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Pressure-density-temperature (p-p-T) relations of CHF3, N2O, and C3H6 in the critical region
78
The Journal of Supercritical
Fluids 1990,3,
78-84
Pressure-Density-Temperature (p-PT) Relations of CHF3, N20, and C3H6 in the Critical Region Kazunari Ohgaki, * Shyoji Umezono, and Takashi Katayama Department of Chemical Engineering, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan Received March 30, 1990; accepted in revised form June 7, 1990
The pressureAensity-temperature @-~7’) relations in the critical region (0.996 < T,.c 1.004 and 0.6 c pr < 1.4) are determined for CHF3, N20, and C3H6 by means of a direct method. The critical properties are determined from a logarithmic plot based on the power law for the vapor-liquid coexistence curve and the p-p relation on the critical isotherm for each system. The restricted linear model, one of the simplest scaled equations-of-state, is applied to describe the p-pT relations in the critical region. Keywords:
critical properties, scaled equation-of-state, p-p-T relation
INTRODUCTION In the last two decades, the studies on thermodynamic properties of near critical and supercritical fluids have significantly increased. This is primarily because of their practical importance for high pressure technology such as supercritical fluid extraction. Another reason is a theoretical one which is concerned with the anomalous behavior of thermodynamic properties in the vicinity of the critical point. The critical anomalies in thermodynamic and transport properties based on density fluctuations were first investigated by Omstein and Zemike.’ Studies on the scaled equation-of-state by Widom* and Landau3 and on the Ising model4 created a new field concerning the critical exponents. Josephson5 and Schofield6” proposed a parametric representation by a transformation from physical to scaling variables. Also, special attention was given to the universality law8g9for the critical exponents in the second order phase transitions such as the Curie point. In experimental studies, the critical exponents were obtained from pressure-density-temperature (p-p-0 relations in the critical region for several substances.‘O-I5 Furthermore, the precise p-FT data in the critical region were correlated by means of nonclassical equations-of-state based on the power law.16 In the present study, the p-p-T relations for N20, CHF3, and C3H6 are measured in the critical region. Critical properties are evaluated from the data using the power law. Finally, the applicability of the universality law for p-p-T relations in the critical region is discussed by means of the universal restricted linear model. 0896-8446/90/0302-0078$4.00/O
v Figure 1. Schematic diagram of experimental apparatus for determining p-p-T relations. (A), piston gauge; (B), pressure transducer; (C), oil hand pump; (D), oil reservoir; (E), oil manometer; (F), oil dead-weight gauge; (G), quartz Bourdon gauge; (H), N, gas hand pump; (I), compressor; (J), N, gas cylinder; (K), sample gas cylinder; (L), high pressure pump; (M), pressuretransducer; (N), high pressurecell; (P), trapping cell; (Q), liquid N, bath; (R), vacuum pump; (S), Pirani gauge. EXPERIMENTAL Experimental apparatus and procedures. A schematic diagram of the experimental apparatus used in the measurement of the pressure+lensity-temperature (pp-7’) relations is shown in Figure 1. The apparatus consists of a high pressure cell and sampling part, a pressure measurement part, a temperature measurement part, and a vacuum line. The high pressure cell (N) of stainless steel (the internal volume of ca. 90 cm3 calibrated with compressed CO*” and C3H6 I8 fluids at several pressures) was evacuated together with the sampling part (between the 0 1990 PRA Press
Pressure-Density-Temperature
The Journal of Supercritical Fluids, Vol. 3, No. 2, 1990 valves 11 and 17) to lower than 10e3mmHg by a vacuum pump (R). By using a high pressure pump, the sample fluid was charged up to the desired pressure. After thermal equilibrium was reached over a period of about one hour, the equilibrium pressure was measured with an accuracy of f5 x 10e4 MPa by use of a combination of a null-type pressure transducer (M) and a quartz Bourdon-tube gauge (G) calibrated by an oil dead-weight gauge (G) and an air piston gauge (A) manufactured by Ruska Instrument Corp. The effects of working pressure and temperature on the null-position of the diaphragm in the pressure transducer were corrected previously. The temperature of the oil bath (used in the experiments with C3H6) and water bath (for N20 and CHF3) was measured within an accuracy of ti.004 K (in the oil bath) and ti.001 K (in the water bath) by a set of a platinum resistance and a resistance thermometer bridge as defined by the IPTS of 1968. The platinum resistance was calibrated by the National Research Laboratory of Metrology, Japan. The procedures for pressure and temperature measurements are essentially the same as reported previously.‘9 After the pressure measurement, a small amount of fluid (ca. 2% of the total sample) was expanded into the sampling part (valves 1l-17) in order to change the bulk density. Then, the content of the sampling part was introduced to a trapping cell (P) which was evacuated beforehand to about 10e3 mmHg. The evaporated sample was solidified in the trapping cell maintained in a liquid nitrogen bath (Q). The complete solidification was confirmed through a vacuum gauge (S). As the volume of line (from valve 11 to the trapping cell) is small enough, the amount of gas retained at the sublimation pressure can be neglected. The amount in the trapping cell was determined by weighing on a precision balance (max. 3 kg) of Cho Balance Corp. with an accuracy of 50.3 mg. Both the accurate pressure measurement and the determination of amount of sample trapped were required in every expansion. After several expansions with about one hour intervals, the fluid retained in the high pressure cell was solidified completely in the trapping cell in order to determine the total amount of sample, using the same procedure described above. The total weights of sample fluid were about 50-60 g for CHF3 and N20 and 20-30 g for C3H6. The densities were determined within an accuracy of f0.03% from the amount of sample and the known cell volume. Finally, the several series of the p-p relations were obtained at different temperatures in the critical region. Materials. CHF3 which was obtained from Daikin Kagaku Corp. and N20 from Seitetu Kagaku Corp. had special minimum purity of 99.999 mol%. Research grade C3H6 obtained from Sumitomo Kagaku Corp. had purity of 9.96 mol%. The substances were used in this experiment without further purification.
Relations
79
RESULTS AND DISCUSSION p-p relation for the one-phase
region. Table I gives the p-p-T relations obtained in the one-phase region. This table does not contain the data in the twophase region. Saturation properties. When the pressure was not changed by an expansion, we regarded it as the saturation pressure at that temperature. Also, the density where the isothermal p-p line crossed the saturation pressure at a given temperature can be taken as the saturation density. The saturation pressures are used to determine the parameters, co and cl, of a regular term of the Helmholtz free energy in the restricted linear model discussed later. The saturation pressures and densities are obtained with accuracies of +0.03% and ti.2%, respectively. The saturation densities of gases are generally less reliable than those of liquids. As it is difficult to obtain data for gas density, some isotherms do not contain the data on saturated gas density. The main reason is that the amount of mass expanded is too small to give a precise density for the gas phase. Furthermore, the adsorption effect in the gas phase causes significant errors of density measurements. Table II summarizes the saturation properties obtained for each fluid. Critical
properties
and
critical
exponents.
The critical anomalies are described in terms of logarithmic scale by
4
0~ [AT,?
