4377
THERMODYNAMICS OF BINARY LIQUIDMIXTURES On the basis of estimated limiting slopes through the 25" pKz value, we conclude that AH" is uncertain t o ' about 15% and AS" to about 6%. The unusual strength of squaric acid, however, is in the first dissociation, and a clearer understanding of this phenomenon awaits more accurate measurements probably by some
other technique. I n this laboratory we are attempting to measure the first ionization conductometrically.
Acknowledgment. We thank Professor Robert I. Gelb for his many helpful discussions and the Research Corporation for partially supporting this work.
Thermodynamics of Binary Liquid Mixtures by Total Intensity Rayleigh Light Scattering. I1 by Harvie H. Lewis, Raymond L. Schmidt, and H. Lawrence Clever* Chemistry Department, Emory University, Atlanta, Georgia 90922 (Received March 26, 1970)
Rayleigh scattering and depolarization were measured for benzene-methanol, benzene-propanol-2, and benzene-n-dodecane solutions. The gradient of the concentration dependence of refractive index, An/ AX, was directly measured as a function of composition for the solutions by differential refractometry. The measurements were combined with total intensity light scattering and depolarization data to obtain activity coefficients and excess Gibbs free energies of mixing which were compared with results from vapor pressure measurements for the alcohol-containing solutions. The excess free energy of mixing from light scattering a t 0.5 mole fraction was 4% below the accepted vapor pressure value for benzene-methanol and 16% below for benzene-propanol-2. The benzene-n-dodecane solutions appear to have a negative excess free energy of mixing over part of the composition range. The light scattering method, a t best, can give only approximate thermodynamic values for solutions with a negative excess free energy of mixing.
Rayleigh scattering from liquids and liquid solutions is an active field of study with a constant reevaluation of the theoretical equations as they apply t o liquids.l-* Over the past several years there have been several studies of Rayleigh scattering from binary nonelectrolyte solutions of small molecules. I n these studies either the Rayleigh scattering and depolarization measured for the solution were treated to obtain activity coefficients of the components and excess Gibbs free energy of mixing6-' or properties of the solutions were used to calculate a predicted Rayleigh scattering which was compared with experiment.*bg Studies of the second type have also been carried out on aqueous electrolyte solutions.10 This work is a further study of the relationship between Rayleigh scattering and the thermodynamic properties of binary nonelectrolyte solutions, with particular attention to the contribution of the gradient of t h e concentration dependence of refractive index t o the relationship. The total Rayleigh scattering, RgO, can be separated into an isotropic contribution, Ris, and an anisotropic contribution, R,,, by the Cabannes relation. In pure liquids the isotropic scattering is due to density fluctua-
tions, Rd. I n solutions two further terms contribute to the isotropic scattering, a concentration fluctuation, R,, and a density-concentration fluctuation cross term, R#. The total isotropic Rayleigh scattering from a solution is the sum R i s = Rd R, R#with
+ +
Rd
=
(a2/2b4)k TKt [N(be/bN)t 1'
(1)
* T o whom correspondence should be addressed. (1) D. J. Coumou, E. L. Mackor, and J. Hijmans, Trans. Faraday Soc., 60, 1539 (1964).
(2) A. Litan, J . Chem. Phys., 48, 1052, 1058 (1968). (3) M . Kerker, "The Scattering of Light and Other Electromagnetic Radiation," Academic Press, New York, N. Y., 1969. (4) D. McIntyre and J. V. Sengers in "Physics of Simple Liquids," H. N. V. Temperley, J. 5. Rowlinson, and G. S. Rushbrooke, Ed., North-Holland Publishing Co., Amsterdam, 1968, Chapter 11. (5) D. J. Coumou and E . L. Mackor, Trans. Faraday Soc., 60, 1726 (1964). (6) R. L. Schmidt and H. L. Clever, J . Phys. Chem., 72, 1529 (1968). (7) R. 6 . Myers and H . L. Clever, J . Chem. Thermodynamics, 2, 53 (1970). (8) G. D. Parfitt and J. A . Wood, Trans. Faraday SOC.,64, 805 2081 (1968). (9) M. 6. Malmberg and E. R . Lippincott, J . CoZZoid Interface Sci., 27, 591 (1968). (10) B. A. Pethica and C. Smart, Trans. Faraday Soc., 62, 1890 (1966).
