J. Phys. Chem. 1980, 84, 2883-2887
be used to determine qualitatively or quantitatively relative internuclear distances in systems of unknown structure. The inverse problem of obtaining uij and pi values from a series of observed curves appears soluble in principle. A successive iteration method, similar to that used in obtaining shifts and coupling constants from high-resolution NMR spectra, would be expected to converge. In a series of trials, curves geinerated from small arbitrarily parameterized spin systems could be reproduced, and the parameters found to good precision by hand iteration (Le., guessing parameters and making logical adjustments). Once a satisfactory fit is obtained, it can be tested by performing further experiments (perturbation of new spins, simultaneous inversion of two spins, nonselective perturbations, etc.). The interpretation of the aij and p* thus obtained in terms of structure obviously requires care. Since the aij depend both on internuclear distance and correlation time, the possibility of internal motions or anisotropic reorientation would have to be carefully considered. In the experiments above, brucine, a rigid, nearly spherical molecule, was chosen to avoid this difficulty. To the extent that the unknown molecule is rigid and rotates isotropically, the ui‘ will be pro,portional to rij+, and adjacent nuclei shouli be identifiable. Further examination of some real cases will help decide this question.
2883
Acknowledgment. This work was supported by grant AM-16532 from the National Institutes of Health. Spectra were obtained at the NMR Facility for Biomedical Studies, supported by grant RR-00292, also from the National Institutes of Health. References and Notes P. Balaram, A. A. Bothner-By, and J. Dadok, J. Am. Chem. Soc., 94, 4015 (1972). W. E. Hull and 8. D. Sykes, J. Chem. Phys., 63, 867 (1975). A. Kaik and H. J. C. Berendson, J. Magn. Reson., 24, 343 (1976). B. M. Harlna, A. A. Bothner-By, and T. J. GIII, 111, Biochemistry, 16, 4504 (1977). A. A. Bothner-By and P. E. Johner, Biophys. J., 24, 779 (1978). K. Akasaha, M. Konrad, and R. S. Goody, FEBS Left., 96,267 (1978). J. D. Stoesz, A. 0. Redfield, and D. Mallnowski, FEBSLett., 91,320 (1978). S. L. GordonandK. W W i , J. Am. Chem. Soc.,100,7094(1978). 0. Wagner and K. WMhrich, J . Magn. Reson., 33, 675 (1979). A. Dubs, 0. Wagner, and K. WLithtich, Bbchim. Biophys. Acta, 577, 177 (1979). M. VaJk, K. Nagayama, K. Wuthrich, M. L. Mertens, and J. H. R. Kagi, Biochemistry, 18, 5050 (1979). A. A. Bothner-By and J. H. Noggle, J . Am. Chem. Soc., 100,5152 (1979). J. Carter, G. W. Luther, 111, and T. C. Long, J. Magn. Reson., 15, 122 (1974). J. H. Noggle and R. E. Schirmer, “The Nuclear Overhauser Effect”, Academic Press, New York, 1971, p 25. “Development of Process and Equlpment for Production of Phosphoric Acid”, published by the Tennessee Valley Authority, 1948.
