T = temperature, OK uo = volumetric feed rate, ft3/sec u = gas velocity, ft/sec D =
average gas velocity, ft/sec
V = reactor volume, ft3 xA = fractional reactor conversion = density of carbon, lb/ft3 T = reactor space time, sec
PC
Literature Cited Andrews. A. J.. Pollock, L. W., Ind. Eng. Chem., 51 ( 2 ) , 125-128 (1959). Bird, R. B . , Stewart, W. E., Lightfoot, E. N . , "Transport Phenomena," p 23, Wiley, New York, N.Y., 1960. Bonner, J. S., Honeycutt, J. M., Adv. Pet. Chem. Refin.. 5 , 83-114 (1962). Burk, R . E., Laskowski, L., Landelma. t i P., J. Am. Chem. SOC.. 63, 48-50 (1941) Feigin, E. A,, Klim. Prom., 5, 19-26 (1963). Frey. F. E., Hepp, H. J., "The Science of Petroleum." p 1994, Oxford University Press, London, 1964. Himmeiblau, "Basic Principles of Calculations of Chemical Engineering," 2nd ed, p 444, Prentice-Hall, Englewood Cliffs, N.J., 1967. Tables of Chemical Kinetics, Circular 510, "Homogeneous Reaction Kinetics Pyrolysis of Aliphatic Hydrocarbons." National Bureau of Standards, Nationai Research Council, 507-517 (1951) Kevorkian. V . , A d v . Pet. Chem. Refin., 5, 369-438 (1962).
Lindahl, A.. Chem. Eng. Prog., 61 ( 4 ) , 77-81 (1965). Myers, P. S . . Watson, K. M.. Nat. Pet. News, 38 ( 1 8 ) , 9388-R442 (1946). Lindsay, H. F., Wulzen. R . , "Refinery Studies of Digital Computer," World Petroleum Congress, Proc. 5th N.Y., 1959, 3, 251-60 (1960). Noguchi. T.. Suzuki, J . , Seki, M., Sekiyu Gakkai Shi, 12 ( 2 ) . 107-112 (1969). Palmer. H . B., "Kinetics and Mechanism of Carbon Deposition During Gaseous Pyrolysis," Fuel SOC.J.. (1962). Robertson, R . W. J , , M.S. Thesis. Newark College of Engineering, Newark, N.J.. 1972. Rossini, F. D., "Selected Values of Chemicai Thermodynamic Properties," National Bureau of Standards, Series 111, Table 23, Mar 31, 1947. Rossini, F. D.. "Selected Values of Chemicai Thermodynamic Properties." National Bureau of Standards, June 30, 1946. Schneider, V., Froiich, P. K., Ind. Eng. Chem.. 23, 1405 (1931). Shah, M. J . , "An Approach to Model Simplification in Process of Optimization and Control," American Institute of Chemical Engineering-lnstitute of Chemical Engineering Symposium, No. 4, 4:47-4:52, 1965. Shah, M. J . , "Computer Control and Ethyiene Production," Ind. Eng. Chem.. 59 ( 5 ) , 70-87 (1967). Tayyabkhan. M . T.,Petro;Chem. €ng.. 58, 54-56 (1966). Towell, D.G.. Martin, J . J.. A.1.Ch.E.J., 7 ( 4 ) . 693-698 (1961). Waiker. H. M.. Chem. Eng. Prog., 65 ( 8 ) ,53-58 (1969). Wilke, C. R., J, Chem. Phys., 18, 517-554 (1950)
Receivedfor reuieu M a r c h 1, 1974 Accepted January 2,1975
Estimation of U nlike-Pair Potential Parameter in Single-Parameter Wilson Equation Mitsuyasu Hiranuma* and Kiyoshi Honma Tomakomai Technicai Coiiege, Nishfkfoka, Tomakomai, Hokkafdo,Japan
The concept of three-dimensional energy of vaporization is applicable to the estimation of the unlikepair potential parameter X1z in the single-parameter Wilson equation when the following combination rules are used: A 1 2 = - ( 2 / z ) u 1 2 ; u12 = u l z d u 1 2 l for nonpolar-polar systems; u12 = ulzd -I- (u1P u ~ P ) l / ~~~2~ for polar systems, where u l z d = (harmonic mean of molar volumes) X (harmonic mean of dispersion cohesive energy densities), u L P is dipole energy of substance i, u t 2 1 or u l z h is estimated from experimental data for a system similar in characteristics to the system in question. A factor 2/z, which accounts for a fraction of u, is chosen to be '13. The proposed method results in almost as good a representation of the vapor-liquid equilibria as the single-parameter Wilson equation does, if a base system is as close as possible in characteristics to the system in question.
