The speed of sound and derived thermodynamic properties of ethane at temperatures between 220 K and 450 K and pressures up to 10.5 MPa

J. Chem. Thermodynamics 1997, 29, 991–1015

The speed of sound and derived thermodynamic properties of ethane at temperatures between 220 K and 450 K and pressures up to 10.5 MPa A. F. Estrada-Alexanders a and J. P. M. Trusler b Department of Chemical Engineering and Chemical Technology, Imperial College, London SW7 2BY, UK The speed of sound u in gaseous ethane of mole-fraction purity 0.9999 has been measured along 17 isotherms at temperatures between 220 K and 450 K. At temperatures of 300 K and above, the greatest pressure on each isotherm was chosen to correspond to that at approximately one half of the critical density. Below T = 300 K, the greatest pressure on each isotherm was limited to that at which the estimated density was about 0.8 of the saturated vapour density. The measurements were made using a spherical acoustic resonator and have an estimated total uncertainty not greater than about 1·10−4·u; this uncertainty is dominated by the possibility that the sample contained undetected impurities. Second and third acoustic virial coefficients have been obtained from the results and used to determine parameters in model two- and three-body intermolecular potential-energy functions for ethane. Ordinary second and third virial coefficients calculated from these models are reported. Moreover, a numerical integration of the equations that link u with the other thermodynamic properties of the fluid has been performed inside the region where the speed of sound was measured. The maximum uncertainty of the thermodynamic properties derived in this way is estimated to be 0.04 per cent for the compression factor and 0.4 per cent for the isobaric and isochoric heat capacities. 7 1997 Academic Press Limited

KEYWORDS: ethane; speeds of sound; compression factors; heat capacity; virial coefficients

1. Introduction With current techniques, the speed of sound in compressed gases may be measured with outstanding precision and accuracy.(1–5) The results may be used to determine other thermodynamic properties through a process of numerical integration(2,6–8) or to determine parameters in semi-realistic models of the two- and three-body intermolecular potential-energy functions.(9–13) In the present paper we report new measurements of the speed of sound u in gaseous ethane at temperatures between 220 K and 450 K and with pressures up to 10.5 MPa. The compression factor Z and the isobaric Cp,m and isochoric Cv,m molar heat capacities have been obtained from the results by numerical integration. We also report perfect-gas heat capacities and second and third acoustic virial coefficients determined from our sound-speed measurements. The latter have been used to determine the parameters in simple a Permanent address: Departmento de Fı´ sica, Universidad Auto´noma Metropolitana, Apdo. 55-534, Me´xico, D.F., CP 09340. b To whom correspondence should be addressed.

0021–9614/97/090991 + 25 $25.00/0/ct970217

7 1997 Academic Press Limited

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A. F. Estrada-Alexanders and J. P. M. Trusler

model potential-energy functions from which ordinary second and third virial coefficients have been calculated.

2. Experimental All measurements were performed with the spherical-resonator apparatus described previously.(2,5) Speeds of sound were determined from each of the lowest five radial resonances of the gas-filled spherical resonator. The mean temperature of the gas was inferred from the readings of a pair of capsule-type platinum-resistance thermometers attached to the outer surface of the spherical resonator. These were calibrated by comparison with a standard thermometer which had itself been calibrated on ITS-90 at the U.K. National Physical Laboratory. Temperatures are estimated to be accurate to better than 25 mK over the whole temperature range investigated. Pressures above 1.4 MPa were measured with an uncertainty of 20.5 kPa by means of a Paroscientific Digiquartz transducer (model 43KT, 20 MPa full scale) located in an oven where, in order to prevent condensation of the sample, the temperature was maintained at 310 K. Pressures below 1.4 MPa were measured with an uncertainty of 20.2 kPa by means of a second Digiquartz transducer (model 2900AT, 1.4 MPa full scale) which was also located in the external tubing. Since condensation was not a problem at the lower pressures, this second transducer was operated at ambient temperature. Research grade ethane was supplied by Linde Gas UK with a claimed mole-fraction purity of 0.9999. The purity of the material was investigated by g.c. with flame ionization detection. The sensitivity of the analysis towards hydrocarbon impurities was about 1·10−5 in mass fraction, and a range of column temperatures was investigated in an attempt to resolve impurities; none were found. At the time of these analyses, a satisfactory alternative detector for the g.c. was unavailable and the possibility therefore remains that the material contained impurities to which the ionization detector was insensitive. Before use, the material was transferred to a previously baked and evacuated sample cylinder where it was degassed by repeated cycles of freezing, evacuation and thawing. No other method of purification was attempted. In order to fill the apparatus to pressures in excess of the vapour pressure at the ambient temperature, the temperature of the sample cylinder was raised. The tubing and valves were maintained at T = 310 K to prevent condensation of the ethane. After use in one set of measurements, the ethane was condensed back into the sample cylinder, degassed once more and re-used on subsequent isotherms. Seventeen isotherms were studied, 10 of them below the critical temperature, Tc = 305.33 K, and seven above. For T e 300.65 K the greatest pressure on each isotherm was chosen to correspond to rn 1 r nc /2, where rn is the amount of substance density and r nc = 6.875 mol·dm−3, while for T Q 300.65 K, the greatest pressure was chosen to correspond to approximately 0.8 of the saturated vapour density. Measurements started at the greatest pressure and proceeded downwards in predetermined decrements until the density was approximately 0.1 of the initial value; a final measurement at between 0.001 and 0.003 of the initial density was also made.

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Speeds of sound and thermodynamic properties of ethane

On the isotherm at T = 300.65 K, the greatest pressure corresponded to about 0.96 of the saturated vapour density and, contrary to reported results at lower reduced temperatures,(14) no evidence of pre-condensation phenomena was observed. At each state point, the temperature, pressure and the resonance frequencies of the five radial modes were measured. The pressure was corrected for the hydrostatic head between the equatorial plane of the resonator and the mid-point of the pressure transducer as described previously.(5) At most, this correction amounted to 1.9 kPa.

3. Results The speed of sound u was determined from the frequencies f0n of the radial modes by analysis in terms of the standard acoustic model in which (u/a0 ) is given by:(1,2,5,15)

0

(u/a0 ) = (2pa/a0 ) f0n − s Dfj j

1>

n0n .

(1)

Here, a is the radius at pressure p, a0 is the mean radius of the resonator at the temperature in question but at zero pressure and Sj Dfj is the sum of small ‘‘correction’’ terms which are described in detail elsewhere.(1,15) These corrections take into account the thermal-boundary layer, coupling between the motions of the gas and the shell, openings in the shell and geometric imperfections. Thermodynamic properties of the gas required in these calculations were obtained from the Lee–Kesler equation of state.(16) Transport properties were obtained from two sources: the viscosity was calculated from the correlation due to Younglove and Ely,(17) while the thermal conductivity was obtained from the correlation reported by Vesovic et al.(18) The zero-pressure radius of the resonator was taken from our recent calibration measurements.(5) The possibility of dispersion arising from vibrational relaxation in the gas was also investigated. Mean vibrational relaxation times tvib for ethane (assuming series excitation of all internal modes) were estimated from the observed resonance line widths by the method described elsewhere.(2) The results of this analysis, which are presented in table 1, indicate that dispersion effects were always negligible in the present work. Except for a few cases in which a resonance mode was rejected on account of strong coupling to a shell resonance, the speeds of sound derived from the five radial TABLE 1. Mean vibrational relaxation times t of ethane at r = 1 kg·m−3 T/K

