11239
J. Phys. Chem. 1995, 99, 11239- 11247
Correlation of Partial Molar Heats of Transfer at Infinite Dilution by a Linear Solvation Energy Relationship Steven R. Sherman, David Suleiman, Mitchell J. Hait, Martin Schiller, Charles L. Liotta, and Charles A. Eckert* Schools of Chemical Engineering and Chemistry and the Specialty Separations Center, Georgia Institute of Technology, Atlanta, Georgia 30332-0100
Jianjun Li and Peter W. Carr Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455
Russell B. Poe and Sarah C. Rutan Department of Chemistry, Virginia Commonwealth University, Richmond, Virginia 23284 Received: February 20, 1995; In Final Form: April 17, 1995@
Linear solvation energy relationships and solvatochromic parameters have been used to develop an improved relationship for the partial molar heats of transfer of an infinitely dilute solute in nonionic solutions which appears to be accurate to f 4 % for the database tested. This result has been coupled with a thermodynamic - in order to develop a corresponding correlation for the partial molar excess enthalpy at infinite dilution, cycle hE". -
However, because this result represents the difference of large numbers, the accuracy of prediction of
hE" is only accurate to about f28%. The correlation presented provides the insight for understanding, on a chemical and physical basis, the temperature effect on the phase equilibria of dilute solutions.
Introduction The study of the temperature-dependent phase equilibria of dilute solutions has been of interest to thermodynamicists over the last several decades. One way of studying this behavior is through the -use of partial molar excess enthalpies at infinite dilution, hE". Recently a sizable body of accurate data nonionic solutions has begun to appear in the literature.' Such results are complimentary to limiting activity coefficients, y"; RT In y" is a measure of the excess Gibbs free -energy for solute-solvent interactions in the mixture,and hE" represents the corresponding enthalpy. Physically, hE" is the enthalpy, on a molar basis, associated with removing a single solute molecule from a reference state of pure solute and placing it in a solvent at infinite dilution. The experimental methods for obtaining such data are reviewed by Fuchs and Stephenson et al.,la-IJ and by Trampe and Eckert.Ik In addition to experimental measurements, hE" can also be determined indirectly from the thermodynamic cycle depicted in Figure 1. The thermodynamic cycle is used to separate hE" into two separately-determined components. The first component is the heat of vaporization, Ahv, defined as the energy required, on a molar basis, to vaporize the liquid, an endothermic process. These values are available in the literature and are easily measured or estimated.The second component is the partial molar heat of solvation, hs, defined as the energy required, on a molar basis, to solvate gas phase solute molecules into a liquid solvent at infinite dilution. These values are not widely available. For most systems, & is an exothermic process. Thermodynamically, hE" can be calculated by summing Ahv and & for the given solute. ~~
~
* Author to whom correspondence should be addressed. @Abstractpublished in Advance ACS Abstracts, June 15, 1995.
TiE," Solute
Solvent
Figure 1. Thermodynamic cycle to model hE". hEm is described as the difference between two steps, the vaporization of a single solute molecule (Ah,), and the solvation of the gas phase molecule in a given solvent (hs).
One important application of hE" is for the extrapolation of limiting activity coefficients to other temperatures, through the exact relationship.
In principle, one could use eq 1 to obtain values of hE" from the measured variations of y" values with temperature. In practice, although hundreds of values of y" exist over a temperature range, the experimental methods used to measure y" allow data to be collected only over a narrow temperature range, and differentiation - of these data invariably leads to large uncertainties in hE". A number of authors have assumed some temperature dependence in the prediction or correlation of ym.2 Because of the limited number of y"(T) data and their corresponding limited temperature range, these in general have given a good representation within the experimental uncertainty of the y" data (generally 10-20%). However, as we shall discuss below, this
'
0022-365419512099-11239$09.00/0 0 1995 American Chemical Society
Sherman et al.
11240 J. Phys. Chem., Vol. 99, No. 28, 1995
does not ensure good values of hE"; the y" data available are simply insufficient to test the models. In fact, it has been shown that attempts to obtain excess enthalpies from excess Gibbs energy data often give inaccurate result^.^ This loss in precision is due to the use ofan integral property (y") to predict a derivative property (hE"). It is always statistically superior to measure directly a derivative property rather than to obtain it by differentiating other data, as long as the precision of the two experimental techniques are comparable. The many practical advantages of using y" have already been well demonstrated. Entire binary vapor-liquid equilibrium (VLE) curves can be constructed from two y" data points, or multicomponent vLE4 can be predicted from ymdata only. Also good estimation techniques for this property are available (ASOG,5 UNIFAC,2a-2emodified UNIFAC,2f-2h MOSCED,6 SPACE7) and excellent measurement techniques have been developed to specifically determine ym.* It is the goal of this work to compile - an improved correlative and predictive framework for hE" which will provide a more effective understanding of the temperature effect on the phase equilibria of dilute solutions.
