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Excess and unlike-Interaction second virial coefficients and excess molar enthalpy of (0.500CO + 0.500CO2)
M-3068 J. Chem. Thermodynamics 1995, 27, 267–271
Excess and unlike-interaction second virial coefficients and excess molar enthalpy of (0.500CO + 0.500CO2 ) Peter J. McElroy a Chemical & Process Engineering Department, University of Canterbury, Christchurch, New Zealand
and Josef Moser Faculta¨t fu¨r Chemische Verfahremstechnik, Universita¨t Karlsruhe, Germany
(Received 9 July 1994; in revised form 22 August 1994) The excess second virial coefficient of (0.500CO + 0.500CO2 )(g) has been determined at temperatures 303.15 K, 313.14 K, 323.15 K, 333.15 K, and 343.15 K. The pressure-change-ofmixing method was used. The excess molar enthalpy was derived as well as unlike-interaction second virial coefficients which were compared with predictive equations.
1. Introduction Carbon monoxide and carbon dioxide are often significant components of natural gas and of ‘‘coke-oven gas’’. For custody transfer and general design calculations accurate (p, V, T) information is required. This work is part of an ongoing study of gas mixtures of industrial importance.(1, 2) Measurement of the pressure change on mixing of (0.5A + 0.5B)(g), initially at equal pressures p, allows calculation of the excess second virial coefficients o:(3) o = 2RTDp/{p 2 (1 + Dp/p)}.
(1)
The Dp is the pressure change which occurs when equal amounts of substance of the two gases A and B initially both at pressure p are mixed at constant temperature and constant total volume. Since the excess second virial coefficient relates to the unlike-interaction second virial coefficient B12 as o = B12 − (B11 + B22 )/2,
(2)
B12 values may be estimated when B11 and B22 are known. a
To whom correspondence should be sent.
0021–9614/95/030267 + 05 $08.00/0
7 1995 Academic Press Limited
268
P. J. McElroy and J. Moser
Knoester et al.(4) have shown that at pressures where the virial equation truncated after the second coefficient is applicable, the excess molar enthalpy of HmE relates to the excess second virial coefficient by HmE = 2y(1 − y)p(o − T ·do/dT),
(3)
Where y is the mole fraction of A in the mixture. In this work then pressure change on mixing at five temperatures has been measured so that o and do/dT and subsequently B12 and HmE can be determined.
2. Experimental The basic apparatus and recent modifications have been described previously(1, 5) and the reported procedure has been followed here. The pressure change on mixing was measured on a differential pressure gauge (MKS Instruments Inc, Baratron model 315 BD Type 270B) with a claimed accuracy of 0.25 per cent of full scale. Pressure was measured using a 100 kPa piezoelectric-element pressure gauge (Digiquartz model 1015 A-01) with a claimed accuracy of 0.01 per cent of full scale and which was calibrated against a mercury-in-glass manometer. Temperature was measured with an Automatic Systems Laboratory resistance bridge (Model F26) and a platinum resistance thermometer with 20.005 K accuracy. Temperatures were cross-checked with a Model RT200 Reference Thermometer developed by the New Zealand Department of Science and Industrial Research. The thermometers had been calibrated using the IPTS-68 and were converted to ITS-90 using standard tables. The carbon monoxide was ‘‘ultra-high-purity’’ grade supplied by Matheson gas products and certified with a minimum purity of 99.8 moles per cent of CO, N2 being the major impurity. The carbon dioxide was supplied by New Zealand Industrial Gases with purity determined as 99.98 moles per cent of CO2 using gas chromatography as described previously.(1)
3. Results and discussion The measured values of temperature T, pressure change Dp on mixing, and the derived excess second virial coefficient o, are listed in table 1. An equation quadratic in reciprocal temperature fitted to the measurements to give least-square deviations has the form: o/(cm3 ·mol−1 ) = −37.875 + 21021/(T/K) − 1383380/(T/K)2.
