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Effect of pressure on the coexistence curve of methanol+n-heptane in the near critical region
Chemical Physics 173 (1993) 457-466 North-Holland
Effect of pressure on the coexistence curve of methanol + rt-heptane in the near critical region Arturo G. Aizpiri, Francisco Monroy, Arturo G. Casielles, Ramon G. Rubio I and Francisco Ortega Departamento de Quimlca Fisica. Facultad de Quimica. Universidad Complutense, 28040 Madrid, Spam
Received 9 July 1992; in final form 26 February 1993
The coexistence curve of methanol+ n-heptane has been measured in the pressure range 0. I MPa
1. Introduction
M=~R-AL=BOt~+B,t~+d+B2t~+2*+...
Near the critical point (CP) of a fluid mixture, various thermodynamic quantities along special thermodynamic paths vary as power laws with critical exponents which, according to the universality hypothesis, depend only on the symmetry of the Hamiltonian and the dimensionality of the system, and not on its chemical nature or the specific intermolecular potential among its molecules, unless the latter is long ranged [ 11. This means that all systems belonging to the same universality class should exhibit the same critical exponents. Pure fluids near the gas-liquid CP, mixtures near liquid-gas or liquidliquid (LL) CPs, and ferromagnets fall into the universality class of three-dimensional Ising-like systems. However, the simple power laws are valid only asymptotically close to CP, and in general such a critical region is rather small [ 2 1. The analysis of data measured outside that region is only possible when effective critical exponents are used, or when correction-to-scaling terms are used [ 3 1. For a binary mixture near an upper critical solution temperature (UCST), the shape of the coexistence curve can be described as [ 41 ’ To whom correspondence should be addressed. 0301-0104/93/$06.00
,
(1)
where/3=0.325?0.001 and A=O.50+0.02 are critical exponents; t = ( T- T,) / T,; T, is the critical temperature; Bo, B, and B2 are critical amplitudes; and M is the order parameter, which is the difference between the compositions of the coexisting phases, R and L referring to both branches of the coexistence curve [5]. Besides the universality of the critical exponents, the renormalization group predicts certain universal relationships between the critical amplitudes of different thermophysical properties, and as a consequence only two of them are independent [ 61. There exists substantial experimental support for the validity of such relationships for the amplitudes of the simple scaling terms. Nevertheless, the situation is far from satisfactory for the amplitudes of the correction-to-scaling terms [ 7 1. Pressure is not expected to affect the critical exponents and the dimensionless amplitude ratio referred to above. This is a consequence ofi (a) p is not a field conjugate to the order parameter, and (b) p does not change the symmetry of the Hamiltonian. The invariance of some of these universal quantities has been observed in several systems [ 81. However, there seems to be no theoretical reason to expect that an
0 1993 Elsevier Science Publishers B.V. All rights reserved.
458
A.G. Aizpwi et al. /Chemical Physics I73 (1993) 45 7-466
external thermodynamic field, such as the pressure, will not affect the size of the region around the CP in which simple scaling is valid, and therefore, the critical amplitudes. In fact the amplitude of the first correction-to-scaling terms for the thermal coefficient and the superfluid density in 3He are pressure dependent [9]. In a previous work [ 10 ] we measured the coexistence curve of a binary mixture, methanol+cyclohexane, at different pressures. The results seemed to indicate that the amplitudes in eq. ( 1) were pressure independent; however, their uncertainties were too high. Recently, we have elaborated on the determination of correction-to-scaling amplitudes and of the range of validity of simple scaling [ 111. This might lead to an improvement in the determination of the amplitudes and therefore we have considered it interesting to obtain further experimental evidence on the possible effect of pressure on the parameters characterizing the coexistence curve near a UCST. To this end we have chosen the system methanol+nheptane for which a well defined curve exists at 1 bar
illI. The rest of this paper is organized as follows: section 2 gives some details about the experimental procedure. The results are presented in section 3, and discussed in section 4. Finally, section 5 summarizes the conclusions.
2. Experimental The experimental method is almost the same as described in ref. [ lo]. Essentially the experimental setup consists of a thick-walled glass capillary tube (4 mm I.D. and 10 mm O.D. ) in which a sample of less than 10 mm high is trapped with mercury. The tube is attached to the pressure system, in which paraffin oil acts as pressure fluid in contact with the mercury. A water bath is used as thermostat. The phase separation temperatures are measured visually and always going from a homogeneous state to the twophase one. Minor changes have been introduced in the method of preparing the mixtures and of trapping them with mercury in the measuring cell, in order to decrease the scattering of the data. The substances were the same used in ref. [ 111. The pressure was measured using a strain-gage
transducer calibrated against a dead-weight balance. Pressure was known within + 0.0 1 MPa through the whole range of our study. The temperature stability of the bath was + 1 mK over 20 h periods, and was measured with a quartz thermometer calibrated against a gallium melting point standard, and frequently compared with a calibrated Pt resistance thermometer. The temperature scale agrees with the IPTS-90 within 10 mK in the range of our data. The mixtures were prepared by weight, the mole fractions being known within + 0.00005.
3. Results Table 1 shows the experimental isopleths (phase separation versus pressure curves at constant composition). Each one has been fitted to a polynomial of the form T/K-273.15=a+b(p/MPa)+c(p/MPa)‘.
(2)
The parameters a, b and c have been obtained using a regression method based on the maximum likelihood principle [ 12 1. The relative weights of the variables were chosen according to their experimental uncertainties. As in a previous work [ lo] a(T) = 5 mK and a(p) ~0.01 MPa were adequate to obtain estimated variances of the fits x2 < 1. The values of a, b and c for each isopleth are given in table 1 together with standard deviations of the variables. Table 2 gives the coexistence curves at round values of p, interpolated from eq. (2 ). Extrapolations of the fits to eq. (2) to p= 0.1 MPa are in very good agreement with the coexistence curve previously reported and measured with a different experimental technique [ 111. Therefore, in the analysis of the influence of pressure in the coexistence curve we will use the data at 0.1 MPa obtained from eq. (2) and the parameters given in table 1.