Liquid-Liquid Critical Phenomena. The Coexlstence Curve of n-Heptane-Acetic Anhydride N. INagarajan, Anll Kumar, E: S. R. Gopal, Department of Physics, Indian Institute of Science, Bangalore 560 0 12, Indk
andl S. C. Greer” Department of Chemlsty, University of Maryland, Coiiege Park, Myland 20742 (Received: April 14, 1980)
The coexistence curve of the binary liquid mixture n-heptane-acetic anhydride has been determined by the observation of the transition temperatures of 76 samples over the range of compositions. The functional form of the difference in order parameter, in terms of either the mole fraction or the volume fraction, is consistent with theoretical predictions invoking the concept of universality at critical points. The average value of the order parameter, the diameter of the coexistence curve, shows an anomaly which can be described by either an exponent 1- a,as predicted by various theories (where a is the critical exponent of the specific heat), or by an exponent 20 (where P is the coexistence curve exponent), as expected when the order parameter used is n o t the one the diameter of which diverges asymptotically as 1 - a. Introduction The concept of critical point universality states that all systems with the same spatial dimensionality and with the same dimensionaliity of the order parameter should be in the same “universality class” and should have the same “critical exponents” which describe the behavior of thermodynamic properties as the critical temperature, T,,is approached.1-2 Thus the Ising model, pure fluids at gasliquid critical points, and binary liquid mixtures at lisuid-liquid critical points all belong to the same lass.^-^ The critical exponent P defines the shape of the coexistence curve as t := (T,- T ) / T , nears zero Ap = Bt@
(1)
where Ap is the difference in order parameter between coexisting phases. Over a wide range in temperature, it is necessary to add other terms to eq 1; Wegner6 and Ley-Koo and Green’ suggest an expansion of the form Ap = Et@+ Bltfl+A1 + B2t@+2A1 + ... (2) where A1 is a correction exponent. Recent renormalization group calculation^^^^ yield ,f3 = 0.325 and A1 = 0.50. Equation 1has been applied at temperatures very near T, to pure fluids3 and to binary mixtures,l@-13and values of ,f3 are obtained which agree quite well with the theoretical value. Equation 2 has been applied over wider temperature ranges to pure fluids7J4and binary mixturesl1J5with like success. Thus the universality of the exponent @ is 0 1980 American Chemical Society
2884
The Journal of Physical Chemistry, Vol. 84, No. 22, 1980
confirmed within experimental error. In this paper we test this universality for new coexistence curve data obtained by visual observation of transition temperatures for a number of samples over a broad temperature range for the system n-heptane-acetic anhydride. We find that this coexistence curve is likewise consistent with theoretical predictions. Various theoretical approaches,16J7including the renormalization group a p p r ~ a c halso , ~ predict that the diameter of the coexistence curve, which is the average value of the order parameter in coexisting phases, should behave as (pi + p J / 2 = A1 + Azt + A3t1-a + A4 tl-a+Al+ ... (3) where CY is the same critical exponent which describes the divergence of the specific heat at the critical point. This anomaly is small, so experimental evidence for this divergence is s c a r ~ e . ~ J ~The - ~ Oissue is complicated by the fact that a 2P divergence can be expected as a “trivial result” if the order parameter selected is not the proper one.20i21Thus, with every order parameter but the “best” one, we expect (pi + pz)/2 A1 + Azt + A3t2’ + ... (4) which may mask the 1- CY divergence. For binary liquids, we do not know a priori which order parameter is to be preferred4~22~23-molefraction, volume fraction, mass density, etc. We examine here the diameter of the coexistence curve for n-heptane plus acetic anhydride in variables mole fraction and volume fraction. We fiid evidence for an anomaly in the diameter but cannot differentiate between a 1 - CY anomaly and a 20 anomaly for either variable. Experimental Section Sample Preparation, Temperature Control, and Measurement. Analytical reagent grade materials supplied by M/S Riedel and Co. were used for all measurements without further purification. Vapor-phase chromatography showed no impurities greater than 100 ppm. For all of the experiments, the same stock of samples was used, so that the level of impurity is common to all of the measurements. The effect of impurities in binary liquid mixtures seems to be confined to causing a shift in T,, having little effect on critical e x p o n e n t ~ . ~ * ~ ~ ~ ~ 5 To reduce the effects of gravity26and of the pressure on T,, we used sample holders of small height ( 1 cm). They were small Pyrex bulbs with a capillary stem attached at the top. To avoid loss of liquid over long periods, the top of the capillary stem was flame-sealed. Measured volumes of the liquids were introduced into each bulb by using a hypodermic syringe. The weight of each liquid was measured with a precision of 0.05 mg. Using these weights we calculated the exact composition in terms of mole fractions. A three-stage thermostat employing paraffin oil as the thermostatic fluid was used for all of +he measurements. The stability of the bath was 1 mK over periods of 10-12 h. Thermally cycled thermistors were used as temperature sensors and were operated at low dissipation levels to avoid self-heating and instabilities. The thermistors were calibrated against standard liquids in an ebulliometer and against a platinum resistance thermometer. The measured temperatures are accurate to f50 mK, while the precision of the measurements is f l mK. Temperature gradients were less than 1 mK over the region in the bath where observations are made. Procedure. The method employed to determine the coexistence curve is a point-by-point determination method. It consists of measuring the phase separation temN
Nagarajan et al.
