Bulk modulus for some simple molecular fluids

Chemical Physics 130 (1989) 177-186

North-Holland, Amsterdam

BULK MODULUS FOR SOME SIMPLE MOLECULAR FLUIDS

A. COMPOSTIZO, A. CRESPO COLIN, M.R. VIGIL, R.G. RUB10 and M. DIAZ PENA Departamento de Quimica Fisica, Facultad de Quimicas, Universidad Complutense, 28040 Madrid, Spain

Received 13 July 1988

Dedicated to Professor F. Kohler on the occasion of his 65th birthday

The bulk modulus B of several molecular fluids composed of rigid molecules has been calculated from pp-T data obtained

with a high-pressure vibrating tube densitometer. The data of all the substances studied, including Ar, can be described by a single

master curve when plotted versus the reduced density p*, in agreement with the predictions of the Gubbins-Gray perturbation

theory for fluids with the same reference system. Combination of p-p-T and C,. data with the virial theorem has allowed the

calculation of the exponent characterizing the repulsive branch of the intermolecular potential n. The different values of n suggest

that different reference systems should be used for each substance, in contradiction with the conclusions obtained from the B

versus p* curves. This indicates that p-p-T data are less sensitive to the details of the intermolecular potential than their combi-

nation with other thermal properties like CV, internal energy or residual entropy.

1. Introduction

Even though the degree of understanding of the structure and thermodynamic properties of simple fluids is quite complete [ 11, the situation is far from satisfactory for the case of molecular fluids. Pertur- bation theories have been frequently used for predicting the thermodynamics of fluids. The theory of Gubbins and Gray [ 2 ] uses a spherical reference system, defined by the orientational average of the full intermolecular potential, and may be applied to fluids with multipolar interactions, provided that the anisotropy of shape of the molecular core is small. This theory has been tested in predicting phase equilibria of pure substances and mixtures [ 31 and also gives reasonable results for the p-p-T data of pure fluids in wide ranges of p and T ( = 1% in p) [4]. Neverthe- less, the tests have been performed for relatively simple molecules, e.g. ethylene [4], and only very recently extended to more complicated molecules like haloderivatives of methane, though again the anisotropy of shape was kept small, and no permanent dipole moment was present [ 5 1.

As has been shown by Kohler’s group [ 6,7 1, anisotropies of shape can be better dealt with using a perturbation theory based on a spherical reference

system defined through the orientational average of the Boltzmann factor of the full intermolecular potential. The theory has recently been extended to n- center Lennard-Jones molecules, and it has been tested against virial coefficients of pure substances [ 8 1, and against low-pressure VE, HE and GE data for binary mixtures [9]. Thus the test has been less exhaustive than for the Gubbins-Gray theory. The main drawback of Kohler’s perturbation theory is that multipolar interactions are not allowed, hence it has been applied to fluids with no permanent dipoles (though molecules with quadrupoles and other higher electrical moments have been considered [ 91) .

Recently, Shukla et al. [ lo] have extended the Gubbins-Gray theory calculating the spherical reference system from a site-site potential model, while keeping the possibility of having multipolar interactions. This way one could expect that larger anisotropies of shape might be considered. Again, the theory has been mainly applied to the prediction of vapor-liquid equilibrium at low pressures, and a more systematic study of its ability for predicting p-p-T properties of pure fluids with different shapes, sizes and polarities remains to be done.

One of the main problems in comparing theoretical calculations with experimental data is the lack of

0301-0104/89/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

178 A. Compostrzo et al. /Bulk modulus for molecularJluids

knowledge about the intermolecular potential in real fluids. Besides the comparison of simulation results with experiments [ 111, Powles et al. [ 121 have sug- gested that good effective intermolecular potentials could be obtained from p-p-T data of fluids. Hence this type of data for fluids of rigid molecules seems useful for testing the different versions of perturbation theory. Haloderivatives of methane seem to be ideal for this purpose since one can change in quite systematic way the size, shape and electrical moments of the molecule. In addition, interesting orientational correlations have been pointed out in some of them [ 13 1, and good predictions have been obtained for the structure of non-polar tetrahedral molecules and their binary mixtures [ 14 ] using the RISM formalism. Also, site-site intermolecular potentials have recently been reported for methane derivatives and other simple molecules [ 15 1. The potentials arise from a combination of ab initio calculations and virial coefficients and when used in computer simula- tions reproduce quite accurately both static and dy- namic properties, though comparisons have been done for very small regions of the p-p-T surface. Since the site-site Ornstein-Zemike equation has recently been solved for polar fluids [ 16 1, p-p-T data for this type of molecules could also be helpful in testing the predictive ability of this integral equation theory.

