3485
J. Phys. Chem. 1992, 96, 3485-3489 4
r lr:l\ ' \ ' \ ' \
3 1
Appendix
In order to calculate the dielectric friction in the bulk liquid we will follow the cavity approach.12J6q'8 Consider a dielectric sphere of radius R and dielectric constant el at the center of which a dipole is located. Outside the sphere there is a liquid with dielectric function e(k,w), eq 1. For this model the electrostatic potential, 4, can be written by expanding the solution of the Laplace equation in spherical coordinates. The potential in the liquid, r > R , should vanish at infinite distance, and the homogeneous part of the potential inside the sphere, r < R , has to be analytic at the origin. The nonhomogeneous part of the potential in the sphere arises from the point dipole with the dipole moment p at the center. Thus we have
'
T
Figure 4. Temperature dependencies of the normalized rotation relaxation time, 7 i Rat a given frequency; y = 2R'kTIw/p2. The temperatures TIL&)and T2*(L2) are the transition temperatures in eq 36 for the pores with different sizes, L, > L2;To= I , cI = 1, cb = 80,w i = 0.1, c. = 3. Curves 1 and 1' are obtained at R / & = IO; curves 2 and 2' at R / & = 4.
interval around the transition point, T*(L),the dielectric friction and correspondingly the orientational relaxation time change from the value €6") to the value €6")' determined by eq 29b. The temperature and size dependencies of the rotational relaxation time at different values of parameters are shown in Figure 4. It should be emphasized, however, that eqs 36-39 present an example rather than an analysis of a real system. A different temperature dependence will be obtained using the same methodology but assuming different behaviors of A and T . In conclusion, we have studied the role of a boundary in modifying the relaxation behavior of a dipole embedded in a liquid. The formalism presented here which is based on the continuum approach extends previous works by introducing a nonlocal dielectric description of the liquid. Although the relaxation may be nonexponential in time we have calculated the dielectric friction for a given frequency as a function of zo, the distance of the dipole from the boundary. The effect of the boundary has been shown to be small unless the properties of the liquid itself are drastically changed due to the presence of the interface. The nonlocal nature of dielectric function modifies the dielectric friction derived within the local approximation and allows to introduce temperature dependence into the relaxation process through the characteristic length,
4=
('42) The coefficients A, B, and C are obtained from the boundary conditions at r = R which are similar to the conditions 8,9, and 10 used for the plane interface. As a result the field induced at the surface of dielectric sphere, r = R, has the form
€I )
4R/ A) I I)
(A3 1
withf(R/A) given by eq 28. A similar approach for the calculation of the dipole damping in the liquid (with the same boundary conditions) was adopted by van der Zwan and Hynes in ref 12. Our results, however, differ from theirs.
Refractive Index and Density Variations in Pure Liquids. A New Theoretical Relation A. Proutiere,* E. Megnassan, and H. Hucteau Laboratoire de Spectrochimie des Ions, UniversitE de Nantes, 2, rue de la Houssinigre, 44072 Nantes Cedex 03, France (Received: July 17, 1991)
A new theoretical expression of the density fluctuation of the refractive index (dn2/ddp or DFRI has been deduced from a general expression of the Lorentz-Lorenz equation and from the calculation of the partial derivatives of n2. The square of the refractive index in each point of the liquid has been considered as a function of the number N of molecules per volume unit, of the temperature T and of the pressure P. The final expression of the DFRI is the sum of the large isothermal term (an2/adP,, and of the smaller one (an2/ar),,/(dd/dT),. Seventy-five organic solvents have been tested and the results have been compared to the experimental values deduced from (dn/dT), and (dd/dT), measurements. This new DFRI expression gives numerical results in better agreement with experimental data than the empirical Eykman's rule.
