3879
J. Phys. Chem. 1991.95, 3879-3884
A Study of Solvent Effect on the Optical Rotation of Chlral Biaryls Giovanni Gottarelli,**+ Michail A. Osipov,***and Gian Piero Spadat Dipartimento di Chimica Organica "A. Mangini". Universitci di Bologna, via S. Donato 15. I-401 27 Bologna, Italy. and the Institute of Crystallography, Academy of Science of the USSR, 1 1 7333 Moscow, Leninsky prosp. 59, USSR (Received: June 18, 1990)
We report that axially chiral biaryls, bridged or unbridged, show remarkable variations in their optical rotatory power in passing from ordinary to biphenyl-type solvents. These variations are in the sense expected if some molecules of the solvent adopt the same helicity as the solute. Molecular statistical calculations of the optical activity of the fluids were developed by using the density functional approach. The chiral part of the correlation function was estimated considering the steric repulsion term as the predominant factor. The results show that there is an excess of solvent molecules, with the same helicity as the solute, of the same order of magnitude as the concentration of the optically active solute. This is confirmed by an empirical comparison of the observed optical rotation with that of a resolved model of the chiral solvent.
Although the optical rotation in isotropic media is strictly related to the molecular chirality, its magnitude and, sometimes, sign depend largely on the influence of the solvent and temperature. The importance of these effects has been fully recognized since the work of Biot on tartaric acid.' The phenomena were soon recognized to be comlicated? since chemical and conformational effects as well as physical influences are at work. As a consequence, several approaches to the problem were proposed, ranging from the classical idea of Rule3 of the decrease of [CY] with increasing of the dipole moment of the solvent to the quantum theory by Weigang' where the solvent field correction for the electric dipole and rotatory strength of the chiral solute molecule is considered. In this approach, a nonzero correction exists even for randomly oriented solute and solvent molecules; furthermore, the dependence of the optical activity of the solute on the permanent dipole of the solvent is considered. In previous papers,>' we have studied the ability of chiral solutes to induce cholesteric structures in nematic solvents (helical twisting power, HTP). The value of HTP is very sensitive to the nature of the nematic ~ l v e n t In . ~ the case of axially chiral biaryl systems, which are characterized by high values of HTP, we proposed a mechanism of cholesteric induction in which the solute promotes preferentially solvent conformations of the same chirality6 (see Figure 1). This mechanism gives a major contribution in the case of nematics of the biphenyl type. If the nematic has a rigid core (as for perhydrophenanthrene derivatives), the value of HTP is substantially lower and seems to be connected to chiral correlations of the solvent molecules induced by the solute.' The twisting of the director in a nematic, induced by chiral dopants, is determined by chiral anisotropic interactions which, in ordinary cases, are limited to the direct chiral solute-achiral solvent interactions.* Instead, in the case of chiral biaryl systems, we assume that the high values of HTP are determined mainly by the large additional contribution from the interaction between solvent molecules that possess an induced chirality. These anisotropic interactions vanish after the averaging in the isotropic phase and hence they are not effective there, with the exclusion of the pretransitional region, where the short-range nematic or cholesteric order still remains. On the other hand, the induction of chiral conformations in the neighbor solvent molecules should be present also in the isotropic phase and manifest itself in large optical activity contributions roughly proportional to the effective concentration of chiral molecules. In connection with these ideas, we were interested to see whether these phenomena were effectively present in isotropic solutions and to obtain in this way an indirect confirmation of the induction of chirality in nematic liquid crystals. To whom correspondence should be addressed.
'UniversitH di Bologna, Italy.
'Academy of Science of the USSR.