(4)
Using this equation and equation (3) values of {HmE /2py(1 − y)} and of HmE can be generated and are listed in table 1. The values at 303.15 K are of a similar magnitude to those reported by Martin et al.(6) for {0.500CO2 + 0.500(Ar or N2 or O2 or CH4 )}, their values ranging from (71 to 82) cm3 ·mol−1 at T = 300 K. To determine unlike-interaction second virial coefficients using equation (2) pure-component B values must be selected from previously published values. Jaeschke et al.(7) (referred to as the GERG group) have reviewed the published measurements on carbon dioxide and concluded that the studies of Michels and
269
B12 and HmE for (0.500CO + 0.500CO2 )(g)
TABLE 1. Excess second virial coefficients and excess molar enthalpies of (0.500CO + 0.500CO2 )(g) T K
p kPa
Dp Pa
o12 cm3 ·mol−1
2HmE /p cm3 ·mol−1
HmE (p = 104 kPa) J·mol−1
B12 cm3 ·mol−1
303.15 303.16 313.14 313.14 323.15 323.14 333.15 333.15 343.15 343.15
103.937 105.071 103.639 103.554 103.734 103.951 103.901 103.896 103.958 102.514
35.12 35.53 32.23 31.53 27.44 27.68 24.69 24.75 21.96 21.80
16.38 2 0.5 16.22 2 0.5 15.62 2 0.5 15.31 2 0.5 13.70 2 0.5 13.76 2 0.5 12.67 2 0.5 12.70 2 0.5 11.60 2 0.5 11.83 2 0.5
55.6 2 1.0 55.5 2 1.0 54.5 2 1.0 54.2 2 1.0 52.3 2 1.0 52.3 2 1.0 50.8 2 1.0 50.9 2 1.0 49.4 2 1.0 49.6 2 1.0
2.89 2 0.05 2.91 2 0.05 2.83 2 0.05 2.81 2 0.05 2.71 2 0.05 2.72 2 0.05 2.64 2 0.05 2.64 2 0.05 2.57 2 0.05 2.54 2 0.05
−46.9 2 3 −47.0 2 3 −42.3 2 3 −42.6 2 3 −39.5 2 3 −38.9 2 3 −36.4 2 3 −36.4 2 3 −33.9 2 3 −33.7 2 3
Michels(8) and of Michels et al.(9) represented the most reliable and extensive measurements. Brugge et al.(10) compared their two more recent measurements (at 300 K and 320 K) with earlier work and demonstrated excellent agreement with Michels and Michels.(8) Jaeschke et al.(7) reviewed the reported measurements for CO2 and generated the best-fit equation: B(CO2 )/(cm3 ·mol−1 ) = −86.834 + 4.0376·T/K − 5.1657·10−3 ·(T/K)2.
(5)
Holste et al.(11) also reported a critical study of the virial coefficients of CO2 and generated a best-fit equation with the form: B(CO2 )/(cm3 ·mol−1 ) = 23.02991 − 2.455297·103/(T/K) − 1.226275·107/(T/K)2. (6) The GERG equation is in excellent agreement with this and so has been used in this work to derive B12 values. The second-virial-coefficient measurements on CO were reviewed by Goodwin.(12) Jaeschke et al.(7) who also briefly reviewed the reported experimental measurements, concluded that the 1952 measurements of Michels et al.(9) were the best available and so they generated a fitting equation for those results. In this temperature range their equation is in excellent agreement with the Goodwin equation which has the form: B(CO)/(cm3 ·mol−1 ) = 55.9560 − 201.6749/(T/K) + 66.554·104/(T/K)2 − 109.7726·106/(T/K)3,
(7)
and so we have employed the Goodwin equation to generate the B(CO) values needed for B12 estimation. The unlike-interaction B12 values generated using equations (2), (6) and (7) are listed in table 1, and B12 values along with the only literature values at similar temperatures are plotted in figure 1. The Tsonopoulos(13) modification of the Pitzer and Curl(14) equation has been reproduced often and widely used to predict second virial coefficients. In common with many reduced equations of state when applied to unlike-interaction terms it uses the mixing rules:
270
P. J. McElroy and J. Moser
FIGURE 1. Unlike-interaction second virial coefficient B12 for CO–CO2 : W, this work; w, Cottrell et al.;(19) q, Menon;(17) r, Brewer;(18) – – , Tsonopoulos correlation;(13) - - - - , Mak and Lielmezs correlation;(15) ——, B12 from equations (4), (5), and (7).
