The properties of gases and liquids, fifth edition poling, prausnitz, o’connell

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The Properties of Gases and Liquids, Fifth Edition Bruce E Poling, John

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CHAPTER ONE THE ESTIMATION OF PHYSICAL

knowl-The physical properties of every substance depend directly on the nature of themolecules of the substance Therefore, the ultimate generalization of physical prop-erties of fluids will require a complete understanding of molecular behavior, which

we do not yet have Though its origins are ancient, the molecular theory was notgenerally accepted until about the beginning of the nineteenth century, and eventhen there were setbacks until experimental evidence vindicated the theory early inthe twentieth century Many pieces of the puzzle of molecular behavior have nowfallen into place and computer simulation can now describe more and more complexsystems, but as yet it has not been possible to develop a complete generalization

In the nineteenth century, the observations of Charles and Gay-Lussac were

combined with Avogadro’s hypothesis to form the gas ‘‘law,’’ PV ⫽ NRT, which

was perhaps the first important correlation of properties Deviations from the gas law, though often small, were finally tied to the fundamental nature of themolecules The equation of van der Waals, the virial equation, and other equations

ideal-of state express these quantitatively Such extensions ideal-of the ideal-gas law have notonly facilitated progress in the development of a molecular theory but, more im-portant for our purposes here, have provided a framework for correlating physicalproperties of fluids

The original ‘‘hard-sphere’’ kinetic theory of gases was a significant contribution

to progress in understanding the statistical behavior of a system containing a largenumber of molecules Thermodynamic and transport properties were related quan-titatively to molecular size and speed Deviations from the hard-sphere kinetic the-ory led to studies of the interactions of molecules based on the realization thatmolecules attract at intermediate separations and repel when they come very close.The semiempirical potential functions of Lennard-Jones and others describe attrac-tion and repulsion in approximately quantitative fashion More recent potentialfunctions allow for the shapes of molecules and for asymmetric charge distribution

in polar molecules

Although allowance for the forces of attraction and repulsion between molecules

is primarily a development of the twentieth century, the concept is not new Inabout 1750, Boscovich suggested that molecules (which he referred to as atoms)are ‘‘endowed with potential force, that any two atoms attract or repel each otherwith a force depending on their distance apart At large distances the attractionvaries as the inverse square of the distance The ultimate force is a repulsion whichincreases without limit as the distance decreases without limit, so that the two atomscan never coincide’’ (Maxwell 1875)

From the viewpoint of mathematical physics, the development of a sive molecular theory would appear to be complete J C Slater (1955) observedthat, while we are still seeking the laws of nuclear physics, ‘‘in the physics ofatoms, molecules and solids, we have found the laws and are exploring the deduc-tions from them.’’ However, the suggestion that, in principle (the Schro¨dinger equa-tion of quantum mechanics), everything is known about molecules is of little com-fort to the engineer who needs to know the properties of some new chemical todesign a commercial product or plant

comprehen-Paralleling the continuing refinement of the molecular theory has been the velopment of thermodynamics and its application to properties The two are inti-mately related and interdependent Carnot was an engineer interested in steam en-gines, but the second law of thermodynamics was shown by Clausius, Kelvin,Maxwell, and especially by Gibbs to have broad applications in all branches ofscience

de-Thermodynamics by itself cannot provide physical properties; only moleculartheory or experiment can do that But thermodynamics reduces experimental ortheoretical efforts by relating one physical property to another For example, theClausius-Clapeyron equation provides a useful method for obtaining enthalpies ofvaporization from more easily measured vapor pressures

The second law led to the concept of chemical potential which is basic to anunderstanding of chemical and phase equilibria, and the Maxwell relations provide

ways to obtain important thermodynamic properties of a substance from PVTx lations where x stands for composition Since derivatives are often required, the

re-PVTx function must be known accurately.

The Information Age is providing a ‘‘shifting paradigm in the art and practice

of physical properties data’’ (Dewan and Moore, 1999) where searching the WorldWide Web can retrieve property information from sources and at rates unheard of

a few years ago Yet despite the many handbooks and journals devoted to lation and critical review of physical-property data, it is inconceivable that all de-sired experimental data will ever be available for the thousands of compounds ofinterest in science and industry, let alone all their mixtures Thus, in spite of im-pressive developments in molecular theory and information access, the engineerfrequently finds a need for physical properties for which no experimental data areavailable and which cannot be calculated from existing theory

compi-While the need for accurate design data is increasing, the rate of accumulation

of new data is not increasing fast enough Data on multicomponent mixtures areparticularly scarce The process engineer who is frequently called upon to design

a plant to produce a new chemical (or a well-known chemical in a new way) oftenfinds that the required physical-property data are not available It may be possible

to obtain the desired properties from new experimental measurements, but that isoften not practical because such measurements tend to be expensive and time-consuming To meet budgetary and deadline requirements, the process engineeralmost always must estimate at least some of the properties required for design

1-2 ESTIMATION OF PROPERTIES

In the all-too-frequent situation where no experimental value of the needed property

is at hand, the value must be estimated or predicted ‘‘Estimation’’ and ‘‘prediction’’are often used as if they were synonymous, although the former properly carriesthe frank implication that the result may be only approximate Estimates may bebased on theory, on correlations of experimental values, or on a combination ofboth A theoretical relation, although not strictly valid, may nevertheless serve ad-equately in specific cases

For example, to relate mass and volumetric flow rates of air through an

air-conditioning unit, the engineer is justified in using PV⫽NRT Similarly, he or she

may properly use Dalton’s law and the vapor pressure of water to calculate themass fraction of water in saturated air However, the engineer must be able to judgethe operating pressure at which such simple calculations lead to unacceptable error.Completely empirical correlations are often useful, but one must avoid the temp-tation to use them outside the narrow range of conditions on which they are based

In general, the stronger the theoretical basis, the more reliable the correlation.Most of the better estimation methods use equations based on the form of anincomplete theory with empirical correlations of the parameters that are not pro-vided by that theory Introduction of empiricism into parts of a theoretical relationprovides a powerful method for developing a reliable correlation For example, the

van der Waals equation of state is a modification of the simple PV⫽NRT; setting

N⫽1,

a

冉 冊V2

Equation (1-2.1) is based on the idea that the pressure on a container wall, exerted

by the impinging molecules, is decreased because of the attraction by the mass ofmolecules in the bulk gas; that attraction rises with density Further, the availablespace in which the molecules move is less than the total volume by the excluded

volume b due to the size of the molecules themselves Therefore, the ‘‘constants’’ (or parameters) a and b have some theoretical basis though the best descriptions

require them to vary with conditions, that is, temperature and density The

corre-lation of a and b in terms of other properties of a substance is an example of the

use of an empirically modified theoretical form

Empirical extension of theory can often lead to a correlation useful for estimationpurposes For example, several methods for estimating diffusion coefficients in low-pressure binary gas systems are empirical modifications of the equation given bythe simple kinetic theory for non-attracting spheres Almost all the better estimationprocedures are based on correlations developed in this way

1-3 TYPES OF ESTIMATION

An ideal system for the estimation of a physical property would (1) provide reliablephysical and thermodynamic properties for pure substances and for mixtures at anytemperature, pressure, and composition, (2) indicate the phase (solid, liquid, or gas),(3) require a minimum of input data, (4) choose the least-error route (i.e., the best

estimation method), (5) indicate the probable error, and (6) minimize computationtime Few of the available methods approach this ideal, but some serve remarkablywell Thanks to modern computers, computation time is usually of little concern.

