Experimental and theoretical study of the equation of state of

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J . Phys. Chem. 1991, 95, 3351-3357

3351

Experimental and Theoretical Study of the Equation of State of CHF, in the Near-Critical Region Arturo G. Aizpiri, Antonio Rey, Jorge Dbvila, Ramdn G. Rubio,* Departamento de Quimica- Fhica, Facultad de Quimicas, Universidad Complutense. 28040 Madrid, Spain

John A. Zollweg, and William B. Streett School of Chemical Engineering, Olin Hall, Cornell University, Ithaca, New York 14853 (Received: August 2, 1990; In Final Form: October 25, 1990)

The pressure-density-temperature (p-p-T) surface of CHF3 has been measured in the near-critical region by using an expansion-type apparatus. The data overlap with other previously published data for the classical region and seem compatible with results already reported for the critical region. The abilities of two revised and extended scaling equations of state for predicting the data have been tested. The equation of Senger et al. describes the data satisfactorily, though in a quite limited region around the critical point. The results obtained with the equation of Anisimov et al. lead to similar results, though its applicability is restricted to a smaller region around the critical point. A classical equation with modified thermodynamic fields, leading to the correct scaling asymptotic behavior, describes the near-critical data as well as the equation of Sengers et at. while giving a better description of the crossover region.

Introduction The classical region of the liquid-vapor phase transition has been studied in the framework either of liquid-state theory' or of renormalization group (RG).2 It has become clear that perturbative approaches and integral equations are unable to give a realistic description of the critical region. On the other hand, the usual momentum space RG techniques provide information only about the universal quantities characterizing the scaling behavior near the critical point (CP). Some attempts have concentrated upon the idea of combining separate classical and nonclassical equations of state (EoS),each describing its appropriate part of the phase diagram.3 The crossover between these two regions is then smoothly switched by a third "switching" function. An important problem associated with this method is that of choosing a switching function so that the resulting behavior, in particular that of the second-derivative properties, is free from irregularities on the boundary between classical and nonclassical regions. The RG approach leads to expressions for the thermodynamic potential in terms of two scaling fields4 One of these fields is conjugate to the order parameter, and the other plays the role of the temperature. The relationship between these scaling fields and the thermodynamic field variables accessible to experiment is not universal and is not known from theory for any real fluid system. Even though latticeholes or parametric models6 can help in dealing with this problem, they allow one to describe only a very restricted class of fluid properties (those that are strongly divergent), and even then the description is confined to a very small neighborhood of the critical point. Moreover, the above-mentioned models lead to a symmetric phase diagram, which is in disagreement with the situation found in real fluids. A way to extend the range of validity of this type of EoS's has been the use of effective critical exponents7 that differ from the universal values predicted by RG or from series expansions of the three-dimensional Ising model8 Of course, this is a nonsatisfactory procedure, which can be improved by extending the asymptotic equation so as to include correction-tescaling terms's9 and by incorporating scaling fields that are combinations of the two mentioned above. Such mixing of fields arises from the fact that the fluctuations of particles An and of energy AE at constant density p = p c for a liquid are not statistically independent, and in consequence An and hE can be represented as linear combinations of the corresponding quantities for a symmetric system.I0 The field mixing arises in a natural way in the decorated lattice gas" and also in *Towhom correspondence should be, addressed. 0022-3654/91/2095-3351$02.50/0

lattice models that include triplet interactions.l* In the past few years, the p-p-T surfaces of several pure fluids (steam,13 isobutane,14 C02,15and CH4I6)have been analyzed in terms of an extended and revised scaling E& proposed by Sengers and Levelt S e n g e r ~ .Furthermore, ~ steam has also been analyzed by using a similar EoS developed by Anisimov et aI.l7 in terms of extended scaling and Prokovskii's transformation.1° Even though these revised and extended scaling EoS's give very satisfactory results for describing thermodynamic variables in the near-critical region, they are applicable only to a small region around the CP, and they do not reduce to other well-known representations of fluid data away from the CP, such as the virial expansion. (1) Hansen, J. P.; McDonald, 1. R. Theory of Simple Liquids; Oxford University Press: London, 1976. (2) Ma, S.-K. Modern Theory of Critical Phenomena; Benjamin Cummings: Reading, MA, 1976. (3) Fox, J. R. Fluid Phase Equil. 1983,14,45. Erickson, D. D.; Leland, T. W. Inr. J. Thermoohvs. 1983. 7.91 1. Pitzer. K. S.:Schreiber. D. R.Fluid Phase Equil. 1988,41,i. Levelt Gngers, J. M. H. Pure Appl. Chem. 1983, 55.437. Erickson, D. D.; Leland, T. H.; Ely, J. F. Fluid Phase Equil. 1987, 37, 185. (4) Ley-Koo, M.; Green, M. S. Phys. Reu. A 1977,16,2483. Sengers, J. V.; Levelt Sengers, J. M. H. Annu. Reu. Phys. Chem. 1986, 37, 189. ( 5 ) Wheeler, J. C. Annu. Reu. Phys. Chem. 1977, 28, 411. (6) Schofield, P. Phys. Rev. Lerr. 1969, 22, 606. (7) Stanley, H. E. Introduction to rhe Phase Transitions and Crirical Phenomena; Oxford University Press: Oxford, 1987. Kumar, A,; Krishnamhrthy, H. R.;Gopal, E. S . R. Rep. Prog. Phys. 1983, 98, 57. (8) Fisher, M. E. Lecrure Nora in Physics; Springer Verlag: Berlin, 1983; Vol. 186. (9) Balfour, F. W. Ph.D. Thesis, University of Maryland, 1982. (10) Patashiskii, A. Z.; Pokrovskii, V. L. Flucruarion Theory of Phase Transitions; Pergamon Press: London, 1979. (1 1) Wheeler, J. C.; Andersen, G.T. C. J. Chem. Phys. 1980,73, 2082. Andersen, G. R.;Wheeler, J. C. J . Chem. Phys. 1978, 69, 2082. (12) Goldstein, R.E.;Parola, A.; Ashcroft, N. W.; Pestak, M. W.; Chan, M. H. W.; de Bruyn, J. R.;Balzarini, D. A. Phys. Rev. Lerr. 1987, 58, 41. Pestak, M. W.; Goldstein, R. E.;Chan, M. H. W.; de Bruyn, J. R.;Balzarini, D. A.; Ashcroft, N. W. Phys. Reu. B 1987,36, 599. Goldstein, R.E.; Parola, A. J. Chem. Phys. 1988,88, 7059. (13) Levelt Sengers, J. M. H.; Kamgar-Parsi, B.; Balfour, F. W.; Sengers, J. V. J. Phys. Chem. Ref. Dara 1983, 12, 1. (14) Levelt Sengers, J. M. H.; Kamgar Parsi, B.; Sengers, J. V. J. Chem. Eng. Dara 1983, 28, 354. (15) Albright, P. C.; Edwards, T. J.; Chen, Z. Y.;Sengers, J. V. J. Chem. Phys. 1983,87, 1717. (16) Kurumov, D. S.;Olchowy, G. A.; Sengers, J. V. Inr. J . Thermophys. 1988. 9. 73. (17)'Anisimov, M. A.; Kiselev, S.B.;Kostukova, I. G. Inr. J . Thermophys. 1985, 6, 465. ~

