Pressure dependence, phase equilibria

The pressure dependence of equilibrium constants in this work are estimated with Eq. 2.29, which requires knowledge of the partial molar volumes and compressibilities for ions, water, and solid phases. For ions and water, molar volumes and compressibilities are known as a function of temperature (Table B.8 Eqs. 3.14 to 3.19). Molar volumes for solid phases are also known (Table B.9) unfortunately, the isothermal compressibilities for many solid phases are lacking (Millero 1983 Krumgalz et al. 1999). 

This overall flow pattern in a distillation column provides countercurrent contacting of vapor and hquid streams on all the trays through the column. Vapor and liquid phases on a given tray approach thermal, pressure, and composition equilibriums to an extent dependent upon the efficiency of the contac ting tray. 

In porous media, liquid-gas phase equilibrium depends upon the nature of the adsorbate and adsorbent, gas pressure and temperature [24]. Overlapping attractive potentials of the pore walls readily overcome the translational energy of the adsorbate, leading to enhanced adsorption of gas molecules at low pressures. In addition, condensation of gas in very small pores may occur at a lower pressure than that normally required on a plane surface, as expressed by the Kelvin equation, which relates the radius of a curved surface to the equilibrium vapor pressure [25],... 

At present there are two fundamentally different approaches available for calculating phase equilibria, one utilising activity coefficients and the other an equation of state. In the case of vapour-liquid equilibrium (VLE), the first method is an extension of Raoult s Law. For binary systems it requires typically three Antoine parameters for each component and two parameters for the activity coefficients to describe the pure-component vapour pressure and the phase equilibrium. Further parameters are needed to represent the temperature dependence of the activity coefficients, therebly allowing the heat of mixing to be calculated. 

An interesting example of a one-component systems is SiOa, which can exist in five different crystalline forms or as a liquid or a vapor. As C = 1, the maximum number of phases that can coexist at equilibrium is three. Each phase occupies an area on the T P diagram the two-phase equilibria are represented by curves and the three-phase equilibria by points. Figure 13.1 (2, p. 123), which displays the equUi-brium relationships among the sohd forms of Si02, was obtained from calculations of the temperature and pressure dependence of AG (as described in Section 7.3) and from experimental determination of equUibrium temperature as a function of equilibrium pressure. 

To explore Young s equation still further, suppose we distinguish between ysv and ySo, where the former describes the surface of a solid in equilibrium with the vapor of a liquid and the latter a solid in equilibrium with its own vapor. Since Young s equation describes the three-phase equilibrium, it is proper to use ysv in Equation (44). The question arises, however, what difference, if any, exists between these two y s. In order to account for the difference between the two, we must introduce the notion of adsorption. In the present context adsorption describes the attachment of molecules from the vapor phase onto the solid surface. All of Chapter 9 is devoted to this topic, so it is unnecessary to go into much detail at this point. The extent of this attachment depends on the nature of the molecules in the vapor phase, the nature of the solid, and the temperature and the pressure. 

Zeolite crystallization represents one of the most complex structural chemical problems in crystallization phenomena. Formation under conditions of high metastability leads to a dependence of the specific zeolite phase crystallizing on a large number of variables in addition to the classical ones of reactant composition, temperature, and pressure found under equilibrium phase conditions. These variables (e.g., pH, nature of reactant materials, agitation during reaction, time of reaction, etc.) have been enumerated by previous reviewers (1,2, 22). Crystallization of admixtures of several zeolite phases is common. Reactions involved in zeolite crystallization include polymerization-depolymerization, solution-precipitation, nucleation-crystallization, and complex phenomena encountered in aqueous colloidal dispersions. The large number of known and hypo-... 

Sometimes it is possible to evaluate the range of validity of measurements and correlations of physical properties, phase equilibrium behavior, mass and heat transfer efficiencies and similar factors, as well as the fluctuations in temperature, pressure, flow, etc., associated with practical control systems. Then the effects of such data on the uncertainty of sizing equipment can be estimated. For example, the mass of a distillation column that is related directly to its cost depends on at least these factors ... 

Activation parameters for the high pressure gas-phase approach of 1,2-d2-cyclopropanes to cis, trans equilibrium (equation 1) have been reported as log A, a(kcal mol"1) of 16.0,64.2 and 16.4,65.176,77. From pressure-dependent measurements of rate constants and calculations based on RRKM theory, the threshold energy E for the cis, trans isomerization has been estimated to be 61.1 kcal mol"1 and 61.3 kcal mol"11 16 1, s. 

