While confining our present assess-ment to an index of accuracy, we havementioned that the ROC is the properbasis for an evaluation of the usefulnessof a diagnostic system, which would in-clude elements of medical efficacy, risk,and cost. In such an evaluation one pro-ceeds from probabilities of responsesbased on the diagnostic system understudy, through a further decision-thera-peutic tree, to health outcomes. TheROC is a means of determining the re-sponse probabilities appropriate to thebest available estimates of the values,costs, and event probabilities that inherein the relevant diagnostic and therapeu-tic context-and not merely the responseprobabilities associated with whateverdecision criterion might have been em-ployed in a test of the system (13). Costsand benefits may, therefore, be deter-mined for a system operating at its best.

References and Notes

1. The materials and methods of this study are de-scribed in more detail, along with additional re-sults and discussion, in Technical Report No.3818 from Bolt Beranek and Newman Inc., tothe National Cancer Institute (1979).

2. W. W. Peterson et al., Trans. IRE Prof.Group Inf. Theory PGIT-4, 171 (1954).

3. D. M. Green and J. A. Swets, Signal DetectionTheory and Psychophysics (Wiley, New York,1966; reprinted by Krieger, New York, 1974).

4. J. A. Swets, Science 182, 990 (1973).5. __ and D. M. Green, in Psychology: From

Research to Practice, H. L. Pick, Jr., H. W.Leibowitz, J. E. Singer, A. Steinschneider, H.W. Stevenson, Eds. (Plenum, New York, 1978),pp. 311-331.

6. L. B. Lusted, Introduction to Medical DecisionMaking (Thomas, Springfield, Ill., 1968); B. J.McNeil, B. Keeler, Jr., S. J. Adelstein, N. Engl.J. Med. 293, 211 (1975); J. A. Swets, Invest. Ra-diol. 14 (No. 2), 109 (1979).

7. C. E. Metz, Semin. Nucl. Med. 8, 283 (1978).8. Principal investigators and institutions partici-

pating in the collaborative study were H. L.Baker, Jr., Mayo Clinic; D. 0. Davis, GeorgeWashington University Medical Center; S. K.Hilal, Columbia University; P. F. J. New, Mas-lachusetts General Hospital; and D. G. Potts,Cornell University Medical Center. Data coordi-pation for the collaborative study was suppliedby C. R. Buncher, University of CincinnatiMedical Center.

9. D. D. Dorfman and E. Alf, Jr., J. Math. Psy-chol. 6, 487 (1969).

10. S. J. Starr, C. E. Metz, L. B. Lusted, D. J.Goodenough, Radiology 116, 533 (1975).

11. M. G. Kendall, Rank Correlation Methods (Haf-ner, New York, 1962).

12. R. F. Jarrett and F. M. Henry, J. Psychol. 31,175 (1951).

13. J. A. Swets and J. B. Swets, in Proceedings ofthe Joint American College ofRadiology-IEEEConference on Computers in Radiology (IEEEComputer Society, Silver Spring, Md., in press).

14. Actively representing the sponsor to the projectwere R. Q. Blackwell, W. Pomerance, J. M. S.Prewitt, and B. Radovich. We are indebted toDr. Prewitt for recognizing the need for a gener-al protocol and for suggesting the present appli-cation. Members of an advisory panel, who con-tributed extensively to the protocol develop-ment and the design of this study, were S. J.Adelstein, H. L. Kundel, L. B. Lusted, B. J.McNeil, C. E. Metz, and J. E. K. Smith. J. A.Schnur participated throughout the project; con-sultants for the study reported here were W. B.Kaplan and G. M. Kleinman. We are indebtedalso to the principal investigators in the collabo-rative study (8). We thank the following radiol-ogists, who participated as readers in the testsconducted at Bolt Beranek and Newman Inc.:computed tomography-L. R. Altemus, R. A.Baker, A. Duncan, D. Kido, J. Lin, and T. P.Naidrich; radionuclide scanning-H. L. Chan-dler, J. P. Clements, T. C. Hill, L. Malmud,F. D. Thomas, and D. E. Tow.

Molecular Thermodynamics forChemical Process Design

The properties of fluid mixtures must be understoodfor economic manufacture of chemical products.

J. M. Prausnitz

Chemical engineering design is con- quires quantitative knowledge of thecerned with finding economic and effi- properties of the materials to be pro-cient methods for producing on a large cessed. Usually, these materials are mix-scale what the chemist or materials sci- tures.