Linear Solvation Energy Relationships Linear solvation energy relationships (LSERs), such as those described by Kamlet and Taftg were developed to explore a variety of solvent effects and have since been extended to characterize solute effects. They provide a very useful method for the correlation and prediction of solution behavior and especially for the description of configurational properties (properties relative to the ideal gas at the same temperature). These have been reviewed recently by Carr.Io It is now known that very many solvent-dependent measurable properties of a dilute solution (e.g., reaction rate constants, equilibrium constants, and chromatographic capacity factors) can be expressed as linear expressions of the following form:
XYZ = XYZ,,
+ cavity formation term + C(solute-solvent interactions) (2)
where XYZo is an intercept, the cavity formation term accounts for the energy needed to form a space in the solvent to accommodate the solute molecule, and the last term accounts for the polar forces and the hydrogen bond forces between unlike molecules. The parameters used to scale the various solvent properties and their related intermolecular interactions x*,a, and /3 are now well established for many hundreds of liquid compounds," and this method has been widely applied to many applications in the chemical literature, including, but not limited to, predictions of dipole moments, fluorescence lifetimes, reaction rates, NMR shifts, gas-liquid partition coefficients, solubilities in water and blood, and biological toxicities.I0-l2 Approaches based on these scales not only have the advantage of separating the contributions from various types of forces but also have the support of many thousands of good data values for solute and solvent (even supercritical fluid solvent) properties. While the original parameter scales were set up to describe pure bulk species, i.e. solvents, it has now become apparent that molecules may behave very differently at infinite dilution. For example, an associating fluid is highly structured as a solvent, but a lone molecule of solute at infinite dilution experiences quite a different environment and consequently has different properties. Currently, different scales for the param-
eters n*2,a 2 , and ,& which represent the corresponding solute properties are being developed for many species. Fuchs and Stephenson et al.,ia-'J attempted to predict &, which they called the heat of solvation, using an LSER formulation. They modeled the & by using LSERs to describe the difference in & of the actual solute versus that of a model compound. They defined their model compound as the corresponding alkane or nonfunctional aromatic having a chemical backbone similar to the solute molecule. For example, the model compound for pentanol was pentane; for nitrobenzene it was toluene. They then correlated the difference of the & between the solute and its model compound in the same solvent using an LSER. They used a separate LSER for each solute and related the difference in % between the particular solute and the model compound to each solute's solvent (Z*KT, a K T , and BKT)solvatochromic parameter^.^ Typical errors for their prediction of the difference in & were h0.29-0.37 kcaYmol.'esli In order to predict & for a particular compound, one also needs the value of & for the corresponding model compound. Thus a separate experimental measurement must be made, and any experimental error adds to the uncertainty of the prediction. Alternatively, Fuchs and Stephenson'g reported a method of prediction of % based on a LSER and a group contribution method for alkanes, which has a mean error of fit of 10.35 kcaYmo1. In predictive mode, this method gives errors on the order of 10.5 kcal/mol or greater. Although our ultimate goal is to increase our understanding of the temperature effect on -the phase equilibria of dilute solutions throughthe use of hE" information, this investigation did -not correlate hE" but rather partial molar heats of transfer, hm (see Figure l), which is defined as a matter of convenience to be equal to -&. The motivationfor studying and developing LSERsfor G rather than for hE" rests upon the fact that although hE" and G are similar configurational properties, ~ T R is more easily represented by an LSER because it correlates only one step in thethermodynamic cycle (Figure 1) as the LSER in its opposed to two steps for hE". As a result, present form was better able to correlate hE" data indirectly through the use of G data. Although we use a similar thermodynamic cycle and approach in this work as employed by that of Fuchs and Stephenson et al.,Ia-lJ our method differs in four respects. First, we correlate ~ T R directly instead of the difference in & with respect to a model compound. Second, we now have solute scales available for the solvatochromic parameters, which should give a superior and more realistic representation of dilute solution properties. Third, there are simply more reliable experimental data now available. The resulting method is as accurate (the mean error of fit for the chosen data set is 10.35 kcal/mol) but is based on a more sound physical-chemical representation of the intermolecular interactions, so that it offers the framework for improvement, with the addition of more data and better parameter scales. Fourth, we include in the regression a new explanatory variable (LI6, see description below) which has proven to be very successful in rationalizing the gas to condensed phase free energy of transfer obtained in gas chromatography. I 3a,13b,13r
Implementation According to Figure 1, hE" is equal -to the Ahv of the pure solute minus the G. However, the hE" calculated this way is the difference of two large quantities with similar magnitudes.
Partial Molar Heats of Transfer at Infinite Dilution
J. Phys. Chem., Vol. 99, No. 28, 1995 11241
As a result, the relative error associated with the hE“ is much greater than the relative error for either the G or the Ahv due to propagation of errors. In order to predict G, an LSER (eq 3) was used for each solvent.
h,, = 1 log L16
+ sjt? + dd2 + uai + b& + intercept
(3)
4,
The LI6, .7t*c2, d2, and /3i are solute properties that will be described below. The solvent coefficients I, s, d, a, b, and the intercept were determined using a multiple linear regression14 against the solute properties. The values usedin this study were calculated from a total of 473 experimental hE” values in 18 solvents. The data and the references are provided in the supplementary material. The selection of the solvents was based upon the availability of published hE” data. These solvents and their relevant physical properties are listed in Table 1A. The 56 solutes used in this study are listed in Table 1B along with their corresponding solute solvatochromic parameters. Although 18 solvents and 56 solutes were examined, data were not available for every solute and every solvent.
