2759
ANALYSES OF HE AND VE DATA
Excess Properties of Some Aromatic-Alicyclic Systems.
11. Analyses of HE
and VE Data in Terms of Three Different Theories of Molecular Solutions’
by I. A. McLure,2J. E. Bennett,a A. E. P. W a t ~ o n and , ~ G.C. Benson Division of Pure Chemistry, National Research Council, Ottawa, Canada
(Received April $7, 1366)
The experimental data for the molar excess enthalpies and volumes of aromatic-alicyclic systems, presented in part I, have been examined from three different points of viewthe Scatchard-Hildebrand equation, a quasi-lattice treatment, and a form of corresponding states theory. None of these approaches is able to provide satisfactory independent estimates of the excess enthalpy but in most cases it is possible to fit the theoretical forms approximately to the data and to obtain information about the molecular interactions. For the eight systems studied, the energy of interaction between unlike molecules is found to be about 3% less than the arithmetic mean of the interactions of like pairs of molecules.
I. Introduction I n part I5of this series, the results of measurements of the molar excess enthalpy (HE) and molar excess volume (VE), both a t 25”, were reported for the binary systems of benzene with cyclopentane, cyclohexane, cycloheptane, and cyclooctane, and of toluene with the same alicyclic compounds. The present paper describes attempts to interpret the data in terms of several current theories of molecular solutions. The three different theoretical approaches considered are the Scatchard-Hildebrand equationle the quasi-lattice theory as developed by Barker,’ and the corresponding states average potential model due to Prigogine and coworkersag The treatment of Scatchard and Hildebrand is essentially an elaboration of that of van Laar and assumes a random distribution of the molecules (both in position and orientation) which is independent of the temperature. This approach should be fairly applicable to nonpolar systems. On the other hand, the quasi-lattice theory was formulated originally to describe the properties of mixtures in which the molecular interactions are strongly directional in character, as in hydrogen-bonded systems. The average potential model is probably best suited for mixtures of small spherically symmetric molecules. The aromatic-alicyclic systems which have been investigated do not fall unambiguously into any of these categories. In the past all three treatments have been applied to the
interpretation of the excess properties of benzenecyclohexane so1utiom6~8-10 but a comparative study of their application to a fairly extensive series of similar systems has not been carried out previously.
II. Scatchard-Hildebrand Theory Scatchardll derived the expression
AUM =
+
(~181~2V2)-4124142
(1)
for the energy change when a mole of binary solution is formed a t constant temperature and volume by mixing z1 and z2 moles of the pure components, having molar volumes VI and V2,respectively. In eq. 1, chi represents (1) Issued as National Research Council No. 8563. (2) National Research Council of Canada Postdoctorate Fellow, 1962-1963. (3) National Research Council of Canada Postdoctorate Fellow, 1961-1963. (4) National Research Council of Canada Postdoctorate Fellow, 1959-1961. (5) A. E. P. Watson, I. A. McLure, J. E. Bennett, and G. C. Benson, J . Phys. Chem., 69, 2753 (1965). (6) J. H. Hildebrand and R. L. Scott, “Regular Solutions,” Prentice Hall Inc., Englewood Clifis, N. J., 1962, see Chapter 7. (7) J. A. Barker, J . Chem. Phys., 2 0 , 1526 (1952). (8) I. Prigogine, “The Molecular Theory of Solutions,” North Holland Publishing Co., Amsterdam, 1957, Chapters 10 and 11. (9) G. Scatchard, S. E. Wood, and J. M. Mochel, J . Phys. Chem., 43, 119 (1939). (10) J. B. Ott, J. R. Goates, and R. L. Snow, ibid., 67, 515 (1963). (11) G. Scatchard, Chem. Rev., 8 , 321 (1931).