Summary. Chemical process design requires quantitative information on the equi-librium properties of a variety of fluid mixtures. Since the experimental effort needed toprovide this information is often prohibitive in cost and time, chemical engineers mustutilize rational estimation techniques based on limited experimental data. The basisfor such techniques is molecular thermodynamics, a synthesis of classical and statis-tical thermodynamics, molecular physics, and physical chemistry.

entist produces in small quantities. Inmeeting this concern, the chemical engi-neer must conceive, design, and buildlarge units of equipment for processinglarge quantities of gases, liquids, and sol-ids. Rational design of this equipment re-SCIENCE, VOL. 205, 24 AUGUST 1979

While test tubes and glass retorts areuseful for manufacturing a few grams ofa chemical substance, entirely differentequipment is needed when manufactur-ing tons of that substance. Similarly, thechemical processes used for large-scale

production are often highly dissimilarfrom those used for small-scale produc-tion. For example, a standard laboratorymethod for producing a few grams of hy-drogen is to react hydrochloric acid withzinc, giving zinc chloride in addition tohydrogen. However, when hydrogen isneeded in large amounts this procedureis inefficient, not only because the reac-tants are too expensive, but also becausethe available supply of zinc is too smallto meet the world's need for hydrogenand because there would be a severeproblem of what to do with vast amountsof zinc chloride. In principle, the zincchloride could be reduced to recoverzinc, but the expense of that operationwould be prohibitive. For large-scaleproduction of hydrogen, the traditionalindustrial method is to react steam withcoal or other hydrocarbons or to oxidizepartially natural gas with air.

In addition to dominant economic fac-tors, there is a significant technical dif-ference between laboratory-scale andindustrial-scale production of typicalchemical products. In the laboratory, thechemist generally uses pure reactants(for example, pure hydrochloric acidfrom one bottle and pure zinc from an-other) and the reaction is often so carriedout that the desired product (gaseous hy-drogen) and the undesired side product(solid zinc chloride) appear as separatephases, thereby avoiding any cimber-some separation operation. In industrial

The author is a professor in the Department ofChemical Engineering, University of California,Berkeley 94720.

0036-8075/79/0284-0759$01.75/0 Copyright X 1979 AAAS 759

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Raw material!

Stages I and III require separation operations (such as distillation,absorption, extraction). In a typical chemical plant, 40-80% ofinvestment is for separation-operation equipment.

Fig. 1. Schematic of a chemical plant.

tures that is encountered in chemical in-dustry and the wide ranges of pressure,

-Product temperature, and composition that aremet in chemical processes, it is not pos-sible to measure experimentally all the

products physical properties needed for the designof efficient separation operations. It isdesirable, indeed necessary, to establishgeneralizations for estimating physicalproperties of mixtures, using only limit-ed experimental information. Molecularthermodynamics is an important tool forproviding such generalizations.

processes, however, separations play amajor role because more often than notthe reactants as well as the products aremixtures, usually fluid mixtures.The bulk raw materials that nature

provides (such as petroleum and air) arerarely in pure form and therefore it is'of-ten necessary to separate desired fromundesired reactants prior to chemical re-action and, similarly, to separate desiredfrom undesired products after chemicalreaction. Further, since a chemical reac-tion is rarely complete, it is usually nec-essary in addition to separate from theproduct stream the unreacted reactants,which are then recycled'. Figure 1 illus-trates schematically the need for separa-tion operations in industrial chemicalprocesses.The heart of a chemical plant is the re-

actor, where reactants are converted toproducts. But in addition to a heart, achemical plant also requires a mouth anda digestive system; purification (separa-tion) operations are often necessary toprepare the reactants for reaction, andextensive purification is almost alwaysnecessary after the reactor to separatethe products wanted from those that arenot and from those reactants that havefailed to react.

Separation of Mixtures

In typical cases, the various chemicalspecies needed for reaction, as well asthose leaving the reactor, are in the samephase (usually a gas or a liquid) andtherefore separation can usually not beaccomplished by a simple mechanicalmethod such as filtration. Instead, it isnecessary to establish separation by dif-fusional operations, such as distillation,absorption, and extraction. The costs as-sociated with these operations often rep-resent a substantial fraction of the totalcost of the final product. In many chem-ical plants, the investment costs for sep-aration operations account for 40 to 80percent of the total; in some cases (for

760

example, petroleum refineries), that per-centage may be larger.

Design of chemical reactors dependsstrongly on the chemical properties ofthe reactants; that is, the properties thatdetermine the tendency of the reactantmolecules to react chemically to formnew molecules. By contrast, design ofseparation operations depends stronglyon the physical properties of the variousmolecules in the mixture, as on their rel-ative volatilities in absorption or theirrelative solubilities in extraction. Effi-cient design of separation equipmenttherefore requires reliable techniques forestimating the physical properties ofmixtures, not only to minimize the sizeand complexity of the equipment but al-so to minimize the energy requirement(heating, cooling, compressing, expand-ing) for operating it. This requirementhas been of major importance for manyyears in the chemical (and related) indus-tries, and it is becoming much more im-portant now as energy is increasingly ex-pensive and as convenient raw materialsare increasingly scarce, to be replacedby other raw materials of lower grade.The chemical process industries there-fore have a large financial incentive tofind effective methods for determiningthe physical properties of mixtures to fa-cilitate design of economic separationoperations.

Considering the wide variety of mix-

Abstract world:

Real world: Problem

II

111

Answer

Step IProjection of real problem into an abstract (math-

ematical) formulation. (Definition of abstractquantities such as chemical potential.)

Step II

Solution of mathematical equations.Step Ill

Reprojection of mathematical solution into physi-cal reality. This is the hard step.

Fig. 2. Thermodynamics in practice: abstrac-tion and reality.

