Equilibria between Condensed Phases

Indirect methods. These are specially used for the plasticizers and other less volatile substances. The application of the generalized expression of the first and the second laws of thermodynamics to heterogeneous equilibrium between condensed phase and vapor in isobaric conditions is given by the Clausius-Clapeyron equation. It links the enthalpy of vapor formation at the vapor pressure, P, and the temperature, T. In the case of a one-component system the Clausius-Clapeyron expression has the form ... 

The Clausius-CIapeyron equation (5.24) applies to phase equilibrium in singlecomponent systems where one phase is an ideal gas thus, the expression can describe phase equilibrium at evaporation and sublimation. Phase equilibrium between condensed phases is described by the Clapeyron equations (5.19) and (5.21), respectively. 

The two calculation methods in Section 4.2 enable prediction of the three-phase (Lw-H-V) gas mixture region extending between the two quadruple points Qi and Q2 in Figure 4.1. Section 4.3 provides a method to use the techniques of Section 4.2 to locate both quadruple points on a pressure-temperature plot. Section 4.3 also discusses equilibrium of three condensed phases [aqueous liquid-hydrate-hydrocarbon liquid (Lw-H-Lhc)] Determination of equilibrium from condensed phases provides an answer to the question, Given a liquid... 

Equation (26) represents the intersection of two surfaces in p(P, T) space. The intersection of two surfaces is a curve in the three-dimensional space. The projection of this curve on the PT plane is given by P(T). Because P is a function of T, at equilibrium between two phases, the system has been reduced to one degree of freedom by the requirement of Eq. (26). If one of the phases is a gas, P(T) is the vapor pressure curve of the condensed phase. If both phases are condensed, P is the externally applied pressure. Alternatively, we could consider T(P), which gives the temperature at which two phases are at equilibrium as a function of pressure. 

Reasonably accurate (agreement with literature values better than 5%) heats of evaporation of some solvents (water, ether, dioxane) and NH4N03 solutions were obtained by TG methods (99). This technique also was used (100) to study the volatilization rates of some organic compounds that are of interest as environmental contaminants (naphthalene, hexachlorobenzene, 4-chlorobiphenyl, n-decane). Evaporation rates were influenced by the rates of heating (5, 10, and 25 K min-1)-A good representation of behavior was provided by the evaporation model, described in detail, provided that the surface area of the substance was known. It was assumed that equilibrium was established between condensed phase and vapor and that there was convective transport and diffusion to the container outlet. It was concluded that TG methods provide a useful method for studies of the evaporation of organic compounds. 

Phase equilibria between condensed phases, like melting and crystal polymorphic transitions, have no mass-dependent terms (no equilibrium constants) since the activity of pure condensed phases is unity, and hence the equilibrium thermodynamics is represented by the simple relationships ... 

Phase equilibrium between condensed ammonia and gases,... 

We can refer to equilibrium between different phases or chemical species within the system as well. A system is said to be in phase equilibrium if it has more than one phase present with no tendency to change. For example, a two-phase liquid-vapor system is in phase equilibrium when there is no tendency for the liquid to boil or the vapor to condense. To be complete, we must also have mechanical and thermal equilibrium between the liquid (/) and vapor (v) phases, that is. 

These models consider either the thermodynamic or mechanical non-equilibrium between the phases. The number of conservation equations in this case are either four or five. One of the most popular models which considers the mechanical non-equilibrium is the drift flux model. If thermal non-equilibrium between the phases is considered, constitutive laws for interfacial area and evaporation/condensation at the interface must be included. In this case, the number of conservation equations is five, and if thermodynamic equilibrium is assumed the number of equations can be four. Well-assessed models for drift velocity and distribution parameter depending on the flow regimes are required for this model in addition to the heat transfer and pressure drop relationships. The main advantage of the drift flux model is that it simplifies the numerical computation of the momentum equation in comparison to the multi-fluid models. Computer codes based on the four or five equation models are still used for safety and accident analyses in many countries. These models are also found to be useful in the analysis of the stability behaviour of BWRs belonging to both forced and natural circulation type. 

The Clapeyron equation expresses the dynamic equilibrium existing between the vapor and the condensed phase of a pure substance ... 

