(4) du Pont de Nemours & Go., E. I., LYilmington, Del., “Preliminary Properties on Freon 14,” 1961. (5) Eckert, C. A., Ph.D. dissertation, University of California, Berkeley, 1964. (6) Eckert, C. A., Renon, H., Prausnitz, J. M., IND.END.CHEM. FUNDAMENTALS 6, 58 (1967). (7) Eyring, H., Hirschfelder, J., J . Phys. Chem. 41, 249 (1937). (8) Flory, P. J., J . Am. Chem. SOC.87, 1833 (1965). (91 Florv. P. J., Orwoll, R. A., Vrii. A.,Ibid., 86. 3507 (1964). (10) G d y , E. R., Cryogenics 2, 22i(1962). (11) Guggenheim, E. A.,Mol. Phys. 9,43 (1965). (12) Hermsen, R. W,, Prausnitz, J. M., Chem. Eng. Sci. 21,791, 803 (1966). (13) Hijmans, J., Physica 27, 433 (1961). (14) Hildebrand, J. H., Scott, R. L., “Solubility of Nonelectrolvtes.” Reinhold. New York. 1936. (15) Hoge, H. J., J . Res. Natl. Bur. Std. 44, 321 (1950). (16) Holleman, Th., Hijmans, J., Physica 28, 604 (1962). (17) Itterbeek, van A., Staes, K., Verbeke, O., Theeuwes, F. Ibrd., 30, 1896 (1964). (18) Itterbeek, van A . , Verbeke, 0..Ibid., 26, 931 (1960). (19) Kac, M., Uhlenbeck, G. E., Hemmer, P. C., J . Math. Phys. 4, 216 (1963). (20) Keyes. F. G., Taylor, R. S., Smith, L. B., J . Math. Phys. M I T 1, 211 (1922). (21) Knobler, C. M., Van Heijningen, R. J. J., Beenakker, J. J. M., Physica 27, 296 (1961). (22) Longuet-Higgins, H. C., Widom, B., Mol. Phys. 8, 549 (1964). (23) Mathias, E., Crommelin, C. A., Ann. Phys. (Parrs) 5, 137 I
,
(19%) \ - _ - _
(24) MGhias, E., Crommelin, C. A., Bijleveld, W. J., Grigg, Ph. P., Commun. Phys. Lab. Uniu. Leiden 221b (1932). (25) Mathias, E., Crommelin, C. A., Garfit Watt, H., Ibid., 189a (lYL/).
(26) Mathias, E., Crommelin, C. A., Kamerlingh-Onnes, H., Ibid., 162b (1923). (27) Mathias, E., Crommelin, C. A , , Meihuizen, J. J., Ibid., 248b (1 937). (28) Mathias, E., Kamerlingh-Onnes, H., Ibid., 117 (1911). (29) Mathias, E., Kamerlingh-Onnes, H., Crommelin, C. A., Ibid., 145c (1915).
(30) Michels, A., Wassenaar, T., Physica 16, 221 (1950). (31) Ibid., p. 253. (32) Michels, A., Wassenaar, T., de Graaf, W., Prins, Ch. R., Ibid., 19, 26 (1953). (33) Michels, A., Wassenaar, T., Zwietering, Th. N., Ibid., 18, 63 (1952). (34) Ibid., p. 160. (35) Mullins, J. C., Kirk, B. S., Ziegler, W. T., Tech. Rept. 2, Proj. A-663, Engineering Experiment Station, Georgia Inst. Technology, Atlanta, 1963. (36) Mullins; J. C., Ziegler, W.T., Kirk, B. S., Aduan. Cryog. Eng. 8, 126 (1962). (37) Patterson, H. S., Cripps, R. C., Whytlay-Gray, R., Proc. Roy. SOC.(London) A86, 579 (1912). (38) Pitzer, K. S., Lippman, D. Z., Curl, R. F., Jr., Huggins, C. M., Petersen, D. E., J . Am. Chem. SOC. 77, 3433 (1955). (39) Prigogine, I., “The Molecular Theory of Solutions,” Amsterdam. North Holland. 1957. (40) Prigogine, I., Trappeniers, N., Mathot, V., Discussions Faraday SOC.15, 93 (1953). (41) Renon, H.; Ph.D. dissertation, University of California, Berkeley, 1966. (42) Rossini, F. D., “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,” Carnegie Press, Pittsburgh, 1953. (43) Rowlinson, J. S., “Liquids and Liquid Mixtures,” Butterworths, London, 1959. (44) Schmidt, H. H., Oppdycke, J., Clark, R. K., J . Phys. Chem. 68, 2393 (1964). (45) Sherwood, A. E., Prausnitz, J. M., J. Chem. Phys. 41, 429 (1964). 86, 197 (1964). (46) Simha, R., Havlik, A. J., J . Am. Chem. SOC. (47) Wertheim, M. S., J . Chem. Phys. 43,1370 (1965). (48) Ziegler, W. T., Mullins, J. C., Tech. Rept. 1, Proj. A-663, Engineering Experiment Station, Georgia Inst. Technology, Atlanta, 1963. RECEIVED for review February 8, 1966 ACCEPTED August 8, 1966
