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Contents

Contributors ................................................................................ vii

Preface ....................................................................................... ix

Part 1. General Principles ........................................................ 1 1. Phase Equilibria ............................................................................ 3

1.1 Introduction ................................................................... 3 1.2 Thermodynamic Framework for Phase Equilibria .......... 4 1.3 Fugacity Coefficients .................................................... 12 1.4 Activity Coefficients ....................................................... 24 1.5 Vapor-Liquid Equilibria and Liquid-Liquid

Equilibria ....................................................................... 34 1.6 Fluid-Solid Equilibrium .................................................. 45 1.7. High Pressure Vapor-Liquid Equilibria .......................... 51 1.8 Summary ...................................................................... 54 Notation .................................................................................. 54 References ............................................................................. 57

2. Mass Transfer Principles ............................................................... 60 2.1 Introduction ................................................................... 60 2.2 Conservation Laws ....................................................... 61 2.3 Molecular Diffusion ....................................................... 70 2.4 Mass Transfer in Turbulent Flow ................................... 100 2.5 Notation ........................................................................ 121 References ............................................................................. 123

3. Phase Segregation ........................................................................ 129 3.1 Basic Mechanisms and Analogies ................................. 129 3.2 Gas-Liquid Segregation ................................................ 132 3.3 Immiscible Liquid Segregation ...................................... 148

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3.4 Liquid-Solid Segregation ............................................... 157 3.5 Gas-Solid Segregation .................................................. 176 Notation .................................................................................. 193 References ............................................................................. 195

4. General Processing Considerations ............................................. 197 4.1 Methods of Operation ................................................... 197 4.2 Process Synthesis ........................................................ 204 4.3 Control of Separation Processes .................................. 218 4.4 Special Problems .......................................................... 220 References ............................................................................. 222

Part 2. Individual Separation Processes ................................ 227 5. Distillation ...................................................................................... 229

5.1 Introduction ................................................................... 229 5.2 Phase Equilibrium ......................................................... 231 5.3 Equilibrium Stages ........................................................ 238 5.4 Specification of Variables .............................................. 258 5.5 Special Distillations ....................................................... 261 5.6 Transfer Units ............................................................... 275 5.7 Tray-Type Distillation Columns ..................................... 276 5.8 Packed-Type Distillation Columns ................................ 295 5.9 Mass Transfer in Tray Columns .................................... 312 5.10 Mass Transfer in Packed Columns ............................... 323 5.11 Distillation Column Control ............................................ 328 Notation .................................................................................. 331 References ............................................................................. 335

6. Absorption and Stripping ............................................................... 340 6.1 Basic Concepts ............................................................. 340 6.2 Multistage Contactors ................................................... 354 6.3 Differential Contactors .................................................. 364 6.4 Predicting Contactor Performance ................................ 379 Notation .................................................................................. 400 References ............................................................................. 402

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7. Extraction Organic Chemicals Processing ................................ 405 7.1 Diffusion and Mass Transfer ......................................... 405 7.2 Equilibrium Considerations ........................................... 414 7.3 Stagewise and Differential Contacting Calculation

Methods ........................................................................ 415 7.4 Stagewise Contact in Perforated Plate Columns ........... 432 7.5 Stagewise Contact in Mixer-Settlers ............................. 434 7.6 Mechanically Agitated Columns .................................... 438 7.7 Performance and Efficiency of Selected

Contactors .................................................................... 441 7.8 Solvent and Process Selection ..................................... 445 Notation .................................................................................. 456 References ............................................................................. 461

8. Extraction Metals Processing ..................................................... 467 8.1 Introduction ................................................................... 467 8.2 Extraction Chemistry and Reagents .............................. 467 8.3 Phase Equilibria ............................................................ 477 8.4 Extraction Kinetics ........................................................ 486 8.5 Contacting Equipment and Design Calculations ........... 490 8.6 Process Design and Engineering .................................. 495 8.7 Summary ...................................................................... 496 References ............................................................................. 496

9. Leaching Metals Applications .................................................... 500 9.1 Introduction ................................................................... 500 9.2 Leaching Practice ......................................................... 504 9.3 Thermodynamics of Leaching ....................................... 522 9.4 Kinetics of Leaching ...................................................... 529 Notation .................................................................................. 536 References ............................................................................. 536

10. Leaching Organic Materials ....................................................... 540 10.1 Definition of Process ..................................................... 540 10.2 Contacting Methods ...................................................... 540 10.3 Industrial Leaching Equipment ...................................... 541

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10.4 Products, Raw Materials, and Extraction Conditions ..................................................................... 550

10.5 Phase Equilibrium ......................................................... 550 10.6 Multistage Equilibrium Countercurrent Extraction ......... 550 10.7 Ficks's Laws ................................................................. 554 10.8 Unsteady-State Batch Extraction .................................. 555 10.9 Solute Diffusivities ........................................................ 557 10.10 Fixed Beds .................................................................... 558 10.11 Axial Dispersion and Flow Maldistribution ..................... 560 10.12 Superposition-Multistage Countercurrent

Extraction ..................................................................... 563 10.13 Continuous Countercurrent Extraction .......................... 564 10.14 Diffusion Batteries Performance ................................... 566 10.15 Differential Extraction .................................................... 566 10.16 Reflux Extractors .......................................................... 566 10.17 Solubilization ................................................................ 567 10.18 Solvent Selection .......................................................... 567 10.19 Equipment Selection ..................................................... 568 10.20 Solids Feed Preparation ............................................... 568 10.21 Spent Solids Treatment ................................................ 569 10.22 Hydrodynamic Considerations ...................................... 569 10.23 Solid-Liquid Heat Exchange .......................................... 569 Notation .................................................................................. 570 References ............................................................................. 573

11. Crystallization Operations ............................................................. 578 11.1 Introductory Comments ................................................. 578 11.2 Fundamentals ............................................................... 580 11.3 Solution Crystallization ................................................. 606 11.4 Melt Crystallization ........................................................ 626 11.5 General Design and Operational Considerations .......... 635 Notation .................................................................................. 638 References ............................................................................. 639 General Bibliography .............................................................. 642

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12. Adsorption ..................................................................................... 644 12.1 Introduction ................................................................... 644 12.2 Adsorbents ................................................................... 645 12.3 Criteria for Adsorption Use ............................................ 653 12.4 Basic Adsorption Cycles ............................................... 654 12.5 Process Flowsheets ...................................................... 659 12.6 Selecting a Process ...................................................... 668 12.7 Process-Design Considerations .................................... 669 12.8 Future Directions for Adsorption Technology and

Uses ............................................................................. 690 References ............................................................................. 691

13. Ion Exchange ................................................................................ 697 13.1 Principles of Ion Exchange ........................................... 697 13.2 Applications of Ion Exchange ........................................ 711 13.3 Equipment for Ion Exchange ......................................... 717 13.4 Recent Developments in Ion Exchange ........................ 726 Notation .................................................................................. 729 References ............................................................................. 730

14. Large-Scale Chromatography ....................................................... 733 14.1 Theory .......................................................................... 733 14.2 Scale-Up of Elution Chromatography ............................ 739 14.3 Countercurrent and Simulated Countercurrent

Systems ........................................................................ 745 14.4 Hybrid Systems ............................................................. 751 14.5 Other Alternatives ......................................................... 753 14.6 System Comparisons .................................................... 756 Acknowledgment .................................................................... 756 References ............................................................................. 757

15. Separation Processes Based on Reversible Chemical Complexation ................................................................................ 760 Summary ................................................................................ 760 15.1 Introduction ................................................................... 761 15.2 Specific Examples ........................................................ 764

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References ............................................................................. 772 16. Bubble and Foam Separations Ore Flotation ............................ 775

16.1 Introduction ................................................................... 775 16.2 Flotation Techniques .................................................... 775 16.3 Physicochemical Principles ........................................... 779 16.4 Flotaids ......................................................................... 792 16.5 Variables in Flotation .................................................... 796 References ............................................................................. 800

17. Bubble and Foam Separations Waste Treatment ..................... 806 17.1 Background .................................................................. 806 17.2 Theory of Separation .................................................... 809 17.3 Laboratory Studies ........................................................ 814 17.4 Role of Column Design ................................................. 818 17.5 Larger-Scale Studies .................................................... 818 17.6 Applications .................................................................. 822 17.7 Conclusions .................................................................. 822 References ............................................................................. 823

18. Ultrafiltration and Reverse Osmosis ............................................. 826 18.1 Ultrafiltration ................................................................. 826 18.2 Reverse Osmosis ......................................................... 836 References ............................................................................. 839 Bibliography ............................................................................ 839

19. Recent Advances in Liquid Membrane Technology ..................... 840 19.1 Introduction ................................................................... 840 19.2 General Description of Liquid Membranes .................... 840 19.3 Principles of Separation ................................................ 841 19.4 Practical Applications for Liquid Membranes ................. 845 19.5 Conclusions .................................................................. 858 References ............................................................................. 858

20. Separation of Gaseous Mixtures Using Polymer Membranes ................................................................................... 862 20.1 Introduction ................................................................... 862

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20.2 General Design Procedures and Application Examples ...................................................................... 872

