SOLUBILITY THERMODYNAMICS IN CHEMICAL ENGINEERING

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640

Vol. (3.3

J. 31. I’RAUSNITB TABLX

Terminated a t the linear term Tcrminated at the quadratic term

1

VIRIAL COEFPICIENTS FOB r i - B u ~ a ~ AT r n 29.88” Coiicn. series Pressure aerica Eq. l a Eq, l b nR?’ = 14,796.7 f 1.5 ~ m . ~ m r n . nRT = 14,801.1 f 2 cm.31n111. UPV = 2.1 uPV = 2.8 B = -715 f 5 cm.3mole-1 9 , = -745 f 6 cm.amole-l

nR2’ = 14,793.2 f 4 cm.8 mm. UPV = 2.2

nR2‘ = 14,793.5 f 4 cm.* mm. UPV = 2.2 B = -695 f 25 cm.s mole-’ C/lV = -4 f 4 cm.6 mole+

thc relevant prcssme rangc, should be used for correcting propcrties of real gases to the hypothetical ideal gas state. We conclude: In tjhe absence of information concerning the magnitudc of the third virial coefficient (either from careful measurcments a t higher pressures or from clearly defensible assumptions which always should be stated) OIIC cannot, from a linear plot of PV us. l / V or PV us. P, dctermiiie the true

B = -691 f 26 mole-’ yRT/l06 = -10 f 5 cm.6 mole-2

(limiting) value of the second virial cocfficient with anywhere near the accuracy which sometimes is claimed. We wish to thank D. B. Myers of the University of California for helpful discussion and for making some of the calculations. Support of the U. S. Atomic Energy Commission (R.L.S.) and the Y’ational Science Foundation (R.D.D.) is gratefully acknowledged.

SOLUBILITY THEltIUODYNAMICS I N CEIEMICAL l!“INEEItISG BY J. &I.PRAUSNTZ University of Calijornia, Berkeley, Calijornia Receised October 87, 1961

The semi-empirical approach of IIildebrand is highly suitable for cnginecring requirements and t,he chcmical engineer who is familiar with Hildebrand’s work is well prepared to attack many typical chemical engincering problems. Two examples are given which indicate the usefulness of some of Hildebrand’s ideas; one of these deals with gas solubilities and the other with extractive distillation. The examples illustrate how a semi-empirical treatment of solution properties provides the necessary concepts for interpretation and correlation of useful solubility data.

Problems in phase equilibria play an important role in chemical engineering. A large part of the chemical engineer’s task is to separate mixtures and the most common technique for performing such separations is by contacting the mixture with another phase which will preferentially disso1.i-e one of the components of the mixture. Hence chemical engineers are interested in correlatiiig and predicting solubilities as required in the design of technical chemical operations and it is not surprising that they turn to the work of Joel H. Hildebrand to seek aid in realizing their goal. The semiempirical a p proach of Hildebrand and his co-workers is highly suitable for engineering requirements and the chemical engineer who is familiar with Hildebrand’s work is well prepared to attack many typical phase equilibrium problems which arise in his professional practice. The great utility of Hildebrand’s approach is that it provides not abstract aiid formal theories but rather concrctc concepts and physical insights which can serve as theoretical foundations for practical interpretation and correlation of phase equilibrium data. The potential applications of Hildebrand’s solubility thermodynamics to chemical engineering problems are numerous. To illustrate how Hildehrnnd’s work can be effectively ut>ilizedin the soliitioil of practical problems two examples are described below. Solubility of Gases in Typical Non-polar Solvents.-It frequently is desirable to estimate the

solubility of a gas in a liquid a t a specified temperature and pressure. While many gas solubility data are in the literature, a significant fraction is of doubtful accuracy and the vast majority of the data were obtained a t room temperature; as one proceeds to temperatures either higher or lower than about 25’ the amount of data falls exponentially. It therefore frequently is necessary to estimate gas solubilities from little or essentially no information. A technique for correlating reliable gas solubilities in non-polar systems’ is provided by a combination of two concepts: the idea of corresponding states and the idea of a regular solution. The isothermal dissolution of a gaseous solute in a liquid solvent can be considered to take place in two consecutive steps: in the first step, the gas is “condensed” to a liquid-like volume and in the second stcp the “condensed” gas (or hypothetical liquid) is dissolved in the solvent. If the gas is initially a t unit fugacity then the free energy for the first step is A G ~= RT hfoL

(1)

and that for the second step is AGII = RT In 72x2

( 2)

where joLis the fugacity of the hypothetical liquid, x2 is the solubility (in mole fraction) of the gaseous solute denoted by subscript 2, and yz is the activity coefficient of the gaseous solute referred to the (1) J. N. Prausniti arid 1 .: 11. Shuir, A.I.Ch.Z. Journal, 7, 682 (1YG1).