t/ns

T/K

t/ns

T/K

t/ns

220.00 235.00 250.00 265.00 275.00 285.00

7.7 9.9 10.6 10.6 11.4 11.5

290.00 295.00 300.65 305.00 315.00 330.00

12.1 12.5 14.7 14.8 14.7 14.5

345.00 365.00 385.00 415.00 450.00

14.4 14.4 14.3 14.3 13.9

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modes were in good agreement. The worst results were obtained on the isotherm closest to Tc where the ‘‘spread’’ between the five radial modes varied from 25·10−6·u at low pressures to 220·10−6·u at the greatest pressure. The final results, after the application of small corrections to reduce each datum to exactly the stated temperature and pressure, are given in table 2. As a check of the internal consistency, the speeds of sound on each isotherm were fitted with a polynomial in density (with densities once again calculated from the Lee–Kesler equation of state(16) ). A four- or five-term fit was required to accommodate the results to within 10−5·u, except at temperatures near to the critical where up to six terms were required to achieve a similar fit. The estimated combined uncertainty of the absolute speeds of sound, excluding the unknown effects of impurities, varies from 23·10−5·u at low pressures to 25·10−5·u at the highest pressure on the isotherm (T = 305 K) which was closest to the critical temperature. These are estimated error bounds (not standard deviations) determined from all known sources of uncertainty in both (u/a0 ) and in the zero-pressure radius a0 . Additional uncertainty, probably not exceeding 21·10−4·u, arises from our incomplete characterization of the purity of the material. Clearly this last factor is dominant; it might be reduced in future work by more TABLE 2. Speed of sound u of ethane at temperatures from 220 K to 450 K and pressures up to 10.5 MPa p/kPa

u/(m·s−1 )

p/kPa

407.40 365.06 326.86

258.377 260.146 261.703

289.51 254.48 209.56

T = 220 K 263.192 169.79 264.562 126.04 266.280 86.75

267.770 269.374 270.787

44.52 13.94

272.277 273.340

704.89 641.67 576.75

258.301 260.751 263.181

513.48 445.63 380.99

T = 235 K 265.475 307.14 267.860 233.02 270.064 156.16

272.510 274.894 277.295

79.32 16.04

279.629 281.509

1088.89 994.91 908.19

257.188 260.580 263.577

806.76 702.11 594.43

T = 250 K 266.939 489.49 270.266 371.62 273.554 251.41

276.641 279.983 283.269

132.23 21.11

286.415 289.263

1699.36 1577.88 1444.69

250.690 255.039 259.523

1298.16 1139.55 973.39

264.165 268.913 273.620

793.68 611.74 422.36

278.447 283.094 287.706

217.98 22.01

292.460 296.830

2164.99 2001.09 1845.92

246.338 252.090 257.144

1666.44 1474.40 1265.74

T = 275 K 262.608 1040.70 268.081 804.51 273.661 554.49

279.335 284.960 290.600

286.62 30.81

296.332 301.550

2793.23 2616.85 2417.77

238.207 244.592 251.148

2208.24 1960.52 1691.63

T = 285 K 257.483 1407.76 264.398 1086.90 271.347 754.92

278.178 285.391 292.390

398.77 39.73

299.455 306.185

u/(m·s−1 )

p/kPa

u/(m·s−1 )

p/kPa

u/(m·s−1 )

T = 265 K

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Speeds of sound and thermodynamic properties of ethane TABLE 2—continued p/kPa

u/(m·s−1 )

p/kPa

3153.39 2975.33 2764.41

233.445 240.042 247.079

2516.63 2266.43 1964.46

3671.99 3526.09 3320.19

222.894 229.072 236.735

4258.65 4183.63 4007.44

u/(m·s−1 )

p/kPa

u/(m·s−1 )

p/kPa

u/(m·s−1 )

T = 290 K 254.573 1646.79 261.506 1294.95 269.225 886.85

276.739 284.477 292.836

456.65 48.21

301.062 308.404

3062.99 2764.62 2418.17

T = 295 K 245.176 2045.74 253.914 1601.67 263.058 1123.54

272.009 281.767 291.418

592.71 60.64

301.319 310.542

211.299 215.319 223.236

3771.18 3424.76 3054.03

T = 300.65 K 232.122 2648.37 243.218 2110.23 253.572 1486.06

263.702 275.775 288.357

812.30 92.16

300.657 312.673

4530.50 4384.52 4165.55

213.491 219.847 227.991

3900.70 3567.98 3166.64

T = 305 K 236.573 2708.49 246.129 2160.93 256.440 1531.70

267.061 278.574 290.605

815.98 84.42

303.091 314.813

5014.61 4775.12 4494.81

225.752 232.130 239.217

4178.00 3803.43 3352.86

T = 315 K 246.778 2857.02 255.181 2256.06 264.634 1599.05

274.340 285.289 296.409

829.14 89.42

308.502 319.322

5474.44 5326.39 5017.05

246.466 248.776 253.736

4578.62 4140.64 3619.63

260.891 268.016 276.334

284.953 294.769 304.959

873.41 93.31

315.972 325.977

6107.42 5876.65 5466.25

261.334 263.637 268.092

5004.81 4489.94 3917.54

273.490 279.769 286.893

3278.37 2565.73 1791.60

294.867 303.656 312.994

926.78 96.36

323.110 332.479

6941.25 6562.17 6085.17

279.852 282.011 285.237

5518.89 4907.50 4217.93

T = 365 K 289.632 3543.74 294.869 2763.91 301.176 1914.67

307.590 315.160 323.451

992.07 102.66

332.405 340.899

7587.35 7348.39 6699.18

297.689 298.456 301.118

6061.39 5368.70 4650.15

304.424 308.615 313.468

T = 385 K 3832.84 2968.95 2041.09

319.432 326.080 333.458

1051.48 109.83

341.463 349.095

8683.06 8433.54 7634.09

321.575 321.803 323.135

6809.96 6011.45 5153.57

T = 415 K 325.360 4251.47 328.212 3262.15 331.904 2232.67

336.355 341.752 347.795

1140.89 117.62

354.533 361.051

10455.70 9658.07 8733.70

347.557 346.872 346.872

7824.03 6777.84 5771.32

T = 450 K 347.641 4721.58 349.374 3602.75 351.792 2440.94

354.985 359.023 363.780

1248.41 124.03

369.136 374.524

T = 330 K 3063.20 2404.47 1688.16

T = 345 K

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FIGURE 1. Fractional deviations Du/u of the speed of sound u of ethane reported by Boyes(19) from these data interpolated from the results of this work: Q, T = 230 K; R, T = 250 K; W, T = 300 K; T, T = 330 K; R, T = 360 K.

careful analysis and/or purification of the material. It is probably true to say generally that sample purity is the largest source of uncertainty in measurements of the speed of sound in organic gases. The speeds of sound are compared in figure 1 with those measured by Boyes with a similar high-precision spherical resonator.(19) In order to effect this comparison, the present results were interpolated with respect to pressure and, where necessary, temperature as described below in connection with the numerical integration. For all isotherms except one, the agreement is within 25·10−5·u which is reasonable in the light of the claimed uncertainties. In the case of the isotherm at T = 300 K, there are deviations of order 2·10−4·u which, after consideration of the derived perfect-gas heat capacities, we shall attribute to a small systematic error in the results of Boyes. It is notable that a comparison of the ratios (u/u0 ), where u0 is the speed of sound extrapolated to the limit of zero pressure, reveals differences that are always within 22·10−5·(u/u0 ).

4. Perfect-gas heat capacities and acoustic virial coefficients pg The perfect-gas isobaric molar heat capacity Cp,m and the second ba and third ga acoustic virial coefficients were obtained by analysis of the speed of sound on each isotherm in terms of the expansion:

u 2 = A0 ·{1 + ba rn + ga r n2 + · · ·}.