0 1991 American Chemical Society

3352 The Journal of Physical Chemistry, Vol. 95, No. 8, 1991 This can be explained in terms of which variables dominate the physical behavior in each of the regions of the phase diagram. In effect, very near the CP the correlation length t diverges, thus taking much larger values than the range of the intermolecular potential. These large-scale fluctuations dominate the physics of critical systems, allowing one to neglect any fluctuation whose range is smaller than a given cutoff length A in the momentum space RG technique.2.8J8 Moreover, since t = at the CP, the value of A becomes unimportant, and any contributions to the thermodynamic properties of order 1 are ignored (Wegner series4J9). According to this, it seems reasonable that such a theory cannot account for the transition from the CP to a region dominated by fluctuations of order A. Recently, Albright and NicollZ0 and Albright et aL2l have presented preliminary results of a crossover theory that does not neglect the cutoff. When applied to the van der Waals equation, the theory allows one to describe both the classical and the critical regions of the phase diagram. On the other hand, important efforts have been made to derive a liquid-state theory with accurate critical-region behavior.22 Parola and ReattoZ3have developed the so-called hierarchical reference theory of liquids, which together with an Ornstein-Zernike closure leads to a nonanalytical EoS in the critical region.24 Nevertheless, the predicted critical exponents are not in agreement with those of the 3D Ising model. These facts indicate that an accurate representation of the crossover between the scaling and the noncritical region is still an open problem in the theory of the fluid state. Moreover, for some of the fluids studied, the thermodynamic data were available in the two separate regions, but they were scarce in the crossover region, the data being obtained in different laboratories and with different samples. This can be an important problem since the coordinates of the critical point and the amplitudes of the scaling and correction-to-scaling terms are strongly sensitive to impurities,4v25v26and the values of T,, pc, and p c play an important role in the ability of the EoS to describe experimental data.3J4J5 Recently, some of us have reported an extensive study of the EoS of CHF3 in the classical region: 0.1 I p/MPa I110, 126 I T/K 5 332.*’ The data have been described by using an empirical multiparameter equation and show good agreement with other previous sets of data2* and with values of second virial coefficient^.^^ In addition, Narger et al.30 have published a detailed study of the coexistence curve. Such data may allow one to determine some of the parameters that appear in the revised and extended scaling EoS. The aim of the present paper is to present a detailed experimental study of the EoS of CHF3 in the near critical region. The set of p-p-T data overlap with those presented in ref 27 for the classical region and approach the CP close enough as for extended scaling to be valid. The experiments will be performed on the same sample used for the classical region.27 In this way, a very detailed p-p-T surface will be available for CHF3 which, as already mentioned, is important for discussing present EoSs and

-

(18) Amit, D. J. Field Theory, The Renormalization Group and Critical Phenomena; McGraw-Hill: New York, 1978. (19) Wegner, F. J. Phys. Rev. B 1972, 5, 4529. (20) Nicoll, J. F.; Albright, P. C. Phys. Rev. E 1985, 31, 4576. (21) Albright, P. C.; Sengers, J. V.; Nicoll, J. F.; Ley-Koo, M. Int. J . Thermophys. 1986. 7. 1 5 . (22) Hoye, J. S.; Stell, G. Inr. J . Thermophys. 1985, 6, 561. (23) Parola, A.; Reatto, L. Phys. Rev. A 1985, 31, 3309. (24) Parola. A.; Reatto, L. Europhys. Lerr. 1987, 3, 1185. (25) Tvekreem, J. L.; Jacobs, D. T. Phys. Reu. A 1983.27.2773. Honcssou, c . ; Guenoun, P.; Gostand, R.; Perrot, F.; Beysens, D. Phys. Reo. A 1985. 32. .- - . . -. 1818. .... (26) Aizpiri, A. G.; Correa, J. A.; Rubio, R. G.; Diaz Pefia, M.Phys. Rev. B 1990. 41. - - - .-, . ., 9003. - - -- . (27) Rubio, R. G.; Zollweg, Z. A.; Street, W. B. Ber. Bunsen-Ges. Phys. Chcm. 1989, 93, 791. (28) Hori, K.; Okazaki, S.; Uematsu, M.; Watanabe, K. hoc. 8rh Symp. Thermophys. Prop.; Sengers, J. V.. Ed.; American Society of Mechanical Engineers: New York, 1982. (29) Dymond, J. H.;Smith, E. 8. The Virial Coefficients of Pure Gases and Mixtures; Clarendon Press: Oxford, 1980. (30) NBrger, U.; de Bruyn, J. R.; Stein, M.; Balzarini, D. A. Phys. Reu. B 1989, 39, 11914. NHrger, U.; Balzarini, D. A. Phys. Rev. B 1989, 39,9330.