(4) The LI6 parameter used in eq 3 is a free-energy parameter and represents the partition coefficient for a solute between the gas phase and liquid hexadecane at 25 OC. The logarithm of LI6 is related to y; as shown by eq 4, where R is the gas constant, T i s the temperature, PFpis the solute vapor pressure at temperature T, and VC16 is the molar volume of hexadecane at temperature T. The value of yy of a solute in general is a measure of the differential dispersion, dipolarity, hydrogenbonding, and cavity formation energies in a solvent and in the pure solute. In hexadecane, however, the contribution of dipole-dipole, and hydrogen-bonding are minimal. As a result, loglo LI6 can be used to characterize the dispersion and cavity formation interactions. The product I log LI6 is very important and is usually the dominant term in the LSER (Table 7). This term simultaneously represents the endothermic cavity formation and the exothermic dispersion (London) solute-solvent interactions. The next four terms ( 7 ~ * ~ 282, , and pi) in eq 3 are solute characteristic solvatochromicparameters. These solvatochromic parameters are free-energy based since they are derived from chromatographic retention measurements. The .7t*c2 parameter represents the dipolarity and polarizability of the solute. The product s 7 ~ * ~ is 2 a measure of the dipolar and induction interactions between the solute and the solvent. An empirical dipolarity and polarizability correction term, 82, is commonly used in conjunction with 7 ~ to~compensate ~ 2 for the fact that the .7t*5parameter does not account for the fact that interactions based on the polarizability and dipolarity of a solute cannot be described by a single linear term. For aliphatic compounds, 8 2 is equal to zero. For polychlorinated aliphatic compounds, 82 is equal to 0.5, and for any aromatic compounds, 8 2 is equal to 1. Therefore, the product d82 represents a sometimes-necessary correction term for the s 7 ~ * ~term. 2 The parameters 4 and 6 represent the hydrogen-bonding acidity and basicity of the solute, respectively. The product u represents the interaction between the solute as a hydrogen bond donor and the solvent as a hydrogen bond acceptor. Analogously, the product b& represents the interactions be-
4,
TABLE 1: Solvents and Solutes and Their Relevant Properties A. Solventsa
solvent cyclohexane heptane dibutyl ether diethyl ether ethyl acetate carbon tetrachloride dichloromethane benzene mesitylene toluene methanol 1-butanol 1-octanol acetonitrile dimethyl sulfoxide NA-dimethylformamide nitromethane triethylamine
J&
0.00 -0.02 0.24 0.27 0.55 0.28 0.82 0.59 0.47 0.55 0.60 0.47 0.40 0.75 1.00 0.88 0.85 0.14
BH,
aKT
PKT
0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.46 , 0.47 0.45 0.00 0.00 0.10 0.13 0.11 0.62 0.88 0.45 0.31 0.76 0.69 0.25 0.71
0.32
0.00 0.00 0.00 0.93 0.79 0.77 0.19
0.00 0.00 0.22
0.00
(ca~cm3)’/* 8.20 7.43 7.76 7.4 9.1 8.55 9.88 9.17 8.80 8.91 14.5 11.60 10.3 12.11 12.0 12.1 12.7 7.42
B. Solutes solute
logL16
n;
acetone acetonitrile acetophenone aniline anisole benzaldehyde benzene 2-butanone 1-butanol butyronitrile carbon tetrachloride chlorobenzene 1-chlorobutane chloroform 1-chloropropane m-cresol cyclohexane cyclohexanone decane dichloromethane diethyl ether ethanol ethyl acetate heptane hexadecane hexane 1-heptene 1-hexene methanol methyl acetate 2-methylpentane nitrobenzene nitroethane nitromethane nitropropane nonane 1-nonene octane 1-octanol pentane 1-pentanol 2-pentanone 1-pentene phenol 1-propanol propionaldehyde propionitrile pyridine tetrahydrofuran toluene triethylamine p-xylene
1.766 1.537 4.458 3.934 3.916 3.935 2.792 2.269 2.539 2.540 2.822 3.630 2.716 2.478 2.223 4.329 2.906 3.580 4.685 1.997 2.066 1.462 2.359 3.173 7.714 2.668 3.063 3.063 0.916 1.946 2.507 4.433 2.313 1.839 2.773 4.176 4.000 3.677 4.619 2.163 3.057 2.726 2.013 3.641 1.975 1.770 1.978 2.969 2.521 3.343 3.008 3.867
0.38 0.62 0.80 0.76 0.52 0.75 0.29 0.39 0.30 0.57 0.16 0.44 0.19 0.27 0.22 0.78
JC*KT, a K T ,
0.00 0.59 -0.11 0.34 0.03 0.27 0.31 -0.14 -0.05 -0.16 -0.05 -0.07 0.35 0.33 -0.14 0.91 0.66 0.67 0.65 -0.12 -0.07 -0.12 0.37 -0.18 0.32 0.40 -0.03 0.77 0.32 0.35 0.64 0.60 0.27 0.29 0.02 0.28
bl 6 dhv, kcaVmol 0.0 0.01 0.52 7.48 0.0 0.05 0.37 7.87 1.0 0.00 0.49 12.76 1.0 1.0 1.0 1.0
0.0 0.0 0.0 0.5 1.0
0.0 0.5
0.0 1.0 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 0.0 1.0 0.0 1.0
KT were obtained from ref
0.20 0.42 0.00 0.22 0.00 0.42 0.00 0.10 0.00 0.48 0.31 0.52 0.00 0.41 0.00 0.04 0.00 0.09 0.00 0.08 0.16 0.04 0.00 0.08 0.66 0.24 0.00 0.00 0.00 0.56
0.00 0.00 0.06 0.06
0.00 0.40 0.29 0.52
0.00 0.49 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.00 0.02 0.35 0.52 0.00 0.47 0.00 0.00 0.00 0.21 0.00 0.17 0.06 0.16 0.00 0.18 0.00 0.00 0.00 0.02
0.00 0.00 0.35 0.00 0.32 0.00 0.00 0.69 0.32 0.00 0.00
0.00 0.00 0.00 0.00 0.00 9.
0.51 0.00 0.52 0.48 0.02 0.23 0.52 0.37 0.41 0.90 0.61 0.11 0.64 0.12
13.35 11.20 12.00 8.09 8.25 12.51 9.40 7.75 9.79 8.01 7.97 6.81 14.75 7.86 10.73 12.28 6.83 6.5 1 10.11 8.51 8.74 19.45 7.54 8.52 7.32 8.95 7.72 7.14 13.15 9.94 9.15 10.37 11.10 10.88 9.92 16.96 6.32 13.61 8.25 6.09 13.82 11.31 7.08 8.61 9.66 7.65 9.08 8.32 10.13
11242 J. Phys. Chem., Vol. 99, No. 28, 1995 TABLE 2: LSER Coefficients and Errors in Fitting A h n solvent intercept 1 2.131f0.037 cyclohexane 1.71f0.19 1.22 f 0.16 2.32 f 0.26 heptane dibutyl ether 1.79 f 0.18 2.219 f 0.030 diethyl ether 1.44 f 0.35 2.294 f 0.070 1.707 0.045 ethyl acetate 2.50 f 0.27 carbon tetrachloride 2.35 f 0.24 2.020 f 0.041 dichloromethane 1.61 f 0.34 1.852 f 0.065 benzene 2.13 f 0.30 1.901 f 0.060 mesitylene 1.73 f 0.16 2.207 f 0.027 toluene 2.14 f 0.21 2.023 f 0.035 methanol 2.49 f 0.44 1.657 f 0.085 1-butanol 0.39 f 0.99" 2.15 f 0.15 1-octanol 0.48 f 0.54" 2.375 f 0.075 acetonitrile 1.51 f 0.41 1.628 rt 0.074 2.97 f 0.73 1.05 f 0.15 dimethyl sulfoxide N,N-dimethylformamide 2.26 f 0.66 1.65 f 0.11 nitromethane 0.66 f 0.48 1.711 f 0.090 2.32 f 0.45 2.150 f 0.078 triethylamine a
Sherman et al.
S
d
-0.75f0.25 -0.94 f 0.23 1.12 f 0.28 3.55 f 0.23 4.47 f 0.44 2.07 f 0.35 5.35 f 0.24 4.28 f 0.40 2.23 f 0.25 3.01 f 0.33 1.41 f 0.46 -1.78 f 0.92" -2.70 f 0.81 6.45 f 0.29 7.3 f 1.0 5.59 f 0.67 7.36 f 0.39 1.73 f 0.76
-0.59i0.19 -0.13 f 0.15" -0.50 f 0.17 -0.29 f 0.32 -0.52 i 0.27 -0.83 f 0.25 -0.34 f 0.32" -1.14 i 0.27 -0.77 f 0.16 -0.61 f 0.23 -0.21 f 0.34" -0.42 f 0.49" 0.55 f 0.46" -0.63 f 0.35 -0.84 3= 0.61" -0.69 f 0.42" -0.61 f 0.43" - 1S O f 0.48
U
b
7.30 i 0.42 9.84 f 0.62 6.07 f 0.54 2.13 f 0.43 -0.94 f 0.53 1.04 f 0.33 0.51 f 0.45" 9.49 f 0.81 11.2 f 1.2 11.1 f 1.1 7.77 f 0.98 8.3 f 1.2 7.49 f 0.89 6.7 f 1.2 12.61 f 0.97
3.35 f 0.62 4.62 f 0.85 3.83 f 0.65 -0.05 f 0.56" -0.57 f 0.91"
Statistical t-test indicated that the coefficient is not significantly different from zero within a 90% probability.