In numerous practical cases, the most accurate method may not be the best forthe purpose Many engineering applications properly require only approximate es-timates, and a simple estimation method requiring little or no input data is oftenpreferred over a complex, possibly more accurate correlation The simple gas law

is useful at low to modest pressures, although more accurate correlations are able Unfortunately, it is often not easy to provide guidance on when to reject thesimpler in favor of the more complex (but more accurate) method; the decisionoften depends on the problem, not the system

avail-Although a variety of molecular theories may be useful for data correlation,there is one theory which is particularly helpful This theory, called the law ofcorresponding states or the corresponding-states principle, was originally based onmacroscopic arguments, but its modern form has a molecular basis

The Law of Corresponding States

Proposed by van der Waals in 1873, the law of corresponding states expresses thegeneralization that equilibrium properties that depend on certain intermolecularforces are related to the critical properties in a universal way Corresponding statesprovides the single most important basis for the development of correlations andestimation methods In 1873, van der Waals showed it to be theoretically valid for

all pure substances whose PVT properties could be expressed by a two-constant

equation of state such as Eq (1-2.1) As shown by Pitzer in 1939, it is similarlyvalid if the intermolecular potential function requires only two characteristic pa-rameters Corresponding states holds well for fluids containing simple moleculesand, upon semiempirical extension with a single additional parameter, it also holdsfor ‘‘normal’’ fluids where molecular orientation is not important, i.e., for moleculesthat are not strongly polar or hydrogen-bonded

The relation of pressure to volume at constant temperature is different for ferent substances; however, two-parameter corresponding states theory asserts that

dif-if pressure, volume, and temperature are divided by the corresponding critical erties, the function relating reduced pressure to reduced volume and reduced tem-perature becomes the same for all substances The reduced property is commonly

prop-expressed as a fraction of the critical property: P r⫽ P / P c ; V r⫽ V / V c ; and T r⫽

T / T c

To illustrate corresponding states, Fig 1-1 shows reduced PVT data for methane

and nitrogen In effect, the critical point is taken as the origin The data for saturatedliquid and saturated vapor coincide well for the two substances The isotherms

(constant T r), of which only one is shown, agree equally well

Successful application of the law of corresponding states for correlation of PVT

data has encouraged similar correlations of other properties that depend primarily

on intermolecular forces Many of these have proved valuable to the practicingengineer Modifications of the law are commonly made to improve accuracy or ease

of use Good correlations of high-pressure gas viscosity have been obtained byexpressing␩/␩c as a function of P r and T r But since␩cis seldom known and noteasily estimated, this quantity has been replaced in other correlations by othercharacteristics such as␩⬚c,␩⬚T,or the groupM1 / 2P2 / 3c T1 / 6c ,where␩⬚cis the viscosity

at T c and low pressure,␩⬚T is the viscosity at the temperature of interest, again at

FIGURE 1-1 The law of corresponding states applied to the PVT

low pressure, and the group containing M, P c , and T cis suggested by dimensionalanalysis Other alternatives to the use of ␩cmight be proposed, each modeled onthe law of corresponding states but essentially empirical as applied to transportproperties

The two-parameter law of corresponding states can be derived from statisticalmechanics when severe simplifications are introduced into the partition function.Sometimes other useful results can be obtained by introducing less severe simpli-fications into statistical mechanics to provide a more general framework for thedevelopment of estimation methods Fundamental equations describing variousproperties (including transport properties) can sometimes be derived, provided that

an expression is available for the potential-energy function for molecular tions This function may be, at least in part, empirical; but the fundamental equa-tions for properties are often insensitive to details in the potential function fromwhich they stem, and two-constant potential functions frequently serve remarkablywell Statistical mechanics is not commonly linked to engineering practice, but there

interac-is good reason to believe it will become increasingly useful, especially when bined with computer simulations and with calculations of intermolecular forces bycomputational chemistry Indeed, anticipated advances in atomic and molecularphysics, coupled with ever-increasing computing power, are likely to augment sig-nificantly our supply of useful physical-property information

com-Nonpolar and Polar Molecules

Small, spherically-symmetric molecules (for example, CH4) are well fitted by atwo-constant law of corresponding states However, nonspherical and weakly polarmolecules do not fit as well; deviations are often great enough to encourage de-velopment of correlations using a third parameter, e.g., the acentric factor,␻ Theacentric factor is obtained from the deviation of the experimental vapor pressure–temperature function from that which might be expected for a similar substance

consisting of small spherically-symmetric molecules Typical corresponding-states

correlations express a desired dimensionless property as a function of P r , T r, andthe chosen third parameter

Unfortunately, the properties of strongly polar molecules are often not torily represented by the two- or three-constant correlations which do so well fornonpolar molecules An additional parameter based on the dipole moment has oftenbeen suggested but with limited success, since polar molecules are not easily char-acterized by using only the dipole moment and critical constants As a result, al-though good correlations exist for properties of nonpolar fluids, similar correlationsfor polar fluids are often not available or else show restricted reliability

satisfac-Structure and Bonding

All macroscopic properties are related to molecular structure and the bonds betweenatoms, which determine the magnitude and predominant type of the intermolecularforces For example, structure and bonding determine the energy storage capacity

of a molecule and thus the molecule’s heat capacity

This concept suggests that a macroscopic property can be calculated from groupcontributions The relevant characteristics of structure are related to the atoms,atomic groups, bond type, etc.; to them we assign weighting factors and then de-termine the property, usually by an algebraic operation that sums the contributionsfrom the molecule’s parts Sometimes the calculated sum of the contributions is notfor the property itself but instead is for a correction to the property as calculated

by some simplified theory or empirical rule For example, the methods of Lydersen

and of others for estimating T cstart with the loose rule that the ratio of the normalboiling temperature to the critical temperature is about 2:3 Additive structural in-crements based on bond types are then used to obtain empirical corrections to thatratio

Some of the better correlations of ideal-gas heat capacities employ theoreticalvalues of C⬚p (which are intimately related to structure) to obtain a polynomialexpressing C⬚p as a function of temperature; the constants in the polynomial aredetermined by contributions from the constituent atoms, atomic groups, and types

of bonds

1-4 ORGANIZATION OF THE BOOK

Reliable experimental data are always to be preferred over results obtained byestimation methods A variety of tabulated data banks is now available althoughmany of these banks are proprietary A good example of a readily accessible databank is provided by DIPPR, published by the American Institute of Chemical En-gineers A limited data bank is given at the end of this book But all too oftenreliable data are not available

The property data bank in Appendix A contains only substances with an uated experimental critical temperature The contents of Appendix A were takeneither from the tabulations of the Thermodynamics Research Center (TRC), CollegeStation, TX, USA, or from other reliable sources as listed in Appendix A Sub-stances are tabulated in alphabetical-formula order IUPAC names are listed, withsome common names added, and Chemical Abstracts Registry numbers are indi-cated

eval-FIGURE 1-2 Mollier diagram for methane The solid lines represent measured data.

dichlorodifluoro-Dashed lines and points represent results obtained by timation methods when only the chemical formula and the normal boiling temperature are known.

es-In this book, the various estimation methods are correlations of experimentaldata The best are based on theory, with empirical corrections for the theory’sdefects Others, including those stemming from the law of corresponding states, arebased on generalizations that are partly empirical but nevertheless have application

to a remarkably wide range of properties Totally empirical correlations are usefulonly when applied to situations very similar to those used to establish the corre-lations

The text includes many numerical examples to illustrate the estimation methods,especially those that are recommended Almost all of them are designed to explainthe calculation procedure for a single property However, most engineering designproblems require estimation of several properties; the error in each contributes tothe overall result, but some individual errors are more important that others For-tunately, the result is often adequate for engineering purposes, in spite of the largemeasure of empiricism incorporated in so many of the estimation procedures and

in spite of the potential for inconsistencies when different models are used fordifferent properties

As an example, consider the case of a chemist who has synthesized a newcompound (chemical formula CCl2F2) that boils at⫺20.5⬚C at atmospheric pressure.Using only this information, is it possible to obtain a useful prediction of whether

or not the substance has the thermodynamic properties that might make it a practicalrefrigerant?

