Ind. Eng. Chem. Res. 1989,28, 315-324 Franjione, J. C.; Ottino, J. M. Feasibility of Numerical Tracking of Material Lines and Surfaces in Chaotic Flows. Phys. Fluids 1987, 30, 3641-3643. Kolodziej, P.; Macosko, C. W.; Ranz, W. E. The Influence of Impingement Mixing on Striation Thickness Distribution and Properties in Fast Polyurethane Polymerization. Polym. Eng. Sci. 1982,22, 388-392. Lee, L. J.; Ottino, J. M.; Ranz, W. E.; Macosko, C. W. Impingement Mixing in Reaction Injection Molding. Polym. Eng. Sci. 1980,20, 868-874. Mehta, R. V.; Tarbell, J. M. An Experimental Study of the Effect of Turbulent Mixing on the Selectivity of Competing Reactions. AIChE J. 1987, 33, 1089-1101. Nguyen, L. T.; Suh, N. P. Effect of High Reynolds Number on the Degree of Mixing in RIM Processing. Polym. Process. Eng. 1985, 3(1,2), 37-56. Ottino, J. M. Description of Mixing with Diffusion and Reaction in terms of the Concept of Material Surfaces. J. Fluid Mech. 1982, 114, 83-103. Ranz, W. E. Analysis of Reaction Processes in Which Microscopic
315
Heterogeneities Appear: Scale-up and Scale-Down of Polymerization Reactions. Ind. Eng. Chem. Fundam. 1986,25, 561-565. Riley, J. J.; Metcalfe, R. W.; Orszag, S. A. Direct Numerical Simulations of Chemically Reacting Turbulent Mixing Layers. Phys. Fluids 1986, 29, 406-422. Sandell, D. J.; Macosko, C. W.; Ranz, W. E. Visualization Technique for Studying Impingement Mixing a t Representative Reynolds Numbers. Polym. Process. Eng. 1985, 3(1,2), 57-70. Tucker, C. L., 111; Suh, N. P. Mixing for Reaction Injection Molding. I. Impingement Mixing of Liquids. Polym. Eng. Sci. 1980, 20, 875-886. Tyagi, A.; Wood, P. E.; Hrymak, A. N. Fluid Mechanics of Jet Impingement Mixing. Presented at the 1987 AIChE Annual Meeting, New York, paper 34b. Wickert, P. D.; Ranz, W. E.; Macosko, C. W. Small-scale Mixing Phenomena During Reaction Injection-Molding. Polymer 1987, 28, 1105-1110. Received for review April 25, 1988 Accepted October 3, 1988
I
A Study of Hydrogen Bonding in Alcohol Solutions Using NMR Spectroscopy Anne M. Karachewski, Marianne M. McNiel,+and Charles A. Eckert" Department of Chemical Engineering, University of Illinois, Urbana, Illinois 61801
The thermodynamics of hydrogen-bonded systems are best represented by chemical or combined physical-chemical theories of solutions. Although this entails the disadvantage of a large number of parameters, this limitation can be overcome by independent measurement of physically meaningful parameters. Previously we have shown how NMR chemical shift data and limiting activity coefficient measurements could make such direct determination for highly solvated systems. Here we extend these methods to associated systems. Data and mathematical models are presented for alcoholhydrocarbon systems. The results demonstrate the use of NMR to predict thermodynamic properties such as VLE and enthalpies. Phase equilibria data are often needed for design purposes in the chemical, petrochemical, and pharmaceutical industries. Usually for complicated or highly nonideal systems, direct experimental measurements are performed over the entire composition range. Predictive methods for phase equilibria would be more useful than direct experimental measurements especially when quick estimates are needed. Prediction of phase equilibria for strongly complexing or hydrogen-bonding mixtures has remained a challenge. Most solution theories for correlating or predicting phase equilibria attempt to explain all solution nonidealities in terms of nonspecific physical intermolecular forces. Such solution theories are not very sensitive to the association models and because of their empirical nature do not allow cross-prediction of thermodynamic properties (Wilson, 1964; Renon and Prausnitz, 1968). Solution theories based on chemical theory have also been used to represent phase equilibria of hydrogen-bonding mixtures and have been more successful (Dolezalek, 1908; Kretschmer and Wiebe, 1954; Renon and Prausnitz, 1967). Physical effects can be included in chemical theory models but usually require one or more additional adjustable parameters (Chen and Bagley, 1978; Nath and Bender, 1983; Zong et al., 1984). The primary disadvantage of such chemical theory models is the large number of adjustable parameters that must
* To whom all correspondence should be addressed. Present address: Air Products and Chemicals, Inc., Allentown, PA 18195.
be obtained especially for systems exhibiting a large degree of association (Acree, 1984). If such parameters can be measured separately and directly, as by spectroscopic methods, the use of these models is much more attractive. Alcohol-hydrocarbon solutions have been chosen as a starting point in the study of associated systems. The large excess properties and immiscibility exhibited by alcohol mixtures is a direct consequence of hydrogen bonding. Hydrogen bonding gives rise to specific interactions between atoms or functional groups. Formation of hydrogen bonds modify many chemical and physical properties including partial molar volumes, viscosities, IR intensities, and NMR chemical shifts (Acree, 1984). Changes in these physical properties can be related to equilibrium constants for complex formation and subsequently used for prediction of phase behavior. In this work, nuclear magnetic resonance (NMR) is used to determine association equilibria for alcohol-hydrocarbon solutions and to relate experimental measurements to equilibrium constants for complex formation (McNiel, 1985, 1987; Eckert et al., 1986; Karachewski, 1988). Equilibrium constants are obtained by fitting chemical shift data to various chemical-physical association models. The use of NMR as a tool for the study of hydrogen bonding has become increasingly more common in recent years (Foster and Fyfe, 1965; Bruno et al., 1983). Formation of a hydrogen bond causes a large change in the shielding of a proton donor by lowering the effective electron density, causing a shift of the resonance signal to lower fields. For aromatic molecules, the signal is shifted
0888-5885/89/2628-0315$01.50/0 0 1989 American Chemical Society
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Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989
to higher fields due to a secondary magnetic field created by the induced ring currents of the n-electron orbital. Although various techniques can be used for studying hydrogen bonding (Coggeshall et al., 1951; Harris and Hobbs, 1954; Liddel and Becker, 1957; Bolles and Drago, 1965; Arnett et al., 1970; Lamberts, 1971; Mullens et al., 1985), NMR offers several distinct advantages. With most commercial NMR instruments, the chemical shift can be easily measured to 0.005 ppm. NMR is a much more sensitive test of an association model than phase equilibria because NMR measurements depend on the relative number and types of hydrogen bonds and not on activity. Fourier transformation has improved the sensitivity of NMR and has allowed the study of very dilute solutions, which is especially importznt for alcohol solutions (Becker, 1980). There are two primary disadvantages with using NMR. First, an association model must be chosen in order to fit the chemical shift data. Second, the time for a resonance measurement is much longer than the exchange rate for hydrogen bond formation, and thus separate peaks for the various hydrogen-bonded species are not observed. In this paper, four association models are tested on two alcohol-hydrocarbon mixtures: 2-propanol in cyclohexane and 1-butanol in cyclohexane. For both systems, chemical shift data are fit and phase behavior predicted. The model which best fits the experimental data and best predicts phase equilibria is then used to study solvent effects in mixtures of 2-propanol and 1-butanol in various solvents.