4. The shape of the coexistence curve and its diameter The experimental method used does not provide the compositions of the two phases in equilibrium at a given p and T, which are necessary to analyze the coexistence curve at a given pressure using eq. ( 1). In order to carry out such an analysis we have used
459
A.G. Aizpiri et al. /Chemical Physics 173 (1993) 45 7-466 Table 1 Experimental
phase-separation
data
T(K)
P @@a)
T(K)
P WPa)
x=0.3391 318.643 318.899 319.172 319.466 319.741 320.012 320.304 320.656 320.937 32 1.404 321.738 322.057
1.81 2.61 3.55 4.57 5.25 6.17 6.99 8.19 9.05 10.35 11.35 12.43
x=0.3601 320.142 320.460 320.753 321.048 321.295 321.587 321.881 322.143 322.411 322.633 322.909 323.190 323.431
2.67 3.49 4.36 5.21 5.95 6.87 7.69 8.53 9.31 10.05 10.93 11.87 12.55
az44.930 b=0.306 c= - 1.30x 1o-2 u( T/K) =9.0x 1O-3 u(p/MPa) =3.3 X IO-* T(K)
P WPa)
x=0.4554 322.812 323.036 323.538 323.758 324.046 324.312 324.506 324.795 325.093 325.310 325.650 325.768 326.009 326.162 326.381 326.575 326.849
0.43 1.01 2.38 2.97 3.79 4.51 5.07 5.91 6.73 7.35 8.35 8.72 9.45 9.87 10.60 11.14 11.99
a=49.503 bz0.378 c= -2.34x lo-’ u(T/K)=2.5~10-’ u(p/MPa)=9.4~10-’
j”(K)
P (MPa)
x=0.3870 320.774 321.041 321.290 321.569 32 1.839 322.096 322.422 322.740 322.910 323.163 323.475 323.764 324.313 324.516 324.732
0.73 1.43 2.19 2.94 3.73 4.51 5.44 6.33 6.85 7.57 8.59 9.45 11.13 11.77 12.47
T(K)
P WPa)
x=0.4108 324.282 324.473 324.852 325.162 325.432 325.733 325.794 326.062 326.422 326.732
0.92 1.41 2.41 3.25 3.98 4.82 4.97 5.72 6.75 7.63
a=46.044 b=0.368 c= -2.50~ lo-’ u(T/K)=4.9XlO-’ a(p/MPa)=1.8~10-’
a=47.364 bc0.362 c= - 1.93x 1o-3 u(T/K)=3.5XlO-’ u(p/MPa)= 1.3x lo-
az48.255 bz0.369 c= - 1.01 x 1o-3 u(T/K)=5.1~10-3 u(p/MPa)= 1.4~ IO-’
T(K)
P (MPa)
T(K)
P (MPa)
T(K)
P
x=0.4667 323.842 324.018 324.289 324.419 324.924 325.21 I 325.334 325.688 326.089 326.696 326.896 326.953 327.574
1.04 1.56 2.26 2.61 4.02 4.85 5.20 6.18 7.3 1 9.10 9.70 9.86 Il.73
x= 0.4990 324.375 324.45 I 324.543 324.895 325.194 325.414 325.724 325.933 326.214 326.399 326.716 326.880 327.055 327.281 327.673 327.856
2.40 2.61 2.88 3.79 4.61 5.24 6.09 6.70 7.52 7.98 8.91 9.39 9.87 10.57 11.75 12.33
.u=O.5302 324.256 324.604 324.826 325.281 325.674 326.079 326.397 326.770 327.463 327.827
1.35 2.30 2.89 4.12 5.25 6.41 7.31 8.43 10.45 11.60
ac49.763 6=0.376 c= -2.33~ 1O-3 u(T/K)=1.6~10-’ u(p/MPa)=6.4~10-’
a=50.306 bz0.387 c= -2.38x 1O-3 u(T/K)=3.0~10-’ u(p/MPa)= 1.2x 10-l
W’a)
a= 50.599 b=0.379 c=-2.37x10-’ u(T/K)=2.3~10-’ o(p/MPa) =8.9x
lo-’
T(K)
P WPa)
x= 0.4243 32 1.960 322.223 322.423 322.729 323.040 323.609 323.937 324.830 325.113 325.396 325.708 326.037 326.362
0.40 1.01 1.55 2.39 3.29 4.77 5.67 8.06 8.83 9.67 10.61 11.65 12.69
a=48.650 b=0.397 c= -2.80x 1o-3 u(T/K)=8.1~10-3 u(p/MPa) = 3.2~ 10F2
T(K)
P WPa)
x=0.5398 324.133 324.329 324.597 324.814 325.115 325.356 325.699 325.997 326.314 326.595 326.920 327.199 327.5 14 327.756 328.067
0.86 1.36 2.11 2.72 3.56 4.23 5.19 6.05 7.01 7.77 8.82 9.57 10.52 11.26 12.23
a= 50.673 b=0.370 c= - 1.85~ IO-’ u(T/K)=3.4~10-’ u(p/MPa)=1.3x10-2 (contrnued on next page)
460
A.G. Aizpiri et al. /Chemical Physics I73 (1993) 45 7-466
Table 1 (continued) T(K)
P WPa)
x=0.5415 324.053 324.180 324.497 324.781 325.221 325.615 325.900 326.296 326.698 327.222 327.99 1 328.156
0.66 0.98 1.85 2.58 3.82 4.91 5.13 6.91 8.08 9.63 11.99 12.51
T(K)
P WPa)
x=0.5427 324.177 324.413 324.642 324.914 325.167 325.426 325.688 325.962 326.229 326.485 326.149 327.037 321.319 327.596 327.841 328.073
0.90 1.58 2.21 2.93 3.64 4.39 5.19 5.97 6.69 7.50 8.27 9.04 9.87 10.68 Il.45 12.19
az50.662 b=0.376 c= -2.33~ lo-’ u(T/K)=2.6~10-’ u(p/MPa)= 1.0x 1O-2
a=50.701 b=0.363 c= - 1.24x 1O-3 a(T/K)=4.5~10-’ o(p/MPa)= 1.7~ lo-*
T(K)
T(K)
x=0.6139 324.285 324.458 324.563 324.667 324.793 324.878 324.975 325.068 325.161 325.276 325.673 325.764 325.964 326.088
P (MPa)
0.82 1.28 1.56 1.78 2.13 2.36 2.60 2.86 3.09 3.43 4.46 4.12 5.29 5.63
x=0.6139 326.158 326.252 326.377 326.467 326.535 326.8 11 327.085 327.194 327.291 327.397 327.497 327.589 328.06 1 328.275
P (MPa) (cont.) 5.83 6.11 6.44 6.73 6.94 7.72 8.51 8.80 9.11 9.44 9.72 9.97 11.43 12.08
a= 50.820 bz0.392 c= -3.01 x 1o-3 a(T/K)=2.7x 1O-3 u(p/MPa) = 1.1 x lo-’
T(K)
P @@a)
x=0.5439 324.173 324.330 324.578 324.954 325.284 325.813 326.175 326.632 326.946 327.271 327.688 328.266
0.91 1.31 2.00 3.06 4.01 5.45 6.51 7.86 8.80 9.81 11.12 12.81
T(K)
P @@a)
T(K)
P WPa)
x=0.5592 324.252 324.655 324.960 325.257 325.609 325.889 326.184 326.469 326.753 327.056 327.342 327.607 327.884 328.168
0.85 2.07 2.91 3.74 4.72 5.51 6.34 7.17 7.99 8.87 9.73 10.46 11.33 12.17
x=0.5750 324.232 324.456 324.829 325.138 325.472 325.716 325.778 326.268 326.555 327.002 327.178 327.41 I 327.477 327.793 328.061
0.79 1.37 2.36 3.22 4.16 4.88 5.03 6.45 7.24 8.57 9.11 9.82 10.01 10.95 11.79
a= 50.693 bE0.370 c= -2.03x 1O-3 u(T/K)=4.3~ lo-.’ u(p/MPa)= 1.6x lo-’
a=50.775
b=0.362 c= - 1.07x IO-’ u(T/K)=3.7~10-’ u(p/MPa)= 1.4x lo-*
a= 50.195 bE0.376 c= -2.29x 1O-3 o(T/K)=2.2~10-’ u(p/MPa) =9.3x
T(K)
P W’a)
T(K)
P (MPa)
T(K)
P W’a)
x=0.6151 324.086 324.275 324.518 324.794 325.078 325.348 325.650 325.97 1 326.263 326.505 326.77 1 326.954 327.490 327.663 327.972 328.256
0.32 0.81 1.47 2.22 3.02 3.75 4.60 5.53 6.34 7.03 7.79 9.17 9.95 10.44 11.37 12.24
x=0.6312 324.305 324.392 324.633 325.013 325.291 325.631 325.828 325.974 326.212 326.447 326.806 327.043 327.334 327.511 327.556 327.803 328.168 328.237
0.89 1.14 1.75 2.79 3.57 4.53 5.05 5.43 6.14 6.78 1.82 8.52 9.37 9.89 10.05 10.80 11.89 12.09
x=0.6338 323.628 323.849 324.151 324.444 324.581 324.659 325.026 325.325 325.692 325.912 326.790 327.182
1.93 2.53 3.37 4.18 4.57 5.15 5.81 6.68 1.79 8.59 11.07 12.29
a= 50.823 b=0.373 c= - 1.87x 1O-3 u(T/K)=2.2~10-’ u(p/MPa)=8.3~10-3
a=50.817 bz0.380 c= -2.22x 10-3 u(T/K)=2.5~10-’ a(p/MPa) = 8.3 x 10e3
10e3
a=50.816 b=0.373 c= -2.05x 1O-3 u(T/K)=2.5~10-’ u(p/MPa) = 9.8 x 10m3
A.G. Aizpiri et al. /Chemical Table
Phyxs
I 73 (1993) 45 7-466
461
1 (continued)
T(K)
p (MPa)
x=0.6585 324.224 324.427 324.698 324.943 325.364 325.764 326.288 327.272 327.479 327.557 328. I22 327.957
0.81 1.38 2.11 2.15 3.91 5.01 6.52 9.38 9.95 10.20 11.89 11.41
a=50.765 bz0.378 c= -2.07x 1O-3 u(T/K)=2.1~10-’ u(p/MPa)=S.lxIO-’
T(K)
P (MPa)
x=0.6890 324.206 324.372 324.742 325.088 325.597 325.653 325.940 326.196 326.460 326.625 327.016 327.328 327.390 327.708 3527.966 328.052
1.18 1.62 2.60 3.56 4.99 5.13 5.97 6.68 7.46 7.87 9.02 9.95 10.14 11.10 11.87 12.11