SOLUBILITY THERMODYNAMICS IN CHEMICAL ENGINEERING

April, 1962

Fig. 1.-Fugacity

of hypothetical liquid at pressure of 1 atmosphere.

hypothetical pure liquid. At equilibrium the free energy of the gas at unit fugacity is the same as that of the gaseous solute in the solvent and thus

+

A G ~ AGrr = 0

(3)

The advantage of splitting the isothermal dissolution process into two steps is that the first step involves only properties of the pure gas and hence it is reasonable to expect that foL may be correlated within the framework of a corresponding states treatment; and that the second step may be described adequately by the regular solution equation. A correlation of gas solubilities in non-polar systems then may be constructed with the additional relations

P PO = 0 In

(2)-

v21,#I,(* 61

y2 = ___-___

RII'

(4) 62)2

641

( 5)

where P, and I', are the critical pressure and critical temperature of the gaseous solute, is a uriiversal function, V & and are the molar volume and solu-

bility parameter of the hypothetical liquid, and +1 and a1 are the volume fraction and solubility parameter for the liquid solvent. Since xz usually is very small, +1 is equal to unity for most practical purposes. The correlating framework outlined by eq. 1 to 5 contains three unknown properties of the hypothetical liquid: foL, VZLand Bz. All of these are temperature dependent but due to regular solution theory it is not necessary to consider the temperature dependence of VZLand &. According to regular solution theory, In y is inversely proportional to temperature a t constant composition and thus the term VzL+12(61- 62)z may be considered independent of temperature. Accordingly V Z and & must be chosen a t the same temperature as 61 and for convenience this arbitrary temperature as used here is 25'. To obtain numerical values for the parameters, SOL, Vz, and Sz all available, reliable gas solubility data were correlated within the framework described above. The sources and temperature ranges

642

J. M. PRAUSSITZ

Vol. 66

Solvent Selectivity in Extractive Distillation of Hydrocarbons.-The separation of hydrocarbons from their mixtures usually is achieved by distilla2c tion but if two hydrocarbons have very nearly the same volatility ordinary distillation becomes extremely inefficient and, in the limiting case, where I the volatilities of two hydrocarbons become identiIO cal, a n azeotrope is formed and ordinary distillation = a fails completely. I n such cases it has become a common procedure in the petrochemical industry a to add deliberately another substance to the hydrocarbon mixture in the hope that this additional substance will favorably affect the volatility rela9 - n-HEXADECANE 4 A - DICYCLOHEXYL II tions so that distillation may be used after all as a I 0 METHYNAPHTHALENE 3 SOURCE C SOLOMON (27) means of separation. This procedure is called extractive distillation and the added substance ‘is called the solvent or entrainer. For the separation of hydrocarbons by extractive distillation an effective solvent must be a polar organic liquid which can induce strong deviations from Raoult’s law and lists the molar volumes and solubility parameters yet be sufficiently soluble in hydrocarbons without which were obtained. The fugacities of the hypo- forming two liquid phases. When a separation by thetical liquid are correlated according to eq. 4 as extractive distillation is contemplated the immedishown in Fig. 1. Since most available data for ate question which arises is “What solvent is most hydrogen correspond to very high reduced tem- effective?” Or, in other words, “What solvent will peratures they are not shown here but may be have the highest selectivity?” I n trying to find an answer to these questions the chemical engineer found in ref. 1. can be greatly assisted by some of the concepts TABLE I advanced by Joel Hildebrand. The selectivity X is related to the relative volaLIQUIDVOLUMES AND SOLUBILITY PARAMETERS FOR GASEtility a by OUSSVLUTES Gas vaL,cm.s/g. mole 8 , (cal./crn.~)’/z 3

i

I

NZ GO 0 2

Ar

CHI

coz

Kr CzH4 CaHs Rn Cl2

32.4 32.1 33.0 57.1 52.0 55.0 65.0 65.0 70.0 70.0 74.0

2.59 3.13 4.00 5.33 5.68 6.00 6.40 6.60 6.60 6.83 8.7

With the use of the information given in Table I and in Fig. 1 it is possible to compute the solubilities of many common gases in numerous non-polar solvents a t any temperature where the solvent’s vapor pressure is not too large. The solubility a t one atmosphere pressure is calculated by combining cq. 1,2, 3, and 5 to give