Classical and Statistical Thermodynamics

Classical chemical thermodynamics,as developed more than 100 years ago byJ. W. Gibbs, gives a comprehensivemethodology for a broad description ofthe equilibrium properties of materials,including fluid mixtures. Gibbsian ther-modynamics, because of its breadth andelegance, is one of the highlights of clas-sical science, but at the same time it pos-sesses severe limitations for practicalwork. Classical chemical thermodynam-ics gives a sweeping view, but it is notsufficient for quantitative results unlesscoupled with extensive-often prohibi-tive-experimental effort. The positiveand negative features of classical ther-modynamics are easily summarized:

1) 'Positive features: Broad, generaldescription of nature. Applicable (inprinciple) to many phenomena. Rigorousmathematical formulation. Elegant, es-thetically pleasing. Independent of anytheory of matter.

2) Negative features: Uses abstractquantities, difficult to visualize. Inter-relates experimental measurements butcannot predict a priori. Incomplete. Formany practical applications need "meta-thermodynamics" (for example, phys-ical models).

During the last 90 years, a vast litera-ture on thermodynamics has accumu-lated. There are hundreds of textbooksand thousands of journal articles andevery university offers several courseson various aspects of thermodynamics,thereby attesting to its academic respect-ability. Nevertheless, in practice, ther-modynamics is often a disappointment tothe chemical engineer. In textbooks, thereader finds dozens of equations whereinproperty A, which he seeks, is related toproperty B, which he does not know;property B is then related to property Cand so on, until, at last, property Z is re-lated to property A. To the nonexpertreader, texts on chemical thermodynam-ics seem to provide little more than a bu-

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reaucratic runaround. Unfortunately,classical thermodynamics never predictsany property by itself but only relatesone property to another. The chemicalengineer, seeking answers, often feelsfrustrated; thermodynamics has let himdown. At the same time, many articlesand compendiums tabulate large quanti-ties of thermodynamic data, but moreoften than not, the published data do notapply to the practical problem of interest.

Classical thermodynamics is revered,honored, and admired, but in practiceit is inadequate. The place of thermo-dynamics in relation to chemical engi-neering may be illustrated by two fa-mous quotations about Helen of Troy:"'Admired much, and much reviled"(Goethe, Faust, part II) and "Was thisthe face that launched a thousand ships?"(Marlowe, Dr. Faustus).

Application of abstract thermodynam-ics to real problems requires a three-stepprocess, shown in Fig. 2. The real prob-lem is first projected into an abstractmathematical world through the use ofingenious abstract functions (such asthe chemical potential introduced byGibbs) which, through the use of stan-dard mathematical operations, allow theproblem to be solved in the abstractworld. The genius of classical thermody-namics lies in providing the abstractfunctions that enable elegant perform-ance of steps I and II. The difficulty inreducing thermodynamics to practicelies in step III. In performing this laststep, classical thermodynamics, by it-self, is not sufficient. For step III, it isnecessary to link abstract thermodynam-ic concepts with real physical properties.

Classical thermodynamics' indepen-dence of any particular theory of matterprovides its glory, but this independenceis also responsible for its weakness. Inhis later work, Gibbs tried to overcomethis weakness through application of sta-tistical physics as developed by Boltz-mann, leading to statistical thermody-namics: an attempt to describe bulk(macroscopic) properties of matter interms of (microscopic) intermolecularforces and the molecular structure of ma-terials. Although statistical mechanics(the basis of statistical thermodynamics)has made remarkable progress since theearly efforts of Gibbs and Boltzmann, ithas not yet reached a stage where, by it-self, it can yield practical methods forcalculating the properties of real mix-tures.A candid view of statistical mechanics

has been provided by Eugene Wigner,who said, "With thermodynamics, onecan calculate almost everything crudely;24 AUGUST 1979

Fig. 3. Equation ofstate from partitionfunction Q.

P = kT [atnQP TT[aV N

For ideal gas:

I [ N N

A= h(2TrmkT)1/ 2

with kinetic theory, one can calculatefewer things, but more accurately; andwith statistical mechanics, one can cal-culate almost nothing, exactly." A chemi-cal engineer is likely to be in full agree-ment with that view, except that he willbe tempted to omit the word "almost."

Molecular Thermodynamics

Although statistical thermodynamicsalone cannot provide the mixture proper-ties needed for the design of separationoperations, it does suggest techniquesfor constructing sensible physical mod-els which, when coupled with appropri-ate concepts from molecular physics andphysical chemistry, can be used to cor-relate limited experimental data. Suchcorrelations can very much reduce thenecessary experimental effort by pro-viding good estimates of bulk propertiesthrough physically meaningful interpola-tion and extrapolation. These estimatesare often sufficiently reliable for chemi-cal process design.

Molecular thermodynamics is a syn-thesis of classical thermodynamics, sta-tistical thermodynamics, molecular phys-ics, and physical chemistry. It is anapplied science in the sense that it com-bines, toward practical ends, the per-tinent tools that physical science offers

Vf = Free volume0/2 = Potential energy associated with

one molecule

p = p(l) + p(2) + p(3)

p( [= kT [aIni]]

T,N

= 1,2,3

Fig. 4 (left). Basic assumptions of generalizedvan der Waals theory. Fig. 5 (right). Freevolume Vf.