T(K)
P (MPa)
x=0.6980 323.874 323.986 324.341 324.956 325.151 325.247 325.400 325.97 I 326.319 326.597 327. I I I 327.173 327.478 327.870
0.71 1.01 2.01 3.64 4.2 I 4.49 4.92 6.52 7.50 8.32 9.87 10.02 10.97 12.16
a=50.618 b=0.377 c=-1.91x10-3 u(T/K)=2..x 1O-3 u(p/MPa) =9.4x IO-’
a= 50.452 bz0.377 c= -2.17x 1O-3 o(T/K)=2.5~10-3 u(p/MPa)=9.8~
IO--
T(K)
P (MPa)
j’-(K)
P (MPa)
-f(K)
P (MPa)
x=0.7080 324.282 324.327 324.662 325.044 325.448 325.758 325.793 326.05 1 326.344 326.733 327.487 327.529 328.282
0.88 0.95 1.88 2.93 4.07 4.94 5.04 5.80 6.59 7.73 10.0 10.1 12.4
x=0.7322 323.214 323.455 323.747 324.045 324.360 324.906 325.310 325.647 326.019 326.505 326.907 327.134 327.409
0.69 1.27 1.97 2.8 1 3.51 4.99 6.12 7.11 8.09 9.50 10.83 11.59 12.43
x=0.7708 321.673 32 1.920 322.211 322.487 322.744 322.971 323.249 323.539 324.339 324.585 324.853 325.134 325.482 325.775
0.77 1.39 2.15 2.92 3.65 4.32 5.13 5.93 8.11 8.82 9.56 10.44 11.41 12.28
a=50.301 b=0.373 c=-l.85~10-~ u(T/K)=2.2~ lo-’ u(p/MPa)=8.9x10m3
az49.778 b=0.419 c= -4.78X 10-3 u(T/K)=5.7~10-’ u(p/MPa) =2.2x
ac48.255 b=0.369 c= - 1.01 x 10-3 u(T/K)=5.1~10-’ u(p/MPa) =2.0x
10m2
/I=&
(~R+~L)/2=IZc+A,t+Azt’--a!+A3f’-a+A+...)
(3) which combined
with eq. ( 1) leads to
T(K) x=0.6359 324.317 324.508 324.887 325.197 325.467 325.829 326.097 326.457 326.767
0.92 1.41 2.4 I 3.25 3.98 4.97 5.72 6.15 7.63
a=50.814 b=0.389 c= -2.85x 1O-3 u(T/K)=8.9x 1O-4 u(p/MPa)=3.6~10-’
lo-*
rl (tB,,tfl+
tB, tfl+A+ jBztfl+='...)
+&+A,t’-a+...) This method
P @@a)
of analysis
(4) has already
been used by
A.G. Aizpin et al. /Chemical Physics I73 (1993) 45 7-466
462 Table 2 Coexistence
curves at round values of p
0.3601 0.3870 0.7708 0.4108 0.4243 0.4554 0.4667 0.7322 0.7080 0.4990 0.6980 0.5302 0.5415 0.5439 0.5398 0.6890 0.5427 0.5592 0.5750 0.6338 0.6585 0.6151 0.6312 0.6359 0.6139
0.1 MPa
2.5 MPa
5.0 MPa
7.5 MPa
10.0 MPa
319.230 320.550 32 1.442 32 1.442 321.840 322.69 1 322.950 322.970 323.488 323.495 323.640 323.786 323.806 323.850 323.860 323.880 323.887 323.953 323.96 1 323.982 324.003 324.003 324.005 324.009 324.010
320.097 321.408 322.321 322.32 1 322.775 323.583 323.838 323.947 324.372 324.409 324.532 324.682 324.698 324.736 324.739 324.750 324.756 324.824 324.847 324.87 1 324.886 324.893 324.902 324.918 324.932
320.970 322.278 323.223 323.223 323.715 324.483 324.734 324.906 325.27 1 325.331 325.435 325.586 325.603 325.626 325.633 325.636 325.643 325.‘709 325.754 325.168 325.780 325.789 325.810 325.837 325.856
321.811 323.124 324.113 324.113 324.619 325.355 325.602 325.806 326.147 326.224 326.3 11 326.461 326.485 326.493 326.500 326.504 326.505 326.581 326.635 326.636 326.648 326.663 326.69 1 326.721 326.743
322.62 1 323.946 324.991 324.99 1 325.489 326.197 326.440 326.645 326.999 327.086 327.160 327.305 327.336 327.342 327.343 327.343 327.352 327.439 327.476 327.490 327.491 327.513 327.543 327.568 327.593
Greer [ 13 1, Beysens [ 141, Japas and Levelt Sengers [ 151 and Damay and Leclercq [ 161 and when applied to the data of methanol-tcyclohexane has led to results very similar to those of the method previously used [ 10 1. In analyzing the data of table 2 with eq. (4) the order parameter has been expressed in terms of the volume fraction of methanol @,according to the conclusions reached in a previous work [ 111. The density of methanol as a function of p and T has been taken from ref. [ 17 ] and that of n-heptane from ref. [ 18 1. Due to the lack of excess volume data, VE, at p> 0.1. MPa, it has been neglected in calculating the density of mixtures. In order to test the importance of VE we have repeated the calculations of ref. [ 111 neglecting VE; the results are very similar to those including VE, thus confirming the validity of our approach. The analysis of the coexistence curves has also been
carried out using a regression method based on the maximum likelihood method [ 191 as in a previous work [ lo]. More recently, using the same set of data of ref. [ lo] for the methanol +cyclohexane system, Kumar et al. [ 191 have found that this regression method leads to similar results that the one more frequently used based on the CURFIT program of Bevington [ 201. Since there is an important degree of correlation between the amplitudes of the different terms of eq. (4), a carefully statistical analysis has to be carried out to be sure that the amplitude of a given term does not depend either upon the values of those of the other, nor upon the maximum 1T- T,) interval studied. The results indicate that although there is a high correlation between the rl, and also between the B,, the correlation between the A, and the B, is quite small. As an example the variance between A0 and A, raises
A.G. Aizpwi et al. /Chemical Physics I73 (I 993) 45 7-466
to - 0.99, while the covariance between A0 and B. is - 0.46. In a previous work we have described a strategy to determine the critical amplitudes from coexistencecurve data that overcomes the difficulties mentioned above. Even though the number of data points in each isobar is smaller than in the p = 0.1 MPa curve studied in ref. [ 2 11, making the determination qf the amplitudes less reliable (larger uncertainties), we have the advantage that, as already said, the present data extrapolate quite smoothly to the p=O. 1 MPa of ref. [ 201. This has given us an important hint on the initial guesses for the simple-scaling parameters for the p= 2.5 MPa isobar. The first correction-to-scaling amplitude has been added when, after increasing the ( T- T, 1 interval, the x2 [ 12,201 was larger than unity, and the distribution of the residuals was nonrandom. A similar procedure has been followed for the other isobars. Since the analysis methodology has been described in great detail in ref. [ 2 11, we will only comment on the final results for the sake of brevity. Fig. 1 shows the pressure dependence of the critical temperature. As in previous cases [ lo,17 ] a secondorder polynomial in p is necessary for the whole pressure interval. The value of (dT,/dp),=,,,,,= 37.5 + 0.1 mK/bar is similar to that found for methanol + cyclohexane [ 10 ] (32.1 mK/bar ). Fig. 2 shows the pressure dependence of the critical composition. In a recent paper Jacobs [ 22 ] has suggested that there should be a linear correlation between the influence of pressure on T, and on &. As it can be observed in
3231
0
Fig. 1. Pressure
I
I
I
I
I
I
2
4
6 p1 MFh
8
10
12
dependence
of the critical temperature.
