Pressure-dependent phase equilibrium

The Clapeyron equation, derived in Sec. 6.4 for the latent heat of phase transition of pure chemical species, is also applicable to pnre-gas adsorption eqnilibrinm. Here, however, the two-phase equilibrium pressure depends not only on temperature, but on surface coverage or the amount adsorbed. Thus the analogous equation for adsorption is written... 

Consequently, at fixed temperature the two phase compositions and the pressure are fixed. Varying the average mole fraction would change the mass distribution between the two phases, but would not change the composition of either phase or the total pressure. That is, when two liquid phases and a vapor phase exist in a binary mixture, the equilibrium pressure depends only on temperature and not on average composition. 

The isotherms represented in Fig. 1 give a general idea of the equilibria in the Pd-H system under different p-T conditions. Most experimental evidence shows, however, that the equilibrium pressure over a + /3 coexisting phases depends on the direction of the phase transformation process p a-p > pp-a (T, H/Pd constant). This hysteresis effect at 100°... 

Kinetic-molecular theory provides an explanation on a molecular level for this equilibrium. Evaporation from the liquid occurs as fast moving molecules on the surface escape from the liquid. In turn, molecules in the gas phase strike the liquid and condense, As the concentration (pressure) of gas molecules builds up in the gas phase, the rate of condensation increases. Eventually, a pressure is reached where the rate of condensation and rate of evaporation just balance, and equilibrium is achieved. The equilibrium pressure is denoted by p and is known as the vapor pressure. The magnitude ofp depends upon the substance, composition of the liquid, and any two of our thermodynamic variables such as temperature and total pressure. The criteria for equilibrium that we will now derive provide the thermodynamic relationships that will help... 

Once a particular class of unit has been decided upon, the choice of a specific unit depends on initial and operating costs, the space available, the type and size of the product, the characteristics of the feed liquor, the need for corrosion resistance and so on. Particular attention must be paid to liquor mixing zones since the circulation loop includes many regions where flow streams of different temperature and composition mix. These are all points at which temporary high supersaturations may occur causing heavy nucleation and hence encrustation, poor performance and operating instabilities. As Toussaint and Donders(72) stresses, it is essential that the compositions and enthalpies of mixer streams are always such that, at equilibrium, only one phase exists under the local conditions of temperature and pressure. 

However, in any phase transition, the equilibrium pressure does not remain constant as the temperature is varied. Hence, to obtain (dAH,./temperature coefficient of the heat of vaporization, we must hnd the dependence of A/Zm on pressure as... 

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. 

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. 

When two phases of a substance are in equilibrium, their molar free energies are equal. Each molar free energy depends on the pressure and the temperature, so the condition for equilibrium of two phases a and (3 can be written... 

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... 

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 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). 

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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