tween the solute as a hydrogen bond acceptor and the solvent as the hydrogen bond donor. The last term (eq 3), the intercept, is a solvent-specific parameter. This parameter compensates for a variety of factors and interactions not accounted for by the previous terms, or for deficiencies in the abilities of the solvatochromic parameters to describe their intended properties. One example is the arbitrary zero of the solvatochromic dipolarity/polarizability7 ~ scale, which is defined to be zero for cyclohexane. In this case, the zero does not imply that there are no dipolarity- or polarizability-dependent interactions for cyclohexane; it merely means that cyclohexane has been chosen as an arbitrary reference compound for the scale. In this and other such cases, the intercept is needed to compensate for the shortcomings of the other explicitly stated terms in the LSER. The identification of log LI6, 7 ~ * ~ 2 , and pi as the relevant solute properties to be included in the LSER was accomplished by employing factor analysis techniques. Factor analysis is a method used to examine trends in a large data set by evaluating the eigenvectors and eigenvalues of a matrix comprising that data. The specific approach for the factor analysis method has been described in detail;I5 the main components of this method are summarized here. A data matrix comprising of G values for 18 solvents and 56 solutes was assembled from available data. Since there were not enough data to completely fill the matrix, the unknown values were estimated and a missing data factor analysis routineI6 was used to iteratively refine the values estimated until these refined values were consistent with the experimental data in the matrix. During this procedure, any experimental values that were predicted with large errors were flagged as suspect values. If the data points identified by this procedure were subsequently c o n f i i e d to be in error, they were corrected before continuing with the data analysis. Our analysis revealed that at least four and perhaps five eigenvectors are required to describe the experimental data. The eigenvectors obtained from factor analysis of the matrix described above can be fit, using linear regression, to each parameter scale of interest. This is the procedure known as "target testing." A good fit obtained from this procedure indicates that the variation described by the parameter scale is consistent with the variation in the experimental G values. Through this analysis, appropriate scales for terms involving dispersion, dipolarity, hydrogen bonding ability, and cavity formation were identified for inclusion in the LSER. In
4,
~
particular, the chromatographically-based dipolarity and hydrogenbonding scales developed by Li and c o - ~ o r k e r swere ' ~ found to give a somewhat better description of the variations in the experimental data as compared to the values obtained by Abraham et al.13h-13p The same LSER used here has also been applied to for 50 very different solutes as determined by their retention on 8 distinct ~ 2 polymeric gas chromatographic phases. The quality of fits were as good as, and in some cases better than, those reported in this work. These results are physically very pi, 7 ~ * ~ 2and , significant because free-energy parameters (g, log LI6)are successfuUy used to correlate an enthdpic p r ~ p e r t y , ~ ~ ~ ~ T R .This success is unusual since models using these parameters usually can predict free-energy-based properties well but often fail when used to predict enthalpic-based properties.
Results The solvents used in this study are listed along with their corresponding solvent solvatochromic parameters in Table 1A. The solutes and their properties are listed in Table 1B. The coefficients of the LSER (1, s, d, a, b) and the intercept are listed in Table 2. The LSER in eq 3 was used to predict hm, which was then used to calculate hE" and was compared with the 473 experimental values. The mean error for the entire data set was 0.35 kcdmol which corresponds to a 4% relative error for G. The resulting relative error of fit for hE" is 28%. Since the experimental Gvalues were calculated from the experimentally determined hE" values and Ahv, the errors associated with both experimental quantities influence the overall error. The experimental uncertainty in the Ahv values for the solutes is estimated to be about 1%.'* The average value of the Ah" is approximately 10 kcdmol; therefore, the absolute error associated with the overall measurement will be approximately 0.1 -kcal/mol. Similarly, the experimental uncertainty ofthe hE" data is approximately 10%. The mean value for the hE" data is 1.27 kcal/mol, which translates into a 0.13 kcdmol error. Assuming independence of errors, the variances are summed and the combined experimental error from the Ahvand the hE" data is 0.16 kcal/mol. The net result is that the hE" is predicted to a precision of 0.35 kcal/mol, which is attributed to inaccuracies in the experimental measurements and in modeling. The results of the LSER analysis are summarized in Table 3.
Partial Molar Heats of Transfer at Infinite Dilution TABLE 3: Summary of Results
- -
J. Phys. Chem., Vol. 99, No. 28, 1995 11243
-
no. of correl A/ZTR /~~"mean hE" data coeff mean % erroffit, mean % solvent points ? erroffit kcaUmol erroffit cyclohexane 54 0.980 4.3 0.30 12 heptane 22 0.997 1.9 0.16 9 dibutyl ether 22 0.997 1.7 0.17 26 diethyl ether 20 0.981 2.4 0.20 33 ethyl acetate 26 0.991 3.4 0.33 29 carbon tetrachloride 27 0.986 3.8 0.33 28 dichloromethane 20 0.989 2.5 0.19 14 benzene 31 0.981 4.8 0.36 31 mesitylene 0.19 20 25 0.996 2.1 toluene 0.26 28 26 0.993 2.9 40 0.966 5.6 methanol 0.49 63 1-butanol 0.54 37 16 0.970 5.7 1-octanol 4.8 0.45 49 23 0.979 acetonitrile 21 0.990 3.0 0.20 14 dimethyl sulfoxide 25 0.967 7.0 0.67 43 N,N-dimethylformamide 28 0.977 5.6 0.47 43 nitromethane 20 0.987 3.5 0.20 11 triethylamine 27 0.979 6.2 0.62 82 overall results 473 0.984 4.0 0.35 28 In order to verify the applicability of the LSER, the relationship was tested with data that were not included in the original fitting data set. The results of the f i s t test, shown in Table 4A, show measurements for Ahv for each solvent as compared with predictions made by the LSER for the same values. See Table 4B for the corresponding solute parameters for the solvents. If the -solute molecule is identical to the solvent molecule, then the hE" for the cycle in Figure 1 will be equal to zero and the & will be equal to Ahv. For the list of solvents, the average error of prediction was found to be 3.7%. Since the average error of fit for this test is comparable to the average error of fit for the fitted data set, this LSER seems to have some general applicability. The LSER was also validated for this hE" data obtained by extrapolating data. Although - finite concentration heat-of-mixingthese hE" data are generally less accurate than hE" measured in the near infinite dilution concentration range, the data are of sufficient accuracy to test the validity of the LSER as long as the data conform to the following criteria: (1) The heat-ofmixing curve cannot be "S"-shaped. (2) The heat-of-mixing data cannot be appreciably skewed to one side or the other of the 0.5 mole fraction mark. (3) The maximum absolute value of the heat-of-mixing curve needs to be larger than 25 cdmol. Extrapolations to infinite dilution made from data sets that did not meet these requirements tended to be very inaccurate. A subset of heat-of-mixing datalg that fit these criteria were modeled using a multiterm Redlich-Kister expression20using 3-6 terms, depending on the best fit of the data. Extrapolations made by using this expression on well-behaved heat-of-mixing data can be trusted to about 10% accuracy. The comparison of the LSER predictions to these extrapolateddata is shown in Table 5 . The absolute average error of the hE" predictions for various solutes in four different solvents was found to be 0.48 kcdmol, a number comparable to that obtained by the group contribution LSER developed by Fuchs and Stephenson,'g in predictive mode.