463
005
0 03
-
0C0CO OCO
0 0
01
0
-0 01
(I
0
I
(’
I
-L
I 0 008
0 004
0 012
Tc -Tc. Tco fg. 2. Correlation of the effect of pressure upon the critical composition and the critlcal temperature. Subscript “0” stands for the values at p=O. 1 MPa.
fig. 2, the present results agree with Jacobs’ hypothesis although the uncertainty in the critical composition should be decreased at least one order of magnitude to make a good test of Jacob’s hypothesis. Fig. 3 shows the pressure dependence of the critical amplitudes of the coexistence curve. The first point to be noticed is that their values are similar to those obtained for methanol + cyclohexane [ lo], though contrary to what happened in ref. [lo] the uncertainty of the amplitudes is well below the pressure dependence. Fig. 4 shows the order parameter for different isobars, as well as the values calculated with eq. (1) and parameters of table 3, and the best fit with a simple scaling equation. We have analyzed the results with B2 = 0, in general the weight of such term is quite small for the range of 1T- T, 1 studied in this paper. Despite of this, the Student’s t-test indicates that it is statistically significant; in fact, the lack of that term introduces a nonnegligible degree of nonrandomness in the residual of fits for the data points corresponding to the high 1T-T, 1 values, except for p=O.l MPa, in accordance to ref. [ lo]. Nevertheless we have included the term B2 (~~0.1 MPa) in table 3 for the consistency with the rest of the isobars. The values of I T- T,) for which the residuals ob-
464
A.G. Anpiri et al. /Chemical Physics I73 (1993) 45 7-466
318
322
326
330
T/K
Fig. 4. Effect of pressure on the order parameter versus Tcurves: (a) 2.5 MPa; (b) 5.0 MPa; (c) 7.5 MPa; (d) 10MPa. Full lines: best tits to eq. ( 1) with the parameters of table 3. Dashed line: best description with simple scaling.
1.350W
I2c
p/MPa
Fig. 3. Effect of pressure on the critxal amphtudes of the order parameter.
tained with the simple scaling are well above the experimental uncertainty of the order parameter, i.e. the range of validity of simple scaling does not depend, within the uncertainty of Bo, of including B2 in the fit, due to the method used for determining the B, [21]. This allows to conclude quite clearly that the range of simple scaling decreases with increasing pressure for the methanol + n-heptane system. We must remember that for the methanol + cyclohexane system it was not possible to detect any pressure de-
pendence within the precision of the order parameter that led to larger uncertainties in the B,. Fig. 5 shows the pressure dependence of the amplitudes that describe the diameter of the coexistence curve. Even though one might be tempted to quote some pressure dependence, the large uncertainties of AC,and A, do in fact prevent any sound discussion. Moreover, both parameters are highly correlated, and A, might be taken, within its uncertainty, as zero for most pressures. We have included the term ti-” to describe the diameter since the coexistence curve at p = 0.1 MPa [ 2 1] clearly showed the existence of 1 - (I! anomaly in the diameter instead of a 2/I one. Nevertheless, the present results do not allow to discriminate between both types of anomalies. In any case, we have checked that the use of tZB term instead of the t’-a does not change the conclusions related to the influence of pressure in the range of validity of simple scaling. Many more isopleths for near critical compositions would probably be necessary in order to have a rigorous description of the anomaly of the diameter of the coexistence curve.
A.G. Aizpiri et al. /Chemical Physics I73 (1993) 45 7-466
465
Table 3 Characteristics of the tits of the data to eq. (4) P OfPa) 0.1
2.5 5.0 7.5 10.0
@c
T, WI
0.304 0.306 0.308 0.309 0.310
324.008 324.894 325.798 326.672 327.522
A0
-3.4 -2.2 -0.2 1.2 2.2
Al
3.1 2.4 1.1 0.2 -0.04
B
B,
&
1.517 I .465 1.418 1.390 1.402
0.15 1.97 3.90 5.20 4.60
-0.8 - 12.3 -28.0 -38.1 - 32.8
scaling terms may have upon the leading ones. The results lead to two main conclusions. First the range of validity of simple scaling decreases as pressure is raised for the present system. Second, contrary to what happened for the methanol+cyclohexane system [ lo], the present data are precise enough as to allow for the determination of pressure dependence of the critical amplitudes of the order parameter. However, this has not been the case for the amplitudes of the diameter of the coexistence curve, or for concluding whether a 2p or a 1--cy anomaly describes the diameter; many more experimental data near (x,, 7’,) are probably necessary for such a discussion.
20 AI 00
20 Ao
Acknowledgement -3 0
This work was supported in part by DGICYT under grant PB89-0094.
-0O&-+++-dO p/Ml%
Fig. 5. Effect of pressure on the critical amplitudes of the diameter of coexistence curve.
5. Conclusions
The coexistence curve of methanol + n-heptane has been measured in the pressure range 0.1 MPa
References [ 1 ] H.E. Stanley, Introduction to phase transitions and critical phenomena (Oxford Univ. Press, Oxford, 1987). [2] J.M.H. Levelt Sengers and J.V. Sengers in: Studies in Statistical Mechanics, Vol. IX, Ed. E.W. Montroll and J.L. Le-bowitz (North-Holland, Amsterdam, 198 1) . [3] F.J. Wegner, Phys. Rev. B 5 (1972) 4529. [ 41 M. Ley-Koo and M.S. Green, Phys. Rev. A 16 ( 1977 ) 2483; 23 (1981) 2650. [ 51 J.C. LeGuillou and J. Zinn-Justin, Phys. Rev. B 21 (1980) 3976. [6 ] J.V. Sengers and M.R. Moldover, Phys. Letters A 66 ( 1978) 44. C. Honesson, P. Guenonn, R. Gastand, F. Perrot and D. Beysens, Phys. Rev. A 32 ( 1985) 1818. ]7 ] J.M.H. Levelt Sengers and J.V. Sengers, Ann. Rev. Phys. Chem. 37 (1986) 189.
466
A.G. Aizpiri et al. /Chemical Physics 173 (I 993) 45 7-466
[8] R.R. Singh and W.A. van Hook, J. Chem. Phys. 87 (1987)
[ 91
[ IO] [ 111
[ 121
[ 131
6097; G. Ahlers, Rev. Mod. Phys. 52 ( 1980) 489; M. Yousuf and A. Kumar, Phys. Rev. B 39 ( 1989) 7288. G. Ahlers, in: Quantum liquids, eds. J. Ruvalds and T. Regge (North-Holland, Amsterdam, 1978); K.H. Mueller, G. Ahlers and F. Pobell, Phys. Rev. B 14 (1976) 2096. A.G. Aizpiri, R.G. Rubio and M. Diaz Peiia, J. Chem. Phys. 88 (1988) 1934. A.G. Aizpiri, J.A. Correa, R.G. Rubio and M. Diaz Petia, Phys. Rev. B 41 ( 1990) 9003. A.E. Bryson and Y.B. Ho, Applied optimal control. Optimization, estimation and control (Bloisdel, Waltham, 1969). S.C. Greer, Phys. Rev. A 14 ( 1976) 1770.
[ 141 D.J. Beysens, J. Chem. Phys. 71 (1979) 2557. [ 151 M.L. Japas and J.M.H. Levelt Sengers, J. Phys. Chem. 94 (1990) 5361. [ 161 P. Damay and F. Leclercq, J. Phys. (Paris) C5 1 ( 1991) 83. [ 171 J.R.S. Machado and W.B. Street&J. Chem. Eng. 28 (1983) 218. [ 181 R. Malhotra and L.A. Woolf, J. Chem. Thermodyn. 23 (1991) 49. [ l9]A. Kumar, V.C. Vam, A.G. Aizpiri and R.G. Rubio, Pramana J. Phys. 39 (1992) 95. [ 201 P.R. Bevington, Data reduction and error analysis for the physical sciences (McGraw-Hill, New York, 1969). [ 2 I ] A.G. Aizpiri, C. de1 Campo, F. Monroy, R.G. Rubio and M. Diaz Pefia, Chem. Phys. 165 (1992) 31. [22] D.T. Jacobs, J. Chem. Phys. 91 (1989) 560.
