Ind. Eng. Chem. Res. 1987, 26, 2542-2543
2542
van der Waals, J. D. Ph.D. Thesis, Leiden, 1873. Vimalchand; Donohue, M. D. Ind. Eng. Chem. Fundam. 1985,24, 246. Wang, W. C.; Kramer, E. J.; Sachse, W. H. J. Polym. Sci. 1982,20, 1371. Weisphart, J. Thermodynamic equilibria of boiling mixtures; Springer-Verlag: Berlin, 1975.
Zeman, L.; Biros, J.; Delma, G.; Patterson, D. J . Phys. Chem. 1972, 76, 1206. Zeman, L.; Patterson, D. J. Chem. Phys. 1972, 76, 1214. Received for review May 4, 1987 Revised manuscript received August 4, 1987 Accepted September 1, 1987
COMMUNICATIONS Correlation of Second Virial Coefficient with Acentric Factor and Temperature A procedure of stepwise regression referred to as double selection is designed to select the terms of the same power of 1/T, in both B(O)and B(') of the Pitzer relation, B = B(O)+ B(')w,correlating the reduced second virial coefficient B with the acentric factor w. The exDerimenta1 data used in the correlation were carefully selected. Significant improvement in accuracy over the previous correlations in the literature is shown. Linear relation of the reduced second virial coefficient B of nonpolar fluids with the acentric factor w was proposed by Pitzer and Curl (1957): B = BCO) + B(1)" (1) where BCo)and B(l)are functions of reduced temperature TI. Polynomials of powers of l/TI are suitable: B(O) = Boo+ Bol/TI Bo2/T,2 + ... (2)
B(') = B b
+ B',/T,
+ + B5/T12 + ...
(3)
The concrete forms of the functions of Pitzer and Curl are B(O)= 0.1445 - 0.330/T1 - 0.1385/T12 - O.O121/T: (4) B(l' = 0.073 + 0.46/T1 - 0.50/T: O.O97/T:
- 0.0073/T,8
(5)
Since then a great number of experimental data have been reported, readjustment of the correlation being thus necessary. Tsonopoulos (1974) modified the equations into
B(O)= 0.1445 - 0.330/T1 - 0.1385/T: O.O121/T: - O.O00607/T,S (6) B(') = 0.0637 + O.331/TI2 - 0.423/T: - 0.O08/TI8 (7) However, data of only a few number of fluids were involved in his modification. Moreover, the modification is rather limited. There is only an extra term of 1/T: in eq 6 as compared with eq 4, the coefficients of the terms in eq 4 kept unchanged. No new selection of terms appears in eq 7 as compared with eq 5 except deleting the l/TI term. It will be valuable to select least terms in eq 2 and 3 to get the best accuracy, i.e., least residual sum of squares. In ordinary cases, the stepwise regression is a suitable method for selecting the most significant terms. A quantity F , known as the variance ratio, is a measure of the significance: F = ( N - m - l)(Qh-l) - Q(m))/Q(m) (8) where N is the total number of data, Q is the residual s u m of squares, Q(") is the Q value after introducing m terms, Os8a-5as5/87/2626-2542$01.50/0
and Q(m-l)is that after deleting a specified term from the m terms. A critical value F, is prescribed. The term with the greatest F greater than F, is introduced, while the term with the smallest F less than F, is deleted. Our purpose is to select terms in the following function resulting from substitution of eq 2 and 3 into eq 1: B(O)= Boo+ Bol/TI + Bo2/T,2 + ... + Bbw + B',o/T, + B'p/T12 + ... (9) However, we want terms with the same power of l / T r in both B(O)and B(l). A special procedure referred to as double selection is designed for this purpose. In the double selection procedure, terms are grouped into pairs, and a pair of terms is introduced or deleted simultaneously. The measure of significance of a pair of terms should be F = ( N - m - 1)(Q(m-2)- Q("))/2Q(") (10) rather than eq 8. Suitable algorithm for conveniently calculating the variance ratio according to eq 10 has been designed. In our task, the term Bbo in eq 9 is selected unconditionally, as it is paired with the constant term Bt0which is always present in the resulting function of stepwise regression. The terms with the same power of l/TI are paired and selected by the double selection procedure. Selection of experimental data will be important to the development of correlation. Dymond and Smith (1980) collected a great number of data of second virial coefficients up to early 1979 with critical comments. We can see from those data that the deviations between different origins often exceed 10%. The deviations are especially great at low temperature. In our regression, only the recommended or the newest sets of data of n-alkanes and argon in Dymond and Smith's book are adopted. The data cover the range of T, from 0.527 to 6.631 and the range of w up to about 0.4. With regard to the less precision and greater magnitude of the lower temperature data, the residual sum of squares of the relative deviation will be a much better indicator of goodness than the actual deviation. In the vicinity of the Boyle temperature where B = 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 2543 Table I. Results of Regression of the Reduced Second Virial Coefficients T , range rms dev, % substance no. of pts 1.23 0.577-3.148 methane 15 2.14 0.655-1.965 ethane 15 1.48 15 0.649-1.487 propane 2.40 0.588-1.317 butane 17 2.38 12 0.639-1.171 pentane 2.38 12 0.591-1.380 hexane 1.00 0.555-1.296 heptane 12 1.54 12 octane 0.527-1.231 1.41 16 argon 0.537-6.631 ~
