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The Properties of Gases and Liquids, Fifth Edition Bruce E. Poling, JohnM. Prausnitz, John P. O’Connell cover

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1.1

CHAPTER ONETHE ESTIMATION OF PHYSICAL

PROPERTIES

1-1 INTRODUCTION

The structural engineer cannot design a bridge without knowing the properties ofsteel and concrete. Similarly, scientists and engineers often require the propertiesof gases and liquids. The chemical or process engineer, in particular, finds knowl-edge of physical properties of fluids essential to the design of many kinds of prod-ucts, processes, and industrial equipment. Even the theoretical physicist must oc-casionally compare theory with measured properties.

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, whichwe 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 werecombined with Avogadro’s hypothesis to form the gas ‘‘law,’’ PV � NRT, whichwas perhaps the first important correlation of properties. Deviations from the ideal-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 equationsof state express these quantitatively. Such extensions 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 contributionto 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 distributionin polar molecules.

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Source: THE PROPERTIES OF GASES AND LIQUIDS

1.2 CHAPTER ONE

Although allowance for the forces of attraction and repulsion between moleculesis 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 comprehen-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 Schrödinger 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.

Paralleling the continuing refinement of the molecular theory has been the de-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.

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 provideways to obtain important thermodynamic properties of a substance from PVTx re-lations where x stands for composition. Since derivatives are often required, thePVTx function must be known accurately.

The Information Age is providing a ‘‘shifting paradigm in the art and practiceof physical properties data’’ (Dewan and Moore, 1999) where searching the WorldWide Web can retrieve property information from sources and at rates unheard ofa few years ago. Yet despite the many handbooks and journals devoted to compi-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.

While the need for accurate design data is increasing, the rate of accumulationof new data is not increasing fast enough. Data on multicomponent mixtures areparticularly scarce. The process engineer who is frequently called upon to designa 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 possibleto 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.



THE ESTIMATION OF PHYSICAL PROPERTIES

THE ESTIMATION OF PHYSICAL PROPERTIES 1.3

1-2 ESTIMATION OF PROPERTIES

In the all-too-frequent situation where no experimental value of the needed propertyis 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 shemay 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, thevan der Waals equation of state is a modification of the simple PV � NRT; settingN � 1,

aP � (V � b) � RT (1-2.1)� �2V

Equation (1-2.1) is based on the idea that the pressure on a container wall, exertedby 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 excludedvolume b due to the size of the molecules themselves. Therefore, the ‘‘constants’’(or parameters) a and b have some theoretical basis though the best descriptionsrequire 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 theuse 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




1.4 CHAPTER ONE

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 lawis useful at low to modest pressures, although more accurate correlations are avail-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.

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 forall pure substances whose PVT properties could be expressed by a two-constantequation 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 dif-ferent substances; however, two-parameter corresponding states theory asserts thatif pressure, volume, and temperature are divided by the corresponding critical prop-erties, the function relating reduced pressure to reduced volume and reduced tem-perature becomes the same for all substances. The reduced property is commonlyexpressed as a fraction of the critical property: Pr � P /Pc ; Vr � V /Vc ; and Tr �T /Tc .

To illustrate corresponding states, Fig. 1-1 shows reduced PVT data for methaneand 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 Tr), of which only one is shown, agree equally well.

Successful application of the law of corresponding states for correlation of PVTdata has encouraged similar correlations of other properties that depend primarilyon intermolecular forces. Many of these have proved valuable to the practicingengineer. Modifications of the law are commonly made to improve accuracy or easeof use. Good correlations of high-pressure gas viscosity have been obtained byexpressing � /�c as a function of Pr and Tr . But since �c is seldom known and noteasily estimated, this quantity has been replaced in other correlations by othercharacteristics such as or the group where is the viscosity1 / 2 2 / 3 1 / 6�� , �� , M P T , ��c T c c cat Tc and low pressure, is the viscosity at the temperature of interest, again at��T





FIGURE 1-1 The law of corresponding states applied to the PVTproperties of methane and nitrogen. Literature values (Din, 1961): �methane, ● nitrogen.

low pressure, and the group containing M, Pc , and Tc is suggested by dimensionalanalysis. Other alternatives to the use of �c might 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 thatan expression is available for the potential-energy function for molecular interac-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 thereis good reason to believe it will become increasingly useful, especially when com-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.

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




1.6 CHAPTER ONE

consisting of small spherically-symmetric molecules. Typical corresponding-statescorrelations express a desired dimensionless property as a function of Pr , Tr , andthe chosen third parameter.

Unfortunately, the properties of strongly polar molecules are often not satisfac-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.

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 capacityof 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 calculatedby some simplified theory or empirical rule. For example, the methods of Lydersenand of others for estimating Tc start 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 (which are intimately related to structure) to obtain a polynomialC �pexpressing as a function of temperature; the constants in the polynomial areC �pdetermined by contributions from the constituent atoms, atomic groups, and typesof 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 eval-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.





FIGURE 1-2 Mollier diagram for dichlorodifluoro-methane. The solid lines represent measured data.Dashed lines and points represent results obtained by es-timation methods when only the chemical formula andthe normal boiling temperature are known.

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 applicationto 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 andin 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 whetheror 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-




1.8 CHAPTER ONE

thalpy of vaporization, and pressure corrections to ideal-gas enthalpies and entro-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 useof 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 evap-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, butit is found that the constant-entropy line through the point for the low-pressurevapor essentially coincides with the saturated vapor curve. The estimated coefficientof performance (ratio of refrigeration rate to isentropic compression power) is es-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.

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 en-ergies of formation and heat capacities, are covered in Chap. 3. Chapter 4 describesthe PVT properties of pure fluids with the corresponding-states principle, equationsof 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 sub-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.