perature at which the meniscus becomes visible when each mixture is cooled from above the critical solution temperature. When this is done for a number of mixtures over the composition range, then the coexistence curve is mapped out. This method has an advantage over the single sample method11-13in that the critical composition x, and the critical temperature T,can be determined very accurately. The precise knowledge of x, is important for studying those properties that have weak anomalies, such as the refractive index, the specific heat, the thermal expansion, the dielectric constant, etc. Experiments on critical phenomena carried out at somewhat off-critical compositions could complicate the nature of the phase transition. This method, however, suffers from the drawback that the phase separation temperatures of the two coexisting compositions are not determined simultaneously. This can lead to difficulties in the analysis of the coexistence curve data.22127 In principle, measurements can be made while heating or cooling. In view of the fact that equilibrium times are much longer in the t ~ o - p h a s region e ~ ~ compared ~ ~ ~ ~ ~ to those in the one-phase region, data were taken while cooling the samples. In cooling, the samples were frequently shaken to reduce the gravity e f f e ~ t close a ~ ~to~ T, ~ and prevent s u p e r c ~ o l i n for g ~ ~compositions far from the critical value. In the present system the opalescence was not marked, and the formation of the meniscus could be observed easily, although it was faint. In the mixtures richer in acetic anhydride (denser component), the meniscus appeared at the top and moved toward the center; in mixtures richer in n-heptane, the meniscus appeared from the bottom and moved toward the middle of the bulb. For the mixture very close to the critical composition, the meniscus appeared in the middle of the sample and remained relatively stationary. Results The coexistence curve was studied over a wide range of compositions ranging from -0.04 to -0.88 mole fraction of n-heptane. This encompasses the reduced temperature range of 3.0 X lo4 I t I 2 X 10-l. Table I lists the mole fraction x of 76 samples along with their phase separation temperatures. The compositions are expressed in both mole fractions (x? and volume fractions (4’). Compositions in terms of volume fractions (4) are calculated using the relation 1 / 4 = (1- K ) + K / x (5) where K = pAMB/p$MA,p and M being the densities3I and the molecular weights, and the subscripts A and B denoting the two components. A plot of mole fraction of n-heptane vs. phase separation temperature is shown in Figure 1 for two different temperature ranges. The composition x of the phase coexisting with the phase x’ was read from the coexistence curve plotted on an enlarged scale to retain precision. Figure 2 shows the coexistence curve in the variable volume fraction. Three samples with compositions 0.4685,0.4707, and 0.4717 mole fraction of n-heptane had the same highest phase separation temperature of 68.522 “C, which is taken as T, for this system. T, is given in the literature as 68 0C.32 The best wayzo~32 to locate x, is to note experimentally the position of the appearance of the meniscus and its subsequent motion. In the sample with x,, the meniscus appears in the middle of the bulb and remains stationary. The meniscus appeared in the middle of the bulb and remained stationary for the composition 0.4707 mole fraction of n-heptane and did not do so at 0.4685 or 0.4717
The Journal of Physical Chemistry, Vol. 84, No. 22, 1980 2885
Liquid-Liquid Critical Phenomena
TABLE I : Phase Separation Temperatures for Various Compositions of the System n-Heptane-Acetic Anhydridea
-
no.