CHART I
00
%OXO
OCHa (1)
0
:":$HcocH3
CH3
3;;: 4 (3)
Y
(2)
(4)
JD-@x
( 5 ) [K15] X=CN, Y=I-C&, (6)
X=H, Y=4-C,H,,
(7)
X=B. Y=4-C&
(8)
X=H, Y=J-CHs
(9)
X=H, Y-2-CHs
9
(10)
X=H, Y=2-N09
(11)
Experimental Section Derivative 1 was synthesized from the commercial acid with diazomethane? Derivatives 2-4 and the solvents are commercial ( 1 ) Biot, J. B. Mem. Acad. Sci. 1838, 15, 207. (2) Lowry, T. M. Oprical Rorarory Power; Longmans: London, 1935; p 349. (3) Rule, H. G.J . Chem. Soc. 1931, 674 and references therein. (4) Weigang, 0. E., Jr. J . Chem. Phys. 1964, I I , 1435. (5) Gottarelli, G.;Spada, G . P. Mol. Crysr. Liq. Crysr. 1985, 123, 377. (6) Gottarelli, G.;Hibert, M.;Samori, B.; Solladi€, G.;Spada, G. P.; Zimmermann, R. J . Am. Chem. Soc. 1983, 105, 7318. (7) Gottarelli, G.;Spada, G.P.; Varech, D.; Jacques, J. Liq. Crysf. 1986, I , 29. (8) Van der M e a , B. W.; Vertogen, G. In The Molecular Physics of Liquid Crysrals; Luckhurst, G. R., Gray, G . R., Eds.; Academic Press: London, 1979. (9) Jacques, J.; Fauquey, C.; Viterbo. R. Tetrahedron Lerr. 1971,4617.
0022-3654/91/2095-3879%02.50/0 0 1991 American Chemical Society
3880 The Journal of Physical Chemistry, Vol. 95, No. 9, 1991
Gottarelli et al.
TABLE I: Specific Optical Rotation of Compounds 1-4 in Several Solvents, Some of Which (5-10) with Bipbeoyl Structure dopant 1 2 3 4 helicity M P HTP -73 +54* -38 +O. 14
bl;: solvent 5' 6
20 OC -1690 -762
7 8 9 10
PhCN MeCN hexane
20 OC +9 -9
-729 -607 -594
40 OC -890 -723 -756 -702 -593 -594
20 OC -413 -313
-8 8 +13 -1 60
40 OC -76 -33 -72 -98 -6.5 -1 59
-363 -327 -500
40 OC -303 -294 -320 -342 -309 -492
-504 -469
-502 -461
-147 -162
-147 -1 53
-201
-192
20 OC +41.3 +40.1
40 OC +45.5 +47.2
+34
+36
+35.4 +55.0
+36.1 -55.8
"TB = @cr)-', w..:rep is :.I pitch of the induceL :holesteric (in gm), c t..> concentration a :he chiral dopant (mol of solute/mol of solvent), and r its optical purity. The solvent used is E7, a commercial mixture (from BDH) of cyanobiphenyl and cyanoterphenyl components. bFrom ref 6. CDopedwith 5% PhCN to obtain an isotropic solution.
CN
I
R Figure 1. Transfer of chirality from a chiral binaphthyl to a biphenyltype liquid crystal solvent.
products and were used without further purification. Optical rotations were measured with a JASCO DIP 370 digital polarimeter in a cell thermostated with a water jacket. HTP's were determined with a polarizing microscope by observing the Grandjean-Can0 disclination lines1"Vb connected with the cholesteric structure; their signs were obtained by means of the Heppke-Oestreicher methodIk and checked by observing the rotatory power of the cholesteric.Iw Experimental details are reported, for example, in ref 11. Refractive indexes of the solvents used, taken from the literature, are as follows: acetonitrile, 1.343; benzonitrile, 1 S29; 5, 1.566; 6, 1S71; 8, 1.602; 9, 1 S92;2-ethylnaphthalene, 1.600; chloroform, 1.448; benzene, 1.501; toluene, I .492; ethanol, 1.361; hexane, 1.375. Results and Discussion
We have measured the optical rotation of a few biaryl derivatives (1-4)in several isotropic solvents, some of which have a biaryl structure ($lo), and the temperature dependence of their solutions (Table I and Figure 2). We have chosen derivative 1 as it is conformationally rigid and easily available. Derivative 2 is specially interesting because the sign of its optical rotation is negative while the molecular helicity is positive. Derivative 3 is a chiral biaryl of great interest for asymmetric synthesis. Camphor (4)was chosen as a reference molecule which, due to its compact structure, should display only small sterically related solutesolvent interaction with large solvent molecules. (IO) (a) Grandjean, F. C. R. Acad. Sci. 1921, 172. (b) Cano. R. Bull. Soc. Fr. Mineral. Crystallogr. 1968, 91,20. (c) Heppke, G.; Ocstreicher, F. Z . Narurforsch., A 1977, 32, 899;Mol. Cryst. Liq. Cryst. 1978, 41, 245. (d) Berthault, J. P.; Billard, J.; Jacques, J. C. R. Acad. Sci., Ser. C 1977, 155, 284.
( 1 1 ) Gottarelli, G.; Samo;, 1981, 37, 395.