N = No. of moleculesV = Total volume

qr,ve = Molecular partition functionfor rotational, vibrationaland electronic motions

h = Planck's constantm = Molecular massk = Boltzmann's constant

to the design engineer to help him solveimportant engineering problems. It isthus a compromise and a synthesis, asindicated by the following steps:

1) Use statistical thermodynamicswhenever possible, at least as a point ofdeparture.

2) Apply appropriate concepts frommolecular physics and physical chemis-try.

3) Construct physically groundedmodels for expressing (abstract) ther-modynamic functions in terms of (real)measurable properties.

4) Obtain model parameters from veryfew, but representative, experiments.

In the next sections I present a few ex-amples to illustrate how molecular ther-modynamics provides quantitative esti-mates of mixture properties that are di-rectly useful for chemical process de-sign.

Applications of a Generalized

van der Waals Partition Function

The first example indicates how an ap-proximate partition function may be em-ployed to obtain useful expressions forcalculating phase equilibria typically en-countered in the petroleum, natural gas,and polymer industries. To start, Fig. 3reviews some well-known relations in

Free volume = total volume - excluded volume

Dilute region (no overlap)

Vf = V - b

Dense region (extensive overlap)

Vf > V - b

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statistical thermodynamics. The parti-tion function Q gives all thermodynamicproperties through standard procedures:differentiation with respect to composi-tion gives the chemical potential, dif-ferentiation with respect to temperaturegives internal energy or entropy, and dif-ferentiation with respect to volume givesthe equation of state. The last differ-entiation is of particular importance formolecular thermodynamics because itlinks the abstract partition function Qto directly measurable quantities: pres-sure P, volume V, and temperature T.For an ideal gas it is simple to write an

expression for Q, but for real fluids(gases and liquids) we can only write anapproximate expression, following ideasthat go back more than 100 years to vander Waals, as indicated in Fig. 4.

In an ideal gas, molecules are pointsthat do not influence each other. How-ever, molecules in a real fluid have afinite size, leading to repulsive forcesat close molecule-molecule separations,and they have electronic configurations,leading to forces of attraction at inter-mediate separations. In the generalizedvan der Waals model, repulsive forcesare taken into account through the con-cept of free volume Vf and attractiveforces through the concept of a uniformpotential field 4, as briefly explained be-low. But in addition to translational mo-tions, molecules also have rotational andvibrational degrees of freedom, whichcontribute to the partition function.(Electronic degrees of freedom are im-portant only at high temperatures andtherefore are not of concern here.)As shown in Fig. 4, there are three

contributions to the equation of state. Inrecent years there has been much prog-ress in understanding the first two contri-butions, but little is known about thethird. Nearly all theoreticians who workin the area of fluids restrict attention toargon-like molecules where the thirdcontribution disappears; the usual as-sumption is that for small, nearly spheri-cal molecules, rotational and vibrationalmotions depend only on temperature andare independent of density. For largermolecules, especially those that are non-spherical, this simplifying assumption isnot valid.

Figure 5 gives a two-dimensional viewof the free-volume concept. The free vol-ume is that part of the total volumewhich is accessible; because of the finitesize of all molecules, a particular mole-cule, wandering about, does not have ac-cess to every part of the total volumesince some of that volume (the excludedvolume) is occupied by other molecules.

762

0 0

00

Potential energy between two molecules is r(r).Potential energy between central molecule andall others is

=(N/VJf r(r)g(r) 4qrr2 dr

g(r) = Radial distribution function, tells wheremolecules are.

0

co

be

1 2 5 10 20 50 100

Pressure (bars)

Fig. 6 (top). Potential energy 0. Fig. 7(middle). Perturbed-hard-chain theory pro-vides an equation of state for fluids containingsimple or complex molecules, covering allfluid densities. Fig. 8 (bottom). K factorsin the binary system methane-propane.

In Fig. 5, each molecule is representedby a billiard ball, or hard sphere. The fi-nite size of a molecule is indicated by thedark line. Let one molecule approach an-other until the two surfaces are in con-tact; this is the closest possible mole-cule-molecule distance. The center ofone of the molecules cannot penetratethe volume bounded by the dashed linearound the other; that volume is notavailable: it is excluded. When the num-ber of molecules is small (low density),this excluded volume is given by the vander Waals parameter b, which is propor-tional to the actual volume of all the fi-nite-sized molecules.At high densities, it is much more diffi-

cult to calculate the excluded volume be-cause of overlap, as shown by theshaded areas in the lower part of Fig. 5.The excluded volume is less than the pa-rameter b because in the calculation of bthe overlapping regions are now errone-ously counted more than once. Van derWaals and his co-workers were wellaware of overlap but no satisfactory so-lution to this difficult mathematical prob-lem was achieved until the middle1960's. There is no need here to go intodetails; it is sufficient to say that forhard-sphere molecules we now have areliable expression for the free volume,valid for all fluid densities (1, 2).