Introduction From the values of the refractive index n and of the density d of a pure liquid the Lorentz-Lorenz (L-L) expression (n2 l)/(n2 + 2) = NAda/3Mq, gives the mean molecular polarizability This equation is based on the classical internal field ap( I ) Glasstone, S. Texr Book of Physical Chemistry; Van Nostrand: Princeton, NJ, 1940. (2) Partington, J. R. An Aduanced Treatise on Physical Chemiswy; Longmans, Green: London, 1953;Vol. 1V.
proximation (Lorentz field) and it is commonly used for the calculation of molecular polarizabilities at ordinary temperature." Unfortunately, the values of dn2/dd which are deduced from this equation are in disagreement with the experimental results. The (3) Le Fevre, C. G.; Le Fevre, R. J. W. Reo. Pure Appl. Chem. 1955,5, 261. (4) Bottcher, C. J. F. Theory o/ dielectric polarization; Elsevier: Amsterdam, 1973,Vol. I; 1975,Vol. 2. ( 5 ) ProutiZre, A. J . Chim. Phys. 1976,73, 665.
0022-3654/92/2096-3485%03.~0/0 0 1992 American Chemical Society
3486 The Journal of Physical Chemistry, Vol. 96, No. 8, 1992
Proutiere et al.
TABLE I: Examples of Calculated and Experimental Values of the Density Fluctuation Term dn’/dd in d m 3 / k g liquid acetonitrile cyclohexane carbon tetrachloride nitrobenzene
Lorentz-Lorenz (eq 1) 1.31 1.79 0.98 1.72
expt
Maxwell
Eykman
(eq 2) 1.12
(eq 3) 1.03 1.33
(eq 4)
0.71 1.17
0.90 1.53
1.63 0.80 1.44
1.25
1.66
“Sodium light, 293 K and I atm. All measured values from ref 7.
ratio dn2/dd is known as the density fluctuation of the refractive index6 (DFRI) and the direct derivation of the L-L equation relative to d gives the following expression of it n2-1 n 2 + 2 3
The experimental values of dn2/dd are deduced from dn/dT and dd/dT values in a straightforward manner through the equation
As an example the data which concern four typical solvents are reported in Table I. It can be seen that the L-L values are about 18% higher than the experimental ones. This discrepancy is at the origin of many expressions of dn2/dd.8-’3 As most of them are empirical we will limit our discussion to the two of them that are directly related to the present work. The Maxwell e q ~ a t i o n , ~n2 l J-~ 1 a d, leads to the following expression of the DFRI
n2 - 1
(3)
($),=T-
It is based upon the same theoretical equation as the L-L equation by considering that the internal field factor, (n2 + 2)/3, is a mean value which is constant in regard to d fluctuations. But the calculated values (Table I) are now about 15% lower than the experimental ones. The empirical Eykman’s rule,9 (n2 - l)/(n 0.4) a d gives
+
(g)
n2
-1
=- d
2n(n
+ 0.4)
(n2 + 0.8n + 1)
(4)
Eykman’s expression is usually considered as giving the best estimation of #/dd Its agreement with experimental results is especially good for hydrocarbons because it was precisely established for these compounds?*’5 For instance, the deviation in cyclohexane is only 2% (Table I) while the average deviation in the above four liquids is 8%. In this work we intend to establish a new theoretical relation between dn, dd, and dT. The difference between the Lorentz(6) Einstein, A. Ann. Phys., Leiprig 1910, 33, 1275. (7) Riddick, J. A,; Bunger, W. B. Organic Solvents. Techniques of Chemistry, 3rd ed.; Wiley-Interscience: New York, 1970; Vol. 11, and ref-
erences therein. (8) Gladstone, J. H.; Dale, T. P. Philos. Trans. 1863, 153, 317. (9) Eykman, J. F. R e d . Trau. Chim., Pays-Bas 1895, 14, 185. (10) Ward, A. L.; Hurtz, S. S. Ind. Eng. Chem., AMI. Ed. 1938.10, 559. (1 1) Bhagavantam. Scattering of Light and the Raman Effect; Chemical Publishing Company: New York, 1942. (12) Oster, G.Chem. Reo. 1948, 43, 319. (13) Gastaud, R.; Beysens, D.; Zalczer, G. J . Chem. Phys. 1990,93, 3432. (14) Carr, C. I.; Zimm, B. H.J . Chem. Phys. 1950, 18, 1616. (15) Gibson, R. E.; Kincaid, J. F. J . Am. Chem. Soc. 1938, 60, 51 I . (16) Kurtz, S. S.; Lipkin, M. R. J . Am. Chem. SOC.1941, 63, 2158. ( I 7) Dreisbach, R. R. Ind. Eng. Chem. 1948, 40, 2269. (18) Weissberger, A. Physical Methods of Organic Chemistry, 3rd ed.; Interscience: New York, 1959; Vol. I.