-X'
composition &ll
XIt
composition
d''
separation temD."C
no.
x'
0.9230 0.9160 0.9090 0.9050 0.8940 0.8890 0.8830 0.8700 0.8520 0.8400 0.8310 0.8210 0.8150 0.8075 0.7995 0.7930 0.7740 0.7530 0.7335 0.7245 0.6870 0.6775 0.6700 0.6620 0.6550 0.6465 0.6390 0.6325 0.6245 0.6140 0.6080 0.6055 0.6030 0.6010 0.5960 0.5915 0.5915 0.5845
2.5 9.2 17.1 23.1 32.5 37.2 40.612 45.008 50.822 53.501 56.020 57.508 58.435 60.123 61.918 63.217 64.642 65.782 66.972 67.507 68.128 68.245 68.322 68.358 68.390 68.422 68.454 68.475 68.503 68.513 68.515 68.516 68.517 68.518 68.519 68.520 68.520 68.521
39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
0.4682 0.4685 0.4707 0.4718 0.4744 0.4764 0.4783 0.4825 0.4873 0.4925 0.5050 0.5100 0.5190 0.5291 0.5337 0.5451 0.5530 0.5672 0.5921 0.6010 0.6381 0.6584 0.6825 0.7151 0.7253 0.7382 0.7602 0.7758 0.8010 0.8053 0.8240 0.8425 0.8480 0.8520 0.8551 0.8603 0.8675 0.8810
6'
X"
6''
separation temp, "C
0.5774 0.4755 0.5845 68.521 68.522 0.5777 68.522 0.5798 68.522 0.5809 0.5834 0.4665 0.5755 68.521 0.5854 0.4670 0.5760 68.521 0.5872 0.4670 0.5760 68.521 0.5913 0.4650 0.5740 68.520 0.5960 0.4625 0.5720 68.519 0.6010 0.4605 0.5965 68.518 0.6129 0.4550 0.5645 68.512 0.6176 0.4510 0.5605 68.509 0.6261 0.4440 0.5535 68.495 0.6355 0.4370 0.5465 68.468 0.6398 0.4340 0,5435 68.447 0.6503 0.4260 0.5355 68.407 0.6575 0.4210 0.5300 68.382 0.6704 0.4125 0.5215 68.322 0.6926 0.3840 0.4920 68.072 0.7001 0.3720 0.4790 67.926 0.7323 0.3280 0.4310 67.107 0.7494 0.3030 0.4030 66.019 0.7694 0.2775 0.3730 64.668 0.7957 0.2400 0.3290 62.500 0.8038 0.2300 0.3165 61.908 0.8140 0.2120 0.2945 59.008 0.8311 0.1810 0.2550 56.018 0.8430 0.1650 0.2350 53.215 0.8620 0.1370 0.1980 48.016 0.8652 0.1325 0.1920 46.039 0.8790 0.1130 0.1650 43.019 0.8925 0.0900 0.1330 33.1 0.8965 0.0800 0.1190 29.9 0.8993 0.0775 0.1150 26.5 0.9016 0.0725 0.1080 22.5 0.9053 0.0670 0.1000 19.2 0.9104 0.0600 0.0900 15.1 0.9199 0.0475 0.0720 5.0 a Mole fractions of coexisting phmes are denoted by x' and x", volume fractions by 6 and 6''. Critical composition x c = 0.4707 f 0.0010 mole fraction of n-heptane; critical temperature Tc = 341.672 f 0.050 K. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
70
0.0440 0.0549 0.0662 0.0721 0.0879 0.0971 0.1075 0.1241 0.1531 0.1683 0.1812 0.1980 0.2085 0.2232 0.2321 0.2440 0.2691 0.2978 0.3220 0.3412 0.3914 0.4041 0.4125 0.4180 0.4228 0.4293 0.4349 0.4377 0.4467 0.4525 0.4561 0.4572 0.4589 0.4605 0.4624 0.4649 0.4656 0.4675