(1)
on the vapor-liquid coexistence line and by (2) on the critical isotherm. The exponents p and 6 characterize, respectively, the asymptotic behavior of the coexistence curve and the critical isotherm. The term AX, is defined as
Ax,= w - X,)/X,
(X= T,p,p>.
(3)
By use of the above relations, the critical temperature, density, and pressure are determined from the p-p isotherms obtained from experiment data. A method for determining the critical temperature is as follows. By using the saturation properties on several isotherms near the critical temperature, AT, is plotted against Apr on a logarithmic scale as shown in Figure 2. If the temporary critical temperature, ?,I, is extremely close to the true critical temperature, the plot shows a linear relation according to eq 1. As Figure 2 shows, the plot becomes straight for’ 299.300 K, while neither T,’ = 299.181 K (T, = 0.9996) nor T,’ = 299.422 K (T, = 1.0004) give a straight line. As the coexistence curve in p-p plane is almost symmetric in this experimental con-
80
Ohgaki et al.
p-p-T
P (MW
The Journal of Supercritical TABLE I-l Relation for CHFJ
p WoVm3)
T = 298.150 K 4.8014 4.7499 4.7428 4.7191 4.7005 4.6955 4.6745 4.6670 4.6554 4.6441
10.432 10.172 10.126 9.9018 9.6801 9.4616 5.1949 5.0503 4.9080 4.7709
T = 298.345 4.8643 4.8974 4.7697 4.7677 4.7405 4.7259 4.7023 4.6926 4.6756 4.6580
K
T = 298.558 4.9187 4.8539 4.8097 4.7990 4.7571 4.7445 4.7359 4.7285 4.7189 4.7104 4.698 1
K
T = 299.300 5.0440 4.9756 4.9241 4.9156 4.9100 4.904 1 4.8899 4.8866 4.8674 4.8653 4.8599 4.8498 4.8423 4.8320 4.8283 4.8257 4.8255 4.8208 4.8123 4.8019 4.7999 4.7918 4.7899
K
10.529 10.300 10.068 10.055 9.8157 9.635 4.3457 5.1954 4.9653 4.8244 10.566 10.334 10.028 9.8610 9.7969 9.3596 5.9338 5.4797 5.3269 5.1742 5.0301 10.539 10.281 10.032 9.9669 9.9426 9.8886 9.7957 9.7366 9.5681 9.5134 9.4546 9.2996 9.0890 8.4464 7.6768 7.3110 6.4990 6.2267 6.0059 5.7965 5.7260 5.5767 5.5564
p (MW
p-p-T
TABLE I-l Relation for CHF, (continued)
p &mVm3>
T = 298.743 4.9499 4.8871 4.8539 4.8166 4.8139 4.7928 4.7755 4.7699 4.7504 4.743 1 4.7216 4.7108
K
T = 299.045 4.9465 4.8629 4.8327 4.8206 4.8124 4.8076 4.7927 4.7867 4.773 1 4.7564 4.7347
K
10.559 10.315 10.156 9.8746 9.8489 9.6687 9.4056 9.2421 5.5385 5.3894 5.0900 4.9475 10.340 9.8604 9.5573 9.3275 9.1285 8.9206 6.2096 5.8918 5.5892 5.2887 5.0109
T = 299.181 K 4.9949 10.427 4.9347 10.172 4.8780 9.7580 4.8292 9.0725 4.8239 8.8530 4.8207 8.6301 4.8186 8.4182 4.8124 6.5422 4.8099 6.2348 4.8036 5.9083 4.7893 5.5982 T = 299.422 5.0265 4.9789 4.9577 4.9273 4.8928 4.8791 4.8650 4.8560 4.8492 4.8476 4.845 1 4.8427 4.8400 4.8373 4.8346 4.8306 4.8239 4.8213 4.8008 4.7879
Fluids, Vol. 3, Nb. 2, 1990
K 10.383 10.134 10.077 9.9003 9.5956 9.3722 9.1438 8.9320 8.3623 8.1434 7.7559 7.3772 6.8632 6.5108 6.3343 6.1636 5.8849 5.8301 5.4118 5.2569
P WW
p (hoVm3)
4.7823 4.773 1 4.7586
5.4297 5.2846 5.1652
P @@‘a)
p (kmoVm3)
4.7616 4.7420
4.9650 4.825 1’
T = 300.144 K 5.0948 10.133 5.0108 9.6113 4.9769 1 9.1762 4.9647 9.0584 4.9483 8.6445 4.9378 8.2530 4.9318 7.8776 4.9255 7.3483 4.9209 6.9928 4.9097 6.5498 4.9023 6.2201 4.8884 5.8979 4.8735 5.5861
p-p-T P WW
TABLE I-2 Relation for N20
p (hol/m3)
T = 308.270 7.5119 7.2239 7.0854
K
T = 308.575 7.5463 7.2807 7.2002 7.1426 7.1076 7.0521 7.023 1
K
T = 308.739 7.3469 7.1900 7.1474 7.1207 7.0938 7.0783 7.0626
K
14.986 14.311 13.669 14.835 14.178 13.851 13.530 13.214 7.1640 6.7861 14.263 13.601 13.273 12.950 7.4788 7.2734 7.0689
T = 308.861 K 7.4588 14.471 7.2615 13.822 7.2032 13.501 7.1656 13.184 7.1415 12.876 T = 309.558 7.6953 7.6338 7.5510
p (MPa)
p (kmol/m3)
T = 309.040 7.5238 7.3166 7.2161 7.1852 7.1681
K
T = 309.226 7.5202 7.4074 7.328 1 7.2699 7.2327 7.2114 7.1646 7.1447
K
T = 309.365 7.3635 7.3039 7.2666 7.2375 7.2214
K
T = 309.922 7.6246 7.4532 7.3619 7.3234 7.3022
K
14.529 13.885 13.278 12.980 12.690 14.364 14.029 13.704 13.370 13.062 12.761 7.7504 7.3415 13.747 13.430 13.117 12.795 13.473 14.169 13.521 12.875 12.283 11.140
K 14.580 14.433 14.232
T = 310.261 K 7.5980 13.802 7.4296 12.829
The Journal
of Supercritical Fluids, Vol. 3, No. 2, 1990 p-p-T
TABLE I-2 Relation for N20 (continued)
p-p-T
p (kmol/m’)
p (MPa)
p (kmol/m”)
7.4537
13.921
7.3741
7.451 I 7.4192 7.3815 7.3784 7.3560 7.3278 7.31 I5 7.2940 7.2815 7.2695 7.2636 7.257 I 7.2489 7.241 I 7.2380 7.2367 7.2330 7.2276 7.2196 7.2109 7.2088 7.1636 7.1399 7.1219 7.098 I 7.0696
13.899 13.774 13.599 13.580 13.460 13.287 13.142 12.979 12.830 12.665 12.519 12.371 12.078 10.930 IO.421 9.8866 8.4735 8.2413 8.0148 7.8563 7.8022 7.2176 7.0189 6.8285 6.639 1 6.4612
7.3581
I I.955 I I.1 I4
P WV
Pressure-Density-Temperature
T = 310.866 8.0398 7.7628 7.6127 7.5312 7.4903 7.4678 7.4567 7.4550 7.441 I 7.4301 7.4086
K 14.542 13.844 13.188 12.560 I I .964 I I.395 10.836 10.753 9.71 I3 9.2190 8.5301
p (MPa)
4.6724 4.6358 4.6100 4.5962 4.5854 4.5783 4.5619 4.5545
7.4507 7.3300 7.2727 7.1554 7.099 I 6.9847 6.9263 6.8344 6.8168 6.7572 6.6678 6.5062 6:3406 6.1900 6.0334 5.6419 5.1887 5.0632 4.7919 4.5422 4.4228 4.3047 4.1846 4.0727
p-p-T p (MPa)
TABLE I-3 Relation for C3H6
p (kmol/m’)
T = 363.373 K 4.5465 4.5125 4.4922