Bulk liquid
-n. _ N
Local surrounding n,N
Figure 1. Refractive index n and number of molecules per volume unit
N in the bulk liquid and in the local surrounding of the cavity. Lorenz and Maxwell treatments of DFRI turns on the averaging of the density fluctuations and this point is also the key of our new relation which must be of the greatest interest for the Rayleigh light scattering theory.6J4J*21 In this theory the critical factor is the square (dn2/dd)2 of the DFRI term which has never been represented by a good theoretical equation. Some authors use the classical eq 1 while introducing additional terms in order to get a reasonable agreement with experimental re~u1ts.l~Other authors like Abdel-Azim and Munk2*or Rubio et ala2’use Eykman’s rule (4) which has no theoretical basis. In our recent ~ ~ ron electrooptics k ~ ~ we , have ~ ~adopted eq 3 and now we intend to show that it gives a good representation of the DFRI term. The validity of our new expression can be estimated from data available in the standard literature and it will be tested by comparing the results with the experimental data on a series of 75 liquids.
”ry
ceaeral Form of the Lorentz-Lorenz Equation. In the general formalism of the molecular orientation theory which has been used by several a ~ t h o r s , ~ , ~the , ~refraction ~ ? ~ ’ equation is derived as follows. With the Einstein notations the basic expression of the polarization vector Poi of a medium along the direction x of the applied optical field E is written as Pol = eo(n2 - l ) E = N(m&)
(5)
to is the dielectric constant of vacuum, N is the number of molecules per volume unit, mi is the component of the induced molecular moment along the major axis i and 4 is the cosine of the angle between the x and i axes. The brackets ( ) indicate the statistical average calculated over all molecular orientations. The molecular moment induced by the local field F is
mi = aiFi (6) where ai is the principal molecular polarizability along the major axis i. The general expression of the component Fiof F in the spherical cavity model is28-30
where u is the volume of the molecular cavity and the bar indicates a mean value in the bulk liquid which surrounds the cavity (see Figure 1). The spherical model is at the origin of the Onsager equation which has been widely applied to the interpretation of the electric permittivity of pure liquids. At first sight an ellipsoidal cavity (19) Cabannes, J. La Diffusion MolSculaire de la Lumilre; Les Presses Universitaires de France: Paris, 1929. (20) Rocard, Y. J . Phys. Rad. 1933, 4, 165. (21) Rousset, A. Lu Diffusion de la Lumilre par les MolCcules Rigides, Conflrences Rapports C.N.R.S.: Paris, 1947. (22) Abded-Azim. A. A.; Munk, P. J . Phys. Chem. 1987, 91, 3910. (23) Rubio, R. G.; Caceres, M.; Masegosa, R. M.; Andreolli-Ball, L.; Costas, M.; Patterson, D. Ber. Bunsenges Phys. Chem. 1989, 93, 48. (24) Proutibre, A. Mol. Phys. 1988, 65, 499. (25) Proutibre, A. J . Chim. Phys. 1990, 87. 331. (26) Buckingham, A. D.; Pople, J. A. Proc. Phys. Soc. 1955, 68, 905. (27) Kielich, S.Pkysica 1967, 34, 365. (28) Stratton, J. A. Elecrromagnetic Theory 1941. (29) Barriol, J. Les Moments Dipokaires; Gauthier-Villars: Paris, 1957. (30) Durand, E. Electrostatique, Mgthodes de Calculs Diglectriques; Masson: Paris, 1966.
