Eisenberg, D., Ph.D. thesis, Oxford, 1964. Fox, J. J., Martin, A. E., Proc. Roy. SOC.London A174, 234
(1940). Foz, 0.‘R., Colomina, M., Garcia, J. F., Anales FZs. Qulm. (Madrid) Ser. B, 44, 1055 (1948). Foz, 0. R., Vidal, J. M., Anales FZs. Quim (Madrid) 43, 842 (1 947),. \_I-. Glaeser, R. M., Coulson, C. A., Trans. Faraday SOC.61, 389 (1955). Goff, J. A., Gratch, S., “Heating Piping and Air Conditioning,” No. 2, 125 (1946); No. 11, 118 (1949). Gross, E. F., “Hydrogen Bonding,” D. Hadzi, ed., Pergamon Press, New York, 1959. Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” Wiley, New York, 1954. Hirschfelder. J. 0.. McClure. F. T.. Weeks. I. F.. J . Chem. Pays. io, 201 (1942). Itean, F. C., Glueck, A. R., Svehls, R. A., NASA Tech. Note D-481 (1961). ---, Kestin, J., Wang, H. E., Physica 26, 575 (1961). Keyes, F. G., J . Chem. Phys. 17,923 (1949). Kielich, S., Acta Phys. Polon. 20, 433 (1961). Knoblauch, O., Jakob, M., 2. Tier. Deut. Zng. 1907, 31. Knoblsuch, O.,Mollier, H., 2. Ver. Deut. Zng. 1911, 665. Knoblauch, O.,Winkhaus, A., 2. Ver. Deut. Zng. 1916, 374, 401. Kretschmer, C. B., Wiebe, R., J . Am. Chem. SOC.76, 2569 (1954). Krieger, F. J., Project RAND, Res. Mem. AM-646 (1951). Lambert, J. D., Proc. Roy. SOC.London 206A, 247 (1951). Lambert, J. D., Roberts, G. A. H., Rowlinson, J. S., Wilkinson, V. J., Proc. Roy. SOC.London A196, 113 (1947). Lawley, K. P., Smith, E. B., Trans. Faraday SOC.69,301 (1963). McBride, B. J., Gordon, S., J . Chem. Phys. 36, 2198 (1961). McCullough, J. P., Pennington, R. F., Waddington, G., J . Am. Chem. SOC.74, 4439 (1952). RIargenau, H., Myers, V. W., Phys. Rev. 66, 307 (1944). Mason, E. A., Monchick, L., J . Chem. Phys. 36, 1622 (1962). \
Monchick, L., Mason, E. A., J . Chem. Phys. 36, 1676 (1961). Morokuma, K., Pedersen, R., J. Chem. Phys. 48, 3275 (1968). Moszynski, J. R., J . Heat Transfer 83, 111 (1962). O’Connell, J. P., Prausnitz, J. M., “Thermodynamic and Transort Properties of Extreme Temperatures and Pressures,’’ . Gratch, ed., . 19, ASME, New York, 1965. Osborne, N. S., &imson, H. F., Ginnings, D. C., J. Res. Natl. Bur. Std. 23, 261 (1939). Pimentel, G. C., McClellan, A. L., “The Hydrogen Bond,” W. C. Freeman, San Francisco, 1962. Pople, J. A., Proc. Roy. SOC.London A221, 508 (1954). Rivkin, S. L., Levin, A. Y., Teploenergetika 13, No. 4, 79 (1966). Rowlinson, J. S., Trans. Faraday Soc. 46, 974 (1949); 47, 120
E
(1951).
Saran, A,, Singh, Y., Barua, A. K., J . Phys. SOC.Japan 22, 77 (1967). Schafer. K.. FOR.0. R., Snales FZz. Quinz. 38, 316 (1942); Phys.’ Chem. B62: 299 (1942). Schneider, W. G., J . Chem. Phys. 23, 26 (1955). Shifrin, A,, Teploenergetika 6, No.9, 22 (1959).
Z.
Singh, Y., Deb, S. K., Barua, A. K., J . Chem. Phys. 46, 4036 (1 067) \I.,”’,.