Figure 6 gives a two-dimensional viewof the uniform potential field concept.Consider the molecule at the center; wewant to calculate the attractive-forcefield experienced by that molecule. Thepotential energy for any pair of mole-cules is designated by F(r) where r is thecenter-to-center distance of separation.To sum up the contributions of all mole-cules interacting with the one at thecenter, we perform an integration, asshown. Imagine a sphere of radius r; thearea of that sphere is 4rr2. We now builda thin shell on that area; the thickness ofthe shell is dr and its volume is 47rr2dr.As a first estimate, we calculate the num-ber of molecular centers in the shell bymultiplying 4rr2dr by the overall densityNIV. But that is a crude calculationwhich erroneously assumes that the localdensity (the density at position r) is al-ways equal to the overall (or average)density. The structure of a fluid is notcompletely random; because of the finitesize of the molecules, there is short-range order. When the magnitude of r issimilar to that of the molecular diameter,the local density is very much differentfrom the average density and therefore itis necessary to put into the integral somestructural information given by the radialdistribution function g(r), which is the

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ratio of local density to average density.The notation g(r) is deceptive; it errone-

ously implies that g is a function only ofr, whereas in fact it is a strong functionof density and, to a lesser extent, of tem-perature.The last 20 years have produced signif-

icant progress in our understanding ofthe radial distribution function, espe-cially because of the molecular dynamicsstudies of Alder et al. (3). Again, there isno need to go into details; it is sufficientto say that we now have useful theoreti-cal methods for calculating as a func-tion of density and temperature.

Large Molecules

For large molecules we must saysomething about the contributions fromrotational and vibrational degrees offreedom or, more precisely, about the ef-fect of volume on rotational and vibra-tional motions. To do so, contributionsto the partition function from rotationaland vibrational degrees of freedom are

factored into an internal part (a functiononly of temperature) and an externalpart, which depends primarily on volume

Q = N! xp1[exp(-2kT)l [qext]N [qi,t]N

where qint is independent of density andtherefore does not contribute to theequation of state.

For the external part we assume a

simple relation, suggested by Bazua andBeret (4), which, in effect, is an inter-polation between the known result at ze-ro density and the high-density approxi-mation of I. Prigogine, who assumed thatthose rotations and vibrations which areexternal (dependent on volume) can betreated as equivalent translations (5).Bazua and Beret suggest

qext = (V)

where 3c is the number of external (den-sity-dependent) degrees of freedom: low-frequency, high-amplitude vibrationsand rotations. For argon, c = 1. At liq-uid-like densitities, V constant. Then

this partition function is the same as

Prigogine's.The domain of the resulting partition

function, here called perturbed-hard-chain (PHC) theory, is shown in Fig. 7,where molecular complexity is one vari-able and density is the other. The PHCtheory provides a smooth interpolation:At low densities, we have theoreticalknowledge for simple and complex mole-24 AUGUST 1979

Ett'yieneo 00EO

103

3 lo Aett"t'

° Experimental

102

150 200 250 300

Temperature (°C)

Fig. 9 (left). Weight-fraction Henry's con-

stants in low-density polyethylene. Fig.

10 (right). Local compositions and the con-

cept of local mole fractions.

cules; for simple molecules we have the-

oretical knowledge at all fluid densities

(perturbed-hard-sphere theory); and at

high densities, we have at least some

theoretical knowledge if we adopt Prigo-

gine's assumption. This theory is surely

not rigorous, but it covers a wide region

of practical interest by meeting the ap-

propriate boundary conditions.The previous discussion has been con-

fined to pure fluids but extension to mix-

tures is relatively simple through theone-fluid theory (6). This theory says

that the thermodynamic properties of afluid mixture are the same as those of a

Pure Hypothetical

Pureliquid

1

15 of type 1

15 of type 2

Overall mole fractions: xl = x2 = 1/2

Local mole fractions:

Molecules of 2 about a central molecule 1

x2l = Total molecules about a central molecule 1

x21 + xl = 1, as shown

x2 + x22 = 1

xll - 3/8

x21 - 5/8

pure hypothetical fluid; the molecularparameters which characterize that hy-pothetical fluid are composition averagesof the parameters which characterize themixture's components.Two examples illustrate possible ap-

plications of the PHC theory. First, Fig.8 shows K factors for the binary mixturemethane-propane, which is representa-tive of mixtures found in a natural-gastechnology. TheK factor is a distributioncoefficient, the ratio of the mole fractionin the vapor phase to that in the liquidphase. The top part of Fig. 8 (whereK > 1) shows K factors for methane,

Umixture = x1UM11 + X2U(2)

U = Internal energy

U(1) = xllull + x21u21

U(2) = x22u22 + x12u12

Liquid(1)

Purei ( N

liquid Liquid

2 --'(2 )

Gibbs-Helm holtz equation Physical assumption

AE/T = 1/T (/T) X21/xll = X2/X1 x exp - (u21 - u11)/RT0

AE = Excess free energy x12/x22 = x1/x2 x exp - (u12 - u22)/RT

T = Temperature

Fig. 11. Local compositions and the two-liquid theory of mixtures.