Volume 63, Number 8 August 1366
I. A. MCLURE,J. E. BENNETT,A. E. P. WATSON, AND G. C. BENSON
2760
the volume fraction of component i a n i is defined with reference to the unmixed state as
4t =
ZtVJ(ZlV1
+
(2)
ZZVZ)
The CoefficientAIz is a constant having the form
Aiz
= cii
- 2~12+ czz
(3)
in which the quantities ct, characterize the interactions between various pairs of molecules 11, 12, and 22. In particular, c , ~is the cohesive energy density of component i and can be calculated from the standard molar heat of vaporization AH; (for vaporization to the ideal gas state) using the expression tit = (AH:
- RT)/V,
(4)
where R and T are the gas constant and absolute temperature. Energy densities for the aromatic and alicyclic compounds considered in this paper are listed in Table I. The references12-18indicate the source of the heat of vaporization data; molar volumes were based on the experimental densities given in part I.
and values of AUMare compared with heats of mixing a t constant pressure. In the present work, experimental values of UVE,the molar excess energy a t constant volume, were obtained from the excess enthalpies and volumes given in part I by using the thermodynamic relation
UvE = H E - T(a/@)VE
(7) Values of a, the coefficient of thermal expansion, and @,the isothermal compressibility, were estimated for the mixtures a t 25" from data for the pure components (summarized in Table I) by assuming additivity on a volume fraction basis. The experimental excess energies for the benzene and toluene systems are plotted in Figures 1and 2. In order to separate the curves for the different alicyclic compounds, the origin of the ordinate axis has been displaced for each curve.
Table I : Values of Cohesive Energy Densities ( c ) , Thermal Expansion Coefficients (cy), and Isothermal Compressibilities ( p ) of Pure Component Liquids a t 25"
Component
C6H.5 C7H8 CsHio C6H12
CTHi4 CsHie
c, joules em. -8
10% deg. -1
1068, atm. -1
351" 332" 276" 281" 296' 303f
1.217"'* 1.071" 1.390" 1.261" 1.00" 0.99"
98.12" 95. Od 135. le 155.1" 98.0 (estimated) 81.2g
'
Reference 12. Reference 14. Reference 15. d Reference 16. ' Reference 17. Reference 13. Reference 18. a
'
BENZENE I 0
In the case of dispersive forces between molecules with approximately equal ionization potentials the Berthelot relation C1Z2
= Cllcn
(5)
may be used to eliminate c12 from eq. 3, giving
Aiz
=
( 6 1-
v'CZZ)~
(6)
The term solubility parameterig is commonly used to indicate the quantity f i i . The assumptions involved in the derivation of eq. 1 have been discussed in detail by Hildebrand and Scott.lg For the present purpose it is important to note that the mixing process considered is one carried out a t constant volume. This restriction is frequently ignored The Journal of Physical Chemistry
0.1
I
0.2
- ALICYCLIC
SYSTEMS AT 25'C
I
I
I
0.3
0.4
0.5
I
I
0.6 0.7 x, mole fraction of benzene.
I
I
0.8
0.9
1.0
Figure 1. Excess energy a t constant volume for benaenealicyclic systems a t 25' : curves experimental, points calculated from quasi-lattice theory. (12) American Petroleum Institute, Research Project 44, 'Is+ lected Values of Properties of Hydrocarbons and Related Compounds," Carnegie Press, Pittsburgh, Pa., 1950,and later revisions. (13) H.L. Finke, D. W. Scott, M. E. Gross, J. F. Messerly, and G. Waddington, J . Am. Chem. SOC.,78, 5469 (1956). (14) S. E. Wood and A. E. Austin, ibid., 67, 480 (1945). (15) G. A. Holder and E. Whalley, Trans. Faraday SOC.,58, 2095 (1962). (16) B.Jacobson, Acta Chem. Scand., 6 , 1485 (1952). (17) A. Weissler, J. Am. Chem. Soc., 71, 419 (1949). (18) E.Butta, Ric. Sci., 26, 3643 (1956). (19) J. H. Hildebrand and R. L. Scott, "Solubility of Noneleotrolytes," 3rd Ed., Reinhold Publishing Corp., New York, N. Y . , 1950.
2761
ANALYSES OF HE AND VE DATA
Table ll : Summary of Calculations for Scatchard-Hildebrand Theory
*
System
OLUENE-ALICYCLIC
0
I
I
0.1
0.2
I
I
0.3
0.4
S Y S T E M S AT 2 5 ' C
I 0.5
I
I
I
I
0.6
0.7
0.8
0.9
1.0
Figure 2. Excess energy at constant volume for toluene-alicyclic systems at 25' : CUNW experimental, points calculated from quasi-lattice theory.