Effect of pressure on the coexistence curve of methanol + rt-heptane in the near critical region Arturo G. Aizpiri, Francisco Monroy, Arturo G. Casielles, Ramon G. Rubio I and Francisco Ortega Departamento de Quimlca Fisica. Facultad de Quimica. Universidad Complutense, 28040 Madrid, Spam
Received 9 July 1992; in final form 26 February 1993
The coexistence curve of methanol+ n-heptane has been measured in the pressure range 0. I MPa
1. Introduction
M=~R-AL=BOt~+B,t~+d+B2t~+2*+...
Near the critical point (CP) of a fluid mixture, various thermodynamic quantities along special thermodynamic paths vary as power laws with critical exponents which, according to the universality hypothesis, depend only on the symmetry of the Hamiltonian and the dimensionality of the system, and not on its chemical nature or the specific intermolecular potential among its molecules, unless the latter is long ranged [ 11. This means that all systems belonging to the same universality class should exhibit the same critical exponents. Pure fluids near the gas-liquid CP, mixtures near liquid-gas or liquidliquid (LL) CPs, and ferromagnets fall into the universality class of three-dimensional Ising-like systems. However, the simple power laws are valid only asymptotically close to CP, and in general such a critical region is rather small [ 2 1. The analysis of data measured outside that region is only possible when effective critical exponents are used, or when correction-to-scaling terms are used [ 3 1. For a binary mixture near an upper critical solution temperature (UCST), the shape of the coexistence curve can be described as [ 41 ’ To whom correspondence should be addressed. 0301-0104/93/$06.00
,
(1)
where/3=0.325?0.001 and A=O.50+0.02 are critical exponents; t = ( T- T,) / T,; T, is the critical temperature; Bo, B, and B2 are critical amplitudes; and M is the order parameter, which is the difference between the compositions of the coexisting phases, R and L referring to both branches of the coexistence curve [5]. Besides the universality of the critical exponents, the renormalization group predicts certain universal relationships between the critical amplitudes of different thermophysical properties, and as a consequence only two of them are independent [ 61. There exists substantial experimental support for the validity of such relationships for the amplitudes of the simple scaling terms. Nevertheless, the situation is far from satisfactory for the amplitudes of the correction-to-scaling terms [ 7 1. Pressure is not expected to affect the critical exponents and the dimensionless amplitude ratio referred to above. This is a consequence ofi (a) p is not a field conjugate to the order parameter, and (b) p does not change the symmetry of the Hamiltonian. The invariance of some of these universal quantities has been observed in several systems [ 81. However, there seems to be no theoretical reason to expect that an
0 1993 Elsevier Science Publishers B.V. All rights reserved.
458
A.G. Aizpwi et al. /Chemical Physics I73 (1993) 45 7-466
external thermodynamic field, such as the pressure, will not affect the size of the region around the CP in which simple scaling is valid, and therefore, the critical amplitudes. In fact the amplitude of the first correction-to-scaling terms for the thermal coefficient and the superfluid density in 3He are pressure dependent [9]. In a previous work [ 10 ] we measured the coexistence curve of a binary mixture, methanol+cyclohexane, at different pressures. The results seemed to indicate that the amplitudes in eq. ( 1) were pressure independent; however, their uncertainties were too high. Recently, we have elaborated on the determination of correction-to-scaling amplitudes and of the range of validity of simple scaling [ 111. This might lead to an improvement in the determination of the amplitudes and therefore we have considered it interesting to obtain further experimental evidence on the possible effect of pressure on the parameters characterizing the coexistence curve near a UCST. To this end we have chosen the system methanol+nheptane for which a well defined curve exists at 1 bar
illI. The rest of this paper is organized as follows: section 2 gives some details about the experimental procedure. The results are presented in section 3, and discussed in section 4. Finally, section 5 summarizes the conclusions.
2. Experimental The experimental method is almost the same as described in ref. [ lo]. Essentially the experimental setup consists of a thick-walled glass capillary tube (4 mm I.D. and 10 mm O.D. ) in which a sample of less than 10 mm high is trapped with mercury. The tube is attached to the pressure system, in which paraffin oil acts as pressure fluid in contact with the mercury. A water bath is used as thermostat. The phase separation temperatures are measured visually and always going from a homogeneous state to the twophase one. Minor changes have been introduced in the method of preparing the mixtures and of trapping them with mercury in the measuring cell, in order to decrease the scattering of the data. The substances were the same used in ref. [ 111. The pressure was measured using a strain-gage
transducer calibrated against a dead-weight balance. Pressure was known within + 0.0 1 MPa through the whole range of our study. The temperature stability of the bath was + 1 mK over 20 h periods, and was measured with a quartz thermometer calibrated against a gallium melting point standard, and frequently compared with a calibrated Pt resistance thermometer. The temperature scale agrees with the IPTS-90 within 10 mK in the range of our data. The mixtures were prepared by weight, the mole fractions being known within + 0.00005.
3. Results Table 1 shows the experimental isopleths (phase separation versus pressure curves at constant composition). Each one has been fitted to a polynomial of the form T/K-273.15=a+b(p/MPa)+c(p/MPa)‘.
(2)
The parameters a, b and c have been obtained using a regression method based on the maximum likelihood principle [ 12 1. The relative weights of the variables were chosen according to their experimental uncertainties. As in a previous work [ lo] a(T) = 5 mK and a(p) ~0.01 MPa were adequate to obtain estimated variances of the fits x2 < 1. The values of a, b and c for each isopleth are given in table 1 together with standard deviations of the variables. Table 2 gives the coexistence curves at round values of p, interpolated from eq. (2 ). Extrapolations of the fits to eq. (2) to p= 0.1 MPa are in very good agreement with the coexistence curve previously reported and measured with a different experimental technique [ 111. Therefore, in the analysis of the influence of pressure in the coexistence curve we will use the data at 0.1 MPa obtained from eq. (2) and the parameters given in table 1.