Figure 1-2 shows portions of a Mollier diagram developed by prediction methodsdescribed in later chapters The dashed curves and points are obtained from esti-mates of liquid and vapor heat capacities, critical properties, vapor pressure, en-

thalpy of vaporization, and pressure corrections to ideal-gas enthalpies and pies The substance is, of course, a well-known refrigerant, and its known propertiesare shown by the solid curves While environmental concerns no longer permit use

entro-of CCl2F2, it nevertheless serves as a good example of building a full descriptionfrom very little information

For a standard refrigeration cycle operating between 48.9 and⫺6.7⬚C, the orator and condenser pressures are estimated to be 2.4 and 12.4 bar, vs the knownvalues 2.4 and 11.9 bar The estimate of the heat absorption in the evaporator checksclosely, and the estimated volumetric vapor rate to the compressor also shows goodagreement: 2.39 versus 2.45 m3/ hr per kW of refrigeration (This number indicatesthe size of the compressor.) Constant-entropy lines are not shown in Fig 1-2, but

evap-it is found that the constant-entropy line through the point for the low-pressurevapor essentially coincides with the saturated vapor curve The estimated coefficient

of performance (ratio of refrigeration rate to isentropic compression power) is timated to be 3.8; the value obtained from the data is 3.5 This is not a very goodcheck, but it is nevertheless remarkable because the only data used for the estimatewere the normal boiling point and the chemical formula

es-Most estimation methods require parameters that are characteristic of single purecomponents or of constituents of a mixture of interest The more important of theseare considered in Chap 2

The thermodynamic properties of ideal gases, such as enthalpies and Gibbs ergies of formation and heat capacities, are covered in Chap 3 Chapter 4 describes

en-the PVT properties of pure fluids with en-the corresponding-states principle, equations

of state, and methods restricted to liquids Chapter 5 extends the methods of Chap

4 to mixtures with the introduction of mixing and combining rules as well as thespecial effects of interactions between different components Chapter 6 covers otherthermodynamic properties such as enthalpy, entropy, free energies and heat capac-ities of real fluids from equations of state and correlations for liquids It also intro-duces partial properties and discusses the estimation of true vapor-liquid criticalpoints

Chapter 7 discusses vapor pressures and enthalpies of vaporization of pure stances Chapter 8 presents techniques for estimation and correlation of phase equi-libria in mixtures Chapters 9 to 11 describe estimation methods for viscosity, ther-mal conductivity, and diffusion coefficients Surface tension is considered briefly inChap 12

sub-The literature searched was voluminous, and the lists of references followingeach chapter represent but a fraction of the material examined Of the many esti-mation methods available, in most cases only a few were selected for detaileddiscussion These were selected on the basis of their generality, accuracy, and avail-ability of required input data Tests of all methods were often more extensive thanthose suggested by the abbreviated tables comparing experimental with estimatedvalues However, no comparison is adequate to indicate expected errors for newcompounds The average errors given in the comparison tables represent but a crudeoverall evaluation; the inapplicability of a method for a few compounds may soincrease the average error as to distort judgment of the method’s merit, althoughefforts have been made to minimize such distortion

Many estimation methods are of such complexity that a computer is required.This is less of a handicap than it once was, since computers and efficient computerprograms have become widely available Electronic desk computers, which havebecome so popular in recent years, have made the more complex correlations prac-tical However, accuracy is not necessarily enhanced by greater complexity

The scope of the book is inevitably limited The properties discussed were lected arbitrarily because they are believed to be of wide interest, especially to

se-chemical engineers Electrical properties are not included, nor are the properties ofsalts, metals, or alloys or chemical properties other than some thermodynamicallyderived properties such as enthalpy and the Gibbs energy of formation.

This book is intended to provide estimation methods for a limited number ofphysical properties of fluids Hopefully, the need for such estimates, and for a book

of this kind, may diminish as more experimental values become available and asthe continually developing molecular theory advances beyond its present incompletestate In the meantime, estimation methods are essential for most process-designcalculations and for many other purposes in engineering and applied science

REFERENCES

Dewan, A K., and M A Moore: ‘‘Physical Property Data Resources for the Practicing Engineer / Scientist in Today’s Information Age,’’ Paper 89C, AIChE 1999 Spring National Mtg., Houston, TX, March, 1999 Copyright Equilon Enterprise LLC.

Din, F., (ed.): Thermodynamic Functions of Gases, Vol 3, Butterworth, London, 1961 Maxwell, James Clerk: ‘‘Atoms,’’ Encyclopaedia Britannica, 9th ed., A & C Black, Edin-

burgh, 1875–1888.

Slater, J C.: Modern Physics, McGraw-Hill, New York, 1955.

CHAPTER TWO PURE COMPONENT

CONSTANTS

2-1 SCOPE

Though chemical engineers normally deal with mixtures, pure component propertiesunderlie much of the observed behavior For example, property models intendedfor the whole range of composition must give pure component properties at thepure component limits In addition, pure component property constants are often

used as the basis for models such as corresponding states correlations for PVT

equations of state (Chap 4) They are often used in composition-dependent mixingrules for the parameters to describe mixtures (Chap 5)

As a result, we first study methods for obtaining pure component constants of

the more commonly used properties and show how they can be estimated if noexperimental data are available These include the vapor-liquid critical properties,atmospheric boiling and freezing temperatures and dipole moments Others such asthe liquid molar volume and heat capacities are discussed in later chapters Valuesfor these properties for many substances are tabulated in Appendix A; we compare

as many of them as possible to the results from estimation methods Though theorigins of current group contribution methods are over 50 years old, previous edi-tions show that the number of techniques were limited until recently when com-putational capability allowed more methods to appear We examine most of thecurrent techniques and refer readers to earlier editions for the older methods

In Secs 2-2 (critical properties), 2-3 (acentric factor) and 2-4 (melting and ing points), we illustrate several methods and compare each with the data tabulated

boil-in Appendix A and with each other All of the calculations have been done withspreadsheets to maximize accuracy and consistency among the methods It wasfound that setting up the template and comparing calculations with as many sub-stances as possible in Appendix A demonstrated the level of complexity of themethods Finally, because many of the methods are for multiple properties andrecent developments are using alternative approaches to traditional group contri-butions, Sec 2-5 is a general discussion about choosing the best approach for purecomponent constants Finally, dipole moments are treated in Sec 2-6

Most of the estimation methods presented in this chapter are of the group, bond,

or atom contribution type That is, the properties of a molecule are usually

estab-lished from contributions from its elements The conceptual basis is that the molecular forces that determine the constants of interest depend mostly on thebonds between the atoms of the molecules The elemental contributions are prin-

inter-cipally determined by the nature of the atoms involved (atom contributions), the bonds between pairs of atoms (bond contributions or equivalently group interaction

contributions), or the bonds within and among small groups of atoms (group tributions) They all assume that the elements can be treated independently of their

con-arrangements or their neighbors If this is not accurate enough, corrections forspecific multigroup, conformational or resonance effects can be included Thus,

there can be levels of contributions The identity of the elements to be considered (group, bond, or atom) are normally assumed in advance and their contributions

obtained by fitting to data Usually applications to wide varieties of species startwith saturated hydrocarbons and grow by sequentially adding different types ofbonds, rings, heteroatoms and resonance The formulations for pure componentconstants are quite similar to those of the ideal gas formation properties and heatcapacities of Chap 3; several of the group formulations described in Appendix Chave been applied to both types of properties

Alternatives to group / bond / atom contribution methods have recently appeared.

Most are based on adding weighted contributions of measured properties such as

molecular weight and normal boiling point, etc (factor analysis) or from

‘‘quan-titative structure-property relationships’’ (QSPR) based on contributions from lecular properties such as electron or local charge densities, molecular surface area,

mo-etc (molecular descriptors) Grigoras (1990), Horvath (1992), Katritzky, et al.