The Journal of Physical Chemistry, Vol. 74, N o . 26, 1970
H. H. LEWIS,R. L. SCHMIDT, AND H. L. CLEVER
4378
R,
( ~ 2 / 2 X ~ 4 ) ~ T V X ~ ( b ~ / b X ~ ) z /(2) ( b p ~calculated / b ~ 2 ) values of Rid, used to form the Rid/& ratio, which came from uncertainties in values of the solution R' = ( ~ ~ / b ~ ) k[N(be/bN)t] TKt [Cz(de/bCz)] (3) refractive index and values of the refractive indexwhere XO is the wavelength of the incident light in a composition slope in the term nbn/bXz. vacuum, V is the molar volume, X 1 and Xz are the To obtain activity coefficients of binary solutions mole fractions of components 1 and 2, respectively, from Rayleigh light scattering data one should measure pz is the chemical potential of component 2, E is the or have available, in addition to the solution Rayleigh optical dielectric constant (e = ?a2, where n is the rescattering and depolarization, the solution isothermal fractive index at wavelength Xo), Kt is the isothermal compressibilities, the optical dielectric constant-number compressibility, CZ is the concentration of component density slope, the solution refractive indices, the re2 in g/cc, N is the number density, k is the Boltzmann fractive index-composition slope, and the solution constant, and T is the absolute temperature. densities. Neither we nor others have measured all In an ideal solution, by definition, bpz/bXz = RT/X2 these solution properties. Many of them have been and R, becomes the ideal concentration scattering, Rid, approximated from theoretical or semiempirical relawhere tions. For example, (1) the pure liquid isothermal compressibilities, when not available in the literature, Rid = (n2/2X04)vX~Xz(~b/bXz)~/Na are found from the pure liquid Rayleigh scattering." = (2+/Xo4) VX1X2(nbn/bX2)2/N, (4) The solution isothermal compressibilities are then assumed to be additive in volume fraction. (2) The numwhere N , is Avogadro's number. ber density dependence of the optical dielectric constant For a real solution the ratio of ideal to real concenis calculated from the Clausius-Mosotti equation multitration scattering is plied by an empirical dense media correction factor.6111 Rid/Rc = (X2/RT)(bp2/bXz) Recently a somewhat improved correction factor has been suggested. l 2 (3) The solution refractive indices = (Xz/RT)(RT/Xz bpzE/bXz) (5) have been estimated from the Lorentz-Lorenz equation. where bpzE= RT b In f z with p z E andfz being the excess The dn/dx and dn/dc values were obtained either by chemical potential and activity coefficient, respectively, differentiation of a quadratic fit to the calculated redue to component 2 in the solvent reference state. fractive indices as a function of composition or alterRearrangement and integration of eq 5 gives natively dn/dx was calculated directly from a differentiated form of the Lorentz-Lorenz equation, which required an accurate density gradient of the solution. I n this work a direct experimental determination of The activity coefficient of component 1 is obtained An/Ax was made, and the resulting values were used from a similar set of equations and the excess Gibbs in the calculation of activity coefficients and excess free energy is calculated from free energies of mixing from the Rayleigh scattering and depolarization values. AGE = RT (XI lnfi X ZI n f ~ ) (7) Experimental Section The relationship between the concentration fluctuaMaterials. Benzene, Mallinckrodt, was distilled on a tion contribution to isotropic scattering, R,, and 20-plate stainless steel helices packed column. nchemical potential is clear and direct, but converting Dodecane, Phillips, 99 mol %, was shaken with concenRayleigh scattering data into activity coefficients of the trated HzS04, washed until neutral to litmus, and dissolution components is a multistepped process with tilled at 63 f 2 mm pressure, 130". A freezing point potential errors in every step, If one accepts the theocurve indicated liquid-soluble, solid-insoluble impurity retical equations some of the possible sources of error at 0.12 mol %. Methanol, Fisher reagent grade, was include ( 1 ) both the reliability of the Cabannes relation distilled from over CaO before use. Propanol-2, Fisher and the experimental uncertainty in the solution depo99 mol %, was distilled twice from over CaO. larization values used to partition RQo into Ri, and Ran; Solution Preparation. Solutions for the measure(2) uncertainties in the calculated values of Rd and R' ment of An/Ax were prepared by weight in special which are required to obtain R, from the difference weighing bottles18 with care taken to correct for any function R, = Ri, - Rd - R f ;(a) uncertainties in the vapor loss. Ax varied from 0.005 to 0.1 with most calculated values of Rd came from uncertainties in the values falling near 0.02. values of solution isothermal compressibilities and in Light scattering measurements were as described in values of the function N ( d e / b N ) t and (b) uncertainties earlier work.6)11 in the calculated values of R' include the same terms reimportant to Rd Plus uncertainties in the (11) R. L. Schmidt, J. Colloid Interface Sei., 27, 516 (1968). fractive index and the gradient of the composition (12) G. H. Meeten, Nature, 218, 761 (1968). (13) R. Battino, J. P ~ Z /Chern., S. 70, 3408 (1966). dependence of refractive index; (3) uncertainties in the =
+
+
The Journal of Physical Chemistry, VoL 74, N o . 26, 1970
4379
THERMODYNAMICS OF BINARY LIQUIDMIXTURES 0.2L
c
Y
I
I
I
I
I
I
I
I
PROPANOL-2
1
IE
-I
I
I
1
I
I
I
I
I
I
“ 9 el 0.1
8-
1
\PROPANOL-~
t METHANOL
6 ,
0
a w a
-e
$1 0.2
- IE 0 3
0.I
I
I
I
I
Figure 2. (An/A2)oalod L-L] 100/(An/Az)ex,tl
BENZENE MOLE FRACTION Figure 1. Experimental An/Ax us. mole fraction, 25O, 546 mp.