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 se-lected arbitrarily because they are believed to be of wide interest, especially to





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 bookof 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 PracticingEngineer /Scientist in Today’s Information Age,’’ Paper 89C, AIChE 1999 Spring NationalMtg., 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.




2.1

CHAPTER TWOPURE 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 oftenused as the basis for models such as corresponding states correlations for PVTequations 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 ofthe 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 compareas 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 boil-ing points), we illustrate several methods and compare each with the data tabulatedin 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 inter-molecular forces that determine the constants of interest depend mostly on thebonds between the atoms of the molecules. The elemental contributions are prin-



Source: THE PROPERTIES OF GASES AND LIQUIDS

2.2 CHAPTER TWO

cipally determined by the nature of the atoms involved (atom contributions), thebonds between pairs of atoms (bond contributions or equivalently group interactioncontributions), or the bonds within and among small groups of atoms (group con-tributions). They all assume that the elements can be treated independently of theirarrangements 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 contributionsobtained 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 asmolecular weight and normal boiling point, etc. (factor analysis) or from ‘‘quan-titative structure-property relationships’’ (QSPR) based on contributions from mo-lecular properties such as electron or local charge densities, molecular surface area,etc. (molecular descriptors). Grigoras (1990), Horvath (1992), Katritzky, et al.(1995; 1999), Jurs [Egolf, et al., 1994], Turner, et al. (1998), and St. Cholakov, etal. (1999) all describe the concepts and procedures. The descriptor values are com-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 areno 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 suchas 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, Tc, pressure, Pc, and volume, Vc, are the pure-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 COMPONENT CONSTANTS

PURE COMPONENT CONSTANTS 2.3

peratures [Teja and Anselme, 1990]. Appendix A contains a data base of propertiesfor all the substances for which there is an evaluated critical temperature tabulatedby 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 NewYork and NIST (see Appendix A for references). There are fewer evaluated Pc andVc than Tc. We use only evaluated results to compare with the various estimationmethods.

Estimation Techniques

One of the first successful group contribution methods to estimate critical propertieswas 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 functional groups, and determined new contributionvalues. His relations for the critical properties are

2 �1

T (K) � T 0.584 � 0.965 N (tck) � N (tck) (2-2.1)� �� c b k kk k

�2

P (bar) � 0.113 � 0.0032N � N (pck) (2-2.2)�� c atoms kk

3 �1V (cm mol ) � 17.5 � N (vck) (2-2.3)�c kk

where the contributions are indicated as tck, pck and vck. The group identities andJoback’s values for contributions to the critical properties are in Table C-1. For Tc,a value of the normal boiling point, Tb, is needed. This may be from experimentor 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 groupmethod.

solution 2-ethylphenol contains one —CH3, one —CH2—, four �CH(ds), oneACOH (phenol) and two �C(ds). Note that the group ACOH is only for the OH anddoes not include the aromatic carbon. From Appendix Table C-1




2.4 CHAPTER TWO

Group k Nk Nk (tck) Nk (pck) Nk (vck)

—CH3 1 0.0141 �0.0012 65—CH2— 1 0.0189 0 56

CH(ds)� 4 0.0328 0.0044 164C(ds)� 2 0.0286 0.0016 64

—ACOH (phenol) 1 0.0240 0.0184 �25

NkFk5�

k�1

0.1184 0.0232 324

The value of Natoms � 19, while Tb � 477.67 K. The Joback estimation method (Sec.2-4) gives Tb � 489.74 K.

2 �1T � T [0.584 � 0.965(0.1184) � (0.1184) ]c b

� 698.1 K (with exp. T ), � 715.7 K (with est. T )b b

�2P � [0.113 � 0.0032(19) � 0.0232] � 44.09 barc

3 �1V � 17.5 � 324 � 341.5 cm molc

Appendix A values for the critical temperature and pressure are 703 K and 43.00bar. An experimental Vc is not available. Thus the differences are

T Difference (Exp. T ) � 703 � 698.1 � 4.9 K or 0.7%c b

T Difference (Est. T ) � 703 � 715.7 � �12.7 K or �1.8%c b

P Difference � 43.00 � 44.09 � �1.09 bar or �2.5%.c

A summary of the comparisons between estimations from the Joback methodand experimental Appendix A values for Tc, Pc, and Vc is shown in Table 2-1. Theresults indicate that the Joback method for critical properties is quite reliable forTc of all substances regardless of size if the experimental Tb is used. When estimatedvalues of Tb are 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 byb in the table, compared to 3.8% for the whole database indicated by a).

For Pc, the reliability is less, especially for smaller substances (note the differ-ence between the a and b entries). The largest errors are for the largest molecules,especially fluorinated species, some ring compounds, and organic acids. Estimatescan be either too high or too low; there is no obvious pattern to the errors. For Vc,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 (seeChap. 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

Property # Substances AAEc A%Ec # Err � 10%d # Err � 5%e

Tc (Exp. Tb)ƒ, K 352a 6.65 1.15 0 345289b 6.68 1.10 0 286

Tc (Est. Tb)g, K 352a 25.01 4.97 46 248290b 20.19 3.49 18 229

Pc, bar 328a 2.19 5.94 59 196266b 1.39 4.59 30 180

Vc, cm3 mol�1 236a 12.53 3.37 13 189185b 13.98 3.11 9 148

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

tested with the method.c AAE is average absolute error in the property; A%E is average absolute percent error.d The number of substances for which the absolute percent error was greater than 10%.e The 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.ƒ The experimental value of Tb in Appendix A was used.g The value of Tb used 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 N (F ) � W M (F ) (2-2.4)� �� k 1k j 2jk j

where ƒ can be a linear or nonlinear function (see Eqs. 2-2.5 to 2-2.7), Nk is thenumber of First-Order groups of type k in the molecule; F1k is the contribution forthe First-Order group labeled 1k to the specified property, F; Mj is the number ofSecond-Order groups of type j in the molecule; and F2j is the contribution for theSecond-Order group labeled 2j to the specified property, F. The value of W is setto zero for First-Order calculations and set to unity for Second-order calculations.