Discussion One goal of this work is to develop an understanding of interactions in dilute solutions. Relationships between the solvent properties and the LSER coefficients can offer insight into the physical interactions of the solvation process. The
TABLE 4: Partial Molar Heats of Transfer and Solute Solvatochromic Parameters for the Solvents Investigated Partial Molar Heats of Transfer When the Solvent Is the Same as the Solute (G= Ah") heat of predicted heat % vaporization, of transfer, error of solventholute kcaVmol kcal/mol prediction cyclohexane 7.86 7.90 0.5heptane 8.74 8.71 0.3 dibutyl ether 10.60 10.61 0.1 diethyl ether 6.51 6.29 3.7 ethyl acetate 8.51 7.92 7.5 carbon tetrachloride 7.75 7.97 2.8 dichloromethane 6.82 7.08 3.7 benzene 8.09 7.55 7.2 mesitylene 11.35 11.41 0.5 toluene 9.08 9.17 1.o methanol 8.95 9.57 6.5 1-butanol 12.51 11.69 7.0 1-0ctanol 16.96 16.31 4.0 acetonitrile 7.87 8.38 6.1 dimethyl sulfoxide 12.64 13.53 6.6 N,N-dimethylformamide 11.36 11.59 2.0 nitromethane 9.15 9.04 1.2 triethylamine 8.32 8.82 5.7 mean % error 3.7 B. Solute Solvatochromic Parameters"
solvent cyclohexane heptane dibutyl ether diethyl ether ethyl acetate carbon tetrachloride dichloromethane benzene mesitylene to1uen e methanol 1-butanol 1atanol acetonitrile dimethyl sulfoxide N,N-dimethylformamide nitromethane triethylamine a
logL'6 2.906 3.173 3.954 2.066 2.359 2.822 1.997 2.792 4.399 3.343 0.916 2.539 4.619 1.537 3.110 2.922 1.839 3.008
x;' 0.0
-0.14 0.04 0.03 0.31 0.16 0.34 0.29 0.33 0.29 0.35 0.30 0.37 0.62 1.00 0.8 1 0.67 0.02
82 0.0 0.0 0.0 0.0 0.0
0.5 0.5 1.0 1.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
8"2 0.00 0.00 0.00 0.00 0.00 0.00 0.06 0.00 0.00 0.00 0.35 0.31 0.35 0.05 0.00 0.00 0.06 0.00
log LI6 parameters were obtained from ref 13h-13p. 17.
4,& parameters were obtained from ref
0.00 0.00
0.29 0.40 0.49 0.04
0.06 0.10 0.13 0.11 0.52 0.52 0.51 0.37 1.54 0.97 0.16 0.64
J C * ~ 62, ~,
coefficients of the LSER should be directly related to the physicochemical properties of the solvent. Lisf demonstrated this by showing, for an LSER correlating stationary phase characteristics with individual solute retention, the existence of a linear relationship between the resulting solute s and a coefficients on various gas chromatographic phases and the solvatochromically determined stationary phase Z*KT and ,&T values, re~pectively.'~~ The sign and magnitude of the coefficients are important in the interpretation of the solvation process. Many of the solvatochromic parameter scales in the present study comprise predominantly positive values. Therefore, negative coefficients indicate an endothermic interaction with the solvent. The magnitude of the different coefficients cannot be directly compared since the corresponding solute parameters have scales with numerical ranges. Analysis of the "l" Coefficient. The 1 coefficient and the corresponding solute parameter log lo L16 describe the dispersion interactions and cavity formation. The correlation between the square of the Hildebrand solubility parameter, &*, which is
Sherman et al.
11244 J. Phys. Chem., Vol. 99, No. 28, 1995
TABLE 5: Comparison of LSER Predictions for hE” to Those Made from Extrapolated Heat-of-Mixing Data Using a Multiterm Redlich-Ester Expression
-
solvent methanol
solute 1-octene 1-propanol 1-hexanol methyl acetate N,N-dimethylformamide toluene dimethyl sulfoxide 2-butanone
LSERhE-, R-KhE”, err ref kcaUmol kcaUmol kcaYmol 16 1.36 0.25 0.91 0.07 -1.58
1.13 0.11 0.52 0.87 -0.80
0.23 0.14 0.39 -0.80 -0.78
0.5 1 -0.92 -0.19
0.42 -0.16 0.57
0.09 a -0.76 a 0.76 b
1-octene carbon tetrachloride chloroform dichloromethane ethanol 1-propanol dimethyl sulfoxide cyclohexane p-xylene 1-nitropropane nitromthane nitroethane
0.99 0.13 0.69 0.03 4.19 4.34 0.60 0.10 0.65 -0.04 0.61 0.40
0.96 0.12 -0.29 -0.07 4.11 3.75 0.88 0.92 0.19 0.13 0.70 0.33
-0.58 -0.72 -0.41 0.83 1.59 0.11 -0.06 0.89 0.62
-0.55 0.20 0.20 1.31 2.34 0.68 0.21 0.33 0.21
0.48 1.47 0.98 5.45 1.96 2.72 4.24 2.48 2.98 3.91 3.58 2.83 1.04
0.15 0.62 0.86 6.07 2.00 3.07 2.57 6.39 2.75 3.71 3.12 2.87 0.70
carbon tetrachloride chloroform methyl iodide ethanol 2-butanone dimethylacetamide benzonitrile benzyl alcohol 1-nitropropane nitromethane nitroethane acetonitrile p-xylene av abs err
0.03 0.01 0.98 0.10 0.08 0.59 -0.18 -0.82 0.46 -0.17 -0.09 0.07
a a a a a a a a a a a a
40
1
I
I
I
1
I
I
I
60
80
100
120
140
160
180
200
220
Figure 2. LSER coefficient analysis, cavity formation term. The LSER coefficient 1 is correlated with the square of the Hildebrand solubility parameter, 6H2.