Excess Properties The equilibrium reactions for associating systems can be represented as follows: A+A=A,
+ A = A, A, + A = A,+i
A,
where i 1 1
where A is one alcohol molecule. Associated with each equilibria in the model is an equilibrium constant, Ki, a Gibbs energy change, Agi, a heat of formation, Ahi, and an entropy change of formation, Asi. These are all related by Agi = -RT In K i= Ahi - TAsi (2) The equilibrium constants in this study have been defined according to Flory's lattice model for polymer solutions in terms of volume fractions (Flory, 1944, 1953):
where u, is the molar volume of hydrogen-bonded species (volume/ mole). In order to calculate excess Gibbs energy and excess enthalpy, it is necessary to know the distribution of the hydrogen-bonded species, particularly the fraction of monomer as a function of composition at the desired temperature. For a given distribution of species, it is possible to define an average number of segments as follows: (4)
In Flory's lattice model (Flory, 1944),the entropy change on mixing can be calculated by filling a pseudolattice with
successive units of an n-mer and then filling the vacant sites with solvent molecules. In this treatment, the addition of cyclic species is equivalent to the addition of linear species. The Gibbs energy is then calculated by adding a regular solution theory expression representing the contribution of physical forces to the entropy upon mixing. Regular solution theory was used to represent the physical contributions because it does an adequate job of representing phase behavior and requires only one interaction parameter, but any physical model could be used. The reference state for the Gibbs energy equation is pure alcohol. Appropriate derivatives of the Gibbs energy expression result in expressions for activity coefficients,
where is the volume fraction of monomer in pure alcohol and f,* is the average number of segments per n-mer in pure alcohol. The excess Gibbs energy is related to the activity coefficients in the usual way, ge = RT(xaIn 7,+ x , In ys) (7) The equations for the excess enthalpy can be determined from the Gibbs-Helmholtz relationship, --he- - a(ge/RT) (8) RT2 aT The excess enthalpy has been evaluated numerically by approximating the derivative in eq 8 with a discreet slope at each mole fraction. Using eq 5-8 to calculate excess properties requires the use of a single physical interaction parameter, 0,which represents the interaction energy density of the solutesolvent pairs (Flory, 1944; Kretschmer and Wiebe, 1954):
P = (6' - 6J2 (9) It has been assumed that the solubility parameters of all complexes are equal and that the excess volume is negligible. The physical interaction parameter, p, could be made an additional model parameter in the chemical-physical models. In this treatment, however, ,L? was determined by using an independently measured thermodynamic property, infinite dilution activity coefficients by differential ebulliometry (Trampe, 1987a,b). Either limiting activity coefficient in each binary mixture could be used to calculate p:
Association Models Gutowsky and Saike (1953) recognized the value of NMR spectroscopy in thermodynamic studies of hydrogen bonding. They showed that the observed chemical shift (observed proton frequency relative to a reference compound, tetramethylsilane) is the weighted average of the chemical shifts of free and complexed protons. The weighting factors are simply the fraction of the total number of hydrogen-bonded protons present in a given state at equilibrium.
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 317 For alcohol-hydrocarbon systems, neglecting solvent effects, a general equation for the observed chemical shift, v, can be written as follows: v, =
(
;)vl
complex v,. The experimental chemical shift data were fit with eq 15, derived from eq 13 by using the above assumptions.
1
2Kaa
+
”=
where Cl is the concentration of monomer (moles/volume), C,, is the concentration of cyclic n-mer, Cnl is the concentration of linear n-mer, Ca is the total concentration of hydrogen-bonding species, v l is the chemical shift of the proton on a monomer (parts per million or hertz), v i is the chemical shift of the free proton on a n-mer, v, is the chemical shift of the bonded protons on a cyclic n-mer, and vnl is the chemical shift of the bonded protons on a linear n-mer. Assuming that the chemical shift of the free proton on an n-mer equals the chemical shift of a monomer and no volume change upon formation of a hydrogen bond, eq 12 can be rewritten as follows:
where A, = Anc
Y, - V I
= vnc - V I
An1 = Vnl
- VI
are volume fractions of cyclic and linear n-mers 9 , and is the total volume fraction of hydrogen-bonding and species. Assuming zero excess volume, a mass balance on the alcohol yields
+ 1 - (4K@,+ 1)’” 2K9,
A,
(15)
The remaining three models are based on spectroscopic evidence (Saunders and Hyne, 1958a,b; Tucker and Becker, 1973) that a trimer appears to be the first important hydrogen-bonded species. One model is the Linear Association with Cyclic Trimer (LACT) model. A second model is the Cyclic Association with Linear Trimer model (CALT), and the third model is the trimer model. The LACT model assumes that the first important hydrogen-bonding species is the cyclic trimer, with an equilibrium constant for hydrogen-bond formation, K3. All higher order species are linear, with a different equilibrium constant K. In fitting the experimental data, the LACT model uses three model parameters: K 3 , K , and v,. Equation 13 was used to derive an equation for the LACT model,
There could be steric hindrance associated with formation of a cyclic trimer. Therefore, the data were also fit to the CALT model. The CALT model is similar to the LACT model. The CALT model assumes that the first important hydrogen-bonded species is a linear trimer with an associated equilibrium constant, K3. All higher order species are cyclic with a different equilibrium constant, K. The three model parameters of the CALT model are K3, K,and v,. By use of eq 13, the equation for the CALT model is
m