Diferential Refractometer. Direct determination of values of An/Ax were made on a Brice-Phoenix Model BP-2000-U differential refractometer. The instrument was calibrated with KC1 and temperature was controlled to 25.0 f 0.1”.
Results and Discussion Values of An/Ax determined experimentally are shown in Figure 1. Values from the smoothed curve are in Table I where a refractive index-composition table is constructed using the 546-mp refractive index of benzene at 25” of 1.5020.16 The values appear reasonable, and the benzene-methanol results are in good agreement with literature values.16 The Lorentz-Lorenz (L-1,) equation is generally accepted €or the calculation of solution refractive indiees.17 However, to obtain the refractive indexcomposition gradient, An/Ax, the use of the L-L equation is found to be unsatisfactory. The direct differentiation of the L-L equation yields an expression which is unduly sensitive to values of the density-composition gradient. The per cent deviation, [(Alz/Az),,,,~ ( A ~ / A ~ ) , , ~ , ~ L - L ] ~ ~ ~ / ( Ais~ shown / A X ) ~in~Figure ~ ~ I , 2. With the exception of the benzene-methanol solutions the errors are quite large, rising to 16% at the composition extremes. Values of the refractive index16at 546 mp and An/Ax calculated from the Lorentz-Lorenz equation agree well with experiment for the methanol-benzene solutions. The refractive index is in error 0.091 and 0.074% at 0.5
I
-I
I
1
0.2 0.4 0.6 0.8 BENZENE MOLE FRACTION Per cent difference [(An/Az)exptl -
I.o
us. mole fraction.
and 0.25/0.75 mole fractions, respectively. The addition of a term ux1xz to the Lorentz-Lorenz equation with a = 5.5 X makes the agreements with experiment almost quantitative (average deviation 0.005%). Both the L-L and the modified L-L equation reproduced An/ AX to 1-3%. The Lorentz-Lorenz equation is not strictly applicable t o the liquid phase. The regular pattern of the deviations of the experimental data from the LorentzLorenz equation suggests we use an equation with corrections for the closeness of approach of the molecules found in the liquid state. Bottcherls suggested that the induced dipole in the molecule would also induce a dipole in its surroundings producing what he called the reaction field which enhances the polarizability of the molecule. The reaction field is a function of the radius of the cavity occupied by the molecule and of the refractive index. A computer program was written to calculate the refractive index of a binary solution from Bottcher’s total molar polarization equation and an assigned value of the cavity radius for the reaction field calculation. The L-L equation was used t o approximate the value of the refractive index for the reaction field, and the cavity radius was calculated assuming the molecules were spheres occupying the total molar volume. The effective polarizabilities of benzene and methanol were increased 10 and 6%, respectively, by the reaction field. The refractive indices calculated (14) A. Kruis, 2.Phys. Chem. (Leipzig),34B, 1 (1936). (15) A. F. Forziati, J . Res. ,?Tat. Bur. Stand., 44,373 (1950). (16) S.E.Wood, S. Langer, and R. Battino, J . Chem. Phys., 32, 1389 (1960). (17) W.Heller, J . Phys. Chem., 69, 1123 (1965). (18) C. J. F.Bottcher, “Theory of Electric Polarization,’’ Elsevier, Amsterdam, 1952, Chapter VI.
