If one considers the extension of the coexistence equation method to ternary systems, one is faced with two partial differential equations involving two independent variables (liquid phase compositions) aind two dependent variables (vapor compositions or partial pressures). I n the present formulation, one has a single partial differential equation in a single dependent variable (the excess free energy). The latter is somewhat more convenient from a computational point of view, and certainly becomes more attractive as one considers solutions with more components. Of the methods reported in the past for calculation of vapor compositions from solution vapor pressure data, only that of Barker ( 7 ) seems easily generalized to ternary and higher ordered systems. Barker's method involves assuming a polynomial form for the excess free energy and then using statistical techniques to estimate the parameters therein so as to get the best fit to the experimental pressure data. T h e present method has a n advantage over Barker's method in that it is not necessary a t the outset to select a model. Though we used polynomials for data smoothing and interpolation, this is not necessary. I n principle, the method can be applied to graphically smoothed data or, if sufficient data are available, to the raw data themselves. Thus, the present method is not limited to the a priori choice of a particular excess free energy functionality. If deviations from ideality in the vapor phase are significant, these deviations can be taken into account rather easily. One first computes vapor phase compositions under the assumption that ideality holds, then uses these compositions to estimate vapor phase activity coefficients by the usual methods. With Equation 1 replaced by
the entire procedure is repeated for the final relaxation. The effect will be to change the coefficients in Equation 9 slightly by the introduction of the gas law correction terms.
Nomenclature = quantity defined by Equation 12 a = parameter in vapor pressure function, Equations 23 to 26 b = quantity defined by Equation 17 B = quantity defined by Equation 13 c = parameter in vapor pressure function C = quantity defined by Equation 14 d = parameter in vapor pressure function = quantity defined by Equation 18 g @ ) ( n ) = nth estimate of correction to G ( ~ ) ( E )
A
f
G GE
=
GE/RT
= excess Gibbs free energy of mixing G ( " ) ( E )= nth estimate of value of G a t lattice point designated
by E number of comDonents in solution = nth estimate of partial pressure of kth component,
=
XkYi;(")Pk0
of kth component in a n ideal solution, xkPko solution vapor pressure vapor pressure of kth component nth estimate of solution vapor pressure a t lattice point designated by E quantity defined by Equation 16 mole fraction of ith component in liquid vector of lattice indices liquid phase activity coefficient of kth component lattice interval elementary vector defined by Equation 3 vapor phase activity coefficient of kth component
= partial pressure = =
= W
=
xi
Yk
= = =
A
=
ak
=
6k
=
E
literature Cited (1) Barker, J. A , Australian J . Chem. 6 , 207 (1953). (2) Bruce, G. H., et al., Trans. Am. Inst. Mining, Met. Petrol. Engrr. 198. 79 (1953). (3) Hala, 'E., Pick, J., Fried, V., Vilim, O., "Vapor-Liquid Equilibrium," Pergamon, New York, 1958. (4) Ljunglin, J. J., Van Ness, H. C., Chem. Eng. Sci. 17, 531 (1962). (5) Scatchard, G., Raymond, C. L., J . Am. Chem. Soc. 60, 1278 (1938). (6)' Sevkrns, \Y. H., ef ai., '4.I.Ch.E. J . 1, 401 (1955). (7) Tao, L. C.. Ind. Eng. Chem. 53, 307 (1961). (8) Torgeson, R. L., "Calculation of Vapor-Liquid Equilibria at Constant Temperature," M.S. thesis, bniver&ty of -Delaware, 1965. RECEIVED for review November 9, 1964 ACCEPTED July 20, 1965
GROUP CONTRIBUTIONS T O ACTIVITY COEFFICIIENTS The CH, and OH Groups W I LL IA M A
.
SC H ELL
E R , The Unnrersity o f x e b r a s k a , Lincoln, A'eb.
The group contribution work reported by Wilson and Deal i s modified and extended. The theory i s developed in terms of thermodynamic instead of empirical quantities. Extensive data for the CH2 and O H group contribultions to the residual partial molal energy of mixing are reported over a CH2 concentration range of 3.0 tlo 99.9%. A method for calculating the standard state group contributions needed for applying the method is presented. Additional information for using the method with diols and triols is provided. Logarithms of activity coefficients for eight binary systems near 50" C. were calculated from group contributions with an absolute average deviation from the experimental values of less than 10%.