For the critical properties, the CG formulations are

T (K ) � 181.128 ln N (tc1k) � W M (tc2j ) (2-2.5)� �� c k jk j

�2

P (bar) � N (pc1k) � W M (pc2j ) � 0.10022 � 1.3705 (2-2.6)� �� c k jk j

3 �1V (cm mol ) � �0.00435 � N (vc1k) � W M (vc2j ) (2-2.7)� �� c k jk j

Note that Tc does not require a value for Tb. The group values for Eqs. (2-2.5) to(2-2.7) are given in Appendix Tables C-2 and C-3 with sample assignments shownin Table C-4.




2.6 CHAPTER TWO

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

solution The First-Order groups for 2-ethylphenol are one CH3, four ACH, oneACCH2, and one ACOH. There are no Second-Order groups (even though the orthoproximity effect might suggest it) so the First Order and Second Order calculations arethe same. From Appendix Tables C-2 and C-3

Group k Nk Nk(tc1k) Nk(pc1k) Nk(vc1k)

CH3 1 1.6781 0.019904 0.07504ACH 4 14.9348 0.030168 0.16860ACCH2 1 10.3239 0.012200 0.10099ACOH 1 25.9145 �0.007444 0.03162

5

N F� k kk�1

52.8513 0.054828 0.37625

T � 181.128 ln[52.8513 � W(0)] � 718.6 Kc

�2P � [0.054828 � W(0) � 0.10022] � 1.3705 � 42.97 barc

3 �1V � (�0.00435 � [0.37625 � W(0)])1000 � 371.9 cm molc

The Appendix A values for the critical temperature and pressure are 703.0 K and 43.0bar. An experimental Vc is not available. Thus the differences are

T Difference � 703.0 � 718.6 � �15.6 K or �2.2%c

�1P Difference � 43.0 � 42.97 � 0.03 kJ mol or 0.1%.c

Example 2-3 Estimate Tc, Pc, and Vc for the four butanols using Constantinou andGani’s group method

solution The First- and Second-Order groups for the butanols are:

Groups/Butanol 1-butanol2-methyl-1-propanol

2-methyl-2-propanol 2-butanol

# First-Order groups, Nk — — — —CH3 1 2 3 2CH2 3 1 0 1CH 0 1 0 1C 0 0 1 0OH 1 1 1 1

Second-Order groups, Mj — — — —(CH3)2CH 0 1 0 0(CH3)3C 0 0 1 0CHOH 0 1 0 1COH 0 0 1 0

Since 1-butanol has no Second-Order group, its calculated results are the same for bothorders. Using values of group contributions from Appendix Tables C-2 and C-3 andexperimental values from Appendix A, the results are:





Property/Butanol 1-butanol2-methyl-1-propanol


Tc, KExperimental 563.05 547.78 506.21 536.05Calculated (First Order) 558.91 548.06 539.37 548.06Abs. percent Err. (First Order) 0.74 0.05 6.55 2.24Calculated (Second Order) 558.91 543.31 497.46 521.57Abs. percent Err. (Second Order) 0.74 0.82 1.73 2.70

Pc, barExperimental 44.23 43.00 39.73 41.79Calculated (First Order) 41.97 41.91 43.17 41.91Abs. percent Err. (First Order) 5.11 2.52 8.65 0.30Calculated (Second Order) 41.97 41.66 42.32 44.28Abs. percent Err. (Second Order) 5.11 3.11 6.53 5.96

Vc, cm3 mol�1

Experimental 275.0 273.0 275.0 269.0Calculated (First Order) 276.9 272.0 259.4 272.0Abs. percent Err. (First Order) 0.71 0.37 5.67 1.11Calculated (Second Order) 276.9 276.0 280.2 264.2Abs. percent Err. (Second Order) 0.71 1.10 1.90 1.78

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 valuequite different from the others; most of this is taken into account by the First-Ordergroups, but the Second Order contribution is significant. Notice that the Second Ordercontributions for the other species are small and may change the results in the wrongdirection so that the Second Order estimate can be slightly worse than the First Orderestimate. This problem occurs often, but its effect is normally small; including SecondOrder effects usually helps and rarely hurts much.

A summary of the comparisons between estimations from the Constantinou andGani method and experimental values from Appendix A for Tc, Pc, and Vc is shownin Table 2-2.

The information in Table 2-2 indicates that the Constantinou/Gani method canbe 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 comparedto Table 2-2A for Tc and Pc but not for Vc. 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 Table2-2. For example, for Tc 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 herefor 335 compounds. Differences for Pc and Vc were also much less than given here.Abildskov (1994) made a limited study of the Constantinou/Gani method (less than100 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




2.8 CHAPTER TWO

TABLE 2-2 Summary of Constantinou /Gani MethodCompared to Appendix A Data Base

A. All substances in Appendix A with data that could betested with the method

Property Tc, K Pc, bar Vc, cm3 mol�1

# Substances (1st)a 335 316 220AAE (1st)b 18.48 2.88 15.99A%E (1st)b 3.74 7.37 4.38# Err � 10% (1st)c 28 52 18# Err � 5% (1st)d 273 182 160

# Substances (2nd)e 108 99 76AAE (2nd)b 17.69 2.88 16.68A%E (2nd)b 13.61 7.33 4.57# Err � 10% (2nd)c 29 56 22# Err � 5% (2nd)d 274 187 159# Better (2nd)ƒ 70 58 35Ave. �% 1st to 2ndg 0.1 0.2 �0.4

B. All substances in Appendix A having 3 or more carbonatoms with data that could be tested with the method

Property Tc, K Pc, bar Vc, cm3 mol�1

# Substances (1st)a 286 263 180AAE (1st)b 13.34 1.8 16.5A%E (1st)b 2.25 5.50 3.49# Err � 10% (1st)c 4 32 10# Err � 5% (1st)d 254 156 136

# Substances (2nd)e 104 96 72AAE (2nd)b 12.49 1.8 17.4A%E (2nd)b 2.12 5.50 3.70# Err � 10% (2nd)c 6 36 15# Err � 5% (2nd)d 254 160 134# Better (2nd)ƒ 67 57 32Ave. �% 1st to 2ndg 0.3 0.1 �0.5

a The number of substances in Appendix A with data that could betested with the method.