MOLECULAR THERMODYNAMICS OF SIMPLE LIQUIDS Mixtures C. A. ECKERT’, H E N R l RENONZ, AND J . M. PRAUSNITZ De$artment of Chemical Engineering, University of California, Berkeley, Calif., and Institute for Materials Research, National Bureau of Standards, Boulder, Colo. The analytical partition function for pure, simple liquids is generalized for liquid mixtures containing any desired number of components. The properties of liquid mixtures are calculated from standard statistical mechanical relations on the basis of Scott‘s two-liquid theory coupled with a three-parameter theorem of corresponding states. The effect of three molecular parameters on solution excess function is investigated, and it is shown that these are very sensitive to the characteristic energy for two unlike molecules; in general, this energy is not sufficiently well approximated by the geometric-mean assumption but must b e determined by some mixture property such as the second virial cross coefficient, Biz. Calculated excess Gibbs energies, enthalpies, and volumes agree very well with experimental results for 17 binary systems containing simple, nonpolar molecules.
treatment of the thermodynamics of liquid mixtures contributes to our fundamental understanding of molecular processes in solutions and is of direct use in the design of typical chemical process equipment. The basis for such a treatment must stem from fundamental molecular considerations ; the methods of statistical mechanics can then be used to provide a link between microscopic molecular
A
THEORETICAL
Present address, University of Illinois, Urbana, Ill. On leave of absence from Institut Fransais du Pttrole, RueilMalmaison, France. 1 2
58
I&EC FUNDAMENTALS
properties and the bulk thermodynamic properties required for practical applications. In the present work we discuss the thermodynamic properties of mixtures, and aim to discover the dependence of these properties on various molecular functions. To do this, we extend to mixtures the statistical thermodynamics of pure components, based on a three-parameter theory of corresponding states, as developed in a previous paper (26). As discussed there, we confine our attention to simple, nonpolar molecules whose intermolecular force fields are nearly spherically symmetric.
Partition Function for Mixtures
T h e cell concept for liquids is applied to mixtures by means In this theory the binary mixture is regarded as consisting of two different types of cells, one type containing each component. The properties of each type of cell differ not only from each other but also from those of the pure liquid; each cell reflects the environment of the molecule ivhich is a t the center of the cell. For the general multicomponent case there are as many types of cells as there x e components. T h e partition function is a generalization of Equation 9 in the previous paper (26)’ o l Scott’s two-fluid theory (28).
between the behavior of like and unlike molecules, calculated results are extremely sensitive to the choice of mixing rules. T h e molecular parameters for which mixing rules must be defined are the parameter, c, the collision diameter, u, and the pair potential, e. For parameter ci, which is related to the noncentral forces of molecule i, we assume that it is the same in the solution as in pure liquid i. Therefore, a relationship between Uim*and Tim*may be chosen such that only t\yo parameters need to be determined for each component in the mixture: (5) T h e common mixing rule for the collision diameters of unlike pairs is that given by the hard-sphere model
where n is the number of components. Substituting the functionsfl,fz, and chosen in the previous paper (26) :
exp
(3v,,*
‘)I
This simple approximation seems to be excellent and is used here. The usual mixing rule for the energy parameter in the pair potential is based on a simplification of the London formula, leading to
(2)
S kT Vi,
T h e notation in Equations 1 and 2 is analogous to that of Equations 11 and 2 3 of (26). Subscript m indicates that the property is the composition-dependent average for the substance in its cell in solution, and not the same as that in the pure fluid, T h e characteristic properties Vzm*, Tam*,and Ut,* are given by mixing rules discussed below. To utilize the partition function for calculating thermodynamic properties of
