sum of squares deviations for a set of parameter values el, e2, , , , ep = predictor variables = calculated value of from a fitted model in uth experiment = observed response value of conversion in uth experiment adsorption equilibrium constants of h and B estimates of adsorption equilibrium constants of A and B experimental error of the response in the uth experiment theoretical conversion (response variable) rate constant parameters; the whole set is called e vector of estimators of elements of a vector parameter e total pressure space time (predictor variable) =
. , 2,
Froment, G. F., Meeaki, R., Chem. Eng. Sn'. 25, 293 (1970). Johnson, W. C., Ph.D. Thesis, University of Wisconsin, Madison, Wis., 1960. Miller, D. N., Kirk, R.S., A.I.Ch.E. J., 8 , 183 (1962).
~
Literature Cited
Bard, Y., Lapitius, L., Catal. Rea. 2, 67 (1968). Draper, h'. li., Smith, H., "Applied Regression Analysis," Wiley, New York, X . Y., 1966.
R E I J I ;\IEZ.4KI*
Chemical Engineering Department S e w York llniversity Bronx, A'. Y . 10453 X 0 R L I . U R. DRAPER RICHARD A. JOHNSON
Statistics Deparfment t 'niversity of TT-isconsin
Madison,Tl'is. 53'706 RECEIVED for review May 17, 1972 ACCEPTED January 18, 1973 Y. R. Draper was artially supported by the United States Navy through the ce of Naval Research under Contract N00014-67A-0128-0017.R. A . Johnson was partially supported by the Air Force Office of Scientific Research under Grant AFOSR69- 1789.
8ffi
Calculation of Phase Equilibria Using the Prigogine Theory of Chain-Molecule Liquids Since the Prigogine theory i s applicable only to dense phases, molecular parameters for pure liquids should be obtained from liquid-phase properties. A fundamental difficulty arises when such parameters are obtained from vapor-pressure data.
T h e Prigogine theory of chain-molecule liquids and t'lieir solutions (Prigogine, 1957; Prigogine et al.,1953a, b, c) has been used by a number of workers to calculate vapor-liquid and liquid-liquid phase equilibria (Flory, et al., 1964; Prigogiiie, et al.,1953a, b, e; Renori, et al., 1967a, b, 1968; Kinnick and Prausnitz, 197la, b ; Yaiig aiid Yeiidall, 1971). In this note we point out some fundamental difficulties that can be eiicountered in such calculations, aiid we refer to some preliminary work toward their resolution. T o determine the adjust'able, pure-component parameters in a st'atistical mechanical model of a pure fluid, it is convenient to fit theoretical vapor pressures to those esperimentally determined. Such fitting, however, is subject' to a fundamental difficulty if the Prigogiiie partition function is used to represent liquid-phase properties. K e explain here the nature of this difficulty, which has not previously been discussed in the literature. -4s suggested by Flory (1965), i t is best to use liquid-phase volumetric properties for data reduct,ion. The Prigogine theory of chain-molecule liquids gives the total caiionical partition function, Z, of a pure substance iii the
z = ZinternalQlatticeQresidual
(1)
form where Zinternal = [jinternal(T) I", t'he partition function determined by internal degrees of freedom (Prigogiiie, e f al., 19.53~)of the molecules; X is the iiumber of molecules; &lattice is the at,hernial configurational partition function (combinatorial contribution); Qresidualis the coiifiguratioiial 254 Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
partition function (due to free-volume aiid energetic effects) determined by external degrees of freedom (Prigogine, et al., 1953~). It is customary to evaluate &lattice using the FloryHuggins combinatorial factor and to evaluate &residual using the Prigogine cell theory (Prigogine, el al., 1953a, b, c ; ~~)]~~~ ~'rigogilie, 1957), where &residual = [ ~ r e s i d ~ a l (p~ is the reduced volume, p is the reduced temperature, and 3c is the number of external degrees of freedom per molecule (Prigogine, 1957). The chemical poteiitial can be calculated from eq 1
- - _I.(
- 111 [ j i n t e r n a l ( T ) I
t
kT
T o calculate phase equilibria between a vapor phase aiid a dense liquid phase, we need to evaluate the chemical potential of the vapor phase, to which the Prigogine theory does not apply. If we use ai1 equation of state for the vapor phase (for example, the virial equation) and equate chemical potentials, we obtain
In eq 3, i t is the first term (in brackets) on the left-hand side which is obtained from the equation of state for the vapor phase. The term p * ( T ) is the chemical potential of an iaolated molecule at temperature T . To use eq 3, it is tempting to assume that