T = 363.612
7.3798 7.2057 7.035 1 6.8693
T = 363.739 K
6.8676 6.7006 6.5353
4.5808 4.5650 4.5575 4.5527
T = 364.371 7.2849 7.1 I52 6.9434 6.7809
T = 363.880 K 4.6178 4.5783 4.5505 4.5363 4.5261
4.5532 4.542 1 4.5378
T = 364.207 K
K
4.5808 4.5478 4.5284 4.5172
p (kmol/m3)
T = 364.057 K 7.3500 7.1813 7.0084
4.5876 4.5473 4.5221 4.5090
p (MPa)
7.3973 7.2082 7.0390 6.8766 6.7186
6.9478 6.7840 6.6430 6.4855 K
4.5899 4.5769 4.5680 4.5638 4.5454 4.5407 4.5334 4.5264
6.8980 6.7297 6.5659 6.3964 4.4678 4.347 1 4.2275 4.1120
T = 364.500 K 4.7315
7.485 I
Saturation
T (K)
7.2878 7.1095 6.9409 6.7752 6.6077 6.443 I 4.6367 4.3958
T = 364.981
K
4.808 I 4.7496 4.7442 4.7039 4.7008 7.6927 4.6736 4.6723 4.6657 4.6472 4.6353 4.6283 4.6222 4.6197 4.6 I80 4.6156 4.6151 4.6107 4.608 I 4.6070 4.6032 4.5955 4.5825
T = 364.690 K 7.5120 7.3189 7.1352 7.1218 6.9586 6.9433 6.7890 6.6026 6.4416 4.5888 4.4658 4.3461 4.2300
p (kmol/m”)
P (MM
T = 364.589 K
4.7688 4.7039 4.660 I 4.6592 4.6324 4.6293 4.6149 4.6033 4.598 I 4.5712 4.5673 4.5622 4.5560
7.4990 7.3346 7.3129 7.1560 7.1353 7.0928 6.9794 6.9660 6.9148 6.7485 6.5861 6.4236 6.2096 6.058 I 5.9075 5.7024 5.6069 5.2896 5.0157 4.881 I 4.7596 4.5165 4.2848
T = 365.927 K 4.7642 4.7283 4.698 I 4.6956 4.6893 4.6839
6.8766 6.5422 5.9035 5.7521 5.4588 5.1774
TABLE II Pressure and Density
ps (MPa) psl WWm3)
psg(hoVm3)
CHF3 298.150 298.345 298.558 298.743 299.045 299.181
81
TABLE I-3 Relation for CJH6 (continued)
p (kmol/m-‘)
4.7364 4.6999 4.6844 4.6577 4.6464 4.6289 4.6201 4.61 I3 4.6097 4.605 I 4.6012 4.5911 4.5877 4.5839 4.5825 4.5791 4.5753 4.5744 4.5701 4.5650 4.5603 4.553 I 4.5452 4.5370
Relations
4.691 4.715 4.740 4.763 4.800 4.816
9.58 9.48 9.29 9.14 8.74 8.47
5.48 5.57 5.78 5.92 6.32 6.59
82 Ohgaki et al.
The Journal
Saturation
of Supercritical
Fluids,
Vol. 3, No. 2, 1990
TABLE II Pressure and Density (continued)
T 6) 308.270 308.575 308.739 308.861 309.040 309.226 309.365
7.041 7.092 7.114 7.133 7.160 7.188 7.210
13.28 13.07 12.84 12.70 12.47 12.18 12.02
7.68 7.87
-2.0 1 2
1 3
1 4
8.29
t 5
I 7
0,
6
-In IAPr I Figure
3. Determination of critical pressure for CHF,.
c3H,
363.373 363.612 363.739 363.880 364.057 364.207 364.37 1
4.479 4.500 4.509 4.518 4.533 4.548 4.558
6.87 6.77 6.70 6.63 6.53 6.43 6.25
Critical
4.80
CHF3
C3H, -6
t
1.25
TABLE III for CHF3, NzO, and C3H6
4.50
N20
I2 I!
Properties
Tc6)
pc WPa)
299.30 299.3 309.56 309.6 309.558 364.59 365.57 364.9
4.828 4.86 7.238 7.24 4.579 4.665 4.60
pc(kmol/m3) -
7.52 7.536 10.3 10.3 10.2697 5.55 5.31 5.525
Ref. * 21 * 21 20 * 18 21
*present study
1.50
1.75
2.00
2.25
-InlAPrl Figure 2. Determination of critical temperature for CHF, by means of the power law on the vapor-liquid coexistence curve. ditions, the critical density can be assumed to be equal to the average value of saturated liquid and vapor densities on each isotherm. The above method is also convenient for predicting the temperature where the p-p relation will be measured in the next run as the critical isotherm. If the discrepancy between the subcritical and supercritical isotherms is less than fO.l K, the critical temperature would be predicted within an accuracy of M.02 K. Once the critical temperature is determined, the critical pressure is obtained as follows. According to eq 2, the logarithmic plot of Apr against Apr should be a straight line for both the pr > I and pr < 1 regions. In other words, the true critical pressure is the pressure which gives a linear relation in the logarithmic plot of Apr and Apr as shown in Figure 3. The line for the pr > 1 region does not exactly coincide with that of pr < I in Figure 3,
although the plot is obtained by means of the optimum value of critical pressure. The main reason is that the isotherm of 299.300 K is not exactly the critical isotherm for CHF3. The plot is quite sensitive to a small temperature difference from the exact critical temperature. Table III summarizes the critical properties obtained in this study accompanied with the literature values. Restricted linear model of scaled equationof-state. As Landau’ has pointed out, no equations-ofstate based on the mean field assumption (e.g., van der Waals type and BWR type equations) can describe critical anomalies. Therefore, the precise p-p-T relations in the critical region have been investigated by using nonclassical equations-of-state. One of the most interesting nonclassical equations-of-state is the scaled equation-of-state proposed by Widom.2 In the present study, we adopted a kind of scaled equation-of-state, the so-called restricted linear model, in order to correlate the p-p-T relation obtained. The scaled equation-of-state by Widom is extended by Schofield, Lister, and Ho’ to the parametric representation of the so-called linear model which has the advantage of being entirely free of nonanalyticities and of being readily integrated. The transformations of Schofield are given by A& = a@1 - @)r@
(4)
The Journal of Supercritical Fluids, Vol. 3, No. 2, 1990 AT, = (1 - b2@)r
Pressure-Density-Temperature
(3
Relations
TABLE IV in the Universal Restricted
Parameters
83
Model
ad StZiIKhd
Apr = k&P.