Using these equations, one can derive an infinite number of models for the chemical shift. In all models, an equilibrium constant is associated with’each complex formed. To eliminate the large number of model parameters, simplifying assumptions are made which include (1) species in solutions (dimers, trimers, tetramers, etc.), ( 2 ) structure of the species (cyclic or linear), and (3) chemical shift of the hydrogen-bonded species, v,. In order to obtain the equilibrium constants for complex formation, the chemical shift data for two alcohol-hydrocarbon systems were fit to four models: CLAM, LACT, CALT, and trimer. A discussion of these association models follows. The Continuous Linear Association Model (CLAM) was derived originally from Flory’s lattice model for polymer solutions and used by Kretschmer and Wiebe (1954) and by Renon and Prausnitz (1967) to represent the behavior of alcohol-hydrocarbon solutions. In this model, linear associated species are present in solution. All protons have an equal probability of reacting irrespective of the size of the hydrogen-bonded species on which they reside. That is, the equilibrium constants, K,+l, for all reactions are equal to a single constant, K. These are the assumptions of the original CLAM model. In all models, the chemical shift of all hydyogen-bonded species are assumed equal in order to reduce the number of model parameters. The CLAM model for hydrogen bonding is very attractive because of its simplicity. There are two model parameters: the equilibrium constant K, and the chemical shift of the
The fourth model tested was a trimer model. In this model, all hydrogen-bonded species are assumed to be trimers, either cyclic or linear. The equilibrium constant for trimer formation is represented as K3. The trimer model has two model parameters, K3 and v,, and is represented by
The assumptions concerning the structure of hydrogen-bonded species (cyclic or linear) must be made for the various models in order to fit the NMR data and to obtain equilibrium constants and heats of formation for hydrogen bonding for the various species. These parameters are needed for predicting phase equilibria and for cross-predieting excess enthalpy.
Experimental Section Studies. Three studies were conducted. In the first study, the behavior of 2-propanol in cyclohexane and 1butanol in cyclohexane was investigated. The purpose of this study was to test a previous model for association (CLAM) and use NMR chemical shift data to develop improvements on this model (LACT, CALT, and trimer). In the second and third studies, the “solvent effect” on alcohol association was investigated. The purpose of the
318
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989
1
I
Mole Fraction 2-Propanol
0.696 0. IO1
N
5 1000
-t
5
BOO
d
4
600
I 0
5
I 4
I 3
2
0
Chemical S h i f t , p p m
0
400
ki m
200
0
Figure 1. NMR spectra for 2-propanol and cyclohexane a t 34 "C.
second study was to use NMR spectroscopy to investigate the association of 2-propanol in four inert solvents: heptane, methylcyclohexane, hexane, and methylcyclopentane. In the third study, the NMR data for 1-butanol and cyclohexane were used to predict phase equilibria for l-butanol in other solvents, in order to determine whether equilibrium constants obtained in one solvent could be used to predict solution behavior in different solvents. Materials, Purification, and Sample Preparation. Aldrich HPLC grade 2-propanol and 1-butanol were dried over Union Carbide 2A molecular sieves in a nitrogen-filled glovebox to remove trace amounts of water, with no further purification. Alcohol purity was 99.9+% with less than 0.01 70water. Aldrich HPLC grade cyclohexane and Aldrich 99.9+ 70 NMR grade tetramethylsilane (TMS) were used without purification. All samples were prepared inside a glovebox flushed with pure dry nitrogen. A positive pressure of nitrogen was maintained to prevent introduction of air inside the box. All samples contained alcohol, inert solvent, and TMS. Each sample contained 0.5 mol % TMS that was used as an internal reference for chemical shift measurements. The samples were prepared in vials and weighed inside the glovebox on a Mettler H72 analytical balance accurate to f O . l mg. Samples with alcohol mole fractions greater than 0.10 were then transferred to an inner tube of a coaxial NMR tube system (3.3 mm 0.d. by 7.5 in.) and sealed with a cap. For dilute samples with an alcohol mole fraction less than 0.10, the NMR coaxial system was not used. Instead, the samples were transferred to 5.0-mm tubes and sealed with a cap. All tubes were flame-sealed to ensure accurate compositions and prevent water contamination. In alcohol solutions, the chemical shift is a strong function of composition. Data must be taken on dilute samples in which the alcohol mole fraction is on the order of 1 x lo4 in order to obtain the correct alcohol monomer chemical shift. Commercial NMR instruments are designed to lock on a deuterium signal. The NMR coaxial tube system avoids using large quantities of expensive deuterated solvents. However, for dilute samples, there is the problem of dynamic range. The dynamic range limits the ratio of maximum to minimum signals that can be processed by the instrument's computer. Deuterium atoms do not appear in 'H NMR spectrum because they have different magnetic moments. Thus, for dilute samples, cyclohexane was replaced with D12 cyclohexane in order to avoid the problem of dynamic range. More detailed information regarding the sample preparation technique is found in McNiel (1985, 1987) and Karachewski (1988). Measurement of Chemical Shift. Experimental data were obtained on a Varian XL-200, 200-MHz FT-NMR (Varian, 1984) that performs automatic variable-temperature operations. Chemical shifts were measured as a
2-PROPANOL
MOLE FRACTION
Figure 2. LACT model fit of the observed chemical shift of the OH proton for 2-propanol-cyclohexane.
[ " ' " ' ' ' ' I -LACT --CLAM
t
s
MODEL MODEL
BOO
-J
4:
0
f 600 W
I 0 0
400
W
2 200 0
2-PROPANOL
MOLE FRACTION
Figure 3. CLAM and LACT model fits of the observed chemical shift of the OH proton in the dilute region for 2-propanol-cyclohexane.
function of composition and temperature. Figure 1 shows how the chemical shift of the OH proton in 2-propanol and cyclohexane varies with composition at 34 "C. The maximum uncertainty in chemical shift is 0.01 ppm. This uncertainty is a result of the broadening and splitting of the proton peak by spin-spin coupling with adjacent protons. The XL-200 was calibrated with a methanol sample for low-temperature experiments (0-40 "C) and with a glycol sample for high-temperature experiments (40-90 "C). The temperature can be calculated from the difference in frequency between CH, and OH groups of methanol and CH2 and OH groups of glycol using Van Geet's equation (Van Geet, 1968, 1970). The temperature accuracy is f 2 "C.