Even though some data are already available for molecular fluids such as halomethanes, the disagreement between the data of different sources prevents any theoretical discussion, especially if they involve derivatives such as compressibility or energy. There- fore, it is the purpose of this paper to present a con- sistent set of p-p--T data for several fluids. In order to increase the range of anisotropy of shape of a set of halomethanes we have also measured p-p-T for benzene and CS,. C6H, will serve as a test of the experimental apparatus, and both CsH6 and CS2 present correlations of molecular order [ 17 1. Since some of the molecules are polar, we will focus our attention on one aspect of the Gubbins-Gray theory. In effect this theory predicts that the bulk modulus of any fluid should follow the same curve when plotted versus the reduced density. We will test whether this conclusion is valid for the fluids studied in this paper, some of which show a noticeable anisotropy of shape.

2. Experimental and results

The p-p-T data were obtained using a high-pressure vibrating tube densitometer. Since the whole apparatus has been described in detail in a previous work [ 18 1, only a brief description will be included here for the convenience of the reader. The experimental setup is similar to that of Matsuo and van Hook [ 191 the main difference being that the pressure is directly applied to the sample by a screw-pump, thus there is no need for oil-sample separators. The strain-gage pressure transducers were calibrated against a dead-weight gage, and the calibration checked after the experiments on each substance were completed; this has allowed the pressure to be measured within _+ 0.0 1 bar. The temperature of the U- tube was kept constant within kO.5 m&L, and it was measured with a quartz thermometer, whose calibration was checked weekly against a Ga melting point standard. The temperature of the whole apparatus was kept constant to &- 0.0 1 K of that of the sample. The densitometer was calibrated with different substances of well-known densities. In contrast with previous works [ 19-2 11, we have found the period of the vibration r versus density relationship to be non- linear, thus we have used p-p-T relationships reported in the literature for N2 [22], CO2 [23] and CCL, [ 181 for estabIishing a calibration curve. The calibration was tested after the experiments on each pure substance were carried out. No hysteresis was found within + 5 x 1 O-’ in r when measuring at low pressure after the vibrating tube was subjected to high pressure. Under these conditionsp could be obtained within +10-4 g cm-” for 0.1~~ (MPa)<40. The substances were of the highest purity available from Fluka except for C6H6 and CHJ which were obtained from Carlo Erba (chromatography grade). CHC13, CHzClz and CH31 were purified according to recommended methods immediately before use [ 241. All of them were dried over 0.4 nm molecular sieves.

Table 1 shows the experimental results for the different liquids studied. The data were fitted to a gen- eralized Tait equation,

p=p,l[l-B,ln(~+p)l(B+Po)l, (1)

where

po=B,+B2T+B3T2, B=B,exp(-B,T).

A. Co~pas~~~o ei al. /Bulk ~odu~us~or ~o[ecu~at~uids 179

Table 1

Experimental results for the substances studied at three temperatures, and isothermal compressibilities, KT, and isobaric thermal expansion coeffkient, ol,, calculated from eq. ( 1)

T (KI

P P l@‘K, 103Lu, T P P IOkT

hem-‘1 10+x@

(bar) (bar-’ f w-‘) (K) (gcm-Jl (bar) (bar-’ 1 (K-l)

CHJI 298.15 2.26640

2.27437

2.27945

2.28355

2.28742

2.29359

2.29802

2.30245

2.30643

2.31087

2.31519

2.31961

2.32504

2.32933

2.33495

2.33977 2.34429

2.34743

1.66 103.8 1.27 35.02 100.7 1.24 57.50 98.7 1.22 75.82 97.2 1.21 94.59 95.7 1.19

122.87 93.5 1.17 144.35 91.9 1.16 165.33 90.3 1.14 184.16 89.0 1.13 207.61 87.5 1.12 228.36 86.1 1.11 250.92 84.7 1.09 278.110 83.0 1.08 301.31 81.7 1.07 330.86 80.0 1.05 354.70 78.8 I .04 378.75 n.5 1.03 393.20 76.8 I .02