-0.03 -0.92 -0.61 -0.48 -0.75 -0.57 -0.27 0.56 0.41
a a a a a a c a
A V
-
Alkanes Ethers and Esters Halogens Aromatics Alcohols Miscellaneous
0.51
av abs err cyclohexane
0
0.30
av abs err carbon tetrahydrofuran tetrachloride cyclohexene methylcyclopentane 1-nitropropane nitromethane dichloromethane methyl iodide methyl acetate butyl acetate
0
a a a a a
0.49
av abs err benzene
A V
-
Alkanes Ethers and Esters Halogens Aromatics Alcohols Miscellaneous
0.33 0.85 0.14 -0.62 -0.04 -0.35 1.67 -3.91 0.23 0.20 0.46 -0.04 0.34
a a a a a a a a a a a a a
0.74
related to the cavity formation, and the LSER coefficient 1 is shown in Figure 2. The correlation between ~ * K T ,which is related to the dispersive interactions, and the LSER coefficient 1 is shown in Figure 3. In Figures 2 and 3, compounds that exhibit strong specific interactions, such as the alcohols, were less well represented by the LSER and appear further from the regression line. The cavity formation process by definition is endothermic, and the dispersive interactions are exothermic. Therefore, a positive coefficient 1 indicates that the energy associated with the dispersion interactions is greater than that associated with the cavity formation process. This indicates one of two things: either the energy required for cavity formation increases as the cohesive energy of the solvent increases, or that the dispersion interactions decrease as the cohesive energy of the solvent increases. Since the coefficient 1 also decreases with increasing KT (see Figure 3), we believe that the former explanation is the most reasonable interpretation of this result. The behavior for the alcohols is not surprising since they are more self-associating than the other solvents in
0
0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
’*
KT
Figure 3. LSER coefficient analysis, cavity formation term. The LSER coefficient 1 is correlated with the solvatochromic polarity and polarizability solvent parameter JC*KT. the study. Also, the 1 coefficients were determined from an enthalpic quantity, G, and not from a related free-energybased parameter which would be more appropriate for dealing with self-associating fluids. Analysis of Dipolarity and Polarizability Contributions. The s coefficient and the corresponding solute parameters n*c2 describe both dipolar interactions and polarizability interactions simultaneously. The predominantly positive coefficients indicate that this interaction is exothermic as expected. There is a reasonable correlation between the s coefficient and Z*KT of the solvent as depicted in Figure 4. The slope indicates that as the solvent becomes more polar, the dipolarity and polarizability interactions become more important. The d coefficient’s corresponding solute parameter 82 represents the polarizability correction2’ for the s ~ r * ~term. 2 The d coefficients are predominantly negative, which compensates for the fact that S Z * ~ Z
Partial Molar Heats of Transfer at Infinite Dilution
J. Phys. Chem., Vol. 99, No. 28, 1995 11245
Ethers and Esters
0
-2
Miscellaneous
0
Ethers and Esters
A
Alcohols
14
10
-I+
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
’*
KT
Figure 4. LSER coefficient analysis, polarity and polarizability term. The LSER coefficient s is correlated with the solvatochromic polarity and polarizability solvent parameter JZ*KT. includes contributions from both dipolarity and polarizability interactions. Therefore the ddz term subtracts at least part of the polarizability contribution from S ~ * ~ Z Several . of the s coefficients are negative, which does not make sense for X*KT values above 0. This phenomenon may be explained by two arguments. The first argument says that the X*KT scale has a significant degree of collinearity with the hydrogen bonding basicity scale and therefore the b& term overestimates the hydrogen bond donation by the solvent and the acceptance of a hydrogen bond by the solute, in which case the coefficient s would be negative to compensate. The second argument would be that the coefficient s in these cases is simply a result of the data and has no real physical significance. An examination of the correlation matrix (Table 8) for all of the solutes involved in the regression reveals some collinearity between the KT and & scales which tends to support the first argument. An examination of the determinant and inverse of the correlation matrix also reveals that some collinearity between terms exists, though not enough to significantly disturb the accuracy of the overall model. This collinearity increases the variance of the regressed coefficients and could account for the negative s coefficients. Analysis of Hydrogen-Bonding Contributions. The hydrogen-bonding term a 4 represents the hydrogen bond accepting ability of the solvent and the hydrogen bond donating ability of the solute. All solvent coefficients are positive with the exception of that for benzene. The positive coefficients represent exothermic interactions which would be consistent with hydrogen bond formation. Benzene is slightly basic, but the coefficient is negative. This could also be due to the covariance between the n* and /3 scales as mentioned before, resulting in the s coefficient being more exothermic than expected. The correlation between the coefficient a and PKTis shown in Figure 5. The positive slope indicates that as the solvent becomes more basic, the hydrogen bond formation term becomes a more important contribution to the solvation process. The term b/3; represents the hydrogen bond donating ability of the solvent and the hydrogen bond accepting ability of the solute. Most of these values are positive, which indicates that the process of hydrogen bond formation is exothermic. The coefficients for acetonitrile and nitromethane, however, are negative. This may be explained by the fact that these solvents
-1 -2
1
A
\ 01
02
I
I
,
03
04
05
06
07
08
09
10
aKT
Figure 6. LSER coefficient analysis, hydrogen-bonding acidity term. The LSER coefficient b is correlated with the solvatochromic acidity solvent parameter a K T . are only slightly acidic according to their a K T values. In addition, these coefficients are not statistically different from zero at the 90% confidence interval and therefore should be considered zero. The correlation between the b coefficient and a K T is shown in Figure 6. The positive slope indicates that as the solvent becomes more acidic, the hydrogen bond formation becomes more important in describing the solvation process. The LSER coefficients as well as the correlation of the coefficients with physicochemical parameters give only a qualitative picture of the solvation process. However, the physical interpretation of the relative importance of the different interactions is important in order to gain an understanding of the solvation process. The slope and the intercepts of the correlations in Figures 2-6 are shown in Table 6. These figures are intended to show the relationship between each solvent’s LSER coefficients and physically measurable properties of the solvent.
11246 J. Phys. Chem., Vol. 99, No. 28, I995
Sherman et al.
TABLE 6: Correlation of LSER Coefficients with Various Solvent Properties LSER
solvent property
coefficient
1 1
8H2
KT KT
S
PKT
a b
intercept
slope
r2
2.5 2.4 -0.84 0.0
-0.023 -0.90 7.7 15.0 4.5
0.52 0.65 0.82 0.56 0.75
0.0
aKT
TABLE 7: Average Percent Contribution of Each LSER Term to the Heat of Transfer for Four Different Classes of Solutes no. of 1 lo datapoints intercept Ll!
class a 4 = 0 and b& = 0 a G f 0 and b& = 0 a G = 0 and b& f 0 a 4 f 0 and bp; # 0
292 92 55 34
23.5 22.2 14.9 11.7
SX;
dd2 aai b&
65.9 9.3 1.4 46.1 16.5 1.6 13.4 59.8 15.8 1.5 8.0 41.7 17.0 1.1 20.2 8.3
TABLE 8: Mean-Centered and Standardized Correlation Matrix for 56 Solutes Used in the LSER Analysis log L'6
log LI6 n*c2 82
a;
/%
1.000 0.684 0.353 0.050 -0.181
J$
0.684 1.000 0.5286 0.332 0.4713
62
as
0.353 0.5286 1.00 0.186 -0.019
0.050 0.332 0.186 1.000 0.206
-0.181 0.4713 -0.019 0.206 1.000
The average contribution of each term in the LSER (eq 3) is shown in Table 7. Each term (except the offset) consists of a solvent coefficient multiplied by its corresponding solute parameter. The absolute value of these quantities for each solvent-solute combination and intercept are taken and summed together, and the relative contribution of each term is calculated. In general, four different situations occur. The f i s t case occurs when the a 4 term is zero because either the PKTparameter for parameter for the solute is zero. the solvent is zero or the The second case occurs when the bp; term is zero because the a K T parameter for the solvent is zero or the parameter for the solute is zero. The third case occurs when case one and case two simultaneously occur and both the a 4 and bp; terms are zero. The fourth case occurs when all the terms are nonzero. The most important term in the LSER is the 1 log LI6 term. This represents both cavity formation and dispersion interactions, where the dispersion terms are predominate. Two trends are apparent from this information. The first trend occurs when the hydrogen-bonding interactions are added. When this happens, the 1 log LI6 term becomes less important. The other trend occurs when the hydrogen-bonding terms are added; the becomes more important. The first trend can be explained by the fact that, as the number of interactions increases, each interaction should contribute less relative to the overall magnitude. The second trend probably results from the fact that polarity and hydrogen-bonding capability are correlated since most compounds capable of hydrogen bonding are somewhat polar.