Table 11. Comparison between Tsonopoulos' and Our Eauation for the Data Not Included in the Regression rms dev, % substance no. of pts T , range Tsonopoulos this work 1.41 10 0.692-1.268 0.70 neopentane 0.26 9 0.850-1.593 1.05 ethylene 0.78 0.85 propylene 8 0.767-1.370 2.82 2.78 isopentane 7 0.608-0.977 7.18 3.27 benzene 13 0.516-1.067 cyclohexane 13 0.542-1.012 6.24 4.36 4.57 2.79 total 60
0, the relative deviation is really unsuitable. Three data (methane at 500 K and argon at 400 and 500 K) are thus excluded in the regression. The number of the included data amounts to 126. Data of critical temperature, critical pressure, and acentric factor are cited from the book of Reid et al. (1977). The terms of eq 9 up to 1/TIl0were regarded as candidates to be selected by the stepwise regression with double selection. The value of F,was set to be 10. The terms of l/Tr, 1/T:, and 1/T: were introduced with high significance (F > 100). The resulting equations are B(O' = 0.1372 - 0.3240/T1 - O.l108/T? - 0.0340/T: (11)
B(l)= 0.9586 - 2.9924/Tr
+ 3.5238/T:
- 1.5477/T:
(12) The detailed deviations for the 126 data are listed in Table I. The root-mean-square (rms) deviation amounts to 1.83%. The root-mean-square deviation of eq 4 and 5 for the same 126 data is 3.79%, while that of eq 6 and 7 is 3.35%. The accuracy of eq 11 and 1 2 is further checked by the data of the rest (not included in the regression) of the nonpolar fluids recommended in Dymond and Smith's book. The comparison with Tsonopoulos' eq 6 and 7 is made in Table 11. The root-mean-square deviation amounts to 2.79% as compared with 4.51% of the latter. The improvement in accuracy is significant, although there are fewer empirical constants in eq 11 and 1 2 than in eq 6 and 7. It is interesting that the term 1/T> or other terms of high power were not introduced during the stepwise regression. The terms of high power are sensitive to low temperature. However, the low-temperature data are less confident, as they show large difference between different
6
i /7 t
1
d
/
-1.01
i4
C t
-
IA Tr
Figure 1. Reduced virial coefficients of heptane: (-) calculated, this work; (--) calculated by Tsonopoulos' equation; (0) Al-Bizreh and Wormald (1978); ( 0 )Hirschfelder et al. (1942); (v)McGlashan and Potter (1962); (A)Belousova and Zaalishvilie (1967); (A)Hajjar e t al. (1969).
sources. Tsonopoulos (1974) illustrated the role of 1/T> by the data of benzene as an example. Just in this very example, the differences between different sources exceed 20% as shown in Figure 2 of his paper, so the importance of the high power term is unclear. In our experience, high power terms were often introduced in the stepwise regression for a set of data of a fluid, but not for data including a series of fluids. This might be due to the error at low temperature. Thus, several sets of data of heptane from different sources are plotted in Figure 1. We can see that the points of lower temperature in each older set deviate from the newest data to the lower side. If we use an older set in a stepwise regression, a high power term will be introduced to fit the points of low temperature.
Literature Cited AI-Bizreh, N.; Wormald, C. J. J. Chem. Thermodyn. 1978,10,231. Belousova, Z. S.; Zaalishvilie, Sh. D. RUSS.J. Phys. Chem. 1967, 41, 1290; Zh. Fiz. Khim. 1967,41, 2388. Dymond, J. H.; Smith, E. B. The Virial Coefficients of Pure Gases and Mixtures; Clarendon: Oxford, 1980; pp 1-183. Hajjar, R. F.; Kay, W. B.; Leverett, G. G. J. Chem. Eng. Data 1969, 14, 377.
Hirschfelder, J. 0.;McClure, F. T.; Weeks, I. F. J.Chem. Phys. 1942, 10, 201.
McGlashan, M. L.; Potter, D. J. B. Proc. R. SOC.London, Ser. A 1962, 267, 478.
Pitzer, K. S.; Curl, R. F. J. Am. Chem. SOC.1957, 79, 2369. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977; Appendix A. pp 631-657. Tsonopoulos, C. AZChE J. 1974,20, 263.
Chen Zhixing,* Yun Fengcun, Wu Yiqin Chemistry Department Zhongshan University Guangzhou, China Received for review September 5, 1984 Revised manuscript received July 20, 1987 Accepted September 21, 1987