20.3 Measurement and Phenomenological Description of Gas Sorption and Transport in Polymers .................. 883

20.4 Fundamentals of Sorption and Transport Processes in Polymers ................................................. 896

20.5 Characterization of Asymmetric Membranes ................. 916 20.6 Modeling and Design Considerations ............................ 920 Notation .................................................................................. 944 References ............................................................................. 944

21. Membrane Processes Dialysis and Electrodialysis ................... 954 21.1 Dialysis ......................................................................... 954 21.2 Electrodialysis ............................................................... 968 Notation for Section 21.1 ........................................................ 977 References ............................................................................. 978

22. Selection of a Separation Process ................................................ 982 22.1 Introduction ................................................................... 982 22.2 Initial Screening ............................................................ 982 22.3 Choosing the Base Case .............................................. 992 22.4 Process Simulation ....................................................... 992 22.5 Process Synthesis ........................................................ 993 Notation .................................................................................. 994 References ............................................................................. 995

Index .......................................................................................... 997

G E N E R A L P R I N C I P L E S

P A R T I

P h a s e E q u i l i b r i a

MICHAEL M. ABBOTTDepartment of Chemical and Environmental EngineeringRensselaer Polytechnic Institute, Troy, New York

JOHN M. PRAUSNlTZDepartment of Chemical EngineeringUniversity of California,Berkeley, California

1.1 INTRODUCTIONMost of the common separation methods used in the chemical industry rely on a well-known observation:when a multicomponent two-phase system is given sufficient time to attain a stationary state called equi-librium, the composition of one phase is different from that of the other. It is this property of nature whichenables separation of fluid mixtures by distillation, extraction, and other diffusional operations. For rationaldesign of such operations it is necessary to have a quantitative description of how a component distributesitself between two contacting phases. Phase-equilibrium thermodynamics, summarized here, provides aframework for establishing that description.

If experimental phase-equilibrium measurements were simple, fast, and inexpensive, chemical engineerswould have little need for phase-equilibrium thermodynamics because in that happy event all comjponent-distribution data required for design would be obtained readily in the laboratory. Unfortunately, however,component-distribution data are not easily obtained because experimental studies require much patienceand skill. As a result, required data are often not at hand but must be estimated using suitable physico-chemical models whose parameters are obtained from correlations or from limited experimental data.

It was Einstein who said that when God made the world, he was subtle but not malicious. The subtletyof nature is evident by our inability to construct models of mixtures which give directly to the chemicalengineer the required information in the desired form: temperature, pressure, phase compositions. Nature,it seems, does not choose to reveal secrets in the everyday language of chemical process design but prefersto use an abstract languagethermodynamics.

To achieve a quantitative description of phase equilibria, thermodynamics provides a useful theoreticalframework. By itself, thermodynamics cannot provide all the numerical information we desire but, whencoupled with concepts from molecular physics and physical chemistry, it can efficiently organize limitedexperimental information toward helpful interpolation and extrapolation. Thermodynamics is not magic; itcannot produce something for nothing: some experimental information is always necessary. But when usedwith skill and courage, thermodynamics can squeeze the last drop out of a nearly dried-up lemon.

The brief survey presented here must necessarily begin with a discussion of thermodynamics as alanguage; most of Section 1.2 is concerned with the definition of thermodynamic terms such as chemicalpotential, fugacity, and activity. At the end of Section 1.2, the phase-equilibrium problem is clearly statedin several thermodynamic forms; each of these forms is particularly suited for a particular situation, asindicated in Sections 1.5, 1.6, and 1.7.

C H A P T E R I

Section 1.3 discusses fugacities (through ftigacity coefficients) in the vapor phase. Illustrative examplesare given using equations of state.

Section 1.4 discusses fugacities (through activity coefficients) in the liquid phase. Illustrative examplesare given using semiempirical models for liquid mixtures of nonelectrolytes.

Section 1.5 gives examples for vapor-liquid equilibria at ordinary pressures and for liquid-liquid equi-libria. Section 1.6 discusses equilibria for systems containing a solid phase in addition to a liquid or gaseousphase, and Section 1.7 gives an introduction to methods for describing fluid-phase equilibria at highpressures.

This brief survey of applied phase-equilibrium thermodynamics can do no more than summarize themain ideas that constitute the present state of the art. Attention is restricted to relatively simple mixturesas encountered in the petroleum, natural gas, and petrochemical industries; unfortunately, limited spacedoes not allow discussion of other important systems such as polymer mixtures, electrolyte solutions,metallic alloys, molten salts, refractories (such as ceramics), or aqueous solutions of biologically importantsolutes. However, it is not only lack of space that is responsible for these omissions because, at present,thermodynamic knowledge is severely limited for these more complex systems.

1.2 THERMODYNAMIC FRAMEWORK FOR PHASE EQUILIBRIA

1.2-1 Conventions and Definitions

Lowercase roman letters usually denote molar properties of a phase. Thus, g, h, s, and v are the molarGibbs energy, molar enthalpy, molar entropy, and molar volume. When it is essential to distinguish betweena molar property of a mixture and that of a pure component, we identify the pure-component property bya subscript. For example, H1 is the molar enthalpy of pure /. Total properties are usually designated bycapital letters. Thus H is the total enthalpy of a mixture; it is related to the molar mixture enthalpy h byH = nh, where n is the total number of moles in the mixture.

Mole fraction is the conventional measure of composition. We use the generic symbol *, to denote thisquantity when no particular phase (solid, liquid, or gas) is implied. When referring to a specific phase, weuse common notation, for example, xt for liquid-phase mole fraction and y, for the vapor-phase molefraction. The dual usage of xf should cause no confusion because it will be clear from the context whetheran arbitrary phase or a liquid phase is under consideration.

The molar residual Junction mR (or mf) is the difference between molar property m (or m;) of a realmixture (or pure substance i) and the value rri (or m,') it would have were it an ideal gas at the sametemperature (T), pressure (P), and composition:

(1.2-1)(1.2-2)

The residual functions (e.g., ^ , hR, and s*) provide measures of the contributions of intermolecular forcesto thermodynamic properties.

The molar excess function mE is the difference between a molar mixture property m and the value mwthe mixture would have were it an ideal solution at the same temperature, pressure, and composition:

(1.2-3)

Excess functions are related to the corresponding residual functions:

(1.2-4)

Thus, the excess functions (e.g., gE, hE, and sE) also reflect the contributions of intermolecular forces tomixture property m. _

Partial molar property m, corresponding to molar mixture property m is defined in the usual way:

(1.2-5)

where subscript n, denotes constancy of all mole numbers except nh All m, have the important featurethat

(1.2-6)

or

(1.2-7)

That is, a molar property of a mixture is the mole-fraction-weighted sum of its constituent partial molarproperties. The partial molar property m, of species i in solution becomes equal to molar property m, ofpure i in the appropriate limit:

(1.2-8)

The chemical potential \it is identical to the partial molar Gibbs energy g,:

(1.2-9)

Thus, the fih when multiplied by mole fractions, sum to the molar Gibbs energy of the mixture:(1.2-10)

or

(1.2-11)

Partial molar properties play a central role in phase-equilibrium thermodynamics, and it is convenient tobroaden their definition to include partial molar residual functions and partial molar excess functions. Hence,we define, analogous to Eq. (1.2-5),

(1.2-12)

and

(1.2-13)

1.2-2 Criteria for Phase Equilibria

Consider the situation shown in Fig. 1.2-1, where two phases a and /3 are brought into contact and allowedto interact until no changes are observed in their intensive properties. The condition where these propertiesassume stationary values is a state of phase equilibrium. It is characterized by temperature T and pressureP (both assumed uniform throughout the two-phase system) and by the sets of concentrations {z") and{r,}, which may or may not be the sets of mole fractions {xf} and {xf}. The basic problem of phaseequilibrium is this: given values for some of the intensive variables (T1 P, and the concentrations), findvalues for the remaining ones.

The route to the solution of problems in chemical and phase equilibria is indirect; it derives from aformalism developed over a century ago by the American physicist J. W. Gibbs.1 Let G be the total Gibbsenergy of a closed, multiphase system of constant and uniform T and P. Equilibrium states are those forwhich G is a minimum, subject to material-balance constraints appropriate to the problem:

Gr>/> = minimum (1.2-14)

Although Eq. (1.2-14) is sometimes used directly for solution of complex equilibrium problems, it ismore often employed in equivalent algebraic forms which use explicitly the chemical potential or otherrelated quantities. Consider a closed system containing x phases and N components. Introducing thechemical potential tf of each component / in each phase p and incorporating material-balance constraints,one obtains as necessary conditions to Eq. (1.2-14) a set of N(ir 1) equations for phase equilibrium:

(1.2-15)

FIGURE 1.2-1 A multicomponent system in two-phase equilibrium.

Thus, temperature, pressure, and the chemical potential of each distributed component are uniform for aclosed system in phase equilibrium. If the system contains chemically reactive species, then additionalequations are required to characterize the equilibrium state.

Equation (1.2-15) is a basis for the formulation of phase-equilibrium problems. However, since thechemical potential has some practical and conceptual shortcomings, it is useful to replace ft, with a relatedquantity, fh tint fugacity. Equation (1.2-15) is then replaced by the equivalent criterion for phase equilibrium,

/ ? = / , ' ( i * l , 2 A f ; ; a , j 5 T - 1) (1.2-16)Equation (1.2-16) is the basis for all applications considered in this chapter. The major task is to representthe dependence of the fugacity on temperature, pressure, and concentration.