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

c The number of substances for which the absolute percent error wasgreater than 10%.

d The number of substances for which the absolute percent error wasless than 5%. The number of substances with errors between 5% and10% can be determined from the table information.

e The number of substances for which Second-Order groups are de-fined for the property.

f The number of substances for which the Second Order result is moreaccurate than First Order.

g The 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 providea 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 threemethods for Tc and Pc that apply to both organic and inorganic species. The Zero-Order method uses factor analysis with boiling point, liquid density and molecularweight as the descriptors. At the First Order, the method uses atomic contributionsalong with boiling point and number of rings, while the Second Order method alsoincludes group contributions. The Zero-Order has not been tested here; it is iterativeand the authors report that it is less accurate by as much as a factor of two or threethan the others, especially for Pc. The First Order and Second Order methods usethe following equations:

0.2

T � T 0.048271 � 0.019846N � N (� tck) � M (� tcj ) (2-2.8)� �� c b r k jk j

P � 0.0186233T / [�0.96601 � exp(Y )] (2-2.9a)c c

Y � �0.00922295 � 0.0290403N � 0.041 N (�pck) � M (�pcj )� �� r k jk j

(2-2.9b)

where Nr is the number of rings in the compound, Nk is the number of atoms oftype k with First Order atomic contributions � tck and �pck while Mj is the numberof groups of type j with Second-Order group contributions � tcj and �pcj. Valuesof the contributions are given in Table 2-3 both for the First Order Atomic Con-tributions and for the Second-Order Group Contributions. Note that Tc requires Tb.Application of the Wilson and Jasperson method is shown in Example 2-4.

Example 2-4 Estimate Tc and Pc for 2-ethylphenol by using Wilson and Jasperson’smethod.

solution The atoms of 2-ethylphenol are 8 �C, 10 �H, 1 �O and there is 1 ring.For groups, there is 1 �OH for ‘‘C5 or more.’’ The value of Tb from Appendix A is477.67 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

Atom k Nk Nk(� tck) Nk(�pck)

C 8 0.06826 5.83864H 10 0.02793 1.26600O 1 0.02034 0.43360

3

N F� k kk�1

— 0.11653 7.53824




2.10 CHAPTER TWO

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

Atom � tck �pck

H 0.002793 0.12660D 0.002793 0.12660T 0.002793 0.12660

He 0.320000 0.43400B 0.019000 0.91000C 0.008532 0.72983N 0.019181 0.44805O 0.020341 0.43360F 0.008810 0.32868

Ne 0.036400 0.12600Al 0.088000 6.05000Si 0.020000 1.34000P 0.012000 1.22000S 0.007271 1.04713Cl 0.011151 0.97711Ar 0.016800 0.79600Ti 0.014000 1.19000V 0.018600 *****Ga 0.059000 *****Ge 0.031000 1.42000As 0.007000 2.68000Se 0.010300 1.20000Br 0.012447 0.97151Kr 0.013300 1.11000Rb �0.027000 *****Zr 0.175000 1.11000Nb 0.017600 2.71000Mo 0.007000 1.69000Sn 0.020000 1.95000Sb 0.010000 *****Te 0.000000 0.43000I 0.005900 1.315930

Xe 0.017000 1.66000Cs �0.027500 6.33000Hf 0.219000 1.07000Ta 0.013000 *****W 0.011000 1.08000Re 0.014000 *****Os �0.050000 *****Hg 0.000000 �0.08000Bi 0.000000 0.69000Rn 0.007000 2.05000U 0.015000 2.04000





TABLE 2-3B Wilson-Jasperson (1996) GroupContributions for Eqs. (2-2.8) and (2-2.9)

Group � tcj �pcj

—OH, C4 or less 0.0350 0.00—OH, C5 or more 0.0100 0.00—O— �0.0075 0.00—NH2, �NH, �N— �0.0040 0.00—CHO 0.0000 0.50�CO �0.0550 0.00—COOH 0.0170 0.50—COO— �0.0150 0.00—CN 0.0170 1.50—NO2 �0.0200 1.00Organic Halides (once /molecule) 0.0020 0.00—SH, —S—, —SS— 0.0000 0.00Siloxane bond �0.0250 �0.50

Thus the First Order estimates are

0.2T � 477.67 / [0.048271 � 0.019846 � 0.11653] � 702.9 Kc

P � 0.0186233(704.1) / [�0.96601 � exp(Y )] � 37.94 barc

Y � �0.0092229 � 0.0290403 � 0.3090678 � 0.2708046

From Table 2-3B there is the ‘‘�OH, C5 or more’’ contribution of Nk� tck � 0.01though for Pc there is no contribution. Thus only the Second Order estimate for Tc ischanged to

0.2T � 477.67 / [0.048271 � 0.019846 � 0.11653 � 0.01] � 693.6 Kc

If the estimated value of Tb is used, the result is 710.9 K. The Appendix A valuesfor the critical properties are 703.0 K and 43.0 bar, respectively. Thus the differencesare

First Order T (Exp. T ) Difference � 703.0 � 702.9 � 0.1 K or 0.0%c b

T (Est. T ) Difference � 703.0 � 719.9 � �16.9 K or �2.4%c b

P Difference � 43.0 � 37.9 � 5.1 bar or 11.9%.c

Second Order T (Exp. T ) Difference � 703.0 � 693.6 � 9.4 K or 1.3%c b

T (Est. T ) Difference � 703.0 � 710.9 � �7.9 K or �1.1%c b

P (� First Order) Difference � 43.0 � 37.9 � 5.1 bar or 11.9%.c

The First Order estimate for Tc is more accurate than the Second Order estimate whichoccasionally occurs.