+
The Wilson equation with the two parameters for each binary pair determined from the infinite dilution activity coefficients, or from vapor-liquid equilibrium data, can represent vapor-liquid equilibrium data over the whole composition range. Even if only one of the infinite dilution activity coefficients is available for a binary pair, the single-parameter Wilson equation gives a good representation of the binary system (Schreiber and Eckert, 1971; Hiranuma, 1972). However, if even one of the infinite dilution activity coefficients is not available, the single-parameter Wilson equation cannot be used. The solubility parameter equation is useful in the estimate of the activity coefficients for nonpolar systems. However, the solubility parameter equation will not apply to the highly polar and/or hydrogen-bonded systems. For such solutions no really satisfactory theory has been developed. It often happens that the data for the system of interest (for an example, the system methanol-acetone) are lacking, but data for a system of similar characteristics (e.g., the system ethanol-acetone) are available. It is our pur-
+
pose here to estimate a parameter A,, in the single-parameter Wilson equation for the system for which estimates are to be made, by use of experimental data for a system similar in characteristics to it. A better understanding of A,, might lead to a better knowledge of liquid mixtures and to prediction of their properties from pure component properties, if possible. Single-Parameter Wilson Equation Wilson (1964) obtained an expression for an excess Gibbs energy by analogy with the Flory-Huggins expression for athermal mixtures, in which he replaced overall volume fractions by local volume fractions. Wilson's equation for activity coefficients of a binary mixture is given as In yi = -In ( x i
+
Ai,xj)
+
A i j = ( v j / v i )exp(-(hij - X i i ) / R T ) Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 3, 1975
( 2) 221
A study by Orye and Prausnitz (1965) indicated that Wilson’s equation, using only two adjustable parameters (A,, - A,,), is useful for representing the vapor-liquid equilibria of a wide variety of liquid mixtures. According to Wilson, A’s characterize intermolecular energies between molecules per pair. Tassios (1971) suggested that the molar energy of vaporization u, was available in order to characterize the like-pair interactions A,,, and the unlike-pair interaction was left as the sole adjustable parameter for the binary pair; i.e., A,, = -u,. Wong and Eckert (1971) modified this approach to retain the physical interpretation of the Wilson parameters; they interpreted A,, as the configurational energy per pair interaction: i.e.
where 2/2 is a term which accounts for a fraction of u, which contributes to A,,. This was tested in describing vapor-liquid equilibria, using the value of each infinite-dilution activity coefficient, y,’, by Schreiber and Eckert (1971). According to Flory and Huggins, the Flory-Huggins equation, which Wilson used as a basis, is supported by the assumption that a polymer molecule in solution behaves like a chain, that is, as if it consists of ( U Z / U ~ ) mobile segments each having the same size as that of a molecule of the solvent 1. Then, the parameter A,, in the Wilson equation may be related to the interaction energy per molecule segment. If one assumes that the like-pair potential parameter A,, relates to the interaction energy per molecule, eq 2 should be replaced by eq 4 (Hiranuma, 1972);
A i j = exp(-(Xij - A,,)/RT)
( 4)
One of the authors showed that eq 4 was somewhat better than eq 2, if A,, was estimated from the energy of vaporization as A,, = -(2/z)u,, and if
4 2 =
A21
( 5)
by several investigators (Hansen, 1967; Blanks and Prausnitz, 1964; Hansen and Skaarup, 1967). (2) The Polarization Energy a n d Hydrogen Bond Energy, u p and uh.The major problem is the determination of contribution up and uh from the remaining energy ua = u - u d . It does not yet appear possible to calculate these quantities individually. Bottcher (1952) has developed a relation for calculating the contribution of the permanent dipoles to the cohesion energy of a liquid. This energy is given as U B in eq 7.
where u is the molar volume, t is the dielectric constant (static value), n D is the index of refraction for D light (sodium), and p is the dipole moment expressed in Debye units (10-ls (esu)(cm)).The model for eq 7 was a spherical molecule with the dipole center a t the center of the molecule. Although errors can be expected for molecules which deviate from the idealized model, the calculated values appear to be reasonable for many compounds (Hansen and Skaarup, 1967). The substances possessing large values of ua departing significantly from Bottcher’s values exhibit the properties of “abnormal” or “associated” liquids. The above considerations were used in the estimation of u p and uh from U a . The energy of U a here is divided into u p and uh by using Bottcher’s values, uB.