8.; Stremmenos, C.; Torre, G. Tefruhedron
With the exclusion of the cyanobiphenyl K15 (5), no other solvents show any mesomorphic state. In particular, we have chosen solvents 6 and 7 as their melting points are different and also their Yvirtual"transition temperatures should be different. In all cases the values of optical rotation in the flat region of the temperature dependence diagram are certainly free from pretransitional phenomena, and these values will be used in our analysis. From a first inspection of Table I, a few regularities emerge: in the three chiral biaryls considered (1-3)one observes a marked variation in passing from benzonitrile to the biaryl-type solvents, and this variation is always correlated to the sign of HTP (and hence to the helicity of the chiral biaryl): i.e., for P helicity (positive HTP) [ a ] D is more positive and for M helicity [ a ] D is more negative. The effect is spectacular in the case of cholchicine 2 where at 20 "C in K15 (5) and in 2-methylbiphenyl (9) the sign is inverted. The temperature effect is specially large for the K15 solvent (doped with a small amount of benzonitrile), which has a clearing point of ca. 17 OC. Obviously, at 20 "C,some pretransitional phenomena are still operative; however, at 40 OC, quite far from the transition temperature, the effect on [aIDis still remarkable. In particular, both 4-pentylbiphenyl (6) and 4-ethylbiphenyl (7) show similar remarkable effects. In the case of biaryl solvents these data can be explained by admitting that the mechanism of transmission of homochirality to solvent molecules, which was considered in the LC phase, operates also in the isotropic solution even if to a lesser extent. An appropriate chiral biphenyl, which can give an indication of the optical activity of the elusive chiral unsubstituted biphenyl, is the chiral bridged derivative 11.12 This derivative with R configuration has M helicity and a value of [ a ] D = -242 in isooctane. This can give a rough estimation of the contribution of a single molecule of "chiral" solvent to the optical activity of the system: solute 1 displays at 40 "C an increment of the optical rotation in biphenyl solvents (non-ortho-substituted) of ca. 250 with respect to benzonitrile: this indicates that an average of about one molecule of solvent is "resolved" for each molecule o f solute. Statistical Theory. To interpret the effect of the solvent on the optical activity of the solute, it is necessary to use a molecular statistical theory for the optical activity of a fluid. Theories o f this type have been developed by several authors," and the most refined version has recently been presented by Kim and Lee,14 who used the general method developed by Bedeaux and MazurI5 for the theory of dielectric susceptibility. The theory of Kim and Lee has been generalized also to the case of anisotropic fluids.16 (12) Mislow, K.; Hopps, H. B. J . Am. Chem. Soc. 1%2,84, 3018. (13) See, for example: Maaskant, W. J. A.; Oosterhoff, L. J. Mol. Phys. 1964,8, 319. Terwiel, R. H.; Mazur, P. Physica 1964, 30,625. (14) Kim, S.K.; Lee, D. J. J. Chem. Phys. 1981, 74, 3591. (15) Bedeaux, D.; Mazur, P. Physica 1973, 67, 23.
The Journal of Physical Chemistry, Vol. 95, No. 9, 1991 3881
Optical Rotation of Chiral Biaryls "
I
I
-4000
A
\
/
-2ooot -3000
ti
-100
4
-200 20
10
30
,
50 T( C) 1
40
-1007
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g=g=gp4+--m
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10
I
30
40
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I
I
/O/A
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-500.-
I
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0
N N, No Y
'c)
m a
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1 0
Y
0
-0.3-
M
2 -0.5-
0
\ 0
0
\n
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.
=0
3
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8
-1.5,
-0.5 i
8
0
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0
-1.54 1.4
0.0 1.3 ~
1.5
1.6
no
7
1.4
1.5
1.6
no
1.7
Figure 3. Plots of the ratio between the optical rotation aD (at the same molar dopant concentration) and the coefficient yo of eq 1 (see text) vs the refractive index of the solvent for the four chiral dopants 1-4. Filled circles represent data at 40 O C from this work while open circles represent data recorded at 20 OC from this work or collected from the literature. (For values of the refractive indexes see Experimental Section.)
According to the results of the statistical theory, one should distinguish between the long-range dielectric effects in a solvent, which manifest themselves in the Lorenz field corrections, and the effect of short-range correlations between the solute and solvent molecules, which results in the renormalization of the solute parameters. As shown in Appendix A, the optical rotation @ of a solution at small concentration of a chiral solute can be written in the formt9 (1)
(Re denoting the real part) where n, is the refractive index of the solvent, po the total number density of molecules, xd the molar fraction of the solute, and perthe effective optical activity of the (16) Osipov, M.A.; Egibyan. A.
