kandliner and Brinkley (1950) and Krieger and White (1948). Such calculations suffer significant disadvantages in that some prior knowledge of potential reactions is often necessary, and it is difficult to analyse the effect of complex reactions involving many species on a particular equilibrium reaction. Furthermore, unless equilibrium constants are defined for all possible chemical reactions, a true equilibrium calculation caiuiot be made and, in the case of a reaction with 50 or 60 substances present, the number of possible reactions is massive. 

In a ternary isothermal section a similar procedure is used where an alloy is stepped such that its composition remains in a two-phase field. The three-phase field is now exactly defined by the composition of the phases in equilibrium and this also provides the limiting binary tie-lines which can used as start points for calculating the next two-phase equilibrium. 

The KTTS depends upon an absolute zero and one fixed point through which a straight line is projected. Because they are not ideally linear, practicable interpolation thermometers require additional fixed points to describe their individual characteristics. Thus a suitable number of fixed points, ie, temperatures at which pure substances in nature can exist in two- or three-phase equilibrium, together with specification of an interpolation instrument and appropriate algorithms, define a temperature scale. The temperature values of the fixed points are assigned values based on adjustments of data obtained by thermodynamic measurements such as gas thermometry. 

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 this section we consider the thermodynamics of micellization from two points of view the law of mass action and phase equilibrium. This will reveal the equivalency of the two approaches and the conditions under which this equivalence applies. In addition, we define the thermodynamic standard state, which must be understood if derived parameters are to be meaningful. 

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 defining features of phase diagrams are the phase boundaries that delineate phase domains and mark the conditions of coexistence with adjacent phases. Theoretical description of a phase diagram is therefore tantamount to finding the equations of coexistence that describe these phase boundaries. For a simple phase equilibrium between phases a and /3, as shown below, the a + /3 coexistence curve is described by an equation of the form P = P(T), whose form we now wish to determine ... 

The need to abstract from the considerable complexity of real natural water systems and substitute an idealized situation is met perhaps most simply by the concept of chemical equilibrium in a closed model system. Figure 2 outlines the main features of a generalized model for the thermodynamic description of a natural water system. The model is a closed system at constant temperature and pressure, the system consisting of a gas phase, aqueous solution phase, and some specified number of solid phases of defined compositions. For a thermodynamic description, information about activities is required therefore, the model indicates, along with concentrations and pressures, activity coefficients, fiy for the various composition variables of the system. There are a number of approaches to the problem of relating activity and concentrations, but these need not be examined here (see, e.g., Ref. 11). 

Here spr is the projected entropy of an ideal mixture. The first term appearing in it, p0 = J dop a), is the zeroth moment, which is identical to the overall particle density p defined previously. If this is among the moment densities used for the projection (or more generally, if it is a linear combination of them), then the term — Tp0 is simply a linear contribution to the projected free energy/pr(p,) and can be dropped because it does not affect phase equilibrium calculations. Otherwise, p0 needs to be expressed—via the A —as a function of the pit and its contribution cannot be ignored. We will see an example of this in Section V. 

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