1.2-3 Behavior of the Fugacity

Table 1.2-1 summarizes important general thermodynamic formulas for the fugacity. Equations (1.2-17)and (1.2-18) define the fugacity/ of a component in solution; Eqs. (1.2-19) and (1.2-20) similarly definethe fugacity/of a mixture. For a component in an ideal-gas mixture, Eq. (1.2-18) implies that

fi^yf (ideal gas) (1.2-26)which leads to the interpretation of a vapor-phase fugacity as a corrected partial pressure. Equations (1.2-21) and (1.2-22) are useful summarizing relationships, which provide by inspection general expressionsfor the temperature and pressure derivatives of the fugacities; note here the appearance of the residualenthalpy hR. Equations (1.2-23) and (1.2-24) are partial-property relationships, and Eq. (1.2-25) is oneform of the Gibbs-Duhem equation.

A pure substance / may be considered a special case of either a mixture or of a component in solution,in the limit as mole fraction Jt1 approaches unity. Thus, formulas for the fugacity/ of pure i are recoveredas special cases of Eqs. (1.2-2I)-(1.2-22). In particular,

(1.2-27)

(1.2-28)

(1.2-29)

Phase 0

Phase a

The fugacity of a pure substance depends on T and P. Absolute values for/ are computed from

(1.2-30)

which follows from Eqs. (1.2-28) and (1.2-29) upon introduction of the compressibility factor Z, ( sPvJRT). Use of Eq. (1.2-30) requires a PVT equation of state, valid from P = 0 to the physical state ofinterest at pressure P. Relative values of / are given by the Poynting correction:

(1.2-31)

which follows from Eq. (1.2-29). Equation (1.2-31) is most often used for calculation of the fugacity of acondensed phase, relative to the fiigacity of the same phase at saturation pressure Pf".

Suppose we require the absolute fugacity of a pure subcooled liquid at some pressure P and that availabledata include the vapor-liquid saturation pressure Pf \ an equation of state for the vapor phase, and molarvolumes i>f for the liquid. Application of Eqs. (1.2-30) and (1.2-31), together with the criterion for pure-fluid vapor-liquid equilibrium,

(1.2-32)

gives the required result, namely,

(1.2-33)

Figure 1.2-2 shows the fugacity of nitrogen at 100 K, as computed from Eqs. (1.2-30), (1.2-32), and(1.2-33). Also shown are several commonly employed approximations. The dashed line/^ = P is the ideal-gas approximation to the vapor fugacity; it is a special case of Eq. (1.2-26) and is a consequence of thedefinition, Eq. (1.2-28). Note that the ideal-gas approximation becomes asymptotically valid as P ap-proaches zero.

The horizontal dashed line/J' = Ff* is the approximation toff employed in Raoult's Law for vapor-

TABLE 1.2-1 Summary of Thermodynamic Relations for Fugacity

(1.2-17)

(1.2-18)

(1.2-19)

(1.2-20)

(1.2-21)

(1.2-22)

(1.2-23)

(1.2-24)

(1.2-25)

P (bar)FIGURE 1.2-2 Pressure dependence of fugacity/of nitrogen at 100 K. Dashed and dotted lines representapproximations to real behavior.

liquid equilibrium. A much better approximation to ff at moderate pressure is afforded by the horizontaldotted line, / f = /f". This approximation involves neglect of the Poynting correction given by Eq. (1.2-31).

Since the molar volume of a condensed phase is frequently insensitive to pressure, Eq. (1.2-31) canoften be approximated by

(1.2-34)

With Vi taken as the molar volume of the saturated liquid, relative fugacities computed from Eq. (1.2-34) for subcooled liquid nitrogen at 100 K produce results nearly identical to those given by the solid curvein Fig. 1.2-2.

The fugacity/ of a component in solution depends on temperature, pressure, and composition. Figure1.2-3 shows the variation/ with JC, for acetone in two binary liquid mixtures (acetone-methanol and acetone-chloroform) at 1 bar and 500C. Although they differ in detail, both/ versus *, curves have certain featuresin common. For example,

(1.2-35)

that is, the fugacity of a component in solution approaches zero as its concentration approaches zero.Moreover,

(1.2-36)

that is, the fugacity of a component in a nonelectrolyte solution asymptotically approaches the linearbehavior represented by the dashed straight line

(1.2-37)

as its mole fraction approaches unity.There is an analogous statement to Eq. (1.2-36) which applies to the limit of zero concentration in a

binary mixture, namely,

(1.2-38)

/(bar)

Afi i n m e t h a n o l

fi (ba

r)

Afi i n c h l o r o f o r m

xi

FIGURE 1.2-3 Composition dependence of fugacity / of acetone in two binary liquid mixtures at 500Cand 1 bar. Dashed and dotted lines represent approximations to real behavior.

where Henry's constant 3C1-j is, for binary nonelectrolyte solutions, a positive definite number that dependson temperature and pressure. Unlike/- in the analogous Eq. (1.2-36), the numerical value of 3C/y alsodepends on the identity of the other component j in the mixture; hence, the double subscript notation onX1J. The dotted straight lines in Fig. 1.2-3 represent the equations

(1.2-39)

which are given by construction as tangent lines drawn to the/ versus X1 curves at JC,- = 0. Henry's constantsare then represented as intercepts of these tangent lines with the vertical axis *, = 1.

Equations (1.2-35), (1.2-36), and (1.2-37) apply without modification to species / in a multicomponentmixture. However, Henry's constant, as defined by Eq. (1.2-38), can assume an infinity of values dependingon the solvent composition. Thus, Henry's constant for a solute species in a multicomponent mixture is afunction of temperature, pressure, and composition. The thermodynamic treatment of this topic is complexand is not considered in this chapter; the reader is referred to an article by Van Ness and Abbott.2

1.2-4 Normalized Fugacities

The group xJP appears as part of the definition of the component fugacity: see Eq. (1.2-18). It followsfrom this definition that ft for a species in a vapor mixture is normally of the same order of magnitude asthe partial pressure y,P: see Eq. (1.2-26). Thus, it is convenient to introduce a normalized fugacity, called

the fugacity coefficient 4>h defined as the ratio of the component fugacity to the pressure-compositionproduct:

(1.2-40)

Similarly, we write for a mixture that

(1.2-41)

and for a pure component / that

(1.2-42)

Fugacity coefficients are dimensionless; they are identically unity for ideal gases. For nonreacting realgases, their values approach unity as pressure approaches zero. Table 1.2-2 summarizes general thermo-dynamic relationships for the fugacity coefficients. Section 1.3 discusses the calculation of fiigacity coef-ficients from PVTx equations of state.

The composition dependence of the component fugacity J1 in condensed phases is conventionally rep-resented through either of two normalized quantities called the activity and the activity coefficient. Themotivation for the definitions of these quantities was provided by Fig. 1.2-3 and the accompanying dis-cussion, where it was shown that for binary nonelectrolyte solutions the limiting / versus JC, behavior is asimple proportionality, given by Eq. (1.2-37) for 1^ -* 1 and by Eq. (1.2-39) for xt -> 0. Either of theselimiting laws, when assumed to apply to all compositions at fixed temperature and pressure, can be usedto define an ideal solution. We generalize this notion by writing

/ i d s *f? (constant T, P) (1.2-48)where superscript id denotes ideal-solution behavior and ft is the standard-state fugacity of species /. Ifthe ideal solution is defined so as to reproduce real behavior for X1 -* 1, then/f = fh and Eq. (1.2-48)becomes

ff (RL) m JC,./ (constant T, P) (1.2-49)

TABLE 1.2-2. Summary of Thermodynamic Relationshipsfor the Fugacity Coefficient

(1.2-40)

(1.2-41)

(1.2-43)

(1.2-44)

(1.2-45)

(1.2-46)

(1.2-47)

where RL indicates that we have chosen a Raoult's-Law standard state. If the ideal solution is defined soas to reproduce real behavior for Jt, - 0, then/? = 3C(J, and Eq. (1.2-48) becomes

}'f (HL) s JC1. K1J (constant 7, P) (1.2-50)

where HL denotes the choice of a Henry's Law standard state. In defining an ideal solution, it is notnecessaiy that one use the same standard-state convention for all components in the mixture.

Equation (1.2-48) is the basis for the definitions of the activity

The transformation of Eq. (1.2-62) from an abstract formulation to one appropriate for engineeringcalculations is accomplished by elimination of the component fugacities ff in favor of the normalizedauxiliary functions ij>; and/or 7,. For two-phase equilibrium, there are three general possibilities:

1. Introduce the activity coefficient for one phase (say a) and the fugacity coefficient for the other.Then, by Eqs. (1.2-52), (1.2-40), and (1.2-62), we obtain

(1.2-63)

2. Introduce activity coefficients for both phases, obtaining

(1.2-64)

3. Introduce fugacity coefficients for both phases, obtaining (since P is uniform)

(1.2-65)

For each of the formulations 1,2, and 3, there are further choices one can make. For example, in Eq.(1.2-63) the choice of standard states for the activity coefficients has been left open.