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while the lower part (where K < 1)shows K factors for propane, bothplotted against pressure, up to the criti-cal pressure where vapor and liquid areidentical (K = 1 for every component).To make the indicated calculations, itwas first necessary to reduce experimen-tal pressure-volume-temperature (PVT)and vapor-pressure data for each of thepure components to obtain three charac-teristic molecular parameters: a hard-core size, a characteristic potential ener-gy, and 3c, the number of external de-grees of freedom. (For methane, c = 1.)The mixing rules for applying one-fluidtheory to the mixture contain only oneadjustable binary parameter, designatedby k,2, which gives the deviation fromthe geometric-mean assumption for cal-culating the characteristic potential ener-gy for an unlike pair: E12 = (E1lE22)1/2(I - k12). All parameters are indepen-dent of temperature, density, and com-position. With only one adjustable bi-nary parameter, PHC theory can repre-sent the experimental mixture data formethane-propane over a wide range ofconditions.

This first application is perhaps notvery dramatic because there is only amodest difference in the molecular sizesof methane and propane. To indicate ap-plication to systems where the molecularsize difference is large, Fig. 9 showsHenry's constants for several volatilesolutes in polyethylene. As defined here,Henry's constant is the reciprocal of thesolubility (weight fraction) when the par-tial pressure of the solute is I atmo-sphere. Again, the calculations requirethree molecular parameters for eachpure component and one binary parame-ter. It is evident that PHC theory canrepresent a wide range of solubilities;note that the (logarithmic) scale on the

120C

100

80 O -Calculatedo Data of Ellis and

X aV Garbett, 1960

EL 60 0 Data reported byE t Stephen and

Stephen, 1964

406

0.0 0.5 1.0xl, Y1

Fig. 12. Temperature-composition diagramfor butanol (1)-water (2) system. Calculationsare based on the UNIQUAC equation withtemperature-independent parameters.

ordinate covers five orders of magnitude.It is particularly gratifying that PHC the-ory can correctly reproduce the effect oftemperature on solubility over an appre-ciable temperature range.Phase equilibrium data, such as those

shown in Fig. 9, are of particular impor-tance in the design of a polymer devola-tilization process wherein volatile, possi-bly toxic, solutes are removed frompolymeric materials used for makingclothing or food containers.

Strongly Nonideal Liquid Mixtures

Another application of molecular ther-modynamics concerns strongly nonidealmixtures of liquids where orientationalforces are appreciable; for example, mix-tures of polar organic fluids, such asaldehydes, chlorinated hydrocarbons,ethers, nitriles, esters, and alcohols, in-cluding water. Such mixtures occur fre-quently in the petrochemical industryand, because of strong preferential inter-actions such as hydrogen bonding, it isnot easily possible to describe such mix-tures by a generalized van der Waalstreatment coupled with the one-fluid the-ory of mixtures.For liquid mixtures with strong orien-

tational forces, we must take into ac-count the tendency of molecules to seg-regate; that is, the existence of local or-der where molecules do not mix atrandom but instead show strong prefer-ences in choosing their immediate neigh-bors. Although many outstanding scien-tists have tried to solve this difficult com-binatorial problem, no satisfactorysolution has been obtained for practicalpurposes. Therefore, it is necessary toconstruct an approximate model as sug-gested in Fig. 10, which introduces theconcept of local composition.The essential idea here is that, because

For example:

Aceton e

of local order, the composition in a verysmall region of the solution is not thesame as the overall composition. To il-lustrate, Fig. 10 shows 30 molecules, 15shaded and 15 white; the overall molefraction, therefore, is 1/2 for each com-ponent. We focus attention on one arbi-trarily selected molecule of component Iand describe the composition in the im-mediate vicinity of that molecule by twolocal mole fractions xI, and x21; here x1lis the number of molecules of type 1about the central molecule divided bythe total number of molecules about thatcentral molecule. An analogous defini-tion holds for x21, and there is the obvi-ous conservation condition xll + x21 =1. As shown in Fig. 10, there are eightmolecules about the central molecule;five are white and three are shaded.Therefore, the local compositions arexll = 3/8 and X21 = 5/8.The local-composition concept can be

used as a point of departure to constructa simple expression for the nonideality ofa liquid mixture, as shown in Fig. 11.The important feature is the boxed equa-tion, which expresses a two-fluid theoryof binary mixtures: the molar internal en-ergy U is given by the mole-fraction av-erage of U'1), the molar internal energy ofhypothetical liquid (1), and U'21, the mo-lar internal energy of hypothetical liquid(2). Hypothetical liquid (1) consists of aregion in which molecule 1 is at the cen-ter and the local mole fractions are xl,and X21; similarly, hypothetical liquid (2)consists of a region in which molecule 2is at the center and the local mole frac-tions are x22 and x12. Parameter uij char-acterizes the potential energy of an ijpair; where i = I or 2 and j = 1 or 2.Given the relations shown, it is easily

possible to derive an expression for AU,the energy change on mixing, in terms ofthe uij and the local mole fractions. Thisexpression for AU is then substituted in-

U NIFAC

Toluene

mny; =Iny;C +.inyiR

lnyiC (as in UNIQUAC) = FC (x,4,O)

tnyiR = FR (X,Q,T,amn)

X = Group mole fraction; Q = Group external surface area

amn = Interaction energy between groups m and n

Fig. 13. Activity coef-ficient yi for molecu-lar species i fromgroup contributions.The configurationalcontribution FC de-pends on (molecular)mole fraction x, sitefraction P, and sur-face fraction 6. Theresidual contributionFR depends on tem-perature and on groupproperties and groupcomposition.