>
uB (Sa)
< uB
(8b)
(1)
up = uB and uh = u, - up, when u,
(2)
up = u, and uh = 0, when ua
It must be emphasized a t this point that “hydrogen bonding” is being used in this sense. Table I presents the values of three-dimensional energies for several compounds. Values for the constants in eq 7 were found in standard references. The value of up is changed greatly by small changes in the assumed dipole moments. Estimation of A l 2 . The quantity A12 is given by the intermolecular energy acting between molecules 1 and 2 in solution
In the following, the single-parameter Wilson equation which is composed of eq 1 , 3 , 4, and 5 is used. Three-Dimensional Energy of Vaporization To understand an interaction energy between molecules, the concept of three-dimensional energy of vaporization is introduced. If it is assumed that the energy of vaporization, u,, arises from contributions from nonpolar interactions, u,d, as well as permanent dipole-permanent dipole interactions, u,P,and hydrogen bonding, uZh,the following equation can be written
(1) The Dispersion Energy, ud. Bondi and Simkin have shown how the energies of vaporization of polar liquids can be divided into polar and nonpolar contributions by using the homomorph idea of Brown (Hansen, 1967). The homomorph of a polar molecule is a nonpolar molecule having very nearly the same size and shape as those of the polar molecule. The energy of vaporization of the homomorph was assumed here to be the dispersion energy. Calculations of ud are performed according to the method of Blanks and Prausnitz (1964). In certain cases the values calculated by this procedure must be modified (Hansen, 1967). The energies of vaporization of the homomorphs for many organic compounds have been tabulated 222
Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 3, 1975
The terms in eq 9 have the following significance: ulzd represents dispersion energy acting between dissimilar molecules. u12p represents the interactions between permanent dipoles in the solute and solvent. The induction contribution, ul$, represents dipole-induced dipole interactions between solute and solvent, ulzh represents the contribution to the interaction energy from any solvation which occurs in solution. (1) The Dispersion Energy, ~ 1 2 (i) ~ . On the basis of the London theory, dispersion energy acting between dissimilar molecules a t large separation, u12d, may be represented by U , Z d = (Upu:)i’2 (10) under the assumption that the ionization energies are equal (cf. Hildebrand and Scott, 1962). This expression is strictly valid only at large separations. (ii) At closer distances, the dispersion energy between a spherical (colloidal) particle with a radius a1 and that with a radius a2 is given as W12 in the literature (Chu, 1969).
Table 1. Three-Dimensional Energy for Various Liquids a t 25”Ca Class
Volume
Halohydrocarbon Chloroform 80.7 Chlorobenzene 102.1 Ethers Diethyl ether 104.8 Ketones Acetone 74 .O Aldehydes Benzaldehyde 101.5 Esters Methyl acetate 79.7 Nitrogen compounds Acetonitrile 52.6 Nit romet hane 54.3 Nit roethane 71.5 Pyridine 80.9 Benzonitri le 102.6 Nitrobenzene 102.7 Formamide 39.8 Alcohols Methanol 40.7 1-Butanol 91.5 Cy clo hexanol 1 0 6 .O Phenols Phenol 87.5 Water 1 8 .O a ud and ( ) = Hansens’ values (1967).
t!.