Tc,12 = (Tc,1Tc,2 )1/2 ·(1 − k12 ),
(8)
1/3 1/3 3 pc,12 = 4·Tc,12 {(pc,1Vc,1 /Tc,1 ) + (pc,2Vc,2 /Tc,2 )}/(Vc,1 + Vc,2 ),
(9)
and v12 = 0.5·(v1 + v2 ),
(10)
where v is Pitzer’s acentric factor and the subscript c indicates a critical value. The coefficient k12 is an association parameter, tabulated values of which can be found, but many such values have been obtained by fitting the equation to measured virial coefficients. Its value for (carbon dioxide + carbon monoxide) is expected to be small and so in figure 1 the Tsonopoulos equation(13) is plotted with k12 = 0. Evidently the relation predicts B12 values approximately 7 cm3 ·mol−1 too positive. Mak and Lielmezs(15) generated an alternative reduced equation to predict second virial coefficients with the form: Bpc /RTc = 0.0778 − 0.45724·A/Tr ,
(11)
A(Tr , v) = A0 (Tr ) + v ·A1 (Tr ),
(12)
A0 (Tr ) = −1.4524905 + 14.360017/Tr−45.000285/Tr2 + 78.907097/Tr3 − 79.449258/Tr4 + 45.841959/Tr5 − 14.078304/Tr6 + 1.7835426/Tr7 ,
(13)
A1 (Tr ) = −4.3816022 + 15.205023/Tr − 20.874489/Tr2 + 12.697209/Tr3 − 2.5851848/Tr4 , Tr = T/Tc .
(14) (15)
This equation is readily extendable to mixtures using equations (9) to (11) just as for the Tsonopoulos equation. If k12 is again set to zero the fit to B12 as indicated by the plot in figure 1 is almost identical to the Tsonopoulos equation. The Mak and Lielmezs equation might be criticised in that it is derived from the Peng and Robinson(16) equation which would not be expected accurately to represent gas-phase
B12 and HmE for (0.500CO + 0.500CO2 )(g)
271
behaviour. The Peng and Robinson equation, however, has merely given form to the equation for B and the parameters are then determined by best fit to a range of systems. Both the Tsonopoulos and the Mak and Lielmezs correlations can be made to fit the experimental observations well if a k12 value of −0.03 is employed. In the published literature measurements of B12 for (CO–CO2) have been reported by Menon,(17) Brewer,(18) and by Cottrell et al.(19) The values are plotted in figure 1 and extrapolation of this work indicates good agreement with the measurement of Brewer, agreement within experimental error with the measurement of Menon, and with the higher-temperature measurements of Cottrell et al. Their measurement at 303.15 K deviates considerably but this is probably indicative of the experimental uncertainty. REFERENCES 1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