0.0670 0.0827 0.0991 0.1.076 0.1301 0.1.430 0.1.574 0.1802 0.2190 0.2390 0.2556 0.2770 0.2902 0.3084 0.3193 0.3337 0.3636 0.3969 0.4243 0.4456 0.4995 0.5128 0.5214 0.5271 0.5320 0.5386 0.5443 0.5471 0.5561 0.5619 0.5655 0.5665 0.5682 0.5698 0.5717 0.5741 0.5748 0.5767
0.8850 0.8750 0.8650 0.8600 0.8450 0.8375 0.8300 0.8120 0.7875 0.7725 0.7600 0.7470 0.7400 0.7300 0.7200 0.7120 0.6880 0.6630 0.6395 0.6290 0.5860 0.5750 0.6670 0.5580 0.5500 0.5410 0.5330 0.5260 0.5175 0.5060 0.5000 0.4975 0.4950 0.4925 0.4873 0.4825 0.4825 0.4755
- n - Heptone + Acetic anhydride
025
045
n- Heptone
+
Acetic anhydride
065
d9
Composition
Molefraction of n - heptane
0 ' 2 d 3 O b 05 0 6 07 0 8 Cornposi tion Volume froction of n - heptone
0
Figure 1. Coexistence curve of n-heptane-acetic anhydride over the temperature range 3 X IO-' C t < 2 X lo-', where t = ( T , - T)/T,, in terms of mole fracticin.
Figure 2. Coexistence curve of n-heptane-acetic anhydride over the temperature range 3 X lo-' < t < 2 X lo-', where t = ( T , - T)/T,, in terms of volume fraction.
mole fraction. Hence, x , is 0.4707 f 0.0010 mole fraction of n-heptane.
program for the numerical differentiation of the function being fitted. Since, as we discuss below, the precision of the data is less than that we anticipated, we fix T,, 6, and A, for all of the fits. T, is taken to be the well-determined experimental value. The exponents 6 and A1 are given the values calculated by renormalization group theory, as discussed above. Thus the constants used are T, = 68.522 "C, = 0.325,A1 = 0.50.
Data Analysis: Coexistence Curve of n -Heptane-Acetic: Anhydride The data analysis was done by using a modified version of Bevington's nonlinear least-squares fitting program CURFIT." The weighting of the data was done by standard methods of propagation of errors,34 using a computer
2886
The Journal of Physical Chemisfry, Vol. 84, No. 22, 1980
NagaraJanet ai.
TABLE I1 : Results of Fits to the Difference in Order Parameterp [Mole Fraction (x) or Volume Fraction (@)I of Coexisting Phases for n-Heptane- Acetic Anhydridea
0 0
~
varino. able
1 2 3 4
5 6
B
1.58 1.89 A x 1.85 A + 1.59 A @ 1.85 A @ 1.77 Ax
AX
OB
B,
0.02 0.01 -1.05 0.02 -0.64 0.02 0.01 -0.86 0.03 -2.0
UB,
B2
OB,
f
0.