Which of the above formulations one adopts for a particular problem is determined not only by thetype of equilibrium (e.g., vapor-liquid, liquid-liquid, or solid-liquid) but also by the type and extent ofthermodynamic data available for evaluation of the auxiliary functions. Representation and evaluation ofthe auxiliary functions is treated in the next two sections.

1.3 FUGACITY COEFFICIENTS

1.3-1 Fugacity Coefficients and the Equation of State

The route to a fugacity coefficient is through a PVTx equation of state. By Eq. (1.2-44), we have for amixture that

(1.3-1)

and thus, as a special case, we obtain for pure component 1 that

(1.3-2)

TABLE 1.2-3. Summary of Thermodynamic Relationshipsfor the Activity Coefficient

(1.2-52)

(1.2-58)

(1.2-59)

(1.2-60)

(1.2-61)

Determination of fugacity coefficients from these equations requires an expression for the compressibilityfactor as a function of temperature, pressure, and (for a mixture) composition. Such an expression, offunctional form

Z = Z (T, P, x)

is called a volume-explicit equation of state, because it can be solved to give the molar volume v as analgebraically explicit function of 7\ P, and JC.

The analogous expression for In 0, follows from Eq. (1.2-43) or, equivalently, from Eq. (1.3-1) viathe partial-property relationship Eq. (1.2-45). Thus,

(1.3-3)

where Z1 is the partial molar compressibility factor:

Determination of ^, therefore requires the same information as that required for the mixture . However,because of the differentiation required to find Z1 and hence h the details of the composition dependenceof Z are crucial here. These details are conventionally expressed in the mixing rules for the equation-of-state parameters.

The above discussion presumes the availability of a volume-explicit equation of state. For applicationsto gases at moderate to high pressures or densities or to vapors and liquids, realistic equations of state arenot volume explicit but are instead pressure explicit. That is, Z is expressed as a function of T3 v, and JCor, equivalently, of T, p (molar density s iT1), and JC:

Z = Z ( r , p , JC)

In this event, Eqs. (1.3-1), (1.3-2), and (1.3-3) are inappropriate; one uses instead the equivalent expres-sions

(1.3-4)

(1.3-5)

(1-3-6)

Here, quantity 2k is a partial molar compressibility factor evaluated at constant temperature and total volume:

(1-3-7)

Again, the details of the composition dependence of the equation of state, as contained in Z,, are crucialto the determination of accurate values for 4>,-.

There is no known PVTx equation of state that is suitable for calculation of fugacity coefficients for allmixtures at all possible conditions of interest. The choice of an equation of state for an engineeringcalculation is therefore often made on an ad hoc basis. Guidelines are available, but they reflect the inevitablecompromise between simplicity and accuracy. We treat in the remainder of this section three popular classesof equations of state commonly employed for practical calculations: the virial equations, used for gases atlow to moderate densities; the cubic equations of state (exemplified by the Redlich-Kwong equations),used for dense gases and liquids; and equations inspired by the so-called "chemical theories," used forassociating vapors and vapor mixtures.

1.3-2 Virial Equations of State

Virial equations of state are infinite-series representations of the gas-phase compressibility factor, witheither molar density or pressure taken as the independent variable for expansion:

(1.3-8)(1.3-9)

Parameters B1 C, D, . . . are density-series virial coefficients, and B', C\ D', . . . are pressure-seriesvirial coefficients. Virial coefficients depend only on temperature and composition; they are defined throughthe usual prescriptions for coefficients in a Taylor expansion. Thus, the second virial coefficients are givenas

Similarly, the third virial coefficients are defined as

Higher virial coefficients are defined analogously as higher-order derivatives of Z, each of them evaluatedat the state of zero density or zero pressure.

The pressure-series coefficients and density-series coefficients are related:

and so on

Thus, the virial expansion in pressure, Eq. (1.3-9), can be written in terms of density-series virial coeffi-cients:

(1.3-10)

This form is preferred to Eq. (1.3-9) because the density-series coefficients are the ones normally reportedby experimentalists, and they are the ones for which correlations (for B and C) are available.

In practice, one must work with truncations of any infinite-series representation and, since virial coef-ficients beyond the third are rarely available, Eqs. (1.3-8) and (1.3-10) are normally truncated after two orthree terms. For low pressures, the two-term truncation of Eq. (1.3-10) is sufficient:

(1.3-11)

For more severe conditions, the three-term truncation of Eq. (1.3-8) is preferred:

(1.3-12)

Equation (1.3-11) should not be used for densities greater than about half the critical value, and Eq.(1.3-12) should not be used for densities exceeding about three-quarters of the critical value. Note that Eq.(1.3-11) can be considered either a volume-explicit or a pressure-explicit equation of state, whereas Eq.(1.3-12) is pressure explicit.

The great appeal of the virial equations derives from their interpretations in terms of molecular theory.Virial coefficients can be calculated from potential functions describing interactions among molecules.More importantly, statistical mechanics provides rigorous expressions for the composition dependence ofthe virial coefficients. Thus, the nth virial coefficient of a mixture is nth order in the mole fractions:

(1.3-13)

and so on (1.3-14)

The subscripted coefficients Bij9 Cijk1 . . . depend only on T, and their numerical values are unaffected onpermutation of the subscripts. Coefficients with identical subscripts (Bn , C222, etc.) are properties of puregases. Those with mixed subscripts {Bn = B2x, C122 = C212, etc.) are mixture properties; they are calledinteraction virial coefficients or cross virial coefficients.

Expressions for fugacity coefficients follow from Eqs. (1.3-2), (1.3-3), and (1.3-11) or from Eqs.(1.3-5), (1.3-6), and (1.3-12). For applications at low pressures, we find for the two-term virial equationin pressure that

(1.3-15)

and

(1.3-16)

Similarly, for conditions requiring the use of the three-term virial equation in density, we obtain

(1.3-17)and

(1.3-18)

In Eqs. (1.3-16) and (1.3-18), quantities S, and C, are partial molar virial coefficients, defined by

and determined from the mixing rules given by Eqs. (1.3-13) and (1.3-14). General expressions for, andC1 and summarized in Table 1.3-1; for components 1 and 2 in a binary mixture, they reduce to

(1.3-19a)(i.3-19b)

TABLE 1.3-1 Expressions for the Partial Molar Virial Coefficients B1 and C,

where

and

where

Source: Van Ness and Abbott.1

and

where(1.3-20)

and(1.3-2Ia)(1.3-2Ib)

where(1.3-22)

For a binary gas mixture at low pressure, Eqs. (1.3-16) and (1.3-19) provide the following frequentlyused expressions for the fugacity coefficients:

(1.3-23a)

(1.3-23b)

Since 6,2 = 2B12 B n B22, t n e details of the composition dependence of #, and ^2 a r e directlyinfluenced by the magnitude of the interaction coefficient B12. The effect is illustrated in Fig. 1.3-1, whichshows values of 0, versus y, computed from Eq. (1.3-23a) for a representative binary system for whichthe pure-component virial coefficients are B n = 1000 cm3/mol and S22 = -2000 cm3/mol. The tem-perature is 300 K and the pressure is 1 bar; the curves correspond to different values of B12, which rangefrom -500 to -2500 cm3/mol. All curves approach asymptotically the pure-component value

as y{ approaches unity, but the infinite-dilution behavior (as yx -* 0) varies from case to case. For thespecial case B12 = -1500 cnvVmol, corresponding to 6,2 0, we see that ^, = constant =

as is parameter b:

(1.3-28)

Equations (1.3-27) and (1.3-28) follow from the classical critical constraints:

The expression for 0, Eq. (1.3-26), was obtained by forcing agreement of predicted with experimentalvapor pressures of pure hydrocarbon liquids. (This procedure is essential if the equation of state is to beused for prediction or correlation of vapor-liquid equilibria.)

Equation (1.3-25) is explicit in pressure; it may be written in the alternative form

(1.3-29)

where p is the molar density. Expressions for the fugacity coefficients then follow on application of Eqs.(1.3-4), (1.3-6), and (1.3-7). The results are

(1.3-30)

and

(1.3-31)

In Eqs. (1.3-30) and (1.3-31), all unsubscripted quantities refer to the mixture. Quantities ~bt and 0,- arepartial molar equation-of-state parameters, defined by

Determination of b{ and B1 requires a set of mixing rules for parameters b and B. The usual procedureis to assume that b and B are quadratic in composition:

(1.3-32)

(1.3-33)

Here, JC is a generic mole fraction and can refer to any phase. When subscripts / and j are identical in Eq.(1.3-32) or (1.3-33), the parameters refer to a pure component. When they are different, the parametersare called interaction parameters and these depend on the properties of the binary i-j mixture as indicatedby the subscripts. To estimate these interaction parameters, we use combining rules, for example,

(1.3-34)(1.3-35)

where Cij and ktj are empirical binary parameters, small compared to unity, that often are nearly independentof temperature over modest temperature ranges. Frequently, c(j is set equal to zero, but it is almost alwaysnecessary to use for k(j some number other than zero. With mixing rules given by Eqs. (1.3-32) and (1.3;33) and combining rules given by Eqs. (1.3-34) and (1.3-35), one finds the following expressions for bfand B1 for components 1 and 2 in a binary mixture:

(1.3-36a)(1.3-36b)

and(1.3-37a)(1.3-37b)

Calculation of fugacity coefficient , for component i in a binary mixture at specified temperature,pressure, and composition is straightforward but tedious and is best done with a computer. First, one findsthe pure-component equation-of-state parameters from Eqs. (1.3-26), (1.3-27), and (1.3-28), and the in-teraction parameters from Eqs. (1.3-34) and (1.3-35). Application of the mixing rules, Eqs. (1.3-32) and(1.3-33), then given parameters b and B for the mixture. Knowing these quantities, one determines themixture p and Z from Eq. (1.3-29). Because the equation of state is cubic in molar density, an analyticalsolution for p (and hence Z) is possible; however, numerical techniques may often be just as fast. Giventhe mixture p and Z, one next finds the mixture from (1.3-30); these quantities, together with the b, and5, as given by Eqs. (1.3-36) and (1.3-37), finally permit calculation of the & from Eq. (1.3-31).