Aizpiri et al. future developments on the crossover problem. Experimental Section The data were obtained by the gas expansion technique, as described in detail by Calado et al.,31with a few minor modifications described elsewhere.32 Further modifications were necessary for the near-critical region. To reduce gravity-induced density gradients, the cell was kept horizontal inside the waterglycol bath; in this way, the sample height was less than 25 mm. In addition the cell, the expansion line, and the zero element of the Ruska dead-weight gauge were kept at almost the same level. Also, due to the behavior of the transport coefficients near the CP,j3 long equilibration times (up to 4 h) were allowed after each expansion for those with IT- T,I I 0.8 K and lp -pel I0.3 MPa. A test similar to that described by Douslin et al.34was done for every point, discarding those for which there could be significant gravity-induced effects. The temperature was controlled to within f l mK and was measured with a platinum resistance thermometer. The temperature measurements have been referred to the IPTS-68 with an accuracy of h0.02 K. Pressures in the cell were measured with a Ruska dead-weight gauge, the accuracy being 0.1%or better and the precision being about 0.01%. Pressures in the expansion volumes were measured with a Texas Instruments quartz Bourdon gauge, with an accuracy of 0.015%and a precision of a few parts in 100000. The main uncertainty in the p-p-T data arises from the calibration of the volumes of the system. From the combined errors in pressure, temperature, and volume, the estimated error is 0.1% for p 2 8 mol dnr3, 0.3% for 2 Ip mol dm-3 I8, and 0.4% below 2 mol d ~ n - ~Most . of our data correspond to p > 9 mol d m 3 . The CHF3 used in this work was donated by the Linde Division of the Union Carbide Corp. It was purified by fractionation in a low-temperature column with a reflux ratio of 19/20. The final purity is estimated to be better than 99.9%. As already mentioned, it was the same sample used for the classical region.27

Results The more than 800 experimental points are collected in Table I. Our isotherms extrapolate quite smoothly to the coexistence curve data of Hori et a1.28 No such direct comparison can be done with the data of Narger et al.,30 since their interferometric technique allows one to calculate only the differences of densities of the phases in coexistence. For the data to be useful for crossover studies, they must extrapolate smoothly to the classical region. To test this point, some isotherms have been measured up to 100 MPa. These data should be described by the Strobridge equation previously proposed:27

+

+

p = RTp + ( A I R T+ A2 + A 3 / T A 4 / F A 5 / F ) p 2+ (AsRT + A7)p3 + A8Tp4 + Aisp6 + [ ( A g / P + A l o / P + A , , / 7 4 ) p 3+ ( A l 2 / P + A , 3 / P + A14/mP51 exp(A1,p2) (1) where the parameters A, are given in Table I1 and are the same as in ref 27. We have included more digits than statistically significant in order to avoid roundoff errors. We have not found statistically significant differences between the high-pressure (p > 15 MPa) data reported in this paper and the data reported in ref 27. Therefore, we can conclude that from the viewpoint of the experimental data there is a smooth crossover on thep-p-T surface of CHF3, and thus the data are valid for discussing the performance of the EoSs available in the literature. (31) Calado, J. C. G.; Clancy, P.; Heintz, A.; Streett, W. B. J. Chem. Eng. Dora 1982, 27, 376. (32) Rubio, R. G.; Calado, J. C. G.; Clancy, P.; Streett, W. B. J . Phys. Chem. 1985,89,4637. (33) Lcvelt Sengers, J. M. H. In High Pressures Technology;Spain, J. L., Paauwe, J., Eds.; Marcel Dekker: New York, 1977; Vol I. (34) Douslin, D. R.; Moore, R. T.; Dawson, J. P.: Waddington, G. J . Am. Chem. Soc. 1958.80, 203 1 .

The Journal of Physical Chemistry, Vol. 95, No. 8. 1991 3353

Equation of State of CHF3

Discussion

/

0.31-

Revised and Extended Scaling EoSs. Ley-Koo and Green4 applied the Wegner expansionI9 to study the critical behavior of fluids. Levelt Sengers et al.I3-l5 have derived an EoS for fluids based on the use of revised scaling fields and Ley-Kooand Green's work. The fundamental equation is expressed in terms of the intensive variables, namely, the pressure p , the temperature T , and the chemical potential p, which are reduced through p T/T, A T =: ( T - T , ) / T c AP = P = Po( f') P = PTcP,/TcP, Po = P(P,,T) P = PT,/TP, (2)

;

\

0.21

'\ \

OlC

I

'.

P

00

67i;

-0 I -02 -0 3

The EoS takes the form

with

+

0

= fi - Po( UT AT CAP (4) where Po(?,p) and Po( 0 are analytic functions represented by truncated Taylor expansions: UH

=

:::1

/'

-0.3L

o 1i300.71 K I--) 0 1 :302.73 K I-.-) 0 1 = 332.L6 K I-)

3

Po(T,P)= 1 + i=C1P j ( A W + AP + PiiATAr

(5)

3

P o ( 0 = P c + jC= 1P j ( A V

(6)

The scaled equation for the singular part AD( p,P) of the potential is specified by the parametric equations corresponding to Schofield's linear model:

p

UH

= Afi = @&at)(1 - 02)

(7)

= r(I - b202)

(8)

UT

P = d d T , ~+) r1-Wk@o(e)+ +k,pl(8)l

(9)

with (10) Pi(9) = poi + P2P2 + P4P4 Equations 7-10 define a transformation from the physical variables AP and AT to the parametric variables r and 8, where r is a measure of the distance from the critical point and 9 determines the location on a contour of constant r, such that 0 = f l corresponds to the two branches of the coexistence curve. The critical exponents p, 6, and A and the coefficients 6, poi, pa, and pu are universal constants, whose values have been taken from Albright et aLIs and are shown in Table 111. The scaling field amplitudes a, ko, k l , and c and the pressure background coefficients p l l ,pl, p2, and p3 depend on the fluid. Since the performance of scaling EoSs depends upon the coordinates of the CP, it is important to have reliable values for those variables. Hori et have reported a detailed study of the vapor-pressure curve and the densities of the coexistence curve near the CP. From their data they report T, = 299.01 K, pc = 4.816 MPa, and pc = 7.556 mol L-I. On the other hand, Ntirger et aLm reported pc = 4.75 MPa, pc = 0.527 g and T, ranging from 298.95 to 299.16 K. They suggest that the drift in T, could be due to the presence of impurities. An experiment performed in our apparatus indicated that presence of two phases at T = 298.05 K and p = 4.78 MPa. Thus, we have chosen Hori's results for the regression of our data. For the fluids studied so Levelt Sengers's EoS has been found to be valid for the intervals -1.005 I 1 -0.94 0.6 Ip 11.3

We have found a similar range for the validity of this E&. Table 111 shows the optimum values of the substance-dependent parameters, and Figure l shows the deviations between experimental and calculated results, together with the residuals of the Strobridge EoS, as a function of the reduced density. For the sake of sim-

:;:I"/,\ -0 3

1.