(6)
The thermodynamic variables are expressed in terms of two scaling variables, r and 8, where r stands for a distance from the critical point and 8 is a location on a contour of the distance: -1 < 0 < 1 for one-phase region; 8 = fl on the vapor-liquid coexistence curve; and 8 = 0 on the critical isochore in one-phase region. From the conditions of the scaling law, the value of b in eq 5 is given by means of the critical exponents p and 6 (restricted condition) b2 = (6 - 3)/(6 - l)(l - 2p) .
a CHF3
k
CO
dev. of p
Cl
12.5
1.31
-6.6443
7.6443
21.4 18.0
1.45 1.30
-5.7102
6.7106
-5.5288
6.5284
0.0015 0.0005 0.0011
(7)
Finally, a and k in eqs 4 and 6 are adopted as adjustable constants in this study. From integration, the reduced form of Helmholtz free energy is easily given as follows: 4 = Ar,oVr) + wr(~J,) ak(l
- 2p)(@ 2w
+
+ f,e* + f2p(6 + p - 2)
+ 1)
(8)
where A,o(T,) is the regular term for the reduced form of Helmholtz free energy described in terms of the reduced temperature alone. The functions f, and f2 are given by eqs9and 10
4.6
1
4
fi = 2(1-P&NP~+ P- 1) f2=(6- 1)2@6-P-
(9)
l)/(S+ l)(p-3)(ps+p-l>.
(10)
Therefore, the reduced form of pressure becomes or = -A,,o(T,) +&(I ak(1
+ 4,)
-
- 2/?)(@ + f,@ + f2)rP(J 2(P6 + p - 2)
+ 1)
(11)
From the saturated properties at each T,., the relation of T, and A,o(T,) is obtained by substituting 8 = fl into eq 11. The description for A,,(T,) is given by eq 12 A,,o(T,) = co + c,Tr .
(12)
By using series expansions of the partition function around the critical point, the classical (the mean-field assumption) exponent p equals l/2, and the critical isotherm is of the third degree (6 = 3). These critical exponents are indeed far from the experimental results. According to the three-dimensional Ising model, the values of the critical
5
6
1
1
1
I
7
8
9
10
P / kmol-ni3 Figure 4. The p-p-T relation for CHF, in the critical region and the calculated value by use of restricted linear model. exponents p and 6 are universally 0.3 125 and 5.0, respectively. It is expected, therefore, that the experimental results can be described rather well by Ising-like critical exponents. According to Levelt Sengers,8 when based on the average values in p-p-T experiments, the universal set of exponents is given as p = 0.355 and 6 = 4.35. Using the universal exponents’ and the scaling law, the p-p-T relations for the three substances investigated in the present study have been correlated with the two adjustable parameters, a and k, for each substance. Table IV gives the optimum parameter set and the average errors. Figure 4 shows the correlation result for CHF, as a typical case. The agreement between the calculated and experimental results is satisfactory in the critical region. + Sengers and Levelt Senger9 also presented a new universal set that is close to the theoretical value of the three-dimensional Ising model.
84 Ohgaki et al.
The Journal of Supercritical
CONCLUSIONS The p-p-T relations of NzO, CHFs, and C3H6 were measured in the critical region by means of a direct method. By using the power law, the critical temperature, pressure, and density were determined for each substance. The universal linear model with the scaling law is also useful for correlating the p-p-T relations for these substances in the critical region. However, the agreement between the calculated and experimental results becomes poor in a region far from the critical point. It is not clear whether the discrepancy in the region is caused by the simplicity of the linear scaled equation-of-state or by the combination of the regular and singular terms for the Helmholtz free energy. ACKNOWLEDGMENTS The authors are grateful to Seitetu Kagaku Corp. and Daikin Ind., Ltd., for supplying the sample gases. They are also grateful to Tatsuya Tohdo for his experimental assistance. This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture of Japan. Nomenclature parameters of the restricted linear model a,k = = b defined in eq 7 coefficient for regular term of Helmholtz ; I function of p and 6 pressure (MPa) P = = r scaled variable temperature (K) 5,s 1 critical exponents = scaled variable chemical potential (J/mol) P = density (kmol/m3) P = Subscripts = = r C
S
=
critical IEdUced
scaled
energy
Fluids, Vol. 3, No. 2, 1990
REFERENCES (1) Omstein, L. S.; Zemike, F. Proc. Acad. Sci. Amst. 1914,14, 793. (2) Widom, B. J. Chem. Phys. 1965,43, 3898. E. M. Statistical Physics; (3) Landau, L.; Lifschitz, Addison-Wesley: New York, 1968; p. 265. (4) Ising, E. Z. Physik. 1925, 31, 253. Josephson, B. D. J. Phys. C. Proc. Phys. Sot. London (3 1969,2, 1113. (6) Schofield, P. Phys. Rev. Lett. 1969,22, 606. (7) Schofield, P.; Lister, J. D.; Ho, J. T. Phys. Rev. Lett. 1969,23, 1098. (8) Levelt Sengers, J. M. H. in High Pressure Technology Vol. 2. Applications and Procedures; Spain, I. L.; Poauwe, J., Eds.; Marcel Dekker: New York, 1977; chap. 5. (9) Senger, J. V.; Levelt Senger, J. M. H. Ann. Rev. Phys. Chem. 1986,37, 189. (10) Kierstead, H. A. Phys. Rev. 1968, 170, 213. (11) Gielen, H. Jansoone, V.; Verbeke, 0. J. Chem. Phys. 1973,59, 5763. (12) Verbeke, 0.; Jansoone, V.; Gielen, H.; de Boelpaep, J. J. Phys. Chem. 1969, 74, 4076. (13) Behringer, R. P.; Doiron, T.; Meyer, H. J. Low Temp. Phys. 1976,24, 315. (14) Levelt Sengers, J. M. H.; Greer, W. L.; Sengers, J. V. J. Phys. Chem. Ref. Data 1976,5, 1. M.; Levelt Sengers, J. M. H.; (15) Vincentini-Missoni, Green, M. S. J. Res. Natl. Bur. Stand. 1969, 73A, 563. (16) Fisher, M. E. J. Math. Phys. 1964.5, 944. S.; Armstrong, B.; de Reuck, K. M. (17) Angus, International Thermodynamic Tables of the Fluid State, Carbon Dioxide; Pergamon: Oxford, 1976. B.; de Reuck, K. M. (18) Angus, S.; Armstrong, International Thermodynamic Tables of the Fluid State Propylene; Pergamon: Oxford, 1980. (19) Ohgaki, K.; Nakamura, Y.; Ariyasu, H.; Katayama, T. J. Chem. Eng. Japan 1982, IS, 85 (20) Levelt Sengers, J. M. H. J. Chem. Phys. 1971,54, 5034. (21) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; 4th ed; MacGraw-Hill: New York, 1987. (22) Le Guillou, J. C.; Zinn-Justin J. Phys. Rev. 1980, B21, 3976.