Results and Discussion Determination of K's. The NMR data for 2-propanol in cyclohexane and 1-butanol in cyclohexane were fit to all four models by using an IMSL nonlinear least-squares minimization routine that uses a finite difference Levenberg-Marquardt algorithm. Figure 2 illustrates the variation in chemical shift over the entire composition region for the 2-propanol-cyclohexane system a t three temperatures. This figure shows that at high alcohol concentrations the chemical shift of the OH proton relative to the chemical shift of TMS is large. This large shift is due to the lowered electron density around the nucleus of the proton upon hydrogen-bond formation. At low concentrations, the chemical shift is smaller due to a lower amount of hydrogen bonding. All four models for both alcohol systems showed a similar fit in chemical shift when the alcohol mole fraction was greater than 0.10.
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 319 Table I. Monomer Chemical Shift of 2-Propanol and 1-Butanol in Cwlohexane chemical shifts, Hz alcohol
CI
~~~~
2-propanol 1-butanol
34 "C 128.5 34 "C 114.7
61 "C 128.1 53 "C 113.5
3 1000
t 5
74 "C 128.2 75 "C 112.6
'33
800
600
I
Table 11. Model Parameters for the CLAM, LACT, CALT, and Trimer Models for 2-Propanol and 1-Butanol in Cyclohexane model alcohol 2'. "C K K, Am CLAM 1025 2-propanol 34 75 1011 61 24 993 74 15 1-butanol 1048 34 80 1046 53 32 1049 14 75 trimer 2-propanol 2500 937 34 441 873 61 169 848 74 2601 965 1-butanol 34 529 53 932 144 894 75 1444 1013 LACT 2-propanol 39 34 225 19 61 926 12 74 894 88 1225 1-butanol 38 34 995 21 53 995 289 11 81 951 75 CALT 2-propanol 43 34 3249 931 11 61 676 946 74 7 256 923 49 34 2601 943 1-butanol 20 53 676 931 11 74 169 880
Differences in the models arise when fitting the dilute data. Figure 3 shows the variation in chemical shift in the dilute alcohol region for the 2-propanol and cyclohexane system. This figure illustrates the fit of dilute data with the CLAM and LACT models. In fitting the dilute data to the CLAM model, the predicted values of the chemical shift were systematically higher. The three models which assume that the trimer rather than the dimer is the first important species in solution fit the data in the dilute region much better than the CLAM model. In the dilute region, the CLAM model has a first-order dependence of chemical shift on alcohol mole fraction, while the LACT, CALT, and trimer models have a second-order dependence. Thus, all models based on the assumption of a predominant trimer will give a similar fit in the dilute region but will differ at higher concentrations of alcohol. The monomer chemical shifts for 2-propanol and l-butanol are listed in Table I and show that the monomer chemical shift is not a function of temperature. Other researchers have reached a similar conclusion (Davis and Deb, 1970; Murthy and Rao, 1968). The difference in the monomer chemical shift of 1-butanol and 2-propanol is
0 400 W
L tt: m
200
0
2-PROPANOL
MOLE FRACTION
Figure 4. LACT model fit of the observed chemical shift of the OH proton for 2-propanol-methylcyclohexane.
attributed to differences in molecular structure, which causes a slight variation in the electron density around the proton, thereby causing a slightly different chemical shift (Davis and Deb, 1970). The experimental chemical shift values for the 2-propanol-cyclohexane and l-butanol-cyclohexane systems are listed as a function of composition and temperature in the supplementary material. Table I1 lists the adjustable parameters for the four models for the 2-propanol-cyclohexane and l-butanolcyclohexane systems. Assuming that the heat of formation for the hydrogen bond does not vary with T, Ahf and Ahn can be obtained from plots of In K versus 1/T and In K3 versus 1/T. All of the values, Ahf, Ahw, Asf, and Asf3, are calculated per hydrogen bond formed. Table I11 lists these values for both alcohol systems. Unusually large Ah values were obtained for the CLAM model, while more reasonable values were obtained for the other three models. Table I11 shows that the low heat of hydrogen-bond formation value for the cyclic trimer in the LACT model indicates that the cyclic trimer is likely to form. The Ahf is very large for cyclic species in the CALT model, and thus the formation of higher order cyclic species is unlikely. Chemical shift data for 2-propanol association were also obtained in heptane, methylcyclohexane, methylcyclopentane, and hexane. As shown below, the LACT model best represents solution behavior in alcohol-hydrocarbon mixtures, and thus the experimental data for this study were fit by using only the LACT model. Table IV lists the values for the model parameters for the four solvents. The experimental chemical shift data for the four solvents are listed in the supplementary material. The chemical shift data for all four solvents resulted in fits similar to that obtained when using cyclohexane as the solvent. Figure 4 shows the fit obtained for 2-propanol in methylcyclohexane. Table V lists the heats and entropies of formation for the four solvents. The values are comparable to those obtained for 2-propanol-cyclohexane. Infinite dilution activity coefficients are listed in Table VI.
Table 111. Gibbs Energy, Heat of Formation, and Entropy of Formation for the Various Complexes in Cyclohexane for the CLAM, LACT, CALT, and Trimer Models &f!