The Journal of Physical Chemistry, Vol. 96, No. 8, 1992 3487
Density Variations in Pure Liquids model should be more suitable but in that case the cavity shape which is an inaccurate parameter comes into the equations. Moreover, the good results obtained from the spherical model seem to be due to random molecular movements which in most cases give a more or less spherical cavity even when the molecules are not spherical. From eqs 5 , 6, and 7 we obtain
the coefficient of thermal expansion ( l/8)(aD/aT)p,N = (1/u). (du/aT)p$N. After the derivations, we assume as usual that u = M/N,d (M = molar mass); consequently (l/u)(au/dT)p,N = -(l/d)(dd/dT),. Hence, by considering after the derivations that N H N , n = ii, and lV = 1/0, we obtain
(g)
P,N
-(
= 2- n 2 - 1 n 2 + 2
+)p) n2-
dT
P
(15)
Then by considering P constant and by reporting d = NM/N, we obtain from the previous dn2 expression where
a = 3A2/(2ii2 + 1) b = 2(ii2 - 1)/(2A2 + 1)3Uto Then, because ef and Ei = E ef are the only terms which depend on the molecular orientation, eq 8 becomes aaiE (m&) = N-( (e;)2) (9) 1 - ba; As the orientational averaging gives 9 lead to
((e)2) = 1/3 eqs 5 and
In order to obtain a practical equation we first assume that 1 - baj 1 - ba (11) where a = (a,+ a2+ a3)/3is the mean molecular polarizability.
Then we take the mean values of all quantities in (10) and especially R N l/a that we use for the evaluation of a; hence (?I
= 3C0(fi2 - 1)/(ii2 + 2 ) n
(12)
Finally eqs 10, 11, and 12 give 3ii2(ii2 2)a N n2- 1 = €0 (2ii2 l)(& + 2) - 2(fi2 - 1)2/lVa
+
+
(13)
This is a more general practical formulation of the L-L equation. Two other ways of making the approximations lead to the usual Lorentz-Lorenz and Maxwell equations which have proved to be unsuitable for our purpose. If it is only assumed that IV N 1/D Maxwell’s equation, n2 - 1 = (N/to)((A2 2)/3) a, is obtained. Moreover, if A N n one gets the L-L equation where it is now written as n: (n2 - l)/(n2 2) = (N/3t0)a. In both cases the first member is proportional to d. The defect of the methods leading to eqs 1 and 3 lies in the application of the approximations before performing the differentiation of n2. On the contrary, here we will first differentiate the general formula (13) and then we will make the necessary approximations. Differentiation of Eq 13. The following exact differentiation of eq 13 takes into account all partial derivatives which correspond to the locally independent variables N, T, and P:
+
+
dn2 =
(g)p,T+ (g)p,N+ ($) dN
dT
N ,T d P
These partial derivatives are calculated from (1 3) as follows. In (dn2/aN)p,n A, 8, and lV are independent of the fluctuations in N because they are mean values in the bulk liquid and they must be considered as constant. a is also constant because it is an intrinsic molecular property. Finally, the only variable in this case in N and hence
(
n2 - 1
$p,T
=
This DFRI term is the sum of an isothermal term, (an2/ aN)p,7(N/d) = (an2/ad)p,T,and of the ratio between an isdensity term (an2/aT)p,Nand the thermal density variation (dd/dT),. Finally, the new expression of the DFRI term, deduced from eqs 14, 15, and 16, is
N
In (an2/aT)p,Nthe only variables are ii and 8. ii depends on T and, from the average value of n taken at different points of the surrounding volume, (aii2/aT)p,N = (an2/aT)p,N. While N and consequently are constant, 0 depends on T according to
Before any detailed numerical application eq 17 seems more consistent than eqs 1 or 3 for two reasons. First, the isothermal term, (n2 - l)/d, was recently used in the interpretation of our light-scattering measurements which are performed at T constant. Then the additional mixed term (15% of the DFRI for n = 1.45) is in agreement with the average discrepancy between the experimental values (eq 2) and those calculated from eq 3 (15%, Table I). Consequently, it is clear that the failure of classical eqs 1 and 3 is due to the use of unsuitable approximations in the derivation of molecular refraction expressions (see previous comments in the first part of this section) together with inadequate differentiations that do not take into account the detailed meaning of the variables in the partial derivatives. Results Numerical applications were performed on 75 organic solvents collected from Organic Soluents.’ The values of dn/dT and of dd/dT or (l/V)(dV/dT) which are directly available from this source have all been taken into consideration for the calculation of the experimental values of the DFRI (dn2/d& through eq 2. They are reported in Table I1 together with the values calculated through eq 4 (Eykman’s rule) and 17 (our equation). As they are used in the calculations, the values of n and d at 20 OC and 1 atm are also reported.