The Journal of Supercritical
Fluids 1990,3,
78-84
Pressure-Density-Temperature (p-PT) Relations of CHF3, N20, and C3H6 in the Critical Region Kazunari Ohgaki, * Shyoji Umezono, and Takashi Katayama Department of Chemical Engineering, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan Received March 30, 1990; accepted in revised form June 7, 1990
The pressureAensity-temperature @-~7’) relations in the critical region (0.996 < T,.c 1.004 and 0.6 c pr < 1.4) are determined for CHF3, N20, and C3H6 by means of a direct method. The critical properties are determined from a logarithmic plot based on the power law for the vapor-liquid coexistence curve and the p-p relation on the critical isotherm for each system. The restricted linear model, one of the simplest scaled equations-of-state, is applied to describe the p-pT relations in the critical region. Keywords:
critical properties, scaled equation-of-state, p-p-T relation
INTRODUCTION In the last two decades, the studies on thermodynamic properties of near critical and supercritical fluids have significantly increased. This is primarily because of their practical importance for high pressure technology such as supercritical fluid extraction. Another reason is a theoretical one which is concerned with the anomalous behavior of thermodynamic properties in the vicinity of the critical point. The critical anomalies in thermodynamic and transport properties based on density fluctuations were first investigated by Omstein and Zemike.’ Studies on the scaled equation-of-state by Widom* and Landau3 and on the Ising model4 created a new field concerning the critical exponents. Josephson5 and Schofield6” proposed a parametric representation by a transformation from physical to scaling variables. Also, special attention was given to the universality law8g9for the critical exponents in the second order phase transitions such as the Curie point. In experimental studies, the critical exponents were obtained from pressure-density-temperature (p-p-0 relations in the critical region for several substances.‘O-I5 Furthermore, the precise p-FT data in the critical region were correlated by means of nonclassical equations-of-state based on the power law.16 In the present study, the p-p-T relations for N20, CHF3, and C3H6 are measured in the critical region. Critical properties are evaluated from the data using the power law. Finally, the applicability of the universality law for p-p-T relations in the critical region is discussed by means of the universal restricted linear model. 0896-8446/90/0302-0078$4.00/O
v Figure 1. Schematic diagram of experimental apparatus for determining p-p-T relations. (A), piston gauge; (B), pressure transducer; (C), oil hand pump; (D), oil reservoir; (E), oil manometer; (F), oil dead-weight gauge; (G), quartz Bourdon gauge; (H), N, gas hand pump; (I), compressor; (J), N, gas cylinder; (K), sample gas cylinder; (L), high pressure pump; (M), pressuretransducer; (N), high pressurecell; (P), trapping cell; (Q), liquid N, bath; (R), vacuum pump; (S), Pirani gauge. EXPERIMENTAL Experimental apparatus and procedures. A schematic diagram of the experimental apparatus used in the measurement of the pressure+lensity-temperature (pp-7’) relations is shown in Figure 1. The apparatus consists of a high pressure cell and sampling part, a pressure measurement part, a temperature measurement part, and a vacuum line. The high pressure cell (N) of stainless steel (the internal volume of ca. 90 cm3 calibrated with compressed CO*” and C3H6 I8 fluids at several pressures) was evacuated together with the sampling part (between the 0 1990 PRA Press
Pressure-Density-Temperature
The Journal of Supercritical Fluids, Vol. 3, No. 2, 1990 valves 11 and 17) to lower than 10e3mmHg by a vacuum pump (R). By using a high pressure pump, the sample fluid was charged up to the desired pressure. After thermal equilibrium was reached over a period of about one hour, the equilibrium pressure was measured with an accuracy of f5 x 10e4 MPa by use of a combination of a null-type pressure transducer (M) and a quartz Bourdon-tube gauge (G) calibrated by an oil dead-weight gauge (G) and an air piston gauge (A) manufactured by Ruska Instrument Corp. The effects of working pressure and temperature on the null-position of the diaphragm in the pressure transducer were corrected previously. The temperature of the oil bath (used in the experiments with C3H6) and water bath (for N20 and CHF3) was measured within an accuracy of ti.004 K (in the oil bath) and ti.001 K (in the water bath) by a set of a platinum resistance and a resistance thermometer bridge as defined by the IPTS of 1968. The platinum resistance was calibrated by the National Research Laboratory of Metrology, Japan. The procedures for pressure and temperature measurements are essentially the same as reported previously.‘9 After the pressure measurement, a small amount of fluid (ca. 2% of the total sample) was expanded into the sampling part (valves 1l-17) in order to change the bulk density. Then, the content of the sampling part was introduced to a trapping cell (P) which was evacuated beforehand to about 10e3 mmHg. The evaporated sample was solidified in the trapping cell maintained in a liquid nitrogen bath (Q). The complete solidification was confirmed through a vacuum gauge (S). As the volume of line (from valve 11 to the trapping cell) is small enough, the amount of gas retained at the sublimation pressure can be neglected. The amount in the trapping cell was determined by weighing on a precision balance (max. 3 kg) of Cho Balance Corp. with an accuracy of 50.3 mg. Both the accurate pressure measurement and the determination of amount of sample trapped were required in every expansion. After several expansions with about one hour intervals, the fluid retained in the high pressure cell was solidified completely in the trapping cell in order to determine the total amount of sample, using the same procedure described above. The total weights of sample fluid were about 50-60 g for CHF3 and N20 and 20-30 g for C3H6. The densities were determined within an accuracy of f0.03% from the amount of sample and the known cell volume. Finally, the several series of the p-p relations were obtained at different temperatures in the critical region. Materials. CHF3 which was obtained from Daikin Kagaku Corp. and N20 from Seitetu Kagaku Corp. had special minimum purity of 99.999 mol%. Research grade C3H6 obtained from Sumitomo Kagaku Corp. had purity of 9.96 mol%. The substances were used in this experiment without further purification.
Relations
79
RESULTS AND DISCUSSION p-p relation for the one-phase
region. Table I gives the p-p-T relations obtained in the one-phase region. This table does not contain the data in the twophase region. Saturation properties. When the pressure was not changed by an expansion, we regarded it as the saturation pressure at that temperature. Also, the density where the isothermal p-p line crossed the saturation pressure at a given temperature can be taken as the saturation density. The saturation pressures are used to determine the parameters, co and cl, of a regular term of the Helmholtz free energy in the restricted linear model discussed later. The saturation pressures and densities are obtained with accuracies of +0.03% and ti.2%, respectively. The saturation densities of gases are generally less reliable than those of liquids. As it is difficult to obtain data for gas density, some isotherms do not contain the data on saturated gas density. The main reason is that the amount of mass expanded is too small to give a precise density for the gas phase. Furthermore, the adsorption effect in the gas phase causes significant errors of density measurements. Table II summarizes the saturation properties obtained for each fluid. Critical
properties
and
critical
exponents.
The critical anomalies are described in terms of logarithmic scale by
4
0~ [AT,?