(2)

Speeds of sound and thermodynamic properties of ethane

997

In this equation, A0 is the square of the speed of sound in the limit of zero density pg and is related to Cp,m by: pg Cp,m /R = {1 − (RT/MA0 )}−1,

(3)

where M is the molar mass and R is the universal gas constant. In the present work we have exploited the limiting-slope analysis, described in detail previously,(5,8) for all T e 275 K. Briefly, this method is based on the quantity Yb which is defined by: Yb = {(u 2/A0 ) − 1}/rn = ba + ga rn + · · ·,

(4)

such that Yb : ba , with slope ga , as rn : 0. Throughout the analysis, A0 was estimated from the speed of sound at the lowest experimental pressure by means of equation (2) truncated after the third acoustic virial coefficient, and rn was calculated at the experimental temperatures and pressures from an equation of state. Crude initial estimates were made for ba and ga so that the starting value of A0 could be calculated. The zero-density intercept and slope of Yb was then obtained by means of a suitable polynomial leading to improved values of ba and ga . The process was iterated to convergence; typically five cycles were required. The deliberate strategy of including one measurement on each isotherm at a very low density greatly facilitated the determination of A0 . It should be noted that, although great weight is placed upon that single measurement, the precision of the instrument in the low-density regime is especially favourable and we are satisfied that the emphasis is justified for T e 275 K. Quadratic polynomials fitted with the lower five or six remaining points were used in the determination of the zero-density intercept and slope of Yb . In the case of the isotherms at T Q 275 K, the density range was restricted and the lowest density was no longer a small fraction of the second lowest. Consequently, pg we preferred to determine Cp,m , ba and ga by direct analysis of the results in terms of equation (2). Four-term fits, restricted to the results of rn E 0.65 mol·dm−3, were used. As a check on consistency, both polynomial and limiting-slope analyses were performed at T = 275 K, and the values of ba so determined differed by only 0.08 cm3·mol−1 (which is about the same as the estimated uncertainty of either value); the corresponding values of ga differed by about 400 cm6·mol−2. In all cases, amount-of-substance densities were calculated from the truncated virial equation: p/rn RT = 1 + Brn + Cr n2

(5)

with virial coefficients B and C determined from the present acoustic virial coefficients by means of the analysis described below (see part 5). It was therefore necessary to perform several iterations of the entire analysis. pg pg Table 3 gives the final values of Cp,m , ba and ga . The imprecision of Cp,m was always smaller than 0.01 per cent but, after considering the total uncertainty assigned to the absolute speeds of sound, we believe that the true overall uncertainty is about 0.05 per cent. Quite large differences between precision and overall accuracy are typical of acoustic determinations of perfect-gas heat capacities in organic gases

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TABLE 3. Acoustic virial coefficients (ba , ga ) and isobaric molar perfect-gas heat capacities Cp,m of ethane T/K

pg C p,m /R

ba /(cm3·mol−1 )

10−3·ga /(cm6·mol−2 )

220.00 235.00 250.00 265.00 275.00 285.00 290.00 295.00 300.65 305.00 315.00 330.00 345.00 365.00 385.00 415.00 450.00

5.3007 2 0.002 5.4715 2 0.002 5.6541 2 0.003 5.8471 2 0.003 5.9822 2 0.003 6.1212 2 0.003 6.1921 2 0.003 6.2639 2 0.003 6.3452 2 0.003 6.4093 2 0.003 6.5566 2 0.003 6.7843 2 0.003 7.0157 2 0.004 7.3275 2 0.004 7.6394 2 0.004 8.1024 2 0.004 8.6341 2 0.004

−458.27 2 0.13 −404.88 2 0.12 −360.18 2 0.11 −322.39 2 0.13 −300.37 2 0.12 −280.25 2 0.11 −270.85 2 0.11 −261.98 2 0.11 −251.99 2 0.11 −244.98 2 0.11 −229.56 2 0.11 −208.63 2 0.11 −189.74 2 0.11 −167.52 2 0.11 −147.91 2 0.11 −123.04 2 0.10 −98.48 2 0.10

32.5 2 1.2 34.9 2 0.7 33.6 2 0.4 31.8 2 0.5 30.8 2 0.5 29.2 2 0.2 28.5 2 0.2 28.1 2 0.2 27.2 2 0.2 26.9 2 0.2 25.8 2 0.2 24.2 2 0.2 22.7 2 0.2 21.0 2 0.2 19.3 2 0.2 18.0 2 0.1 16.4 2 0.1

where purity may be in question; nevertheless, the method compares very favourably with calorimetry. The imprecision of the second acoustic virial coefficients estimated by the method described previously(5) was very small (typically 0.05 cm3·mol−1 ) but we tabulate more conservative figures which were obtained by combining the imprecision with an estimate (20.1 cm3·mol−1 ) of systematic uncertainties. Similarly, the uncertainties of ga include a contribution of 200 cm6·mol−1 to allow for possible systematic errors. The tabulated uncertainties should be interpreted as estimated error bounds. pg The present experimental values of Cp,m are correlated with a standard deviation of 0.0006·R (about 0.01 per cent) by the empirical formula: 4

pg (Cp,m /R) = s ci t i,

(6)

i = −1

where t = T/Tc , Tc = 305.33 K and the coefficients ci are given in table 4. Figure 2 shows the present results, together with those of other workers, as deviations from equation (6). Except at T = 300 K, the agreement with the acoustic results of Boyes(19) pg TABLE 4. Coefficients (i, ci ) in the correlation of isobaric molar perfect-gas heat capacity Cp,m /R {equation (6)}

i

ci

−1 0 1 2 3 4

−2.7850 19.2355 −32.0424 34.8569 −15.4481 2.5970

Speeds of sound and thermodynamic properties of ethane

999

pg pg FIGURE 2. Fractional deviations DC p,m /C p,m of the perfect-gas heat capacities of ethane from equation (6): W, this work; w, reference 19; Q, reference 20; r, reference 21; q, reference 22; t, reference 23.

is always within our claimed overall uncertainty of 20.05 per cent. However, at T = 300 K the value reported by Boyes is about 0.18 per cent below equation (6), consistent with the deviations of u shown in figure 1, and this suggests that his results contain a small error at that temperature. The agreement with the values calculated from spectroscopic data by Chao et al.(20) is always within 0.1 per cent. The calorimetric results of Ernst and Hochberg(21) deviate by between −0.1 per cent and −0.3 per cent, but this is reasonable in the light of their claimed maximum uncertainty of 20.3 per cent. The acoustic results of Esper et al.(22) deviate by between 0.3 per cent and −0.4 per cent; these authors make no statement about their overall uncertainty but clearly the observed deviations greatly exceed the available precision in an acoustic determination. Finally, we mention the results of Bier et al.(23) which have a claimed uncertainty of 0.2 per cent but which actually deviate from equation (6) by as much as −1.3 per cent.

5. Model potential-energy functions and the virial coefficients of ethane The second and third acoustic virial coefficients determined in this work have been used to determine parameters in model intermolecular pair and triplet potential-energy functions from which the ordinary second and third virial coefficients were then calculated. The method of analysis has been described in detail elsewhere and is not therefore repeated here.(12,13)

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It was assumed that an accurate representation of the virial coefficients may be obtained by adopting a model in which the pair potential energy f is treated as a function of the separation r of the molecules only. The specific model considered was the Maitland Smith Potential,(24) f(r)/o = {6/(n − 6)}·(rm /r)n − {n/(n − 6)}·(rm /r)6,

(7)

which contains the following four parameters: o, the depth of the potential well; rm , the separation at which f = −o; and the shape parameters m and n which determine the exponent n through the linear equation: n = m + n{(r/rm ) − 1}.