A summary of the comparisons between estimations from the Wilson and Jas-person method and experimental values from Appendix A for Tc and Pc are shownin Table 2-4. Unlike the Joback and Constantinou/Gani method, there was no dis-




2.12 CHAPTER TWO

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

PropertyTc, K

(Exp. Tb)*Tc, K

(Est Tb)�Pc, bar

(Exp Tc)#Pc, bar

(Est Tc)@

# Substancesa 353 — 348 348AAE (First Order)b 8.34 — 2.08 2.28A%E (First Order)b 1.50 — 5.31 5.91# Err � 10% (First Order)c 0 — 54 66# Err � 5% (First Order)d 220 — 234 220

# Substancese 180 289 23 23AAE (Second Order)b 6.88 16.71 1.82 2.04A%E (Second Order)b 1.22 2.95 4.74 5.39# Err � 10% (Second Order)c 0 15 46 57# Err � 5% (Second Order)d 348 249 245 226# Better (Second Order)ƒ 120 77 19 18Ave. �% First to Second Orderg 0.5 �1.8 8.6 7.9

* Eq. (2-2.8) with experimental Tb.� Eq. (2-2.8) with Tb estimated from Second Order Method of Constantinou and Gani (1994).# Eq. (2-2.9) with experimental Tc.@ Eq. (2-2.9) with Tc estimated using Eq. (2-2.8) and experimental Tb.a The 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.c The number of substances for which the absolute percent error was greater than 10%.d The 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 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.g The 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 propertyso only the overall is given.

The information in Table 2-4 indicates that the Wilson/Jasperson method is veryaccurate for both Tc and Pc. When present, the Second Order group contributionsnormally make significant improvements over estimates from the First Order atomcontributions. The accuracy for Pc deteriorates only slightly with an estimated valueof Tc if the experimental Tb is used. The accuracy of Tc is somewhat less when therequired Tb is 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 itis the simplest of those considered here and applies to all sizes of substancesequally.

Wilson and Jasperson compared their method with results for 700 compoundsof 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-Marejón and Pardillo-Fontdevila(1999) describe a method for Tc, Pc, and Vc that they call a group interactioncontribution 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

2

T � T / 0.5851 � 0.9286 N tcbk � N tcbk (2-2.10)� �� c b k kk k

�2

P � 0.1285 � 0.0059N � N pcbk (2-2.11)�� c atoms kk

V � 25.1 � N vcbk (2-2.12)�c kk

where Natoms is the number of atoms in the compound, Nk is the number of atomsof type k with contributions tcbk, pcbk, and vcbk. Note that Tc requires Tb, butMarrero and Pardillo provide estimation methods for Tb (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 � means2 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 bondsis to F, while C� & F— means that the C atom is doubly bonded to another�atom 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 both�have 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 CFCF3�would 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—, whereasin 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 itor of the double bond were changed, the contributions would change.

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

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




2.14 CHAPTER TWO

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

Pair # Atom/Group Pairs tcbk pcbk vcbk tbbk

1 CH3— & CH3— �0.0213 �0.0618 123.2 113.122 CH3— & —CH2— �0.0227 �0.0430 88.6 194.253 CH3— & �CH— �0.0223 �0.0376 78.4 194.274 CH3— & �C� �0.0189 �0.0354 69.8 186.415 CH3— & CH—� 0.8526 0.0654 81.5 137.186 CH3— & C�� 0.1792 0.0851 57.7 182.207 CH3— & �C— 0.3818 �0.2320 65.8 194.408 CH3— & �CH— [r] �0.0214 �0.0396 58.3 176.169 CH3— & �C� [r] 0.1117 �0.0597 49.0 180.60

