Theory of Dipole Correlation for Chain-Associated Solvents
2941
Hydrogen Bonding in Polar Liquid Solutions. 5. Theory of Dipole Correlation for ChainAssociated Solvents Containing Hydrogen-Bonding Solutes. Application to 1-Octanolla Kee-Chuan Pan and Ernest Grunwald*’b Department of Chemistry, Brandeis University, Waltham, Massachusetts 02 154 (Received June 23, 1976) Publication costs assisted by the Petroleum Research Fund
Assuming a freely rotating hydrogen-bonded chain structure for the solvent, with solute molecules hydrogen-bonded to the chain terminals, and equilibrium constants conforming to the model of sitewise equilibrium, expressions are obtained for dipole correlation factors of solvent and solute and for the solute-induced medium effect, a t c2 = 0. The theory is used to analyze dielectric constant data for the following solutes in 1-octanol: acetone, methyl isobutyl ketone, benzaldehyde, dimethyl sulfoxide, pyridine, and chloroform. The theory leads to values of the pairwise dipole moment ~ 1 (which 2 is analogous to the dipole moment for a 1:l complex) which are in quite reasonable agreement (10% discrepancy) with dipole moments for the corresponding 1:l complexes measured in benzene.
For definiteness, consider a dilute solution of a hydrogenbond acceptor solute (X) in a hydrogen-bond, chain-associdenote i the mean number ated alcohol solvent (ROH). Let ? of solvent molecules per chain. Let f denote the fraction of solute molecules that are associated with solvent chain terminals, and g2a denote the dipole correlation factor of these molecules. For the complementary fraction 1 - f of unsolvated X molecules, gp,, = 1,by hypothesis of the chemical modeL2 Let gI+)and gllo denote the dipole correlation factors of solvent molecules in chains with and without terminal X molecules, respectively. Equation 1-13 then takes the form,2
the auerage dipole correlation factor for the v solvent molecules in chains of u links.
Dipole Correlation Factors
For the general chain model, the distribution function p l ,may be different from p,.’, and dipole correlations such as ( ( p l I PI, ) ) may change upon addition of X. A great simplification results if it is assumed that specific interactions within the chain are limited to nearest neighbors. Such a model is consistent with two explicit features of the forthcoming calculations: (1)Stepwise association constants are independent of chain length, as expressed in (4-4) and ( 4 - l l ) ,with the result that a sitewise equilibrium formulation becomes valid. (2) Each hydrogen-bond strength is independent of the state of the adjacent hydrogen bonds, so that free rotation about hydrogen bonds is permitted. Consequences are that p , ,and pi,’, ?i and E’, and solvent-solvent dipole correlations ( p l , . k l h )and (pli’.plk’)between any pair of sites in chains of any length, all become equal. On introducing these consequences into (5-3), ( 5 - 5 ) , and (5-6),one obtains for the “nearest-neighbor” model
In this section we shall express gZa,g12, and gll as sums of pairwise dipole correlations, first for a general chain model, then for chain models in which specific interactions are limited to nearest neighbors, or to nearest as well as next-nearest neighbors. The scheme for labeling chain sites is shown in Figure 1. The chains are of variable length, the chain-length Y following a probability distribution function p , , such that
c Pi, = 3 P
1’=
1
We shall use primed symbols if the chain contains a terminal X molecule, and unprimed symbols if it does not. On applying the general definition (1-2), we then write for solute molecules attached to chains of v links. g2a”” =
(
PLZ’*(112’
+
2
MI:)
)
1 ~ 2 ’
The average g12 for the ensemble is (Z,,p,’vg12(”)/~uvp,,’), and is obtained from (5-4), recalling that Z,,vp,,‘ = E’.
The average g l l for solvent chains not terminating in X is obtained similarly. The result is,
(5-2)
The average gzafor the solute is given by gza = Z,py’gza(u). Introducing ( 5 - 2 ) ,we obtain:
(gr, - 1 h d = 3g12 - glI)k12 =
((
b2/.
2 ,;) ) i=