(8)

It is also assumed that the non-additive contribution Df3 to the potential energy of a cluster of three molecules is given by the Axilrod–Teller triple-dipole term: 3 3 3 −1 r 13 r 23 ) (1 + 3 cos u1 cos u2 cos u3 ). Df3 = n123 (r 12

(9)

Here, rij is the distance between molecules i and j, ui is the angle subtended at molecule i by molecules j and k, and n123 is a dispersion coefficient which we treated as an adjustable parameter. Initially, all five parameters (rm , o, m, n, and n123 ) were adjusted in a simultaneous weighted fit to the experimental second and third acoustic virial coefficients. The leading translational quantum correction was included in the calculations of both B and C. It was found that the results were not sensitive to both m and n, and that all values of n between 0 and 20 gave virtually identical fits and almost identical values of B and C. Consequently, in the final analysis, we set n = 0 (the pair potential then reduces to the Lennard-Jones n,6 model) and adjusted the remaining four parameters with the following results: rm = 0.43752 nm; o/k = 426.31 K; m = 42; n123 /k = 0.013939 K·nm9. Although translational quantum corrections were included in these calculations, we note that equally good results may be obtained at rather less effort from fully classical calculations; the optimum parameters in that case are: rm = 0.43997 nm, o/k = 415.03 K, m = 39, and n123 /k = 0.014077 K·nm9. The deviations of our second acoustic virial coefficients from the fit are shown, together with the results of Boyes,(19) in figure 3. Although there is evidence of some small systematic deviations, the fit is nevertheless very good. We also note the rather close agreement with the results of Boyes,(19) except at T = 300 K where his ba deviates from the model by about 0.6 cm3·mol−1. The results of Esper et al.(22) are not plotted as they deviate from the model by between 10 cm3·mol−1 and −3 cm3·mol−1. The present third acoustic virial coefficients and those of Boyes(19) are compared with the model in figure 4. In view of the experimental uncertainties, which decrease rapidly with increasing temperature, the agreement between the two sets of experimental data is generally good while the agreement with the model is also good, except below about T = 250 K where there are small discrepancies. The datum of

Speeds of sound and thermodynamic properties of ethane

1001

FIGURE 3. Deviations Dba of the second acoustic virial coefficients of ethane from values calculated from the model intermolecular potential: w, reference 19; W, this work.

Boyes at T = 210 K, ga = (47.4 2 4.4) × 103·cm6·mol−2, is both off the scale of the plot and inconsistent with the model. Ordinary second and third virial coefficients calculated from the model are given in table 5. We estimate that the uncertainty of B declines from 21 cm3·mol−1 at T = 220 K to 20.5 cm3·mol−1 in the interval 250 E T/K E 450. The uncertainty of C is rather difficult to estimate but the agreement between the model and the acoustic virial coefficients suggests that the results are accurate to within about 5 per cent or better in the interval 250 E T/K E 450. In figures 5 and 6, we compare the derived virial coefficients with directly measured values taken from the literature. The present second virial coefficients are in good agreement with those of Douslin and Harrison,(25) Michels et al.(26) and Mansoorian et al.(27) The near agreement with Douslin and Harrison over the full temperature range of their measurements (273.15 E T/K E 623.15) is particularly encouraging and demonstrates the reliability of the model as a means of extrapolation. We also note good agreement with the ‘‘square-well’’ formula derived by Boyes(19) from his second acoustic virial coefficients. The results of Pope et al.(18) are in quite good agreement at temperatures above 240 K but depart by as much as 12 cm3·mol−1 at lower temperatures. The recommended values of Dymond and Smith(29) appear to be insufficiently negative by approximately 10 cm3·mol−1 at 200 K but agree with the model to within 1 cm3·mol−1 at all temperatures above 270 K. In the case of the third virial coefficients, we note particularly good agreement with Douslin and Harrison(25) at temperatures below 500 K and also with Michels et al.(26)

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FIGURE 4. Third acoustic virial coefficients ga of ehtane: w, reference 19; W, this work (experimentally); ——, this work (calculated).

6. Derived thermodynamic properties In order to determine other thermodynamic properties of gaseous ethane from our speeds of sound, we have performed numerical integrations of the differential equations which link these quantities.(6,7) The domain of integration was split into two parts along the isotherm T = T0 at which the initial conditions were imposed. For T e T0 , T and rn were taken as the independent variables and the equation: u2 = (RT/M)[{Z + rn (1Z/1rn )T } + (R/CV,m ){Z + T(1Z/1T )rn }2 ],

(10)

TABLE 5. Second B and third C virial coefficients of ethane T/K

B/(cm3·mol−1 )

C/(cm6·mol−2 )

T/K

B/(cm3·mol−1 )

C/(cm6·mol−2 )

200 210 220 230 240 250 260 280 300 320

−419.7 −379.0 −344.2 −314.2 −288.0 −264.9 −244.6 −210.1 −182.1 −158.9

−12800 −3100 2760 6320 8410 9580 10160 10340 9900 9240

340 360 380 400 425 450 475 500 550 600

−139.4 −122.8 −108.4 −96.0 −82.5 −70.8 −60.7 −51.9 −37.1 −25.2

8540 7880 7290 6760 6200 5730 5340 5010 4520 4170

Speeds of sound and thermodynamic properties of ethane

1003

FIGURE 5. Deviations DB of second virial coefficients of ethane from values calculated from the model intermolecular potential: w, reference 25; q, reference 26; r, reference 27; t, reference 28; ——, reference 19.

was solved simultaneously with the relation: (1CV,m /1rn )T = −(R/rn ){2·T(1Z/1T )rn + T 2(12Z/1T 2 )rn }

(11)

to determine the compression factor Z = p/RTrn and the isochoric molar heat capacity CV,m . The isobaric molar heat capacity was also computed from the relation: (Cp,m /R) = (CV,m /R) + {Z + T(1Z/1T )rn }2/{Z + rn (1Z/1rn )T }.

(12)

For T Q T0 , a transformation of co-ordinates was made such that the independent variables were T and f, where f is defined by: f = (rn /r nc )/f(T )

(13)

and f(T ) is a specified function that we discuss further below. We also made use of a new heat capacity, defined by Cf,m = T(1Sm /1T )f , in terms of which the equations to be solved were: u 2 = (RT/M)[Cf,m {Z + f(1Z/1f)T } + R{Z + ZT(d ln f/dT ) + T(1Z/1T )f }· {Z + T(1Z/1T )f − fT(d ln f/dT )(1Z/1f)T }]· [Cf,m + RT(d ln f/dT ){Z + T(1Z/1T )f − fT(d ln f/dT )(1Z/1f)T }]−1,

(14)

1004

A. F. Estrada-Alexanders and J. P. M. Trusler

FIGURE 6. Third virial coefficients C of ethane: w, reference 25; q, reference 26; r, reference 27; t, reference 28; ——, this work.

and −(f/R)(1Cf,m /1f)T = 2·T(1Z/1T )f + T 2(12Z/1T 2 )f − T(d ln f/dT ){f(1Z/1f)T + Tf(12Z/1T 1f)} − T 2(d2 ln f/dT 2 ){f(1Z/1f)T }.

(15)

The isochoric and isobaric heat capacities were obtained in this case from the relations: (CV,m /R) = (Cf,m /R) + T(d ln f/dT ){Z + T(1Z/1T )f − fT(d ln f/dT )(1Z/1f)T }

(16)

(Cp,m /R) = (Cf,m /R) − T(d ln f(dT )[Z{1 + T(d ln f/dT )} + T(1Z/1T )f ] + [Z{1 + T(d ln f/dT )} + T(1Z/1T )f ]2{Z + f(1Z/1f)T }−1.