4
Conclusions A linear solvation energy relation based on solute solvatochromic scales was used to correlate and indirectly hE". Although the LSER developed here provides estimations of to the same degree of accuracy as the LSER developed by Fuchs and Stephenson et al., our LSER provides better physical and chemical insight into the solvation process, as supported by the correlation of the various LSER coefficients to their
physically measurable counterparts. Upon measurement of new hE" data, solvatochromic parameters, and partition coefficients, this physical significance will provide the foundation on which future improvements in the accuracy and applicability of the LSER will be based.
Acknowledgment. We acknowledge gratefully the financial support of DuPont Chemical Company, Amoco Chemical Company, and the National Science Foundation. Nomenclature a = LSER coefficient for hydrogen-bonding acidity parameter b = LSER coefficient for hydrogen-bonding basicity parameter d = LSER coefficient for the polarity/polarizabilitycorrection parameter 1 = LSER coefficient for the log LI6 parameter s = LSER coefficient for the polarity/polarizability parameter LI6 = gas-hexadecane partition coefficient [mol/L]/[mol/L] Ahv = heat of vaporization [kcaVmol] hE" = partial molar enthalpy [kcaVmol] hs = partial molar heat of solvation [kcaVmol] hTR = partial molar heat of transfer [kcaVmol] Pvap = vapor pressure [ " H g ] v = molar volume [cm3/mol] T = temp [K] R = ideal gas constant P = pressure [ " H g ] VLE = vapor-liquid equilibria
Greek Symbols
a = solvatochromic hydrogen-bonding acidity parameter ,l3 = solvatochromic hydrogen-bonding basicity parameter 6 = solvatochromic polarity/polarizability correction parameter n* = solvatochromic polarity/polarizability parameter 6~ = Hildebrand solubility parameter [ k ~ a l ~ , ~ / c m ' , ~ ] y" = infinite dilution activity coefficient Subscripts/Superscripts
2 = solute c = chromatographic solute parameter'3b KT = Kamlet-Taft solvatochromic solvent parameter9 m = infinite dilution Supporting Information Available: The database of hE" values and references used in this study can be ordered (19 pages). Ordering information is given on any current masthead page. References and Notes (1) (a) Fuchs, R.; Cole, L. L.; Rodewald, R. F. J. Am. Chem. SOC. 1972,94,8645.(b) Fuchs, R.; Young, T. M.; Rodewald, R. F. J. Am. Chem. SOC.1974, 96, 4705. (c) Fuchs, R.; Saluja, P. S. Can. J. Chem. 1976, 54, 3857. (d) Fuchs, R.; Peacock, A,: Stephenson, W. K. Can. J. Chem. 1982, 60, 1953. (e) Stephenson, W. K.; Fuchs, R. Can. J. Chem. 1985, 63, 336. (0 Stephenson, W. K.; Fuchs, R. Can. J. Chem. 1985,63, 342. (8) Fuchs, R.; Stephenson, W. K. Can. J. Chem. 1985, 63, 349. (h) Stephenson, W. K.; Fuchs, R. Can. J. Chem. 1985,63,2529.(i) Stephenson, W. K.; Fuchs, R. Can. J. Chem. 1985,63,2540.(i) Stephenson, W. K.; Fuchs, R. Can. J. Chem. 1985, 63, 2535. (k) Trampe, D. M.; Eckert, C. A. J. Chem. Eng. Data 1991, 36, 112. (2) (a) Fredenslund, A,; Jones, R. L.; hausnitz, J. M. AIChE J. 1975, 21, 1086. (b) Fredenslund, A.; Gmehling, J.; Michelson, M. L.; Rasmussen, P.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1977, 16, 450. (c) Fredenslund, A.; Gmehling, J.; Rasmussen, P. Vapor-Liquid Equilibria Using UNIFAC; Elsevier: New York, 1977. (d) Gmehling, J.; Rasmussen, P.; Fredenslund, A. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 118. (e) Hansen, H. K.; Rasmussen, P.; Fredenslund, A,; Schiller, M.; Gmehling, J.
Partial Molar Heats of Transfer at Infinite Dilution lnd. Eng. Chem. Res. 1991, 30, 2352. (f) Gmehling, J.; Weidlich, U. Ind. Eng. Chem. Res. 1987, 26, 1372. (g) Gmehling, J.; Li, J.; Schiller, M. Ind. Eng. Chem. Res. 1992, 32, 178. (h) Larsen, B. L.; Rasmussen, P.; Fredenslund, A. lnd. Eng. Chem. Res. 1987, 26, 2274. (3) (a) Nicolaides, G. L.; Eckert, C. A. Ind. Eng. Chem. Fundam. 1978, 17 (41, 33 1 . (b) Trampe, D. M.; Eckert, C. A. J. Chem. Eng. Data 1990, 35, 156. (4) Schreiber, L. B.; Eckert, C. A. Ind. Eng. Chem. Process Des. Dev. 1971, IO, 512. (5) (a) De,: E. L.; Deal, C. H. Inst. Chon. Eng. Symp. Ser. 1969, 3, 40. (b) Tochigi, K.; Tiegs, D.; Gmehling, J.; Kojima, K. Chem. Eng. Jpn. 1990, 23, 453. (6) (a) Thomas, E. R.; Eckert, C. A. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 194. (b) Howell, W. J.; Karachewski, A. M.; Stephenson, K. M.; Eckert, C. A.; Park, J. H.; Cam, P. W.; Rutan, S. C. Fluid Phase Equilib. 1989, 52, 151. (7) Hait, M. J.; Liotta, C. L.; Eckert, C. A,; Bergmann, D. L.; Karachewski, A. M.; Dallas, A. J.; Eikens, D. I.; Li, J.; Carr, P. W.; Poe, R. B.; Rutan, S. C. Ind. Eng. Chem. Res. 1993, 32, 2905. (8) (a) Scott, L. S. Fluid Phase Equilib. 