M N hTk
P pk R S S‘Vk
T
Av X
Ax Y AY Zk
Y
e U
= number of data points generated in random error
test cases = number of components in a multicomponent system = percent of points greater than sk in absolute value , y2 where k may be f, y ~ or = total pressure = vapor pressure of component k = gas constant = estimate of standard deviation u = standard deviation on N k where k may be f o r y = absolute temperature = volume change of mixing a t constant T and P per mole of solution formed = liquid phase mole fraction = random error generat,ed in x a t a given datum point = vapor phase mole fraction = random error generated in y a t a given datum point = set of independent variables in eq 3 = activity coefficient defined by eq 4 = pressure imperfection term used in eq 4 = standard deviation
SUBSCRIPTS a, b = data point identification i, j = component identification f , x, y , P , T , N = hr, s, or u applies to variables so indicated
SUPERSCRIPTS r = indicates variable containing random error
Literature Cited
Black, C., Ind. Eng. Chem. 50,391 (1958). Black, C., Derr, E. L., Papadopoulos, M. N., Ind. Eng. Chem. 55(9). 38 (1963). CarlsoG H. C., Colburn, A. P., Ind. Eng. Chem. 34,581 (1942). Chang, S.-D., Lu, B. C.-Y., “International Symposium on Distillation,” Brighton, England, Part 3, p 22, Sept 1969. Li, J. C. M., Lu, B. C.-Y., Can. J. Chem. Eng. 37,117 (1959). McDermott, G., Ellis, S. R. M., Chem. Eng. Sci. 20,293 (1965). Mixon, F. O., Gumowski, B., Carpenter, B. H., IND.ENG. CHEM.,FUNDAM. 4,455 (1965). Prausnitz, J. M., Snider, G. D., A.I.Ch.E. J. 5 , 75 (1959). Ross, J. F., Paper presented at Third Joint Meeting AIChEIMIQ, Denver, Colo, Sept 1970. Scatchard, G., Raymond, C. L., J. Amer. Chem. SOC.60, 1278 (1 0.18 ). \----,.
Stevenson, F. D., Sater, V. E., A.I.Ch.E. J. 12,586 (1966). ENG.CHEM.,FUNDAM. 2, 119 (1962). Tao, L. C., IND. Tao, L. C., Ind. Eng. Chem. 56 (2), 36 (1964). Tassios, D., “Prediction of Binary Vapor-Liquid Equilibria; Members of a Homologous Series in a Common Solvent,” unpublished Ph.D. thesis, University of Texas, Austin, Texas, 1967
Ulrichson, D. L., “Effect of Experimental Error in Vapor-Liquid Equilibrium Data on Thermodynamic Consistency,” unpublished Ph.D. thesis, Iowa State University, Ames, Iowa, 1970. RECEIVED for review March 15, 1971 ACCEPTED February 24, 1972 Work was performed in the Ames Laboratory of the U. S. Atomic Energy Commission. Contribution No. 2797.
Statistical Thermodynamics of Group Interactions in Pure n-Alkane and n-Alkanol-1 Liquids Tsung-Wen lee, Robert A. Greenkorn, and Kwang-Chu Chao* School of Chemical Engineering, Purdue U n i v u d y , Lafayette, Ind. 47907
A partition function i s developed for the description of nonpolar and polar chain molecule liquids b y combining the cell theory and the quasichemical lattice theory. On the basis of this partition function, the interaction properties of methylene and methyl groups are determined from analysis of literature data on n-alkanes. An explicit procedure i s devised and followed for the stepwise decomposition of the molecular properties into group properties, including the interaction energy parameters, the core volumes, and the external degrees of freedom of the methylene and the methyl groups. The interaction properties of the hydroxyl groups are then determined from analysis of literature data on n-olkanol-1 liquids. The hydrogen-bonding energy thus determined agrees with accepted values. The core volume, degrees of freedom, and energy parameters of the hydroxyl groups are evaluated. The densities and heats of vaporization calculated from the theory for the n-alkanol-1 liquids are in good agreement with data.