(17)

The transformation of co-ordinates from (T, rn ) to (T, f) is required on ground of stability when T Q T0 and the essential feature of the function f(T ) is that it satisfies the constraint: (df/dT ) q (1r/1T )S ,

(18)

where S denotes entropy. In this work, where T0 was below the critical temperature, f(T ) was chosen such that paths at constant f were parallel to the saturated vapour density r ns (T ); this automatically satisfies the inequality (18) because the saturated

Speeds of sound and thermodynamic properties of ethane

1005

vapour density rises with increasing temperature more rapidly than an adjacent isentrope. Thus, it was proposed that: f(T ) = r ns (T )/r nc

(19)

so that lines at constant f in the space (T, f) mapped onto lines falling parallel to the saturation curve in the space (T, rn ).(8) The methods of solution for use with either equations (10) and (11) or equations (14) and (15) has been described in detail elsewhere.(2,7,8) Here we will discuss further only the method of interpolation used to obtain u in between experimental points and the initial conditions. The speed of sound required at each grid point used in the solution algorithm was obtained from the table of experimental results by bidimensional interpolation. The method of solution was such that, whenever a speed of sound was required at a state point specified by (T, rn ) or (T, f), the corresponding value of Z (and hence p) had already been determined. In order to permit more reliable interpolation between experimental points, the interpolation routine operated not on u itself but on the ratio (u/u0 )/(u/u0 )eos , where (u/u0 )eos was calculated from an equation of state. In the present case the equation of state of Stewart et al. was used.(30) In the construction of the interpolation table, u0 was set equal to the value of (A0 )1/2 determined by analysis of the isotherm in question. Once a value of (u/u0 ) was found by interpolation, u0 was calculated at the temperature in question from the relation: pg pg u0 = [RTCp,m /{M(Cp,m − R)}]1/2,

(20)

pg with Cp,m determined by means of equation (6). This procedure imposed a smooth temperature dependence upon the interpolated speeds of sound at zero density but returned values that did not agree exactly with the experimental results. In fact, differences not exceeding 2·10−5·u were found. Full details of the interpolation algorithm can be found elsewhere.(7,8) The initial conditions required for the integration in both domains were values of Z and (1Z/1T )rn along the isotherm at T = T0 . These quantities, which we denote by Z0 and Z'0 , respectively, were determined from the prnT data reported by Douslin and Harrison(25) on two closely spaced isotherms: T1 = 298.15 K and T2 = 303.15 K. The initial temperature was chosen to be the mean of T1 and T2 , T0 = 300.65 K, and this also happened to be a temperature at which we had measured u. The experimental compression factors at densities up to 3.5 mol·dm−3 were fitted by fourth-degree polynomials in rn constrained to be unity at zero density. The polynomial approximations, which we denote by Z1 (at T = T1 ) and Z2 (at T = T2 ) were found to fit the data to within 20.007 per cent at T = T1 and to within 20.005 per cent at T = T2 . The required initial values were determined from these smooth polynomials by means of the formulae:

Z0 = (Z1 + Z2 )/2 − (Z00 /2!)(DT )2 + O(DT )4

(21)

1006

A. F. Estrada-Alexanders and J. P. M. Trusler

and (DT )Z'0 = (Z2 − Z1 )/2 + O(DT )3,

(22)

where DT = (T2 − T1 )/2 and Z00 = (1 Z/1T )rn at T = T0 . The terms of O(DT )3 and above were neglected but the term of O(DT )2 in equation (21) was determined iteratively from Z1 , Z2 and u by the following procedure. First, Z00 was neglected entirely and the initial conditions evaluated from equations (21) and (22). Then, a new value of Z00 was obtained from the speed of sound on the isotherm at T = T0 by means of equations (10) and (11); this was returned to equation (21) to start a new iteration; after a few cycles no further changes were found. For T e T0 , the integration was performed along 50 evenly spaced isochores with rn E 3.30 mol·dm−3 and with a uniform temperature increment of 0.5 K. All but one of the experimental isotherms in this region extended to the greatest density used in the calculations. However, the isotherm at T = 415 K only reached rn = 3.19 mol·dm−3 and the interpolation polynomial was therefore used as a means of performing the short extrapolation to rn = 3.30 mol·dm−3. The density of the saturated vapour r ns required to calculate f(T ) and hence f, was obtained from the correlation given by Tester.(31) It is not important that this correlation be accurate although it appears to be correct to within 1 per cent. The integration in the subcritical region took place along 50 lines of constant f; the maximum fraction of the saturated vapour density corresponded to a value of f = 0.8. The temperature increment in this case was −0.01 K. The initial conditions were imposed at T = T0 as before, but with the direction of the partial derivative of Z with respect to T changed from that at constant rn to that at constant f. As with argon,(7) small ripples were found to develop in the heat capacities as the solution proceeded and it was necessary to introduce the numerical filter described previously(7) to prevent this. In the present case, the filter was required at both subcritical and supercritical temperatures. For T Q Tc , the filter routine was applied directly to Cf,m . The filter had very little effect on the derived values of Z and the largest change, just 8·10−5·Z, was found at T = 220 K and f = 0.8. The derived Z, CV,m and Cp,m are given in table 6. Where necessary, the results have been interpolated to exactly the specified densities. The table has been constructed so that values of the thermodynamic properties may be obtained at intermediate temperatures and densities by quadratic interpolations without loss of accuracy. A computer program for that purpose (incorporating the table) may be obtained from the authors upon request. We comment that interpolation within a table is generally much faster computationally than enumerating a complicated correlating equation and that storage requirements are trivial for a pure substance. Questions of thermodynamic inconsistency do arise when quantities such as Z and CV are tabulated separately as they are here. However, in the present case, inconsistencies are smaller than the claimed uncertainties. On the basis of studies of the propagation of errors, we expect the uncertainty of Z to be dominated by that of the initial conditions. In the case of Z0 , this is taken 2

2

1007

Speeds of sound and thermodynamic properties of ethane

TABLE 6. Compression factor Z, heat capacities CV,m and Cp,m of ethane in the temperature range from 220 K to 450 K and densities rn up to 3.3 mol·dm−3 a rn /(mol·dm−3 )

Z

CV,m /R

Cp,m /R

rn /(mol·dm−3 )

0.009 0.018 0.027 0.036 0.045 0.054 0.063 0.072 0.081 0.090 0.099 0.108 0.117

0.996952 0.993901 0.990847 0.987790 0.984731 0.981669 0.978605 0.975538 0.972469 0.969398 0.966325 0.963250 0.960172

4.3075 4.3139 4.3203 4.3268 4.3334 4.3401 4.3469 4.3538 4.3608 4.3678 4.3750 4.3823 4.3897

T = 220.0 K 5.3196 5.3382 5.3571 5.3765 5.3961 5.4161 5.4365 5.4573 5.4784 5.4999 5.5219 5.5442 5.5669

0.016 0.032 0.048 0.064 0.080 0.096 0.112 0.128 0.144 0.160 0.176 0.192 0.208

0.995237 0.990472 0.985705 0.980935 0.976164 0.971393 0.966620 0.961847 0.957074 0.952301 0.947528 0.942756 0.937985

4.4796 4.4885 4.4976 4.5069 4.5164 4.5262 4.5362 4.5464 4.5569 4.5676 4.5786 4.5898 4.6013

0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325

0.993411 0.986825 0.980244 0.973669 0.967099 0.960535 0.953979 0.947431 0.940891 0.934359 0.927837 0.921324 0.914821