308.15 2.23173 1.67 112.6 1.28

2.24117 15.01 111.1 1.21 2.24632 36.19 108.8 1.25 2.25642 78.16 104.7 1.22 2.26150 99.77 102.6 1.20

2.268 19 128.83 100.0 1.18 2.27363 152.83 98.0 1.16 2.27857 175.78 96.1 1.16 2.28336 197.97 94.4 1.13

2.28825 221.24 92.6 1.11 2.293 10 243.58 91.0 1.10 2.298 1 I 268.28 89.3 I .09

2.30331 293.47 87.6 1.07

Z-30793 316.92 86.1 1.06 2.3126% 340.89 84.6 1.05 2.31718 364.60 83.2 1.03

2.32163 389.16 81.8 I .02 2.32541 404.21 80.9 1.01

318.15 2.20833 1.07 122.2 1.30 2.21527 26.22 119.0 1.27 2.22158 48.57 116.3 1.25 2.23127 85.59 112.0 1.22 2.23719 110.04 109.4 1.20 2.24276 132.35 107.2 1.1% 2.23813 155.00 1050 1.16 2.25284 176.53 102.9 1.15 2.25730 196.47 101.1 1.14 2.26133 215.89 99.5 1.12 2.27315 267.44 95.3 1 .OP

2.27852 291.95 93.4 1.08 2.28350 315.25 91.7 1.06 2.28787 336.85 90.2 1.05 2.29178 357.85 88.8 1.04 2.29592 377.79 87.5 1.03

Cc& 298.15 1.58368

1.59104

1.59447

1 s9794

1.60076

1.60383

1.60687 1.60979

I.61255

1.61517

1.61767

1.62082

1.62348 1.62585

1.62855

1.63082 1.63355

1.63613

1.63868

1.64236

0.36 108.0 1.24 41.67 103.1 1.20 61.66 100.9 1.19 82.97 98.7 1.17

100.78 96.9 1.16 121.31 94.9 1.14 142.2 93.0 1.13 161.93 91.2 1.11 181.61 89.6 t.to 199.63 88. I 1.09 218.49 86.6 1 .os 242.11 84.8 1.07 261.86 83.4 1.06 278.76 82.2 1.05 299.99 80.7 1.04 318.08 79.5 1.04 337.41 78.3 I .02 357.58 77.0 1.01 376.92 75.9 1 .oo 401.76 74.4 0.99

308.15 1 S6440 0.43 117.3 1.27 1S6801 22.67 114.1 1.25, 1.57121 39.18 111.9 1.23 1.57393 55.29 109.8 1.22 1.57718 73.55 107.5 1.20 1.58100 94.21 105.1 1.t8 1.58450 115.4% 102.7 1.17 1.58790 136.114 100.4 1.15 I.59115 157.60 98-3 1.14 1.59479 181.44 95.9 1.12 1.59816 205.58 93.7 1.11 1.60183 228.61 91.6 1.09 1.60515 251.%2 89.7 1.08 I .60850 275.94 87.7 1.07 1.61114 295.27 86.2 1.06 1.61441 319.98 84.4 1.04 1.61771 344.13 82.6 1.03 1.62016 361.83 81.4 1.02 1.62323 384.29 79.9 1.01 1 A2669 407.01 78.5 1.00

(continued on next page)

180

Table 1 (continued)

A. Compostizo et ai. /Buik modulus for molecularjhids

P P 1 O&K, 1 O%xD T P P IObKT 1 O”cY,

&cm-‘) (bar) (bar-‘) 0-l) (Kf (g cm-“) (bar) (bar-‘) (K-‘1

318.51 I.54461 0.49 127.4 1.30 1.55075 33.49 122.0 1.26 1.55416 50.31 119.4 1.25 1.55739 67.97 116.8 1.23 1.56092 87.48 114.1 1.22 1.56453 108.73 111.2 1.20 1.56803 129.43 108.6 1.18 1.57115 150.00 106.2 1.17 1.57492 170.79 103.8 1.15 1 S7822 192.05 101.5 1.14 1.58183 213.35 99.2 1.12 1.58501 233.87 97.2 I.11 1.58843 255.56 95.1 1.10 1.59151 276.46 93.2 1.08 1.59452 298.20 91.3 1.07 1.59784 320.4 1 89.5 1.06 1.60083 341.63 87.8 1.05 1.60353 361.60 86.2 1.04 1.60634 383.95 84.6 1.03 1.60960 404.14 83.1 1.02