A group is an identifiable structural unit of a molecule, such
as a methyl group or a hydroxyl group. A few kinds of groups make up a large number of the molecules of interest in chemical processes. This fact has provided the motivation for numerous investigations of pure substance and solution properties from the viewpoint of group contributions. The success of these efforts confirms that the intermolecular forces, being short-ranged, can be validly considered, for a large class of molecules, to be of a local nature between groups that come into close proximity due to molecular motion.
Langmuir (1925) suggested the premise that the force field of a group is independent of the nature of the rest of the molecule. The electronic structure of molecules suggests that Langmuir’s principle of independent action cannot be strictly valid, but is a good approximation for many molecules. The degree of approximation can be improved by refined classification to reflect the nature of the rest of the molecule. Notable developments of group contribution to solution properties have been made by Pierotti, et al. (1956, 1959), Deal, et al. (1962), Wilson and Deal (1962), Hermsen and Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
293
Prausnitz (1966), and Kuo, e l al. (1970) and were recently reviewed by Deal and Derr (1968). Since then Derr and Deal (1969) have described the activity coefficients of groups in terms of the Wilson equation. The objective of this work is to relate the group interactions to the configurational properties of liquids by means of statistical thermodynamics. An important advantage to be gained in such an effort is that one is no longer restricted to the correlation of any one thermodynamic property such as the excess free energy of solutions. Indeed, in principle, all of the configurational properties are described a t the same time in a statistical mechanical formulation. The degree of practical success of such a description will necessarily vary from one property to the other in a manner that reflects the various imperfections of the physical model employed. The same types of groups and their interactions are usually found in pure liquids as well as in liquid solutions. I n fact, from the group viewpoint, pure liquids and liquid solutions are not distinctly different. The concept of group interaction can be regarded as providing a unifying link between pure liquids and their solutions. Since the methylene and methyl groups form the backbone of a large class of chain molecules, the interaction properties of these two groups are first determined from analysis of literature data on n-alkanes. The interaction properties of the hydroxyl groups are then evaluated from analysis of literature data on the pure n-alkanol-1 liquids and their solutions. Properties of the pure n-alkane and n-alkanol-1 liquids are calculated and compared with data in the literature.
the total potential energy, and v represents the volume occupied by a group. Equation 1 implies that the chain segments are of the same kind, and such a chain is said to be homogeneous. As our present interest encompasses heterogeneous chains made up of several kinds of segments, we need to generalize eq 1. The generalization is restricted to segments of not greatly different sizes so that the basic assumptions of the cell theory of chain molecules remain valid. Consider a heterogeneous chain made up of two kinds of groups, referred to by subscripts 1 and 2, respectively. (Further generalization to an arbitrary number of segment species will become obvious.) The part of equation 1 that describes the motion of the segments for a heterogeneous chain is 20 =
[,&(v1'/8
- vl*l/3 ) 18zlcl [ ~ ~ ( v Z ' ' ~ - v ~ * ' / ~ )
1 3 ~ 2 ~ 2
In terms of the reduced variables BL = we have
where g stands for the combinatorial factor of placement of the chain molecules in the regularly spaced cells, Eo denotes 294 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
and BZ = vz/vz*,
If we assume the volume occupied by a group is proportional to its core volume, then BI = BZ and we regroup the factors, and drop the subscript from B to obtain
+
where 3zc = 321Cl ~ Z Z C Zand represents the total external degrees of freedom of the heterogeneous chain. Accordingly, the volume occupied by a group becomes simply related to the liquid volume, e.g., for group 1 (5)
The Cell Theory for Chain Molecules