0.033 0.066 0.099 0.132 0.165 0.198 0.231

0.991954 0.983921 0.975903 0.967900 0.959914 0.951945 0.943993

Z

CV,m /R

Cp,m /R

0.126 0.135 0.144 0.153 0.162 0.171 0.180 0.189 0.198 0.207 0.216 0.225

0.957093 0.954012 0.950930 0.947845 0.944760 0.941672 0.938584 0.935494 0.932402 0.929310 0.926216 0.923121

4.3971 4.4047 4.4124 4.4202 4.4281 4.4361 4.4442 4.4524 4.4607 4.4692 4.4777 4.4864

5.5900 5.6136 5.6375 5.6619 5.6868 5.7120 5.7378 5.7640 5.7906 5.8178 5.8454 5.8735

T = 235.0 K 5.4985 5.5269 5.5560 5.5860 5.6168 5.6484 5.6809 5.7144 5.7487 5.7840 5.8203 5.8575 5.8958

0.224 0.240 0.256 0.272 0.288 0.304 0.320 0.366 0.352 0.368 0.384 0.400

0.933215 0.928446 0.923679 0.918913 0.914149 0.909387 0.904626 0.899868 0.895112 0.890358 0.885606 0.880856

4.6131 4.6251 4.6375 4.6501 4.6630 4.6762 4.6897 4.7034 4.7175 4.7318 4.7465 4.7614

5.9352 5.9756 6.0172 6.0599 6.1037 6.1488 6.1952 6.2428 6.2918 6.3421 6.3938 6.4471

4.6642 4.6754 4.6871 4.6992 4.7118 4.7248 4.7381 4.7518 4.7659 4.7804 4.7952 4.8103 4.8258

T = 250.0 K 5.6906 5.7292 5.7694 5.8112 5.8545 5.8993 5.9458 5.9938 6.0434 6.0947 6.1477 6.2025 6.2591

0.350 0.375 0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625

0.908328 0.901846 0.895375 0.888914 0.882464 0.876026 0.869598 0.863182 0.856777 0.850383 0.844000 0.837628

4.8417 4.8579 4.8744 4.8914 4.9087 4.9264 4.9445 4.9631 4.9821 5.0016 5.0216 5.0422

6.3175 6.3779 6.4404 6.5049 6.5717 6.6408 6.7124 6.7867 6.8636 6.9435 7.0266 7.1130

4.7939 4.8070 4.8207 4.8352 4.8502 4.8657 4.8818

T = 260.0 K 5.8265 5.8736 5.9230 5.9746 6.0285 6.0845 6.1428

0.462 0.495 0.528 0.561 0.594 0.627 0.660

0.888886 0.881098 0.873333 0.865589 0.857869 0.850171 0.842496

5.0065 5.0259 5.0458 5.0661 5.0868 5.1080 5.1298

6.6187 6.6974 6.7792 6.8643 6.9528 7.0451 7.1413

1008

A. F. Estrada-Alexanders and J. P. M. Trusler TABLE 6—continued

rn /(mol·dm−3 )

Z

CV,m /R

Cp,m /R

rn /(mol·dm−3 )

Z

CV,m /R

Cp,m /R

0.264 0.297 0.330 0.363 0.396 0.429

0.936060 0.928147 0.920253 0.912380 0.904527 0.896696

4.8984 4.9154 4.9328 4.9506 4.9689 4.9875

6.2034 6.2664 6.3317 6.3995 6.4698 6.5429

0.693 0.726 0.759 0.792 0.825

0.834843 0.827213 0.819606 0.812020 0.804457

5.1522 5.1751 5.1988 5.2232 5.2484

7.2417 7.3467 7.4566 7.5718 7.6928

0.044 0.088 0.132 0.176 0.220 0.264 0.308 0.352 0.396 0.440 0.484 0.528 0.572

0.990055 0.980141 0.970260 0.960413 0.950602 0.940826 0.931088 0.921388 0.911726 0.902104 0.892523 0.882982 0.873483

4.9290 4.9448 4.9614 4.9790 4.9973 5.0163 5.0360 5.0562 5.0770 5.0983 5.1202 5.1452 5.1653

T = 270.0 K 5.9699 6.0286 6.0906 6.1557 6.2240 6.2956 6.3706 6.4489 6.5308 6.6163 6.7056 6.7989 6.8964

0.616 0.660 0.704 0.748 0.792 0.836 0.880 0.924 0.968 1.012 1.056 1.100

0.864025 0.854610 0.845238 0.835908 0.826621 0.817378 0.808177 0.799020 0.789906 0.780835 0.771807 0.762821

5.1886 5.2124 5.2367 5.2616 5.2871 5.3133 5.3402 5.3680 5.3966 5.4263 5.4570 5.4889

6.9984 7.1052 7.2170 7.3343 7.4575 7.5870 7.7234 7.8673 8.0193 8.1803 8.3510 8.5325

0.059 0.118 0.177 0.236 0.295 0.354 0.413 0.472 0.531 0.590 0.649 0.708 0.767

0.987616 0.975296 0.963042 0.950856 0.938738 0.926690 0.914714 0.902810 0.890980 0.879224 0.867544 0.855941 0.844414

5.0693 5.0886 5.1092 5.1308 5.1534 5.1769 5.2011 5.2261 5.2518 5.2781 5.3051 5.3326 5.3608

T = 280.0 K 6.1210 6.1953 6.2741 6.3576 6.4459 6.5391 6.6373 6.7408 6.8499 6.9647 7.0858 7.2134 7.3481

0.826 0.885 0.944 1.003 1.062 1.121 1.180 1.239 1.298 1.357 1.416 1.475

0.832965 0.821594 0.810302 0.799089 0.787955 0.776901 0.765926 0.755030 0.744214 0.733477 0.722819 0.712240

5.3895 5.4189 5.4491 5.4799 5.5116 5.5442 5.5777 5.6124 5.6483 5.6856 5.7245 5.7650

7.4904 7.6408 7.8002 7.9693 8.1491 8.3405 8.5447 8.7632 8.9974 9.2492 9.5206 9.8140

0.078 0.156 0.234 0.312 0.390 0.468 0.546 0.624 0.702 0.780 0.858 0.936 1.014

0.984774 0.969666 0.954678 0.939811 0.925068 0.910451 0.895961 0.881600 0.867368 0.853269 0.839301 0.825468 0.811769

5.2139 5.2373 5.2623 5.2885 5.3159 5.3444 5.3737 5.4040 5.4350 5.4667 5.4992 5.5323 5.5662

T = 290.0 K 6.2786 6.3716 6.4711 6.5773 6.6904 6.8108 6.9389 7.0751 7.2199 7.3740 7.5380 7.7129 7.8995

1.092 1.170 1.248 1.326 1.404 1.482 1.560 1.638 1.716 1.794 1.872 1.950

0.798205 0.784779 0.771489 0.758336 0.745322 0.732446 0.719708 0.707109 0.694648 0.682326 0.670142 0.658096

5.6008 5.6362 5.6725 5.7096 5.7478 5.7872 5.8279 5.8700 5.9137 5.9593 6.0069 6.0568

8.0990 8.3127 8.5419 8.7885 9.0543 9.3417 9.6532 9.9920 10.3615 10.7661 11.2108 11.7014

0.091 0.182

0.982863 0.965888

5.2883 5.3148

T = 295.0 K 6.3620 1.274 6.4681 1.365

0.775583 0.760891

5.7233 5.7630

8.5304 8.7967

1009

Speeds of sound and thermodynamic properties of ethane TABLE 6—continued rn /(mol·dm−3 )

Z

CV,m /R

Cp,m /R

rn /(mol·dm−3 )