model CLAM trimer (cyclic) LACT CALT
alcohol 2-propanol 1-butanol 2-propanol 1-butanol 2-propanol 1-butanol 2-propanol 1-butanol
kcal/mol; 25 "C -2.8 -2.9 -2.5 -2.4 -2.4 -2.5
hgn, kcal/mol; 25 "C
-1.7 -1.7 -1.6 -1.4 -2.4 -2.3
Ah,
kcal/mol -8.6 -9.2 -6.1 -6.6 -9.7 -7.7
Ahn, kcal/mol -4.7 -5.0 -4.9 -4.7 -6.7 -7.2
cal/(mol K) -19 -21
Asn, cal/(mol K) -10
-12 -14 -24 -18
-11 -11 -1 1
-13 -16
320 Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989
I.. 1973
o NACATA
c
z wu G 2.0 LL
-I
0 0.6
I
V
0
Ya
LL
EL
0.4
t 1.0
0
z
0
3
F 0.2 0
a K LL 0.0 0.0
0.2
0.4
0.6
0.8
0.0 0.0
1.0
ALCOHOL VOLUME FRACTION
0.2
0.4
0.6
0.1
1.0
CYCLOHEXANE MOLE FRACTION
Figure 5. LACT model distribution of hydrogen-bonded species a t 25 "C. Table IV. LACT Model Parameters for the 2-Propanol-Inert Solvent Study solvent T."C K K, heptane 34 43 1838 48 26 491 61 18 194 hexane 34 41 1709 48 25 452 61 17 170 methylcyclohexane 34 40 1560 48 26 483 61 17 176 38 1457 methylcyclopentane 34 48 23 389 61 16 152
Figure 6. Trimer model prediction of activity coefficients for 2propanol-cyclohexane a t 60 "C. 3.0 I
A-
io13 974 932 1015 977 937 1016 976 936 1018 981 942
Discussion. Figure 5 shows a typical distribution of hydrogen-bonded species at 25 OC for the LACT model. This figure as well as others shows that there is little change for the distribution of hydrogen-bonded species above an alcohol volume fraction of 0.2. The primary changes occur in the dilute alcohol region. Fitting of experimental NMR data for the 2-propanolcyclohexane and 1-butanol-cyclohexane systems indicated that the models based on trimer formation fit dilute chemical shift data better than the models based on dimer formation. To further test the solution theories, partial pressures, activity coefficients, and excess enthalpy were predicted without any new model parameters. The gas phase was assumed ideal. The ge and he expressions for the CALT and LACT models do not differ from one another, and consequently the CALT model was not used in phase behavior predictions. In predictions of partial pressures and activity coefficients for the two alcohols in cyclohexane, LACT model predictions were the best, CLAM model predictions were intermediate, and the trimer model predictions were the least accurate. The trimer model overpredicted activity coefficients and partial pressures. These results indicate that higher order complexes must be included. For the trimer model, a large p value compensated for the small chemical contribution to T~~ or -yam. Figures 6-8 illustrate typical predictions of
Y&2?2%J 0.00.0
0.2
0.4
0.6
0.8
1.0
CYCLOHEXANE MOLE FRACTION
Figure 7. LACT model prediction of activity coefficients for 2propanol-cyclohexane a t 60 "C.
L I-
-=
:
0.00.0
0.2
0.4
0.6
0.B
1.0
CYCLOHEXANE MOLE FRACTION
Figure 8. LACT model prediction of activity coefficients for l-butanol-cyclohexane a t 50 "C.
activity coefficients for the trimer and LACT models. Excess enthalpy was predicted for both alcohol systems using the CLAM and LACT models. Typical excess enthalpy predictions are shown in Figures 9-11. LACT
Table V. Gibbs Energy, Heat of Formation, and Entropy of Formation for the 2-Propanol Complexes in Various Solvents for the LACT Model ~~~~~
&f
solvent heptane hexane methylcyclohexane methylcyclopentane
9
kcal/mol; 25 O C -2.4 -2.4 -2.4 -2.3
&a, kcallmol; 25 OC -1.6 -1.6 -1.7 -1.6
Aha kcal/mol -6.4 -7.0 -6.6 -6.6
Aha, kcal/mol -5.5 -5.8 -5.7 -5.7
b f ,
cal/(mol K) -13.4 -15.5 -14.0 -14.3
Asn, cal/(mol K) -13.0 -14.1 -13.5 -13.8
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 321 Table VI. Limiting Activity Coefficients for 2-Propanol and 1-Butanol in Various Inert Solvents system 2-propanol-cyclohexane
T,OC 25 35 45 60 25 35 60 25 35 45 50 25 35 100
2-propanol-methylcyclohexane 1-butanol-cyclohexane
1-butanol-toluene
7;
5.44 5.25 5.07 4.84 7.24 6.76 5.80 3.84 3.75 3.67 3.63 3.54 3.38 2.66
1
c -I
SMIRNOVA
fi
i
=I.,1 9 6 9
0
I
> v
BOO
>. -I L q
E z W
In
400
v1 W
0 W X
L 00.0
1
,
I
I
0.2
0.4
v
I
0.6
I
, - Y 1.0
0.8
1 -BUTANOL MOLE FRACTION
Figure 11. LACT model prediction of excess enthalpy for l-buta-
2ooo
7nol-cyclohexane. VESELY $
2 1600
i.,1 9 7 9 I
0
z
>
3'0 I-
aB
v
1200
44 I 2 W
G
LL W
2.0
0 0
800
>.
c
v)
W v1
g
2
G4 1.0
400
W
l
,
00.0
I
0.2
,
1
,
"
0.6
0.4
2-PROPANOL
'
0.8
'
Y
w --I
fi
tt
o NAGATA
B_t
"a
ALCOHOL lg7A
SOLVEN
1.0
MOLE FRACTION
Figure 9. CLAM model prediction of excess enthalpy for 2propanol-cyclohexane. 1600
9
7
0.0 0.0
0.2
0.4
0.6
0.8
1.0
METHYLCYCLOHEXANE MOLE FRACTION
Figure 12. LACT model prediction of activity coefficients for 2at 60 "C.
7propanol-methylcyclohexane
1
@., 1 9 7 9
VESELY
g 1200 L
1
> v
t 800
E
a v)
400 0
E
'
0.;
0 . ;
2-PROPANOL
'
O.b
O.b
?O
MOLE FRACTION
Figure 10. LACT model prediction of excess enthalpy for 2propanol-cyclohexane.
model predictions for 2-propanol-cyclohexane are similar to those obtained for 1-butanol-cyclohexane. The LACT model results are superior to the CLAM model predictions. The LACT model predicts correctly that excess enthalpy for 1-butanol-cyclohexane should be smaller than the excess enthalpy values for 2-propanol-cyclohexane. Results of earlier hydrogen-bond studies on gaseous alcohols postulated that the dominant hydrogen-bonded species could be the cyclic tetramer (Weltner and Pitzer, 1951). Thus, the experimental chemical shift data were also fit to a model which assumed the formation of cyclic tetramers. The fit of experimental chemical shift for the tetramer model was very similar to the fits obtained for the trimer models. However, prediction of phase equilibria showed that the trimer models better represented solution behavior.