CHC13

298.15 1.47164

1.4788 1 I.48276

1.48612

1.49013

1.49298

1.49562

I .49969

1.50233

1.50490

1.50726 1.51024

1.51268

1.51730

1.51958

1.52357

i .52689

1.86 107.6 1.29

39.26 103.7 1.27

62.62 101.4 1.25

84.38 99.4 1.24 112.10 96.9 1.22 132.20 95.2 1.21 151.24 93.6 1.20 182.40 91.2 1.18 202.67 89.6 1.17 222.61 88.2 1.16 241.33 86.9 1.15 264.33 85.3 1.14 286.10 83.9 1.13 318.46 81.9 1.11 338.19 80.7 1.10 372.91 78.7 1.09

396.67 77.4 I ‘OS

308.15 1.45376 1.55 115.4 1.28 1.45922 33.89 111.5 1.25 I .46323 56.94 i08.9 1.24 1.46730 81.58 106.3 1.22 1.47090 104.92 103.9 1.20 1.47439 128.31 101.6 1.19 1.47778 151.45 99.5 1.18 1.48097 173.98 97.5 1.16 1.48399 195.43 95.8 1.15 1.48704 217.67 93.8 1.14 1.49044 242.15 91.9 1.13

1.49351 264.47 90.2 1.11

1.49660 287.88 88.5 1.10

1.49963 310.81 86.9 1.09

1 SO245 332.51 85.4 1.08

1.50513 354.02 84.1 1.07 1.50777 376.00 82.7 1.06 1.51077 398.70 81.3 1.05

318.15 1.43616 0.75 123.9 1.27

1.43925 20.38 121.2 1.25

1.44512 56.56 116.4 1.22

1.45145 95.18 Ill.8 1.19

I .45?28 132.25 107.7 i.17

1.46197 162.65 104.5 1.15 1.46718 197.78 101.1 1.13

1.47180 229.09 98.3 1.1 I

1.4763 1 259.79 95.7 1.09

1.4807 1 290.63 93.2 1.08

1.4849 1 319.85 90.9 1.06

1.48901 353.80 88.5 1.05

1.49267 379.47 86.7 1.04

1.49738 408.36 84.8 1.02

CHzClz

298. I5 1.31533

1.31925

1.32259 1.32629 1.32936 1.33185 1.33412

1.33659

1.33887 1.34122

1.34344

1.34565

1.34782

1.34997

I.35235 1.35432

1.35662

1.35858

1.36075

1.36309

2.21 106.6 1.40

28.71 103.5 I .38

51.72 100.9 1.36

78.85 98.1 1.33 103.75 95.6 1.31

123.33 93.7 1.30

142.87 92.0 1.29 163.51 90.2 1.27

183.48 88.5 1.26

203.99 86.8 1.24

223.41 85.3 1.23

243.39 83.8 1.22

263.39 82.4 1.21

282.95 81.0 1.20

304.49 79.6 1.19

323.08 78.4 1.18

344.75 77.0 1.17

363.31 75.9 1.16

383.32 74.7 1.15

400.92 73.7 1.14

308.15 1.29631 1.75 116.6 1.44

1.29902 18.88 114.2 1.42

1.30224 38.96 111.5 1.40

1.30598 63.69 108.3 1.38 I .30999 91.08 105.0 1.35

Table I (continued)

A. Compostizo et al. /Bulk modulus for molecular fluids 181

T P P 1 O%cT 1 O’a, P P 1 06Kr 1 O’cu,

(K) (g cm-‘) (bar) (bar-‘) (K-l) TK) (g cmm3) (bar) (bar-‘) (K-l)

1.31353 116.26 102.1 1.33 1.31704 142.43 99.3 1.31 1.31998 164.70 97.1 1.29 1.32290 188.38 94.8 1.28