Z

CV,m /R

Cp,m /R

0.273 0.364 0.455 0.546 0.637 0.728 0.819 0.910 1.001 1.092 1.183

0.949077 0.932432 0.915957 0.899652 0.883521 0.867564 0.851784 0.836182 0.820759 0.805518 0.790459

5.3429 5.3725 5.4034 5.4355 5.4685 5.5025 5.5374 5.5730 5.6094 5.6466 5.6845

6.5823 6.7048 6.8361 6.9767 7.1271 7.2881 7.4603 7.6449 7.8428 8.0553 8.2840

1.456 1.547 1.638 1.729 1.820 1.911 2.002 2.093 2.184 2.275

0.746384 0.732063 0.717928 0.703980 0.690218 0.676644 0.663257 0.650057 0.637044 0.624217

5.8036 5.8454 5.8884 5.9327 5.9787 6.0265 6.0763 6.1284 6.1830 6.2406

9.0852 9.3984 9.7395 10.1122 10.5207 10.9703 11.4669 12.0176 12.6314 12.3186

0.108 0.216 0.324 0.432 0.540 0.648 0.756 0.864 0.972 1.080 1.188 1.296 1.404

0.980377 0.960981 0.941818 0.922888 0.904197 0.885745 0.867535 0.849571 0.831853 0.814385 0.797168 0.780203 0.763492

5.3638 5.3939 5.4260 5.4600 5.4955 5.5323 5.5702 5.6092 5.6491 5.6899 5.7315 5.7739 5.8171

T = 300.0 K 6.4493 6.5726 6.7065 6.8513 7.0078 7.1767 7.3589 7.5556 7.7679 7.9975 8.2461 8.5159 8.8092

1.512 1.620 1.728 1.836 1.944 2.052 2.160 2.268 2.376 2.484 2.592 2.700

0.747037 0.730839 0.714898 0.699216 0.683793 0.668630 0.653727 0.639084 0.624701 0.610579 0.596717 0.583114

5.8613 5.9065 5.9528 6.0005 6.0498 6.1008 6.1539 6.2094 6.2676 6.3290 6.3939 6.4627

9.1290 9.4786 9.8619 10.2835 10.7491 11.2649 11.8389 12.4804 13.2006 14.0134 14.9359 15.9890

0.132 0.264 0.396 0.528 0.660 0.792 0.924 1.056 1.188 1.320 1.452 1.584 1.716

0.976861 0.954062 0.931607 0.909501 0.887747 0.866350 0.845312 0.824637 0.804328 0.784387 0.764818 0.745622 0.726801

5.4427 5.4789 5.5174 5.5579 5.6001 5.6437 5.6886 5.7346 5.7815 5.8293 5.8781 5.9278 5.9786

T = 305.0 K 6.5451 6.6944 6.8581 7.0370 7.2322 7.4454 7.6780 7.9322 8.2104 8.5154 8.8505 9.2197 9.6277

1.848 1.980 2.112 2.244 2.376 2.508 2.640 2.772 2.904 3.036 3.168 3.300

0.708357 0.690290 0.672603 0.655297 0.638371 0.621826 0.605662 0.589879 0.574477 0.559455 0.544813 0.530549

6.0305 6.0837 6.1386 6.1952 6.2541 6.3155 6.3799 6.4478 6.5196 6.5959 6.6774 6.7647

10.0799 10.5829 11.1445 11.7739 12.4821 13.2828 14.1921 15.2299 16.4204 17.7939 19.3874 21.2493

0.132 0.264 0.396 0.528 0.660 0.792 0.924 1.056 1.188 1.320 1.452 1.584 1.716

0.977640 0.955617 0.933933 0.912592 0.891598 0.870953 0.850661 0.830724 0.811144 0.791924 0.773066 0.754571 0.736441

5.5155 5.5500 5.5865 5.6246 5.6642 5.7050 5.7469 5.7897 5.8333 5.8777 5.9227 5.9684 6.0146

T = 310.0 K 6.6149 6.7587 6.9154 7.0857 7.2708 7.4719 7.6902 7.9275 8.1856 8.4667 8.7732 9.1081 9.4747

1.848 1.980 2.112 2.244 2.376 2.508 2.640 2.772 2.904 3.036 3.168 3.300

0.718678 0.701283 0.684256 0.667599 0.651311 0.635395 0.619849 0.604673 0.589868 0.575433 0.561368 0.547672

6.0616 6.1092 6.1577 6.2070 6.2574 6.3090 6.3620 6.4166 6.4731 6.5317 6.5928 6.6567

9.8768 10.3186 10.8053 11.3425 11.9370 12.5964 13.3296 14.1467 15.0591 16.0804 17.2256 18.5122

1010

A. F. Estrada-Alexanders and J. P. M. Trusler TABLE 6—continued

rn /(mol·dm−3 )

Z

CV,m /R

Cp,m /R

0.132 0.264 0.396 0.528 0.660 0.792 0.924 1.056 1.188 1.320 1.452 1.584 1.716

0.978386 0.957104 0.936156 0.915545 0.895275 0.875347 0.855765 0.836529 0.817642 0.799107 0.780924 0.763094 0.745620

5.5887 5.6216 5.6561 5.6920 5.7292 5.7675 5.8067 5.8468 5.8875 5.9288 5.9706 6.0128 6.0554

T = 315.0 K 6.6852 6.8238 6.9739 7.1365 7.3124 7.5025 7.7080 7.9303 8.1707 8.4311 8.7132 9.0192 9.3514

0.330 0.660 0.990 1.320 1.650

0.948364 0.898790 0.851314 0.805962 0.762756

5.7104 5.7960 5.8876 5.9830 6.0809

0.330 0.660 0.990 1.320 1.650

0.951698 0.905378 0.861066 0.818781 0.778535

0.330 0.660 0.990 1.320 1.650

rn /(mol·dm−3 )

Z

CV,m /R

Cp,m /R

1.848 1.980 2.112 2.244 2.376 2.508 2.640 2.772 2.904 3.036 3.168 3.300

0.728502 0.711741 0.695337 0.679292 0.663605 0.648277 0.633308 0.618697 0.604443 0.590548 0.577009 0.563827

6.0983 6.1415 6.1851 6.2289 6.2732 6.3178 6.3629 6.4085 6.4548 6.5019 6.5498 6.5988

9.7126 10.1057 10.5340 11.0013 11.5115 12.0694 12.6798 13.3482 14.0804 14.8826 15.7612 16.7230