Solvent Effect Studies. "Solvent effects" are changes in the chemical shift that are not due to hydrogen bonding. The function of the solvent is to control and vary the amount of complex formation. Solvents are chosen such that they are inert and do not participate in complex formation. The purpose of the 2-propanol study was to determine the extent to which the solvent affects the association of 2-propanol in methylcyclohexane, heptane, methylcyclopentane, and hexane. In the 2-propanol study, three experimental data points were used to obtain the LACT model parameters. The 2-propanol monomer chemical shift value was determined previously and assumed independent of solvent. The experimental data were fit to the LACT model only, since the results above indicate that this model best represents the alcohol association behavior. The absolute value of chemical shift varies slightly from solvent to solvent, with the greatest deviations occurring at low alcohol mole fractions and at high temperatures due to a lower degree of hydrogen bonding. The activity coefficient and excess enthalpy predictions in all four solvents agreed well with experimental data (see, for example, Figures 12 and 13). Although the activity coefficients depend most strongly on the chemical contributions, the excess enthalpy predictions are more sensitive to the value used for calculation of the physical interaction parameter, 0. /3 may be calculated from a single 7-determination but can vary by as much as 15070,depending on which limiting activity coefficient was used. The fi value was always greater when calculated using yam.The sensitivity of the (3 parameter may be attributed to the strong temperature dependence
322 Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 1600 h
5 n
1200
NAGATA
& 4..
I I I I I I I I I I I MRAZEK
1978
p 1200
5 >
h
1
BOO
1 -
;
et d.,1961
n
o
o
o
o 0
0
0
O
I
35°C 250c
0
0
0
400 0
E
8.0
2-PROPANOL
MOLE FRACTION
Figure 13. LACT model prediction of excess enthalpy for 2propanol-methylcyclohexane.
2'o L
z
0.2
0.4
0.6
0.8
1.0
1 -BUTANOL MOLE FRACTION
Figure 15. LACT model prediction of excess enthalpy for 1-butanol-toluene.
calculated using experimental ysmvalues. The results of the 1-butanol study indicate that the solvent is not completely inert and may influence the extent of hydrogen bonding. The 2-propanol study results confirm this fact since slightly different equilibrium constants are obtained for different solvents, resulting in different heats and entropies of complex formation. However, as the 2-propanol study shows, remarkably good predictions of activity coefficients and excess enthalpies can be achieved by measuring the proton chemical shift in the concentrated alcohol region for each solvent of interest at alcohol mole fractions of approximately 0.15,0.45, and 0.70 and then fitting these data to the LACT model. Application of the NMR Technique. Results of the alcohol-hydrocarbon studies indicate that NMR can be used successfully to predict solution properties including partial pressure, activity coefficients, and excess enthalpy. To obtain the equilibrium constants for complex formation and subsequently the heats and entropies of complex formation needed for the prediction of solution behavior, it is necessary to determine experimentally (1) the monomer chemical shift and (2) the OH chemical shift for a minimum of three alcohol mole fractions, for example, 0.15, 0.45, and 0.70. In data analysis, it can be assumed that the monomer chemical shift is independent of solvent. In the calculation of the heats and entropies of formation, experimental data should be obtained at a minimum of three temperatures, chosen such that the degree of hydrogen bonding changes significantly over reasonable composition ranges (Deranleau, 1969; Person, 1965). This will result in reliable equilibrium constants. The authors recommend using the LACT model to predict solution behavior and ysmto evaluate the physical interaction parameter, /3.
I 1 + 1 o GORBUNOV
4..
1968
w
0
LL W
0
ALCOHOL
A
'A
0.00.0
0.2
0.4
0.6
0.8
1.0
TOLUENE MOLE FRACTION
Figure 14. LACT model prediction of activity coefficients for 1butanol-toluene a t 100 "C.
of yamand the use of a pure alcohol reference state in the activity coefficient equations. In general, better phase behavior predictions result when ysmis used to calculate @, and there are two possible explanations. ysmis easier to measure experimentally and has a weaker temperature dependence. An alcohol is highly hydrogen bonded in its pure state but virtually unbonded when dilute in an inert solvent. Relative to its pure state, the alcohol is in a greatly different environment (the reason for such large values for Y ~ ~ On ) . the other hand, the inert solvent does not associate, so even dilute in alcohol, it is in a similar environment to its reference state (Thomas and Eckert, 1984). In the third study, phase behavior was predicted in a variety of 1-butanol solutions based only on experimental results from the 1-butanol-cyclohexane system. Solution behavior in other solvents could be predicted without gathering additional experimental data. The solvents tested were benzene, heptane, hexane, pentane, and toluene. For the various 1-butanol systems, the prediction of activity coefficients agreed with experimental results. Figure 14 illustrates the prediction for 1-butanol-toluene. The prediction of activity coefficients is a less stringent test of a solution theory than prediction of a derivative property, and excess enthalpy calculations do reveal a dependence on the inert solvent. In aromatic solvents, excess enthalpy was underpredicted, while in aliphatic hydrocarbons excess enthalpy was overpredicted. Figure 15 shows the prediction of excess enthalpy for 1-butanol-toluene. The best phase behavior predictions were obtained when the physical interaction parameter was
Conclusions Chemical-physical models have been successful in representing phase behavior of associating mixtures. In this work, NMR has been used to measure directly equilibrium constants for complex formation. Chemical shift measurements can be routinely measured on commercial NMR instruments. The OH chemical shift has been measured as a function of sample composition and temperature for various alcohol mixtures. The chemical shift data were fit to various chemical-physical models, with the equilibrium constants for complex formation as the adjustable parameters. Analysis of dilute NMR data suggests that the trimer (not dimer) is the dominant species a t dilute concentrations. The results of further data analysis and phase be-
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 323 havior predictions for the alcohol mixtures show that the LACT model best represents association in alcohol-hydrocarbon mixtures. Better predictions of activity coefficients and excess enthalpy are obtained when the physical interaction parameter is calculated using Y~~ rather than yam.