V. Khim. Fir. 1983, 1. 325.
chiral solute embedded in ther solvent. It should be noted that the value of Ber is not equal to the optical activity of the isolated molecule but is affected by correlations with solvent molecules. The second term in eq 1 represents the effect of the chiral induction in the biphenyl solvent. Here Axs = xSR- x t is the equilibrium difference between the molar fractions of the right-handed and left-handed enantiomers of the biphenyl solvent in the presence of the chiral dopant, and per is the effective optical activity of 0 when xd the solvent enantiomer in excess. Obviously Ax, 0. Equation 1 is sufficiently general, and it can readily be seen that it describes at least three different solvent effects. The first effect is related to the coefficient vo = (lo2+ 2)2/9n,. In this effect the solvent manifests itself as a continuous dielectric medium characterized only by its average refractive index. The second effect is the chiral induction in the originally "racemic" solvent. This effect is described by the second term in the bracket of eq
-
-
3882 The Journal of Physical Chemistry, Vol. 95, No. 9, 1991 1. Finally, the third effect is determined by short-range correlations between solute and solvent molecules and results in the renormalization of the optical activity of the chiral solute. In the present discussion we are mainly interested in the last two effects and, in particular, in the chiral induction. One can make a quick estimate of the relative importance of the first effect by plotting the values of the optical rotation @ divided by the coefficient uo for different solvents as a function of the solvent refractive index (Figure 3). It can readily be seen that the real situation is far from the constant value expected if the contributions of the first type were dominant. On the contrary, the variations of the effective optical activities in various solvents become larger after they have been divided by vo. Therefore, the effects of chiral induction and perhaps that of intermolecular correlations are likely to be dominant for the solutesolvent pairs being considered. Let us consider the problem of the chiral induction in more detail: the most important problem is to determine the average number of solvent molecules that become "resolved" under the influence of the chiral solute, i.e., to estimate the ratio Lix,/xd which determines the order of magnitude of the effect. The change in the number of right-handed and left-handed enantiomers in the solvent can be estimated by using the so-called density functional method, which has been used successfully in the description of various molecular liquids, liquid crystals, and so1ids.l' In this theory the free energy is represented as a function of the one-particle density, and the difference between the free energies of two states of the medium with slightly different density contributions can readily be expressed in terms of the direct correlation functions. The density functional theory is discussed in more detail in Appendix B. As shown in Appendix B, the difference between the free energy of the biphenyl solution doped with chiral molecules F,and the free energy of the pure solvent Fs0 can be written approximately in the form (F, - F,o)/kTV = -PdAPsGds/2 + (APA~PPo-'- G,1/4 + Pd2.const (2)
where Pd is the concentration (density number) of the dopant and Ap, is the equilibrium difference between the concentrations of the right- and left-handed enantiomers of the biphenyl solvent induced by the chiral dopant. Quantities Gdsand G, are determined by the direct correlation functions between the molecules of the solute and solvent, cdL(1,2) and cdR( 1,2), and between the chiral enantiomers of the solvent, CRR(1,2) = CLL(l,2) and C R L ( I , ~ ) Gds = Jdr12 del de2 CdR(rIZ,elre2) Jdr12
del de2
cdL(r121e1,(32) (3)
G,, = S d r 1 2 d e l de2 CRL(r12,@1,e2) -
Jdrl2 del de2 CRR(rIZ,elre2) (4) where the variables 8, and 8, determine the orientation of the anisotropic molecules "1" and "2". The quantity Gd, describes the difference in the short-range correlations between the dopant and the left-handed or right-handed solvent enantiomers; it is this effect which breaks the symmetry between the opposite enantiomers of the solvent and produces a nonzero difference of their equilibrium concentration. On the other hand, the quantity G, describes the difference in similar short-range correlations between solvent molecules of equal and opposite chiralities. Minimization of the free energy in eq 2 with respect to Aps gives the following expression for the equilibrium concentration of the chiral molecules of the solvent: APs = PdPOGds/ (2 - POGss) (5) (17) (a) Lebowitz, J. L.; Percus, J. K. J . Math. Phys. 1963, 4, 116. (b) Sluckln, T. J.; Shukla. P. J . Phys. A 1983, 16, 1539. (c) Singh, Y. Phys. Rev. A 198430,583. (d) Haymet, A. D. J.; Oxtoby, D. W. J . Chem. Phys. 1985, 84. 1769.