The behavior of Soave-Redlich-Kwong fugacity coefficients is best illustrated by numerical example.In Fig. 1.3-2 we show computed values of ^1 for / = H2S in the H2S-ethane system at 300 K. Two pressurelevels are represented: 15 bar, for which states of superheated vapor are obtained at all compositions, and50 bar, for which all states are subcooled liquids. In this example, interaction parameter c,2 is set equal tozero; for each pressure level, the different curves correspond to different values of kn-> which varies from-0.20 to +0.20. For the vapor mixtures, behavior similar to that illustrated in Fig. 1.3-1 is observed:variations in Jt12 are reflected qualitatively in the shapes of the ^H2S curves but, for these conditions, thequantitative effects are not large. The situation is dramatically different for the liquid mixtures. Here, smallchanges in kx2 promote large changes in #H2S; typically, the effect on , is greatest for mixtures dilute incomponent /. Analysis of vapor-liquid equilibrium data for this system shows that kxl is about 0.10 at300 K; comparison of the curves in Fig. 1.3-2 illustrates the substantial effect of this apparently small

Mole fraction H2S

FIGURE 1.3-2 Composition dependence of fugacity coefficient of hydrogen sulfide in binary mixtureswith ethane at 300 K. Curves labeled V are for superheated vapors at 15 bar; those labeled L are forsubcooled liquids at 50 bar. All curves are computed from the Soave-Redlich-Kwong equation, with valuesof interaction parameter kX2 as shown.

V

L

quantity. This example demonstrates an extremely important feature of applied equation-of-state thermo-dynamics: the implications of mixing rules and combining rules are seen most dramatically in fugacitycalculations for dense phases. Application of an equation of state to vapor-liquid equilibrium calculationsvia formulation 3 of Section 1.2-5 therefore requires mixing rules of appropriate flexibility. Developmentand testing of such rules is a major area of research in chemical engineering thermodynamics.

1.3-4 Chemical Theories of Vapor-Phase Nonideaiity

It may happen that the nonideal behavior of gases results wholly or partly from stoichiometric effectsattributable to the formation of extra chemical species. When this is the case, a "chemical theory" can beused to develop an equation of state from which fugacity coefficients may be determined. A generaltreatment of chemical theories is beyond the scope of this chapter; to illustrate the principles involved, wedevelop instead, by way of example, the procedure for treating strong dimerization in gases at low pressure.

For orientation, consider the following simple thought emperiment. A gas mixture, of total apparentnumber of moles /?, is contained in a vessel of known total volume V which is submerged in a thermostatedbath at known temperature T. A measurement of the equilibrium pressure of the gas permits calculation ofthe apparent compressibility factor Z:

(1.3-38)

Suppose now that the apparent number of moles n is not the correct value; thatfor whatever reasonthetrue value is n'. Then the true compressibility factor Z' is

(1.3-39)

Now quantities 7, P, and K are the same in Eqs. (1.3-38) and (1.3-39); they are values obtained by directmeasurement or by calibration. Combination of the two equations thus produces the relation

(1.3-40)

Equation (1.3-40) is one of the fundamental equations for the chemical theory of vapor-phase non-idealities. It asserts that the apparent, or observed, compressibility factor Z differs from the true value Z'because of differences between the apparent, or assumed, mole number n and the true value n'. In achemical theory, such differences are assumed to obtain because of the occurrence of one or more chemicalreactions. If the reactions are at equilibrium, then one finds the following relationship for the apparentfugacity coefficient }. Todo this, one must propose a reaction scheme: this provides relationships for n'ln and y/Ay, in terms ofequilibrium conversions. One must also assume an expression for Z', which in turn implies an expressionfor the ;. The true fugacity coefficients ,', when incorporated into the criteria for chemical-reactionequilibrium for the true mixture, permit determination of the equilibrium conversions, and hence, finally,via Eqs. (1.3-40) and (1.3-41), expressions for Z and 0, as functions of 7, P, and the set of apparentcompositions {y,}.

The simplest cases (the only ones considered here) obtain for pressures sufficiently low that the truemixtures can be considered ideal-gas mixtures. In this event, Z' = 1 and J^ = 1, and Eqs. (1.3-40) and(1.3-41) reduce to

(1.3-42)

and

(1.3-43)

Suppose that n'ln > 1, as would occur, for example, as the result of a dissociation reaction undergone bya nominally pure chemical species. Then, according to (1.3-42), the apparent compressibility factor isgreater than unity. On the other hand, suppose that n'ln < 1, as would occur, for example, if a nominallypure substance underwent association. Then, by (1.3-42), the apparent compressibility factor is less thanunity. In both casesdissociation and association of a nominally pure substance at low pressuretheapparent fugacity coefficient is also different from the expected value of unity.

The simplest example of self-association in the vapor phase is dimerization, as exemplified by hydrogenbonding in carboxylic acids. Consider the dimerization of acetic acid:

where the dots denote hydrogen bonds. Evidence for vapor-phase association of acetic acid is provided bythe PVT data of MacDougall,12 shown in Fig. 1.3-3 as a plot of Z versus P for a temperature of 400C.Even though the pressure level is extremely low, the apparent compressibility factor is small (0.7 or lessfor P > 0.005 atm); at this pressure any normal vapor would exhibit a Z very close to unity.

If we write the acetic acid dimerization reaction as

then reaction stoichiometry provides the following material balance equations:

P/10- 3 (a tm)

FIGURE 1.3-3 Compressibility factor Z for acetic acid vapor at 400C, Circles are data; curve is computedfrom chemical theory, assuming dimerization, with K = 380.

Z

Here nA is the apparent number of moles of monomer, the primed quantities are true mole numbers, ande is the number of moles of dimer formed. The material balance equations produce expressions for the truemole fractions:

where is a dimensionless extent of reaction:

By Eq. (1.3-42), the apparent compressibility factor is

and it remains to determine . We do this by assuming that the true mixture is at chemical-reactionequilibrium:

or

from which

(1.3-44)

where K is the chemical-reaction equilibrium constant. Thus, we obtain finally the following expressionfor Z:

(1.3-45)

A test of the usefulness of the dimerization model is provided by the ability of Eq. (1.3-45) to representMacDougall's volumetric data for acetic acid vapor. The solid line in Fig. 1.3-3, generated from Eq.(1.3-45) with K = 380, provides an excellent fit of the data; one concludes that the dimerization model isconsistent with the observed PVT behavior at 400C. The apparent fugacity coefficient for acetic acid vapor,found from Eq. (1.3-43) with 0, = A is small (0.6 or less for P > 0.005 atm) ata pressure level where we would expect it to be very nearly unity.

So far we have only considered the behavior of apparently "pure" acetic acid vapor. Dimerization alsooccurs in vapor mixtures containing carboxylic acids. The effect on component fugacity coefficients is easilyillustrated for binary vapor mixtures containing acetic acid and an inert substance I. For this example, thetrue mixture contains three species: monomer, dimer, and inert. A development similar to that just presentedproduces a similar expression for the dimensionless extent of reaction:

(1.3-47)

P/l

FIGURE 1.3-5 Composition dependence of fugacity coefficients at 400C and 0.025 atm in binary gasmixture containing acetic acid and an inert component. Curves are computed from chemical theoiy, as-suming dimerization of acetic acid, with K = 380.

1.4 ACTIVITYCOEFFICIENTS

1.4-1 Activity Coefficients and the Excess Gibbs Energy

The route to an activity coefficient is through an expression for the dimensionless excess Gibbs energy,gE/RT, to which In 7,- is related as a partial molar property:

(1.2-59)

For a binary mixture, In 7, and In y2 are conveniently expressed in terms of gE and its mole-fractionderivative:

(1.4-la)

(1.4-lb)

Acetic acid

Inert

Hence, gE is a generating function for the activity coefficient; given an expression for the compositiondependence of gE, expressions for the 7, follow. Conversely, by (1.2-60), gEIRT is the mole-fraction-weighted sum of the In 7,; given values for the 7,, values for gE follow:

(1.2-60)

If gE is known as a function of temperature, pressure, and composition, then the other excess functionscan be derived from it. For example, the excess entropy is proportional to the temperature derivative ofgE'

(1.4-2)

Similarly, the excess volume ("volume change of mixing'*) is equal to the pressure derivative of gE:

(1.4-3)

The excess enthalpy ("heat of mixing") can be related either to gE and sE or, through Eq. (1.4-2), to thetemperature derivative of gEIRT:

(1.4-4a)

(1.4-4b)

Equation (1.4-4b) is an example of a Gibbs-Helmholtz equation.Equations (1.2-59), (1.2-60), and (1.4-1)-(1.4-4) are valid regardless of the standard-state conventions

adopted for the components of the mixture. If Raoult's Law standard states are used for all components(the usual procedure), then

and

In particular,

Thus, for a binary mixture, the excess functions are identically zero at the composition extremes; for anymixture, the activity coefficient of a component approaches unity as that component approaches purity.