Discussion and Conclusion At first sight the examination of Table I1 shows that eq 17 gives (dn2/dd)pvalues as suitable as the values deduced from Eykman’s rule 4. As already mentioned, Eykman’s rule is most efficient for hydrocarbons because it has been precisely fitted on these compounds, but this does not prove its general validity. Furthermore, in many other compounds eq 17 gives better results than Eykman’s rule. A precise and detailed comparison between the two equations is difficult because the experimental uncertainties of dn/dT and dd/dT data of Table I1 are not well known and sometimes high. For instance, in the case of 2-butanone (nD = 1.3788), the experimental value 2.16 dm3/kg is evidently erroneous in comparison to the calculated values 1.26 and 1.37 dm3/kg with eqs 17 and 4, respectively. This failure is certainly due to a wrong dd/dT value, -61.2 X because if we use measured d values at 20 and 25 “Cwe deduce d d l d T = -104 X kg dm” K-’ and obtain the more realistic experimental value (dn2/ddp = 1.27 dm3/kg. Seven other compounds are concerned with similar doubtful data: 2-ethoxyethyl acetate ( n D = 1.4044), 2-ethylhexyl acetate (1.4204), 1-decene (1.4215 ) , butyl phosphate (1.4250), piperidine (1.4525), ethylenediamine (1.4568), and ethyl benzoate (1.5057). Moreover, some discrepancies can be explained by specific interactions not included in eq 17 or 4. In the particular case of water (nD =
Proutiere et al.
3488 The Journal of Physical Chemistry, Vol. 96, No. 8, 1992
TABLE 11: Comparison between Values of Density Fluctuation (dn*/dd), from Different Equations. Measured Values of Refractive Index nD (Sodium Light), Density d (kgldm’), dn/dT, and dd/dT (20 OC, 1 atm) Collected from Ref 7
compound‘ acetaldehyde ( I 34) water (0). acetonitrile (265) dimethoxymethane (132) n-pentane (2) ethyl formate (165) ethanol (59) propionaldehyde ( I 35) propionitrile (266) formic acid (1 48) acetic acid (1 49) ethyl acetate (172) n-hexane (7) n-butanone (142) butyraldehyde (136) nitromethane (261) ethyl propionate (184) propyl acetate (1 75) propionic acid ( I 50) n-heptane ( 1 3) I-chloropropane (221) I-hexane (49) acetic anhydride (161) acrylonitrile (276) nitroethane (261) ethyl butyrate (185) butyl acetate (177) 2-nitropropane (263) 2-methyl-I-propanol (64) butyric acid (1 5 I ) I-butanol (62) I-nitropropane (262) acrolein ( I 39) I-chlorobutane (223) 2-ethoxyethyl acetate (346) nonane (22) 3-methyl- 1 -butanol (70) ethyl oxalate (202) decane (27) butyric anhydride (163) ethyl malonate (203) 1.1-dichloroethane (233) 2-ethylhexyl acetate (182) I-decene (53) dodecane (29) dichloromethane (230) butyl phosphate (211) cyclohexane (6) I-octanal (90) crotonaldehyde ( 1 40) 1, I , I -trichloroethane (235) 1,2-dichloroethane (234) chloroform (231) cis-l,2-dichloroethylene(243) methyl oleate (193) piperidine (297) 2-aminoethanol (334) ethylenediamine (290) cyclohexylamine (284) oleic acid (160) carbon tetrachloride (232) trans-decahydronaphthalene(26) trichloroethylene (245) bicyclohexyl (28) cis-decahydronaphthalene(25) I , 1,2,2-tetrachloroethane (236) ethyl benzoate (195) tetrachloroethylene (246) methyl benzoate (194) anisole ( I 27) chlorobenzene (228) benzonitrile (275) styrene ( 5 5 ) o-dichlorobenzene (238) nitrobenzene (264)
nD
1.3311 1.3330 1.3441 1.3534 1.3575 1.3599 1.3614 1.3619 1.3658 1.3714 1.3719 1.3724 1.3749 1.3788 1.3791 1.3812 1.3839 1.3844 1.3865 1.3876 1.3879 1.3879 1.3904 1.3915 1.3919 1.3928 1.3940 1.3943 1.3959 1.3980 1.3993 1.4016 1.4017 1.4021 1.4044 1.4054 1.4071 1.4102 1.4119 1.4127 1.4136 1.4164 1.4204 1.4215 1.4216 1.4242 1.4250 1.4262 1.4295 1.4373 1.4379 1.4448 1.4458 1.4490 1.4521 1.4525 1.4539 1.4568 1.4593 1.4599 1.4602 1.4693 1.4775 1.4799 1.4810 1.4939 1.5057 1.5057 1.5168 1.5170 1.5248 1.5282 1.5468 1.5515 1.5523
d 0.7780 0.9982 0.7822 0.8601 0.6262 0.9225 0.7894 0.7970 0.7818 1.2203 1.0493 0.9006 0.6594 0.8049 0.8016 1.1382 0.8901 0.8884 0.9934 0.6838 0.8909 0.6732 1.0811 0.8060 1.0506 0.8792 0.88 14 0.9884 0.8016 0.9582 0.8097 1.0014 0.8389 0.8862 0.9730 0.7176 0.8104 1.0785 0.7300 0.9668 1.0551 1.1758 0.87 18 0.7408 0.7487 1.3258 0.9806 0.7786 0.8256 0.8516 1.3376 1.2531 1.4892 1.2837 0.8738 0.8601 1.0155 0.8965 0.8668 0.8905 1.5940 0.8697 I ,4680 0.8862 0.8967 1.5946 1.0465 1.6229 1.0886 0.9940 1.1063 1.0050 0.9060 1.3059 1.2033
-dn/dT X IO5 56.3 8.8 45 49 53 44 40 51.8 45 38 38 49 54.2 48 50.5 45 46 48 38 50.8 54 57 41 53.9 43.9 47 47 41.1 39 38.2 39 40.5 57 51 42 46.3 37 42 44.7 41.6 39 52 62 51 42.5 55 48 54 40 55.6 52 51 59 59 37.7 48 36 50 58 35.4 55 43.4 56.8 45.4 44 51 40 53 46 50 54 48 51.9 49 46
-dd/dT X IOs 132.5 20.5 107.8 126.7 97.5 130 86.8 116.5 I03 124 116.5 I20 89. I 61.2 104 137.7 112 116.4 I06 84 123 94.3 121 110.6 118.7 104.6 102 109.9 76.2 98.7 76.1 107.1 116.1
‘In parentheses is the number of the compound in ref 7. *Doubtful value (see Discussion)
Ill
79 77.4 74.6 1 I6 75.1 96.7 105.3 I56 87 77.7 71.9 180 91.2 94.7 81 104.7 165.7 I44 185.7 160 72 1 15.7 79 70 100.9 68 202 74.9 164.9 74 16 159 93.1 164.6 95.4 93.2 108.4 88 87.4 Ill
98.9
expt (eq 2) 1.13 1.14 1.12 I .05 1.48 0.92 I .25 1.21 1.19 0.84 0.89 1.12 1.67 2.16b 1.34 0.90 1.14 1.14 0.99 1.68 1.22 1.68 0.94 1.36 I .03 I .25 I .28 1.04 1.43 1.08 1.43 1.06 1.38 1.29 1.49b 1.68 I .40 1.02 1.68 1.22 1.05 0.94 2.026 1 .87b 1.68 0.87 1s
o b
1.63 1.41 1.53 0.90 1.02 0.92 1.07 1.52 1.2lb 1.33 2.Ogb 1.68 I .52 0.80 1.70 1.02 I .82
1.71 0.96 1 .29b 0.97 I .46 1.63 1.52 I .67 I .a4 I .37 1.44