Representation of the experimental results for UvE by two different empirical forms
+
22v2)4142
Ci"Cp(42 - 41)'-l
(8)
P
and
UVE=
2 1 2 2 Cl0CP'(X2
- 21)"-1
(9)
P
was investigated. The values of the coefficients C p and C p in these were determined by least-square calculations. For n = 1 it was generally possible to obtain a closer fit in terms of the volume fractions; the resulting values of A12 (ie., C1for n = 1) are summarized in Table I1 along with UU, the standard deviation between the experimental and calculated values of UvE. The superiority of eq. 8 over eq. 9 for n = 1 appears to indicate that the use of the volume fraction takes account of spatial considerations in these systems more realistically. However, the best fits to the experimental data were obtained by employing three or four constants in eq. 8 and 9. The values of rU were then very nearly the same as those given for uEiin part I and there was no significant difference in the ability of either form to represent the results. An examination of the differences of shows that eq. 6 is not valid and that the Berthelotrelationcannot be applied to the present systems. Optimum values of c12 obtained for each system by inserting in
d%
CsHsCsHio CsHsCeHiz CeH&?Hu CeHgCsHis
22.87 23.65 20.35 21.34
7.4 9.5 7.9 8.0
C?H&LHIO C7H,C& C7Hs-C7Hii C7Hs-CsHis
13.Q 16.02 14.67 14.24
2.6 6.2 8.5 8.9
G
z
joules cm.-8
,
Basad on eq. 10 cis, A~P, joules joules om.-' cm.-a
302 304 314 316
312 314 323 326
310 311 317 317
298 299 307 310
330 306 314 317
303 305 312 314
7 10 14 20 3 3 5
8
eq. 3 the value of A12and of c11 and c z 2 for the pure components are given in Table I1 along with values of 1/cllc22 for comparison. Kohler20 has described a method of calculating the interaction between unlike molecules which leads to the expression
x, mole fraction of toluene.
UvE = (ZIVI
-Least-square fi+ Aiz, m, CIZ, joules joules joules em.-' cm.-a mole-'
where CY( is the molecular polarizability of species i and Pt = cttVt3/a?
(11)
Equation 10 is based on the London formula for dispersive forces but has the advantage that the ionization potentials of the molecules do not appear explicitly. This approach has been used by Munn21in discussing the relative deviations of the interaction energies between unlike molecules from a geometric mean combining rule in a number of hydrocarbon systems. In the present work polarizabilities were estimated from the Lorenz-Lorentz equation and the refractive in) in part I. dices of the pure components ( P ~ ~ Dgiven The values of c12 calculated from eq. 10 and the corresponding results for AB are given in the last two columns of Table 11. The Kohler formula leads to values of c12which are somewhat less than those given by the Berthelot relation, but the values obtained for A12 are still much too small for all the systems with the exI n deriving eq. 10 the interception of C~HG-C~HIG. actions between the molecules are assumed to occur between point centers. The use of this crude approximation for fairly large molecules may be responsible for the failure of eq. 10 to provide a satisfactory esimate of c12. (20) F. Kohler, Monatsh., 88, 857 (1957). (21) R. J. Munn, Trans. Faraday SOC.,57, 187 (1961).
Volume 69,Number 8 August 1966
I. A. MCLURE,J. E. BENNETT, A. E. P. WATSON,AND G. C. BENSON
2762
+ XZ f 723x3) = 5-52 x3(rjdi + + X3) = 1.52
III. Quasi-Lattice Theory
x2(7]12xl
I n the quasi-lattice model, the molecules of a solution are distributed over the sites of a h e d lattice which is usually assumed to have a coordination number of 4. Different molecular sizes are considered by allowing the molecules to occupy different numbers of sites. Various contact areas are recognized on each molecular species and the configurational energy of the mixture is expressed as a sum of contributions from interactions between different areas on molecules occupying nearestneighbor sites. For details of the theory, reference should be made to Barker’s original publication’; only the special forms of the equations needed for the present application will be outlined here. An alicyclic molecule of ring size p is assumed to have 2p similar contact areas which will be called type 1. Benzene possesses 12 aromatic (type 2) areas; toluene in addition to 11 type 2 areas also has three areas of a differentkind (type 3) around the methyl group. The energy of interaction (per mole) for a contact between areas of types i and j is denoted by Ui, and is defined in such a way that 2Uij is the increase in energy (per mole) when two i-j contacts are formed from a pair of i-i and j-j contacts. Due to the restriction imposed by the fixed underlying lattice, the excess energy calculated from the model again refers to a mixing process at constant volume. In the case of the benzene systems, the theoretical expression for the molar excess energy is