T = 320.0 K 6.9610 7.3574 7.8341 8.4074 9.0983

1.980 2.310 2.640 2.970 3.300

0.721710 0.682832 0.646120 0.611569 0.579168

6.1799 6.2793 6.3783 6.4768 6.5750

9.9335 10.9460 12.1755 13.6667 15.4644

5.8571 5.9345 6.0156 6.0994 6.1847

T = 330.0 K 7.0930 7.4567 7.8852 8.3895 8.9828

1.980 2.310 2.640 2.970 3.300

0.740332 0.704170 0.670039 0.637921 0.607795

6.2704 6.3551 6.4375 6.5163 6.5898

9.6799 10.4969 11.4500 12.5534 13.8146

0.954769 0.911436 0.870023 0.830543 0.793003

6.0055 6.0754 6.1475 6.2211 6.2956

T = 340.0 K 7.2280 7.5636 7.9517 8.3999 8.9167

1.980 2.310 2.640 2.970 3.300

0.757401 0.723728 0.691970 0.662102 0.634097

6.3701 6.4435 6.5148 6.5825 6.6451

9.5103 10.1888 10.9585 11.8228 12.7792

0.330 0.660 0.990 1.320 1.650

0.957606 0.917028 0.878284 0.841383 0.806329

6.1560 6.2202 6.2850 6.3502 6.4155

T = 350.0 K 7.3667 7.6784 8.0329 8.4355 8.8913

1.980 2.310 2.640 2.970 3.300

0.773119 0.741741 0.712178 0.684405 0.658392

6.4803 6.5440 6.6058 6.6649 6.7202

9.4048 9.9798 10.6181 11.3188 12.0767

0.330 0.660 0.990 1.320 1.650

0.962677 0.927017 0.893030 0.860724 0.830101

6.4613 6.5172 6.5721 6.6258 6.6779

T = 370.0 K 7.6512 7.9249 8.2280 8.5624 8.9296

1.980 2.310 2.640 2.970 3.300

0.801155 0.773877 0.748251 0.724258 0.701873

6.7283 6.7768 6.8231 6.8672 6.9087

9.3303 9.7642 10.2295 10.7227 11.2376

0.330 0.660 0.990 1.320 1.650

0.967072 0.935672 0.905810 0.877490 0.850716

6.7681 6.8177 6.8654 6.9112 6.9550

T = 390.0 K 7.9406 8.1843 8.4487 8.7343 9.0412

1.980 2.310 2.640 2.970 3.300

0.825482 0.801783 0.779608 0.758942 0.739770

6.9965 7.0356 7.0722 7.1063 7.1375

9.3684 9.7143 10.0762 10.4500 10.8304

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Speeds of sound and thermodynamic properties of ethane TABLE 6—continued rn /(mol·dm−3 )

Z

CV,m /R

Cp,m /R

0.330 0.660 0.990 1.320 1.650

0.970910 0.943237 0.916989 0.892170 0.868782

7.0756 7.1213 7.1641 7.2044 7.2422

T = 410.0 K 8.2333 8.4536 8.6883 8.9375 9.2006

0.330 0.660 0.990 1.320 1.650

0.974284 0.949898 0.926844 0.905127 0.884750

7.3793 7.4225 7.4626 7.4998 7.5343

0.330 0.660 0.990 1.320 1.650

0.977267 0.955794 0.935583 0.916635 0.898953

7.6721 7.7103 7.7476 7.7833 7.8169

rn /(mol·dm−3 )

Z

CV,m /R

Cp,m /R

1.980 2.310 2.640 2.970 3.300

0.846825 0.826296 0.807189 0.789496 0.773210

7.2777 7.3108 7.3416 7.3699 7.3956

9.4764 9.7628 10.0574 10.3567 10.6563

T = 430.0 K 8.5240 8.7256 8.9376 9.1596 9.3905

1.980 2.310 2.640 2.970 3.300

0.865713 0.848018 0.831665 0.816652 0.802981

7.5661 7.5955 7.6224 7.6471 7.6695

9.6291 9.8733 10.1210 10.3693 10.6151

T = 450.0 K 8.8055 8.9882 9.1806 9.3812 9.5885

1.980 2.310 2.640 2.970 3.300

0.882539 0.867398 0.853534 0.840952 0.829661

7.8480 7.8765 7.9024 7.9258 7.9469

9.8004 10.0150 10.2300 10.4431 10.6515

a In order to permit accurate interpolations, Z, CV,m and Cp,m are each given to more significant figures than strictly justified by the uncertainties claimed.

to increase roughly linearly with density from zero at rn = 0 to 1.5·10−4 (about 0.03 per cent) at rn = 3.3 mol·dm−3. The uncertainty in T0 Z'0 , which we estimate from the precision of the experimental prnT data at T1 and T2 , may reach 22·10−3. Errors in u/u0 are thought to be less than 5·10−5·(u/u0 ), and this will make only a small additional contribution to the uncertainty in Z. However, the heat capacities are sensitive to errors in the absolute speed of sound to which we have assigned an overall uncertainty of 1·10−4·u. The combined effects of these uncertainties propagate into uncertainties not exceeding 20.04 per cent in Z and 20.4 per cent in CV,m and Cp,m . The possibility of obtaining initial conditions from a smooth equation of state, rather than from experimental isotherms, was investigated. The recent equation of state for ethane proposed by Friend and Ely(32) was considered but was found to give almost unchanged results. Although a large amount of prnT data has been reported for ethane, only a few sets have an accuracy comparable to that achieved in the numerical integration of the speed of sound. The present compression factors are compared with those of Douslin and Harrison(25) and Mansoorian et al.(27) in figures 7 and 8. First, it should be noted that the integration procedure reproduced the smooth polynomial approximations Zj used in the initial conditions almost exactly so that the deviations showed in figure 7 at T = 298.15 K and T = 303.15 K corresponded to the scatter of the prnT data. Second, from a total of 122 points considered only four deviate more than 0.06 per cent and the r.m.s.d. was just 0.02 per cent. This situation is certainly satisfactory in the light of the uncertainties reported by Douslin and

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A. F. Estrada-Alexanders and J. P. M. Trusler

FIGURE 7. Fractional deviations DZ/Z of compression factors Z of ethane reported by Douslin and Harrison(25) (open symbols) and by Mansoorian et al.(27) (solid symbols) from the present results at temperatures between 273.15 K and 323.15 K: q, T = 273.15 K; w, T = 298.15 K; r, T = 303.15 K; t, T = 323.15 K.

FIGURE 8. Fractional deviations DZ/Z of the compression factors Z of ethane reported by Douslin and Harrison(25) (open symbols) and by Mansoorian et al.(27) (solid symbols) from the present results at temperatures between 348.15 K and 448.15 K. q, T = 348.15 K; w, T = 373.15 K; r, T = 398.15 K; t, T = 423.15 K; r, T = 448.15 K.

Speeds of sound and thermodynamic properties of ethane

1013

FIGURE 9. Fractional deviations DCV,m /CV,m of the isochoric molar heat capacities of ethane reported by Roder(33) along two isochores; W, rn = 1.58 mol·dm−3; Q, rn = 3.27 mol·dm−3.

Harrison of between 0.02 per cent and 0.3 per cent.(25) Finally, we note that the compression factors of Mansoorian et al.,(27) which have a claimed uncertainty of 0.033 per cent and extend up to rn = 1.5 mol·dm−3, all lie within 20.04 per cent of the present results. As the initial conditions for our integrations came from a different source, this agreement is encouraging. The isochoric heat capacities are compared in figure 9 with values from the literature.(33) However, this comparison is not very sensitive as the uncertainty in directly-measured heat capacity is typically 2 per cent. The observed differences lie between −1.2 per cent and 0.4 per cent. Finally, in figure 10 we compare the isobaric heat capacities with those of Ernst and Hochberg(21) which have claimed uncertainties of between 0.2 to 0.4 per cent. The agreement is within 0.5 per cent in all cases.

7. Conclusions We believe that the speeds of sound and derived thermodynamic properties reported here make a valuable contribution to our knowledge of the thermodynamic properties of ethane. The work highlights once more the high precision of acoustic measurements, but also shows that the questions of sample purity limit somewhat the absolute accuracy that can be achieved. We expect that, at the cost of much greater efforts in purification and analysis, this source of uncertainty could be substantially reduced for light hydrocarbons such as ethane.

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A. F. Estrada-Alexanders and J. P. M. Trusler

FIGURE 10. Fractional deviations DCp,m /Cp,m of the isobaric molar heat capacities of ethane reported by Ernst and Hochberg:(21) Q, T = 303.15 K; R, T = 333.15 K; W, T = 363.15 K; T, T = 393.15 K.

One of us (A.F.E.A.) thanks Consejo Nacional de Ciencia y Tecnologia (CONACyT) for financial support. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

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(Received 4 December 1996; in final form 21 February 1997)

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