1986, 26, 149. (b) Hussam, A.; Can, P. W. Anal. Chem. 1985, 57, 793. (c) Kikic, I.; Renon, H. Sep. Sci. 1976, 11, 45. (d) Landau, I.; Belfer, A. J.; Locke, D. C. Ind. Eng. Chem. Res. 1991, 30, 1900. (e) Thomas, E. R.; Newman, B. A,; Long, T. C.; Wood, D. A.; Eckert, C. A. J. Chem. Eng. Data 1993,32,2905.(f) Li, J. Ph.D. Thesis, University of Minnesota, Minneapolis, 1992. (g) Trampe, D. M.; Eckert, C. A. J. Chem. Eng. Data 1990, 35, 156. (9) (a) Kamlet, M. J.; Taft, R. W. J. Am. Chem. SOC.1976, 98, 377. (b) Kamlet, M. J.; Taft. R. W. J. Am. Chem. Soc. 1976,98,2886. (c) Kamlet, M. J.; Taft, R. W. J. Chem. Soc., Perkin Trans. 2 1979, 349. (d) Kamlet, M. J.; Taft, R. W. J. Chem. Soc., Perkin Trans. 2 1979, 1723. (e) Kamlet, M. J.; Abboud, J. L. M.; Taft, R. W. J. Am. Chem. Soc. 1977,99,6027.(f) Kamlet, M. J.; Jones, M. E.; Taft, R. W.; Abboud, J. L. M. J. Chem. Soc., Perkin Trans. 2 1979, 342. (g) Kamlet, M. J.; Solomonovici, A.; Taft, R. W. J. Am. Chem. Soc. 1979, 101, 3734. (h) Kamlet, M. J.; Hall, T. N.; Boylun, J.; Taft, R. W. J. Org. Chem. 1979, 44, 2599. (i) Kamlet, M. J.; Abboud, J. L. M.; Taft, R. W. J. Am. Chem. Soc. 1981, 103, 1080. (i) Kamlet, M. J.; Abboud, J. L. M.; Abraham, M. H.; Taft, R. W. J. Org. Chem. 1983, 48, 2877. (10) Carr, P. W. Microchem. J . 1993, 48, 4. ( 1 1 ) Kamlet, M. J.; Abraham, M. H.; Carr, P. W.; Doherty, R. M.; Taft, R. W. J. Chem. Soc., Perkin Trans. 2 1988, 2087. (12) Taft, R. W.; Abboud, J. L. M.; Kamlet, M. J.; Abraham, M. H. J. Solution Chem. 1985, 14, 153. (13) (a) Li, J.; Zhang, Y.; Dallas, A. J.; Carr, P. W. J. Chromatogr. 1991, 550, 101. (b) Li, J.; Zhang, Y.; Carr, P. W. Anal. Chem. 1992, 64, 210. (c) Sadek, P. C.; Cam, P. W.; Doherty, R. M.; Kamlet, M. J.; Taft, R.
J. Phys. Chem., Vol. 99, No. 28, 1995 11247 W.; Abraham, M. H. Anal. Chem. 1985,57, 2971. (d) Leahy, D. E.; Carr, P. W.; Pearlman, R. S.; Taft, R. W.; Kamlet, M. J. Chromatographia 1986, 21, 473. (e) Carr, P. W.; Doherty, R. M.; Kamlet, M. J.; Taft, R. W.; Melander, W.; Horvath, C. Anal. Chem. 1986, 58, 2674. (f) Park, J. H.; Carr, P. W.; Abraham, M. H.; Taft, R. W.; Doherty, R. M.; Kamlet, M. J. Chromatographia 1988,25,373. (g) Park, J. H.; Carr, P. W. J. Chromatogr. 1989,465, 123. (h) Abraham, M. H.; Duce, P. P.; Schulz, R. A.; Moms, J. J.; Taylor, P. J.; Barratt, D. G. J. Chem. Soc., Faraday Trans. 1 1987, 83, 2867. Q) Abraham, M. H.; Duce, P. P.; Prior, D. V.; Schulz, R. A,; Moms, J. J.; Taylor, P. J. J. Chem. Soc., Faraday Trans. 1 1988, 84, 865. (k) Abraham, M. H.; Grellier, P. L.; Prior, D. V.; Moms, J. J.; Taylor, P. J.; Maria, P. C.; Gal, J. F.,J.Phys. Org. Chem. 1989, 2, 243. (1) Abraham, M. H.; Duce, P. P.; Grellier, P. L.; Prior, D. V.; Moms, J. J.; Taylor, P. J. Tetrahedron Lett. 1988, 30, 2571. (m) Abraham, M. H.; Grellier, P. L.; Prior, D. V.; Moms, J. J.; Taylor, P. J.; Laurence, C.; Berthelot, M. Tetrahedron Lett. 1989,30,2571. (n) Abraham, M. H.; Grellier, P. L.; Prior, D. V.; Duce, P. P.; Moms, J. J.; Taylor, P. J. J. Chem. Soc., Perkin Trans. 2 1989, 699. (0)Abraham, M. H.; Buist, G. J.; Grellier, P. L.; McGill, R. A.; Prior, D. V.; Oliver, S.; Tumer, E. J. Phys. Org. Chem. 1989, 2, 540. (p) Abraham, M. H.; Duce, P. P.; Prior, D. V.; Barratt, D. G.; Moms, J. J.; Taylor, P. J. J. Chem. Soc., Perkin Trans. 2 1989, 1355. (9)Weimer, R. F.; Prausnitz, J. M. Hydrocarbon Process. Petrol. Rejin. 1965, 44, 237. (r) Abraham, M. H.; Whiting, G. S. J. Chem. Soc., Perkin Trans 2 1990, 291. (14) Neter, J.; Wasserman, W.; Kutner, M. H. Applied Linear Regression Models, Irwin: Burr Ridge, IL, 1989. (15) Poe, R. B.; Rutan, S. C.; Hait, M. J.; Eckert, C. A,; Carr, P. W. Anal. Chim. Acta 1993, 277, 223. (16) Dempster, A. P.; Laird, N. M.; Rubin, D. B. J. R. Statist. Soc, 1977, 39, 1. (17) (a) Li, J.; Zhang, Y.; Dallas, A. J.; Can, P. W. J. Chromatogr. 1991, 550, 101. (b) Li, J.; Zhang, Y.; Carr, P. W. Anal. Chem. 1992, 64, 210. (c) Li, J. Ph. D. Thesis, University of Minnesota, Minneapolis, 1992. (18) Majer, V.; Svoboda, V. Enthalpies of Vaporization of Organic Compounds, International Union of Pure and Applied Chemistry, Chemical Data Series No. 32; Blackwell Scientific Publications: Oxford, 1985. (19) (a) Gmehling, J.; Holderbaum, T.; Christensen, C.; Rasmussen, P.; Weidlich, U. Heats of Mixing Data Collection, 4 parts, DECHEMA Chemistry Data Series; DECHEMA: FrankfudMain, 1984. (b) Chao, J. P.; Dai, M. Thennochim. Acta 1991, 179, 257. (c) Gmehling, J.; Meents, B. In?. DATA Ser. Sel. Data Mixtures, Ser. A. 1992, 153. (20) Redlich, 0.;Kister, A. T. Ind. Eng. Chem. 1948, 40, 345. (21) Laurence, C.; Nicolet, P.; Dalati, M. T.; Abboud, J.-L. M.; Notario, R. J. Phys. Chem. 1994, 98, 5807. JF950479V
