Vapor-liquid equilibria subcritical

Ttlis rigorous equation for subcritical vapor/liquid equilibrium, given by Eq. (14.1), reduces to Eq. (10.1) when the two listed assumptions are imposed. 

FIGURE 1.5-1 Isothermal phase diagrams for subcritical binaty vapor-liquid equilibrium. Case (a) represents Raoult s-Law behavior. Cases negative deviations from Raoulfs Law cases (d) and (e) illustrate positive deviations from Raoult s Law. 

In Chapter 8 we discussed the mechanical stability of a pure fluid in terms of the behavior of a subcritical isotherm on a Pv diagram. A sample isotherm is shown at the top of Figure 9.3, computed using the van der Waals equation of state. Also in Chapter 8 we showed that pure-fluid vapor-liquid equilibrium states are found by solving the equilibrium conditions (9.1.8). The equality of chemical potentials in (9.1.8) can also be expressed as an equality of fugacities in the case of pure-fluid vapor-liquid equilibria. 

The essential features of vapor-liquid equilibrium (VLE) behavior are demonstrated by the simplest case isothermal VLE of a binaiy system at a temperature below die critical temperatures of both pure components. For this case ( subcritical VLE), each pure couqionent has a well-defln vapor-liquid saturation pressure Ff , and VLE is possible for the fiiU range of liquid and vsqior compositions X/ and y,. Figure l.S-1 illustrates several pes of behavior riiown by such systems. In each case, the upper solid curve ( bubble curve ) represents states of saturated liquid the lower solid curve ( dew curve") represents states of saturated vapor. 

The volumes in this series are compilations of vapor-liquid equilibrium data on subcritical binary homogeneous (single-phase) or heterogeneous (two-phase) liquid-liquid systems. 

The lines labeled T and T2 are for subcritical temperatures, and consist of three segments. The horizontal segment of each isothemi represents all possiblemixturesofliqnid and vapor in equilibrium, rangingfrom 100% liquid at the left end to 100% vapor at the right end. The locns of these end points is the dome-shaped curve labeled BCD, the left half of which (from B to C)... 

The simplest application of an equation of state for VLE calculations is to a pure species to find its saturation or equilibrium vapor pressure at given temperature T. As discussed in Sec. 3.5 with respect to cubic equations of state for pure species, a subcritical isothenu on a P V diagram exliibits a smooth transitionfrom liquid to vapor tliis is shown on Fig. 3.12 by the curve labeled Ti < T. Independent knowledge was there assumed of vapor pressures. In fact, tliis infomiation is implicit in an equationof state. Figure 14.7 illustrates a realistic subcritical isotherm owa PV diagram as generatedby an equation of state. One of the features of such an isothenu for temperahires not too close to fr is tliat the ininimum lies below the F = 0 axis. 

For gases, such as methane, which are supercritical at hydrate forming temperatures, there is one quadruple point, as indicated by point Q1 in Fig. 1. At this point, ice, liquid water, gas and hydrate are in equilibrium. For gases that are subcritical at hydrate forming temperatures, such as ethane, ° there are two quadruple points (Q1 and Q2 in Fig. 2). While Q1 lies at approximately the freezing point of water, Q2 is at approximately the intersection of the hydrate-water-gas three-phase equilibrium curve with the vapor pressure curve. At this latter point, liquid water, gas, hydrate, and liquid hydrate former are all in equilibrium. As seen in Fig. 2, the hydrate... 

For each subcritical temperature T < 1, dynamic System 3 has a simple equilibrium point, i.e. dp/dA = 0 = dP/dA, at (pG(T), Pa(T)), (pd(T), Pa(T)), and (pL(T), P<7(T)). Using well-established methods of dynamic system theory—for simple equilibrium points it suffices to examine the eigenvalues of System 3—one then determines that both the saturated vapor point (po(T), Po-(T)) and the saturated liquid point (Pl(T), PV(T)) are (orbitally stable with respect to A) dicritical nodes, i.e. that each solution path p(A), P(A) approaches the equilibrium point from a definite direction and, conversely, each direction corresponds to exactly one path (see Figure 2). 

We will find in 9.1 that, for a pure substance in two-phase equilibrium, only one property is needed to specify the intensive state in (8.2.20) we have used temperature. However, even after we set a value for the subcritical temperature, (8.2.20) remains implicit in three unknowns the vapor pressure P plus the molar volumes of the liquid and vapor phases, and v. To close the problem we need another equation, typically, a PvT equation of state that relates P to both saturated volumes at Ae specified T. Therefore, we must choose an equation of state that is sufficiently complicated that it bifurcates and provides multiple roots for the volume over some range of states. Such equations of state are explicit in the pressure [P = P(T, v) and then we would compute 9 from... 

The goal of a subcritical VLE calculation is to quantitatively predict or correlate the various kinds of behavior illustrated by Fig. 1.5-1 or by its isobatic or multicomponent counterparts. The basis for the calculation is phase-equilibrium formulation 1 of Section 1.2-5, where liquid-phase fugacities are eliminated in favor of liquid-phaW activity coefficients, and vapor-phase fugacities in favor of vapor-phase fugacity coefficients. Raouit s Law standard states are chosen for all components in the liquid phase hence, (ff f = /f, and Eq. (1.2-63) becomes... 

The state (of aggregation) is the same for all the chemical systems of this volume each pure component is a liquid in equilibrium with vapor (subcritical fluid), flie binary is a single-phase liquid or two-phase liquid-liquid system. Two-phase regions are clearly marked on flie PDF tables and the SELF and ELDATA files. 

If a porous solid is in equilibrium with bulk vapor at a vapor relative pressure P/Po the adsorbate consists of a capillary condensed liquid filhng the pores with radii smaller than the Kelvin radius, rK (subcritical pores, r rx) and an adsorbed layer of thickness t covering the walls of the supercritical pores (r > rx). For the classic case of N2 sorption on a mesoporous medium at 77 K, the following expressions have been employed [12-14]... 

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