The state of infinite solution is also of special interest. We define the activity coefficient of componenti at infinite dilution by

(1.4-5)

Usually, but not always, 7,* is the extreme (maximum or minimum) value assumed by 7, for a componentin a binary mixture. Hence, the 7" are often used as measures of the magnitudes of nonidealities of binaryliquid mixtures. Another measure is provided by gE or gE/RT for the equimolar mixture; for many binarysolutions, this is near to the maximum (or minimum) value. For liquid solutions exhibiting positive devia-tions from ideal-solution behavior (activity coefficients greater than unity), a 7,* of about 5 or an equimolargEIRTof about 0.5 is considered "large."

For binary solutions, the auxiliary function

(1.4-6)

is convenient for displaying and smoothing experimental data. The activity coefficients are obtained fromit via the expressions

(1.4-7a)

(1.4-7b)

which are analogues of Eqs. (1.4-la) and (1.4-Ib). In particular, the y are simply related to the limitingvalues of 9:

The diversity of behavior exhibited by real liquid solutions is illustrated by Fig. 1.4-1, which showsplots of 8 and the In 7,- versus mole fraction for six binary mixtures at 5O0C and low pressures. Of the sixmixtures, acetone-methanol shows the simplest behavior, with nearly symmetrical curves for the In 7, andan essentially horizontal straight line for S. The magnitudes of the deviations from ideality are small: the7J* are only about 1.9. Methyl acetate-1-hexene is an example of a simple nonsymmetrical system: the7" are again of modest size, but here S is described by a straight line of nonzero slope. Ethanol-chloroformis highly asymmetrical, and the 7, exhibit interior extrema with respect to composition; here, the activitycoefficient of chloroform shows a pronounced maximum at an ethanol mole fraction of about 0.85. Ethanol-/i-heptane exhibits extremely large positive deviations from ideality, so large that the equimolar mixture isvery close to a condition of instability with respect to liquid-liquid phasesplitting. Acetone-chloroformshows negative deviations from ideality, and benzene-hexafluorobenzene is a rare example of a systemthat shows negative and positive deviations: for low benzene concentrations gE is negative, whereas forlarge concentrations it is positive.

One of the goals of applied solution thermodynamics is to develop expressions for gE of minimalcomplexity but of sufficient flexibility to represent the various types of behavior illustrated by Fig. 1.4-1.It is desirable for such expressions to have a sound physicochemical basis, so that the numerical values ofthe parameters in the expressions are susceptible to correlation and estimation. Moreover, one would liketo be able to reliably estimate gE for multicomponent mixtures with parameter values determined frombinary data only. Finally, a desirable expression for gf would incorporate a "built-in" temperature depen-dence accurate enough to permit estimation of gE at one temperature from parameters determined at another.By Eq. (1.4-4b), this requires that the expression for gE imply realistic values for the excess enthalpy hE.

No known model for gE meets all the above criteria. As with PVTx equations of state, the choice ofan expression for gE for an engineering calculation is frequently made on an ad hoc basis. We discuss inthe next two sections two classes of expressions commonly employed for practical work: classical empiricalexpressions for ^ and more recent expressions for gE based on the local-composition concept.

1.4-2 Empirical Expressions for gE

The simplest procedure for generating an expression for gE of a binary mixture is through a power-seriesexpansion of the function g in mole fraction. For example, one can write

(1.4-8)

where parameters /412, A2x 9 D12, Z)2I are functions of temperature and (in principle) pressure. However,for most liquid mixtures the pressure dependence of gE is small; it is usually ignored or else incorporatedthrough the excess volume via Eq. (1.4-3). Equation (1.4-8) is the generalized Margules equation.

The Margules equation is rarely used in a form containing more than four parameters. By Eq. (1.4-7),the four-parameter Margules equation gives the following expressions for the activity coefficients:

(1.4-9a)

(1.4-9b)

FIGURE 1.4-1 Composition dependence of 9 ( s ^IxxX2RT) and of activity coefficients for six binaryliquid mixtures at 500C and low pressures: (a) acetone (l)-methanol(2); (b) methyl acetate(l)-l-hexene(2);(c) ethanol(l)-chloroform(2); (d) ethanol(l)--heptane(2); (e) acetone(l)-chloroform(2); and if) ben-zene(l)-hexafluorobenzene(2).

(b)(a)

(c) (d)

(e)

(f)

Because they require four binary parameters, Eqs. (1.4-9a) and (1.4-9b) are infrequently used for processcalculations. However, they are required for the precise representation of the 7, for binary mixtures inwhich 9 exhibits a reversal in curvature, for example, the ethanol-chloroform system shown in Fig.1.4-1.

Lower-order Margules equations follow from Eqs. (1.4-8) and (1.4-9) on appropriate assignments ofthe parameters. Thus, with D12 = D2x D1 w e obtain the three-parameter Margules equation, for whichgEfRT and the activity coefficients are

(1.4-10)

and

(1.4-1 Ia)(1.4-Ub)

Equations (1.4-10) and (1.4-11) are appropriate for binary mixtures in which 9 exhibits modest curvature,such as the acetone-chloroform system shown in Fig. 1.4-1.

With D12 D2\ = 0 , one obtains the two-parameter Margules equation, for which

(1.4-12)

and(1.4-13a)(1.4-13b)

Here, 9 is linear with mole fraction: see the methyl acetate-1-hexene system in Fig. 1.4-1. Finally, ifD\i ~ D2x = 0 and An = A21 = A, then Eq. (1.4-8) reduces to the one-parameter Margules equation, orPorter equation. Here,

(1.4-14)

and

(1.4-15a)(1.4-15b)

The Porter equation is the simplest realistic expression for gE. It is appropriate for "symmetrical" binarymixtures showing small deviations from ideality, for example, the acetone-methanol system depicted inFig. 1.4-1.

An alternative procedure to the series expansion of Eq. (1.4-8) is a power-series representation in molefraction of the reciprocal of 9- This expansion may be written

(1.4-16)

where again parameters AJ2, A2U DJ2, D 2 1 , . . . depend on temperature and (in principle) pressure. Equation(1.4-16) is a generalized van Laar equation. The van Laar equation is almost always applied in its two-parameter form, for which

(1.4-17)

and

(1.4-18a)

(1.4-18b)

If A\2 - A2x = Ay then Eq. (1.4-17) reduces to the Porter equation.

The infinite-dilution activity coefficients given by the generalized Margules and van Laar equations are

In y? = /i,2(Margules) = /t|2(van Laar) (1.4-19a)In

yf /i2J(Margules) = /I2* ,(van Laar) (1.4-19b)

These expressions justify the notation adopted for the parameters: subscript 12 denotes the state of infinitedilution of component 1 in component 2; similarly, subscript 21 denotes the state of infinite dilution ofcomponent 2 in component 1.

The Margules and van Laar equations have different correlating capabilities. These differences are mosteasily illustrated for the two-parameter expressions, Eqs. (1.4-12) and (1.4-17). Figure 1.4-2 shows plotsof 9 and the In 7, generated from these equations for four cases. In each case, we have taken In 7J0 = 0.5and have fixed A2x and A2x by Eq. (1.4-19b); the different cases represent different values of In 7J8, with/4,2 and A\2 given by Eq. (1.4-19a). For the smallest value of the ratio Rn (= AyA2x = AnIA2x), thegenerated curves differ only in minor detail; were this a data-fitting exercise, the two equations would showsimilar correlating abilities for a real mixture exhibiting modest deviations from ideality and modest asym-metry. However, differences in the behavior of the equations become apparent as Rn increases. Withincreasing Rn, the van Laar 9 exhibits increasingly greater curvature, and the Margules activity coefficientseventually (for Rn > 2) show interior extrema. Which (if either) of these two expressions one wouldchoose for a highly asymmetrical mixture (e.g. , as represented by Fig. 1.4-2d) would be determined by a

(C) (d)

FIGURE 1.4-2 Composition dependence of 9 (^gE/xxx2RT) and of activity coefficients for binary mix-tures, as represented by the two-parameter Margules equation (solid curves) and by the van Laar equation(dashed curves). For all cases, In 7 " = 0.5; different cases correspond to different values for In 7* .

(a) (b)

careful examination of the available data; both kinds of behavior illustrated in Fig. 1 A-2d are observed forreal mixtures.

The two-parameter van Laar equation cannot represent maxima or minima in the activity coefficients,nor can it represent the mixed deviations from ideality exemplified by the benzene-hexafluorobenzenesystem of Fig. 1.4-1. However, it is superior to the Margules equation for some extremely nonideal systemssuch as alcohol-hydrocarbon mixtures, for example, the ethanol-/i-heptane system of Fig. 1.4-1. For suchmixtures the two-parameter Margules equation often incorrectly predicts liquid-liquid phase splitting. Higher-order Margules equations can sometimes be used for these systems, but at the expense of many additionalparameters.