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to the (classical) thermodynamic equa-tion (shown at the lower left of Fig. I 1) toobtain an expression for the excessHelmholtz energy, which in turn gives usthe activity coefficients and thereby thedeviations from ideality for the liquidmixture.The remaining requirement is to say

something useful about the local molefractions, which are conceptual quan-tities and are not easily measured. To re-late the local mole fractions to the mea-sured overall mole fractions, we postu-late a simple relation (shown at the lowerright of Fig. 11) suggested by statisticalmechanics: for each hypothetical liquid,the ratio of local mole fractions is as-sumed to be equal to the ratio of overallmole fractions, multiplied by a Boltz-mann factor. This relation is in no senserigorous, but it provides a reasonable ap-proximation consistent with what statis-tical mechanics suggest about the prop-erties of an assembly of molecules thatdiffer appreciably in their intermolecularattractive forces.

Figure 11 illustrates the essential fea-tures of molecular thermodynamics ofmixtures. At the lower left, we have clas-sical thermodynamics; at the top, wehave a physical model that attempts toreflect our limited knowledge of molecu-lar behavior; and at the lower right, wehave a relation suggested by statisticalmechanics. Synthesis of these figuresyields an expression for calculating mix-ture nonideality in terms of two adjust-able parameters u21 - u1l and u12 - u22.For simplicity, Figs. 10 and 11 repre-

sent only mixtures of equisized sphericalmolecules. However, the proceduresoutlined in Figs. 10 and 11 can be gener-alized to molecules of different size andshape by using local surface fractions(instead of local mole fractions), whichare based on geometric properties of themolecules obtained from x-ray and simi-lar data (7).

Figure 12 shows a temperature-com-position diagram for a strongly nonidealmixture, n-butanol and water, based onthe local-composition concept. By usingonly two adjustable parameters, it is pos-sible to obtain an almost quantitativerepresentation of the experimental dataat a constant pressure of 1 atmosphere;at low temperatures, there are two liquidphases (partial miscibility) and at highertemperatures the vapor-liquid equilibriaproduce an azeotrope.The phase diagram shown in Fig. 12

was calculated with a particular equa-tion, called UNIQUAC for universalquasi-chemical (8, 9), based on the local-composition concept.24 AUGUST 1979

Activity Coefficients from

Group Contributions

The local-composition concept is eas-ily reconciled with an old idea (10) in ap-plied thermodynamics: group contribu-tions. The idea is that some thermody-namic properties of fluids containingpolyatomic molecules can be calculatedfrom interactions, not between the mole-cules, but between functional groupsthat constitute the molecules. This pos-tulate can also be used for calculating ac-tivity coefficients, as shown in Fig. 13. InFig. 13 an acetone molecule is divided in-to two functional groups and a toluenemolecule into six functional groups ofwhich five are identical. An extension ofUNIQUAC yields simple expressionsfor activity coefficients in terms of sur-face fractions 0, site fractions 'D, andgroup-group interaction parameters amn,where m and n refer not to molecules,but to functional groups. This expressionis called UNIFAC, an abbreviation foruniversal functional activity coefficient(11, 12). It is advantageous to use agroup-contribution method for petro-chemicals, in which the number of dis-tinct functional groups is much smallerthan the number of distinct molecules.Almost all petrochemicals, includingpolymers, can be constituted from per-haps 30 to 40 functional groups. If the in-teractions between these groups can becharacterized, it is possible to estimateactivity coefficients for a very large num-ber of mixtures for which no experimen-tal data have been obtained.Group interaction parameters amn are

calculated by data reduction for binarymixtures for which good experimentaldata are at hand. This has been done for

"I 1.0c

0.6E

C0.2

._cr

20

Feed Phenol feed

xI = x2 = 0.5 x3 = 1.0

(x3 = 0) (xl = x2 = 0)

23.26 mole/hour 76.74 mole/hour

Fig. 14. Continuous extractive distillation ofmethyl cyclohexane (I)-toluene (2) with phe-nol (3) in 21 stages.

a substantial number of group pairs. TheUNIFAC equations are easily pro-grammed for computer calculations, andthe resulting equilibria for separation-process design. The accuracy of theseestimates is not high but it is often suf-ficient for design purposes. Especiallyfor preliminary calculations, UNIFACprovides a useful tool since it is appli-cable to a very large variety of mixturesfor which no experimental information isavailable. It is not surprising that UNI-FAC is popular with practicing chemicalengineers, since it provides useful designdata with little effort.An engineering application of UNI-