UvE = - ~ R T X I XIn~912~ ~ ~ The quantities X1 and equations
X2
(12)
are solutions of the pair of
923x2
(18) (19)
In eq. 16 X2’ and X3’ denote the values of X2 and X3 obtained from eq. 18 and 19 for the special case 2 = 1. Since the equations which define UvE as a function of the contact energies are quite complicated, a leastsquare determination of the best values of these parameters was not attempted. Instead, the use of the above equations was essentially empirical and values of the contact energies were adjusted by a trial and error procedure in an attempt to fit the theoretical forms to experimental results. Numerical calculations were done on an IBM 1620 computer using programs developed previously. 22 The values of the aromatic-alicyclic interaction energy for the benzene systems were chosen to fit the experimental UTEdata around z = 0.5. Treatment of the toluene systems was based on the assumption that the difference between methyl (aliphatic) and alicyclic contact areas could be neglected. I n this approximation UIZ= u 2 3 , U13 = 0, and the value of U12 was again selected to fit the results for an equimolar solution. The interaction energies obtained in this way are summarized in Table I11 and results for UvE calculated from the quasi-lattice theory are plotted as points in Figures 1 and 2. In general, the theoretical values tend to be skewed more to small mole fractions of the aromatic component than is observed experimentally. Values of the standard deviations, uu, between the experimental and theoretical results (also included in Table 111) show a greater variation in goodness of fit than that found in section 11.
Table Ill: Values of the Contact Energies used in the Quasi-Lattice Theory Calculations (Energies in joules mole-’)
and
(14) I n these expressions R represents the gas constant, T the absolute temperature, 2 the mole fraction of the aromatic species, and rjij
=
+
+
UvE = -2RT[X1Xm In 912 XIX3913 In 913 (x2X3 - %Xz’Xa’)r123In 9231 (16) where now the quantities XI, X2,and X3 must satisfy the conditions 912x2
f 913x3) = P(1
The J o u r d of Phyaical Chemistry
Uia
el7
202 202 171 177
25.5 13.5 7.5 3.2
System
V I P= Uza
su
C7Hs-CsHlo C7Hs-CeHu C?Hs-C7Hir C7Hs-CsHie
200 230 203 198
15.6 9.9 4.6 14.6
(15)
exp(-U,,/RT)
For toluene systems the formula corresponding to eq. 12 is
xl(x1 f
System
CscCsH10 Ck&C& CeHsC7Hi4 C6Hcc&j
- 2)
(17)
It is evident that the data for all the systems cannot be fitted by a single value of UIZ. An energy of about 200 joules mole-’ gives fairly reasonable representations for five of the systems but a variation of rt 14% . . in this energy is needed to cover the other three systems. One of the drawbacks of the quasi-lattice theory is the (22)
B. Dacre and G . C. Benson, Can. J . C h m . , 41, 278 (1963).
ANALYSES OF HE AND VE DATA
fact that the assignment of contact energies is often somewhat arbitrary and several different schemes may be found which fit the experimental excess energy equally well. The benzene systems studied here are of some interest since they involve only one kind of contact and hence, within the framework of the treatment outlined above, each system leads to a unique value for the aromatic-alicyclic interaction energy. The average value found for this energy (188 joules mole-l) is only about half the magnitude of the aromatic-aliphatic contact energy obtained by Ott, Goates, and Snow.1° This difference reflects the different approach used by the latter authors, in particular their neglect of volume changes. I n the case of the present systems the volume term in eq. 7 provides a sizable correction and has the general effect of making the values of UVEmore nearly the same for the various systems of both series than were the original enthalpy results. Other energy schemes in which the over-all variation of Ulzis somewhat less than for that shown in Table I11 can be obtained by assigning U23 a small nonzero value. However, the fit between experiment and theory does not appear to be improved sufficiently to warrant the introduction of extra parameters a t this time. It should be noted that the computation of UVEdoes not use the lattice coordination number or require a knowledge of the number of sites occupied by the various species of molecules. Some calculations of the excess free energy and entropy of mixing, which do depend on these geometric factors, were also carried The excess entropies obtained are all small and negative; the excess free energies appear to be too 2s but the experimental information is meager. largelz4~ If reliable free energy data were available, it might be possible to assign the contact energies in a less arbitrary fashion.