Multicomponent extensions of the Margules and van Laar equations are available. The simplest is theexpression

(1.4-20)

which asserts that g* for a multicomponent mixture is just the sum of the gE values for the constituentbinary mixtures. Thus, for a ternary mixture containing components 1,2, and 3

Although acceptable performance is sometimes obtained, neither Eq. (1.4-20) nor other formulationsincorporating only binary parameters have proved generally reliable for estimating gE and activity coeffi-cients for moderately to highly nonideal multicomponent mixtures. This shortcoming derives ultimatelyfrom the inherent empiricism of Eqs. (1.4-8) and (1.4-16); these series expansions have little physicalsignificance. Reasonable extrapolation from binary to multicomponent behavior requires models for g witha sound basis in molecular theory.

1.4-3 Local Composition Expressions for gE

The inability of the classical expressions for g8 to provide adequate descriptions of multicomponent behaviorhas inspired the search for molecularly based models. The most popular class of models is based on theconcept of local composition, which explicitly recognizes that the local environment of a molecule insolution with molecular species of other types is not the same as that provided by the overall compositionof the solution. Local mole fractions are semiempirically related to overall mole fractions through Boltzmannfactors incorporating characteristic energies of interaction; these terms appear in the parameters in thecorrelating expressions for gE. Well-defined assumptions regarding the molecular interactions permit ex-tension of binary local-composition expressions to the multicomponent case, without the introduction ofadditional parameters. Moreover, use of Boltzmann factors produces a built-in temperature dependence forthe parameters which, while not generally precise, is sometimes acceptable for engineering applications.

The prototypical local-composition expression forg is the Wilson equation? which for a binary mixtureis

(1.4-21)

Parameter A1-,- is given by

(1.4-22)

where v( and u, are the molar volumes of the pure components and AX(> is a characteristic energy. Equation(1.4-21) yields the following expressions for the activity coefficients:

(1.4-23a)

(1.4-23b)

Infinite-dilution activity coefficients are related to parameters A12 and A2\ via

(1.4-24a)(1.4-24b)

Wilson's equation is particularly useful for homogeneous mixtures exhibiting large positive deviations fromideality, for example, alcohol-hydrocarbon systems. Like the van Laar equation, it cannot represent interiorextrema in the activity coefficients; moreover, it is incapable of predicting liquid-liquid phase splitting. Toremedy the latter shortcoming, the following three-parameter modification of Eq. (1.4-21) has been pro-posed:

(1.4-25)

Here, C is an adjustable positive constant; when C = 1, the original Wilson equation is recovered.A significant increase in flexibility over the Wilson equation is afforded by the NRTL equation of Renon

and Prausnitz.2 For a binary mixture, this three-parameter local-composition expression reads

(1.4-26)

hence,

(1.4-27a)

(1.4-27b)

and

(1.4-28a)(1.4-28b)

Parameters T1^ and G0 are related:

(1.4-29)

and

(1.4-30)

where Ag1J is a characteristic energy and a,y is a nonrandomness parameter. If a,-, = 0, then G,-,- 1, andEq. (1.4-26) reduces to the Porter equation. The NRTL equation is capable of reproducing most of thefeatures of real-mixture behavior, but at the expense of a third parameter.

The UNIQUAC equation (Abrams and Prausnitz,3 Anderson and Prausnitz,4 Maurer and Prausnitz5)provides an example of a two-parameter local-composition equation of great flexibility. In its original form,it is written

(1.4-31)

where, for a binary mixture,

(1.4-32)

(1.4-33)

Here, z is a coordination number (usually set equal to 10), the $, are molecular segment fractions, and the$i are molecular area fractions. Quantities $, and B1 are related to pure-component molecular-structureconstants r, and qk by the prescriptions

(1.4-34a)

and

(1.4-34b)

Parameter Ty is given by

(1.4-35)

where Auy is a characteristic energy. Since structure constants r, and q-, can be computed once and for allfor each pure component i (see Prausnitz et al.6 for a tabulation of values), the UNIQUAC equation is atwo-parameter expression for gE; characteristic energies AM12 and AM2, (and hence parameters T12 and T21)must be found by analysis of binary mixture data, usually vapor-liquid equilibrium data. UNIQUAC activitycoefficients for a binary mixture are given by

(1.4-36a)

(1.4-36b)

where

(1.4-37a)

(1.4-37b)

Hence,

(1.4-38a)

(1.4-38b)

Sometimes insufficient binary data are available for evaluation of parameters Ti} in the UNIQUACequation. In this event, parameters may be estimated by the UNIFAC correlation,7 a group-contributiontechnique. The UNIFAC method is based on the notion that functional groups (e.g., methyl groups,hydroxyl groups), rather than entire molecules, are the key units of interaction in a mixture. In an exhaustiveand continuing effort, Aa. Fredenslund, J. Gmehling, and their coworkers have evaluated and reducedavailable vapor-liquid equilibrium data to obtain interaction energy parameters for many of the functionalgroups present in substances of commercial importance. The UNIFAC procedure is treated in a monograph,8and updated tables of parameter values appear periodically in the open literature.9"11 A special set ofparameter values is available for liquid-liquid equilibrium calculations.12

Local-composition equations for gE are readily extended to multicomponent mixtures. The multicom-ponent expressions contain parameters obtainable in principle from binary data only; they provide descrip-tions of gE of acceptable accuracy for many engineering calculations of multicomponent vapor-liquidequilibria at subcritical conditions. Listed below are the multicomponent versions of the Wilson, NRTL,and UNIQUAC equations.

Mukicomponent Wilson Equation

(1.4-39)

hence,

(1.4-40)

The A1J are defined by Eq. (1.4-22).

Mukicomponent NRTL Equation

(1.4-41)

hence,

(1.4-42)

Quantities ri} and Gkj are defined by Eqs. (1.4-29) and (1.4-30).

Mukicomponent UNIQUAC Equation

(1.4-31)

(1.4-43)

(1.4-44)

Here,

(1.4-45a)

(1.4-45b)

where r{ and q, are pure-component molecular-structure constants and parameter 7,-, is defined by Eq.(1.4-35). The activity coefficient 7, is given by

(1.4-46)

where

(1.4-47)

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1.5 VAPOR-LIQUID EQUILIBRIA AND LIQUID-LIQUID EQUILIBRIA1.5-1 Subcritical Vapor-Liquid EquilibriaThe essential features of vapor-liquid equilibrium (VLE) behavior are demonstrated by the simplest case:isothermal VLE of a binary system at a temperature below the critical temperatures of both pure components.For this case ("subcriticaT' VLE), each pure component has a well-defined vapor-liquid saturation pressurePf1, and VLE is possible for the full range of liquid and vapor compositions x{ and yh Figure 1.5-1illustrates several types of behavior shown by such systems. In each case, the upper solid curve ("bubblecurve") represents states of saturated liquid; the lower solid curve ("dew curve") represents states ofsaturated vapor.

Figure 1.5- Ia is for a system that obeys Raoult's Law. The significant feature of a Raoult's Law systemis the linearity of the isothermal bubble curve, expressed for a binary system as

(15-1)

(a)Constant TConstant T

Constant T

(b)Constant T

(C)Constant T

(d) (e)

FIGURE 1.5-1 Isothermal phase diagrams for subcritical binary vapor-liquid equilibrium. Case (a) rep-resents Raoult's-Law behavior. Cases (b) and (c) illustrate negative deviations from Raoult's Law; cases(d) and (e) illustrate positive deviations from Raoult's Law.

Previous Page

Although Raoult's Law is rarely obeyed by real mixtures, it serves as a useful standard against which realVLE behavior can be compared. The dashed lines in Figs. 1.5-16-1.5-1* are the Raoult's Law bubblecurves produced by the vapor pressure Fj* and Pf1.

Figures 1.5-16 and 1.5-lc illustrate negative deviations from Raoult's Law: the actual bubble curveslie below the Raoult's Law bubble curve. In Fig. 1.5-16 the deviations are moderate, but in Fig. 1.5-lcthe deviations are so pronounced that the system exhibits a minimum-pressure (maximum-boiling) homo-geneous azeotrope.

The systems of Figs. 1.5-1*/ and 1.5-1* show positive deviations from Raoult's Law, for which thetrue bubble curves lie above the Raoult's Law line. In Fig. 1.5-k/, the deviations are modest; in Fig.1.5-Xe they are large, and a maximum-pressure (minimum-boiling) homogeneous azeotrope occurs.

The goal of a subcritical VLE calculation is to quantitatively predict or correlate the various kinds ofbehavior illustrated by Fig. 1.5-1 or by its isobaric or multicomponent counterparts. The basis for thecalculation is phase-equilibrium formulation 1 of Section 1.2-5, where liquid-phase fugacities are eliminatedin favor of liquid-phase activity coefficients, and vapor-phase fugacities in favor of vapor-phase fugacitycoefficients. Raoult's Law standard states are chosen for all components in the liquid phase; hence,(fff = / f and Eq. (1.2-63) becomes

(1.5-2)

where liquid-phase activity coefficients are defined by the Raoult's Law convention (see Section 1.2-4).AU quantities in Eq. (1.5-2) refer to the actual conditions of VLE. For applications it is convenient to

eliminate the fugacity / f in favor of the vapor pressure Pf* of pure i, and to refer the activity coefficientsTf to a fixed reference pressure Pr. Equation (1.5-2) then becomes

(1.5-3)

where we have deleted superscripts identifying a phase: here and henceforth in this subsection fugacitiesand activity coefficients are for the liquid phase, and fugacity coefficients are for the vapor phase. Quantities^f1 and/-8' are evaluated for pure component i at the vapor pressure Pf1.

