Vapor phase standard state

For the liquid-phase standard state we usually choose the pure liquid at the system temperature and its vapor pressure P, (T),... 

Here p. is the vapor pressure of the ith substance over the solution, p. is the vapor pressure it would exert in its standard (pure hquid) state, and x. is its mole fraction in the solution. We can now define an absolute activity (not really absolute, but relative to the gas phase standard state on the pressure scale as earlier) measured by p., assuming that the vapor may be treated as an ideal gas or by the fugacitf if necessary. We shall always make the ideal gas assumption, without restating it. 

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

Biphenyl, terphenyl, and their alkyl or hydrogenated derivatives generally serve markets where price and performance, rather than composition, is the customer s primary concern. Performance standards for heat-transfer appHcations are usually set by the fluid suppHer. The biphenyl—diphenyl oxide eutectic (26.5% biphenyl, 73.5% DPO) represents a special case. This composition has become a widely recogni2ed standard vapor-phase heat-transfer medium. It is sold throughout the world under various trademarks. In the United States, Dow (Dowtherm A) and Monsanto (Therminol VP-1) are the primary suppHers. Alkylated biphenyls and partially hydrogenated terphenyls serving the dielectric and carbonless copy paper dye solvent markets likewise are sold primarily on the basis of price and performance characteristics jointly agreed on by producer and user. 

Since Eqs. (5) and (6) are not restricted to the vapor phase, they can, in principle, be used to calculate fugacities of components in the liquid phase as well. Such calculations can be performed provided we assume the validity of an equation of state for a density range starting at zero density and terminating at the liquid density of interest. That is, if we have a pressure-explicit equation of state which holds for mixtures in both vapor and liquid phases, then we can use Eq. (6) to solve completely the equations of equilibrium without explicitly resorting to the auxiliary-functions activity, standard-state fugacity, and partial molar volume. Such a procedure was discussed many years ago by van der Waals and, more recently, it has been reduced to practice by Benedict and co-workers (B4). 

When the standard states for the solid and liquid species correspond to the pure species at 1 atm pressure or at a low equilibrium vapor pressure of the condensed phase, the activities of the pure species at equilibrium are taken as unity at all moderate pressures. Consequently, the gas phase composition at equilibrium will not be... 

The NPT + test particle method [8, 9] aims to determine phase coexistence points based on calculations of the chemical potentials for a number of state points. A phase coexistence point is determined at the intersection of the vapor and liquid branches of the chemical potential versus pressure diagram. The Widom test particle method [7] of the previous paragraph or any other suitable method [10] can be used to obtain the chemical potentials. Corrections to the chemical potential of the liquid and vapor phases can be made, using standard thermodynamic relationships, for deviations... 

Fig. 12.4. Vapor-to-water transfer data for saturated hydrocarbons as a function of accessible surface area, from [131]. Standard states are 1M ideal gas and solution phases. Linear alkanes (small dots) are labeled by the number of carbons. Cyclic compounds (large dots) are a = cyclooctane, b = cycloheptane, c = cyclopentane, d = cyclohexane, e = methylcyclopentane, f = methylcyclohexane, g = cA-l,2-dimethylcyclohexane. Branched compounds (circles) are h = isobutane, i = neopentane, j = isopentane, k = neohexane, 1 = isohexane, m = 3-methylpentane, n = 2,4-dimethylpentane, o = isooctane, p = 2,2,5-tri-metbylhexane. Adapted with permission from [74], Copyright 1994, American Chemical Society...

This illustrates the statement made earlier that the most convenient choice of standard state may depend on the problem. For gas-phase problems involving A, it is convenient to choose the standard state for A as an ideal gas at 1 atm pressure. But, where the vapor of A is in equilibrium with a solution, it is sometimes convenient to choose the standard state as the pure liquid. Since /a is the same for the pure liquid and the vapor in equilibrium... 

The production of toluene from benzene and xylenes was studied by Johanson Watson (National Petroleum News, 7 Aug 1946) in a standard 1-inch pipe reactor with a silica-alumina catalyst. At the reaction temperature of 932 F (773 K) the reaction mixture was vapor phase, but the feeds were measured as liquids. The feed consisted of an equimolal mixture of reactants. The stated LHSV is (ml feed at 60 F/h)/(ml reactor). The reactor contained 85 g catalyst packed in a volume of 135 ml. The densities of benzene and xylenes at 60 F are 0.879 and 0.870, respectively. 

It is observed in table 7.2 that the most important terms in the reduction to standard states are the decomposition of aqueous nitric acid (AU (,), the compression of the initial gaseous phase (02+H20) from a negligibly small pressure to the initial pressure (A 6/7), the decompression of the final gaseous phase (02 + N2 + C02 + H20) from the final pressure to a negligibly small pressure (Af/19), and the evaporation of C02 from the final aqueous phase (AU o). The terms relative to the vaporization and condensation of liquid water (A f/3 — AIJ24) almost cancel out. 