Gottarelli et al. According to our expectations, the concentration Aps is proportional to that of the dopant and to the parameter Gh, which is a measure of the induction effect in the solvent due to short-range chiral correlations. The ratio Aps/pd has the physical meaning of the susceptibility of the solvent with respect to the chiral solute which plays the role of the external field. From the practical point of view, the most important question is to know how many optically active molecules can be induced in the solvent by one molecule of the chiral solute. Note that the number of such molecules is just equal to the "susceptibility" Ap,/pd. Equation 5 is general, but at present there is no hope of calculating the coefficients Gdsand G,, more or less accurately since very little is known about the correlation functions of chiral liquids. However, it is possible to draw some conclusions directly from eq 5. First, it is reasonable to assume that poG, < 2. Indeed, when poG, = 2, the susceptibility &/Pd goes to infinity and the system becomes unstable with respect to the concentration fluctuations of chiral enantiomex20 The critical point with poG, = 2 is the point of a phase transition which is called spontaneous chiral discrimination. However, spontaneous chiral discrimination has never been observed in liquids. Therefore, one can assume that generally we are far from the critical point and that poC, < 2. On the other hand, the chiral interaction between the dopant and the solvent (which determines the coefficient Gb) should be stronger than the corresponding interaction between the solvent enantiomers (which determines C,) due to the rigid structure of the chiral dopant and its larger volume. It follows, then, that Gds
'Further information about the coefficients GSS
Gh and G, can be obtained only with the help of the molecular theory. However, this kind of theory cannot be developed readily since the direct correlation function is known only for simple liquids, composed of molecules with primitive model shape (mainly, hard spheres). Moreover, as far as we know, the chiral part of the correlation function was not considered in the statistical theory of liquids. Thus, the only thing that can be done at the moment is to perform some simple order of magnitude estimation. In the molecular field approximation the direct correlation is written in the form function C(il2,O1,e2)
The first term in eq 6 represents the contribution from steric repulsion. Here Tl2 is the closest distance of approach between the centers of mass of the two molecules and Q(r12- t12)is a step function. The function fI2 = {12(312,tj1,e2) depends on the relative orientation of the two molecules and determines the excluded volume for them: V, = J!?(r12 - {12)r122drI2dii12d e , de2, where iI2= rI2iil2. The second term in eq 6 is simply the energy of attractive intermolecular interaction which is mainly determined by dispersion forces. Note also that the parameters G, and Gdsare determined by the chiral part of C(1,2). In the present molecular theory of simple liquids steric repulsion is considered to be the predominant factor in the formation of the short-range order, and the intermolecular attraction is taken into account as a perturbation. In the present case of biphenyl-type solvents doped with chiral molecules there is an additional argument to assume that the steric effect is the m a t important one. Indeed, the corresponding dopant molecules possess a strongly chiral nonplanar rigid shape with planar fragments similar to the ones of the solvent enantiomers. However, here we need a weaker assumption: Le., it is sufficient to assume that the two terms in eq 6 give the contributions of the same order of magnitude (and sign). Then the whole effect can be estimated taking into consideration only the steric repulsion between chiral molecules. Thus, the parameters Gh and G, can be estimated by the difference in the corresponding excluded volumes for different pairs of chiral molecules:
The Journal of Physical Chemistry, Vol. 95, No. 9, I991 3883
Optical Rotation of Chiral Biaryls
D
(a)
(6)
Figure 4. Top: the geometrical model for a chiral biaryl. Bottom: approaches between two molecules of a chiral biaryl with the same (a)
and opposite (b) configurations;The dark segment indicates the aromatic ring closer to the observer. The difference of the excluded volumes formed by two molecules of equal and opposite chirality is mainly determined by the difference in the minimum distances between the centers of the molecules. One can see that, for two molecules of the same chirality (depicted on Figure 4), the closest distance of approach between the centers of mass f l z is approximately equal to the thickness of the phenyl ring ( f L L = f R R = h i= 3.5 A), while for two molecules with opposite chirality, the distance tLiR is related to the breadth of the ring D = 5 A, f L R i= h + Dlsin 281, where 8 is the absolute value of the dihedral angle between the planar fragments of the molecule. Note that for the right-handed molecule 8 = eoand for the left-handed one 8 = -eo. When the interacting molecules posses the different dihedral angles (as in the case of dopant and solvent molecules) e d and e,, one can readily see that fLLdS = tRRdS i= h + Dlsin (e, - es)l and S;R& FJ h + D)sin (e, + Now the differences in the excluded volumes (7) can be estimated as