this work (ep 17) 1.09 0.85 1.14 1.07 1.49 1.02 I .20 1.19 1.23 0.8 1 0.94 1.10 1.51 1.26 1.26 0.90 1.16 1.16 1.05 1.53 1.17 1.55 0.98 1.31 1.01 1.21 1.21 1.08 1.34 1.13 1.35 1.10 1.31 1.24 1.14 1.55 1.38 1.05 1.56 1.18 1.09 0.98 1.35 1.59 1.57 0.90 1.22 1.54 1.47 1.46 0.93 1.02 0.86 1.01 1.49 1.52 1.29 I .48 I .54 1s o 0.84 1.59 0.97 1.61 1.60 0.94 1.48 0.96 1.47 1.62 1.49 1.66 1.94 1.37 I .49
Eykman (eq 4) 1.19 0.94 1.25 1.17 1.63 1.12 1.32 1.31 1.35 0.88 1.03 1.20 1.65 1.37 1.38 0.98 1.26 1.27 1.14 1.66 1.28 1.69 1.06 1.43 1.10 1.32 1.32 1.18 1.46 1.23 1.46 1.19 1.42 1.35 1.24 1.68 1.50 1.14 1.69 1.28 1.18 I .06 1.45 1.72 1.70 0.97 1.31 1.66 1.58 1.57 I .oo I .09 0.92 1.08 1.60 1.63 1.38 1.58 I .65 1.61 0.90 1.69 1.02 1.71 1.69 0.99 1.55 1 .oo
1.54 1.69 1.55 1.72 2.00 1.40 1.53
J. Phys. Chem. 1992,96, 3489-3504
1.3330)both eqs 17 and 4 give poor results (0.85and 0.94instead of 1.14)and this must be due to the particular structure of water which is sensitive to temperature effects. This qualitative analysis has been completed by the calculation of the relative deviations A = (Y(calcu1ated) - Y(experimental))/Y(experimental), where Y = (dn2/dd)p For the 75 compounds of Table I1 the mean values A of A are +4.1% (Eykman’s rule (4)) and -3.3% (our eq 17). The absolute mean relative deviations given in the same order are equal to 8.3% and 6.6%. After removal of the nine doubtful cases (seeabove) the A values for the 66 remaining solvents become equal to +6.1% and -1.6% and the values to 6.4% and 4.1%. These results show that the uncertainties are better distributed (A) in the case of the new equation, and the individual values are also in better agreement with the experimental results. A part of the residual discrepancies could be explained by taking into account polarizability anisotropy effects related to nonspherical molecular shapes. Unfortunately,
3489
such calculations need more accurate and reliable values on dn/dT and dd/dT values than those which are usually available in the literature. In conclusion, through a more general and consistent derivation than those which are usually performed, we have deduced a new expression of the DFRI term. First, the new equation gives results which are better distributed and with slightly lower deviations than those deduced from the empirical Eykman’s rule. But its main interest lies in its theoretical derivation which has been made by using the same assumptions as for the isothermal DFRI term in the Rayleigh light scattering t h e ~ r y . ~ ‘ JThese ~ good results give a new direct m n f i i t i o n of the validity of our theoretical method. Similar studies on dielectric constants and attempts to explain residual discrepancies are in progress. Acknowledgment. We express our gratitude to Prof. Martial Chabanel for his helpful comments.