(1995; 1999), Jurs [Egolf, et al., 1994], Turner, et al (1998), and St Cholakov, et

al (1999) all describe the concepts and procedures The descriptor values are puted from molecular mechanics or quantum mechanical descriptions of the sub-stance of interest and then property values are calculated as a sum of contributionsfrom the descriptors The significant descriptors and their weighting factors arefound by sophisticated regression techniques This means, however, that there are

com-no tabulations of molecular descriptor properties for substances Rather, a molecularstructure is posed, the descriptors for it are computed and these are combined inthe correlation We have not been able to do any computations for these methodsourselves However, in addition to quoting the results from the literature, since sometabulate their estimated pure component constants, we compare them with the val-ues in Appendix A

The methods given here are not suitable for pseudocomponent properties such

as for the poorly characterized mixtures often encountered with petroleum, coal andnatural products These are usually based on measured properties such as averagemolecular weight, boiling point, and the specific gravity (at 20⬚C) rather than mo-lecular structure We do not treat such systems here, but the reader is referred tothe work of Tsonopoulos, et al (1986), Twu (1984, Twu and Coon, 1996), andJianzhong, et al (1998) for example Older methods include those of Lin and Chao(1984) and Brule, et al (1982), Riazi and Daubert (1980) and Wilson, et al (1981)

2-2 VAPOR-LIQUID CRITICAL PROPERTIES

Vapor-liquid critical temperature, T c , pressure, P c , and volume, V c, are the component constants of greatest interest They are used in many correspondingstates correlations for volumetric (Chap 4), thermodynamic (Chaps 5–8), andtransport (Chaps 9 to 11) properties of gases and liquids Experimental determi-nation of their values can be challenging [Ambrose and Young, 1995], especiallyfor larger components that can chemically degrade at their very high critical tem-

pure-peratures [Teja and Anselme, 1990] Appendix A contains a data base of propertiesfor all the substances for which there is an evaluated critical temperature tabulated

by the Thermodynamics Research Center at Texas A&M University [TRC, 1999]plus some evaluated values by Ambrose and colleagues and by Steele and col-leagues under the sponsorship of the Design Institute for Physical Properties Re-search (DIPPR) of the American Institute of Chemical Engineers (AIChE) in New

York and NIST (see Appendix A for references) There are fewer evaluated P cand

V c than T c We use only evaluated results to compare with the various estimationmethods

Estimation Techniques

One of the first successful group contribution methods to estimate critical properties

was developed by Lydersen (1955) Since that time, more experimental values havebeen reported and efficient statistical techniques have been developed that allowdetermination of alternative group contributions and optimized parameters We ex-amine in detail the methods of Joback (1984; 1987), Constantinou and Gani (1994),Wilson and Jasperson (1996), and Marrero and Pardillo (1999) After each is de-scribed and its accuracy discussed, comparisons are made among the methods,including descriptor approaches, and recommendations are made Earlier methodssuch as those of Lyderson (1955), Ambrose (1978; 1979; 1980), and Fedors (1982)are described in previous editions; they do not appear to be as accurate as thoseevaluated here

Method of Joback. Joback (1984; 1987) reevaluated Lydersen’s group

contribu-tion scheme, added several new funccontribu-tional groups, and determined new contribucontribu-tion

values His relations for the critical properties are

where the contributions are indicated as tck, pck and v ck The group identities and

Joback’s values for contributions to the critical properties are in Table C-1 For T c,

a value of the normal boiling point, T b, is needed This may be from experiment

or by estimation from methods given in Sec 2-4; we compare the results for both

An example of the use of Joback’s groups is Example 2-1; previous editions giveother examples, as do Devotta and Pendyala (1992)

Example 2-1 Estimate Tc, Pc, and Vc for 2-ethylphenol by using Joback’s group method.

solution 2-ethylphenol contains one —CH 3 , one —CH 2 —, four ⫽CH(ds), one

does not include the aromatic carbon From Appendix Table C-1

Appendix A values for the critical temperature and pressure are 703 K and 43.00

T Difference (Exp T ) c b ⫽ 703 ⫺ 698.1 ⫽ 4.9 K or 0.7%

T Difference (Est T ) c b ⫽ 703 ⫺ 715.7 ⫽ ⫺12.7 K or ⫺1.8%

P Difference c ⫽ 43.00 ⫺ 44.09 ⫽ ⫺1.09 bar or ⫺2.5%.

A summary of the comparisons between estimations from the Joback method

and experimental Appendix A values for T c , P c , and V cis shown in Table 2-1 Theresults indicate that the Joback method for critical properties is quite reliable for

T c of all substances regardless of size if the experimental T bis used When estimated

values of T bare used, there is a significant increase in error, though it is less forcompounds with 3 or more carbons (2.4% average increase for entries indicated by

bin the table, compared to 3.8% for the whole database indicated bya)

For P c, the reliability is less, especially for smaller substances (note the ence between the aandbentries) The largest errors are for the largest molecules,especially fluorinated species, some ring compounds, and organic acids Estimates

differ-can be either too high or too low; there is no obvious pattern to the errors For V c,the average error is several percent; for larger substances the estimated values areusually too small while estimated values for halogenated substances are often toolarge There are no obvious simple improvements to the method Abildskov (1994)did a limited examination of Joback predictions (less than 100 substances) andfound similar absolute percent errors to those of Table 2-1

A discussion comparing the Joback technique with other methods for criticalproperties is presented below and a more general discussion of group contributionmethods is in Sec 2-5

Method of Constantinou and Gani (CG). Constantinou and Gani (1994)

devel-oped an advanced group contribution method based on the UNIFAC groups (see

Chap 8) but they allow for more sophisticated functions of the desired properties

TABLE 2-1 Summary of Comparisons of Joback Method with Appendix A Database

aThe number of substances in Appendix A with data that could be tested with the method.

bThe number of substances in Appendix A having 3 or more carbon atoms with data that could be tested with the method.

cAAE is average absolute error in the property; A%E is average absolute percent error.

dThe number of substances for which the absolute percent error was greater than 10%.

eThe number of substances for which the absolute percent error was less than 5% The number of substances with errors between 5% and 10% can be determined from the table information.

ƒThe experimental value of T bin Appendix A was used.

g The value of T bused was estimated by Joback’s method (see Sec 2-4).

and also for contributions at a ‘‘Second Order’’ level The functions give moreflexibility to the correlation while the Second Order partially overcomes the limi-tation of UNIFAC which cannot distinguish special configurations such as isomers,multiple groups located close together, resonance structures, etc., at the ‘‘First Or-

der.’’ The general CG formulation of a function ƒ[F] of a property F is

F⫽ƒ冋 冘N (F ) k 1k ⫹W冘M (F ) j 2j册 (2-2.4)

where ƒ can be a linear or nonlinear function (see Eqs 2-2.5 to 2-2.7), N k is the

number of First-Order groups of type k in the molecule; F 1kis the contribution for

the First-Order group labeled 1k to the specified property, F; M jis the number of

Second-Order groups of type j in the molecule; and F 2jis the contribution for the

Second-Order group labeled 2j to the specified property, F The value of W is set

to zero for First-Order calculations and set to unity for Second-order calculations.For the critical properties, the CG formulations are

Example 2-2 Estimate T c , P c , and V cfor 2-ethylphenol by using Constantinou and Gani’s group method.

solution The First-Order groups for 2-ethylphenol are one CH3, four ACH, one ACCH2, and one ACOH There are no Second-Order groups (even though the ortho proximity effect might suggest it) so the First Order and Second Order calculations are the same From Appendix Tables C-2 and C-3

The Appendix A values for the critical temperature and pressure are 703.0 K and 43.0

Property/Butanol 1-butanol

1-propanol

T c, K

V c, cm 3 mol ⫺1

The First Order results are generally good except for 2-methyl-2-propanol

(t-butanol) The steric effects of its crowded methyl groups make its experimental value quite different from the others; most of this is taken into account by the First-Order groups, but the Second Order contribution is significant Notice that the Second Order contributions for the other species are small and may change the results in the wrong direction so that the Second Order estimate can be slightly worse than the First Order estimate This problem occurs often, but its effect is normally small; including Second Order effects usually helps and rarely hurts much.