Smith, C. J., Proc. Roy. SOC.London A106, 63 (1924). Stockmayer, W. €I., J . Chem. Phys. 9, 398 (1941). Stogryn, D. E., Hirschfelder, J. O., J . Chem. Phys. 31, 1531, 1545 (1959).
Thiel, M. van, Becker, E. U., Pimentel, G. C., J . Chem. Phys. 27, 486 (1957). Vargaftik, N. B., Zaitseva, L. S., Znzh. Fiz. Zh. Akud. Nauk Belorussk. S S R 6 (5), 3 (1963). T’iktorova, A. C., Zsz. Vyssh. uch. zav. Radio$zika 7, 415, 424 (1964).
Weltner, W., Pitzer, K. S., J . Am. Chem. SOC.73,.2606 (1951). Zimmerman, R., Pimentel, G. D., “Advances in Molecular Spectroscopy,” p. 726, Pergamon, New York, 1962.
Moelwyn-Hughes, E. A., “Physical Chemistry,” Pergamon, London, 1957.
RECEIVED for review January 29, 1968 A C C E P T E D December 5, 1968
INTERMOLECULAR FORCES IN AQUEOUS VAPOR M I X T U R E S M . R I G B Y , ’ J . P. O ’ C O N N E L L , 2 A N D J . M . P R A U S N I T Z Department of Chemical Engineering, University of California, Berkeley, Calif.94720 Experimental data for second virial cross coefficients and for vapor-phase diffusivities of binary aqueous mixtures were reduced with a potential function of the Kihara type. The second component in the aqueous mixtures was argon, nitrogen, methane, or oxygen. A good fit of all the data can be obtained using the following parameters for woter: &/k = 170’ K., u (core-to-core distance) = 2.65 A., and 20 (core diameter) = 0.265 A. These parameters are consistent with those obtained from a study of vapor-phase properties of pure water.
OLECULAR interpretation of equilibrium and transport properties of water vapor at low and moderate pressures requires a reasonable potential function for water-water interactions. The large dipole moment of water and the tendency of water to dimerize by hydrogen bonding complicate the analysis of experimental data. I n an earlier paper (O’Connell and Prausnitz, 1969) an attempt was made to interpret the available data using a Stockmayer-Kihara potential,
r > 2a =
QD,
r
(1)
7 2a and all e, cp
Present address, Queen Elizabeth College, London, England. Present address, University of Florida, Gainesville, Fla. 460
l&EC
FUNDAMENTALS
+
where u 2a is the center-to-center separation for which the angle-independent potential is zero, U,is the maximum energy of attraction due to nonpolar forces, and 2a is the diameter of the spherical, hard core (a* = 2 a / u ) . The final term represents the dipole-dipole interaction, where f(0, c p ) is a known function of the angles between the dipole vectors. For water dimer, an open-rhain linear model was assumed (O’Connell and Prausnitz, 1969) and it was possible to determine the range of possible parameters for the Kihara potential which was compatible with various values for the heat of formation of the dimer. It was concluded that the value of U,/k is less than 300°K., and that the distance parameter, u 2a, is between 2.68 and 3.00 A. I n a n attempt to determine these values more precisely, recently obtained experimental data leading to information about the interactions of water with simple nonpolar molecules were studied. Interpretation of these interactions is
+
much simplified by the absence of the large angle-dependent dipoledipole terms which dominate the properties of pure water. The general form of the (orientation-averaged) potential between a molecule of water (molecule l ) and a molecule of a nonpolar gas (molecule 2 ) of mean polarizability a2 is assumed to hare the form
where the characteristic parameter9 now relate to the 1-2 interaction and g(&) is a knoirn function of orientation (Buckingham and Pople, 1955). These parameters may be related to those of the 1-1 and 2-2 interactions by the combining rules a12 = (a11 a22)/2
certainty in the experimental solubility data is small and leads to errors in the virial cross coefficients of probably no more than k 5 ml. per mole. Unfortunately, with the exception of the methane-water data, the available temperature range is rather narrow, limiting the precision with which the characteristic parameters for the interaction may be determined. The second virial coefficient of a system of molecules interacting via a potential of the form given in Equation 2 may be written as a sum of two terms: the second virial coefficient calculated on the basis of the Kihara potential (the term in square brackets in Equation 2), for which wellestablished expressions are available (Kihara, 1953) ; and the contribution due to the induction energy (the final term in Equation 2 ) . The angle-independent induction term was taken into account by writing the intermolecular potential