Molecular Theory of Chromatography for Blocklike Solutes In Anisotropic Stationary Phases and Its Application Chao Yant and Daniel E.Martire* Department of Chemistry, Georgetown University, Washington, D.C. 20057-2222 (Received: September 26, 1991)
To develop a more complete and informative molecular theory of chromatography,DiMarzio’s lattice model, based on statistical thermodynamics, is extended and used to describe the equilibrium partitioning of blocklike molecules between an isotropic mobile phase and an anisotropic stationary phase. Both repulsive and attractive interactions are taken into account. The configurational partition functions for both the mobile and stationary phase systems are rigorously derived, and from them thermodynamic properties such as the Helmholtz free energy, the chemical potential, the entropy, and the enthalpy can be expressed in terms of state variables (temperature and density) and molecular parameters (interaction energies and molecular dimensions). A retention equation, applicable to gas, liquid, and supercritical fluid chromatography (GC, LC, and SFC), for blocklike solutes in an isotropic mobile phase and an anisotropic stationary phase is obtained. The solute distribution coefficient, K,is well represented by In K = QIAmin+ Q2Acf+ Q3Vw,where A ~ , ,Act, , and V, are, respectively, the minimum area, the effective contact area, and the van der Waals volume of the solute molecules, and Ql, Q2, and Q3 are related to the state variables and molecular parameters. A linear relationship between In K and for isomeric, polycyclic aromatic hydrocarbon solutes in anisotropic stationary phases is predicted. The theory is successfully applied to the interpretation and analysis of GC, LC, and SFC data.
1. Introduction
The separation and identification of polycyclic aromatic hydrocarbons (PAHs) are important and challenging tasks. The pioneering work of Kelker’ and Dewar and Schroeder* on the use of liquid crystals as stationary phases in gas chromatography (GC) opened a realm of new possibilities for the truly effective separation of structural isomers. It has been shown that liquid crystals are very effective stationary phases for the separation of isomers in complex PAH m i ~ t u r e s . ~ -In~ liquid chromatography (LC), immobilized bonded p h a w with some anisotropic character, such as (4,4’-dipentylbiphenyl)dimethylsiloxane (55B),6 and octadecylsilane (C18)polymeric phase,’ also exhibit good shape selectivity for isomeric polycyclic aromatic hydrocarbons that is similar to the shape selectivity observed with bulk liquid crystalline phases. The retention mechanism of PAHs, especially on anisotropic phases such as liquid crystalline and cl8 bonded phases, is still not fully understood. Sleight8 studied the structure-retention ‘Present address: Analytical Research and Development, Building 360/ 1034, Sandoz Pharma Ltd.. CH-4002 Basel, Switzerland. Corresponding author.
relationship of PAHs in LC on c18 columns and derived a simple linear relationship between the retention and the number of carbon atoms. Lockeg suggested that the basis of reversed-phase LC selectivity for PAHs is the relative solubility of the PAHs in the mobile phase. However, the solubility data alone do not adequately explain the unique chromatographic selectivity of many PAHs. In GC, Janini et al.lOJ’noted that the retention of PAHs on liquid-crystal stationary phases correlated with the shape of the (1) Kelker, H. 2.Anal. Chem. 1963, 198, 254. (2) Dewar, M. J . S.; Schroeder, J. P. J . Am. Chem. SOC.1964,86, 5235. (3) Janini, G. M.; Muschik, G.M.; Zielinski, W. L., Jr. Awl. Chem. 1976, 48, 1879. (4) Kelker, H. Advances in Liquid Crystals; Brown, G . H., Ed.; Academic Press: New York, 1978; Vol. 3, p 237. ( 5 ) Witkiewicz, Z.; Mazur, J. LC-GC 1990, 8, 224. (6) Lochmiiller, C . H.; Hunnicutt, M . L.; Mullaney, J. F. J . Phys. Chem. 1985.89, 5770. (7) Wise, S. A.; Sander, L. C.; Chang, H. C. K.; Markides, K. E.; Lee, M . L. Chromafographia 1988. 25, 473. (8) Sleight, R. B. J . Chromatogr. 1973, 83, 31. (9) Locke, D. C. J . Chromarogr. Sci. 1974, 12, 433. (10) Janini, G.M.; Johnston, K.; Zielinski, W. L., Jr. Anal. Chem. 1975, 47, 670. (11) Zielinski, W. L.;Janini, G.M. J . Chromatogr. 1979, 186, 237.
0022-365419212096-3489$03.00/00 1992 American Chemical Society