(7)

where subscripts 1 and 2 refer to the hydrocarbons to be separated, Po is the vapor pressure, and y the activity coefficient. A good solvent is characterized by a value of S well removed from unity; if a solvent has a selectivity S = 1 it is worthless because it affects the volatility of both hydrocarbons equally and therefore does not facilitate separation. At constant temperature, the selectivity will be a maximum when the polar solvent is present in excess and therefore it is most convenient to try to compute y1 and y~ for a solution where hydrocarbons 1 and 2 are infinitely dilute in the polar solvent designated by subscript 3. Consider now a solution of hydrocarbon 1 (or 2) in polar solvent 3. The heat of mixing can be divided into two contributions: one from changes in potential energy and the other from changes in vibrational and rotational energy. Thus AH = AH(pot)

The large advantage of Fig. 1 is that it utilizes solubility data primarily obtained near room temperature to predict the solubility of gases a t temperatures well removed from room temperature. This correlation scheme enables the chemical engineer to make predictions of gas solubilities over a wide temperature range. The accuracy is about 10%. Such predictions are of value in numerous common problems such as the preliminary design of absorption equipment. One example of such an application is given in Fig. 2, which compares calculated and observed K values for methane in several petroleum solvents.

+ AH(rot,vib)

(8)

If now the simplifying assumptions are made that the excess entropy is given by T ASE = AH(rot,vib) (9) and that AH(pot) is given by a quadratic function of the volume fractions, then the excess free energy is given by

+

aGE = (xlV1 z3Vd +143A~3 (10) where A13 = cll c33 - 2cI3,the exchange energy per unit volume, and c stands for cohesive energy

+

density. The selectivity now can be expressed as Sl,2

= exp

(VlA13-&V

!z- 4 23)

(11)

SOLUBILITY '~"URMWYNAMICS

April, 1962

IN

r

80

643

CHEMICAL ENGINEEHING I

I

1

I

IC

Ilm

I

I

I

5

10

I 15

I

I

I

20

25

30

tk

Fig. 4.-Effect of unsaturation of the hydrocarbon on heats of mixing for nitroethane-hydrocarbon systems a t 45

.

C A R B O N NO. OF P A R A F F I N

Fig. 3.-Size effect in solvent selectivity: activity coefficients of normal paraffins in polar solvents at infinite dilution.

Equation 10 is, of course, the Hildebrand-Scatchard solubility equation but the c terms now include contributions from polar as well as non-polar forces. Techniques for estimating quantitatively the strength of these forces are given elsewhere.2 Induction Effects.-As a first thought, one might expect that induction forces may be responsible for selectivity; if the polar solvent can induce a dipole in hydrocarbon 1 which is significantly difYerent from that induced in hydrocarbon 2 then clearly a basis for good selectivity is established. But a moment's reflection indicates that this possibility is not promising. When a polar rnoleculc of radius a having a dipolc momeiit p is surrounded by a non-polar fluid having diclcctric constant E the energy of induction is given by Thus plcfercntJial induction can occiir only if thc two hydrocarbons have considerably different dielectric constants. Howcver, there is, in fact, very little variation in the dielectric constants of hydrocarbons and thus it appears that induction is not the key to good selectivity. Contributions to Selectivity : Effect of Molecular Size.-If appropriate estimates of the c terms are made, it can be shown2 that the selectivity can be expressed as the sum of three contributions: a polar, a non-polar, and induction term RT'lnSlz = P -+ D $. I (13) where 1'

-~

=

- Vz) - Vz(83" -

= fi3P'(Vl

- 61)'

62)'

(14) (15)

(2) J. M.Piairsriitl; and R. Anderson.A.1.Ch.W. 810urna2,7,96 (1961).

k'ig. S.-Voluinc

change on mixing nitrocthane with hydrocarbons at 25'.

111 eq. 14 to 16 83pz stands for thc polar ciiergy dcnsity, 811*for the non-polar energy density, and { for the induction energy per unit volume. Typical phase equilibrium data indicate that the polar term P is by far the most important term whenever the volumes of hydrocarbons 1 and 2 are not very near to each other. The contributions from D and I are then always small. It appears then that it is possible to separate hydrocarbons by extractive distillation on the basis of size difference; the selectivity rises as the difference in molar volumes of the two hydrocarbons increases and the most effective solvent is one having a high polar encrgy and a small volumc. Thlis, for ex-