McElroy, P. J.; Battino, R; Dowd, M. K. J. Chem. Thermodynamics 1990, 22, 505. McElroy, P. J.; Ji, F.; Ababio, B. D. J. Chem. Eng. Data 1993, 38, 410. Knobler, C. M. Rev. Sci. Instr. 1967, 38, 184. Knoester, M.; Taconis, K. W.; Beenakker, J. J. M. Physica 1967, 33, 389. McElroy, P. J. J. Chem. Thermodynamics 1994, 26,00. Martin, M. L.; Trengrove, R. D.; Harris, K. R.; Dunlop, P. J. Aust. J. Chem. 1982, 35, 1525. Jaeschke, M.; Audibert, S.; Canegham, P. van; Humphreys, A. E.; Janssen Rosmalen, R. van; Pellei, Q.; Michels, J. P. J.; Schouten, J. A.; Seldam, C. A. High Accuracy Compressibility Factor Calculation for Natural Gases and Similar Mixtures by Use of a Truncated Virial Equation. GERG Technical Monograph 2, 1988. Michels, A.; Michels, C. Proc. Roy. Soc. 1036, A153, 201. Michels, A.; Michels, C.; Wouters, H. Proc. Roy. Soc. 1936, A153, 214. Brugge, H. B.; Hwang, C. A.; Rogers, W. I.; Holste, J. C.; Hall, K. R.; Lemming, W; Esper, G. J.; Marsh, K. N.; Gammon, B. E. Physica 1989, A156, 382. Holste, J. C.; Hall, K. R.; Eubank, P. T.; Esper, G.; Watson, M. Q.; Warony, W.; Bailey, D. M.; Young, J. G.; Bellamy, M. T. J. Chem. Thermodynamics 1987, 19, 1233. Goodwin, R. D. Cryogenics 1983, 23, 403. Tsonopoulos, C. A.I.Ch.E.J. 1974, 20, 263. Pitzer, K. S.; Curl, R. F. J. J. Am. Chem. Soc. 1957, 79, 2369. Mak, P. C.; Lielmezs, J. J. Ind. Eng. Chem. Res. 1989, 28, 127. Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59. Menon, P. G. Ind. J. Pure Appl. Phys. 1965, 3, 334. Brewer, J. Contract No AF 49/638/-1620 AFOSR No. 67-2795 Arlington, Virginia, U.S.A. Cottrell, T. L.; Hamilton, R. A.; Taubinger, R. P. Trans. Faraday Soc. 1956, 52, 1310.
Excess and unlike-interaction second virial coefficients and excess molar enthalpy of (0.500CO + 0.500CO2 ) Peter J. McElroy a Chemical & Process Engineering Department, University of Canterbury, Christchurch, New Zealand
and Josef Moser Faculta¨t fu¨r Chemische Verfahremstechnik, Universita¨t Karlsruhe, Germany
(Received 9 July 1994; in revised form 22 August 1994) The excess second virial coefficient of (0.500CO + 0.500CO2 )(g) has been determined at temperatures 303.15 K, 313.14 K, 323.15 K, 333.15 K, and 343.15 K. The pressure-change-ofmixing method was used. The excess molar enthalpy was derived as well as unlike-interaction second virial coefficients which were compared with predictive equations.
1. Introduction Carbon monoxide and carbon dioxide are often significant components of natural gas and of ‘‘coke-oven gas’’. For custody transfer and general design calculations accurate (p, V, T) information is required. This work is part of an ongoing study of gas mixtures of industrial importance.(1, 2) Measurement of the pressure change on mixing of (0.5A + 0.5B)(g), initially at equal pressures p, allows calculation of the excess second virial coefficients o:(3) o = 2RTDp/{p 2 (1 + Dp/p)}.
(1)
The Dp is the pressure change which occurs when equal amounts of substance of the two gases A and B initially both at pressure p are mixed at constant temperature and constant total volume. Since the excess second virial coefficient relates to the unlike-interaction second virial coefficient B12 as o = B12 − (B11 + B22 )/2,
(2)
B12 values may be estimated when B11 and B22 are known. a
To whom correspondence should be sent.