FAC is shown in Fig. 14 which presentsthe composition profile of an extractive-distillation column. This equipment isused to separate a mixture of toluene andmethyl cyclohexane (MCH). Since theboiling points of these liquids are close,ordinary distillation is difficult, requiringa large number of stages and hence avery tall (and expensive) distillation col-umn. To facilitate separation, phenol isadded to the liquid mixture because po-lar phenol depresses the volatility of(aromatic) toluene and raises the vol-atility of (saturated) MCH. Since an aro-matic hydrocarbon has pi electrons and asaturated hydrocarbon does not, phenol"likes" toluene but "dislikes" MCH.Thus, phenol acts as an extractive agent,which in engineering language is calledthe entrainer. The presence of phenoldrastically reduces the number of stagesrequired for separation.For proper design, it is necessary to

calculate the equilibrium composition oneach stage in the distillation column. Thecomposition profile (Fig. 14) is calcu-lated by using UNIFAC and materialbalances. Nearly pure MCH is obtainedat the top of the column, while the bot-tom of the column yields a mixture of tol-uene and phenol, virtually free of MCH.The latter binary mixture must be sepa-rated in another distillation column, notshown here; however, that separation issimple, requiring only a few stages be-cause the boiling point of phenol is muchhigher than that of toluene.

Conclusion

A few examples have been presentedto illustrate the use of molecular ther-modynamics to obtain a quantitative rep-resentation of mixture properties from aminimum of experimental data. Theseexamples stress the important featuresof molecular thermodynamics. We startwith the abstract concepts of thermody-

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namics, in particular the chemical poten-tial, and then use whatever physical con-cepts we have to reduce abstract ther-modynamics to reality.

Figure 2 shows the three-step ther-modynamic process. The focus of molec-ular thermodynamics is on the last step:for useful results, the answer to the prob-lem solved easily in the abstract worldmust be reprojected into the real world.To achieve this reprojection, we must gobeyond classical thermodynamics, utiliz-ing physical models based on molecularscience.

In performing this reprojection stepwe must make compromises because rig-orous procedures are hardly ever avail-able for real situations. The last step re-quires some knowledge of appropriatephysical concepts. But an even more im-portant requirement is courage, willing-ness to make bold simplifying assump-

tions, to try them out and, if they fail, tochoose some other simplifying assump-tions and try again. Molecular thermody-namics is guided by Whitehead's famousadvice to scientists: "Seek simplicity,but distrust it." But it is propelled for-ward by another, less well known sen-tence of Whitehead's: "Panic of error isthe death of progress."For chemical process design, molecu-

lar thermodynamics provides a usefultool for estimating, with a minimum ofexperimental effort, those equilibriumproperties of mixtures which are re-quired for efficient design of separationoperations (for instance, distillation, ex-traction, and absorption). Since separa-tion operations represent large invest-ment and operating costs in the chemicaland related industries, molecular ther-modynamics contributes to the econom-ic manufacture of chemical products.

References and Notes1. N. F. Carnahan and K. E. Starling, J. Chem.

Phys. 51, 635 (1969).2. C. A. Croxton, Liquid State Physics-A Statis-

tical Mechanical Introduction (CambridgeUniv. Press, Cambridge, England, 1974), p. 73.

3. B. J. Alder, D. A. Young, M. A. Mark, J. Chem.Phys. 56, 3013 (1972).

4. S. Beret and J. M. Prausnitz, AIChE J. 21, 1123(1975).

S. P. J. Flory, Discuss. Faraday Soc. 49, 7 (1970).6. M. D. Donohue and J. M. Prausnitz, AIChE J.

24, 849 (1978).7. A. Bondi, Physical Properties of Molecular

Crystals, Liquids and Glasses (Wiley, NewYork, 1968).

8. G. Maurer and J. M. Prausnitz, Fluid PhaseEquilibria 2, 91(1978).

9. T. F. Anderson and J. M. Prausnitz, Ind. Eng.Chem. Process Des. Dev. 17, 526 (1978); ibid.,p. 551.

10. I. Langmuir, Third Colloid Symposium Mono-graph (Chemical Catalog Co., New York, 1925).

11. Aa. Fredenslund et al., Ind. Eng. Chem.Process Des. Dev. 16, 450 (1977).

12. Aa. Fredenslund, J. Gmehling, P. Rasmussen,Vapor-Liquid Equilibria Using UNIFAC (Elsev-ier, Amsterdam, 1977).

13. Supported by grants from the National ScienceFoundation, the Gas Processors' Association,the Gas Research Institute, and the Donors ofthe Petroleum Research Fund, administered bythe American Chemical Society.

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The AAAS-Newcomb Cleveland Prize is awarded annually tothe author of an outstanding paper published in Science from Au-gust through July. This competition year starts with the 3 August1979 issue ofScience and ends with that of 25 July 1980. The valueof the prize is $5000; the winner also receives a bronze medal.

Reports and Articles that include original research data, theo-ries, or synthesis and are fundamental contributions to basicknowledge or technical achievements of far-reaching consequenceare eligible for consideration for the prize. The paper must be afirst-time publication of the author's own work. Reference to per-tinent earlier work by the author may be included to give per-spective.

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Molecular Thermodynamics for Chemical Process DesignJ. M. Prausnitz

DOI: 10.1126/science.205.4408.759 (4408), 759-766.205Science 

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