I I

effects; however, chemical effects also play a role in determining solution properties. Hildebrand’s classic studies on iodine solubility show that complex formation is a common occurrence in solutions and therefore it is reasonable to suspect that complex formation plays an important role in determining solvent selectivity. The Lewis concept of acids and bases suggests that unsaturated hydrocarbons with pi electrons could serve as electron donors (bases) while typical polar solvents can act as electron acceptors (acids). If a hydrocarbon can - - 0 285 9 7 1 74 70 0 022 form a complex its volatility is, of course, reduced A 280 1840 8 0 4 0 0 022 and thus if a polar solvent forms a complex with one 0.6 I I hydrocarbon and not with another a clear basis of 0 20 40 60 8 0 100 120 140 selectivity is established. [TOLdENE] gmol / l i t e r . There is considerable evidence to show that polar Fig. 6.-The evaluation of the equilibrium constant for the solvents do, in fact, complex with unsaturated nitroethane-toluene system a t 25’. hydrocarbons. Figure 4 shows some heat of mixing data for various Ce hydrocarbons with n i t r ~ e t h a n e . ~ PARAFFINS-PARAFFINS (25°C) PARAFFINS-PENTENE -1125” Complex formation makes an exothermic (negative) 0 FURFURAL 0 FURFURAL contribution to the heat of mixing and Fig. 4 shows A METHYL ETHYL KETONE CYCLOHEXANE -BENZENE(Z5’( that as the unsaturation of the hydrocarbon inPHENOL A METHYL ETHYL KETONE creases the heat of mixing falls. Similarly, complex PARAFFINS-BENZENE ( 2 5 ° C ) PHENOLr -. formation makes a negative contribution to the volume change on mixing and as showii by the data A METHYL ETHYL KETONE FURFURAL in Fig. 5 the volume change on mixing nitroethane 0 PHENOL / (BENZENE) k I with a saturated hydrocarbon is positive whereas that with urisaturated hydrocarbons is negative.3 Finally, the presence of a complex frequently can be seen in the ultraviolet spectrum and Fig. 6 shows some spectral data for nitroethane-toluene which have been reduced according to the method of Benesi and EIildebrand.4 From these data it has 1 FURFURAL €NE-I) been possible to calculate an equilibrium constant for the nitroethane-toluene complex. Similar data I / NO COMPLEX for other unsaturated hydrocarbons6 indicate that, ( CALCULATED 1 as expected, complex stability rises as the ionization potential of the hydrocarbon falls. It is evident, then, that chemical as well as physical effects play a role in determining solvent selectivity. In order to correlate solvent selectivity data a useful correlating equation has the form I

hiTROEiHANE

I

- TOLuENt

I

@ 25’C

401

‘

,

~

~

Fig. 7.--Effect of complex formation on solvent selectivity.

ample, cq. 13 predicts that acetone should be a more selective solvent than diethyl ketone and this prediction is in agreement with experiment. Yurther, eq. 13 says that in the absence of chemical effects the larger hydrocarbon has the larger activity coefficient and this too is supported by experimental evidence as shown in Fig. 3. The effect of a polar molecule on a non-polar is to cause positive deviations from Raoult’s law. The larger non-polar molecule presents a larger crosssection for interaction with the polar and thus a mechanical analog of the state of t,he solution is to think of the polar molecule wandcririg about “kicking” the hydrocarbon molecules; the larger hydrocarbon gets “kicked” more frequently than the smaller one and thus the larger hydrocarbon has the higher activity coefficient. Chemical Effects.-The analysis of solvent selectivity given so far has considered only physical

where \k is a collision volume which replaces the molar volume as given by eq. 14. The molar volume of a hydrocarbon is only a very crude measure of its molecular size as “seen” by the polar species and it is desirable to replace it by another term which more truly reflects the cross-sectioii for interaction which the hydrocarbon possesses. The collision volume is dcfined by Sf =

[V,’/a

+ Vf’/qa - kSPlV

(18)

where V ~ V is the van der Waals volume, Vf is the free volume, IC is Illeares’ constant, p is the compressibility, and P I is the internal pressure. This definition is quite arbitrary and is justified primarily by its utility. The function 0 is not specified but it depends on the equilibrium constants of the two hydrocarbons (3) R. Anderson, R. Carnbio, and J. &I. Prausnits. A.1.Ch.E. Journd, in press. (4) H. Benesi and J. II. IIildebrand, J . Am. Chem. Soc.. 71, 2703 (1949). ( 5 ) R. Anderson, Dissertation, University of California, Berkeley, 1961.