FOR
process evaluation and design calculations it is highly desirable to account for solution nonidealities because of their effect on equilibrhm composition and equipment requirements. However, it is not always feasible to determine activity coefficients experimentally for this purpose. As a result attempts have been made to develop generalized methods
for predicting activity coefficients, especially for solutions of nonelectrolytes. Hildebrand's method (5) for regular solutions requires the use of solubility parameters. These can be calculated from heats of vaporization. In the method described by Pierotti, Deal, and Derr ( 7 7 ) , activity coefficients a t infinite dilution VOL. 4
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459
are estimated from a consideration of the individual interactions between pairs of structural groups making up the solvent and solute molecules. A relationship such as the van Laar equation may then be used to obtain values for the activity coefficients a t concentrations other than infinite dilution. Wilson and Deal (76) consider a solution to be a mixture of structural groups. Activity coefficients for the groups in solution are calculated from molecular activity coefficients and plotted against group concentration. These group activity coefficients can then be used to calculate molecular activity coefficients for other systems containing these same structural groups. I t is this work by Wilson and Deal that is modified and extended in this paper.
intermolecular forces. Such considerations led Hildebrand (5) to the use of soiubility parameters in estimating the energy of mixing. Because the solubility parameter depends o n the' heat of vaporization, which can be estimated from critical properties (74) which in turn can be estimated from molecular it appears that the partial energy of group contributions (8), mixing might be estimated directly from group contributions. Redlich, Derr, and Pierotti (72) proposed a model for the heat of mixing based on the assumption that the interaction between molecules can be represented as the sum of contributions of pairs of interacting groups. Papadopoulos and Derr ( 9 ) successfully tested Redlich's theory against experimental data for a number of binary hydrocarbon systems.
Theory
Calculations
The activity coefficient for component i in a solution is related to the excess partial free energy of solution by the equation
Experimental isothermal low pressure vapor-liquid equilibrium data can be used to calculate the liquid phase activity coefficient from the relationship
I t is apparent from this equation that if the excess partial molal entropy and the partial heat of mixing can be estimated, the activity coefficient can be calculated. The excess entropy of mixing is defined by the equation
The residual partial molal energy of mixing is then calculated by combining and rearranging Equations l and 6.
ASiE = where
Asi'
ASi.%r - AS,'
(2)
-R In x i
(3)
=
I n the following development the expression used for entropy of mixing is based on a derivation by Hildebrand (4) for athermal binary systems of nonelectrolytes, assuming that the mixing process may be regarded as equivalent to the sum of two expansions into available free volumes. Further, if one assumes that the ratio of the free volumes is proportional to the molar volumes of the pure components, then
Asl."'
=
+ az(1
-R [In a1
-
91
(4)
T h e excess entropy of mixing for component i in a multicomponent solution may be obtained by substituting Equations 3 and 4 in Equation 2 and replacing with at and with 1 - ai, and rearranging to give
A s i E = -R where
[
+( 3 :< 1
In f
1
-
-
J
This relationship is identical to the Flory-Huggins equation for excess entropy of mixing. Wilson and Deal (76) also used the Flory-Huggins equation, but in evaluating @%/xi they used the number of atoms (other than hydrogen) per molecule rather than molar volume. Thus they assume that all atoms regardless of type have the same molar volume. For certain systems-e.g., polar or associated systemsEquation 5 gives a poor estimate of Because of this, excess partial entropies calculated from Equation 5 will be designated as AS,FH. The actual excess partial entropy can then be related to ASiFH with the aid of a residual excess partial entropy A s i R as follows: A number of investigators have proposed methods for estimating partial molal heats of mixing from a consideration of 460
l&EC FUNDAMENTALS
A n i R = Aat:
- T d i R = RT In y i + TAsiFH
(8)
I t is assumed that A n i R is a linear combination of the group contributions. AniR =
k
nk,i
(2,'- a h G * )
(9)
It is further assumed that the group contributions to the residual partial molal energy of mixing are a function only of group concentration, Z , in the solution. These last two assumptions agree with those of Wilson and Deal (76). The term is the standard state energy for group k and is independent of the molecular concentration of the solution. Its value depends on the groups present in the pure components and on the molecules that make up the system. The standard state used here is that for which = 0 when = 1.0. A literature search was made and five papers containing original low pressure isothermal (50' C.) binary vapor-liquid equilibrium data were found for six systems containing CH2 and O H groups. These systems and their maximum pressures are tabulated in Table I. From these data, values of y i , A S i F H ,A n i R , RrG, and Z, were calculated. T h e values of y i were tested for thermodynamic consistency by the methods of Redlich and Kister (73) and any obviously bad data were discarded. There remained 56 sets of x-y-P data which provided 112 activity coefficients and an equal number of partial entropies, residual partial energies of mixing, and group contributions to the residual partial molal energy of mixing over a C H ? group concentration range of 3.0 to 99.9%. In processing the alcohol-water data, water was assumed to be equivalent to 1.6 OH groups of the type contained in normal primary alcohols. T h e justification for this assumption is discussed below.