A summary of the comparisons between estimations from the Constantinou and

Gani method and experimental values from Appendix A for T c , P c , and V cis shown

in Table 2-2

The information in Table 2-2 indicates that the Constantinou / Gani method can

be quite reliable for all critical properties, though there can be significant errors forsome smaller substances as indicated by the lower errors in Table 2-2B compared

to Table 2-2A for T c and P c but not for V c This occurs because group additivity isnot so accurate for small molecules even though it may be possible to form themfrom available groups In general, the largest errors of the CG method are for thevery smallest and for the very largest molecules, especially fluorinated and largerring compounds Estimates can be either too high or too low; there is no obviouspattern to the errors

Constantinou and Gani’s original article (1994) described tests for 250 to 300substances Their average absolute errors were significantly less than those of Table

2-2 For example, for T c they report an average absolute error of 9.8 K for FirstOrder and 4.8 K for Second Order estimations compared to 18.5K and 17.7 K here

for 335 compounds Differences for P c and V cwere also much less than given here.Abildskov (1994) made a limited study of the Constantinou / Gani method (less than

100 substances) and found absolute and percent errors very similar to those of Table2-2 Such differences typically arise from different selections of the substances anddata base values In most cases, including Second Order contributions improved the

TABLE 2-2 Summary of Constantinou / Gani Method

Compared to Appendix A Data Base

A All substances in Appendix A with data that could be

tested with the method

B All substances in Appendix A having 3 or more carbon

atoms with data that could be tested with the method

10% can be determined from the table information.

eThe number of substances for which Second-Order groups are fined for the property.

de-fThe number of substances for which the Second Order result is more accurate than First Order.

gThe average improvement of Second Order compared to First Order.

A negative value indicates that overall the Second Order was less

accu-rate.

results 1 to 3 times as often as it degraded them, but except for ring compoundsand olefins, the changes were rarely more than 1 to 2% Thus, Second Order con-tributions make marginal improvements overall and it may be worthwhile to includethe extra complexity only for some individual substances In practice, examiningthe magnitude of the Second Order values for the groups involved should provide

a user with the basis for including them or not

A discussion comparing the Constantinou / Gani technique with other methodsfor critical properties is presented below and a more general discussion is found inSec 2-5

Method of Wilson and Jasperson. Wilson and Jasperson (1996) reported three

methods for T c and P cthat apply to both organic and inorganic species The

Zero-Order method uses factor analysis with boiling point, liquid density and molecular weight as the descriptors At the First Order, the method uses atomic contributions

along with boiling point and number of rings, while the Second Order method also

includes group contributions The Zero-Order has not been tested here; it is iterative

and the authors report that it is less accurate by as much as a factor of two or three

than the others, especially for P c The First Order and Second Order methods usethe following equations:

type k with First Order atomic contributions⌬tck and⌬pck while M jis the number

of groups of type j with Second-Order group contributions⌬tcj and⌬pcj Values

of the contributions are given in Table 2-3 both for the First Order Atomic

Con-tributions and for the Second-Order Group ConCon-tributions Note that T c requires T b.Application of the Wilson and Jasperson method is shown in Example 2-4

Example 2-4 Estimate T c and P cfor 2-ethylphenol by using Wilson and Jasperson’s method.

solution The atoms of 2-ethylphenol are 8 ⫺C, 10 ⫺H, 1 ⫺O and there is 1 ring.

477.67 K; the value estimated by the Second Order method of Constantinou and Gani (Eq 2-4.4) is 489.24 K From Table 2-3A

TABLE 2-3A Wilson-Jasperson (1996) Atomic Contributions for Eqs (2-2.8) and (2-2.9)

TABLE 2-3B Wilson-Jasperson (1996) Group Contributions for Eqs (2-2.8) and (2-2.9)

From Table 2-3B there is the ‘‘⫺OH, C5 or more’’ contribution of Nk⌬tck ⫽ 0.01

changed to

0.2

for the critical properties are 703.0 K and 43.0 bar, respectively Thus the differences are

occasionally occurs.

A summary of the comparisons between estimations from the Wilson and

Jas-person method and experimental values from Appendix A for T c and P care shown

in Table 2-4 Unlike the Joback and Constantinou / Gani method, there was no

dis-TABLE 2-4 Summary of Wilson / Jasperson Method Compared to Appendix A Data Base

* Eq (2-2.8) with experimental T b.

⫹Eq (2-2.8) with T bestimated from Second Order Method of Constantinou and Gani (1994).

# Eq (2-2.9) with experimental T c.

@ Eq (2-2.9) with T c estimated using Eq (2-2.8) and experimental T b.

aThe number of substances in Appendix A with data that could be tested with the method.

bAAE is average absolute error in the property; A%E is average absolute percent error.

cThe number of substances for which the absolute percent error was greater than 10%.

dThe number of substances for which the absolute percent error was less than 5% The number of substances with errors between 5% and 10% can be determined from the table information.

eThe number of substances for which Second-Order groups are defined for the property.

ƒ The number of substances for which the Second Order result is more accurate than First Order.

gThe average improvement of Second Order compared to First Order A negative value indicates that overall the Second Order was less accurate.

cernible difference in errors between small and large molecules for either property

so only the overall is given

The information in Table 2-4 indicates that the Wilson / Jasperson method is very

accurate for both T c and P c When present, the Second Order group contributionsnormally make significant improvements over estimates from the First Order atom

contributions The accuracy for P cdeteriorates only slightly with an estimated value

of T c if the experimental T b is used The accuracy of T cis somewhat less when the

required T bis estimated with the Second Order method of Constantinou and Gani(1994) (Eq 2-4.4) Thus the method is remarkable in its accuracy even though it

is the simplest of those considered here and applies to all sizes of substancesequally

Wilson and Jasperson compared their method with results for 700 compounds

of all kinds including 172 inorganic gases, liquids and solids, silanes and siloxanes.Their reported average percent errors for organic substance were close to thosefound here while they were somewhat larger for the nonorganics The errors fororganic acids and nitriles are about twice those for the rest of the substances.Nielsen (1998) studied the method and found similar results

Discussion comparing the Wilson / Jasperson technique with other methods forcritical properties is presented below and a more general discussion is in Sec 2-5

Method of Marrero and Pardillo. Marrero-Marejo´n and Pardillo-Fontdevila

(1999) describe a method for T c , P c , and V c that they call a group interaction

contribution technique or what is effectively a bond contribution method They give

equations that use values from pairs of atoms alone, such as ⬎C⬍ & —N⬍, orwith hydrogen attached, such as CH3—& —NH2 Their basic equations are

where Natomsis the number of atoms in the compound, N k is the number of atoms

of type k with contributions tcbk, pcbk, and vcbk Note that T c requires T b, but

Marrero and Pardillo provide estimation methods for T b(Eq 2-4.5)

Values of contributions for the 167 pairs of groups (bonds) are given in Table2-5 These were obtained directly from Dr Marrero and correct some misprints inthe original article (1999) The notation of the table is such that when an atom isbonded to an element other than hydrogen,—means a single bond,⬎or⬍means