\\ \

Figure 1. Differences between experimental and calculated pressures for several isotherms as a function of the reduced density. Symbols correspond to the extended and revised equation of state of Sengers et al., using the parameters of Table 111. Lines correspond to the classical Strobridge equation of state, with the parameters of Table I1 (see text).

plicity, these residuals have been represented by lines in Figure 1, since they have been calculated for a larger number of data points and they show a very smooth behavior. Even though the residuals are not randomly distributed, most of them are within the experimental uncertainty. Small changes in the critical point coordinates do not improve the results. The improvement with respect to the classical EoS is obvious near the CP, though one must emphasize that the residuals increase significantly outside the p, p , T region shown. This fact shows again the poor convergency properties of the Wegner expansion, and the need for the inclusion of medium-range fluctuations of the order of the cutoff parameter in the RG analysis. Similar conclusions were reached in the analysis of other fluids. Equations 3-10 provide expressions for the amplitudes of the leading terms of the extended scaling equations for different thermodynamic properties in terms of the universal and the fluid-dependent parameters mentioned above:

p - 1 = *B(AQ@ 2 = I'+IAqT

for p =p,,, T 5 T, for

AP = &Dlp - lib

p

= pc, T 2 T,

for T = T,

(1 1) (12)

(13)

where B, I'+, and D are defined through

B = k0/(b2 - 1)'

(14)

r+= k o / a o

(15)

D = a(b2 - I)bh3/ko*

(16)

subscript u referring to the coexistence curve. Using the critical amplitudes obtained by Narger et aI.'O an eqs 11-1 3, we have obtained a. = 22.7 and ko = 0.22, which are in reasonable agreement with the values of Table 111, especially considering the slight discrepancy in T, and pc referred to above. In any case, the discrepancy is similar to that obtained by Albright et al.15 for C 0 2 . Further test of the EoS cannot be done until

3354 The Journal of Physical Chemistry, Vol. 95, No. 8, 1991

Aizpiri et ai.

TABLE I: Experimental p - p T Data for CHFJ

PI

PI

PI

PI

PI

PI PI MPa moldm-’ T = 285.13 K 16.190 71.370 16.1 16 68.628 16.042 66.407 15.968 64.287 15.895 6 1.708 15.823 59.263 15.751 57.047 15.608 55.150 15.542 52.904 15.477 50.757 15.413 49.083 15.349

PI

PI

PI

PI

PI

moldm-’

MPa

moldm-)

MPa

moldm-3

19.012 18.919 18.842 18.768 18.675 18.584 18.498 18.424 18.332 18.240 18.168

28.892 27.705 26.576 25.453 24.399 23.368 22.424 2 1.453 20.597 19.749 18.944

17.070 16.986 16.903 16.821 16.738 16.652 16.573 16.496 16.418 16.342 16.266

10.871 10.129 9.450 9.144 8.593 8.104 7.491 6.883 6.504 5.953 5.368

15.287 15.164 15.050 14.988 14.885 14.789 14.654 14.512 14.405 14.240 14.080

11.153 10.809 10.492 9.893 9.356 8.871 8.43 I 8.056 7.712 7.429 7.149 6.826

T = 293.10 K 14.777 67.799 14.704 64.956 14.635 62.398 14.569 57.137 14.504 54.902 14.376 52.753 14.253 48.716 14.140 46.801 14.023 44.954 13.912 43.092 13.801 41.414 13.695 39.790 13.608 38.245 13.495 36.718

18.617 18.513 18.415 18.204 18.109 18.015 17.830 17.737 17.646 17.550 17.459 17.370 17.281 17.191

19.681 18.923 18.234 17.566 16.949 16.356 15.771 15.178 14.657 14.159 13.633 13.165 12.713 12.297

15.838 15.758 15.679 15.602 15.525 15.448 15.372 15.293 15.217 15.144 15.067 14.992 14.919 14.846

6.642 6.493 6.319 6.157 5.968 5.849 5.733 5.489 5.287 5.106 4.870 4.760

13.426 13.361 13.279 13.198 13.137 13.058 12.983 12.892 12.760 12.654 12.485 12.380

14.596 14.524 14.451 14.383 14.316 14.250 14.186 14.122 14.06 1 14.004 13.943

7.296 7.193 7.089 6.993 6.896 6.828 6.745 6.676 6.610 6.547 6.277

T = 295.56 K 13.349 17.283 13.310 16.707 13.276 16.122 13.208 15.591 13.178 15.068 13.144 14.572 13.120 14.129 13.094 13.681 13.071 13.261 13.033 12.847 12.887 12.462

15.426 15.349 15.273 15.196 15.120 15.046 14.961 14.886 14.813 14.740 14.668

9.027 8.821 8.635 8.442 8.270 8.104 7.953 7.801 7.663 7.533 7.409

13.886 13.830 13.774 13.721 13.668 13.618 13.568 13.522 13.476 13.431 13.389

6.201 6.139 5.891 5.772 5.726 5.567 5.450 5.280 5.071 4.964

12.830 12.772 12.650 12.561 12.516 12.461 12.352 12.179 1 1.985 11.855

54.131 52.036 50.027 48.110 44.461 42.759 39.547 38.024 36.587

17.934 17.837 17.744 17.651 17.466 17.374 17.193 17.103 17.014

20.776 17.201 16.074 15.020 14.067 13.221 12.421 11.718 1 1.043

T = 296.59 K 15.729 74.041 7 1.274 15.31 1 15.154 68.509 15.001 65.905 14.850 63.434 14.706 60.979 14.556 58.577 14.429 56.317 14.289