4. The shape of the coexistence curve and its diameter The experimental method used does not provide the compositions of the two phases in equilibrium at a given p and T, which are necessary to analyze the coexistence curve at a given pressure using eq. ( 1). In order to carry out such an analysis we have used
459
A.G. Aizpiri et al. /Chemical Physics 173 (1993) 45 7-466 Table 1 Experimental
phase-separation
data
T(K)
P @@a)
T(K)
P WPa)
x=0.3391 318.643 318.899 319.172 319.466 319.741 320.012 320.304 320.656 320.937 32 1.404 321.738 322.057
1.81 2.61 3.55 4.57 5.25 6.17 6.99 8.19 9.05 10.35 11.35 12.43
x=0.3601 320.142 320.460 320.753 321.048 321.295 321.587 321.881 322.143 322.411 322.633 322.909 323.190 323.431
2.67 3.49 4.36 5.21 5.95 6.87 7.69 8.53 9.31 10.05 10.93 11.87 12.55
az44.930 b=0.306 c= - 1.30x 1o-2 u( T/K) =9.0x 1O-3 u(p/MPa) =3.3 X IO-* T(K)
P WPa)
x=0.4554 322.812 323.036 323.538 323.758 324.046 324.312 324.506 324.795 325.093 325.310 325.650 325.768 326.009 326.162 326.381 326.575 326.849
0.43 1.01 2.38 2.97 3.79 4.51 5.07 5.91 6.73 7.35 8.35 8.72 9.45 9.87 10.60 11.14 11.99
a=49.503 bz0.378 c= -2.34x lo-’ u(T/K)=2.5~10-’ u(p/MPa)=9.4~10-’
j”(K)
P (MPa)
x=0.3870 320.774 321.041 321.290 321.569 32 1.839 322.096 322.422 322.740 322.910 323.163 323.475 323.764 324.313 324.516 324.732
0.73 1.43 2.19 2.94 3.73 4.51 5.44 6.33 6.85 7.57 8.59 9.45 11.13 11.77 12.47
T(K)
P WPa)
x=0.4108 324.282 324.473 324.852 325.162 325.432 325.733 325.794 326.062 326.422 326.732
0.92 1.41 2.41 3.25 3.98 4.82 4.97 5.72 6.75 7.63
a=46.044 b=0.368 c= -2.50~ lo-’ u(T/K)=4.9XlO-’ a(p/MPa)=1.8~10-’
a=47.364 bc0.362 c= - 1.93x 1o-3 u(T/K)=3.5XlO-’ u(p/MPa)= 1.3x lo-
az48.255 bz0.369 c= - 1.01 x 1o-3 u(T/K)=5.1~10-3 u(p/MPa)= 1.4~ IO-’
T(K)
P (MPa)
T(K)
P (MPa)
T(K)
P
x=0.4667 323.842 324.018 324.289 324.419 324.924 325.21 I 325.334 325.688 326.089 326.696 326.896 326.953 327.574
1.04 1.56 2.26 2.61 4.02 4.85 5.20 6.18 7.3 1 9.10 9.70 9.86 Il.73
x= 0.4990 324.375 324.45 I 324.543 324.895 325.194 325.414 325.724 325.933 326.214 326.399 326.716 326.880 327.055 327.281 327.673 327.856
2.40 2.61 2.88 3.79 4.61 5.24 6.09 6.70 7.52 7.98 8.91 9.39 9.87 10.57 11.75 12.33
.u=O.5302 324.256 324.604 324.826 325.281 325.674 326.079 326.397 326.770 327.463 327.827
1.35 2.30 2.89 4.12 5.25 6.41 7.31 8.43 10.45 11.60
ac49.763 6=0.376 c= -2.33~ 1O-3 u(T/K)=1.6~10-’ u(p/MPa)=6.4~10-’
a=50.306 bz0.387 c= -2.38x 1O-3 u(T/K)=3.0~10-’ u(p/MPa)= 1.2x 10-l
W’a)
a= 50.599 b=0.379 c=-2.37x10-’ u(T/K)=2.3~10-’ o(p/MPa) =8.9x
lo-’
T(K)
P WPa)
x= 0.4243 32 1.960 322.223 322.423 322.729 323.040 323.609 323.937 324.830 325.113 325.396 325.708 326.037 326.362
0.40 1.01 1.55 2.39 3.29 4.77 5.67 8.06 8.83 9.67 10.61 11.65 12.69
a=48.650 b=0.397 c= -2.80x 1o-3 u(T/K)=8.1~10-3 u(p/MPa) = 3.2~ 10F2
T(K)
P WPa)
x=0.5398 324.133 324.329 324.597 324.814 325.115 325.356 325.699 325.997 326.314 326.595 326.920 327.199 327.5 14 327.756 328.067
0.86 1.36 2.11 2.72 3.56 4.23 5.19 6.05 7.01 7.77 8.82 9.57 10.52 11.26 12.23
a= 50.673 b=0.370 c= - 1.85~ IO-’ u(T/K)=3.4~10-’ u(p/MPa)=1.3x10-2 (contrnued on next page)
460
A.G. Aizpiri et al. /Chemical Physics I73 (1993) 45 7-466
Table 1 (continued) T(K)
P WPa)
x=0.5415 324.053 324.180 324.497 324.781 325.221 325.615 325.900 326.296 326.698 327.222 327.99 1 328.156
0.66 0.98 1.85 2.58 3.82 4.91 5.13 6.91 8.08 9.63 11.99 12.51
T(K)
P WPa)
x=0.5427 324.177 324.413 324.642 324.914 325.167 325.426 325.688 325.962 326.229 326.485 326.149 327.037 321.319 327.596 327.841 328.073
0.90 1.58 2.21 2.93 3.64 4.39 5.19 5.97 6.69 7.50 8.27 9.04 9.87 10.68 Il.45 12.19
az50.662 b=0.376 c= -2.33~ lo-’ u(T/K)=2.6~10-’ u(p/MPa)= 1.0x 1O-2
a=50.701 b=0.363 c= - 1.24x 1O-3 a(T/K)=4.5~10-’ o(p/MPa)= 1.7~ lo-*
T(K)
T(K)
x=0.6139 324.285 324.458 324.563 324.667 324.793 324.878 324.975 325.068 325.161 325.276 325.673 325.764 325.964 326.088
P (MPa)
0.82 1.28 1.56 1.78 2.13 2.36 2.60 2.86 3.09 3.43 4.46 4.12 5.29 5.63
x=0.6139 326.158 326.252 326.377 326.467 326.535 326.8 11 327.085 327.194 327.291 327.397 327.497 327.589 328.06 1 328.275
P (MPa) (cont.) 5.83 6.11 6.44 6.73 6.94 7.72 8.51 8.80 9.11 9.44 9.72 9.97 11.43 12.08
a= 50.820 bz0.392 c= -3.01 x 1o-3 a(T/K)=2.7x 1O-3 u(p/MPa) = 1.1 x lo-’
T(K)
P @@a)
x=0.5439 324.173 324.330 324.578 324.954 325.284 325.813 326.175 326.632 326.946 327.271 327.688 328.266
0.91 1.31 2.00 3.06 4.01 5.45 6.51 7.86 8.80 9.81 11.12 12.81
T(K)
P @@a)
T(K)
P WPa)
x=0.5592 324.252 324.655 324.960 325.257 325.609 325.889 326.184 326.469 326.753 327.056 327.342 327.607 327.884 328.168
0.85 2.07 2.91 3.74 4.72 5.51 6.34 7.17 7.99 8.87 9.73 10.46 11.33 12.17
x=0.5750 324.232 324.456 324.829 325.138 325.472 325.716 325.778 326.268 326.555 327.002 327.178 327.41 I 327.477 327.793 328.061
0.79 1.37 2.36 3.22 4.16 4.88 5.03 6.45 7.24 8.57 9.11 9.82 10.01 10.95 11.79
a= 50.693 bE0.370 c= -2.03x 1O-3 u(T/K)=4.3~ lo-.’ u(p/MPa)= 1.6x lo-’
a=50.775
b=0.362 c= - 1.07x IO-’ u(T/K)=3.7~10-’ u(p/MPa)= 1.4x lo-*
a= 50.195 bE0.376 c= -2.29x 1O-3 o(T/K)=2.2~10-’ u(p/MPa) =9.3x
T(K)
P W’a)
T(K)
P (MPa)
T(K)
P W’a)
x=0.6151 324.086 324.275 324.518 324.794 325.078 325.348 325.650 325.97 1 326.263 326.505 326.77 1 326.954 327.490 327.663 327.972 328.256
0.32 0.81 1.47 2.22 3.02 3.75 4.60 5.53 6.34 7.03 7.79 9.17 9.95 10.44 11.37 12.24
x=0.6312 324.305 324.392 324.633 325.013 325.291 325.631 325.828 325.974 326.212 326.447 326.806 327.043 327.334 327.511 327.556 327.803 328.168 328.237
0.89 1.14 1.75 2.79 3.57 4.53 5.05 5.43 6.14 6.78 1.82 8.52 9.37 9.89 10.05 10.80 11.89 12.09
x=0.6338 323.628 323.849 324.151 324.444 324.581 324.659 325.026 325.325 325.692 325.912 326.790 327.182
1.93 2.53 3.37 4.18 4.57 5.15 5.81 6.68 1.79 8.59 11.07 12.29
a= 50.823 b=0.373 c= - 1.87x 1O-3 u(T/K)=2.2~10-’ u(p/MPa)=8.3~10-3
a=50.817 bz0.380 c= -2.22x 10-3 u(T/K)=2.5~10-’ a(p/MPa) = 8.3 x 10e3
10e3
a=50.816 b=0.373 c= -2.05x 1O-3 u(T/K)=2.5~10-’ u(p/MPa) = 9.8 x 10m3
A.G. Aizpiri et al. /Chemical Table
Phyxs
I 73 (1993) 45 7-466
461
1 (continued)
T(K)
p (MPa)
x=0.6585 324.224 324.427 324.698 324.943 325.364 325.764 326.288 327.272 327.479 327.557 328. I22 327.957
0.81 1.38 2.11 2.15 3.91 5.01 6.52 9.38 9.95 10.20 11.89 11.41
a=50.765 bz0.378 c= -2.07x 1O-3 u(T/K)=2.1~10-’ u(p/MPa)=S.lxIO-’
T(K)
P (MPa)
x=0.6890 324.206 324.372 324.742 325.088 325.597 325.653 325.940 326.196 326.460 326.625 327.016 327.328 327.390 327.708 3527.966 328.052
1.18 1.62 2.60 3.56 4.99 5.13 5.97 6.68 7.46 7.87 9.02 9.95 10.14 11.10 11.87 12.11