ax'*
a,'
z,
n,'*,
Discussion of Results
Figure 1 is a plot of the group contributions divided by RT as a function of the CH2 group concentration. Table I1 contains a smoothed set of group contributions read from Figure 1. The values of nxG*/RTthat were used to establish Figure 1 are tabulated in Table I. Values of nkG*/RTfor any system made up of CH2 and OH groups may be calculated with the
Table 1. Sources of Vapor-liquid Equilibrium Data and Standard State Group Contributions for Systems Containing CH2 and O H Groups nb. o f Max. Data HkG*/RT Temp., Pressure, Points System O C. M m . Hg Used Ref. CHz OH 0 7 ( 7 ) 1.23 1. Methanol-water 49.76 404.6 0 8 (6) 0 . 8 9 2. Ethanol-water 5 0 . 4 4 225.0 0 9 (77) 0 . 7 2 3. 1-Propanol-water 49.92 1 4 0 . 4 1.75 9 (75) 0 4. Ethanol-n50.0 311.3 heptane 12 (7) 0 1.75 50.0 5. Ethanol-iso318.75 octane 11 (75) 0 2.28 50.0 152.3 6. 1-Butanol-nheptane Total
5.51 4.5
-
4Q
-
METHANOL-WATER
D ETHANOL-WATER
A I -PROPANOL-WPTER 0 ETHANOL-n-HEPTANE V ETHANOL-iso-OCTANE 0 I -BUTANOL-n-HEPTANE SO'C.
5.0
-
3,5
c
3 3.0-
l r
2.5
Table II.
0
Smoothed Group Contributions from Figure 1
-
2.0
RCH2G ZCH2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.97 1 .00
RT 2.23 1.85 1.50 1.20 0.94 0.73 0.53 0.37 0.22 0.10 0.05 0.03 0.00
1.5
0.00 0.02 0.09 0.20 0.33 0.50 0.73 1 .OS 1.49 2.22 2.93 3.39 5.35
1.0
0.5
0.1
O'%O
0.2
0.3
04
0.5
0.6
01
0.8
0.9
IO
GROUP FRACTION CH2 IN SOLUTION
Figure 1. CH2 and OH group contributions to residual partial molal energy of mixing at 50" C.
aid of Figure 1 if it is recognized that ARiR= 0 when x i = 1.O. Substituting this fact into Equation 9, dividing by RT, and rearranging gives 120110
When calculating residual partial molal energies of mixing with the aid of Equation 9 and Figure 1, it is not really necessary to calculate the individual values of g k G * / R Tif it is noted that Equation 9 can be rearranged thus:
-
90 -
100
W
J O
f1
80-
3
w 70-
5-t
0
> 60a
T h e standard state terms in Equation 11 are simply read from , ~ "each compound. Figure 1 a t Z C H ~ for I t was assumed in processing the alcohol-water data that water contained the equivalent of 1.6 OH groups of the type found in normal primary alcohols. This relationship was arrived a t by plotting the reciprocal of the fraction of OH groups contained in the normal primary alcohols-Le., the total number of g r o u p in the molecule per OH groupagainst their molar volumes a t 50" C. T h e straight line resulting (Figure 2) wa; extrapolated to the molar volume of water and the fraction of OH groups was found to be 1.6. T h e low molar volume of water when compared with normal primary alcohols and hence the high apparent OH group concentration in water are the result of a greater degree of hydrogen bonding in w'3ter (two bonds per molecule) than in alcohol (one bond).
a
-I
0
z
50-
40
-
20 10-
30
01
0.0
I 1.0
I I I I 2.0 30 40 5.0 RECIPROCAL OF FRACTION OF OH G R O U P S IN THE MOLECULE
0
Figure 2. Molar volumes of normal primary alcohols, 1,2diols, and 1,2,3-triols VOL. 4
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461
5'0 -CALCULATED 0
mental data. Lack of isothermal data whose thermodynamic consistency can be checked makes it difficult to evaluate the correlation for heavy materials such as glycerol. However, the correlation is usable for these components if appropriate use is made of Figure 2 to determine the equivalent OH fractions in the molecules.
50'C.
EXPERIMENTAL (31 45%.
Acknowledgment
The author expresses his appreciation to E. I. d u Pont de Nemours and Co. for partial support of this research through a faculty fellowship for the summer of 1964. Nomenclature
A C i E = excess partial molal free energy of mixing for com-
ARi."
Figure 3. Calculated and measured activity logarithms for n-hexane-methanol system
coefficient
Table 111.