The statistical thermodynamical development of group interaction in this work is based on the cell theory of liquids. Eyring and Jhon (1969) postulated that the nearest neighbors of a "central" molecule form a cell (or a cage) in which the central molecule wanders. The resultant force field of the neighbors governs the motion of the central molecule. The arbitrary but convenient identification of a central molecule makes it possible to factor the partition function of a liquid into cell partition functions that describe the motions of the individual molecules in their cells. Prigogine and coworkers (1957) made a significant contribution to the theory of chain molecules with their recognition of two types of degrees of freedom of molecular motion in liquids : the intramolecular (internal), and the intermolecular (external) degrees of freedom. The former are associated with the higher frequency motions such as bond stretching, which is unaffected by the environment. The latter degrees are associated with the lower energy motions of the molecule, such as rotation around a bond, and these interact with the surroundings. These external degrees of freedom enter into the configuration properties. Prigogine achieved remarkable success with this concept in the development of a principle of corresponding states for chain molecules. Flory and coworkers (1964) incorporated the concept of Prigogine's external degrees of freedom into the cell theory and developed partition function for chain molecules. For a chain molecule of 2: segments with 32c total external degrees of freedom, Flory, et al., suggested
vl/v1*
(2)
where -yz denotes the core volume of group 2 relative to group 1, and equals v z * / v l * ; ni stands for number of i groups in a chain molecule. Equation 5 is expressed in terms of group 1 being the base group of heterogeneous chain. I n the present work methylene serves as the base group. I n terms of the zo of equation 4, we express the configurational partition function of a liquid of N heterogeneous chain molecules by
where i denotes a group species. The Quasichemical Relation
The interaction of a group varies in energy, depending on the nature of a partner group. For instance, a hydroxyl group can form a hydrogen bond with another hydroxyl group; the energy of this interaction is much higher than that of the same hydroxyl group in contact with a methylene. The higher energy represents enhanced attraction. As a result, the groups are preferentially paired, and the molecules are ordered in orientation; however, the ordering is incomplete because of thermal motion. The quasichemical relation of Guggenheim (1944) is used in this work to represent the orderdisorder effects. We visualize the cells occupied by the groups to form a quasilattice structure. A group i has z i nearest neighbor contacts, all of which are assumed to be satisfied. Let e i j denote the interaction energy between a group i and group j . Let N i j denote the number of ij pairs. Then the total interaction energy in the liquid is given by
(7)
I n the absence of preferential attraction of groups, the values of N i j t h a t we would obtain are referred to as N i l * and are given by the formulas for i # j
where N i j denotes the number of pairs of nearest neighbor groups of the species i and j . The energy eri depends on the volume per group in a manner similar to eq 15, where ' I r j
are the constant energy parameters for ij interaction. The interaction energy of the hydrogen bond is represented by Equations 8 and 9 are explicit in the N*'s; however, where preferential interaction exists, the Nij's are no longer given by any explicit equations. (The rigorous theory for three-dimensional lattices has not been developed.) I n this work we use the approximate solution due to Guggenheim, called the quasichemical relation, which is expressed
where the exchange energy w i j is defined by
I n the evaluation of the Nij's eq 10 is solved simultaneously with the stoichiometry of coordination numbers 2Nit
+
j#i
Nij = Nnizi
(12)
I n a liquid containing k types of segments there are k ( k 1)/2 equations like eq 10, and k equations like eq 12, making up a total system of k ( k 1)/2 equations, corresponding to exactly the same number of Nij's. Guggenheim's quasichemical theory also leads to an expression for the combinatorial factor g in eq 6.
where the subscript HB denotes hydrogen bond, and subscript B refers to the base group of the chain, i.e., methylene. Equation 18 expresses the hydrogen bond energy as a sum of a dispersive energy and a chemical reaction energy. Our selection of this function is influenced by the work of Allen, et al. (1960a,b), and Bagley, et al. (1970). They observed that the hydrogen bonds appear to remain intact in a small isothermal expansion; the volume change occurs largely due to increases in the spacing of the hydrocarbon portion of the molecules. The inclusion of a constant, volume-independent term in eq 18 lowers the internal pressure energy (piV) relative to the cohesive energy. Such a lowering effect was observed experimentally by Allen, et al. (1960a,b), for the alcohols. Summing the pairwise energies according to eq 16 and substituting into the partition function of eq 6
+
g* denotes the value of g in the absence of preferential interactions evaluated by Flory (1953). (We will not be concerned about g* until we encounter solution behavior.) I n