2 single bonds,⫽means a double bond and ⬅means a triple bond, [r] meansthat the group is in a ring such as in aromatics and naphthenics, and [rr] means thepair connects 2 rings as in biphenyl or terphenyl Thus, the pair⬎C⬍& F—meansthat the C is bonded to 4 atoms / groups that are not hydrogen and one of the bonds

is to F, while⫽C⬍ & F— means that the C atom is doubly bonded to anotheratom and has 2 single bonds with 1 of the bonds being to F Bonding by multiplebonds is denoted by both members of the pair having [⫽] or [⬅]; if they bothhave a⫽or a⬅without the brackets [ ], they will also have at least 1 — and thebonding of the pair is via a single bond Therefore, the substance CHF⫽CFCF3would have 1 pair of [⫽]CH— & [⫽]C⬍, 1 pair of⫽CH— & F—, 1 pair of

C⬍& —F, 1 pair of C⬍and⬎C⬍, and 3 pairs of⬎C⬍& —F The location

of bonding in esters is distinguished by the use of [ ] as in pairs 20, 21, 67, 100and 101 For example, in the pair 20, the notation CH3— & —COO[—] meansthat CH3— is bonded to an O to form an ester group, CH3—O—CO—, whereas

in the pair 21, the notation CH3— & [—]COO— means that CH3— is bonded tothe C to form CH3—CO—O— Of special note is the treatment of aromatic rings;

it differs from other methods considered in this section because it places single anddouble bonds in the rings at specific locations, affecting the choice of contributions.This method of treating chemical structure is the same as used in traditional Hand-books of Chemistry such as Lange’s (1999) We illustrate the placement of sidegroups and bonds with 1-methylnaphthalene in Example 2-5 The locations of thedouble bonds for pairs 130, 131, and 139 must be those illustrated as are the singlebonds for pairs 133, 134 and 141 The positions of side groups must also be care-fully done; the methyl group with bond pair 10 must be placed at the ‘‘top’’ of thediagram since it must be connected to the 131 and 141 pairs If the location of it

or of the double bond were changed, the contributions would change

Example 2-5 List the pairs of groups (bonds) of the Marrero / Pardillo (1999) method for 1-methylnaphthalene.

solution The molecular structure and pair numbers associated with the bonds from Table 2-5 are shown in the diagram.

TABLE 2-5 Marrero-Pardillo (1999) Contributions for Eqs (2-2.10) to (2-2.12) and (2-4.5)

TABLE 2-5 Marrero-Pardillo (1999) Contributions for Eqs (2-2.10) to (2-2.12) and (2-4.5)

TABLE 2-5 Marrero-Pardillo (1999) Contributions for Eqs (2-2.10) to (2-2.12) and (2-4.5)

TABLE 2-5 Marrero-Pardillo (1999) Contributions for Eqs (2-2.10) to (2-2.12) and (2-4.5)

133 130

130 139 141 134 134

Pair # 10 130 131 133 134 139 141

Atom / Group Pair

Other applications of the Marrero and Pardillo method are shown in Examples2-6 and 2-7 There are also several informative examples in the original paper(1999)

Example 2-6 Estimate T c , P c , and V cfor 2-ethylphenol by using Marrero and Pardillo’s method.

solution The chemical structure to be used is shown The locations of the various

—

N k tck

⫺ 0.0227 0.1012

⫺ 0.2246

⫺ 0.7172 0.4178 0.2318 0.0931

⫺ 0.1206

N k pck

⫺ 0.0430

⫺ 0.0626 0.1542 0.2980

The estimates from Eqs (2-2.10) to (2-2.12) are

The Appendix A values for the critical temperature and pressure are 703.0 K and 43.0

experimental Tb.

Example 2-7 Estimate Tc, Pc, and Vcfor the four butanols using Marrero and Pardillo’s method.

solution The Atom / Group Pairs for the butanols are:

1-propanol

a Calculated with Eq [2-2.10] using T bfrom Appendix A.

b Calculated with Eq [2-2.10] using T bestimated with Marrero / Pardillo method Eq (2-4.5).

TABLE 2-6 Summary of Marrero / Pardillo (1999) Method Compared to Appendix A Data Base

A All substances in Appendix A with data that could be tested with the method

* Calculated with Eq [2-2.10] using T bfrom Appendix A.

# Calculated with Eq [2-2.10] using T bestimated with Marrero / Pardillo method Eq (2-4.5).

aThe number of substances in Appendix A with data that could be tested with the method.

b AAE is average absolute error in the property; A%E is average absolute percent error.

cThe number of substances for which the absolute percent error was greater than 10%.

dThe number of substances for which the absolute percent error was less than 5% The number of substances with errors between 5% and 10% can be determined from the table information.

e The number of substances for which the T c result is more accurate when an estimated T bis used than with an experimental value.

The results are typical of the method Notice that sometimes the value with an

A summary of the comparisons between estimations from the Marrero and dillo method and experimental values from Appendix A for critical properties isshown in Table 2-6 It shows that there is some difference in errors between smalland large molecules

Par-The information in Table 2-6 indicates that the Marrero / Pardillo is accurate for

the critical properties, especially T c The substances with larger errors in P c and V c

are organic acids and some esters, long chain substances, especially alcohols, andthose with proximity effects such as multiple halogens (including perfluorinatedspecies) and stressed rings

A discussion comparing the Marrero and Pardillo technique with other methodsfor the properties of this chapter is presented in Sec 2-5

Other methods for Critical Properties. There are a large number of other group /

bond / atom methods for estimating critical properties Examination of them

indi-cates that they either are restricted to only certain types of substances such asparaffins, perfluorinated species, alcohols, etc., or they are of lower accuracy thanthose shown here Examples include those of Tu (1995) with 40 groups to obtain

T c for all types of organics with somewhat better accuracy than Joback’s method;

Sastri, et al (1997) treating only V cand obtaining somewhat better accuracy than

Joback’s method; Tobler (1996) correlating V cwith a substance’s temperature anddensity at the normal boiling point with improved accuracy over Joback’s method,but also a number of substances for which all methods fail; and Daubert [Jalowkaand Daubert, 1986; Daubert and Bartakovits, 1989] using Benson groups (see Sec.3.3) and obtaining about the same accuracy as Lydersen (1955) and Ambrose (1979)for all properties Within limited classes of systems and properties, these methodsmay be more accurate as well as easier to implement than those analyzed here

As mentioned in Sec 2.1, there is also a great variety of other estimation

meth-ods for critical properties besides the above group / bond / atom approaches The techniques generally fall into two classes The first is based on factor analysis that

builds correlation equations from data of other measurable, macroscopic propertiessuch as densities, molecular weight, boiling temperature, etc Such methods includethose of Klincewicz and Reid (1984) and of Vetere (1995) for many types of sub-stances Somayajulu (1991) treats only alkanes but also suggests ways to approachother homologous series However, the results of these methods are either reducedaccuracy or extra complexity The way the parameters depend upon the type ofsubstance and their need for other input information does not yield a direct oruniversal computational method so, for example, the use of spreadsheets would bemuch more complicated We have not given any results for these methods

The other techniques of estimating critical and other properties are based on

molecular properties, molecular descriptors, which are not normally measurable.

These ‘‘Quantitative Structure-Property Relationships’’ (QSPR) are usually obtainedfrom on-line computation of the structure of the whole molecule using molecularmechanics or quantum mechanical methods Thus, no tabulation of descriptor con-tributions is available in the literature even though the weighting factors for thedescriptors are given Estimates require access to the appropriate computer software

to obtain the molecular structure and properties and then the macroscopic propertiesare estimated with the QSPR relations It is common that different methods usedifferent computer programs We have not done such calculations, but do comparewith the data of Appendix A the results reported by two recent methods We com-

ment below and in Sec 2.5 on how they compare with the group / bond / atom

meth-ods The method of Gregoras is given mainly for illustrative purposes; that of Jurs

shows the current status of molecular descriptor methods.