18.715 18.615 18.516 18.417 18.331 18.224 18.126 18.030

35.195 33.898 31.411 29.120 28.036 26.01 1 24.078 22.358

16.926 16.838 16.665 16.493 16.406 16.235 16.055 15.891

10.185 9.427 8.788 8.242 6.986 5.560 5.120 4.851

14.087 13.901 13.719 13.544 13.057 12.187 1 1.775 1 1.383

15.442 15.363 15.285 15.207 15.131 15.055 14.98 1 14.907 14.834 14.761 14.688

9.287 9.082 8.882 8.510 8.194 8.035 7.886 7.760 7.505 7.392 7.287

13.829 13.770 13.711 13.601 13.493 13.442 13.391 13.340 13.247 13.203 13.124

6.469 6.377 6.301 6.230 6.163 6.104 6.049 5.994 5.939 5.891 5.840

T = 296.95 K 12.742 12.413 12.722 12.048 1 1.697 12.673 1 1.373 12.622 1 1.077 12.582 12.536 10.781 12.490 10.498 12.446 10.223 12.404 9.978 12.358 9.509 12.315

14.544 14.475 14.407 14.339 14.271 14.206 14.141 14.078 14.014 13.889

7.186 7.089 7.001 6.917 6.839 6.765 6.697 6.641 6.580 6.522

13.082 13.043 13.005 12.969 12.935 12.903 12.877 12.837 12.800 12.769

5.790 5.742 5.698 5.652 5.608 5.567 5.526 5.487 5.447

12.274 12.232 12.188 12.182 12.140 12.098 12.065 12.041 12.005

5.106 5.149 5.616 5.699 5.774 5.843

10.943 1 1.045 11.711 1 1.805 11.889 1 1.958

7.485 7.705 7.843 8.097 8.242 8.442

12.949 13.052 13.105 13.211 13.270 13.327

9.537 9.771 10.012 10.251 10.508 10.78 1

T = 298.76 K 13.708 6.270 13.773 6.3 18 13.839 6.373 13.905 6.552 13.972 7.076 14.042

12.303 12.335 12.365 12.474 12.763

8.566 8.743 8.929 9.1 17 9.323

13.392 13.454 13.517 13.580 13.644

11.071 1 1.374 1 1.686 12.021

14.108 14.178 14.248 14.320

4.903 4.91 1

9.619 9.734

5.994 6.017

1 1.978

8.31 1 8.469

T = 298.97 K 13.264 4.919 13.324 4.93 1

9.849 9.979

6.045 6.067

12.019 12.043

8.637 8.816

13.384 13.445

MPa

mol dm-’

MPa

mol dm-)

MPa

102.000 100.460 98.227 96.137 92.549 90.066 87.75 1 85.687 82.598 77.398 75.460 73.269

19.899 19.861 19.804 19.749 19.652 19.585 19.520 19.458 19.368 19.209 19.147 19.075

47.145 45.278 43.468 41.689 40.056 38.437 36.862 35.333 33.870 32.555 3 1.204 30.036

18.081 17.986 17.898 17.812 17.728 17.644 17.557 17.471 17.386 17.306 17.221 17.146

18.172 17.435 16.735 16.060 15.419 14.806 14.218 13.144 12.643 12.166 11.711 1 1.284

120.470 1 15.930 1 1 1.540 107.250 103.020 99.055 95.180 9 I .393 86.898 83.439 80.001 76.397 73.435 70.537

20.093 19.997 19.899 19.797 19.693 19.590 19.486 19.380 19.251 19.143 19.034 18.917 18.816 18.716

35.313 33.886 32.543 31.260 30.057 27.802 26.734 25.715 24.730 23.789 22.902 22.032 21.203 20.43 1

17.103 17.014 16.925 16.836 16.750 16.579 16.493 16.408 16.323 16.238 16.157 16.076 15.997 15.917

1 1.902 1 1SO7

25.446 24.88 1 23.960 23.106 22.252 21.432 20.679 19.938 19.239 18.558 17.910

16.257 16.211 16.133 16.055 15.977 15.897 15.818 15.739 15.661 15.583 15.504

12.090 1 1.739 1 1.408 1 1.084 10.781 10.499 10.210 9.957 9.702 9.440 9.243

103.730 102.370 98.476 94.670 89.819 86.5 I9 83.205 80.036 76.986

19.605 19.570 19.467 19.361 19.221 19.122 19.019 18.917 18.815

18.213 17.600 17.023 16.466 15.93 1 15.414 14.916 14.448 13.991 13.563 13.164

12.000

Equation of State of CHF3 TABLE I (Continued) PI PI

PI

The Journal of Physical Chemistry, Vol. 95, No. 8, 1991 3355

PI

PI

PI

MPa 4.946 4.963 4.979 5.000 5.023 5.049 5.080 5.1 12 5.147 5. I83 5.638 5.674 5.709

moldm-) 10.093 10.205 10.316 10.357 10.475 10.591 10.696 10.797 10.903 10.997 1 1.659 1 1.698 1 I .736

MPa 6.092 6.1 18 6.146 6.177 6.203 6.239 6.276 6.31 1 6.354 6.398 6.442 6.495 6.547

moldm-) 12.067 12.092 12.117 12.142 12.174 12.200 12.229 12.262 12.295 12.302 12.336 12.369 12.406

MPa 8.998 9.188 9.397 9.613 9.836 10.074 10.322 10.584 10.857 11.139 1 1.442 1 1.759 12.090

mold" 13.507 13.571 13.635 13.700 13.766 13.832 13.898 13.963 14.043 14.102 14.169 14.238 14.308

19.336 18.694 18.085 17.490 16.934 16.397 15.881 15.385 14.912 13.619 10.375 10.129

15.429 15.349 15.269 15.190 15.112 15.035 14.959 14.884 14.810 14.569 13.893 13.825

7.799 7.679 7.565 7.457 7.348 7.158 7.071 6.988 6.912 6.836 6.773 6.705

13.027 12.972 12.917 12.864 12.829 12.727 12.678 12.629 12.583 12.538 12.492 12.451