This is a simple and important result. It equates VPIE to the isotopic difference of standard state free energies on phase change, plus a small correction for vapor phase nonideality, here approximated through the second virial coefficient. Therefore Equation 5.8 is limited to relatively low pressure. As T and P increase third and higher virial corrections may be needed, and at even higher pressures the virial expansion must be abandoned for a more accurate equation of state. 

The standard state Helmholtz free energy difference, 8AA°, was introduced in Equations 5.9 and 5.11 to show the connection between VPIE and molecular structure and dynamics. Molecular properties are conveniently expressed using standard state canonical partition functions for the condensed and vapor phases, Qc° and Qv° remember A0 = —RT In Q°. The Q s are 3nN dimensional, n is the number of atoms per molecule and N is Avogadro s number. For convenience we have now dropped the superscript o s on the Q s. The o s specify standard state conditions, now to be implicitly understood. For VPIE and a respectively, not too close to the critical region,... 

The JANAF tables specify a volatilization temperature of a condensed-phase material to be where the standard-state free energy A Gf approaches zero for a given equilibrium reaction, that is, M/fyl), M/)y(g). One can obtain a heat of vaporization for materials such as Li20(l), FeO(l), BeO(l), and MgO(l), which also exist in the gas phase, by the differences in the All" of the condensed and gas phases at this volatilization temperature. This type of thermodynamic calculation attempts to specify a true equilibrium thermodynamic volatilization temperature and enthalpy of volatilization at 1 atm. Values determined in this manner would not correspond to those calculated by the approach described simply because the procedure discussed takes into account the fact that some of the condensed-phase species dissociate upon volatilization. 

It is convenient, therefore, to choose the pure condensed phase at the temperature of the solution at the equilibrium vapor pressure of the pure condensed phase as the standard state for the component in the solution (see Ref. 1). Thus, Equation (14.6) can also be written... 

Jaques and Furter (37,38,39,40) devised a technique for treating systems consisting of two volatile components and a salt as special binaries rather than as ternary systems. In this pseudo binary technique the presence of the salt is recognized in adjustments made to the pure-component vapor pressures from which the liquid-phase activity coefficients of the two volatile components are calculated, rather than by inclusion of the salt presence in liquid composition data. In other words, alteration is made in the standard states on which the activity coefficients are based. In the special binary approach as applied to salt-saturated systems, for instance, each of the two components of the binary is considered to be one of the volatile components individually saturated with the... 

The same reference (standard) state, f is chosen for the two phases, so that it cancels on both sides of equation 39. The products stffi and y" are referred to as activities. Because equation 39 holds for each component of a liquid—liquid system, it is possible to predict liquid—liquid phase splitting when the activity coefficients of the individual components in a multicomponent system are known. These values can come from vapor—liquid equilibrium experiments or from prediction methods developed for phase-equilibrium problems (4,5,10). Some binary systems can be modeled satisfactorily in this manner, but only rough estimations appear to be possible for multicomponent systems because activity coefficient models are not yet sufficiendy developed in this area. 

The thermodynamic reaction equilibrium constant K, is only a function of temperature. In Equation 4.18, m, the activity of the guest in the vapor phase, is equal to the fugacity of the pure component divided by that at the standard state, normally 1 atm. The fugacity of the pure vapor is a function of temperature and pressure, and may be determined through the use of a fugacity coefficient. The method also assumes that an, the activity of the hydrate, is essentially constant at a given temperature regardless of the other phases present. 

Henry s law is also used in some cases as a limiting expression of the behavior of a component in a single condensed phase at a given temperature and an arbitrary constant pressure P0 with no concept of an equilibrium between the condensed phase and a vapor phase. In order to obtain the concept of Henry s law under these conditions, two different expressions are required for the chemical potential of the component with the use of two different standards states. In terms of mole fractions, the usual practice is to... 

The differences between the standard states of the component in the condensed and gaseous states at the two temperatures cannot be evaluated without knowledge of the composition of standard states in the condensed phases. However, this requires prior knowledge of the excess chemical potentials. The experimental study of the isothermal vapor-liquid equilibria is therefore more convenient and yields the same information. 

In order to evaluate each of the derivatives, such quantities as (V" — V-), (S l — Sj), and (dfi t/x t)T P need to be evaluated. The difference in the partial molar volumes of a component between the two phases presents no problem the dependence of the molar volume of a phase on the mole fraction must be known from experiment or from an equation of state for a gas phase. In order to determine the difference in the partial molar entropies, not only must the dependence of the molar entropy of a phase on the mole fraction be known, but also the difference in the molar entropy of the component in the two standard states must be known or calculable. If the two standard states are the same, there is no problem. If the two standard states are the pure component in the two phases at the temperature and pressure at which the derivative is to be evaluated, the difference can be calculated by methods similar to that discussed in Sections 10.10 and 10.12. In the case of vapor-liquid equilibria in which the reference state of a solute is taken as the infinitely dilute solution, the difference between the molar entropy of the solute in its two standard states may be determined from the temperature dependence of the Henry s law constant. Finally, the expression used for fii in evaluating (dx Jdx l)TtP must be appropriate for the particular phase of interest. This phase is dictated by the particular choice of the mole fraction variables. 