The Journal of Physical Chemistry, Vol. 74, No. 26, 1970
H. H. LEWIS,R. L. SCHMIDT, AND H. L. CLEVER
4380
Table I: Smoothed Values of An/AX and Calculated Refractive Indices of 546 mp and 25" Benzene mole
fraotion
XZ
----Propanol-2-benzene--An/ AX
(0. 1246)d
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
...
0.138
*..
0.132
...
0.128
...
0.125
...
0.122
...
0.121
...
0.120
...
0.121
...
0.122
*..
0.124
...
--Methanol-benzene---
----n-Dodecane-benzene---7
niisi
An/ AX
niiei
AnlAX
n w
(1.3774y 1.3767
(0. 1741)d
(1. 3279)b 1.3278
(0.0805)d
(1.4215)" 1.4226
...
...
... *.. 1.4165 ...
...
...
...
*..
0.141
...
0.123
...
1.4653
...
0.099
...
1.4896
...
...
The Journal of Physical Chemistry, Vol. 74, No. 36,1070
...
...
1.4481
...
0.085
*..
...
0.110
1.4827
I . .
...
1,4566
...
1.4676
...
0.146
1.4926
1.4822
,..
...
...
0.198
...
1.50200
Reference 16.
Refractive index
1.5020"
0.0
...
1.4416
0.065
a Pure liquid refractive index at 5461 8 calculated from n-X and an/&! data in Timmermans. of benzene at 25" and 5461 8 from ref 15. d Slope if refractive index is linear in mole fraction.
from Bottcher's total molar polarization equation agreed with experiment within 0.035%, which is a substantial improvement over the L-L equation agreement. However, the Bottcher equation was very sensitive to small changes in the cavity radius. Although promising, the Bottcher approach was pursued no further at this time. The light scattering measurements are reported in Table 11. Values of Ria were calculated using smoothed values of the experimental An/Ax values. Thermodynamic results for benzene-methanol are given in Table I11 where the excess free energies are compared with the accepted literature values1e and with values obtained using An/Ax values approximated from the Lorentz-Lorenz refractive indices of the solution. The use of the experimental An/Ax values did not alter appreciably the free energy of mixing of benzenemethanol at 0.5 mole fraction. However, the overall shape of the free energy-composition curve was improved with values being consistently 4% low except in the dilute solutions where differences approached 10%. Measurements on benzene-propanol-2 solutions were made at temperatures of 25 and 43.7". Activity coefficients and excess free energies of mixing are given in Table IV. The free-energy results parallel but are 16% below values obtained in a careful vapor pressure study.2O
1.4364
...
*
0.094
1.50200
*..
0.052
1.4718
...
1.4252
0.044
... 1 4454 ... 1.4595 ...
0.109
1.4774 4 . .
...
1.4292
0.162
1.4533
...
1.4105
...
1.4412
*..
... ...
0.187
1.4290
... 0.031 ... 0.037 ...
1.3886
0.219
... ... 1.4283 ... 1.4320 ...
0.026
1.3618
0.268
1.4037
...
...
0.340
1.3905
...
0.2
0.4
0.6
0.8
1.0
%ENZENf, MOLE FRACTION Figure 3. Propanol-2-benzene solutions; a comparison of excess free energy of mixing vs. mole fraction: upper curve, 45' from vapor pressure measurements, ref 20; lower curves, 25 and 43.7" from light scattering measurements.
4381
THERMODYNAMICS OF BINARY LIQUIDMIXTURES Table I1 : Total Intensity Rayleigh Scatt,ering and Depolarization Mole fraction Xa
436 mp
546 m p
Mole
Reo X 106,
Reo X 10-8, om-1
fraction Xa
om-1
pp
PP
8.60 12.85 18.20 47.95 59.70 84.50 95.85 62.15 49.70 47.75 46.50
0.093" 0.119" 0.131" 0,125" 0.124 0.133 0.277 0.385 0.393 0.420
3.10 4.50 6.45 16.30 21 .OO 28.90 32.55 21.10 16.85 16.15 16.10
0,047" 0.093" 0.119" 0. 131" 0,125" 0.120 0.134 0.271 0.377 0.393 0,410
Propanol-2 (1)-benzene (2), 25" 0.000 0.043 0.155 0.157 0.218 0.319 0.395 0.461 0.477 0.645 0.681 0.834 0.912 0.935 0,971 1,000
a
9.75
...