6.180 6.152 6.097 6.044 5.994 5.946 5.899 5.819 5.760 5.723 5.666 5.614

T = 299.21 12.107 12.081 12.039 12.000 11.926 11.887 11.854 11.787 11.730 11.689 1 1.633 11.572

103.570 99.922 96.392 94.035 91.061 52.229 26.369 25.432 24.530 23.67 I 22.845 22.053

19.505 19.416 19.325 19.261 19.177 17.731 16.124 16.040 15.955 15.872 15.789 15.708

14.786 14.335 13.908 13.498 13.103 12.730 12.372 12.028 1 1.746 1 1.387 11.091 10.809

14.762 14.687 14.612 14.537 14.463 14.388 14.314 14.241 14.167 14.098 14.028 13.959

7.719 7.560 7.344 7.223 7.126 6.959 6.802 6.669 6.552 6.394 6.304 6.201

10.285 10.047 9.799 9.620 9.417 9.213 8.841 8.674 8.499 8.355

13.774 13.710 13.646 13.587 13.525 13.462 13.340 13.279 13.221 13.162

7.147 7.067 6.982 6.908 6.836 6.767 6.698 6.635 6.580 6.526

12.633 12.574 12.530 12.484 12.441 12.399 12.361 12.325 12.288 12.253

17.111 16.703 16.308 15.833 15.470 15.006 1 1.004

15.082 15.009 14.937 14.848 14.778 14.687 13.926

9.344 9.147 8.961 8.782 8.614 8.454 8.306

11.560 1 I .264 10.98 1 10.710 10.459 10.215 9.985

14.030 13.958 13.887 13.816 13.747 13.678 13.610

54.475 50.523 48.66 I 45. I33 43.469 41 3 6 8

17.697 17.509 17.415 17.228 17.135 17.036

PI

PI

PI

PI

PI

PI

MPa 5.727 5.751 5.764 5.786 5.806 5.824 5.863 5.884 5.905 5.922 5.950 5.971

moldm-) 11.756 11.771 1 1.796 11.813 11 3 3 3 11.854 11.859 11.879 11.898 11.922 11.935 11.957

MPa 6.602 6.665 6.731 7.112 7.293 7.396 7.510 7.625 7.749 7.886 8.014 8.157

moldm-3 12.442 12.479 12.517 12.736 12.828 12.879 12.930 12.984 13.039 13.094 13.150 13.207

MPa 12.434 12.806 13.171 13.571 13.984 14.421 14.877 15.355 19.322 20.7 14

moldm-3 14.378 14.449 14.522 14.596 14.671 14.747 14.824 14.902 15.445 15.604

K 9.892 9.670 9.450 9.247 9.054 8.868 8.692 8.524 8.369 8.209 8.066 7.929

13.758 13.692 13.627 13.563 13.499 13.437 13.376 13.316 13.255 13.197 13.139 13.092

6.644 6.588 6.538 6.488 6.439 6.398 6.357 6.3 18 6.281 6.246 6.21 3

12.443 12.403 12.362 12.324 12.289 12.253 12.220 12.221 12.191 12.160 12.133

5.566 5.520 5.477 5.439 5.329 5.294 5.262 5.214 5.177 5.145 5.1 12

11.513 1 1.452 11.390 11.330 11.141 1 1.078 11.014 10.912 10.818 10.732 10.629

T = 299.48 K 12.949 21.299 12.879 20.564 12.775 19.867 12.716 19.198 12.665 18.572 12.567 17.965 12.477 17.366 12.390 16.811 12.308 16.266 12.206 15.757 12.144 15.254 12.052

15.625 15.544 15.463 15.381 15.302 15.223 15.146 15.068 14.991 14.914 14.839

10.540 10.247 9.799 9.578 9.365 9.158 8.959 8.607 8.435 8.125 7.980

13.891 13.827 13.696 13.630 13.564 13.500 13.436 13.309 13.243 13.122 13.062

6.105 6.026 5.960 5.899 5.844 5.789 5.718 5.665 5.602 5.538

11.969 1 1.905 I 1.842 11.780 11.719 11.691 11.592 11.533 1 1.439 11.371

6.175 6.146 6.118 6.089 6.067 6.040 6.017 5.330 5.273 5.232

T = 299.79 K 8.206 8.054 11.939 7.925 11.917 7.792 11.890 7.675 11.868 7.560 11.845 7.453 10.902 7.347 10.768 7.245 10.650

13.104 13.048 12.992 12.938 12.884 12.831 12.778 12.728 12.680

6.478 6.432 6.391 6.348 6.3 IO 6.274 6.240 6.205

12.219 12.185 12.153 12.123 12.093 12.065 12.037 12.015

5.191 5.153 5.117 5.090 5.065 5.041 5.019 5.000

10.521 10.398 10.259 10.126 9.983 9.834 9.673 9.496

13.456 13.392 13.329 13.266 13.204 13.143 13.081

7.416 7.314 7.214 7.124 7.034 6.952 6.8 14

T = 300.34 K 12.684 10.740 12.628 10.471 12.577 10.225 12.527 9.994 12.479 9.767 12.433 9.551 12.347

13.856 13.787 13.720 13.653 13.587 13.521

8.150 8.009 7.875 7.756 7.636 7.526

13.022 12.964 12.906 12.849 12.792 12.737

6.687 6.571 6.479 6.389 6.3 18 6.242

12.265 12.183 12.116 12.045 1 1.979 1 1.925

8.474 8.323 8.183 8.040 7.913 7.790 7.672

13.093 13.032 12.973 12.915 12.857 12.800 12.745

6.924 6.851 6.784 6.719 6.617 6.514 5.902

T = 300.71 12.344 12.299 12.253 12.207 12.129 12.048 1 I .489

9.764 9.545 9.347 9.158 9.016 8.798 8.635

13.543 13.477 13.412 13.346 13.278 13.218 13.155

7.560 7.454 7.351 7.257 7.167 7.080 7 .OOO

12.69 1 12.638 12.587 12.530 12.482 12.436 12.390

5.804 5.730 5.659 5.591 5.537 5.479 5.420

11.353 11.236 11.121 10.999 10.890 10.756 10.609

30.063 29.003 27.98 1 27.003 26.066 24.303

16.256 16.170 16.085 16.000 15.914 15.744

6.819 6.605 5.966 4.996 4.458 4.238

T = 302.73 K 11.832 40.325 1 1.693 38.858 10.849 37.434 4.710 36.091 '3.231 34.768 2.884 33.519

16.948 16.859 16.771 16.684 16.598 16.512

23.472 21.942 20.510 8.883 8.419 8.146

15.658 15.489 15.308 12.968 12.779 12.657

4.085 3.974 3.855 3.629 3.380

2.676 2.538 2.399 2.158 1.917

1 1.989 1 1.964

K

33% The Journal of Physical Chemistry, Vol. 95, No. 8, 1991

Aizpiri et al.