11
UP
6108.2 8830.6
161.8 ( 181.6) 504.9 ( 450.3)
652.4 ( 632.7) 47.5 ( 102.1)
5283 .O
186.0 ( 205.4)
1686.5 (1667.1)
4274.2
2361.3 (1924.7)
418.9 ( 855.4)
9160.4
1777.3 (1315.4)
224.3 ( 686.1)
4603.5
620.3 ( 976.3)
1447.1 (1091.1)
2958.8 3219.4 4350.1 6997 .O 7412.9 9863.3 2808.3
4546.7 4935.3 4021.9 1362.7 2249.0 2222.4 8402.5
2228.7 5566.9 7658.5
1494.6 (1465.2) 650.3 ( 717.4) 744.9 ( 424.0)
4806.2 (4835.6) 5492.1 (5425.0) 4296.5 (4617.4)
6776.0 648 .O
482.2 ( 735.9) 4667.3 (4213.6)
4916.6 (4662.9) 4532.4 (5020.0)
where R is the distance between the centers of the particles and A* is a n energy parameter characteristic of interacting particles. Equation 11 shows that the dispersion energy between a spherical particle with a radius a 1 and m ) is twice the an infinitely iarge particle (a radius a2 dispersion energy between similar spherical particles with each radius al a t very small separation ( R = al az; that is the liquid state.). (iii) Equation 10 is not consistent with this result unless one assumes that the dispersion energy for unlike molecules is equal to the harmonic mean of the dispersion energies of the individual molecules; i.e., ulzd = 2ulduzd/ (uld + u 2 d ) 2 u l d when uzd > uld. However, when one assumes that u12d is a harmonic mean of uld’s, the value compoof ulzd for the system of component 1 ( u l d , u I ) nent 2 ( u ~ ~ , visz )not distinguishable from that for the system of component 1* ( u I * ~= u l d , but u 1 * # u 1 ) + component 2 . It is necessary to separate the effect of cohesive energy density from the effect of volume difference on
-
+
-
+
U12d.
(iv) Equation 11 has the form A* x F(al,az,R), where F is a function of only distance and radius parameters. Let cLd ( = u l d / u l )be the nonpolar cohesive energy density. ~ is expressible When cld = czd = cd, 2 ~ l d u 2 ~ / ( u+l uzd) as the product of two parts, cd x 2 u l u z / ( u l + U Z ) by analogy with the form A* x F . (v) When cld # c ~ d how , is the value of c12d given? In equations of state for gas mixtures and in theories of liquid solutions, the energy parameters of the potential function e * 1 2 4 ( r / r * 1 2 ) correspond with A* or c12d. The mixing rule is often given by e * 1 2 = { ( e * 1 1 e * 2 2 ) 1 z , where e* is an energy parameter characteristic of interacting pairs and experimental values of { reported to data are less than unity (cf. Jones and Rowlinson, 1963). Approximately, { ( e * l l e * z z ) l with j- 5 1 corresponds to Z e * l l e * z z / ( e * l l + e * z z ) , since 2 e * l l e * z z / ( e * l l + *22) 5 ( e * l 1 e * 2 2 1 z . A simi-
(4073.3) (4596.0) (4129.8) (1495.8) (1986.3) (1811.6) (6520.8)
0.0 0.0 454.1 813.5 0.0 0.0 1560.6
( 473.4) ( 339.4) ( 346.1) ( 680.4) ( 262.7) ( 410.8) (3442.3)
lar analysis is useful in the estimation of clzd. We assume that a dispersion energy, for any two species
uiZd = (harmonic mean of c i d ’ s ) x (harmonic mean of u I ’ s ) (12) (2) The Polarization Energy, u ~ z p For . polar molecules which may be represented by spherical force fields with small ideal dipoles at their center, the interactions between permanent dipoles in the solute and solvent, u1zp may be represented by (cf. Hildebrand and Scott, 1962)
We assume that u 1 z p is given by the geometric mean of up’s, because Bottcher’s value is alloted to u l p . (3) The permanent dipole-induced dipole interaction energy, ulzl. It is generally small as compared with the other energies. The Debye theory of induction suggests that ulzl should depend on the quantity (u,Puld)l (Hildebrand et al., 1971).We assume that ulzl is given by
where the constant K must be approximately determined from experimental vapor-liquid equilibrium data for nonpolar-polar systems. (4) The Solvation Energy, ~ 1 2 Very ~ . little theoretical knowledge is available to characterize an energy of solvation which occurs in solution. When specific interactions occurs in solution, u1zh must be estimated from experimental data for a system similar in characteristics to the system in question, by assuming the near-constancy of u1zh for the same functional group interaction
Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 3, 1975
223
where the values with an asterisk are those for a base system and u12* is determined from experiniental data (infinite-dilution activity coefficienis or vapor-liquid equilibrium data), the terms u12*d and u 1 2 * p , respectively, are calculated from eq 12 and 13. When specific interactions do not occur, ulzh may be represented by for the following cases (cf. Pimente1 and McClellan, 1960) (a) mixtures of nonpolar substances and hydrogen bonded substances (in this case, ulzh = 0); (b) mixtures of solvents which act as proton donors such as chloroform and pentachlorethane; (c) mixtures of solvents which act as proton acceptors, such as ketones, aldehydes, ethers, and esters. ( 5 ) The Factor 2 / z . The value of 2/2 must be selected so that the single-parameter Wilson equation can represent data for a base system as well as possible. The variation of 2/2 in the single-parameter Wilson equation is not so- sensitive to the results in many cases. However, for very nonideal systems where one of the infinite-dilution activity coefficients was of order 10 or greater, 2/z = 3s gives significantly better results (Hiranuma, 1972). Then, 2/z = y3 is chosen in common to all systems. Consequently, the unlike-pair potential parameter is written by xi2
ui2 =
ul2d
ui2 = u l t
= -(2/z)u*z
(16)
+
uiZi for nonpolar-polar s y s t e m s
+
uI2*+ uIZhfor polar-polar s y s t e m s
(17a)
(17b) The parameter ulzl or ulzh , respectively, is left as the sole adjustable parameter for nonpo!ar-polar systems or for polar-polar systems. However, ulzl and ~ 1 are 2 pa~ rameters for each system, which can estimate from a system similar to the system in question. Effects of Temperature on Parameter (A,, - A L L ) The energies of vaporization of liquids are clearly functions of temperature. It is desirable to calculate the three-dimensional energy of vaporization and to determine uL,under the same temperature as the system in question. However, these data are not always available. Even if they are available, the equilibrium data are not always available for a base system a t the same temperature as the system in question. Fortunately, the difference between the energies of vaporization of substances is often nearly temperature independent. For example, the energies of vaporization of acetone at 25 and 95"C, respectively, are 7019.1 and 5900.5 cal/mol, and those of methanol are 8327.8 and 7137.1 cal/ mol (Bagley et al., 1971). The increase in energy of vaporization for acetone as the temperature falls by 70" is 1118.6 cal/mol, whereas that for methanol is 1190.7, very similar values. The parameters in the proposed equation are always given by the difference between two energies. Moreover, the Wilson equation has a feature which makes it particularly useful for engineering applications; to a good approximation one may consider the energy difference ( A L , - A L L ) to be independent of temperature, at least over a modest temperature interval, which means that parameters obtained from data a t one temperature may be used with reasonable confidence to predict activity coefficients a t some other temperature not too far away (Orye and Prausnitz, 1965). Consequently, the energy difference (u,, - u,)is assumed to be independent of temperature, at least over a modest temperature interval. When uL'sat 25°C are used for the values a t some other temperature and the energy difference (u,, - u , ) is assumed to be independent of temperature, the value ulzh 224
Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 3, 1975
/ /
/
/'
/ / /
/
/
/
// 0
I 6
10
I
20
Number of carbon
Figure 1. Ut2 determined for ethanol-paraffin systems: 0 , u12 determined from Pierotti equation at 60°C ( I d . Eng. Chem., 51, 95 (1955)); -, u12d calculated from eq 12; - - -, ulzd calculated from eq 10.