IV. Corresponding States Average Potential Model The corresponding states approach is based upon the assumption that the potential functions for molecular interaction between the various species of the system can be reduced to a common form by the introduction of suitable scale factors E* and r* for the energy and molecular separation. In the average potential model, the average interaction energy of a particular molecule in the solution is set equal to the mean of its interactions with neighboring molecules. These may be either of the same or different species, and their relative numbers are proportional to their mole fractions. The formulas which have been derived8Iz6for the molar excess enthalpy and molar excess volume are
2763
+ o.56).]
{e(xl -
(21)
In these equations component 1 (the aromatic compound) is chosen as a reference standard and VI,hl, and c,, and c,, indicate the molar volume, configurational enthalpy, and configurational heat capacities of this material, respectively, as a pure substance at temperature T. The three parameters 6, 0, and p in eq. 20 and 21 are defined by the equations
E12*
= T22*
Ell*(
1
+ 6 + e)
= r11*(1
The values of 6, 8, and rule
p
+
P)
(24)
together with the combining
rl2* = (rn*
+ r22*)/2
(25)
can be used to specify the energy and distance parameters for 12 and 22 interactions in terms of those occurring in the reference compound. It is implicit in the
(23) Formulas for these functions can be found in ref. 7. A lattice coordinationnumber of 4 was used and it WBS assumed that the number of sites covered by a molecule was equal to the number of carbon atoms contained in it. (24) J. S. Rowlinson, “Liquids and Liquid Mixtures,” Butterworth and Co. Ltd., London, 1959, see Section 4.8. (25) R. W. Hermsen and J. M. Prausnits, Chem. Eng. Sei., 18, 485 (1963). (26) We believe several typographical errors have occurred in the printing of eq. 10.7.8 in ref. 8. The result given above for VE WBS obtained by the methods outlined in Chapter X of Prigogine’streatise.
Volume 69,Number 8 August 1966
I. A. MCLURE,J. E. BENNETT, A. E. P. WATSON, AND G. C. BENSON
2764
derivation of eq. 20 and 21 that the magnitudes of 6, 8, and p are small. The contributions of the last terms in eq. 20 and 21 are generally relatively small and, in the present numerical calculations, both were neglected. The data used for the properties of the aromatic compounds are summarized in Table IV. Values of 6 and p for the
Table 1V: Values of Properties of the Reference Compounds at 25'"
e=
Compound-
CsHs
hl, joules mole--' cp,, joules mole-' deg. -1 dCpi -, dT
CiHa
-31,370 57.7
joule mole-' deg.-2
VI, cm.* mole-' d Vi cm.8 mole--' deg.-l dT' d2Vi 108 x -, cm.8 mole-1 deg.-2 dT2
-35,510 63.6
-0.2
DO
89,385
106.857
-
0.1072
0.1114
0.285
0.311
Most of the data are derived from ref. 12.
alicyclic compounds referred to benzene and toluene were obtained from molar volumes of the pure substances as functions of temperature.8112 For cycloheptane and cyclooctane the data available in the A.P.I. Tables cover only a small temperature range and this was extended by density measurements from 30 to 60". The values of 6 and p which lead to a reasonable superposition of the molar volume curves are listed in Table V. As far as the molar volumes are concerned it appears that all six compounds obey the law of corresponding states fairly well; this is evidenced by the fact that the ratios of 1 6 and 1 p for the same alicyclic
+
+
Table V : Summary of Values Used for the 8, p, and 0 Parameters and the Excess Volumes Calculated from Eq. 21 ux, joules
6
P
-0.056 0.0 0.085 0.150
0.0120 0.0680 0.1183 0.1645
-0.144 -0.084 -0.001 0.061
-0.057 - 0.0037 0.0442 0.0874
The Journal of Physical Chemistry
B
mole-'
VE at
ZI = 0.6, cm.: mole-' Calcd. Found
9.1 0.41 -0.0253 -0.0124 13.2 0.98 0.0318 21.4 2.10 0.0879 17.4 3.82
0.30 0.64 0.67 0.58
0.0012 -0.0229 -0.0128 0.0117
0.08 0.57 0.53 0.51
10.2 14.5 6.8 8.9
compound referred to different aromatic components show a variation of only 1.7 and 0.2%, respectively. It should be noted, however, that this behavior is not sufficient to establish the applicability of the corresponding states treatment and that, in fact, other experimental data (e.g., second virial coefficientsand critical constants2') indicate that the law of corresponding states does not hold for these compounds. Calculation of HE and VE from eq. 20 and 21 requires a value of 0 for each system. Use of the relation
0.70 0.36 0.59 1.41
- ~ / 8
(26)
which corresponds to the geometric mean combining rule for ~12*,in general failed to reproduce the experimental results. As an alternative approach, calculations were carried out for a series of e values and the results were compared with experimental data. For most of the systems it was not possible to fit both HE and VE with the same value of e; in general, a value of 0 which gave agreement with the enthalpy data led to excess volumes which were too large and vice versa. A reasonable compromise was possible for several systems but the determination of 8, which is an energy parameter, from enthalpy data should provide a fairer basis for comparison. Values of e chosen to fit H E a t 21 = 0.5 are given in Table V along with the calculated and experimental results for VE a t this concentration. Column 5 of the table lists standard deviations, uE, between the experimental and theoretical excess enthalpies. From these it again appears that there is a wider variation in the goodness of fit than that obtained in section 11. It is interesting tonote that sizable deviations between experimental and calculated excess volumes for the benzene series of solutions occur for the two larger alicyclic compounds and that the range of agreement shifts to higher members in the toluene series. These observations can be explained qualitatively in terms of the relative sizes of the component molecules since the range of validity of eq. 20 and 21 is restricted to small values of the parameter p (ie., the molecules must be of nearly equal size.)