(1)
on the vapor-liquid coexistence line and by (2) on the critical isotherm. The exponents p and 6 characterize, respectively, the asymptotic behavior of the coexistence curve and the critical isotherm. The term AX, is defined as
Ax,= w - X,)/X,
(X= T,p,p>.
(3)
By use of the above relations, the critical temperature, density, and pressure are determined from the p-p isotherms obtained from experiment data. A method for determining the critical temperature is as follows. By using the saturation properties on several isotherms near the critical temperature, AT, is plotted against Apr on a logarithmic scale as shown in Figure 2. If the temporary critical temperature, ?,I, is extremely close to the true critical temperature, the plot shows a linear relation according to eq 1. As Figure 2 shows, the plot becomes straight for’ 299.300 K, while neither T,’ = 299.181 K (T, = 0.9996) nor T,’ = 299.422 K (T, = 1.0004) give a straight line. As the coexistence curve in p-p plane is almost symmetric in this experimental con-
80
Ohgaki et al.
p-p-T
P (MW
The Journal of Supercritical TABLE I-l Relation for CHFJ
p WoVm3)
T = 298.150 K 4.8014 4.7499 4.7428 4.7191 4.7005 4.6955 4.6745 4.6670 4.6554 4.6441
10.432 10.172 10.126 9.9018 9.6801 9.4616 5.1949 5.0503 4.9080 4.7709
T = 298.345 4.8643 4.8974 4.7697 4.7677 4.7405 4.7259 4.7023 4.6926 4.6756 4.6580
K
T = 298.558 4.9187 4.8539 4.8097 4.7990 4.7571 4.7445 4.7359 4.7285 4.7189 4.7104 4.698 1
K
T = 299.300 5.0440 4.9756 4.9241 4.9156 4.9100 4.904 1 4.8899 4.8866 4.8674 4.8653 4.8599 4.8498 4.8423 4.8320 4.8283 4.8257 4.8255 4.8208 4.8123 4.8019 4.7999 4.7918 4.7899
K
10.529 10.300 10.068 10.055 9.8157 9.635 4.3457 5.1954 4.9653 4.8244 10.566 10.334 10.028 9.8610 9.7969 9.3596 5.9338 5.4797 5.3269 5.1742 5.0301 10.539 10.281 10.032 9.9669 9.9426 9.8886 9.7957 9.7366 9.5681 9.5134 9.4546 9.2996 9.0890 8.4464 7.6768 7.3110 6.4990 6.2267 6.0059 5.7965 5.7260 5.5767 5.5564
p (MW
p-p-T
TABLE I-l Relation for CHF, (continued)
p &mVm3>
T = 298.743 4.9499 4.8871 4.8539 4.8166 4.8139 4.7928 4.7755 4.7699 4.7504 4.743 1 4.7216 4.7108
K
T = 299.045 4.9465 4.8629 4.8327 4.8206 4.8124 4.8076 4.7927 4.7867 4.773 1 4.7564 4.7347
K
10.559 10.315 10.156 9.8746 9.8489 9.6687 9.4056 9.2421 5.5385 5.3894 5.0900 4.9475 10.340 9.8604 9.5573 9.3275 9.1285 8.9206 6.2096 5.8918 5.5892 5.2887 5.0109
T = 299.181 K 4.9949 10.427 4.9347 10.172 4.8780 9.7580 4.8292 9.0725 4.8239 8.8530 4.8207 8.6301 4.8186 8.4182 4.8124 6.5422 4.8099 6.2348 4.8036 5.9083 4.7893 5.5982 T = 299.422 5.0265 4.9789 4.9577 4.9273 4.8928 4.8791 4.8650 4.8560 4.8492 4.8476 4.845 1 4.8427 4.8400 4.8373 4.8346 4.8306 4.8239 4.8213 4.8008 4.7879
Fluids, Vol. 3, Nb. 2, 1990
K 10.383 10.134 10.077 9.9003 9.5956 9.3722 9.1438 8.9320 8.3623 8.1434 7.7559 7.3772 6.8632 6.5108 6.3343 6.1636 5.8849 5.8301 5.4118 5.2569
P WW
p (hoVm3)
4.7823 4.773 1 4.7586
5.4297 5.2846 5.1652
P @@‘a)
p (kmoVm3)
4.7616 4.7420
4.9650 4.825 1’
T = 300.144 K 5.0948 10.133 5.0108 9.6113 4.9769 1 9.1762 4.9647 9.0584 4.9483 8.6445 4.9378 8.2530 4.9318 7.8776 4.9255 7.3483 4.9209 6.9928 4.9097 6.5498 4.9023 6.2201 4.8884 5.8979 4.8735 5.5861
p-p-T P WW
TABLE I-2 Relation for N20
p (hol/m3)
T = 308.270 7.5119 7.2239 7.0854
K
T = 308.575 7.5463 7.2807 7.2002 7.1426 7.1076 7.0521 7.023 1
K
T = 308.739 7.3469 7.1900 7.1474 7.1207 7.0938 7.0783 7.0626
K
14.986 14.311 13.669 14.835 14.178 13.851 13.530 13.214 7.1640 6.7861 14.263 13.601 13.273 12.950 7.4788 7.2734 7.0689
T = 308.861 K 7.4588 14.471 7.2615 13.822 7.2032 13.501 7.1656 13.184 7.1415 12.876 T = 309.558 7.6953 7.6338 7.5510
p (MPa)
p (kmol/m3)
T = 309.040 7.5238 7.3166 7.2161 7.1852 7.1681
K
T = 309.226 7.5202 7.4074 7.328 1 7.2699 7.2327 7.2114 7.1646 7.1447
K
T = 309.365 7.3635 7.3039 7.2666 7.2375 7.2214
K
T = 309.922 7.6246 7.4532 7.3619 7.3234 7.3022
K
14.529 13.885 13.278 12.980 12.690 14.364 14.029 13.704 13.370 13.062 12.761 7.7504 7.3415 13.747 13.430 13.117 12.795 13.473 14.169 13.521 12.875 12.283 11.140
K 14.580 14.433 14.232
T = 310.261 K 7.5980 13.802 7.4296 12.829
The Journal
of Supercritical Fluids, Vol. 3, No. 2, 1990 p-p-T
TABLE I-2 Relation for N20 (continued)
p-p-T
p (kmol/m’)
p (MPa)
p (kmol/m”)
7.4537
13.921
7.3741
7.451 I 7.4192 7.3815 7.3784 7.3560 7.3278 7.31 I5 7.2940 7.2815 7.2695 7.2636 7.257 I 7.2489 7.241 I 7.2380 7.2367 7.2330 7.2276 7.2196 7.2109 7.2088 7.1636 7.1399 7.1219 7.098 I 7.0696
13.899 13.774 13.599 13.580 13.460 13.287 13.142 12.979 12.830 12.665 12.519 12.371 12.078 10.930 IO.421 9.8866 8.4735 8.2413 8.0148 7.8563 7.8022 7.2176 7.0189 6.8285 6.639 1 6.4612
7.3581
I I.955 I I.1 I4
P WV
Pressure-Density-Temperature
T = 310.866 8.0398 7.7628 7.6127 7.5312 7.4903 7.4678 7.4567 7.4550 7.441 I 7.4301 7.4086
K 14.542 13.844 13.188 12.560 I I .964 I I.395 10.836 10.753 9.71 I3 9.2190 8.5301
p (MPa)
4.6724 4.6358 4.6100 4.5962 4.5854 4.5783 4.5619 4.5545
7.4507 7.3300 7.2727 7.1554 7.099 I 6.9847 6.9263 6.8344 6.8168 6.7572 6.6678 6.5062 6:3406 6.1900 6.0334 5.6419 5.1887 5.0632 4.7919 4.5422 4.4228 4.3047 4.1846 4.0727
p-p-T p (MPa)
TABLE I-3 Relation for C3H6
p (kmol/m’)
T = 363.373 K 4.5465 4.5125 4.4922