and implicitly, this is what some previous authors have done. However, this assumptioii is erroneous, because some of the degrees of freedom included in p* ( T ) become external degrees of freedom (and therefore contribute to &residual) in the dense liquid (l'rigogine, 1957; Prigogiiie, et ai., 1953a, b, e ) . Since [ j i n t e r n a 1 ( T ) ] l i q u i d is uiiknowii, it is iiot possible to carry out the phase equilibriuni calculation indicat,ed in eq 3. The Prigogine theory is based 011 the assumption that 3c, the riuniber of external degrees of freedom per molecule, is a molecular constant, independent of volunie and temperature. This is approximately t,rue a t high densities. .Ispointed out by several authors (I3oiiner, et al., 1972; Scott aiid van Koynenberg, 1970; Simha aiid Somcyiisky, 1969), when this assumption is erroneously applied to the vapor phase, it' leads to the incorrect result
Pv lim L.-m RT
= c
where v is the molar volume. Equation 4 is erroneous because the Prigogine theory (mit'li e const'arit and different from unity) is applicable to deiise fluids but iiot to vapnrp. Since the difference between [p*(T)/'kT] and [hi j j i n t e r n a l ( T ) ) ] l i q u i d is not zero, it is difficult to assign physical significance to 1)artitioii-fuiictioii parameters obtained from reduction of vapor-pressure data. Some previous publicat'ions implicitly circumvented the problem through a modification of eq 1. Renon, et al. (1967a, b ; 1968), and Winnick and Prausnitz (1971a, b) have multiplied 2 by ail empirical function of reduced temperature, f(T). This function is intimately related to
However, even when this implicit circumventioii is used, special care must, be exercised t80assure dimensional homogeneity. I n calculating vapor-liquid equilibria in a binary or a multi-component mixt,ure, the difficu1t.y described above caii be avoided by using t,he well-knowii coiicept of excess functions. The activity coefficieiit', y i , is related to the excess Gibbs energy by
culate GE. W11en using eq 5, the terms In [ j i n t e r n a l ( T ) j l l i q u i d cancel. However, in order to perform t'he vapor-liquid equilibrium calculation, one must know the reference fugacity of each component in t'he liquid phase. I n a typical applicat,ion, this reference fugacity (usually bhe vapor pressure of the pure liquid) is a known experimental quantity and is not obtained from the part'ition function. To calculate phase equilibrium between two dense liquid phases, we can assume that' the Prigogine partition function applies to bot,li phases. In that event, the parameter c for each component is the same in bot'h phases, and j i n t e r n a l ( T ) iii one phase is the same as that in the other. The internal contrihutioiis cancel, allowing the calculation to be performed. The difficult'y in using vapor-pressure data to obtain parameters is due t o the fact that the Prigogine partition funct,ioii is applicable only to the dense liquid region. Recently, the Prigogine theory has been modified (Ronner, et al., 1972; Simha and Somcyiisky, 1969) by usiiig the hole theory of liquids to give a part,ition fuiict.iori for chain molecules applicable to both gas and liquid densities. Use of such a partition function eliminates the difficulty described above. literature Cited
Bonner, D. C., Bellemans, *4., Prausnit'z, J. AI., J . Polym. Sci., Part C 39, 1 (1972). Flory, P. J., J . Amer. Chem. SOC.87, 1833 (1965). Flory, P. .J., Orwoll, 11. A,, Vrij, A , , J . Amer. Chenz. SOC.86, 3515 ( 1964)
.
Prigogine, I., "The 1Iolecular Theory of Solutions," Nort,hHolland Piibiishiiig Co., Ainsterdain, 1957. Prigogine, I., Trappeniers, S . , IIathot, V., J . Chem. Phys. 21, N., Mathot, V., J . Cheiri. Phys. 21,
N., AIathot, V., Discuss. Faraday
Itenon, H.. Eckert, C. A , , Piausnith, J. >I., IXD.ESG. CHLM., F c > D . ~6v,.5 2 (196ia). Itenon. EKG.CHEM.. --,H.. Eckert. C. A,. Prausnitz. J. 11..IND. FUND.IM. 6 , 38 (i96ib).' Cenon, H., Eckert,, C. A , , Prausnitx, J. IND.ENG.CHEM., FUND.IY. 7, 33,j (19681. Scott, 11. L., van Koynenberg, P. H., Discuss. Furarlay SOC.49, 87 (1970). Simha. R.. Sorncvnskv. T.. Mncrontoiecules 2. 342 11969). Winnick, J , j Prausniti,'J. il., Chem. Eng. J . 2 , 233'(1971a). Winnick, J., Prausnitz, J. II., Cheni. Eng. J . 2, 241 (1971b). Yang, C.-L., Yendall, E. F., A.Z.Ch.E. S.17, 396 (1971). I
D. C. BONX'ER*' E. R. BAZUA J. 11. PRAUSXITZ Chemical L'ngineering Department rniversity of California, Berkeley Berkeley, Calif. 34720 Address correspondence t o this aut,hor a t the l)epart>ment of Chemical Engineering, Texas Tech University, Lubbock, Tx. 79409. for review June 6, 1972 ACCCPTED January 15, 1973
h3CEIVED
where n, is the number of moles of component i. Since GE is the exceSs Gibbs energy oi the real liquid mixture minus that of the ideal liquid mixture a t the same temperature, pressure, and composition, the Prigogine theory can be used to cal-
The authors are grateful to the National Gas Producers Association, t o Gulf Research and 1)evelopment Company, and to the National Science Foundation for financial support. E. It.B. acknowledges with thanks the scholarship provided by Consejo Racional de Ciencia y Tecnologia, 1Ii.xico.
Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
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