10 CH3— & C� [r]� 0.0987 �0.0746 71.7 145.5611 CH3— & F— �0.0370 �0.0345 88.1 160.8312 CH3— & Cl— �0.9141 �0.0231 113.8 453.7013 CH3— & Br— �0.9166 �0.0239 ***** 758.4414 CH3— & I— �0.9146 �0.0241 ***** 1181.4415 CH3— & —OH �0.0876 �0.0180 92.9 736.9316 CH3— & —O— �0.0205 �0.0321 66.0 228.0117 CH3— & �CO �0.0362 �0.0363 88.9 445.6118 CH3— & —CHO �0.0606 �0.0466 128.9 636.4919 CH3— & —COOH �0.0890 �0.0499 145.9 1228.8420 CH3— & —COO[—] 0.0267 0.1462 93.3 456.9221 CH3— & [—]COO— �0.0974 �0.2290 108.2 510.6522 CH3— & —NH2 �0.0397 �0.0288 ***** 443.7623 CH3— & —NH— �0.0313 �0.0317 ***** 293.8624 CH3— & �N— �0.0199 �0.0348 76.3 207.7525 CH3— & —CN �0.0766 �0.0507 147.9 891.1526 CH3— & —NO2 �0.0591 �0.0385 148.1 1148.5827 CH3— & —SH �0.9192 �0.0244 119.7 588.3128 CH3— & —S— �0.0181 �0.0305 87.9 409.8529 —CH2— & —CH2— �0.0206 �0.0272 56.6 244.8830 —CH2— & �CH— �0.0134 �0.0219 40.2 244.1431 —CH2— & �C� �0.0098 �0.0162 32.0 273.2632 —CH2— & CH—� 0.8636 0.0818 50.7 201.8033 —CH2— & C�� 0.1874 0.1010 24.0 242.4734 —CH2— & �C— 0.4160 �0.2199 33.9 207.4935 —CH2— & �CH— [r] �0.0149 �0.0265 31.9 238.8136 —CH2— & �C� [r] 0.1193 �0.0423 ***** 260.0037 —CH2— & C� [r]� 0.1012 �0.0626 52.1 167.8538 —CH2— & F— �0.0255 �0.0161 49.3 166.5939 —CH2— & Cl— �0.0162 �0.0150 80.8 517.6240 —CH2— & Br— �0.0205 �0.0140 101.3 875.8541 —CH2— & I— �0.0210 �0.0214 ***** 1262.8042 —CH2— & —OH �0.0786 �0.0119 45.2 673.2443 —CH2— & —O— �0.0205 �0.0184 34.5 243.3744 —CH2— & �CO �0.0256 �0.0204 62.3 451.2745 —CH2— & —CHO �0.0267 �0.0210 106.1 648.7046 —CH2— & —COOH �0.0932 �0.0253 114.0 1280.3947 —CH2— & —COO[—] 0.0276 0.1561 69.9 475.6548 —CH2— & [—]COO— �0.0993 �0.2150 79.1 541.2949 —CH2— & —NH2 �0.0301 �0.0214 63.3 452.3050 —CH2— & —NH— �0.0248 �0.0203 49.4 314.71





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


51 —CH2— & �N— �0.0161 �0.0170 32.7 240.0852 —CH2— & —CN �0.0654 �0.0329 113.5 869.1853 —CH2— & —SH �0.0137 �0.0163 93.3 612.3154 —CH2— & —S— �0.0192 �0.0173 57.9 451.0355 �CH— & CH— �0.0039 �0.0137 18.3 291.4156 �CH— & �C� 0.0025 �0.0085 8.6 344.0657 �CH— & CH—� 0.8547 0.0816 48.9 179.9658 �CH— & C�� 0.1969 0.1080 4.3 249.1059 �CH— & �CH— [r] 0.0025 �0.0168 ***** 295.3360 �CH— & C� [r]� 0.1187 �0.0556 ***** 132.6661 �CH— & F— �0.0200 �0.0147 37.7 68.8062 �CH— & Cl— �0.0142 �0.0131 68.6 438.4763 �CH— & —OH �0.0757 �0.0093 45.6 585.9964 �CH— & —O— �0.0162 �0.0155 23.7 215.9465 �CH— & �CO �0.0194 �0.0112 39.3 434.4566 �CH— & —CHO �0.0406 �0.0280 92.2 630.0767 �CH— & [—]COO— �0.0918 �0.2098 72.3 497.5868 �CH— & —COOH �0.1054 �0.0358 110.2 1270.1669 �CH— & —NH2 �0.0286 �0.0212 39.2 388.4470 �CH— & —NH— �0.0158 �0.0162 ***** 260.3271 �C� & �C� 0.0084 0.0002 22.7 411.5672 �C� & CH—� 0.8767 0.0953 23.4 286.3073 �C� & C�� 0.2061 0.1109 8.8 286.4274 �C� & �C� [r] 0.0207 0.0213 ***** 456.9075 �C� & �CH— [r] 0.0049 �0.0111 ***** 340.0076 �C� & C� [r]� 0.1249 �0.0510 ***** 188.9977 �C� & F— �0.0176 �0.0161 30.0 �16.6478 �C� & Cl— �0.0133 �0.0129 63.7 360.7979 �C� & Br— �0.0084 �0.0121 85.7 610.2680 �C� & —OH �0.0780 �0.0094 40.6 540.3881 �C� & —O— �0.0156 �0.0103 40.8 267.2682 �C� & �CO �0.0114 �0.0085 62.1 373.7183 �C� & —COOH �0.1008 �0.0455 89.0 1336.5484 [ ]CH2 & [ ]CH2� � �0.9129 �0.0476 105.3 51.1385 [ ]CH2 & —CH[ ]� � �0.8933 �0.1378 77.4 205.7386 [ ]CH2 & �C[ ]� � �0.4158 �0.2709 99.2 245.2787 [ ]CH2 & C[ ]� � � �0.0123 �0.0239 68.4 183.5588 —CH[ ] & —CH[ ]� � �1.7660 �0.2291 47.8 334.6489 —CH[ ] & �C[ ]� � �1.2909 �0.3613 73.6 354.4190 —CH[ ] & C[ ]� � � �0.8945 �0.1202 43.6 316.4691 CH— & CH—� � 1.7377 0.1944 42.1 174.1892 CH— & C�� 1.0731 0.2146 16.6 228.3893 CH— & �C—� 1.2865 �0.1087 ***** 174.3994 CH— & C� [r]� � 0.9929 0.0533 ***** 184.2095 CH— & F—� 0.8623 0.0929 41.4 5.5796 CH— & Cl—� 0.8613 0.0919 68.7 370.6097 CH— & —O—� 0.8565 0.0947 36.4 204.8198 CH— & —CHO� 0.8246 0.0801 ***** 658.5399 CH— & —COOH� 0.7862 0.0806 107.4 1245.86

