Single-Component Phase Equilibrium

1 SINGLE-COMPONENT PHASE EQUILIBRIUM 6.1a Phase Diagrams 

What the phase diagram means and what can be done with it are illustrated by a hypothetical experiment in which pure water is placed in a leakproof evacuated cylinder fitted with a movable piston, as shown in the diagram below. Heat can be added to or withdrawn from the cylinder, so that the temperature in the chamber can be adjusted to any desired value, and the absolute pressure of the cylinder contents [which equals F + W)/ A, where W is the weight of the piston] can similarly be adjusted by varying the force F on the piston. 

Notice that the phase transitions—condensation at point B and evaporation at point D—take place at boundaries on the phase diagram the system cannot move off these boundaries until the transitions are complete. 

Several familiar terms may be defined with reference to the phase diagram. 

If r and P correspond to a point on the vapor-liquid equilibrium curve for a substance, P is the vapor pressure of the substance at temperature T, and T is the boiling point (more precisely, the boiling point temperature) of the substance at pressure P. 

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There are seven thermodynamics parameters characterizing this equilibrium (p, T, rr, ya, yB, xa, andxB). In that case, the application of the phase rule of Gibbs leads to a variance equal to 3. The representation of the binary adsorption equilibrium is obviously more complicated than for the adsorption of a single component. Coadsorption equilibrium could be represented in a three dimensional space by plotting for example the amounts adsorbed as a function of the pressure in one direction and the composition of the adsorbate in another one. However, this kind of plot is not common because it requires a lot of data measured at different composition and pressure. We prefer to represent the coadsorption equilibria in a two-dimension space by maintaining constant certain variables. The representations that are the more often used and easily understandable are ... 

Thermodynamics of Liquid—Liquid Equilibrium. Phase splitting of a Hquid mixture into two Hquid phases (I and II) occurs when a single hquid phase is thermodynamically unstable. The equiUbrium condition of equal fugacities (and chemical potentials) for each component in the two phases allows the fugacitiesy andy in phases I and II to be equated and expressed as ... 

The working capacity of a sorbent depends on fluid concentrations and temperatures. Graphical depiction of soration equilibrium for single component adsorption or binary ion exchange (monovariance) is usually in the form of isotherms [n = /i,(cd or at constant T] or isosteres = pi(T) at constant /ij. Representative forms are shown in Fig. I6-I. An important dimensionless group dependent on adsorption equihbrium is the partition ratio (see Eq. 16-125), which is a measure of the relative affinities of the sorbea and fluid phases for solute. 

Let us next consider the equilibrium of a heterogeneous complex of two phases, numbered 1 and 2, of a single component. The conditions for equilibrium are ... 

With a suitable equation of state, all the fugacities in each phase can be found from Eq. (6), and the equation of state itself is substituted into the equilibrium relations Eq. (67) and (68). For an A-component system, it is then necessary to solve simultaneously N + 2 equations of equilibrium. While this is a formidable calculation even for small values of N, modern computers have made such calculations a realistic possibility. The major difficulty of this procedure lies not in computational problems, but in our inability to write for mixtures a single equation of state which remains accurate over a density range that includes the liquid phase. As a result, phase-equilibrium calculations based exclusively on equations of state do not appear promising for high-pressure phase equilibria, except perhaps for certain restricted mixtures consisting of chemically similar components. 

Having considered single component systems, multicomponent systems need to be addressed now. If a closed system contains more than one phase, the equilibrium condition can be written as ... 

Since the pressure drop in two-phase flow is closely related to the flow pattern, most investigations have been concerned with local pressure drop in well-characterized two-phase flow patterns. In reality, the desired pressure drop prediction is usually over the entire flow channel length and covers various flow patterns when diabatic condition exist. Thus, a summation of local Ap values is necessary, assuming the phases are in thermodynamic equilibrium. The addition of heat in the case of single-component flow causes a phase change along the channel consequently, the vapor void increases and the phase (also velocity) distribution as well as the momentum of the flow vary accordingly. 

Stabilization of Ru based oxides by valve metal oxides has not been studied in such detail using photoelectron spectroscopy. The most common compositions, however, with relatively high valve metal content, are not in favor of formation of a solid solution. Studies of the phase formation in Ru/Ti mixed oxides has shown [49] that homogeneous solutions are formed for compositions with Ru < 2% or Ru > 98% (see Section 3.1.1). Therefore electrodes with other compositions are better described as physical mixtures and the electrochemical behaviour is most likely that of a linear superposition of the single components. It has to be considered, however, that the investigations performed by Triggs [49] concern thermodynamic equilibrium conditions. If, by means of the preparation procedure, thermodynamic equilibrium is... 

Characterization of Equilibria Phase equilibrium between fluid and sorbed phases for one or many components in adsorption or two or more species in ion exchange is usually the single most... 

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... 

For a single-component systemp and T can be varied independently when only one phase is present. When two phases are present in equilibrium, pressure and temperature are not independent variables. At a certain pressure there is only one temperature at which the two phases coexist, e.g. the standard melting temperature of water. Hence at a chosen pressure, the temperature is given implicitly. A point... 

In this first example, a single-component system consisting of a liquid and a gas phase is considered. If the surface between the two phases is curved, the equilibrium conditions will depart from the situation for a flat surface used in most equilibrium calculations. At equilibrium the chemical potentials in both phases are equal ... 

We will consider a reversible change in the system at constant temperature and pressure of the liquid phase pl. Furthermore, we will assume that the equilibrium concentration of all components in the liquid, except for the single component, i, of the solid phase, is fixed. The equilibrium condition yields... 

Winsor [15] classified the phase equilibria of microemulsions into four types, now called Winsor I-IV microemulsions, illustrated in Fig. 15.5. Types I and II are two-phase systems where a surfactant rich phase, the microemulsion, is in equilibrium with an excess organic or aqueous phase, respectively. Type III is a three-phase system in which a W/O or an O/W microemulsion is in equilibrium with an excess of both the aqueous and the organic phase. Finally, type IV is a single isotropic phase. In many cases, the properties of the system components require the presence of a surfactant and a cosurfactant in the organic phase in order to achieve the formation of reverse micelles one example is the mixture of sodium dodecylsulfate and pentanol. 

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