T = 363.612
7.3798 7.2057 7.035 1 6.8693
T = 363.739 K
6.8676 6.7006 6.5353
4.5808 4.5650 4.5575 4.5527
T = 364.371 7.2849 7.1 I52 6.9434 6.7809
T = 363.880 K 4.6178 4.5783 4.5505 4.5363 4.5261
4.5532 4.542 1 4.5378
T = 364.207 K
K
4.5808 4.5478 4.5284 4.5172
p (kmol/m3)
T = 364.057 K 7.3500 7.1813 7.0084
4.5876 4.5473 4.5221 4.5090
p (MPa)
7.3973 7.2082 7.0390 6.8766 6.7186
6.9478 6.7840 6.6430 6.4855 K
4.5899 4.5769 4.5680 4.5638 4.5454 4.5407 4.5334 4.5264
6.8980 6.7297 6.5659 6.3964 4.4678 4.347 1 4.2275 4.1120
T = 364.500 K 4.7315
7.485 I
Saturation
T (K)
7.2878 7.1095 6.9409 6.7752 6.6077 6.443 I 4.6367 4.3958
T = 364.981
K
4.808 I 4.7496 4.7442 4.7039 4.7008 7.6927 4.6736 4.6723 4.6657 4.6472 4.6353 4.6283 4.6222 4.6197 4.6 I80 4.6156 4.6151 4.6107 4.608 I 4.6070 4.6032 4.5955 4.5825
T = 364.690 K 7.5120 7.3189 7.1352 7.1218 6.9586 6.9433 6.7890 6.6026 6.4416 4.5888 4.4658 4.3461 4.2300
p (kmol/m”)
P (MM
T = 364.589 K
4.7688 4.7039 4.660 I 4.6592 4.6324 4.6293 4.6149 4.6033 4.598 I 4.5712 4.5673 4.5622 4.5560
7.4990 7.3346 7.3129 7.1560 7.1353 7.0928 6.9794 6.9660 6.9148 6.7485 6.5861 6.4236 6.2096 6.058 I 5.9075 5.7024 5.6069 5.2896 5.0157 4.881 I 4.7596 4.5165 4.2848
T = 365.927 K 4.7642 4.7283 4.698 I 4.6956 4.6893 4.6839
6.8766 6.5422 5.9035 5.7521 5.4588 5.1774
TABLE II Pressure and Density
ps (MPa) psl WWm3)
psg(hoVm3)
CHF3 298.150 298.345 298.558 298.743 299.045 299.181
81
TABLE I-3 Relation for CJH6 (continued)
p (kmol/m-‘)
4.7364 4.6999 4.6844 4.6577 4.6464 4.6289 4.6201 4.61 I3 4.6097 4.605 I 4.6012 4.5911 4.5877 4.5839 4.5825 4.5791 4.5753 4.5744 4.5701 4.5650 4.5603 4.553 I 4.5452 4.5370
Relations
4.691 4.715 4.740 4.763 4.800 4.816
9.58 9.48 9.29 9.14 8.74 8.47
5.48 5.57 5.78 5.92 6.32 6.59
82 Ohgaki et al.
The Journal
Saturation
of Supercritical
Fluids,
Vol. 3, No. 2, 1990
TABLE II Pressure and Density (continued)
T 6) 308.270 308.575 308.739 308.861 309.040 309.226 309.365
7.041 7.092 7.114 7.133 7.160 7.188 7.210
13.28 13.07 12.84 12.70 12.47 12.18 12.02
7.68 7.87
-2.0 1 2
1 3
1 4
8.29
t 5
I 7
0,
6
-In IAPr I Figure
3. Determination of critical pressure for CHF,.
c3H,
363.373 363.612 363.739 363.880 364.057 364.207 364.37 1
4.479 4.500 4.509 4.518 4.533 4.548 4.558
6.87 6.77 6.70 6.63 6.53 6.43 6.25
Critical
4.80
CHF3
C3H, -6
t
1.25
TABLE III for CHF3, NzO, and C3H6
4.50
N20
I2 I!
Properties
Tc6)
pc WPa)
299.30 299.3 309.56 309.6 309.558 364.59 365.57 364.9
4.828 4.86 7.238 7.24 4.579 4.665 4.60
pc(kmol/m3) -
7.52 7.536 10.3 10.3 10.2697 5.55 5.31 5.525
Ref. * 21 * 21 20 * 18 21
*present study
1.50
1.75
2.00
2.25
-InlAPrl Figure 2. Determination of critical temperature for CHF, by means of the power law on the vapor-liquid coexistence curve. ditions, the critical density can be assumed to be equal to the average value of saturated liquid and vapor densities on each isotherm. The above method is also convenient for predicting the temperature where the p-p relation will be measured in the next run as the critical isotherm. If the discrepancy between the subcritical and supercritical isotherms is less than fO.l K, the critical temperature would be predicted within an accuracy of M.02 K. Once the critical temperature is determined, the critical pressure is obtained as follows. According to eq 2, the logarithmic plot of Apr against Apr should be a straight line for both the pr > I and pr < 1 regions. In other words, the true critical pressure is the pressure which gives a linear relation in the logarithmic plot of Apr and Apr as shown in Figure 3. The line for the pr > 1 region does not exactly coincide with that of pr < I in Figure 3,
although the plot is obtained by means of the optimum value of critical pressure. The main reason is that the isotherm of 299.300 K is not exactly the critical isotherm for CHF3. The plot is quite sensitive to a small temperature difference from the exact critical temperature. Table III summarizes the critical properties obtained in this study accompanied with the literature values. Restricted linear model of scaled equationof-state. As Landau’ has pointed out, no equations-ofstate based on the mean field assumption (e.g., van der Waals type and BWR type equations) can describe critical anomalies. Therefore, the precise p-p-T relations in the critical region have been investigated by using nonclassical equations-of-state. One of the most interesting nonclassical equations-of-state is the scaled equation-of-state proposed by Widom.2 In the present study, we adopted a kind of scaled equation-of-state, the so-called restricted linear model, in order to correlate the p-p-T relation obtained. The scaled equation-of-state by Widom is extended by Schofield, Lister, and Ho’ to the parametric representation of the so-called linear model which has the advantage of being entirely free of nonanalyticities and of being readily integrated. The transformations of Schofield are given by A& = a@1 - @)r@
(4)
The Journal of Supercritical Fluids, Vol. 3, No. 2, 1990 AT, = (1 - b2@)r
Pressure-Density-Temperature
(3
Relations
TABLE IV in the Universal Restricted
Parameters
83
Model
ad StZiIKhd
Apr = k&P.