1.32309 192.91 94.4 1.27

1.32566 209.20 92.9 1.26

1.32625 218.89 92.0 1.26

1.32930 244.62 89.8 1.24

1.33241 270.45 87.7 1.23

1.33515 294.31 85.8 1.21 1.33816 321.30 83.8 1.20 1.34106 346.32 82.0 1.18

1.34408 373.11’ 80.2 1.17

1.34764 400.97 78.4 1.16

318.15 1.27841 1.21 127.6 1.47 1.28494 42.06 120.9 1.43

1.28930 70.60 116.6 1.40 1.29272 94.96 113.2 1.38

1.29577 116.70 110.3 1.36

1.29921 140.90 107.3 1.34

1.30217 163.01 104.7 1.32

1.30560 189.23 101.8 1.30

1.30897 214.84 99.1 1.28

1.31215 239.93 96.6 1.27

1.31510 263.79 94.3 1.25

1.31784 285.41 92.4 1.24

1.32110 310.78 90.2 1.22

1.32435 337.90 88.0 1.21

1.32764 368.90 85.6 1.19

1.33114 399.32 83.3 1.18

1.33261 407.47 82.8 1.17

CS, 298.15 1.25393

1.25694

1.26014

1.26271

1.26556

1.26776 1.27014

1.27227

1.27476

1.27750

1.28031

1.28267

1.28475

1.28751

1.29056

1.29189

1.29357

1.29704

1.06 94.7 1.18

20.25 93.3 1.17

41.13 91.8 1.16

60.29 90.4 1.15

83.79 88.8 1.13

103.23 87.6 1.12

125.26 86.2 1.11

145.17 85.0 1.10

169.44 83.6 1.09

197.15 82.0 1.08

226.13 80.4 1.06

249.66 79.2 1.05

271.46 78.1 1.04

300.61 76.7 1.03

331.90 75.2 1.02

345.61 74.6 1.01

362.67 73.8 1.01

395.76 72.4 0.99

308.15 1.24037 1.11 101.1 1.20

1.24446 33.49 98.4 1.18

1.24824 62.53 96.1 1.17

1.25115 86.85 94.3 1.15

1.25399 110.83 92.6 1.14

1.25692 136.10 90.8 1.12

1.25970 161.64 89.1 1.11

1.26223 184.92 87.6 1.10

1.26508 212.04 85.9 1.08

1.26778 237.30 84.4 1.07

1.27017 259.58 83.1 1.06

1.27268 284.03 81.7 1.05

1.27540 310.63 80.3 1.04

1.27777 335.29 79.0 1.03

1.28054 362.63 77.7 1.02

1.28262 384.05 76.6 1.01

1.28526 405.60 75.6 1.00

318.15 1.22612 1.17 108.0 1.23

1.22983 32.12 105.1 1.20

1.23304 58.81 102.7 1.19

1.23648 86.44 100.3 1.17

1.23956 112.87 98.1 1.15

1.24287 143.74 95.7 1.14

1.24643 171.09 93.7 1.12

1.24964 198.16 91.8 1.11

1.25280 227.15 89.8 1.09

1.25560 252.75 88.2 1.08

1.25815 278.60 86.6 1.07

1.26162 304.55 85.0 1.06

1.26436 329.44 83.6 1.04

1.26674 351.88 82.3 1.04

1.26985 382.69 80.6 1.02

1.27399 413.00 79.1 1.01

W-h 298.15 0.87378

0.87575

0.87738 0.87928

0.88072

0.88227

0.88365 0.88502

0.88639

0.88783

0.88917

0.89053 0.89195

0.89330

0.89458

0.89590

1.40 94.7 1.23

23.25 92.6 1.22

41.01 91.0 1.20

64.27 88.9 1.19

82.97 87.3 1.18

103.03 85.7 1.16

122.73 84.2 1.15

143.05 82.7 1.14

162.61 81.2 1.13

183.62 79.8 1.11

202.75 78.5 1.10

222.77 77.2 1.09

244.33 75.8 1.09

263.51 74.7 1.07

282.83 73.6 1.06

301.72 72.5 1.06

(continued on next page)

T P P lobK7 1 O’er, T P P 1 ohK7 I 03a,

(K) (g cm+) (bar) (bar-‘) (K-l) (K) (gem-‘) (bar) (bar-‘) (K-l)