0021–9614/95/030267 + 05 $08.00/0
7 1995 Academic Press Limited
268
P. J. McElroy and J. Moser
Knoester et al.(4) have shown that at pressures where the virial equation truncated after the second coefficient is applicable, the excess molar enthalpy of HmE relates to the excess second virial coefficient by HmE = 2y(1 − y)p(o − T ·do/dT),
(3)
Where y is the mole fraction of A in the mixture. In this work then pressure change on mixing at five temperatures has been measured so that o and do/dT and subsequently B12 and HmE can be determined.
2. Experimental The basic apparatus and recent modifications have been described previously(1, 5) and the reported procedure has been followed here. The pressure change on mixing was measured on a differential pressure gauge (MKS Instruments Inc, Baratron model 315 BD Type 270B) with a claimed accuracy of 0.25 per cent of full scale. Pressure was measured using a 100 kPa piezoelectric-element pressure gauge (Digiquartz model 1015 A-01) with a claimed accuracy of 0.01 per cent of full scale and which was calibrated against a mercury-in-glass manometer. Temperature was measured with an Automatic Systems Laboratory resistance bridge (Model F26) and a platinum resistance thermometer with 20.005 K accuracy. Temperatures were cross-checked with a Model RT200 Reference Thermometer developed by the New Zealand Department of Science and Industrial Research. The thermometers had been calibrated using the IPTS-68 and were converted to ITS-90 using standard tables. The carbon monoxide was ‘‘ultra-high-purity’’ grade supplied by Matheson gas products and certified with a minimum purity of 99.8 moles per cent of CO, N2 being the major impurity. The carbon dioxide was supplied by New Zealand Industrial Gases with purity determined as 99.98 moles per cent of CO2 using gas chromatography as described previously.(1)
3. Results and discussion The measured values of temperature T, pressure change Dp on mixing, and the derived excess second virial coefficient o, are listed in table 1. An equation quadratic in reciprocal temperature fitted to the measurements to give least-square deviations has the form: o/(cm3 ·mol−1 ) = −37.875 + 21021/(T/K) − 1383380/(T/K)2.
(4)
Using this equation and equation (3) values of {HmE /2py(1 − y)} and of HmE can be generated and are listed in table 1. The values at 303.15 K are of a similar magnitude to those reported by Martin et al.(6) for {0.500CO2 + 0.500(Ar or N2 or O2 or CH4 )}, their values ranging from (71 to 82) cm3 ·mol−1 at T = 300 K. To determine unlike-interaction second virial coefficients using equation (2) pure-component B values must be selected from previously published values. Jaeschke et al.(7) (referred to as the GERG group) have reviewed the published measurements on carbon dioxide and concluded that the studies of Michels and
269
B12 and HmE for (0.500CO + 0.500CO2 )(g)
TABLE 1. Excess second virial coefficients and excess molar enthalpies of (0.500CO + 0.500CO2 )(g) T K
p kPa
Dp Pa
o12 cm3 ·mol−1
2HmE /p cm3 ·mol−1
HmE (p = 104 kPa) J·mol−1
B12 cm3 ·mol−1
303.15 303.16 313.14 313.14 323.15 323.14 333.15 333.15 343.15 343.15
103.937 105.071 103.639 103.554 103.734 103.951 103.901 103.896 103.958 102.514
35.12 35.53 32.23 31.53 27.44 27.68 24.69 24.75 21.96 21.80
16.38 2 0.5 16.22 2 0.5 15.62 2 0.5 15.31 2 0.5 13.70 2 0.5 13.76 2 0.5 12.67 2 0.5 12.70 2 0.5 11.60 2 0.5 11.83 2 0.5
55.6 2 1.0 55.5 2 1.0 54.5 2 1.0 54.2 2 1.0 52.3 2 1.0 52.3 2 1.0 50.8 2 1.0 50.9 2 1.0 49.4 2 1.0 49.6 2 1.0
2.89 2 0.05 2.91 2 0.05 2.83 2 0.05 2.81 2 0.05 2.71 2 0.05 2.72 2 0.05 2.64 2 0.05 2.64 2 0.05 2.57 2 0.05 2.54 2 0.05
−46.9 2 3 −47.0 2 3 −42.3 2 3 −42.6 2 3 −39.5 2 3 −38.9 2 3 −36.4 2 3 −36.4 2 3 −33.9 2 3 −33.7 2 3
Michels(8) and of Michels et al.(9) represented the most reliable and extensive measurements. Brugge et al.(10) compared their two more recent measurements (at 300 K and 320 K) with earlier work and demonstrated excellent agreement with Michels and Michels.(8) Jaeschke et al.(7) reviewed the reported measurements for CO2 and generated the best-fit equation: B(CO2 )/(cm3 ·mol−1 ) = −86.834 + 4.0376·T/K − 5.1657·10−3 ·(T/K)2.