0.06
...
20.9 23.3 26.7 37.0 44.0 51.0 51.0 59.3 61.4 55.6 49.5 49.9
0.17 0.19 0.185 0.175 0 * 185 0.175 0.185 0.21 0.21 0.295 0.365 0.385
46.5
0.42
.*.
*..
546 m p
PP
Reo x 10-0, om-'
PP
Propanol-2 (1)-benzene (2), 43.7"
Methanol (1)-benzene (2), 25' 0.0000 0.0275 0,0585 0,2235 0.2905 0,4400 0,5740 0.8250 0.9155 0,9500 1* 000
436 mp Roo X 108, om-1
0.05 0.10 0.17 0.16 0.16 0.165 0.18 0.17 0.175 0.20 0.215 0.29 0.35 0.39 0.425 0.41
3.51 4.63 7.1 8.1 9.1 12.0 14.9 16.6 16.8 19.7 20.3 18.6 16.7 '17.3 16.7 16.1
0.000 0.026 0.038 0.043 0.155 0.218 0.319 0.395 0.461 0.477 0,645 0.681 0.834 0.912 0.935 0.971 1.000
10.95 11.4
...
...
0.06 0.10
*.. ...
21.7 25.8 35.3 41.2 46.8 46.6 55.3 56.5 54.5 51.1 50.9
0.145 0.185 0.185 0.185 0.185 0.185 0.21 0,235 0,305 0,365 0.39
50.1
0.43
*..
...
3.90
0.05
4.4 4.9 7.5
0.08 0.10 0.13
... ...
11.6 13.3 15.4 15.2 18.1 19.1 18.4 17.5 17.6 17.5 17.5
...
...
0.18 0.18 0.185 0.175 0.20 0,235 0.295 0.35 0.385 0.405 0,425
n-Dodecane (1)-benzene (2), 30' 0.000 0,040 0,397 0,419 0.592 0.708 0.831 0.884 0,920 0.964 0.990 1.000
18.0 18.7 28.1 29.1 35.3 42.0 49.2 50.0 50.3 50.0 48.3 49.5
0.24 0.27 0.315 0.31 0.30 0.315 0.32 0,325 0.34 0.38 0.40 0,415
5.75 5.95 9.05 9.35 11.5 13.9 16.4 16.7 16.9 16.8 16.5 16.8
0.175 0.23 0.28 0,275 0.32 0.305 0.315 0.30 0.315 0.34 0.375 0.41
From data reported in ref 5.
Table I11 : Excess Free Energy of Mixing and Activity Coefficients of Methanol (1)-Benzene (2) Solutions a t 25'
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0 0,0174 0,0698 0.1545 0.269 0.414 0,603 0.864 1.24 1.81
...
... 1.502 1.205 0,950 0.738 0.560 0.406 0.266 0.142 0.0431 0.0
0 98 176 233 270 289 287 264 214 130
0 106 197 238 274 293 289 267 220 140
0 108 188 246 283 301 299 274 223 137
0
0
0
(I), R i d calculated using experimental An/AX values; (2), calculated using An/AX values approximated from LorentzLorenz equation; (3), ref 19. a Rid
It was disappointing to find that use of the smoothed experimental An/Ax values actually made the agree-
7 w-108.0
0 . 0.2 BENZENE,0.4 MOLE 0.6 FRACTION0.8
I.o
0
Figure 4. n-Dodecane-benzene solutions, 30"; excess free energy and activity coefficients ua. composition.