TABLE I (Continued)

PI

PI

PI MPa

PI mol dm”

MPa

12.379

13.661

3.763

2.207

5.974

5.302

4.603

2.833

108.070 104.280 100.730 97.261 93.898 90.717 87.628 84.613

18.960 18.855 18.749 18.644 18.539 18.435 18.331 18.227

62.122 60.055 56.1 72 54.310 51.329 48.130 46.553 45.065

17.317 17.213 17.023 16.926 16.762 16.572 16.478 16.383

28.493 26.920 25.455 24.110 22.868 17.160 16.104 15.171

106.950 101.900 98.571 95.324 77.860

18.647 18.498 18.392 18.287 17.664

50.633 49.083 47.241 44.736 42.931

16.316 16.219 16.098 15.928 15.797

18.957 17.127 15.474 14.786 10.719

PI mol dm-’

MPa 2.897

PI PI moldm-) MPa T = 306.17 K 1.486 10.657

PI

MPa

PI

PI moldm-’

16.195 16.102 16.008 15.913 15.729 15.544 15.363 15.183

14.218 12.593 11.374 10.191 9.436 3.494 2.801 2.092

12.521 11.939 11.370 10.627 9.971 1.675 1.263 0.892

15.634 15.286 15.141 14.987 14.836

9.847 8.821 7.898 7.028

8.518 7.160 5.784 4.570

PI mol dm”

MPa

PI moldm-)

13.183

3.001

1.561

T = 322.90 K 15.005 81.758 14.828 79.115 14.653 76.336 14.479 73.697 14.305 71.191 13.290 68.819 13.043 66.476 12.799 64.274

18.124 18.020 17.918 17.817 17.716 17.615 17.515 17.416

42.291 40.973 39.678 38.438 36.132 34.001 32.017 30.201

T = 332.46 K 74.399 12.896 70.805 12.438 1 1.944 68.529 53.923 11.702 52.256 9.434

17.508 17.354 17.252 16.510 16.413

40.793 36.642 35.062 33.457 31.997

T = 313.21 K

TABLE 11: A, Coefficients of the Strobridge Equation of State for

CHF,

i

4

1 2 3 4 5 6 7 8 9

0.0454 57 f 0.003 dm3 mol-’ 6.9009 f 0.07 bar dm6 mol-2 -5684.3 f 2 bar dm6 K mol-2 (5.8935 f 0.01) X IO’ bar dm6 K2 mol-2 -(2.6234 f 0.009) X IO9 bar dm6 K4 mol-2 -(6.9026 f 0.5) X IO” dmd molq2 0.213 56 f 0.007 bar dm9 mol-’ (2.2642 f 0.1) X IO” bar dmI2 K-l mol4 -339 30 f 80 bar dm9 K2m o P (2.971 1 f 0.004) X IO7 bar dm9 K3 mol-’ -(3.4292 f 0.009) X IO9 bar dm9 K4 mol-) 47.361 f 0.3 bar dmI5 K2 mol-15 -646 99 f 2 bar dm15 K’ mol-5 (7.8646 f 0.008) X IO‘ bar dm” K4 mol-5 (2.0224 f 0.007) X IO-’ bar dmI* mold -0.0040000 f 0.001 dm6 mol-2

IO II 12 13 14 15 16

TABLE I V Substance-Dependent Panmeters in Anisimov et aI.’s Equation of State a = 22.007 d = 13.650 f -7 1.732 c = 3.4640 k = 1.3298 fi = -7.8339 = 57.171 S, = -519.32 fi = 2559.1

The chemical potential term is defined through Ar(r,0) = ApRG(f,e)

+ P*+’[b,+ (62 - blb2)e2 + (6, - b2b2)@] (21)

where AA and the 6,‘s are functions of the critical exponents, and A&O(r,e) = afidO(l- d2)

+ cF+AB

(22)

On the other hand

+ k?3r+3fl-lt9{d+ y3V- 2d(e - fl)]b2e2+

\k(r,e) = \kRG(r,B)

(1

TABLE III: Universal Constants and Panmeters in the Equrtion of

State of Scnpers et al.

- 2/3)/(5 - 2e)[del +fez]b4@) (23)

where a, c, d, and f are new nonuniversal constants, and the e,’s are again combinations of the critical exponents. Finally

Universal Constants fa = 0.586535 Pol = 0.103246 6 = 1.17290 = 0.325

f20 f21

-1.026243

fa

= 0.160322

f41

= 0.1085 d = 4.820 CY

0.612903

= -0.169860 A = 0.500

Substance-Dependent Parameters ko = 1.2428 kl 0.311 14 c = -0.012471 fill = -0.07871 fl1 6.3382 f l 2 = -21.669 f l j = O.Oo0 a = 23.046

derivatives of the pp-T surface (speed of sound and heat capacity) are available. Anisimov et al.” have reported an EoS based on extended scaling theory and Prokovskii’s transformation.I0 They express the potential as

- \k(r,e) - d~“f)

where the j;.Is are substance-dependent parameters, and the parametric variablesr and 0 are obtained through the transformation

A T = r(1 - b202) Ap

= kr%

+ B3AT

with k and B3 being also substance-dependent.