T(K)
P (MPa)
x=0.6980 323.874 323.986 324.341 324.956 325.151 325.247 325.400 325.97 I 326.319 326.597 327. I I I 327.173 327.478 327.870
0.71 1.01 2.01 3.64 4.2 I 4.49 4.92 6.52 7.50 8.32 9.87 10.02 10.97 12.16
a=50.618 b=0.377 c=-1.91x10-3 u(T/K)=2..x 1O-3 u(p/MPa) =9.4x IO-’
a= 50.452 bz0.377 c= -2.17x 1O-3 o(T/K)=2.5~10-3 u(p/MPa)=9.8~
IO--
T(K)
P (MPa)
j’-(K)
P (MPa)
-f(K)
P (MPa)
x=0.7080 324.282 324.327 324.662 325.044 325.448 325.758 325.793 326.05 1 326.344 326.733 327.487 327.529 328.282
0.88 0.95 1.88 2.93 4.07 4.94 5.04 5.80 6.59 7.73 10.0 10.1 12.4
x=0.7322 323.214 323.455 323.747 324.045 324.360 324.906 325.310 325.647 326.019 326.505 326.907 327.134 327.409
0.69 1.27 1.97 2.8 1 3.51 4.99 6.12 7.11 8.09 9.50 10.83 11.59 12.43
x=0.7708 321.673 32 1.920 322.211 322.487 322.744 322.971 323.249 323.539 324.339 324.585 324.853 325.134 325.482 325.775
0.77 1.39 2.15 2.92 3.65 4.32 5.13 5.93 8.11 8.82 9.56 10.44 11.41 12.28
a=50.301 b=0.373 c=-l.85~10-~ u(T/K)=2.2~ lo-’ u(p/MPa)=8.9x10m3
az49.778 b=0.419 c= -4.78X 10-3 u(T/K)=5.7~10-’ u(p/MPa) =2.2x
ac48.255 b=0.369 c= - 1.01 x 10-3 u(T/K)=5.1~10-’ u(p/MPa) =2.0x
10m2
/I=&
(~R+~L)/2=IZc+A,t+Azt’--a!+A3f’-a+A+...)
(3) which combined
with eq. ( 1) leads to
T(K) x=0.6359 324.317 324.508 324.887 325.197 325.467 325.829 326.097 326.457 326.767
0.92 1.41 2.4 I 3.25 3.98 4.97 5.72 6.15 7.63
a=50.814 b=0.389 c= -2.85x 1O-3 u(T/K)=8.9x 1O-4 u(p/MPa)=3.6~10-’
lo-*
rl (tB,,tfl+
tB, tfl+A+ jBztfl+='...)
+&+A,t’-a+...) This method
P @@a)
of analysis
(4) has already
been used by
A.G. Aizpin et al. /Chemical Physics I73 (1993) 45 7-466
462 Table 2 Coexistence
curves at round values of p
0.3601 0.3870 0.7708 0.4108 0.4243 0.4554 0.4667 0.7322 0.7080 0.4990 0.6980 0.5302 0.5415 0.5439 0.5398 0.6890 0.5427 0.5592 0.5750 0.6338 0.6585 0.6151 0.6312 0.6359 0.6139
0.1 MPa
2.5 MPa
5.0 MPa
7.5 MPa
10.0 MPa
319.230 320.550 32 1.442 32 1.442 321.840 322.69 1 322.950 322.970 323.488 323.495 323.640 323.786 323.806 323.850 323.860 323.880 323.887 323.953 323.96 1 323.982 324.003 324.003 324.005 324.009 324.010
320.097 321.408 322.321 322.32 1 322.775 323.583 323.838 323.947 324.372 324.409 324.532 324.682 324.698 324.736 324.739 324.750 324.756 324.824 324.847 324.87 1 324.886 324.893 324.902 324.918 324.932
320.970 322.278 323.223 323.223 323.715 324.483 324.734 324.906 325.27 1 325.331 325.435 325.586 325.603 325.626 325.633 325.636 325.643 325.‘709 325.754 325.168 325.780 325.789 325.810 325.837 325.856
321.811 323.124 324.113 324.113 324.619 325.355 325.602 325.806 326.147 326.224 326.3 11 326.461 326.485 326.493 326.500 326.504 326.505 326.581 326.635 326.636 326.648 326.663 326.69 1 326.721 326.743
322.62 1 323.946 324.991 324.99 1 325.489 326.197 326.440 326.645 326.999 327.086 327.160 327.305 327.336 327.342 327.343 327.343 327.352 327.439 327.476 327.490 327.491 327.513 327.543 327.568 327.593
Greer [ 13 1, Beysens [ 141, Japas and Levelt Sengers [ 151 and Damay and Leclercq [ 161 and when applied to the data of methanol-tcyclohexane has led to results very similar to those of the method previously used [ 10 1. In analyzing the data of table 2 with eq. (4) the order parameter has been expressed in terms of the volume fraction of methanol @,according to the conclusions reached in a previous work [ 111. The density of methanol as a function of p and T has been taken from ref. [ 17 ] and that of n-heptane from ref. [ 18 1. Due to the lack of excess volume data, VE, at p> 0.1. MPa, it has been neglected in calculating the density of mixtures. In order to test the importance of VE we have repeated the calculations of ref. [ 111 neglecting VE; the results are very similar to those including VE, thus confirming the validity of our approach. The analysis of the coexistence curves has also been
carried out using a regression method based on the maximum likelihood method [ 191 as in a previous work [ lo]. More recently, using the same set of data of ref. [ lo] for the methanol +cyclohexane system, Kumar et al. [ 191 have found that this regression method leads to similar results that the one more frequently used based on the CURFIT program of Bevington [ 201. Since there is an important degree of correlation between the amplitudes of the different terms of eq. (4), a carefully statistical analysis has to be carried out to be sure that the amplitude of a given term does not depend either upon the values of those of the other, nor upon the maximum 1T- T,) interval studied. The results indicate that although there is a high correlation between the rl, and also between the B,, the correlation between the A, and the B, is quite small. As an example the variance between A0 and A, raises
A.G. Aizpwi et al. /Chemical Physics I73 (I 993) 45 7-466
to - 0.99, while the covariance between A0 and B. is - 0.46. In a previous work we have described a strategy to determine the critical amplitudes from coexistencecurve data that overcomes the difficulties mentioned above. Even though the number of data points in each isobar is smaller than in the p = 0.1 MPa curve studied in ref. [ 2 11, making the determination qf the amplitudes less reliable (larger uncertainties), we have the advantage that, as already said, the present data extrapolate quite smoothly to the p=O. 1 MPa of ref. [ 201. This has given us an important hint on the initial guesses for the simple-scaling parameters for the p= 2.5 MPa isobar. The first correction-to-scaling amplitude has been added when, after increasing the ( T- T, 1 interval, the x2 [ 12,201 was larger than unity, and the distribution of the residuals was nonrandom. A similar procedure has been followed for the other isobars. Since the analysis methodology has been described in great detail in ref. [ 2 11, we will only comment on the final results for the sake of brevity. Fig. 1 shows the pressure dependence of the critical temperature. As in previous cases [ lo,17 ] a secondorder polynomial in p is necessary for the whole pressure interval. The value of (dT,/dp),=,,,,,= 37.5 + 0.1 mK/bar is similar to that found for methanol + cyclohexane [ 10 ] (32.1 mK/bar ). Fig. 2 shows the pressure dependence of the critical composition. In a recent paper Jacobs [ 22 ] has suggested that there should be a linear correlation between the influence of pressure on T, and on &. As it can be observed in
3231
0
Fig. 1. Pressure
I
I
I
I
I
I
2
4
6 p1 MFh
8
10
12
dependence
of the critical temperature.