(6)
The thermodynamic variables are expressed in terms of two scaling variables, r and 8, where r stands for a distance from the critical point and 8 is a location on a contour of the distance: -1 < 0 < 1 for one-phase region; 8 = fl on the vapor-liquid coexistence curve; and 8 = 0 on the critical isochore in one-phase region. From the conditions of the scaling law, the value of b in eq 5 is given by means of the critical exponents p and 6 (restricted condition) b2 = (6 - 3)/(6 - l)(l - 2p) .
a CHF3
k
CO
dev. of p
Cl
12.5
1.31
-6.6443
7.6443
21.4 18.0
1.45 1.30
-5.7102
6.7106
-5.5288
6.5284
0.0015 0.0005 0.0011
(7)
Finally, a and k in eqs 4 and 6 are adopted as adjustable constants in this study. From integration, the reduced form of Helmholtz free energy is easily given as follows: 4 = Ar,oVr) + wr(~J,) ak(l
- 2p)(@ 2w
+
+ f,e* + f2p(6 + p - 2)
+ 1)
(8)
where A,o(T,) is the regular term for the reduced form of Helmholtz free energy described in terms of the reduced temperature alone. The functions f, and f2 are given by eqs9and 10
4.6
1
4
fi = 2(1-P&NP~+ P- 1) f2=(6- 1)2@6-P-
(9)
l)/(S+ l)(p-3)(ps+p-l>.
(10)
Therefore, the reduced form of pressure becomes or = -A,,o(T,) +&(I ak(1
+ 4,)
-
- 2/?)(@ + f,@ + f2)rP(J 2(P6 + p - 2)
+ 1)
(11)
From the saturated properties at each T,., the relation of T, and A,o(T,) is obtained by substituting 8 = fl into eq 11. The description for A,,(T,) is given by eq 12 A,,o(T,) = co + c,Tr .
(12)
By using series expansions of the partition function around the critical point, the classical (the mean-field assumption) exponent p equals l/2, and the critical isotherm is of the third degree (6 = 3). These critical exponents are indeed far from the experimental results. According to the three-dimensional Ising model, the values of the critical
5
6
1
1
1
I
7
8
9
10
P / kmol-ni3 Figure 4. The p-p-T relation for CHF, in the critical region and the calculated value by use of restricted linear model. exponents p and 6 are universally 0.3 125 and 5.0, respectively. It is expected, therefore, that the experimental results can be described rather well by Ising-like critical exponents. According to Levelt Sengers,8 when based on the average values in p-p-T experiments, the universal set of exponents is given as p = 0.355 and 6 = 4.35. Using the universal exponents’ and the scaling law, the p-p-T relations for the three substances investigated in the present study have been correlated with the two adjustable parameters, a and k, for each substance. Table IV gives the optimum parameter set and the average errors. Figure 4 shows the correlation result for CHF, as a typical case. The agreement between the calculated and experimental results is satisfactory in the critical region. + Sengers and Levelt Senger9 also presented a new universal set that is close to the theoretical value of the three-dimensional Ising model.
84 Ohgaki et al.
The Journal of Supercritical
CONCLUSIONS The p-p-T relations of NzO, CHFs, and C3H6 were measured in the critical region by means of a direct method. By using the power law, the critical temperature, pressure, and density were determined for each substance. The universal linear model with the scaling law is also useful for correlating the p-p-T relations for these substances in the critical region. However, the agreement between the calculated and experimental results becomes poor in a region far from the critical point. It is not clear whether the discrepancy in the region is caused by the simplicity of the linear scaled equation-of-state or by the combination of the regular and singular terms for the Helmholtz free energy. ACKNOWLEDGMENTS The authors are grateful to Seitetu Kagaku Corp. and Daikin Ind., Ltd., for supplying the sample gases. They are also grateful to Tatsuya Tohdo for his experimental assistance. This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture of Japan. Nomenclature parameters of the restricted linear model a,k = = b defined in eq 7 coefficient for regular term of Helmholtz ; I function of p and 6 pressure (MPa) P = = r scaled variable temperature (K) 5,s 1 critical exponents = scaled variable chemical potential (J/mol) P = density (kmol/m3) P = Subscripts = = r C
S
=
critical IEdUced
scaled
energy
Fluids, Vol. 3, No. 2, 1990
REFERENCES (1) Omstein, L. S.; Zemike, F. Proc. Acad. Sci. Amst. 1914,14, 793. (2) Widom, B. J. Chem. Phys. 1965,43, 3898. E. M. Statistical Physics; (3) Landau, L.; Lifschitz, Addison-Wesley: New York, 1968; p. 265. (4) Ising, E. Z. Physik. 1925, 31, 253. Josephson, B. D. J. Phys. C. Proc. Phys. Sot. London (3 1969,2, 1113. (6) Schofield, P. Phys. Rev. Lett. 1969,22, 606. (7) Schofield, P.; Lister, J. D.; Ho, J. T. Phys. Rev. Lett. 1969,23, 1098. (8) Levelt Sengers, J. M. H. in High Pressure Technology Vol. 2. Applications and Procedures; Spain, I. L.; Poauwe, J., Eds.; Marcel Dekker: New York, 1977; chap. 5. (9) Senger, J. V.; Levelt Senger, J. M. H. Ann. Rev. Phys. Chem. 1986,37, 189. (10) Kierstead, H. A. Phys. Rev. 1968, 170, 213. (11) Gielen, H. Jansoone, V.; Verbeke, 0. J. Chem. Phys. 1973,59, 5763. (12) Verbeke, 0.; Jansoone, V.; Gielen, H.; de Boelpaep, J. J. Phys. Chem. 1969, 74, 4076. (13) Behringer, R. P.; Doiron, T.; Meyer, H. J. Low Temp. Phys. 1976,24, 315. (14) Levelt Sengers, J. M. H.; Greer, W. L.; Sengers, J. V. J. Phys. Chem. Ref. Data 1976,5, 1. M.; Levelt Sengers, J. M. H.; (15) Vincentini-Missoni, Green, M. S. J. Res. Natl. Bur. Stand. 1969, 73A, 563. (16) Fisher, M. E. J. Math. Phys. 1964.5, 944. S.; Armstrong, B.; de Reuck, K. M. (17) Angus, International Thermodynamic Tables of the Fluid State, Carbon Dioxide; Pergamon: Oxford, 1976. B.; de Reuck, K. M. (18) Angus, S.; Armstrong, International Thermodynamic Tables of the Fluid State Propylene; Pergamon: Oxford, 1980. (19) Ohgaki, K.; Nakamura, Y.; Ariyasu, H.; Katayama, T. J. Chem. Eng. Japan 1982, IS, 85 (20) Levelt Sengers, J. M. H. J. Chem. Phys. 1971,54, 5034. (21) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; 4th ed; MacGraw-Hill: New York, 1987. (22) Le Guillou, J. C.; Zinn-Justin J. Phys. Rev. 1980, B21, 3976.