(19) (20)

““’1 Table IV collects the values of the substance dependent parameters for the EoS of Anisimov et al. As in the case of Sengers’s EoS, some parameters can be traced back to the amplitudes of the leading scaling terms of thermodynamic properties. The results of Narger et aL30 lead to k = 1.2545 and c/a = 0.5849. While this value of k does not differ significantly from the value of Table IV, the ratio c/a is quite different ( c / a = 0.1574 from Table IV). In addition, the diameter of the coexistence curve datag0leads to B3 = 1.40; however, we have found no statistical differences between the results with B3 = 0 and B3 = 1.40; probably, the lack of data in the very near proximity of the CP makes B, not significant. Finally, it must be emphasized that there is a strong correlation between the/; parameters, the statistical significance of f g and f4 being quite small. Figure 2 shows the ratio of the residuals of the E d ’ s of Levelt Sengers et ai. and of Anisimov et al. for several near critical isotherms in the proximity of the CP, as a function of the reduced density. The results show that the performance of both equations

The Journal of Physical Chemistry, Vol. 95, No. 8, 1991 3351 1.0

7

0500.

-0.5 -1.0

t

P -

o a

I 12 0

a

Pc I

I

I

146

ILL0

T = 299.21 K 1-3CO.71 K

0

0 0 0

-2

-4

a

0

0

0

T =299.21 K T E 299.U K

0

0 0

-6

0 0

a

a

OQ

a

a

-'[ &r

-2

T1298.97 K 1.299.79 K

0

a

0

-'[ -L

I

"

0

0

P

0

0

0

1

-6

0

2-

0'

ka 41

0

0

I

" 1.2

0.8

1.302.73 K T.332.G K

0

PC

1.6

a

T=298.97K

0

~ = z g g . n ~

2 0 -2

a

-'

Figure 2. Ratio of the residuals of the equations of state of Anisimov et

-6

0 0

0

~

9

2;"

a

0

1f0

-; -

T a300.71 K T ~202.73K T=332.&6K

0

al. and of Sengers et al. as a function of the reduced density for several isotherms.

Figure 3. Ratio of the residuals of Strobridge equation modified according to Erickson et al. and of the extended and revised equation of Sengers et al. as a function of the reduced density for several isotherms.

is quite similar to the near-critical region; though the residuals of Anisimov's EoS are slightly smaller than those of Levelt Sengers's EoS, serious problems of convergence in the fitting procedure limits the applicability of Anisimov's EoS to a more restricted region around the CP. Modified Classical EoS's. Fox3 showed that the description of thermodynamic properties can be significantly improved in the critical region by transforming an analytical EoS into a nonanalytical EoS in a manner that preserves the classical behavior far from the CP but obeys scaling laws near the CP. This transformation is based on a state function 9,which measures the distance from the CP and is defined as 9 = P - P c - P A P - P c ) - 3,(T - T,) (25) where P = ( p - p * ) + RT In (puo) (26) uo being the unit volume, the asterisk referring to the ideal gas behavior, and 5 = p((s - s*) + R[1 - In (uop)]] (27)

0.227 70, w = 7.007 679 X 1WS, X = 0.096 776 8; Figure 3 shows the ratio of the residuals of the Erickson-Leland-Fox (ELF) EoS to those of the Sengers one in the near-critical region. The results indicate that the performance of the ELS EoS is very satisfactory in the near-critical region and can be compared to the Sengers E& in its ability to describe p p - T data, even improving the results for low values of the reduced density. This is an important result, since the ELF equation reduces to eq 1 far from the CP, and thus it gives quite satisfactory results in regions of the phase diagram for which revised and extended scaling EoS's cannot be applied. Even though it remains to be known whether, much closer to the CP, the ELF EoS still gives good results, it is clear that the EoS would be satisfactory for many engineering applications. In spite of the good description of the p p - T surface, again a more extensive analysis of speed of sound or C,, data is needed before a firm conclusion about the general performance of ELF EoS can be drawn. Recently, Pitzer and Schreiber' (PS) have proposed a classical EoS that is valid except in the very critical region. We have used eq 1 plus the fluctuation contribution proposed by Pitzer and Schreiber3 (see eq 3 of ref 3). The results indicate that there is no improvement over the results of the Strobridge E&, and thus we do not describe the PS EoS here. The results agree with the conclusions of Pitzer and Schreiber that their correction term is important only in a very small region around the CP, where the data of Table I are scarce.

s being the molar entropy.

The nonclassical EoS can be defined in terms of the following set of parametric equations: T ' = Tc ( T - T , ) 8 (28) P = P J T 3 + [ P ( P l T ") - P (29) P'= P p C ( P ' - E) + 3,(T'- T,) Q [ ( T ' - T,)*- ( T - T,)'] (30) where e = 2&/(2 - a) (31) r#J = 1 - 28[1 4 2 -a)] (32) being a and 8 the critical exponents. The prime stands for scaled variables, while terms without it are unscaled, or experimental, ones. Finally, g is the "damping" function proposed by Erickson and Leland:3

+

+

+

+

where Y, r, w, and X are adjustable parameters. We have used the above transformation of variables together with the Strobridge equation, eq 1. Using the A, constants of Table 11, we have fitted Y, r, w, and X to the whole set of p-p-T data of Table I. The best set of parameters was Y = - 1 . 1 0 6 0 7 , r =

Conclusions We have obtained an extensive set of p-p-T data in the near-critical region of CHF3. The results are consistent with other previously obtained for the classical region. The results have been analyzed in terms of two revised and extended scaling EoS's and show that while both of them lead to similar results near the critical point, the EoS of Sengers et al. seems to be applicable in a p , p , T range larger than that of Anisimov et al. Also, two classical EoSs with modified thermodynamic fields have been tested. The results indicate that the Strobridge equation, modified according to the method of Erickson et al., gives results comparable to those of Sengers's EoS in the near critical region, while it reduces to the Strobridge EoS far from the critical point. However, firmer conclusions about this kind of EoS would require C, and speed-of-sound data, which are not available for CHF3. The EoS proposed by Pitzer and Schreiber has been found to give results that are essentially equivalent to the Strobridge EoS in the region for which pp-Tdata are available, the switch to scaling behavior taking place only closer to the critical point. Registry No. CHF3, 75-46-7.