100 CH— & —COO[—]� 0.8818 0.2743 55.2 423.86




2.16 CHAPTER TWO



101 CH— & [—]COO—� 0.7780 �0.1007 64.1 525.35102 CH— & —CN� 0.8122 0.0771 107.4 761.36103 �C[ ] & �C[ ]� � �0.8155 �0.4920 93.7 399.58104 �C[ ] & C[ ]� � � �0.4009 �0.2502 58.1 321.02105 C� & C� [r]� � 0.3043 0.0705 ***** 250.88106 C� & F—� 0.1868 0.1064 14.6 �37.99107 C� & Cl—� 0.1886 0.1102 43.3 367.05108 C[ ] & O[ ]� � � �0.0159 �0.0010 51.4 160.42109 CH[�] & CH[�] �0.0288 �0.0226 87.6 120.85110 CH[�] & —C[�] �0.4222 0.1860 73.1 222.40111 —C[�] & —C[�] �0.7958 0.3933 64.3 333.26112 —CH2— [r] & —CH2— [r] �0.0098 �0.0221 47.2 201.89113 —CH2— [r] & �CH— [r] �0.0093 �0.0181 47.5 209.40114 —CH2— [r] & �C� [r] �0.1386 0.0081 49.9 182.74115 —CH2— [r] & CH— [r]� 0.0976 �0.1034 42.5 218.07116 —CH2— [r] & C� [r]� 0.1089 �0.0527 ***** 106.21117 —CH2— [r] & —O— [r] �0.0092 �0.0119 29.2 225.52118 —CH2— [r] & �CO [r] �0.0148 �0.0177 50.7 451.74119 —CH2— [r] & —NH— [r] �0.0139 �0.0127 38.8 283.55120 —CH2— [r] & —S— [r] �0.0071 ***** ***** 424.13121 �CH— [r] & �CH— [r] �0.0055 �0.0088 33.9 210.66122 �CH— [r] & �C� [r] �0.1341 0.0162 ***** 220.24123 �CH— [r] & �CH— [rr] ***** ***** ***** 254.50124 �CH— [r] & �C[ ] [rr]� ***** ***** ***** 184.36125 �CH— [r] & —O— [r] �0.0218 �0.0091 19.2 169.17126 �CH— [r] & —OH �0.0737 �0.0220 597.82127 �C� [r] & �C� [r] 0.0329 �0.0071 36.2 348.23128 �C� [r] & C� [r]� ***** ***** ***** 111.51129 �C� [r] & F— �0.0314 �0.0119 18.4 �41.35130 —CH[ ] [r] & —CH[ ] [r]� � �0.2246 0.1542 36.5 112.00131 —CH[ ] [r] & �C[ ] [r]� � �0.3586 0.1490 34.4 291.15132 —CH[ ] [r] & —N[ ] [r]� � 0.3913 0.1356 8.3 221.55133 CH— [r] & CH— [r]� � 0.2089 �0.1822 39.3 285.07134 CH— [r] & C� [r]� � 0.2190 �0.1324 29.8 237.22135 CH— [r] & —O— [r]� 0.1000 �0.0900 40.3 171.59136 CH— [r] & —NH— [r]� 0.0947 ***** ***** 420.54137 CH— [r] & N— [r]� � �0.4067 �0.1491 65.9 321.44138 CH— [r] & —S— [r]� 0.1027 �0.0916 40.8 348.00139 �C[ ] [r] & �C[ ] [r]� � �0.4848 0.1432 37.8 477.77140 �C[ ] [r] & —N[ ] [r]� � 0.2541 ***** ***** 334.09141 C� [r] & C� [r]� � 0.2318 �0.0809 20.6 180.07142 C� [r] & C� [rr]� � 0.2424 �0.0792 51.7 123.05143 C� [r] & —O— [r]� 0.1104 �0.0374 �0.3 134.23144 C� [r] & N— [r]� � �0.3972 �0.0971 35.6 174.31145 C� [r] & F—� 0.1069 �0.0504 23.7 �48.79146 C� [r] & Cl—� 0.1028 �0.0512 60.3 347.33147 C� [r] & Br—� 0.1060 �0.0548 83.2 716.23148 C� [r] & I—� 0.1075 �0.0514 110.2 1294.98149 C� [r] & —OH� 0.0931 �0.0388 8.5 456.25150 C� [r] & —O—� 0.0997 �0.0523 ***** 199.70







151 C� [r] & �CO� 0.1112 �0.0528 46.3 437.51152 C� [r] & —CHO� 0.0919 �0.0597 ***** 700.06153 C� [r] & —COOH� 0.0313 �0.0684 100.2 1232.55154 C� [r] & [—]COO—� 0.0241 �0.2573 55.2 437.78155 C� [r] & —NH2� 0.0830 �0.0579 33.2 517.75156 C� [r] & —NH—� 0.0978 �0.0471 ***** 411.29157 C� [r] & �N—� 0.0938 �0.0462 ***** 422.51158 C� [r] & —CN� 0.0768 �0.0625 ***** 682.19159 Cl— & �CO �0.0191 �0.0125 84.0 532.24160 [—]COO— & [—]COO— �0.1926 �0.0878 ***** 1012.51161 —O— [r] & N— [r]� �0.5728 ***** ***** 382.25162 �CO & —O— �0.3553 �0.0176 ***** 385.36163 —H & —CHO �0.0422 �0.0123 ***** 387.17164 —H & —COOH �0.0690 ***** ***** 1022.45165 —H & [—]COO— �0.0781 �0.1878 51.2 298.12166 —NH— & —NH2 �0.0301 ***** ***** 673.59167 —S— & —S— �0.0124 ***** ***** 597.59

CH3

H

HH

10

130

133

131

HH

H

H

133

130

130

139

141

134

134

Pair #

10130131133134139141

Atom/Group Pair

CH3— & C� [r]�—CH[ ] [r] & —CH[ ] [r]� �—CH[ ] [r] & �C[ ] [r]� �

CH— [r] & CH— [r]� �CH— [r] & C� [r]� �

�C[ ] [r] & �C[ ] [r]� �C� [r] & C� [r]� �

Nk

1312311

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 Tc, Pc, and Vc for 2-ethylphenol by using Marrero and Pardillo’smethod.

solution The chemical structure to be used is shown. The locations of the variousbond pairs are indicated on the structure shown. The value of Natoms is 19.