~~
i
xu2
30 1.1
0.03 0.19 -0.74
0.32
0.04 0.20 -1.2
21 1.5 0.35 1.1
O D 0
1.0
is one standard deviation, t is (Tc - T ) / T , . x v 2 is a measure of goodness of fit (see ref 34). a a
The order parameters volume fraction, 4, and mole fraction, x, were both considered in the analysis. The standard deviation for the temperature was taken to be the least significant digit in the temperature measurement for each point, which varies from 0.1 to 0.001 O C (see Table I). Initially the standard deviations u for A4 or Ax were likewise taken as the least significant %git, Le., 0.001. However, we found that fits with this choice of uAP gave residuals with a slight systematic oscillatory pattern outside the error 0.001 (see Figure 3). This pattern did not disappear on adding more terms (up to four) in expansion 2; we concluded that it is due to a systematic error in the measurements. We then used = 0.008 to include this systematic error in the uncertainty. A second complication arose with the data closest to the critical point. It is known4tZ3@ that near a liquid-liquid point, gravity effects can cause significant distortions of experimental measurements. Estimates of the range and extent of such effects are hard to make because the effects depend on the particular system and on time periods during the measurements. We found that for these heptaneacetic anhydride measurements the data at t < 0.0025 (T, - T < 0.853) deviated systematically from eq 2. We attribute this deviation to gravity effects and disregard these data. Thus the fits to expansion 2 are for 38 data points for 0.0025 < t < 0.2 with uaP = 0.008 for both Ax and A4. These fits are summarized in Table 11. The residuals to the two-term (fit 2 in Table 11) for Ax are plotted in Figure 3; a plot of the residuals to a three-term fit for A 4 (fit 6 in Table 11) is indistinguishable from this plot. On considering Table 11, we can make the following observations: (a) The Wegner expansion as given in eq 2 is consistent with the measurements, within the experimental error.
0
-2
1
-2.5
-2
0.
a -1.5
I
-I
LOG t Figure 3. Residuals of fit of the differencein order parameter ( x ” x’) for the coexistence of n-heptane-acetic anhydride. The fitting equation is (x”- x’) = Bf@-t Bt@+’f,with /3 = 0.325 and A, = 0.50 and where 0.0025 < t = ( T , T ) / T , < 0.2. See Table 11, fit 2.
-
(b) Ax can be fitted over the range 0.0025 < t < 0.2 with two terms, while A 4 requires three terms. Such better symmetry in Ax than in A+ is in contrast to other binary l i q ~ i d s $where ~ ~ ~ A~4~is more symmetric, and may be related to the fact that the critical mole fraction (0.47) is so close to 0.50. (c) For some flUids,’Jl B1is positive and Bznegative. For polystyrene-cyclohexane,15B1and B2 are both positive. For n-heptane-acetic anhydride, both are negative. We cannot at this time attach a physical significance to these coefficients or to their signs,
Data Analysis: Diameter of Coexistence Curve for n -Heptane-Acetic Anhydride On examining the behavior of the diameter of the coexistence curve, (pl + p2)/2,as a function o f t , we find no evidence of the systematic experimental error which forced us to increase the error estimates for the coexistence curve. It seems that the error in (pl- p2) somehow cancels in (pl + pp)/2,which is quite conceivable. We are able, then, to use for our analysis r((3c1+ x2)/2) = 0.001 and a((& 4J/2) f 0.0013. We have fitted the data to eq 3, with T,,/3, and AI constant as above and with a constant a = 0.12.8 We use the same set of 38 data points, 0.0025 < t < 0.2, as used for the coexistence curve fits. The results of the fits are given in Table 111. From Table 111, we find: (a) A simple linear equation in t will not fit the data. (b) A constant term plus an anomalous term will not fit the data. (c) A constant plus a linear term plus an anomalous term will fit the data very nicely. A typical residual plot, that for fit 10, is shown in Figure 4.
+
TABLE 111: Results of Fits t o the Average Values of the Order Parameter for Coexisting Phases of *HeptaneAcetic Anhydridea @, + p , ) / 2 = A , t A 2 t t A 3 t z Z = 0.66 z = 0.88 no.