463
005
0 03
-
0C0CO OCO
0 0
01
0
-0 01
(I
0
I
(’
I
-L
I 0 008
0 004
0 012
Tc -Tc. Tco fg. 2. Correlation of the effect of pressure upon the critical composition and the critlcal temperature. Subscript “0” stands for the values at p=O. 1 MPa.
fig. 2, the present results agree with Jacobs’ hypothesis although the uncertainty in the critical composition should be decreased at least one order of magnitude to make a good test of Jacob’s hypothesis. Fig. 3 shows the pressure dependence of the critical amplitudes of the coexistence curve. The first point to be noticed is that their values are similar to those obtained for methanol + cyclohexane [ lo], though contrary to what happened in ref. [lo] the uncertainty of the amplitudes is well below the pressure dependence. Fig. 4 shows the order parameter for different isobars, as well as the values calculated with eq. (1) and parameters of table 3, and the best fit with a simple scaling equation. We have analyzed the results with B2 = 0, in general the weight of such term is quite small for the range of 1T- T, 1 studied in this paper. Despite of this, the Student’s t-test indicates that it is statistically significant; in fact, the lack of that term introduces a nonnegligible degree of nonrandomness in the residual of fits for the data points corresponding to the high 1T-T, 1 values, except for p=O.l MPa, in accordance to ref. [ lo]. Nevertheless we have included the term B2 (~~0.1 MPa) in table 3 for the consistency with the rest of the isobars. The values of I T- T,) for which the residuals ob-
464
A.G. Anpiri et al. /Chemical Physics I73 (1993) 45 7-466
318
322
326
330
T/K
Fig. 4. Effect of pressure on the order parameter versus Tcurves: (a) 2.5 MPa; (b) 5.0 MPa; (c) 7.5 MPa; (d) 10MPa. Full lines: best tits to eq. ( 1) with the parameters of table 3. Dashed line: best description with simple scaling.
1.350W
I2c
p/MPa
Fig. 3. Effect of pressure on the critxal amphtudes of the order parameter.
tained with the simple scaling are well above the experimental uncertainty of the order parameter, i.e. the range of validity of simple scaling does not depend, within the uncertainty of Bo, of including B2 in the fit, due to the method used for determining the B, [21]. This allows to conclude quite clearly that the range of simple scaling decreases with increasing pressure for the methanol + n-heptane system. We must remember that for the methanol + cyclohexane system it was not possible to detect any pressure de-
pendence within the precision of the order parameter that led to larger uncertainties in the B,. Fig. 5 shows the pressure dependence of the amplitudes that describe the diameter of the coexistence curve. Even though one might be tempted to quote some pressure dependence, the large uncertainties of AC,and A, do in fact prevent any sound discussion. Moreover, both parameters are highly correlated, and A, might be taken, within its uncertainty, as zero for most pressures. We have included the term ti-” to describe the diameter since the coexistence curve at p = 0.1 MPa [ 2 1] clearly showed the existence of 1 - (I! anomaly in the diameter instead of a 2/I one. Nevertheless, the present results do not allow to discriminate between both types of anomalies. In any case, we have checked that the use of tZB term instead of the t’-a does not change the conclusions related to the influence of pressure in the range of validity of simple scaling. Many more isopleths for near critical compositions would probably be necessary in order to have a rigorous description of the anomaly of the diameter of the coexistence curve.
A.G. Aizpiri et al. /Chemical Physics I73 (1993) 45 7-466
465
Table 3 Characteristics of the tits of the data to eq. (4) P OfPa) 0.1
2.5 5.0 7.5 10.0
@c
T, WI
0.304 0.306 0.308 0.309 0.310
324.008 324.894 325.798 326.672 327.522
A0
-3.4 -2.2 -0.2 1.2 2.2
Al
3.1 2.4 1.1 0.2 -0.04
B
B,
&
1.517 I .465 1.418 1.390 1.402
0.15 1.97 3.90 5.20 4.60
-0.8 - 12.3 -28.0 -38.1 - 32.8
scaling terms may have upon the leading ones. The results lead to two main conclusions. First the range of validity of simple scaling decreases as pressure is raised for the present system. Second, contrary to what happened for the methanol+cyclohexane system [ lo], the present data are precise enough as to allow for the determination of pressure dependence of the critical amplitudes of the order parameter. However, this has not been the case for the amplitudes of the diameter of the coexistence curve, or for concluding whether a 2p or a 1--cy anomaly describes the diameter; many more experimental data near (x,, 7’,) are probably necessary for such a discussion.
20 AI 00
20 Ao
Acknowledgement -3 0
This work was supported in part by DGICYT under grant PB89-0094.
-0O&-+++-dO p/Ml%
Fig. 5. Effect of pressure on the critical amplitudes of the diameter of coexistence curve.
5. Conclusions
The coexistence curve of methanol + n-heptane has been measured in the pressure range 0.1 MPa
References [ 1 ] H.E. Stanley, Introduction to phase transitions and critical phenomena (Oxford Univ. Press, Oxford, 1987). [2] J.M.H. Levelt Sengers and J.V. Sengers in: Studies in Statistical Mechanics, Vol. IX, Ed. E.W. Montroll and J.L. Le-bowitz (North-Holland, Amsterdam, 198 1) . [3] F.J. Wegner, Phys. Rev. B 5 (1972) 4529. [ 41 M. Ley-Koo and M.S. Green, Phys. Rev. A 16 ( 1977 ) 2483; 23 (1981) 2650. [ 51 J.C. LeGuillou and J. Zinn-Justin, Phys. Rev. B 21 (1980) 3976. [6 ] J.V. Sengers and M.R. Moldover, Phys. Letters A 66 ( 1978) 44. C. Honesson, P. Guenonn, R. Gastand, F. Perrot and D. Beysens, Phys. Rev. A 32 ( 1985) 1818. ]7 ] J.M.H. Levelt Sengers and J.V. Sengers, Ann. Rev. Phys. Chem. 37 (1986) 189.
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6097; G. Ahlers, Rev. Mod. Phys. 52 ( 1980) 489; M. Yousuf and A. Kumar, Phys. Rev. B 39 ( 1989) 7288. G. Ahlers, in: Quantum liquids, eds. J. Ruvalds and T. Regge (North-Holland, Amsterdam, 1978); K.H. Mueller, G. Ahlers and F. Pobell, Phys. Rev. B 14 (1976) 2096. A.G. Aizpiri, R.G. Rubio and M. Diaz Peiia, J. Chem. Phys. 88 (1988) 1934. A.G. Aizpiri, J.A. Correa, R.G. Rubio and M. Diaz Petia, Phys. Rev. B 41 ( 1990) 9003. A.E. Bryson and Y.B. Ho, Applied optimal control. Optimization, estimation and control (Bloisdel, Waltham, 1969). S.C. Greer, Phys. Rev. A 14 ( 1976) 1770.
[ 141 D.J. Beysens, J. Chem. Phys. 71 (1979) 2557. [ 151 M.L. Japas and J.M.H. Levelt Sengers, J. Phys. Chem. 94 (1990) 5361. [ 161 P. Damay and F. Leclercq, J. Phys. (Paris) C5 1 ( 1991) 83. [ 171 J.R.S. Machado and W.B. Street&J. Chem. Eng. 28 (1983) 218. [ 181 R. Malhotra and L.A. Woolf, J. Chem. Thermodyn. 23 (1991) 49. [ l9]A. Kumar, V.C. Vam, A.G. Aizpiri and R.G. Rubio, Pramana J. Phys. 39 (1992) 95. [ 201 P.R. Bevington, Data reduction and error analysis for the physical sciences (McGraw-Hill, New York, 1969). [ 2 I ] A.G. Aizpiri, C. de1 Campo, F. Monroy, R.G. Rubio and M. Diaz Pefia, Chem. Phys. 165 (1992) 31. [22] D.T. Jacobs, J. Chem. Phys. 91 (1989) 560.