Method of Grigoras. An early molecular structural approach to physical ties of pure organic substances was proposed by Grigoras (1990) The concept was

proper-to relate several properties proper-to the molecular surface areas and electrostatics as erated by combining quantum mechanical results with data to determine the properform of the correlation For example, Grigoras related the critical properties tomolecular properties via relations such as

T c⫽0.633A⫺1.562A⫺⫹0.427A⫹⫹9.914A HB⫹263.4 (2-2.14)

where A is the molecular surface area, A⫺ and A⫹ are the amounts of negatively

and positively charged surface area on the molecule and A HB is the amount ofcharged surface area involved in hydrogen bonding Examples of values of thesurface area quantities are given in the original reference and comparisons are madefor several properties of 137 compounds covering many different types This is theonly example where a tabulation of descriptors is available

TABLE 2-7 Summary of Grigoras and Jurs Methods

Compared to Appendix A Data Base

All substances in Appendix A with data that could be tested

with the method

Method Property

10% can be determined from the table information.

These relationships can be used to obtain other properties such as P cby lations such as

Method of Jurs. Jurs and coworkers have produced a series of papers describingextensions and enhancements of molecular descriptor concepts (see, e.g., Egolf, etal., 1994; Turner, et al., 1998) Compared to the early work of Grigoras, the quan-tum mechanical calculations are more reliable and the fitting techniques more re-fined so that the correlations should be much better In particular, the descriptorsultimately used for property estimation are now sought in a sophisticated mannerrather than fixing on surface area, etc as Grigoras did For example, in the case of

T c, the descriptors are dipole moment,␮, area A⫹, a connectivity index, number of

oxygens, number of secondary carbon bonds of the sp3type, gravitation index, afunction of acceptor atom charge, and average positive charge on carbons A com-

pletely different descriptor set was used for P c Since the descriptor values must beobtained from a set of calculations consistent with the original fitting, and every-thing is contained in a single computer program, the particular choice of descriptors

is of little importance to the user

Turner, et al (1998) list results for T c and P cwhich are compared in Table 2-7with data of Appendix A It can be seen that these new results are very good and

are generally comparable with the group / bond / atom methods The principal

diffi-culty is that individual access to the computational program is restricted

A discussion comparing the QSPR techniques with other methods for the erties of this chapter is presented below and in Sec 2-5.

prop-Discussion and Recommendations for Critical Properties. The methods of back (1984; 1987), Constantinou and Gani (1994), Wilson and Jasperson (1996)and Marrero and Pardillo (1999) were evaluated Summaries of comparisons withdata from Appendix A are given in Tables 2-1, 2-2, 2-4, and 2-6 A few resultsfrom QSPR methods are given in Table 2-7 Overall, the methods are all comparable

Jo-in accuracy

A useful method for determining consistency among T c , P c , and V cis to use Eq

(2-3.2) relating the critical compressibility factor, Z c ⬅ P c V c / RT c, to the acentricfactor (Sec 2-3) The theoretical basis of the acentric factor suggests that except

for substances with T c⬍100 K, Z cmust be less than 0.291 When Eq (2-3.2) was

tested on the 142 substances of Appendix A for which reliable values of Z cand␻

are available and for which the dipole moment was less than 1.0 debye, the average

absolute percent error in Z cwas 2% with only 9 substances having errors greaterthan 5% When applied to 301 compounds of all types in Appendix A, the averagepercent error was 5% with 32 errors being larger than 10% Some of these errorsmay be from data instead of correlation inadequacy In general, data rather thanestimation methods should be used for substances with one or two carbon atoms

Critical Temperature, T c The methods all are broadly applicable, describingnearly all the substances of Appendix A; the average percent of error is around 1%

with few, if any substances being off by more than 10% If an experimental T bisavailable, the method of Marrero and Pardillo has higher accuracy than does that

of Wilson and Jasperson On the other hand, for simplicity and breadth of stances, the Wilson / Jasperson method is best since it has the fewest groups totabulate, is based mostly on atom contributions, and treats inorganic substances aswell as organics Finally, Joback’s method covers the broadest range of compoundseven though it is somewhat less accurate and more complex

sub-However, if there is no measured T b available and estimated values must beused, the errors in these methods increase considerably Then, if the substance hasfewer than 3 carbons, either the Wilson / Jasperson or Marrero / Pardillo method ismost reliable; if the substance is larger, the Constantinou / Gani approach generallygives better results with Second Order calculations being marginally better thanFirst Order The Joback method is somewhat less accurate than these

The molecular descriptor method of Jurs is as accurate as the group / bond / atom

methods, at least for the substances compared here, though the earlier method ofGrigoras is not While the method is not as accessible, current applications showthat once a user has established the capability of computing descriptors, they can

be used for many properties

Critical Pressure, P c The methods all are broadly applicable, describing nearlyall the substances of Appendix A All methods give average errors of about 5%with about the same fraction of substances (20%) having errors greater than 10%.The Wilson / Jasperson method has the lowest errors when an experimental value

of T c is used; when T c is estimated the errors in P care larger than the other methods.All show better results for substances with 3 or more carbons, except for a fewspecies The Constantinou / Gani Second Order contributions do not significantlyimprove agreement though the Second Order contributions to the Wilson / Jaspersonmethod are quite important Thus, there is little to choose among the methods Thedecision can be based less on accuracy and reliability than on breadth of applica-bility and ease of use

The molecular descriptor method of Jurs is as accurate as the group / bond / atom

methods, at least for the substances compared here, though the earlier method ofGrigoras is not

Critical Volume, V c The methods of Joback, Constantinou / Gani and Marrero /Pardillo are all broadly applicable, describing nearly all the substances of Appendix

A The Joback method has the lowest error, around 3% with better results for largersubstances (3 or more carbons) The Constantinou / Gani results averaged the highest

error at about 4.5% for all compounds and 4% for larger ones For V c estimationwith the CG method, second-order contributions often yield higher error than usingonly first-order contributions About 5% of the estimates were in excess of 10%for all methods There is a little better basis to choose among the methods here,but still a decision based on breadth of applicability and ease of use can be justified

There have been no molecular descriptor methods applied to V c

The particular definition of Eq (2-3.1) arose because the monatomic gases (Ar,

Kr, Xe) have␻ ⬃0 and except for quantum gases (H2, He, Ne) and a few others

(e.g., Rn), all other species have positive values up to 1.5 To obtain values of␻

from its definition, one must know the constants T c , P c , and the property Pvpat the

reduced temperature, T / T c ⫽ 0.7 Typically, as in Appendix A, a value of ␻ is

obtained by using an accurate equation for Pvp(T) along with the required critical

where P c is in bars while T b and T c are both absolute temperatures The functions

ƒ(0) and ƒ(1) are given in Eqs (7-4.2) and (7-4.3), respectively Equation (2-3.3)results from ignoring the term in ␻2 in Eq (7-4.1) and solving for ␻ For 330compounds of Appendix A, the average absolute deviation in␻is 0.0065 or 2.4%

with only 19 substances having an error greater than 0.02 Retaining the␻2termmakes almost no difference in the error because ƒ(2)of Eq (7-4.4) is close to zero

for most values of T br

Example 2-8 Estimate ␻ for benzene using Eq (2-3.3).

solution Properties for benzene from Appendix A are T b ⫽ 353.24 K, Tc⫽ 562.05

It is also possible to directly estimate␻via group / bond / atom contribution

meth-ods Only the Constantinou and Gani method (Constantinou, et al., 1995) attempts

to do this for a wide range of substances from group contributions only The basic

relation of the form of Eq (2-2.4) is

(1 / 0.5050)

␻ ⫽0.4085 ln再 冋 冘N (w1k) k ⫹W冘 M (w2j ) j ⫹1.1507册冎 (2-3.4)

where the contributions w1k and w2j are given in Appendix Table C-2 and C-3 and

the application is made in the same way as described in Sec (2-2) and (3-4).Example 2-9 shows the method and Table 2-8 gives a summary of the results for

Second-TABLE 2-8 Summary of Constantinou / Gani Method for ␻ Compared to Appendix A Data Base

* All Substances in Appendix A with data that could be tested with the method.