p
A,
O.4 1
A2
1
A3
O.4 3
A3
0.43
xu2
0.568 0.002 -0.46 0.02 51 0.572 0.002 -0.39 0.02 39 -0,284 0.009 18 0.582 0.002 0.592 0.001 3.04 0.08 - 2.90 0.07 0.97 @ 0.599 0.001 0.65 0.03 -0.61 0.02 1.1 6 x 0.478 0.001 -0.097 0.009 9.2 x 0.478 0.001 -0.082 0.007 7.8 7 8 x 0.481 0.001 -0.060 0.004 5.2 9 x 0.486 0.005 1.00 0.07 - 0.91 0.06 1.1 10 x 0.488 0.006 0.26 0.02 -0.21 0.01 1.0 a Volume fraction is 6, mole fraction is x, (I is one standard deviation, t = (Tc - T)/T,, x u z is a measure of goodness of fit (see ref 34). The exponent 2 is either 1 - 01 = 0.88 or 20 = 0.65. 1
2 3 4 5
@
+ @ +
The Journal of Physical Chemistry, Vol. 84, No. 22, 1980 2887
Liquid-Liquid Critical Phenomena
Guiiiou and J. ZinnJustin, phys. Rev. Lett., 39, 95 (1977). 1J.G.C.A. LeBaker. B. G. Nickel. M. S. Green. and D. I. Meiron. Phvs. Rev.
+2r-+I
1
e
e
e
e
I.
-2.5
-1.5
-2
J
-I
I
-1
LOG t
Figure 4. Residuals of fi of the diameter of the coexistence curve (x” x’)/2 for n-heptane-aicetic anhydride. The fitting equation is (x” x 7 / 2 = A, A * t A 3t0,85,where 0.0025 < t = T)/ < 0.2. See Table 111, fit 10.
+ +
+
+
(r,-
r,
(d) The three-term fits for q5 and x (fits 4,5,9, and 10) are all equivalent. There is no statistical indication that either variable is to he preferred. For either variable, the difference between a1 1 - a anomaly and a 2p anomaly is statistically insignificant.
Conclusions The coexistence curve of n-heptane-acetic anhydride for 0.0025 C t C 0.2 is consistent with the functional form suggested by renormalization group theory (eq 1 and 2) and with the predicted value of the critical exponent P. The diameter of the coexistence curve of n-heptane plus acetic anhydride shows a critical anomaly, but this anomaly is consistent with either an exponent 1- a or an exponent 2p. Thus we cannot claim confirmation of the predicted16J7 divergence of the diameter as 1 - a, as claimed by other w~rkers.~J”~~ This coexistence curve is slightly more symmetric in the variable mole fraction than in volume fraction. This result complicates further the issue of the proper choice of order parameter for binar:y liquid for volume fraction has more often been seen to be the preferred variable. Acknowledgment. We thank H. Meyer for sending us the preprints of his work and M. R. Moldover for helpful discussions. One of the authors (A.K.) appreciates encouragement from W. I. Goldburg. In the early stages of this work, S. C. Grew was a staff member at the US. National Bureau of Standards and benefited from facilities there. Financial support in India from a PL480 scheme and a DST (NTPP) scheme is gratefully acknowledged. The computer time for this project was provided by the Computer Science Center of the University of Maryland. References and Notes (1) L. P. Kadanoff, Phys:ics,2, 263 (1966); L. P. Kadanoff in “Phase Transitions and Critical Phenomena”, Vol. 5A, M. S. Green and C. Domb, Eds., Academic Press, New York, 1976, Chapter 1. (2) K. G. Wilson, Phys. Rev. 6 , 4, 3174 (1971); K. G. Wilson, Ibid., 4, 3184 (1971); K. 0. Wilson and J. Kogut, phys. Rep., 12C, 75 (1974). (3) R. J. Hocken and M. R. Moldover, Phys. Rev. Left., 37, 29 (1976). (4) S. C. Greer, Acc. Chem. Res., 11, 427 (1976). (5) J. V. Sengers and J. M. H. Levelt Sengers in “Progress in Liquid Physics”, C. A. Croxton, Ed., Wiley, New York, 1978, Chapter 4. (6) F. J. Wegner, Phys. Rev. 6, 5 , 4529 (1972). (7) M. Ley-Koo and M. 5;. Green, Phys. Rev. A , 18, 2483 (1977).
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