The procedure used to define an equilibrium model is to (1) define all the variables and (2) define independent equilibria as a function of phase equilibria. The variables are defined as the chemical parameters typically measured in water chemistry. For the major constituents and some of the more important minor constituents, these are calcium, magnesium, sodium, potassium, silica, sulfate, chloride, and phosphate concentrations as well as alkalinity (usually carbonate alkalinity) and pH. To this list we would also add temperature and pressure. The phase equilibria are defined by compiling well-known equilibria between gas-liquid phases and solid-liquid equilibria for the solids commonly found forming in nature in sedimentary rocks. Within this framework, one can construct different equilibrium models depending upon the mineral chosen actual data concerning the formation of specific minerals therefore must be ascertained to specify a particular model as valid. 

The Gibbs phase rule is the basis for organizing the models. In general, the number of independent variables (degrees of freedom) is equal to the number of variables minus the number of independent relationships. For each unique phase equilibria, we may write one independent relationship. In addition to this (with no other special stipulations), we may write one additional independent relationship to maintain electroneutrality. Table I summarizes the chemical constituents considered as variables in this study and by means of chemical reactions depicts independent relationships. (Throughout the paper, activity coefficients are calculated by the Debye-Hiickel relationship). Since there are no data available on pressure dependence, pressure is considered a constant at 1 atm. Sulfate and chloride are not considered variables because little specific data concerning their equilibria are available. Sulfate may be involved in a redox reaction with iron sulfides (e.g., hydrotroilite), and/or it may be in equilibrium with barite (BaS04) or some solid solution combinations. Chloride may reach no simple chemical equilibrium with respect to a phase. Therefore, these two ions are considered only to the... 

As there exists a phase equilibrium both phases must have reached in the internal thermodynamic equilibrium with respect to the arrangement and distribution of the molecules the measuring time. Therefore, no time effects or path dependencies of the thermodynamic properties in the liquid crystalline phase should be expected. To check this point for the l.c. polymer, a cut through the measured V(P) curves at 2000 bar has been made (Fig. 6) and the volume values are inserted at different temperatures in Fig. 7, which represents the measured isobaric volume-temperature curve at 2000 bar 38). It can be seen from Fig. 7 that all specific volumes obtained by the cut through the isotherms in Fig. 6 he on the directly measured isobar. No path dependence can be detected in the l.c. phase. From these observations we can conclude that the volume as well as other properties of the polymers depend only on temperature and pressure. The liquid crystalline phase of the polymer is a homogeneous phase, which is in its internal thermodynamic equilibrium within the normal measuring time. 

Fig. 4.13 Dependence of equilibrium mole fraction of C60H2n (x) on total pressure and temperature in the reaction of gas-phase C60H2n decomposition...

The equilibrium crystallographic form of ice depends on temperature and pressure. A phase diagram showing some of these is shown in Figure 6.9. 

The melting point of a compound is the temperature at which the solid and liquid phases are in equilibrium at one atmosphere pressure is specified because the melting process involves a change in volume and is therefore pressure dependent. Since the melting point can be determined easily experimentally, it is the most commonly reported physical property for organic compounds. However, in the absence of a rigorous theory of fusion, it is one of the most difficult to predict. 

The individual rates vq and v i are affected by temperature, pressure, and the concentrations of the species in Eq. (5.36). At equilibrium, the left side of Eq. (5.37) will disappear and v]eq/v ]eq where eq is the equilibrium condition will be a function of temperature, pressure, and the equilibrium composition of the exchanger and aqueous solution phase. Because the activities of the species in Eq. (5.36) have an identical dependence, vleq/v leq depends on temperature, pressure, and the species activities (Denbigh, 1981). But this same relationship applies to the quotient of the right and left sides of Eq. (5.38) for the determination of the exchange equilibrium constant (Ksx) for the reaction in Eq. (5.36), which can be expressed as,... 

As in the case for adsorption (see Section 2.2), in equilibrium, the quantity N of a given solute which is dissolved in a given solvent depends on its gas phase partial pressure (fugacity) P and on the temperature T, and a basic phenomenological description of the equilibrium is specification of the functional relationship between N, P, and T. At sufficiently low pressures, it is expected that the pressure dependence is linear (Henry s law) ... 

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