OH

CH CH32H

HH

H

149

130

133

133

141 2

37131

131

Pair #

237

130131133141149

—

Atom / Group Pair

CH3— & —CH2——CH2— & C� [r]�—CH[ ] [r] & —CH[ ] [r]� �—CH[ ] [r] & �C[ ] [r]� �

CH— [r] & CH— [r]� �C� [r] & —OH�

7

N F� k kk�1

Nk

1112211

—

Nk tck

�0.02270.1012

�0.2246�0.7172

0.41780.23180.0931

�0.1206

Nk pck

�0.0430�0.0626

0.15420.2980

�0.3644�0.0809�0.0388

�0.1375

Nk vck

88.652.136.568.878.620.6

8.5

353.7




2.18 CHAPTER TWO

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

T � 477.67 / [0.5851 � 0.1120 � 0.0145] � 699.8 Kc

�2P � [0.1285 � 0.1121 � 0.1375] � 42.2 barc

3 �1V � 25.1 � 353.7 � 378.8 cm molc

The Appendix A values for the critical temperature and pressure are 703.0 K and 43.0bar. An experimental Vc is not available. Thus the differences are

T Difference � 703.0 � 699.8 � 3.2 K or 0.5%c

P Difference � 43.0 � 42.2 � 0.8 bar or 1.8%c

If Marrero and Pardillo’s recommended method for estimating Tb is used (Eq. 2-4.5),the result for Tc is 700.6, an error of 0.3% which is more accurate than with theexperimental Tb.

Example 2-7 Estimate Tc, Pc, and Vc for the four butanols using Marrero and Pardillo’smethod.

solution The Atom/Group Pairs for the butanols are:

Pair # Atom/Group Pair 1-butanol2-methyl-1-propanol


2 CH3— & —CH2— 1 0 0 13 CH3— & �CH— 0 2 0 14 CH3— & �C� 0 0 3 0

29 —CH2— & —CH2— 2 0 0 030 —CH2— & �CH— 0 1 0 142 —CH2— & —OH 1 1 0 063 �CH— & —OH 0 0 0 180 �C� & —OH 0 0 1 0

Using values of group contributions from Table 2-5 and experimental values fromAppendix A, the results are:

Property/Butanol 1-butanol2-methyl-1-propanol


Tc, KExperimental 563.05 547.78 506.21 536.05Calculated (Exp Tb)a 560.64 549.33 513.80 538.87Abs. percent Err. (Exp Tb)a 0.43 0.28 1.50 0.53Calculated (Est Tb)b 560.40 558.52 504.56 533.93Abs. percent Err. (Est Tb)b 0.47 1.96 0.33 0.40

Pc, barExperimental 44.23 43.00 39.73 41.79Calculated 44.85 45.07 41.38 43.40Abs. percent Err. 1.41 4.81 4.14 3.86

Vc, cm3 mol�1

Experimental 275.0 273.0 275.0 269.0Calculated 272.1 267.2 275.0 277.8Abs. percent Err. 1.07 2.14 0.01 3.26

a Calculated with Eq [2-2.10] using Tb from Appendix A.b Calculated with Eq [2-2.10] using Tb estimated with Marrero / Pardillo method Eq. (2-4.5).





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

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

Property Tc*, K Tc#, K Pc, bar Vc, cm3 mol�1

# Substancesa 343 344 338 296AAEb 6.15 15.87 1.79 13.25A%Eb 0.89 2.93 5.29 3.24# Err � 10%c 1 22 47 18# Err � 5%d 336 288 228 241# Better Est Tbe 83

B. All substances in Appendix A having 3 or more carbon atoms with data that could betested with the method

Property Tc*, K Tc#, K Pc, bar Vc, cm3 mol�1

# Substancesa 285 286 280 243AAEb 5.78 15.53 1.68 14.72A%Eb 0.94 2.62 5.38 3.28# Err � 10%c 1 14 39 15# Err � 5%d 282 248 188 200# Better with Est Tbe 68

* Calculated with Eq [2-2.10] using Tb from Appendix A.# Calculated with Eq [2-2.10] using Tb estimated with Marrero / Pardillo method Eq. (2-4.5).a The 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.c The number of substances for which the absolute percent error was greater than 10%.d The 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 Tc result is more accurate when an estimated Tb is used than

with an experimental value.

The results are typical of the method. Notice that sometimes the value with anestimated Tb is more accurate than with the experimental value. As shown in Table2-4, this occurs about 1⁄4 of the time through coincidence.

A summary of the comparisons between estimations from the Marrero and Par-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.

The information in Table 2-6 indicates that the Marrero/Pardillo is accurate forthe critical properties, especially Tc. The substances with larger errors in Pc and Vcare 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




2.20 CHAPTER TWO

Tc for all types of organics with somewhat better accuracy than Joback’s method;Sastri, et al. (1997) treating only Vc and obtaining somewhat better accuracy thanJoback’s method; Tobler (1996) correlating Vc with 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. Thetechniques generally fall into two classes. The first is based on factor analysis thatbuilds 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 onmolecular 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 softwareto 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 Jursshows the current status of molecular descriptor methods.

Method of Grigoras. An early molecular structural approach to physical proper-ties of pure organic substances was proposed by Grigoras (1990). The concept wasto relate several properties to the molecular surface areas and electrostatics as gen-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

V � 2.217A � 93.0 (2-2.13)c

T � 0.633A � 1.562A � 0.427A � 9.914A � 263.4

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