or ulzl obtained from data a t various temperatures is the same as the value at 25°C. Test of the Assumptions To verify the temperature independence of the quantity (uL, - u,),~ 1 2 ' sin the single-parameter Wilson equation
were determined from vapor-liquid equilibrium data at different temperatures for the same system. If the quantit y (u,, - u,) is approximately temperature-independent, ~12's which are determined from data at different temperatures by using ut's a t one temperature (for convenience 25°C) should agree with one another. The examples follow: the values of u12 obtained for the system ethanol-benzene a t 35, 60, and 80°C are, respectively, 5805.6, 5739.0, and 5965.2 cal/mol; those for the system carbon tetrachloride-benzene at 20 and 95°C are, respectively, 7161.6 and 7347.3 cal/mol. Therefore, the assumption that the quantity (u,, - u , ) is approximately temperature -independent is reasonable. In the followings, Hansen's data at 25°C (Hansen and Beerbower, 1971) were used for the three-dimensional energy of vaporization, since they have been calculated for many organic compounds by a method similar to that described in this paper. (1) The Dispersion Energy, u 1 2 d . The experimental vapor-liquid equilibrium data for nonpolar mixtures can be used for testing eq 12. For seven nonpolar systems with large differences in u , or eld the values ulzd were calculated respectively by using eq 10 and 1 2 and also were determined from equilibrium data. The ranges of uL and c,d were respectively 89.4-163.5 cc/mol and 53.29-81.0 cal/cc. When ulzd were calculated from eq 10, the average deviation inul2d ( S , ~ ( u 1calcdd 2 - u12 e x p t l d ( / N )was 322 cal/ mol. On the other hand, the average deviation in ~ 1 was 2 114 cal/mol when u12d values were calculated from eq 12. For ethanol-paraffin systems, eq 12 has also been examined by comparing with u12 determined from the infinite-dilution activity coefficients, as shown in Figure 1. The value u12 ( = u12d + u l $ ) obtained for these systems is considered to be nearly equal to u12d, since ulzl is gen. quantities u12 erally small as compared with 2 ~ 1 2 ~The
~
Table 11. Hydrogen-Bonded Energy of Solvation Determined from Vapor-Liquid Equilibrium Data a t 760 m m H e
II
9 '
L - ,IL
/-
Figure 2. Correlation of dipole-induced dipole interactions t o (uiPujd)l12:0 , for mixtures containing benzene .(phenyl); 0, for mixtures without benzene. determined from the Pierotti equation are quite similar to u12d calculated from eq 12. Therefore, it is reasonable to use eq 12. (2) The Induction Energy, ~ 1 2 Since ~ . u12l is calculated as the difference between u12 e x p t ~and u12 c a ~ c d d ,all errors are grouped into ~ 1 2 The ~ . constant K in eq 14 was roughly determined from experimental vapor-liquid equilibrium data for nonpolar-polar systems as shown in Figure 2; K = 0.28 for mixtures containing phenyl group and K = 0.1 for other mixtures. When u12' for a similar system is not available, the K value can be used instead, without introduction of significant error. I t is considered that there is a special interaction between a phenyl group and a polar substance. (3) The Solvation Energy, ~ 1 2 Table ~ . I1 presents the energies of u12h for several kinds of functional group interactions as determined from vapor-liquid equilibrium data a t 760 mmHg. Although all errors are grouped into ulzh, the near-constancy of u1zh is observed for the same group interactions. Therefore. the assumption of the near-constancy of ulzh is valid. I t is of interest to observe that ulzh is common by hydroxyl-ketone and ester group interactions, as well for active hydrogen (chloroform)-hydroxyl, ester, and ketone group interactions. Performance of the Proposed Method We illustrate the procedure of estimating u12 for a system acetone( 1)-methanol(2), by using u12* determined from equilibrium data for a base system acetone( 1)-ethano1(2*). By using the single-parameter Wilson equation, u12* was determined from data; that is, u12* = 7593.0 cal/mol. When the parameter u12* in the single-parameter Wilson equation is obtained from the vapor-liquid equilibrium data, a nonlinear least-square fitting method was used which minimizes the sum of squares of deviations in activity coefficients for all data points. The data (Hansen and Beerbower, 1971) for the pure liquids a t 25°C are given in Table 111. By eq 12, u12*d = 2 u ~ u 2 * / ( u+~ u 2 * ) X 2cldcz*d/(cld C Z * ~ ) = 2 X 74.0 X 58.5/(74.0 58.5) X 2 X (7.6)2 (7.7)2/((7.6)2 (7.7)2) = 3823.57. By eq 13, u12*P = (u1P = 1442.89. By eq 15, u l p h uz*P)I -- U12* -= ~( U1I U2Z-** ) ~~u ~ Xz + 61P62*P P = 7593.0 - 3823.57 - 1442.89 = 2326.54. Similarly, ulzd = 2952.45; u 1 2 P = 1679.32; u12 = 7 ~ 1 2+ ~ UIZP + u l p h = 2952.45 1679.32 2326.5 = 6958.31, where c L = 6,2 and u, = clur = 6 1 2 u 1The . estimated value u12 is consistent with the value determined from vapor-liquid equilibrium data a t 55°C for the system acetone-methanol (i.e., 7053.6 cal/mol). By using ~ 1 2 '=~ 2326.5, for the systems acetone-methanol, acetone-2-propanol, and methyl ethyl ketone-l-butanol, the vapor-liquid equilibria were predicted and compared with the experimental data (Table IV). As another example, the vapor-liquid equilibria for the systems benzene-methanol, benzene-1-propanol, benzene-ethanol,
+
+
+
+
+
Donor
Acceptor
uilh
Active hydrogen (chloroform)
Hydroxyl
1658.1
System
(Chloroform1-butanol) (Chloroform1974 .O methanol) (Chloroform1858.4 ethanol) (ChloroformEster 1810.3 ethyl acetate) 1706.3 (Chloroformmethyl acetate) Ketone ~ 0 3 . 4 (Acetonechloroform) 2645.1 (Ethyl acetateHydroxyl Ester methanol) (alcqhol) 2562.6 (Ethyl acetateethanol) 2219.5 (AcetoneKetone ethanol) 2485.1 (Acetonemethanol) 2535.3 (Methanolmethyl ethyl ketone) Phenyl 1428.5 (Benzene-lpropanol) (Benzene-l1490.7 butanol) 1498.1 (Toluene-lbutanol) 1299.3 (Benzeneethanol) a Experimental data (Bussei Jyosu, 1963-1970; Kagaku Benran, 1966'.
