Prausnitz, J. RI., A.I.Ch.E.J. 5 , 3 (1959). Prausnitz, J. AI., “RIolecular Thermodynamics of Fluid-Phase Equilibria,” Prentice-Hall, Englewood Cliffs, N. J., 1969. Prausnitz, J. AI., Chueh, P. L., “Computer Calculations for HighPressure Vapor-Liquid Equilibria,’’ Prentice-Hall, Englewood Cliffs, Pi. J., 1968. Prausnitz, J. AT., Eckert, C. A,, Orye, R. V., O’Connell, J. P., “Computer Calculations for Rlulticomponent Vapor-Liquid Equilibria,” Prentice-Hall, Englewood Cliffs, N. J., 1967. Price, A. R., Kobayashi, R., J . Chem. Eng. Data 4,40 (1959). Reid, 11. C., Sherwood, T. K., “Properties of Gasea and Liquids,” 2nd ed., RIcGraw-Hill, New York, 1966. Roberts, L. R., Wang, R . H., RIcKetta, J. J., J . Chem. Eng. Data 7, 484 (1962). Ruhemann, hl., Proc. Roy. SOC.171A, 121 (1939). Sahgal, P. N., Geist, J. RI., Jambhekar, il., Wilson, G. RI., Int. Adaan. Cryog. Eng. 10, 224 (1965). Sprow, F. B., Prausnitz, J. RI., A.I.Ch.E.J. 12,780 (1966).
Strobridge, T. R., “Thermodynamic Properties of Nitrogen from 144’ to 540OR between 1.0 and 3000 psia, ” National Bureau of Standards, Tech. Note 129A (1963). Taff, W. O., “Selected Values of Properties of Hydrocarbons and Related Compounds,” American Petroleum Institute Research Project 44, TexaT A&lI University, 1966. Volk, H., Halsey, G. D., J . Chem. Phys. 33, 1132 (1960). Williams, R. B., Katz, 1).L., Ind. Eng. Chem. 46, 2512 (1954). Wilson, G. RI., Int. Aduan. Cryog. Eng. 9, 168 (1964a). Wilson, G. XI., J . Amer. Chem. SOC.8 6 , 127 (196413). Wohl, K., Chem. Engr. Progr. 49, 218 (1953). Wohl, K., Trans. A.I.Ch.E. 42, 215 (1946). RECEIVED for review February 24, 1970 ACCEPTED August 19, 1970 Division of Industrial and Engineering Chemistry, Symposium on Enthalpy of RIixtures, 159th Meeting, ACS, Houston, Tex., February 1970.
Quasilattice Theory and Paraffin-Alcohol Systems Chia-Ming Kuo,’ R. L. Robinson, Jr., and Kwang-Chu Chao2 Oklahoma State University, Stillwater, Okla. 740’74
The quasilattice theory advanced b y Guggenheim and Barker, re-expressed in canonical partition function of group interactions, i s applied to the representation of excess thermodynamic properties of n-alcoholn-paraffin mixtures. Six exchange energies of the groups involved are determined b y fitting the equations of excess enthalpy to experimental data on eight binary systems over the entire composition range. The exchange energies thus determined are employed to predict excess free energy and excess entropy for comparison with experimental data on two binary systems. Reasonable agreement i s obtained for all three excess properties for the systems studied.
T h e representation of excess thermodynamic properties in liquid systems coiitaiiiing polar substances presents a challenging problem in solution thermodynamics. Recent data on heat:: of mixing (Savini et al., 1965) and phase equilibria (Van S e s s et al., 1967a,b) for a number of n-alcohol-n-paraffin binary systems provide a n opportuiiity for critical comparison;; of various theories that, describe the behavior of these solutioiis. Previous investigators have applied group solution (Chao et al., 1967) and association-type (Renon and Prausnitz, 1967; Kiehe and Bagley, 1967) models to describe alcohol-paraffiii systems. However, 110 similar results have used the quasilattice theory (Barker, 1952; Guggeiiheini, 1944). T h e present work examines the quasilattice model in its ability to describe t’he excess enthalpy and free energy of alcohol-paraffin systems. The quasilattice theory advanced by Guggenheim (1944) and Barker (1952) re-expreqsed in canonical partition function of group interactions, is used in the present study. The present, equations serve as a convenient basis for further development and use in cell-theory calculations. Quasilattice Theory
Basically, the quasilattice theory considers each molecule -4 in solution to be composed of a number, rA, of segments (or Present address, Esso Production Research Co., Houston, Tex. 7T001
2 Present address, Purdue Universitj-, Lafayette, Ind. 47907 To whom correspondence should be sent.
564 Ind. Eng. Chem. Fundam., Vol. 9 No. 4, 1970
groups) placed on well-defined lattice sites. Each type of segment i possesses a number, zit of contact points, where it interacts with adjacent segments-for example, a pentane molecule could be considered t o coil of two methyl segments and three methylene segments. The configuration energy of the solution is the sum of contributions from interaction3 bet,weeii pairs of adjacent segments. Guggenheini (1944) developed the quasilattice theory of molecules of different sizes and occupying different numbers of lattice sites. Segment:: of the same molecule interact in the same n a y . Barker (1952) extended Guggenheini’s theory to allow one molecule to have different kinds of segments. Earlier’.; results are iii terms of grand partitioil functions. I n the following we re-express Guggenheini’s results to allow one molecule to have more than one kind of segmeiit. Our prereiit results are in the caiioiiical form and appear to be well suited for further developnieiit into the language of the cell theories. The configurational energy, E! of a lattice solution is completely determined by a set of numbers, S i j , that represent the numbers of contacts bettTeen segments of types i and j. The caiioiiical partition function is given by Q
=
Ni,
g exp ( - E I k T )
(1)
where g denotes the degeneracy of the configuration described by the set of S i j ’ s . Let e i j stand for t,he energj- of interaction between an i segment and a j segment. Then,
E
=
Yij€i3 z
33i
(2)
Degeneracy g has so far not been evaluated rigorously for three-dimensional lattires. However, several approximations have been developed. Following Guggenheini (1944), we have
Table 1, Number and Type of Contact Points, Sites, and Coordination Numbers Component
The values of g and S,, are related to the corresponding valuei (denoted by asterisks) in the athermal theories of Flory and Huggins (Flory, 1953). Thus,
nHza
Et'haiiol 1 1 Propanol 1 Butanol 1 Pentanol Octanol 1 n-Hexane ... n-Heptane ... n-Sonane ... 4 Coordination number, z,
nozo
nIzr
2 2 2 2 2 ... ...
5
... ...
...
, . .
nszs , . .
... ...
7 9 11 17
given by Zn,zi
...
... 14 16 20 =
rz - (2r
r
L
4 . 3 4 4 4 5 4 6 4 9 4 6
4
7
4
9
-
2).
Table II. Interaction Energy Parameters. at 3 0 ° C
where niA denotes the number of i groups in a molecule A . As usual in statistical thermodynamics, the sum of Equation 1 is replaced by its maximum term. I n order to pick the maximum, we differentiate with respect to Si, aiid set the derivative equal t o zero,
a ~
bSij
[gexp ( - E / k T ) ] = 0
Set
A
WO-H
-2756 B -3173 C -3161 D -3175 E -3748 Cal/g mole.
WH-8
0 (-224 (-194 -91 -251
WH-I
0 = -224) = -194) 0 -253
wo-s
wo-I
WI-8
0 (-284 (-317 216 -257
0 = -284) = -317) 0 -465
2 0