VcdR- VcdLFJ 41[(h
+ DJsin OS1)* - h2]
(84
where I is the distance between the centers of the two planar fragments composing the molecule, and e d and 8, are the dihedral angles between the planar fragments in the molecule of the dopant and solvent, respectively. We assume for simplicity that parameters I, h, and D are the same for the solute and solvent. The differences in excluded volumes (8a) and (8b) vanish when e d = r/2 or 8, = r/2,Le., when one of the interacting molecules is achiral. The parameter Gds is maximum when e d = e,, since in this case one has the most favorable condition for the close packing of the two chiral molecules. Also, the total maximum of the parameters Gdsand G, is achieved at e d = 8, = eo= n/4; the actual value of the angle for the molecules under consideration being not far from this value: biphenyl derivatives without ortho substituents have in fact a dihedral angle of 4 0 - 4 5 O in solution;I* the dihedral angle in the bridged binaphthyl 1 was estimated to be ca. 5 5 O from standard molecular mechanics calculations. The numerical values of the parameters p0C& and pOCscan be estimated with the help of eqs 7 and 8. Assuming po = 1O2I ~ m - I ~=, 5 A, D = 5 A, h = 3 A, and ed = 8, = r / 4 , one arrives a t the estimation p0C& = 1 and poG, i= 1. Thus Aps = pd and (18) Bates. R.;Camou. F. A.; Kane, V. V.; Mishra, P.K.; Suvannachut, K.;White, J. J. J . Org. Chem. 1989, 54, 31 1 and references therein. (19) In this and the following equations, apices or indices s and d refer to solvent and dopant, respectively, while R and L refer to two opposite chiralities. (20) When poGs > 2, q 5 is not applicable b u s e the approximations
introduced in deriving it from the free energy expansion are not valid.
Ax, FJ xd since Ap, = Axgo. This means that the equilibrium difference between the concentrations of the left-handed and right-handed enantiomers of the biphenyl solvent is of the same order of magnitude as the concentration of the chiral solute: roughly speaking, one molecule of chiral solute of the binaphthyl type induces, on the average, one resolved molecule in the biphenyl solvent. This simple semiquantitative prediction compares very well with the empirical estimate given above, based on the value of the rotatory power of a model molecule. The contribution of the second term in eq 1, Le., the chiral induction, seems to explain the main feature of the sometimes spectacular solvent effects observed for chiral biaryls in biphenyl solvents. The effect of short-range chiral correlations of solvent molecules induced by the chiral solute (the third effect in eq 1) has not been considered here and is not likely to be important for the couple biaryl solute/biphenyl solvents; it could possibly be responsible for the variations observed in passing from the acetonitrile to the benzonitrile solvents and should manifest to a greater extent in solvents containing larger polarizable aromatic nuclei.
Acknowledgment. We thank Professors R. G. Weiss (Washington) and C. Zannoni (Bologna) for the helpful discussion and CNR (Italy) and MURST (Italy) for financial support. Appendix A Optical Rotation ofa Solution. The statistical theory of dielectric susceptibility and optical activity of simple isotropic and anisotropic liquids"I6 result in the following general expression for the dielectric susceptibility tensor of a two-component solution [?(Z,W)
- l][?(Z,W)
+ 21-1 = (4/ 3)SPo[Xs@&,W)
+ Xd&der(?,W)] (A 1)
where po is the total number density, x, and xd are the solvent and solute molar fractions, respectively, adefis the averaged effective polarizability of the dopant molecule embedded in the solvent, and itsefis that of the solvent molecule. The effective polarizability includes also the molecular optical activity (A21 = ?ef + iSef(iiZ) where vug., is the antisymmetric Levi Civita tensor and (TeJu@= &ef(CW)
YCrSols.
It should be noted that the effective molecular polarizability Ydef and optical activity pdef of the dopant molecule in solution
are not equal to the corresponding values for isolated molecules (i.e., molecules in the rare gas state) but are renormalized by short-range correlations between the neighbor molecules; moreover, we assume that the chiral dopant shifts the equilibrium between the right- and left-handed enantiomers of the biphenyl solvent and produces an extra amount of chiral molecules which also contribute to the total optical activity of the solution. In the case of small dopant concentrations xd