V. Conclusion It has not been possible to establish unambiguously which of the approaches considered above provides the best basis for an analysis of the data. None of the theories leads to satisfactory a priori calculations of the excess enthalpies (or energies), but in most cases the theoretical forms can be fitted approximately to the experimental data by assigning to the various param(27) J. S. Rowlinson, Nature, 194, 470 (1962).
ANALYSESOF HE AND VE DATA
2765
eters values which are reasonable in magnitude. The corresponding states treatment appears to be of less general utility than the other theories but the requirement of fitting both HE and VE separately is more severe than that of fitting UvEin the other approaches. Estimates of the relative weakness of the interaction between aromatic and alicyclic molecules can be obtained from the empirical values of the parameters in all three theories. I n the four systems for which the corresponding states treatment leads to consistent results, e is negative, indicating that the aromaticalicyclic interaction energy is less than the arithmetic mean of the aromatic-aromatic and alicyclic-alicyclic interactions. It can be shown in the case of the Scatchard-Hildebrand theory that
-’
0 = (2~11V1) [ 16~1zV1V2( V;”
+ Vi/’)-* CIIVI-
~22V21
(27)
A reasonable dehition of 0 in the quasi-lattice theory is
e
=
- 1 2 u 1 2 [ 2 ( ~ 1 v- R T ) ~
(28)
for the benzene systems, and
e
=
- (11u12+ 3UI3)[ 2 ( m I v - RT)]-1
(29)
for those which contain toluene. I n both instances AHlv is the latent heat of vaporization of the aromatic compound. Values of 0 calculated from eq. 27-29, using the least-square values of c12in Table I1 and the contact energies in Table 111,are collected in Table VI.
Table Vl: Values of 0 Estimated According to the Three Theories
System
ScatchardHildebrand theory
C6HsCsHio CsHgC& CsHgC7Hlt C&-c&
-0.0281 -0.0309 -0.0360 -0.0509
C7Hs-CsHio C7&CsH12 C7HgC7Hu C7Hs-C8His
-0.0240 -0.0226 -0.0223 -0.0277
Quasi-lattice theory
Corresponding states theory
-0.0386
-0.0253
-0.0386
-0.0124
-0.0339
... ...
- 0.0327
...
-0.0313
-0.0360
-0.0229 -0.0128
-0.0318 0.0310
-
...
Estimates of e for the eight systems are all roughly similar in magnitude and indicate that the interaction between aromatic and alicyclic molecules is about 3% less than the mean of the interactions between like pairs. Partington, et a1.,28 reached a similar conclusion from a study of gas-liquid critical temperatures for a number of aromatic-aliphatic (both cyclic and noncyclic) systems. Values of e calculated by different methods do not show parallel trends and evidently reflect to some degree the various assumptions and approximations inherent in the different approaches. It appears unwise a t present to attribute differences in e entirely to variations in the interaction energies. ~~
~~
~
~~~
(28) E. J. Partington, J. S. Rowlinson, and J. F. Weston, Trans. Faraday SOC.,56, 479 (1960).
Volume 68,Number 8 August 1966