Equation (1.5-3) can be written in the equivalent form

(1.5-4)

where t- is a composite function defined by

(1.5-5)

Although % is conveniently viewed as a correction factor of order unity, one or more of the ratios on theright-hand side of Eq. (1.5-5) may in particular applications differ appreciably from unity. Thermodynamicsprovides the following exact expressions for these ratios:

(1.5-6)

(1.5-7)

(1.5-8)

Equation (1.5-6) follows from Eq. (1.2-58); it represents the effect of pressure on the activity coefficientand requires liquid-phase excess molar volume data. Equation (1.5-7) is a Poynting correction [see Eq.(1.2-31) and the accompanying discussion], which requires volumetric data for pure liquid i. Equation(1.5-8) represents the contributions of vapor-phase nonidealities, which are represented by a PVTx equationof state. Note here that the effects of vapor-phase nonidealities enter through both 0, and 0"1. Consistentdescription of subcritical VLE via Eq. (1.5-4) requires that ,- and 4>f" be evaluated in a consistentfashion.

Equation (1.5-4) is rarely used in its exact form; approximations are made reflecting the conditions andthe nature of the system and also reflecting the nature and extent of available thermodynamic data. Vaporpressures Pf1 are the single most important quantities in Eq. (1.54): they characterize the states of pure-fluid VLE (the "edges" of the phase diagrams); without them, even qualitative prediction of mixture VLEis impossible. Given good values for the Pf \ one introduces approximations by making statements about

the ratios appearing on the right-hand side of Eq. (1.5-5). In roughly decreasing order of reasonableness,some common approximations are:

1. Assume that the activity coefficients are independent of pressure. By Eq. (1.5-6) this requires thatvflRT be small.

2. Assume that liquid fugacities are independent of pressure. By Eq. (1.5-7), this requires thatVfIRT be small.

3. Assume that vapor-phase corrections are negligible. By Eq. (1.5-8), this occurs if both Z1 andZ1 are close_to unity (the ideal-gas approximation, valid for sufficiently low pressures), but also ifboth Z1 and Z, and P and Pf" are of comparable magnitudes. In the latter case, simplification resultsbecause of a fortuitous cancellation of effects.

4. Assume that the liquid phase is an ideal solution, that is, 7, = 1 for all temperatures, pressures,and compositions.

Unless the reduced temperature of component 1 is high (say about 0.85 or greater), approximation 1 isnearly always reasonable. On the other hand, approximation 4 is rarely realistic: liquid-phase nonidealitiesare normally present and must be taken into account. If one invokes approximations 2, 3, and 4, thenRaoult's Law is obtained:

(1.5-9)

Raoult's Law, however, is useful mainly as a standard of comparison. The simplest realistic simplificationof Eq. (1.5-4) follows from approximations 1, 2, and 3:

(1.5-10)

Here, superscript r has been dropped from 7,- because the 7, are assumed independent of pressure.Equation (1.5-10), unlike Eq. (1.5-9), can reproduce all the qualitative features of subcritical VLE

illustrated in Fig. 1.5-1. If we take (1.5-10) as representing actual behavior, then subtraction of (1.5-9)from (1.5-10) and summation over all components gives

(1.5-11)

Equation (1.5-11) is an expression for the difference between the actual bubble pressure P given by Eq.(1.5-10) and the Raoult's Law bubble pressure PRL given by Eq. (1.5-9). To the extent that the approximateEq. (1.5-10) is valid, Eq. (1.5-11) asserts that deviations from Raoult's Law result from liquid-phasenonidealities: liquid-phase activity coefficients greater than unity promote positive deviations from Raoult'sLaw, and liquid-phase activity coefficients less than unity promote negative deviations from Raoult's Law.

Calculations at moderate pressure levels (say up to 5 atm) require the inclusion of at least some of thecorrections represented by Eq. (1.5-5). A suitable formulation is developed as follows. First we ignore theeffect of pressure on the activity coefficients; thus, we set 7,77, = 1 and write Eq. (1.5-4) as

(1.5-12)

where superscript r has been dropped from 7,. Additionally, we assume that the liquid molar volume ofpure i is independent of pressure and equal to its saturation value: vt ufl. Equation (1.5-7) becomes

which is a special case of Eq. (1.2-34). Finally, we assume that the vapor phase is described by the two-term virial equation in pressure, Eq. (1.3-11). Then, by Eqs. (1.3-15) and (1.3-16) we have

and the working expression for ^, becomes

(1.5-13)

where (see Table 1.3-1)

(1.5-14)

and

(1.5-15)

Equations (1.5-12)-(1.5-15) together constitute the most common formulation for predicting or corre-lating subcritical VLE at low to moderate pressures. When using the formulation for VLE predictions, onerequires data or correlations for pure-component vapor pressures (e.g., Antoine equations), for the activitycoefficients (e.g., the UNIQUAC equation or the UNIFAC correlation), for the second virial coefficients(e.g., one of the correlations referenced in Section 1.3-2), and for the molar volumes of the saturated liquid(e.g., the Rackett equation12 for v?1). The actual VLE calculations are iterative and require the use of acomputer; details are given in the monograph by Prausnitz et al.3

In VLE correlation (data reduction), one again requires values for the P^\ B0, and yf"1, but here thegoal is to determine from VLE data best values for the parameters in an assumed expression for gE: ineffect, to find liquid-phase activity coefficients from VLE data. Data reduction procedures combine VLEcalculations with nonlinear regression techniques and again require use of a computer; this topic is discussedby Van Ness and Abbott4 and by Prausnitz et al.3

Figure 1.5-2 shows experimental and correlated binary VLE data for three dioxane-n-alkane systemsat 800C.5*6 The pressure levels are modest (0.2-1.4 atm); liquid-phase nonidealities are sufficiently largeto promote azeotropy in all three cases. Equations (1.5-12)-( 1.5-15) were used for the data reduction, withexperimental values for the Pf1 and if"; virial coefficients were estimated from the correlation of Tsono-poulos.7 Activity coefficients were assumed to be represented by the three-parameter Margules equation,and the products of the data reduction were sets of values for parameters A12* A2\, and D in Eqs. (1.4-10)and (1.4-11). The parameters so determined produce the correlations of the data shown by the solid curvesin Fig. 1.5-2. For all three systems, the data are represented to within their experimental uncertainty.

1.5-2 Supercritical Vapor-Liquid Equilibria

When the system temperature T is greater than the critical temperature TCi of component , then pure /cannot exist as a liquid. The procedure of Section 1.5-1, which incorporates the vapor-liquid saturationpressure Pf1, is therefore inappropriate for representing VLE for mixtures containing "supercritical" com-ponent /. Several methods are available for the quantitative description of such cases; the most powerfulof them is that using an equation of state as discussed briefly in Section 1.7. Alternatively, one may useEq. (1.5-2),

incorporating for the supercritical components special correlations for/f (or f) for the hypothetical liquidstate. Finally, one may employ a variant of the procedure of Section 1.5-1, in which Henry's Law standardstates are adopted for the supercritical components in the liquid phase. We consider in this subsection thelast of these techniques.

For simplicity, we restrict our discussion to binary systems containing a single supercritical component1 (the "solute" species) and a single subcritical component 2 (the "solvent" species); rigorous extensionsto more complicated situations (e.g., mixed solvents) are complex: see Van Ness and Abbott8 for adiscussion. The basis for the procedure is phase-equilibrium formulation 1 of Section 1.2-5. Subcriticalcomponent 2 is treated as in Section 1.5-1. However, we write for supercritical component 1 that

where 3Ci2 is Henry's constant for 1 dissolved in 2. This choice of standard-state fugacity accommodatesthe fact that the full/f versus *, curve (see Fig. 1.2-3) is undefined for component 1: 5C1,2 is an experi-mentally accessible quantity, whereas f\ is not. The equilibrium equation for component 1 is then

(1.5-16)

where the liquid-phase activity coefficient 7f as 7,(HL) is normalized by the Henry's Law convention (seeSection 1.2-4).

All quantities in Eq. (1.5-16) refer to the actual conditions of VLE. For applications it is convenient

(b) (C)

FIGURE 1.5-2 Correlation of subcritical vapor-liquid equilibria via Eqs. (1.5-12)(1.5-15): (a) 1,4-dioxane(l)-n-hexane(2) at 800C; (b) l,4-dioxane(l)-n-heptane(2) at 800C; and (c) l,4-dioxane(l)-/i-oc-tane(2) at 800C. Circles are data; curves are correlations.

to refer both yf and JC, 2 to a single reference pressure Pr, often taken as the vapor pressure Pf* of the

solvent. Equation (1.5-16) then becomes

(1.5-17)

where V1 is the partial molar volume of the solute in the liquid phase. Note that the term containing V1represents two