+
c12 =
(u11+ u22)/2
(3 1
The first of these rules is exact, the second is probably of high accuracy, and the third is known to be an approximation which generally leads to calculated values of UOLz exceeding the correct values by a few per cent (Hudson and RkCoubry, 1960). In view of the unavoidable uncertainties in this work these rules are probably sufficiently precise for the purpose. Parameters have been characterized for the 1-2 interactions from data for the second virial cross coefficient, B12,and from data for the low-pressure binary diffusion coefficient, D12. Knowing the parameters for the nonpolar substance, it is possible to determine the Kihara parameters for the water interaction, using the rules given above. The uncertainties inherent in this procedure are large for any sifigle case, but by studying jointly experimental data for two different properties and for a variety of systems, it is possible to determine the parameters for water within fairly close limits. Second Virial Cross Coefficients
Experimental second virial cross coefficients may be determined from measurements of B,, second virial coefficients of binary mixtures of known composition, using the relation
where y1 and y~ are the mole fractions of components 1 and 2, whose second virial coefficients are B11 and B22,respectively. However, this approach is not suitable a t normal temperatures, when one component of the mixture is water; in that event, the partial pressure of water must be maintained below its saturation pressure and the available pressure range is severely limited. In addition, this method requires accurate values of the second virial coefficients of the pure components, and these are not established with high accuracy for water. .4n alternative source of second virial cross coefficients is provided by measurement of the solubility of a liquid in a compressed gas (Prausnitz and Benson, 1959). Such measurements can be used to obtain the fugacity coefficient of the liquid component in the gaseous phase, from which the second virial crash coefficient may be accurately calculated. The experimental virial coefficients used here have been derived from such measurements for water and the nonpolar gases, argon, nitrogen, methane, and ethane (Olds et al., 1942; Reamer et al., 1943; Rigby and Prausnitz, 1968). The un-
where the induction term has been included by the modification of the parameters U,,,, u12. For the case of zero-core size (Lennard-Jones 12-6 model) , the modifications are well established (Hirschfelder et al., 1954), but for the Kihara potential the choice is not unique. The expressions
have been used. This complicated function of distance parameters was chosen because the covolume used for the second virial coefficients is proportional to qZ3, and when 2a12 is significant the induction term would be too small if (u12 were used in place of ulZ3in the denominator of Equations 5 and 6. h'ormally, the difference would be negligible, but since the parameters for water are small, the induction contributions are unusually large. For instance, for the methane-water system a t 75' C., the calculated spherically symmetric second virial coefficient using unmodified parameters is - 17.2 cc. per gram mole, while the modified parameters give -32.4 cc. per gram mole. In addition, the angledependent induction terms contribute - 6.4 cc. per gram mole. Differences in calculated values are much larger than the experimental uncertainty, depending on whether or not the core is included in the distance parameter. The contributions from the angledependent induction term were obtained from the expansions of Buckingham and Pople (1955). These were not negligible, and it appears that even higher order terms are important a t low temperatures for the smaller nonpolar substances.
+
Diffusion Coefficients
Experimental values of the diffusion coefficient, 0 1 2 , for water vapor in other gases may be obtained in a number of ways. The Stefan tube has been used (Gillespie, 1967; O'Connell, 1967) to obtain data for the water-nitrogen and water-argon systems a t 1 atm. in the temperature range 282' to 373'K., with an estimated uncertainty of around &2%. Other data for water-nitrogen have been obtained using a variety of experimental methods (Crider, 1956; Nelson, 1956; Schwerz and Brown, 1951). Data for the water-oxygen system have been reported (Walker and Westenberg, 1960) for the temperature range 310' to 1000° K. using the "point-source" technique. VOL.
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