Table 111 2' i
6 id
6 ip
6ih
74. 58.5 40.7
7.6 7.7 7.4
5.1
3.4
4.3 6 .O
9.5 10.9
~~
Acetone(1) Ethanol(2*) Methanol (2)
Table IV. Performance of the Proposed Method in Predicting Vapor-Liquid Equilibria a t Constant Temperature System
Temp, "C
uY x 1030
Acet one-methanol Acet one-et hanolb Acetone-1 -propanol Methyl ethyl ketone-1-butanol Ethanol-benzene Benzene-1 -propanol Benzene-2 -propanol* Ethanol-toluene Met hanol-benz ene
55 18.0 ( 1 l . O ) c 55 22.0 (22.0) 55 27.0 (18.0) 50 37.0 (28.0) 50 18.0 (3.8) 40 32.0 (21.0) 25 13.0 (13.0) 35 15.0 (15.0) 40 23.0 (. . . ) 6, = ( ( V ~ b e x p t l - ycal,d))2/h')1 2 . Experimental data (Bussei Jyosu, 1963-1970; Kagaku Benran, 1966). * A base system. ( ), values from single-parameterWilson equation. Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 3, 1975
225
Table V. Comparison of the Proposed Method with the Single-Parameter Wilson Equation System
o,, x 103b
Base system
(JT ~
Acetone-propyl acetate Ethyl acetate-toluene Methyl acetate-benzene Chloroform-ethyl acetate Carbon tetrachloride2 -propanol Benzene-1 -butanol Toluene-1 -butanol Ethyl acetate-ethanol Acetone-ethanol Methanol-methyl e thy1 ketone
10.9 (3.1)" 13.2 (3.1) 12.5 (12.7) 9.7 (6.2) 4.4 (5.1)
0.97 (1.00) 1.04 (0.21) 0.30 (0.27) 0.75 (0.16) 0.51 (0.63)
20.0 (24.7) 7.9 (10.0) 21.3 (19.9) 29.3 (31.4) 9.4 (6.5)
0.88 (1.18) 0.44 (0.61) 0.92 (0.73) 1.23 (0.43) 0.37 (0.26)
~
~~~
*
Discussion I t has been shown that the proposed method is applicable to systems for which the single-parameter Wilson equation would be applicable. The proposed method results in almost as good a representation of the vapor-liquid equilibria as the single-parameter Wilson equation does, if a base system is as close as possible in characteristics to the system in question. Prediction for aqueous solution is not recommended without modification (cf. Hiranuma, 1972). (1) The value u l p h for one system can be used to predict u12 for some other similar systems. (2) This method is useful to estimate the order of an energy of solvation which is not accurately known. (3) This method is helpful in understanding the properties of highly polar and/or hydrogen-bonded systems to which the regular solution theory will not apply. The second purpose of this study is to confirm whether the Wilson-like model has a theoretical basis and it was concluded that it appears to be reasonable. Nomenclature c = cohesive energy density = a2, cal/cm3 R = gasconstant T = absolute temperature u = energy of vaporization, cal/mol u = molar volume, cm3/mol x = mole fraction
226
Ind. Eng. Chem., Process Des. Dev., Vol. 14, No. 3, 1975
u y x 103 ~
Acetone-ethyl acetate Ethyl acetate-benzene Ethyl acetate-benzene Chloroformlnethyl acetate Carbon tetrachloride-1 butanol Benzene-1 -propanol Benzene-1 -propanol Ethyl acetateinethanol Acetonelnethanol Acetonelnethanol
a ( ) = Values f r o m single-parameter W i l s o n e q u a t i o n . u , = ( S N ( ~ -~ ycaiL,j)2/N)1 ~ ~ ~ I m e n t a l d a t a (Bussei Jyosu, 1963-1970; K a g a k u B e n r a n , 1966).