0.89728 323.77 71.3 1.05 318.15

0.8986i 343.24 70.2 1.04

0.90026 366.07 69.1 1.03

0.90 f 84 386.19 68.1 1.02

308.15 0.86327 1.11 102.8 1.25

0.86504 21.73 100.5 1.23

0.86740 46.95 97.8 1.21

0.86955 72.75 95.2 1.19

0.87136 95.59 93.0 1.18

0.87336 122.47 90.5 1.16

0.87495 145.43 88.6 1.14

0.87676 166.02 86.9 1.13

0.87844 188.58 85.1 I.12 0.88022 213.16 83.2 1.10 0.88201 239.01 81.3 1.09

0.88372 264.79 79.6 1.08

0.88599 293.67 77.7 1.06 0.88776 319.03 76.1 1 .os 0.88947 345.00 74.5 1.04

0.89108 370.61 73.0 1.03 0.89290 394.23 71.7 I .02

0.85214 0.51 111.7 1.26

0.85316 15.10 109.7 1.25

0.85442 28.95 107.9 1.24

0.856 13 33.91 107.3 1.23

0.85827 59.6 I 104.3 1.21

0.86039 84.41 101.3 1.19

0.86233 108.1 1 98.8 1.18 0.86445 134.30 96. I 1.16

0.86660 161.60 93.5 1.14

0.86913 195.88 90.4 1.12

0.87117 222.45 88.i 1.10

0.87309 ‘47.24 86. I 1.09 0.87512 273.35 84.1 1.08 0.87700 298.28 82.3 1.06 0.87901 326.16 80.3 1.05 0.88063 350.80 78.7 1.04

0.88230 374.95 77.1 1.03 0.88392 399.80 75.6 1.02

The parameters of eq. ( 1) and the standard deviations of the fits are shown in table 2. From eq. (1) we have calculated the isothermal compressibilities ICY and the isobaric expansivities oP collected in table 1.

Of all the substances studied only for benzene one can find a critically discussed equation of state in the hterature [ 25 1; fig. I shows the deviations of our data from that equation. The scattering of the points is within the combined uncertainties of the equation of

state of ref. [ 251 and our technique. Similar conclusions have been obtained for NZ. Fig. 1 also shows the deviations obtained for CW,C12 with respect to the data of ref. [ 261 at 298.15 K. However, the discrepancies found when comparing different sets of data available for CCL, and CSL! at 298. I.5 K were appre- ciably larger [ 18 ] thus leading to contradictory conclusions about derived thermodynamic properties [ 27 ] . Though the discrepancies between all the sets of p-p-T data reported in the literature for Ccl, rise

Table 2

Parameters of eq. ( 1) for the different substances, and standard deviation of the tit

C&I cci, CHClz CHzC& CS, CA

& (gcm-3) 3.1304 2.0663 2.2379 I .7834 1.6597 1.1987

&X 10’ (gcm-3deg-E) -0.2928 -0.1276 -0.3226 -0.1295 -0.1226 -0.1099

B,~lO”(gcrn-~deg-~f 0.1006 -1.1431 2.2125 -0.9210 -0.4348 0.0349

B4X 10 1.0177 0.086 1 0.9653 0.854 1 1.062 1 0.8387

&X 10m4 (bar) 1.1164 0.9426 0.7289 1.1519 0.8061 1.0212

&X IO* (deg-I) 0.8162 0.8284 0.7032 0.8950 0.6617 0.8205

o(p) x lOa (gem-3) 0.3 0.2 0.3 0.3 0.4 0.2


I I I I 1

:

i

,” “s, 0

a”

/ I 298 15 K

-I I I I 0 IO 20 30 LO

pIMPa

Fig. 1. Deviations of eq. ( 1) from the equation of state for benzene and for CH2CIZ given in refs. [ 25,261 respectively.

up to + 10d3 g cm-3, the data reported in refs. [ 28- 3 1 ] agree with each other and with those reported in this paper within + 5 x 1 Oe4. The same agreement is found for CS2 with the data of ref. [ 291. All the above make us confident about the accuracy of the present data.