The results of the solvent effect studies indicate that the
OH proton chemical shift is a function of solvent, but solvent effects can be minimized by obtaining a few experimental data in each solvent of interest. Using limited experimental data for 2-propanol association resulted in remarkably good predictions of phase equilibria. The results of the 1-butanol study underline the benefits achieved with minimal experimental data. Acknowledgment The authors gratefully acknowledge financial support from E. I. du Pont de Nemours & Co. Additional support was also received from the Advanced Environmental Control Technology Research Center, supported under cooperative agreement CR 806819 with the Environmental Protection Agency. Nomenclature C = concentration g = Gibbs energy h = enthalpy K = equilibrium constant n = n-mer R = ideal gas constant s = entropy T = temperature V = volume x = apparent mole fraction Greek L e t t e r s 3( = physical interaction
parameter
y = activity coefficient d = solubility parameter A = change u
= chemical shift
@ = volume fraction Subscripts a = alcohol nc = cyclic n-mer
nl = linear n-mer o = observed s = solvent 1 = alcohol monomer Superscripts e = excess
= infinite dilution Registry No. 2-Propanol, 67-63-0; 1-butanol, 71-36-3; cyclohexane, 110-82-7; methylcyclohexane, 108-87-2; toluene, 10849-3. m
Supplementary Material Available: Experimental chemical shift data for 2-propanol-cyclohexane, 1-butanolcyclohexane, 2-propanol-heptane, 2-propanol-methylcyclopentane, 2-propanol-methylcyclohexane, and 2-propanolhexane systems as a function of temperature and composition and equations for calculating the distribution of species for the CLAM, LACT, CALT, and trimer models (6 pages). Ordering information is given on any current masthead page. Literature Cited Acree, W. E. Thermodynamic Properties of Nonelectrolyte Solutions; Academic Press: Orlando, FL, 1984; p 144. Arnett, E. M.; Jorris, L.; Mitchell, E.; Murty, T. S.; Gorrie, T. M.; Von R. Schleyer, P. Studies of Hydrogen-Bonded Complex For-
mation. 111. Thermodynamics of Complexing by Infrared Spectroscopy and Calorimetry. J. Am. Chem. SOC. 1970, 92, 2365. Becker, E. D. High Resolution NMR; Academic Press: New York, 1980; p 37. Bolles, T. F.; Drago, R. S. A Calorimetric Procedure for Determining Free Energies, Enthalpies and Entropies for the Formation of 1965,87,5015. Acid-Base Adducts. J . Am. Chem. SOC. Bruno, T. J.; Martire, D. E.; Harbison, M. W. P.; Nikolic, A.; Hammer, C. F. GasTLiquid Chromatographic and Nuclear Magnetic Resonance Study of Haloform + Di-n-alkyl Ether n-Alkane Mixtures at 30 "C. J. Phys. Chem. 1983,87, 2430. Chen, S.; Bagley, E. B. Thermodynamics of Associated Solutions, Part I and 11. Chem. Eng. SOC. 1978, 33(153), 161. Coggeshall, Norman D.; Saier, Eleanor L. Infrared Absorption Study of Hydrogen Bonding Equilibria. J. Am. Chem. SOC. 1951, 73, 5414. Davis, J. C.; Deb, K. K. Analysis of Hydrogen Bonding and Related Association Equilibria by Nuclear Magnetic Resonance. Adu. Magn. Reson. 1970, 4, 201. Deranleau, D. A. Theory of the Measurement of Weak Molecular 1969, Complexes. I. General Considerations. J. Am. Chem. SOC. 91, 4044. Dolezalek, F. Zur Theorie der binaren Gemische und Konzentrierten Losungen. 2.Phys. Chem. 1908, 64, 727. Eckert, C. A.; McNiel, M. M.; Scott, B. A.; Halas, L. A. NMR Measurements of Chemical Theory Equilibrium Constants for Hydrogen-Bonded Solutions. AIChE J. 1986, 32, 820. Flory, P. J. Thermodynamics of Heterogeneous Polymers and Their Solutions. J. Chem. Phys. 1944, 12, 425. Flory, P. J. Principles of Polymer Chemistry; Cornel1 University Press: Ithaca, NY, 1953; p 495. Foster, R.; Fyfe, C. A. Interaction of Electron Acceptors with Bases: Part 15-Determination of Association Constants of Organic Charge-Transfer Complexes by N.M.R. Spectroscopy. Trans. Faraday SOC. 1965, 61, 1626. Gorbunov, A. N.; Susarev, M. P.; Balashova, I. M. Liquid-Vapor Equilibrium in the System Isobutyl Acetate-n-Butyl AlcoholToluene. Zh. Prikl. Khim. 1968, 41, 312. Gutowsky, H. A.; Saika, A. Dissociation, Chemical Exchange and the Proton Magnetic Resonance in Some Aqueous Electrolytes. J . Chem. Phys. 1953,21, 1688. Harris, James T.; Hobbs, Marcus E. A Study of the Association of Some Organic Acids by Infrared Absorption Measurements. J. Am. Chem. SOC.1954, 76, 1419. Karachewski, A. M. A Study of Hydrogen Bonding in Liquid Mixtures Using NMR Spectroscopy. M.S. Thesis, University of Illinois, Urbana, 1988. Kretschmer, C. B.; Wiebe, R. J. Thermodynamics of Alcohol-Hydrocarbon Mixtures. J. Chem. Phys. 1954, 22, 1697. Lamberts, L. Calorimetric Studies of Molecular Complexes in Solution. Znd. Chim. Belge. 1971, 36, 347. Liddel, Urner: Becker, Edwin D. Infrared Spectroscopic Studies of Hydrogen Bonding in Methanol, Ethanol and &Butanol. Spectrochim. Acta 1957, 10, 70. McNiel, Marianne Marie NMR Measurements of Equilibrium Constants Used in the Chemical Theory of Solutions. M.S. Thesis, University of Illinois, Urbana, 1985. McNiel, Marianne Marie A Study of Hydrogen-Bonded Solutions with NMR Spectroscopy. Ph.D. Thesis, University of Illinois, Urbana, 1987. Mrazek, R. V.; Van Ness, H. C. Heats of Mixing: Alcohol-Aromatic Binary Systems at 25 "C, 35 "C, and 45 "C. AIChE J. 1961, 7, 190-195. Murthy, A. S. N.; Rao, C. N. R. Spectroscopic Studies of the Hydrogen Bond. Appl. Spectrosc. Rev. 1968, 2(1), 69. Mullens, J.; Yperman, J.; Francois, J. P.; Poucke, L. C. Simultaneous Calorimetric Determination of Equilibrium Constant and Enthalpy Change of Hydrogen-Bond Complexes in Dilute Solutions of Phenol with Pyridine in Carbon Tetrachloride. J.Phys. Chem. 1985,89, 2937. Nagata, I.; Asano, H.; Fujiwara, K. Excess Enthalpies for Systems of 2-Propanol-BenzeneMethylcyclohexane. Fluid Phase Equilib. 1978, 1, 211-217. Nagata, I.; Tatsuhiko, 0.;Yoshio, U. Excess Gibbs Free Energies for Binary Systems Isopropanol with Benzene, Cyclohexane and Methylcyclohexane. J. Chem. Eng. Data 1973, 18, 54. Nath, A.; Bender, E. On the Thermodynamics of Associated Solutions, I. An Analytical Method for Determining the Enthalpy and Entropy of Association and Equilibrium Constants of Pure Liquid Substances. Fluid Phase Equilib. 1983, 7, 275.