Comparison of Calculated and Experimental Activity Coefficients 1. Infinite dilution activity coefficients at 50" C. for ethanoln-heptane system In ...
vo
Exptl. Calcd. Solute % Dev. Ref. 3.16 3.26 Ethanol +3.2 (70) 2.56 2.36 n-Heptane -7.8 (70) 2. Infinite dilution activity coefficient of n-decane at 50" C. Solvent Ethanol 3.01 3.08 $2.3 (70) 7.64 Ethylene glycol 8.44 -9.5 (70) 3. Infinite dilution activity coefficients of alcohols in n-heptane at 50" C. Solute Methanol 3.76 3.56 -5.3 (70) 1-Butanol 2.82 2.89 +2.5 (70) 4. Activity coefficient of water in glycerol Mole 7* Component Water In Y TYater 92.75 0.237 0.237 0.0 (2) Temp., C. 58.77 50.0
ponent i, cal./gram mole
i, cal./gram mole ARiR = residual partial molal energy of mixing for component i, cal./gram mole RkG = contribution of a group k to AniR, cal./gram mole RkG* = standard state value of RkG, cal./gram mole = number of groups of type k in component i = system pressure, mm. H g Pio = vapor pressure of pure component i, mm. H g R = gas constant, cal./gram mole-' K . ASiE = excess partial molal entropy of mixing for component i, cal./gram mole-' K. AJiFH = excess partial molal entropy of mixing for component i calculated from Eq. 5, cal,/gram mole-' K. AS,' = ideal partial molal entropy of mixing for component i, cal./gram mole-' K. ASi" = partial molal entropy of mixing for component i, cal./gram mole-' K . ASiR = residual excess partial molal entropy of mixing for component i, cal./gram mole-' K. T = absolute temperature, O K. Vi = molar volume of pure component i, ml./gram mole xi = mole fraction of component i in liquid yi = mole fraction of component i in vapor Zk = fraction of groups of type k i n liquid solution Zk,io = fraction of groups of type k in pure component i yi = liquid phase activity coefficient of component i yio = infinite dilution liquid phase activity coefficient of component i ai = volume fraction of component i in liquid, assuming ideal mixing = partial molal heat of mixing for component
Literature Cited (1) Bredig, G., Bayer, R., 2.Phys. Chem. 130A, 1 (1927). (2) Carr, A. R., Townsend, R. E., Badger, \V. L., Ind. Eng. Chem. 17, 643 (1925). (3) Ferguson, J. B., J . Phys. Chem. 36, 1123 (1932). (4) Hildebrand, J. H., J . Chem. Phys. 15, 225 (1947). (5). Hildebrand, J. H., Scott, R. L., "Regular Solutions," Prentice-Hall. Enclewood Cliffs. N. J.. 1962. (6) Jones, C . Schoenberg, E. M., Colburn, A. P., Ind. Eng. Chem. 35, 666 (1943). (7) Kretschmer, C. B., Nowakowska, J., \Viebe, R., J . Am. Chem. Soc. 70, 1785 (1948). (8) Lyderson, A . L., "Estimation of Critical Properties of Organic ComDounds bv the Method of Grout3 Contributions," Univ. of Wis. hng. Expt. Sta., Madison, Wis.,kept. 3 (1955). . (9) Papadopoulos, M. N., Derr, E. L., J . Am. Chem. Soc. 81, 2285 (1959). (10) Pierotti, G. J., Deal, C. H., Derr, E. L., Document 5782, American Documentation Institute, Library of Congress, FVashington 26, D. C. (11) Pierotti, G. J., Deal, C. H., Derr, E. L., Ind. Eng. Chem. 51, 95 (1959). (12) Redlich, O., Derr, E. L., Pierotti, G. J., J . Am. Chem. Soc. X I . 2287 ---- (1059). - - - ,(13) Redlich, O., Kister, A. T., Ind. Eng. Chem. 40, 345 (1948). (14) Riedel, L., Chem. Eng. Tech. 26, 679 (1954). (15) Smyth, C. P., Enqel, E. W., J . Am. Chem. Soc. 51, 2660 (1929). i16\ \.liiison, G. M., Deal, c . H., IND. ENG. CHEM. FUNDAMENTALS ' 1, 20 (1962). (17) Wrewski, M. S.,J . Russ. Phys. Chem. Sod. 42, 1 (1910).
x.,
Molar volumes fok the 1,2-diols and the 1,2,3-triols are also plotted in Figure 2. Compared with these materials water has apparent OH group fractions of about 1.1 and 1.05, respectively. In using Figure 1 for a solution of alcohol and a 1,2diol it should be assumed that the OH groups in the glycol are equivalent to 1.6/1 .I or 1.45 OH groups of the type contained in the alcohol. T h e method for estimating activity coefficients described here has been tested against experimental data that were not used in developing Figure 1. Figure 3 compares the natural logarithms of the activity coefficients for the n-hexane-methanol system calculated from Equation 1 with the experimental data of Ferguson ( 3 ) . Table I11 contains comparisons for a variety of other systems at infinite dilution and finite concentrations. For the systems tested, it appears that the natural logarithms of the activity coefficients can be estimated with an absolute average deviation of less than 10% from the experi462
I&EC
FUNDAMENTALS
--7
\ -
RECEIVED for review December 7, 1964 ACCEPTED May 24, 1965