3. Discussion

Using Pople’s expansion for the intermolecular po- Despite all the above, Huang and O’Connell [ 341

tential and a reference system characterized by the have been able to find correlations for the bulk mod-

angular average of the full intermolecular potential ulus B

u”(r)=(Nr,wI, w~))~,,~~, (2)

where w, denotes the orientation of molecule i and ( u( r, w,, w2) ) indicates a canonical average. Gub- bins and O’Connell [ 321 have shown that

(pk,T~,)-*=(pk,T~~)-‘(l+pk~T~oTG,), (3)

ke being the Boltzmann constant and rc$ the isothermal compressibility of the reference system. G, in- volves integrals over the anisotropic part of the intermolecular potential and the center pair correlation function of the reference system. For a variety of anisotropic contributions to the potential, e.g. dipole- dipole GK* 1, which leads to

pksTKTzpksT&=f ;(pd, kgTlt) , (da)

where e and u are the parameters of a (n, m) Mie- like potential. Furthermore, f “, was found to show a

B= (pke TK~)-’ , (5)

based on eq. ( 4a), which are followed for fluids rang- ing from Ar to n-C, ,H,,. Fig. 2 shows the results for the systems studied in this paper. It can be observed that when the o values are taken from gas viscosity, the fluids do not follow the same curve (gas viscosity ~7 does not depend upon anisotropic forces). How- ever, describing the fluids by a spherical potential can only be done through the use of effective potential parameters, thus changing slightly the avalues allows one to draw a master curve containing the data of all the substances at the three temperatures considered. The scattering of the data is comparable to that obtained for each substance after mixing the data of the three isotherms, except for CHC13 and Ar for which (tlB/dp)T seems to be slightly smaller than for the other substances; also the scattering for the Ar data

very weak temperature dependence for p>2p,, pc being the critical density, hence

pkgTKT=f :(po3), (4b)

which expresses a law of corresponding states as far as the different fluids may be described by the same reference system.

Two points deserve to be mentioned. On the one hand, the overlap contributions to the potential are very highly dependent upon the anisotropy of shape of the molecules, leading to G,x 1 only for small anisotropies, therefore eq. (4b) might break down for molecules with large anisotropies. On the other hand, in two recent applications of the theory to pure fluids [ 4,5 1, it has been shown that it was necessary to use different values for the exponent n that characterizes the repulsive branch of the Mie potential. The values n = 13 for ethylene [ 41 and n = 20 for CF4 [ 5 ] suggest that even if eq. (4b) holds, several pkBT& versus p* (p*=2pNAo3/3) curves might be found for families of fluids with a given value of n. Moreover, it should be recalled that different n values can be found even for molecules of the same shape but different size. In effect, while the ( 12, 6) potential can be used for Ar, for large spheres the spherical-shell potential [ 331 is a better model, and this can be closely represented by a (28, 7) Mie potential.

184 A. Composti:o et al. /Bulk modulusfor molecularjluids

Fig. 2. Test of the Gubbins-Gray perturbation theory. Bulk mod-

ulus versus reduced density, p*=p2NAu3/ 3.

[ 351 is larger than for the other substances. The results of fig. 2 are compatible with the conclusions of the Gubbins-Gray theory, and using the correlation of Hung and O’Connell [ 341 the present results can be described very accurately.

The above results suggest that for the molecular fluids studied in this work, the p-p-T surface could be described by theories that use a spherical reference system. For molecules that interact through an effective spherical pair potential of the form

u .ff=A(T,p)lr”-B(T,p)lrm, (6)

the use of the virial theorem leads to [ 36,371

C:=(3/n)(aS/aln V),-3R/n, (7)

where Ct is the configurational heat capacity at constant volume, and (X/a In V) T= yV with y= (Y,/K, Fig. 3 shows C, versus y Vfor several fluids along the coexistence curve according to the data of refs. [ 38- 43] as shown in table 3.