+
Ind. Eng. Chem. Res. 1989,28, 324-328
324
Person, W. B. A Criteria for Reliability of Formation Constants of 1965, 87, 167. Weak Complexes. J. Am. Chem. SOC. Renon, H.; Prausnitz, J. M. On the Thermodynamics of AlcoholHydrocarbon Solutions. Chem. Eng. Sci. 1967, 22, 299. Renon, H.; Prausnitz, J. M. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE. J. 1968,14, 135. Saunders, M.; Hyne, J. B. Trimer Association of &Butanol by NMR. J. Chem. Phys. 1958a, 29, 253. Saunders, M.; Hyne, J. B. Study of Hydrogen Bonding in Systems of Hydroxylic Compounds in Carbon Tetrachloride through the Use of NMR. J . Chem. Phys. 1958b, 29, 1319. Smirnova, N. A.; Kurtynina, L. M. Thermodynamic Functions of Mixing for a Number of Binary Alcohol-Hydrocarbon Solutions. Zh. Fiz. Khim. 1969, 43, 1883. Thomas, E. R.; Eckert, C. A. Prediction of Limiting Activity Coefficients by a Modified Separation of Cohesive Energy Density Model and UNIFAC. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 194. Trampe, David M. Measurement and Applications of Limiting Activity Coefficients and Liquid-Liquid Equilibria. M.S. Thesis, University of Illinois, Urbana, 1987a. Trampe, David M. University of Illinois, personal communication, 1987b. Tucker, E. E.; Becker, E. D. Alcohol Association Studies. 11. Vapor Pressure, 220-MHz Proton Magnetic Resonance, and Infrared Investigations of tert-Butyl Alcohol Association in Hexadecane. J . Phys. Chem. 1973, 77, 1783.
Van Geet, A. L. Calibration of the Methanol and Glycol Nuclear Magnetic Resonance Thermometers with a Static Thermistor Probe. Anal. Chem. 1968,40,40. Van Geet, A. L. Calibration of Methanol Nuclear Magnetic Resonance Thermometer at Low Temperature. Anal. Chem. 1970,42, 679. Varian Associates XL-Series NMR Superconducting Spectrometer Systems Basic Operation Manual, 1984. Vesely, Frantisek; Uchytil, P.; Zabransky, M.; Pick, J. Heats of Mixing of Cyclohexane with 1-Propanol and 2-Propanol. Collect. Czech. Chem. Commun. 1979,44, 2869-2881. Vonka, P.; Svoboda, V.; Strubl, K.; Holub, R. Liquid-Vapor Equilibrium. System Cyclohexane-1-Butanol a t 50 "C and 70 "C. Collect. Czech. Chem. Commun. 1971, 36, 18. Weltner, William; Pitzer, Kenneth Methyl Alcohol: The Entropy, Heat Capacity and Polymerization Equilibria in the Vapor, and 1951, Potential Barrier to Internal Rotation. J. Am. Chem. SOC. 73, 2606. Wilson, G. M. Vapor-Liquid Equilibrium. XI. A New Expression for the Excess Free Energy of Mixing. J. Am. Chem. SOC.1964, 86, 127. Zong, Z.; Yang, X.; Zheng, X. Correlation of Vapor-Liquid Equilibria of Associated Solutions. J . Chem. Eng. Jpn. 1984, 17, 71.
Received for review May 24, 1988 Accepted September 26, 1988
A Simple Method for Evaluating the Wilson Constants Alexander Apelblat and Jaime Wisniak* Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel
A simple method is proposed to determine the Wilson constants from binary vapor-liquid equilibrium data. The method is based on a polynomial description of the variation of G E/RTwith composition and finding the maximum value of the curve. The constants are determined by using a hand scientific calculator or a personal computer. Prediction of vapor compositions is as good or better than that obtained by complex optimization techniques. Many equations have been proposed to describe the vapor-liquid equilibrium relationships. For homogeneous systems, the two-parameter Wilson equation (Wilson, 1964) has been shown to be very suitable. In a multicomponent system, it reads
derived from the pertinent binary systems. For a binary system, the activity coefficients are In y1 =
In
where
y1 =
UjL
exp[-(Xij - Xii)/RT]
Ai, =
(2)
ui
To a first approximation the energy terms (Xi,- Xii) are assumed to be independent of temperature, although it is claimed (Nagata and Yamada, 1973) that a better fit is obtained if a polynomial dependency is assumed: Xij
-
Xii
=u
+ bT + cT' + ...
(3)
Obviously application of eq 3 carries the penalty of a larger number of constants. The Wilson equations are attractive because they have a built-in effect of temperature, lacking in previous models, and permit the calculation of multicomponent systems from a combination of parameters A,, and AIi which are
* To whom correspondence should
be addressed.
088S-5885/89/2628-0324$01.50/0
Extensive tables are available (Hudson and Van Winkle, 1970; Hirata et al., 1976; Gmehling and Onken, 1977), reporting the values of either the energy parameters (A,, - A,,) or the constants A, and A,[. The most difficult problem in using Wilson's equations is how to determine the two parameters A12 and Azl from a set of data ( y , , ~ , ) .Equations 4 and 5 are a pair of transcendental equations that can only be solved numerically. Czelej (1987) has developed a mathematical procedure to transform the Wilson equations into a polynomial form that allows an easier determination of the constants. Due to some serious mathematical errors present in Czelej's equations, their use is not recommended. Several methods have been suggested for determining the optimum constants AIz and Azl. According to Hirata et al. (1976), the results of the methods depend upon the 0 1989 American Chemical Society