The equilibrium between a compressed gas and a liquid is outside the scope of this review, since such a system has, in general, two mixed phases and not one mixed and one pure phase. This loss of simplicity makes the statistical interpretation of the behavior of such systems very difficult. However, it is probable that liquid mercury does not dissolve appreciable amounts of propane and butane so that these systems may be treated here as equilibria between a pure condensed phase and a gaseous mixture. Jepson, Richardson, and Rowlinson39 have measured the concentration of... 

Again, if we consider the initial substances in the state of liquids or solids, these will have a definite vapour pressure, and the free energy changes, i.e., the maximum work of an isothermal reaction between the condensed forms, may be calculated by supposing the requisite amounts drawn off in the form of saturated vapours, these expanded or compressed to the concentrations in the equilibrium box, passed into the latter, and the products then abstracted from the box, expanded to the concentrations of the saturated vapours, and finally condensed on the solids or liquids. Since the changes of volume of the condensed phases are negligibly small, the maximum work is again ... 

As discussed in Chapter 6, water forms strong hydrogen bonds and these lead to a number of important features of its atmospheric behavior. All three phases of water exist in the atmosphere, and the condensed phases can exist in equilibrium with the gas phase. The equilibria between these phases is summarized by the phase diagram for water. Fig. 7-9. 

The Volta potential is defined as the difference between the electrostatic outer potentials of two condensed phases in equilibrium. The measurement of this and related quantities is performed using a system of voltaic cells. This technique, which in some applications is called the surface potential method, is one of the oldest but still frequently used experimental methods for studying phenomena at electrified solid and hquid surfaces and interfaces. The difficulty with the method, which in fact is common to most electrochemical methods, is lack of molecular specificity. However, combined with modem surface-sensitive methods such as spectroscopy, it can provide important physicochemical information. Even without such complementary molecular information, the voltaic cell method is still the source of much basic electrochemical data. 

If the no-phase-change restriction does not rigorously apply, a simple design procedure can be formulated based on the results discussed in Section III, where it is shown that thermal equilibrium is quickly achieved in gas-liquid systems because of the large heat effects associated with evaporation or condensation. Although the total mass transfer between the phases may be small, it is not unrealistic to assume that the gas and liquid phases have the same temperature at each axial position. 

Let us now include an additional component to the Fe-0 system considered above, for instance S, which is of relevance for oxidation of FeS and for hot corrosion of Fe. In the Fe-S-0 system iron sulfides and sulfates must be taken into consideration in addition to the iron oxides and pure iron. The number of components C is now 3 and the Gibbs phase rule reads Ph + F = C + 2 = 5, and we may have a maximum of four condensed phases in equilibrium with the gas phase. A two-dimensional illustration of the heterogeneous phase equilibria between the pure condensed phases and the gas phase thus requires that we remove one degree of... 

Chemical separations are often either a question of equilibrium established in two immiscible phases across the contact between the two phases. In the case of true distillation, the equilibrium is established in the reflux process where the condensed material returning to the pot is in contact with the vapor rising from the pot. It is a gas-liquid interface. In an extraction, the equilibrium is established by motion of the solute molecules across the interface between the immiscible layers. It is a liquid-liquid, interface. If one adds a finely divided solid to a liquid phase and molecules are then distributed in equilibrium between the solid surface and the liquid, it is a liquid-solid interface (Table 1). 

For isotope effects on equilibrium constants in both gas and condensed phase the take-home lesson is there is no direct proportionality between measured isotope effects on logarithmic concentration or pressure ratios and isotopic differences in... 

We begin with a discussion of the vapor pressure isotope effect (VPIE). To do so we compare the equilibria between condensed and vapor phase for samples of two isotopomers. At equilibrium, condensed(c) = vapor(v), the partial molar free energies, a(v), and p,(c), of the two phases are equal this, in fact, is the thermodynamic... 

Equation 5.19 relates the molecular energy states of the primed and unprimed isotopomers in condensed and vapor phase to VPIE. The correction terms account for the difference between the Gibbs and Helmholtz free energies of the condensed phase, and vapor nonideality. The comparison is between separated isotopomers at a common temperature, each existing at a different equilibrium volume, V or V, and at a different pressure, P or P, although AV = (V — V) and AP = (P — P) are small. Still, because condensed phase Q s are functions of volume, Q = Q(T,V,N), rigorous analysis requires knowledge of the volume dependence of the partition function, and thus MVIE, since the comparisons are made at V and V. That point is developed later. 

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