It can be observed that in general linearity is re- stricted to a limited range, which is not surmising since eq. (7) was found to hold better in the low-temperature high-density range (low yV and C*,) [ 371. From the slope of the linear parts we have calculated the values of n shown in table 3. Even though these values must be considered as approximated

(lim,,,, Ct# 3Rln in same cases!), we can see that a common n cannot be assigned to all the substances

51 50 100 150

y’l 1 J ml- K-'

Fig. 3. Configurational heat capacity versus V for several pure

substances under orthobaric conditions: (0 ) CS,, ( 0 ) CH,I,

( X ) Ccl,, ( A ) CH*Br,, (0 ) CF,, ( V ) CHCl,.

studied, in agreement with the results for ethylene [ 41 and CF, [ 5 1. Since the repulsive branch of the potential plays a fundamental role in the definition of the reference system, we should conclude that this must

Table 3

References for the heat capacity at constant volume, C, ideal gas

heat capacity, C’$, thermal pressure coefficients, p, and results for the exponent n of the repulsive branch of the Mie potential

Substance Reference n

c,. c;g P

CF, [381 [391 [381 19 CCI, [401 [401 [401 22 cs, [411 ~421 [411 11 CHC& [431 ~421 [431 21 CH41 1431 1421 1431 14 CH*Br, 1431 [421 [431 17


be different for the several substances studied in this paper, in clear disagreement with the conclusions obtained from fig. 2 and by Huang and O’Connell [ 341.

The main difference between the data of figs.. 2 and 3 is that in the latter a thermal property C2 is in- volved, whilst in the former only data involving each separated isotherm (p and Q) are used. This seems to indicate that the details of the i~te~olecular potential are felt with higher intensity on properties which depend simultaneously on (ap/aT), and (ap/ ap) T than when only (ap/ap) r appears, hence, in order to be useful for any discussion on potential models, experimental efforts should be directed to obtain enough isotherms of a quality high enough as to allow the derivation of properties such as enthalpy, entropy, CV, etc. within a given temperature interval.

4. Semi-empirical equations of state

In view of the similar behaviour found for all the substances discussed (fig. 1)) we have found it interesting to check the performance of an equation of state derived for Lennard-Jones spheres from first princi- ples. Peters and Lichtenthaler [ 441 have derived an equation of state from the Percus-Yevick approxi- mation for the radial distribution function. Nicolas et al. 1451 and Adachi et al. [46] have derived their equations from computer simulation results. The three equations contain a set of universal constants and two adjustable parameters D and E/kg per substance. The computer simulation results for LJ fluids are best reproduced by the Adachi et al. equation, for given 0; c/ks and (T, p). In spite of that, in the ap- plication to the real substances both (I and c/kg are

Table 4 Effective potential parameters for the equation of state of Adachi et al. [46]

Substance elk, (K) u (ii)

CH,I 446.6 3.48 cc14 523.3 4.12 CHCl, 492.7 3.85 CH$& 455.6 3.54 CSZ 463.4 3.47 WI6 432.7 4.01

0 10 20 30 LO

P /MPa

Fig. 4. Predictions of an equation of state for an effective Len- nard-Jones potential (--_). Symbols are experimental data: ( 0 ) at318.15K,(x)at308.15Kand(~)at298.15K.

fitted, and the three equations lead to smaller discrepancies. In general the equation of Adachi et al. leads to slightly better results. The equation gives satisfactory results for substances whose complexity seems to be outside the theoretical assumptions of the theory. Table 4 shows the values of o and e/k* found, and fig. 4 shows the predictions for four of the substances of table 4. The results are similar to that found with the Gubbins-Gray perturbation theory for other substances [4,5], typically 1% in p. As expected the values of cr and c/kg do not coincide with those from gas viscosity, however, the discrepancy is not too im- portant if one considers that for Ccl, the values of (T are 0.588 nm from gas viscosity and 0.526 nm when calculated from the critical point coordinates.

5. Conclusions

The p-p-T data of molecular fluids with different multipolar moments and shape anisotropy follow quite closely the corresponding-states law predicted by the Gubbins-Gray perturbation theory for the bulk modulus. However, combination of p-p-T and C, data and the virial theorem leads to conclusions that seem to be in conflict with such a corresponding-states

186 A. Compostizo et al. /Bulk modulus for molecular fluids

law. These results indicate that in order to be useful for discussing any details of the intermolecular potential, the p-p-T should be accurate enough as to allow the calculation of derived properties such as entropy, enthalpy or C,,,

Acknowledgement

This work was supported in part by CAICYT under grant PB-85/0016 and by SEUI under a grant for Formation de Grupos Nuevos.

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Bulk modulus for some simple molecular fluids