(5)
Holste et al.(11) also reported a critical study of the virial coefficients of CO2 and generated a best-fit equation with the form: B(CO2 )/(cm3 ·mol−1 ) = 23.02991 − 2.455297·103/(T/K) − 1.226275·107/(T/K)2. (6) The GERG equation is in excellent agreement with this and so has been used in this work to derive B12 values. The second-virial-coefficient measurements on CO were reviewed by Goodwin.(12) Jaeschke et al.(7) who also briefly reviewed the reported experimental measurements, concluded that the 1952 measurements of Michels et al.(9) were the best available and so they generated a fitting equation for those results. In this temperature range their equation is in excellent agreement with the Goodwin equation which has the form: B(CO)/(cm3 ·mol−1 ) = 55.9560 − 201.6749/(T/K) + 66.554·104/(T/K)2 − 109.7726·106/(T/K)3,
(7)
and so we have employed the Goodwin equation to generate the B(CO) values needed for B12 estimation. The unlike-interaction B12 values generated using equations (2), (6) and (7) are listed in table 1, and B12 values along with the only literature values at similar temperatures are plotted in figure 1. The Tsonopoulos(13) modification of the Pitzer and Curl(14) equation has been reproduced often and widely used to predict second virial coefficients. In common with many reduced equations of state when applied to unlike-interaction terms it uses the mixing rules:
270
P. J. McElroy and J. Moser
FIGURE 1. Unlike-interaction second virial coefficient B12 for CO–CO2 : W, this work; w, Cottrell et al.;(19) q, Menon;(17) r, Brewer;(18) – – , Tsonopoulos correlation;(13) - - - - , Mak and Lielmezs correlation;(15) ——, B12 from equations (4), (5), and (7).
Tc,12 = (Tc,1Tc,2 )1/2 ·(1 − k12 ),
(8)
1/3 1/3 3 pc,12 = 4·Tc,12 {(pc,1Vc,1 /Tc,1 ) + (pc,2Vc,2 /Tc,2 )}/(Vc,1 + Vc,2 ),
(9)
and v12 = 0.5·(v1 + v2 ),
(10)
where v is Pitzer’s acentric factor and the subscript c indicates a critical value. The coefficient k12 is an association parameter, tabulated values of which can be found, but many such values have been obtained by fitting the equation to measured virial coefficients. Its value for (carbon dioxide + carbon monoxide) is expected to be small and so in figure 1 the Tsonopoulos equation(13) is plotted with k12 = 0. Evidently the relation predicts B12 values approximately 7 cm3 ·mol−1 too positive. Mak and Lielmezs(15) generated an alternative reduced equation to predict second virial coefficients with the form: Bpc /RTc = 0.0778 − 0.45724·A/Tr ,
(11)
A(Tr , v) = A0 (Tr ) + v ·A1 (Tr ),
(12)
A0 (Tr ) = −1.4524905 + 14.360017/Tr−45.000285/Tr2 + 78.907097/Tr3 − 79.449258/Tr4 + 45.841959/Tr5 − 14.078304/Tr6 + 1.7835426/Tr7 ,
(13)
A1 (Tr ) = −4.3816022 + 15.205023/Tr − 20.874489/Tr2 + 12.697209/Tr3 − 2.5851848/Tr4 , Tr = T/Tc .