ment 37?* poorer than use of An/Ax values approximated from the Lorentz-Lorenz equation. (19) G. Scatchard and L. B. Ticknor, J . Amer. Chem. SOC.,74, 3724 (1952). (20) I. Brown, W. Fock, and F. Smith, Aust. J . Chem., 9, 364 (1956). The Journal of Physical Chemistry, Vol. 74, No. 26, 1070
H. H. LEWIS,R. L. SCHMIDT, AND H. L. CLEVER
4382
Table IV: Activity Coefficients and Excess Free Energies of Mixing for Propanol-2-Benzene, and n-Dodeoane-Benzene Solutions by Rayleigh Light Scattering Mole fraction benzene XB
AUE, f1
fa
kcal/mol
Propanol-2 (1)-benzene (2), 25" 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.00 1.01 1.05 1.13 1.28 1.53 1.96 2.56 3.86
2.51 2.38 2.13 1.86 1.59 1.38 1.21 1.09 1.02
0.05 0.11 0.15 0.19 0.21 0.22 0.20 0.16 0.09
Propanol-2 (1)-benzene (Z), 43.7' 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.01 1.04 1.10 1.18 1.29 1.50 1.85 2.46 3.46
2.95 2.27 1.88 1.65 1.47 1.32 1.18 1.08 1.01
0.08 0.13 0.16 0.19 0.20 0.205 0.19 0.15 0.08
n-Dodeoane (1)-benzene (2) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.98 0.92 0.85 0.79 0.75 0.75 0.79 0.92 1.18
0.52 0.74 0.94 1.07 1.14 1.15 1.12 1.06 1.02
-0.05 -0.08 -0.08 -0.07 -0.05 -0.02 0,005 0.02 0.02
The temperature coefficient of the excess free energy gave values of TASE and AHmwhich agreed well with literature values at 0.3, 0.4, 0.5, and 0.6 mole fraction alcohol. For example, at 0.5 mole fraction AHm and TASE are 370 and 170, respectively, from our data and 393 and 152, respectively, from the literature.20 However, at high alcohol mole fraction the temperature coefficient of AGE goes negative (Figure 3) while the literature TASE values indicate a positive temperature coefficient of AGE at all compositions at 45". There is ample evidence in the literature of a change in sign of TAXE with composition going to all negative values as temperatures decrease for benzene-ethanoll*~lg toluene-ethanol,20g22 and methylcyclohexane-ethano121r22 solutions. Thus this change in the temperature AGE may be real for the benzene-propanol-2 solutions, but the uncertainty in our AGE values at high alcohol concentration allows no reliable calculation.
The Journal of Physical Chemistry, Vol. 74, N o . 26, 1970
Activity coefficients and excess free energies of mixing of the n-dodecane-benzene solutions are given in Table IV and Figure 4. Over much of the composition range R, < Ria. When this occurs uncertainties in the Ria/ R, ratio become large and the activity coefficients of less than unity that are obtained are not as reliable as the activity coefficients greater than unity. Thus the thermodynamic values by light scattering are at best only an indication of what may be happening in the ndodecane-benzene system. The negative free energy is contrary to what is found in most aromatic-aliphatic hydrocarbon solutions,2a although other hydrocarbon solutions, e.g., n-dodecane-cyclohexane solutions, do show a negative excess free energy.24 It is apparent that the use of experimental An/Ax values does not lead to particularly improved values of activity coefficients and excess free energies of mixing from light scattering measurements. Thus, other approaches must be sought to improve our understanding of the light scattering technique. An obvious place to attack is the de/dN term and to a lesser extent the solution isothermal compressibility. Both terms contribute to the density fluctuation part of the isotropic scattering. An experimental approach is to measure the pressure dependence of refractive index and use equations with N and p as independent variables
R~ E ( ~ Z / ~ X ~ ~ ((ae/ap),2 ~T/KJ A theoretical approachz6would be the use of Onsager's equation for the internal field for the calculation of the density dependence of refractive index
p(ae/ap)=
(E
- 1)(2€2
+ 4 / ( 2 E 2 + 1)
which would result in larger values for R, and improve results for solutions in which the light scattering technique gives low experimental values. The measurement of depolarization needs improvement. The use of laser sourcesZ6 may be the answer to improved depolarization measurements. Acknowledgment. We thank the Emory University Computer Center for use of the computer. The work was supported by National Science Foundation Grant GP-5937 and was presented at the Southeastern Regional Meeting of the American Chemical Society, Richmond, Va., November 1969. (21) C. B. Kretschmer and R. Wiebe, J . Amer. Chem. Soc., 71, 1793, 3176 (1949). (22) S. C. P. Hwa and W. T. Ziegler, J . Phys. Chem., 70, 2572 (1966). (23) J. S. Rowlinson, "Liquids and Liquid Mixtures," Academic Press, New York, N. Y., 1959, p 149. (24) J. D. Gomez-Ibanez, J. J. C. Shieh, and E. M. Thorsteinson, J. Phys. Chem., 70, 1998 (1966). (25) G. Oster, Chem. Rev., 43, 358 (1948). (26) R. C. C. Leife, R. 5.Moore, and S. P. S. Porto, J . Chem. Phys., 40, 3741 (1964).