April, 1962

CELLPOTENTIALS AND GASSOLUBILITY THEORY

for complex formation with the polar solvent. The function @ has the properties

*

If K1, = 10and IG3 = 0 then = If Kl3 = K23 then 0 = If Kzs > Kit then 0 > If K23 < iR13 then @ <

0 0 0 0

Figure 7 shows how eq. 17 may be used to correlate typical solvent selectivity data ( 5 ) . The continuous line is a plot of eq. 17 with @ = 0; that is, the continuous line gives the selectivity in those cases where there are no chemical effects. This calculated line correctly predicts the experimental selectivities for several systems containing only saturated hydrocarbons as indicated by the filled-in black points. The dotted lines show the effect of complexing on selectivity. In the cases shown K23 > 0 whereas K13 = 0; that is, in all these cases the chemical and physical effects act in unison to raise the relative volati1it.y of component 1 and therefore the experimental points lie above the calculated base

645

line for physical effects. Although such cases are not shown in Fig. 7 it is quite possible to have the chemical and physical effects act in opposite directions. Analysis of the role of intermolecular forces in extractive separations cannot as yet eliminate the need for experimental data. However, such analysis can help in a preliminary screening of promising solvents and can contribute to minimizing experimental effort by providing a guide for the interpretation and correlation of phase equilibrium data. Conclusion.--The examples given here illustrate the utility of solubility thermodynamics in typical chemical engineering problems. The semi-empirical treatment of solution properties as pioneered by Joel Hildebrand provides the necessary concepts for perceptive interpretation and subsequent useful correlation of solubility data as required in a variety of chemical engineering operations. Acknowledgment.-The author is grateful to the donors of the Petroleum Research Fund for financial support.

CELL POTENTIALS AND GAS SOLUBILITY THEORY’ BY Y. KOBATAKE AND B. J. ALDER University of California, Lawrence Radiation Laboratory, Livermore, California and University of California, Department of Chemistry, Berkeley, California Received December 8, 1861

A two-parameter cell potential in a free volume type theory is determined from two experimentally obtained thermodynamic quantities. For pure liquids further thermodynamic quantities are predicted quite accurately over a region of temperature by the Lennard-Jones-Devonshire form of the cell potential, although its harmonic oscillator modification gives better results. For a gas dissolved in a liquid, the cell potential yields values of the free volume of the gas molecule that are about 10 times larger than in a typical liquid. Moreover, it can be concluded that the gas molecule is surrounded by about 7 neighbors and that the solvent molecules surrounding the gas contribute importantly to the thermodynamic functions. The gas molecules perturb the solvent significantly over several molecular layers, while in dilute liquid mixtures the effect of the solute on the solvent is mainly confined to one molecular layer.

Introduction Recent developments in the theory of liquids have emphasized the inadequacy of a lattice model even when that theory is optimized.2 The source of failure can be stated in terms of a significant contribution from the communal entropy term, that is, from higher order correlations which were not evaluated. However, it must be emphasized that the exact, single-occupancy problem has not yet been solved !since the solutions so far involved spherical smoothing. There is reason to believe from machine computations on hard sphere syst e m ~that ~ the exact single occupancy solution differs significantly from the one obtained by spherical smoothing. I n any case, these machine computations have shown that the communal entropy of the solid and the dense fluid do not differ very much.4 If the contribution of higher order correlations were small in real dense liquids, the theory of liquids would be greatly simplified since attempts a t estimating even a first correction for (1) This work was performed under the auspices of the U.S. Atomia Energy Commission. (2) J. S. Da,hler and J. 0. Hirschfelder, J . Chem. Phys., 32, 330 (1980). (3) B. J. Alder and T. E. Wainwright, ebid., 33, 1439 (1960). (4) B. J. Alder and T. E. Wainwright, UCRL report 6600 T (1961).

the communal entropy involved such complex mathematics that this so far has been done only approximately with small quantitative improvement.6 Since other theories of the liquid state are mathematically also very complex16~7 it is worthwhile to apply a one-particle theory to some real situations, still in the hope that it will be possible to estimate the contribution of various effects to the thermodynamic behavior. In order to make the one-particle theory as realistic as possible some of the restrictions previously imposed on such theories have been removed. Basic to all existing cell theories is that the volume of the system is spanned by a virtual lattice and that the representative particle moves in the field of the other particles confined to the lattice cells. This imposes an arbitrary order on the particles which depends upon the lattice structure chosen and is particularly poor for a solution, when a lattice of equal sized cells for largely different size molecules is assumed, since that arrangement hardly accounts for the complex packing. The ( 5 ) (a) J. DeBoer, Physica, a i , 137 (1955); (b) J. Pople, Phil. Mag.,

42,459 (1951). (6) J. €3. Dahler and E. G. D. Cohen, Physiea, 26, 81 (1960). (7) J. M. Richardson and S. R. Brinkley, J . Chem. Phys , 33, 1467 (1960).