and toluene-ethanol were predicted by using the value u ~ z determined * ~ from data for a base system benzene( 1)2-propanol(2*) (Table IV). To examine errors in using the proposed method, the performance of the proposed method in predicting vaporphase compositions a t atmospheric pressure is compared with that of the single-parameter Wilson equation with the parameter obtained by using a least-squares fit for all points in reproducing vapor-phase compositions (Table V). The accuracy of the proposed method compares closely with that of the single-parameter Wilson equation for the system in question as shown in Table V.
~
2; UT
UT
~~~
(9.9) (3.0)
(0.16) (0.28)
(7.5) (21.0)
(0.42 (1.27
(23.4)
(1.41
(8.4) (8.1)
(0.37 (0.39
= ( 2 \ ( T p x p-t ~T
~ ~ ~ ~ ~ 2). 2E x/ pNe r)i - 1
Greek Letters y = activity coefficient in the liquid phase
6 = solubility parameter, ~ a 1 1 : ~ / ( c m 3 ) ~ K = constant in eq 14
Subscripts i = for component i ij = forpairij Superscripts d = fordispersion i = for dipole-induced dipole p = fordipole h = for hydrogen bonding = for infinite dilution * = fora basesystem O
Literature Cited Bagley, E. B.,et al., J , Paint Techno/., 43, No. 5 5 5 , 35 (1971). Blanks, R . F., Prausnitz. J. M.. Ind. Eng. Chem.. Fundam., 3, 1 (1964). Bottcher. C. J. F.. "Theory of Electric Polarization," Elsevier, New York, N.Y.. 1952. Chu, B., "Molecular Forces: Based on the Baker Lectures of Peter J. W. Debye," Wiley. New York, N.Y.. 1969. Hansen. C. M., J. Paint Techno/., 39, No. 505, 104 (1967). Hansen. C. M.. Skaaruo. K . : J. Paint Techno/.. 39. No. 51 1, 51 1 (1967) Hansen, C. M.. Ind. Eng. Chem., Prod. Res. Dev..'B, 2 (1969) Hansen, C. M., Beerbower. A . . "Solubility Parameter." in "Encyclopedia of Chemical Technology," Supp. Vol., 2nd ed, p 889, 1971 Hildebrand, J. H., Scott, R. L., "Regular Solution," p 170, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. Hildebrand, J. H., et al., "Regular and Related Solutions." p 191. Van Nostrand-Reinhold, New York, N.Y., 1971 Hiranuma, M., Ind. Eng. Chem.. ProcessDes. Dev., 11, 631 (1972). Jones, I . W., Rowlinson. J. S.. Trans. Faraday Soc., 59, 1702 (1963). Kagaku Kogaku Kyokai, "Bussei Jyosu." Vol. 1-8, Maruzen, 1963-1970. Nippon Kagaku Kay, "Kagaku Benran. Kisohen 1 1 . ' ' Maruzen, 1966. Orye, R. V., Prausnitz, J. M., ind. Eng. Chem., 57, 18 (1965) Pimentel, G. C., McClellan. A . L., "The Hydrogen Bond." W. H . Freeman and Co., San Francisco, Calif., 1960. Schreiber. L. B., Eckert, C. A . , ind. Eng. Chem , Process Des Dev.. 10, S73 f 1 9 7 1 \ Tassios, D.,A./.Ch.E.J., 17, 1367 (1971) Wilson, G M.J. Am. Chem. SOC..86, 127 (1964) Wong, K. F., Eckert, C. A , , Ind. Eng. Chem.. Fundam.. 10, 20 (1971) . I .
Received f o r review J u n e 6, 1973 Resubmitted April 23, 1974 Accepted M a r c h 2, 1975