Physical property data for many of the key components used in the simulation for the ethanol-from-lignocellulose process are not available in the standard ASPEN-Plus property databases (11). Indeed, many of the properties necessary to successfully simulate this process are not available in the standard biomass literature. The physical properties required by ASPEN-Plus are calculated from fundamental properties such as liquid, vapor, and solid enthalpies and density. In general, because of the need to distill ethanol and to handle dissolved gases, the standard nonrandom two-liquid (NRTL) or renon route is used. This route, which includes the NRTL liquid activity coefficient model, Henry s law for the dissolved gases, and Redlich-Kwong-Soave equation of state for the vapor phase, is used to calculate properties for components in the liquid and vapor phases. It also uses the ideal gas at 25°C as the standard reference state, thus requiring the heat of formation at these conditions. 

A sample is placed in a glass vial that is closed with a septum and thermo-stated until an equilibrium is established between the sample and the vapor phase. A known aliquot of the gas is then transferred by a gas-tight syringe to a gas chromatograph and analyzed. The volume of the sample is determined primarily from practical considerations and ease of handling. The concentration of the compound of interest in the gas phase is related to the concentration in the sample by the partition coefficient. The partition coefficient is included in a calibration factor obtained on a standard. The analysis can easily be automated where a series of samples is to be analyzed, resulting in improved precision. 

The standard state of condensed phases, such as liquids, are chosen as the pure substance at 1.0 bar pressure at the temperature of interest. The standard state of an ideal gas is also at 1.0 bar. Over a moderate range of temperature, heats of vaporization and sublimation usually do not vary greatly and, as shown by Eq. (44), can be obtained from plots of the vapor pressure versus 1 /T. An example of this for ethyl acetate is shown in Fig. 5. Note the slight deviation of Fig. 5b from a straight line. 

In order to determine whether a substance will condense or not, one first determines the partial pressure without assuming condensation. If this partial pressure is greater than the vapor pressure, then in an equilibrium situation condensation must have taken place. Because most equilibrium reactions have their K p f s referenced to the elements in the standard states as, for example, carbon cftscussed above, it is difficult to determine the partial pressure of the carbon since the K Pj f s, when carbon is not condensed are not readily available. Say, then, one has carbon as a product and he wishes to determine the physical state. First he calculates the number of moles of carbon as condensed. Then, taking the same number of moles as gaseous, he determines the hypothetical partial pressure these number of moles of gas would exert. This partial pressure must be greater than the vapor pressure for the initial assumption that condensed phase is present to be... 

Although activity and fugacity are closely related, they have quite different characteristics in regard to phase equilibria. Consider, for example, the equilibrium between liquid water and water vapor in the interstices of an unsaturated soil. At a given temperature and pressure, the principles of thermodynamic equilibrium demand that the chemical potentials and fugacities of water in the two phases be equal. However, the activities of water in the two phases will not be the same because the Standard State for the two phases is not j the same. Indeed, f° = 1 atm for the water vapor, so its activity is numerically ]... 

As a conclusion from the Hildebrand/Trouton Rule, the definition of a standard vapor phase in a standard state with a well known amount of disorder can be made. This definition can be used as a starting point for modeling diffusion coefficients of gases and liquids. 

When liquid and gas phases are both present in an equilibrium mixture of reacting species, Eq. (11.30), a criterion of vapor/liquid equilibrium, must be satisfied along with the equation of chemical-reaction equilibrium. There is considerable choice in the method of treatment of such cases. For example, consider a reaction of gas A and water B to form an aqueous solution C. The reaction may be assumed to occur entirely in the gas phase with simultaneous transfer of material between phases to maintain phase equilibrium. In this case, the equilibrium constant is evaluated from AG° data based on standard states for the species as gases, i.e., the ideal-gas states at 1 bar and the reaction temperature. On the other hand, the reaction may be assumed to occur in the liquid phase, in which case AG° is based on standard states for the species as liquids. Alternatively, the reaction may be written... 

Choose as standard states for water and ethylene glycol the pure liquids at 1 bar and for ethylene oxide the pure ideal gas at 1 bar. Assume that the Lewis/Randall rule applies to the water in the liquid phase and that the vapor phase is an ideal gas. The partial pressure of ethylene oxide., river the liquid phase is given by, ... 

Example 6.4 Wetted wall column with a ternary liquid mixture We have the experimental data on distillation of ethanol (1), feri-butanol (2), and water (3) in a wetted wall column reported by Krishna and Standard (1976). As Figure 6.2 shows, the column operates at total reflux with countercurrent flow. Therefore, at steady state, the compositions of the liquid and vapor phases at ary point in the column are equal to each other. The measured compositions of the phases at the bottom are... 

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