⫹ All Substances in Appendix A having 3 or more carbon atoms with data that could be tested with the method.

aThe number of substances in Appendix A with data that could be tested with the method.

bAAE is average absolute error in the property; A%E is average absolute percent error.

cThe number of substances for which the absolute percent error was greater than 10%.

dThe number of substances for which the absolute percent error was less than 5% The number of substances with errors between 5% and 10% can be determined from the table information.

eThe number of substances for which Second-Order groups are defined for the property.

ƒ The number of substances for which the Second Order result is more accurate than First Order.

gThe average improvement of Second Order compared to First Order.

First and Second Order estimates While these are much lower than given in Table 2-8 for all substances, this is not atypical of estimates for normal fluids.

Table 2-8 shows that the errors can be significant though it covers all kinds ofsubstances, not just ‘‘normal’’ ones

Discussion and Recommendations for Acentric Factor. The acentric factor fined in Eq (2-3.1) was originally intended for corresponding states applications

de-of ‘‘normal’’ fluids as defined by Eq (4.3-2) With care it can be used for predictingproperties of more strongly polar and associating substances, though even if the

‘‘best’’ value is used in equations such as Eq (2-3.2) or those in Sec (4-3), there

is no guarantee of accuracy in the desired property

As shown by Liu and Chen (1996), the sensitivity of␻to errors of input mation is very great The recommended procedure for obtaining an unknown value

infor-of ␻ is to use a very accurate correlation for Pvp such as Eqs (7-3.2), (7-3.3) or(7-3.7) directly in Eq (2-3.1) The next most reliable approach is to use accurate

experimental values of T c , P c , T bin Eq (2-3.3) Finally, the method of Constantinouand Gani with Eq (2-3.4) can be used with some confidence

Estimated property values will not yield accurate acentric factors For example,

approximate correlations for Pvp such as the Clausius-Clapeyron Equation (7-2.3)

as used by Edmister (1958) or the Antoine Equation (7-3.1) as used by Chen, et

al (1993) are about as good as (2-3.4) Further, we tried using this chapter’s best

estimated values of T c , P c , T b , or estimated T c and P c with experimental T b, or othercombinations of estimated and experimental data for nearly 300 substances in Ap-pendix A The results generally gave large errors, even for ‘‘normal’’ substances.Earlier methods for␻described in the 4th Edition are not accurate or have limitedapplications

Along these lines, Chappelear (1982) has observed that ‘‘accepted’’ values ofthe acentric factor can change due to the appearance of new vapor pressure orcritical constant data, changing predicted properties In addition, using revised acen-tric factors in a correlation developed from earlier␻values can lead to unnecessaryerrors Chappelear’s example is carbon dioxide In Appendix A, we show ␻ ⫽

0.225; others have quoted a value of 0.267 (Nat Gas Proc Assoc., 1981) Thedifferences result from the extrapolation technique used to extend the liquid region

past the freezing point to T r⫽0.7 Also, Eq (2-3-2) yields␻ ⫽0.213 Yet, in theattractive parameter in the Peng-Robinson equation of state (1976) (see Chapter 4),the value should be 0.225, since that was what was used to develop the equation

of state relations One should always choose the value used for the original lation of the desired property

corre-2-4 BOILING AND FREEZING POINTS

Boiling and freezing points are commonly assumed to be the phase transition whenthe pressure is 1 atm A more exact terminology for these temperatures might be

the ‘‘normal’’ boiling and ‘‘normal’’ freezing points In Appendix A, values for T fp and T bare given for many substances Note that estimation methods of Sec 2-2

may use T b as input information for T c The comparisons done there include testing

for errors introduced by using T bfrom methods of this section; they can be large

A number of methods to estimate the normal boiling point have been proposed

Some were reviewed in the previous editions Several of group / bond / atom methods described in Sec 2-2 have been applied to T fp and T b , as have some of the molecular

descriptor techniques of Sec 2-2 We describe the application of these in a similar

manner to that used above for critical properties

Method of Joback for T ƒp and T b Joback (1984; 1987) reevaluated Lydersen’s

group contribution scheme, added several new functional groups, and determined

new contribution values His relations for T fp and T bare

Joback’s values for contributions to the critical properties are in Table C-1 Example2-10 shows the method.

Example 2-10 Estimate T ƒp and Tb for 2,4-dimethylphenol by using Joback’s group method.

solution 2,4-dimethylphenol contains two —CH 3 , three ⫽ CH(ds), three ⫽ C(ds), and one —OH (phenol) From Appendix Table C-1

the differences are

T Difference b ⫽ 484.09 ⫺ 494.72 ⫽ ⫺10.63 K or ⫺2.2%

Devotta and Pendyala (1992) modified the Joback method to more accurately

treat T bof halogenated compounds They report that the average percent deviationsfor refrigerants and other substances was 12% in the original method; this is con-sistent with our comparison and is much larger than the overall average given below.Devotta and Pendyala did not change Joback’s basic group contribution values; theyonly changed the groups and values for halogen systems Their results showed an

average percent deviation of 6.4% in T b

A summary of the comparisons between estimations from the Joback method

and experimental Appendix A values for T ƒp and T bare shown along with thosefrom other methods in Tables 2-9 and 2-10 below

Method of Constantinou and Gani (CG) for T ƒp and T b Constantinou and Gani

(1994, 1995) developed an advanced group contribution method based on the

UNIFAC groups (see Chap 8) but enhanced by allowing for more sophisticatedfunctions of the desired properties and by providing contributions at a ‘‘SecondOrder’’ level (see Secs 2-2 and 3-3 for details)

For T ƒp and T b, the CG equations are

T ƒp ⫽102.425 ln冋 冘N (tƒp1k) k ⫹W冘M (tƒp2j ) j 册 (2-4.3)

T b⫽204.359 ln冋 冘N (tb1k) k ⫹W冘M (tb2j ) j 册 (2-4.4)

The group values tƒp1k, tƒp2j, tb1k, and tb2j for Eqs (2-4.3) and (2-4.4) are given

in Appendix Tables C-2 and C-3 with sample assignments shown in Table C-4.Examples 2-11 and 2-12 illustrate this method

Example 2-11 Estimate T ƒp and T b for 2,4-dimethylphenol using Constantinou and Gani’s group method.

solution The First-Order groups for 2,4-dimethylphenol are three ACH, two ACCH 3 , and one ACOH There are no Second-Order groups so the First Order and Second Order calculations are the same From Appendix Tables C2 and C3

the differences are

ethyl cyclopentane

dimethyl cyclopentane

cis-1,3- dimethyl cyclopentane First-Order

All of the substances have one or more Second-Order groups Using values of group contributions from Appendix Tables C-2 and C-3 and experimental values from Ap- pendix A, the results are

Property cycloheptane

methyl cyclohexane

ethyl cyclopentane

dimethyl cyclopentane

cis-1,3- dimethyl cyclopentane

errors shown are about average for the method.

A summary of the comparisons between estimations from the Second Order

Constantinou and Gani method and experimental Appendix A values for T fp and T b

are shown along with those from other estimation methods in Tables 2-9 and 2-10

Method of Marrero and Pardillo for T b Marrero-Marejo´n and

Pardillo-Fontdevila (1999) give two equations for estimating T b They call their preferredmethod a group interaction contribution technique; it can also be considered as a

method of bond contributions They tabulate contributions from 167 pairs of atoms

alone, such as⬎C⬍& —N⬍, or with hydrogen attached, such as CH3—&—NH2

(see Table 2-5 and the discussion of section 2-2) For T btheir basic equation is