(14) (15)
This equation is readily extendable to mixtures using equations (9) to (11) just as for the Tsonopoulos equation. If k12 is again set to zero the fit to B12 as indicated by the plot in figure 1 is almost identical to the Tsonopoulos equation. The Mak and Lielmezs equation might be criticised in that it is derived from the Peng and Robinson(16) equation which would not be expected accurately to represent gas-phase
B12 and HmE for (0.500CO + 0.500CO2 )(g)
271
behaviour. The Peng and Robinson equation, however, has merely given form to the equation for B and the parameters are then determined by best fit to a range of systems. Both the Tsonopoulos and the Mak and Lielmezs correlations can be made to fit the experimental observations well if a k12 value of −0.03 is employed. In the published literature measurements of B12 for (CO–CO2) have been reported by Menon,(17) Brewer,(18) and by Cottrell et al.(19) The values are plotted in figure 1 and extrapolation of this work indicates good agreement with the measurement of Brewer, agreement within experimental error with the measurement of Menon, and with the higher-temperature measurements of Cottrell et al. Their measurement at 303.15 K deviates considerably but this is probably indicative of the experimental uncertainty. REFERENCES 1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
McElroy, P. J.; Battino, R; Dowd, M. K. J. Chem. Thermodynamics 1990, 22, 505. McElroy, P. J.; Ji, F.; Ababio, B. D. J. Chem. Eng. Data 1993, 38, 410. Knobler, C. M. Rev. Sci. Instr. 1967, 38, 184. Knoester, M.; Taconis, K. W.; Beenakker, J. J. M. Physica 1967, 33, 389. McElroy, P. J. J. Chem. Thermodynamics 1994, 26,00. Martin, M. L.; Trengrove, R. D.; Harris, K. R.; Dunlop, P. J. Aust. J. Chem. 1982, 35, 1525. Jaeschke, M.; Audibert, S.; Canegham, P. van; Humphreys, A. E.; Janssen Rosmalen, R. van; Pellei, Q.; Michels, J. P. J.; Schouten, J. A.; Seldam, C. A. High Accuracy Compressibility Factor Calculation for Natural Gases and Similar Mixtures by Use of a Truncated Virial Equation. GERG Technical Monograph 2, 1988. Michels, A.; Michels, C. Proc. Roy. Soc. 1036, A153, 201. Michels, A.; Michels, C.; Wouters, H. Proc. Roy. Soc. 1936, A153, 214. Brugge, H. B.; Hwang, C. A.; Rogers, W. I.; Holste, J. C.; Hall, K. R.; Lemming, W; Esper, G. J.; Marsh, K. N.; Gammon, B. E. Physica 1989, A156, 382. Holste, J. C.; Hall, K. R.; Eubank, P. T.; Esper, G.; Watson, M. Q.; Warony, W.; Bailey, D. M.; Young, J. G.; Bellamy, M. T. J. Chem. Thermodynamics 1987, 19, 1233. Goodwin, R. D. Cryogenics 1983, 23, 403. Tsonopoulos, C. A.I.Ch.E.J. 1974, 20, 263. Pitzer, K. S.; Curl, R. F. J. J. Am. Chem. Soc. 1957, 79, 2369. Mak, P. C.; Lielmezs, J. J. Ind. Eng. Chem. Res. 1989, 28, 127. Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59. Menon, P. G. Ind. J. Pure Appl. Phys. 1965, 3, 334. Brewer, J. Contract No AF 49/638/-1620 AFOSR No. 67-2795 Arlington, Virginia, U.S.A. Cottrell, T. L.; Hamilton, R. A.; Taubinger, R. P. Trans. Faraday Soc. 1956, 52, 1310.