T H E
J O U R N A L
OF
PHYSICAL CHEMISTRY 0 Copyright, 1976, b y t h e American Chemical Society
Registered i n U . 9. Patent Office
VOLUME 80, NUMBER 27 DECEMBER 30, 1976
Hydrogen Bonding in Polar Liquid Solutions. 1. Development of Kirkwood’s Dielectric Theory in Terms of a Chemical Modella Ernest Grunwald*’b and Kee-Chuan Pant Department of Chemistry, Brandeis University, Waltham. Massachusetts 02 154 (Received June 23, 1976) Publication costs assisted by the Petroleum Research fund
Kirkwood’s dielectric theory, applied to dilute solutions, is developed in terms of a parameter /qapp, called the apparent dipole moment of the solute. pz,app2can be calculated directly from experimental data and equals gzp22 c10p12 (dgJdc2); p1, p2 denote intrinsic dipole moments, g,, gz dipole correlation,factors, and c l , c2 concentrations of solvent and solute species, respectively. A chemical model is introduced which assumes that dipole correlation between solute and solvent is considerable only in case of molecular complex formation. As a consequence, wz,app2 - p z 2 is dissected neatly into additive contributions from (1)solvation of the solute; (2) transfer of solvent molecules into solvation shells from the bulk solvent; and (3) change in dipole correlation in the bulk solvent.
+
Introduction to Series of Papers The effects of solutes on hydrogen bonding in hydroxylic solvents are so intricate that experimental probes all too often tell us only whether a given solute is “structure-making” or “structure-breaking”. The dielectric constant, when interpreted in terms of Kirkwood’s dielectric theory,’ gives information about dipole alignment and hence can elucidate the specific geometrical structure of hydrogen-bonded complexes. The sharpness of the resulting picture can be further improved if the dielectric measurements are complemented by spectroscopic and other data. In this series we are developing an approach, centered on dielectric measurements, for deducing specific hydrogenbonded structure in solutions in hydroxylic solvents. Although our measurements include a variety of hydrogen-bond donors and acceptors ,and solvents ranging from nonpolar to polar to polar-hydroxylic, the primary substrate will be 1-octanol. The choice of 1-octanol was attractive because (1)the conductivity of the pure liquid is low enough to permit precise measurements of dielectric increments for dilute solutions; (2) dielectric and other properties of the pure liquid are accurately known3 and indicate molecular interactions conforming approximately to the model of a freely rotating hydrogen-bonded Present address: Department of Chemistry, Tamkang College of Arts and Science, Tamsui, Taiwan 251, Republic of China. +
(3) the study is readily extended to isomeric liquid octanols whose dielectric properties are quite different.3 Kirkwood’s exact dielectric theory is so general that, unless the information we seek is already available, simplifying assumptions must be introduced before the theory can be applied. In this paper (part 1 of the series), we shall formulate the theory for convenient application to dilute solutions and introduce a simplifying assumption which we call the chemical model. In part 2, we shall apply this formulation to examine the structure of complexes resulting from the interaction of 1-octanol with various donors and acceptors in nonhydroxylic solvents. In part 3, we shall examine the effects of non-hydrogen-bonding solutes on the dielectric constants of several hydroxylic solvents. In part 4, we shall report data for dilute solutions of various hydrogen-bonding solutes in I-octanol. Finally, in part 5 , we shall analyze the data obtained in part 4 and deduce specific hydrogen-bond structural information for solvent and solute.
Terminology and Definitions When Kirkwood’s theory is applied to one-component the dipole alignment is accounted for by means of the correlation factor g, whose definition is as follows. Let p be the (scalar) molecular dipole moment, p the corresponding molecular dipole vector and the vector sum (Figure 1)of p and the net dipole moment of the surrounding sphere of 2929
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Ernest Grunwald and Kee-Chuan Pan
Application to Dilute Solutions Let component 1 be the solvent and 2 be the solute. Let V1 denote the molar volume of the pure solvent, Vz the apparent molar volume of the solute, and c1 = (1- cZV,)/V, the solvent concentration. In this notation, (1-3a) becomes
f(4 = Pl/Vl
Figure 1. (a)p and localized dipoles around it. (b) 5 is the vector sum of p and all molecular dipoles around it. In practice, only those dipoles within the effective range of interaction from the central molecule need
+ C Z [ P Z - (V,P,/V,)]
(1-5)
where P1 and Pz are functions of c2. To obtain an expression that is accurate up to terms of first order in C Z , we write (i = 1,2)Pi= Pio c2(dPJdcz)c2=0,f(t) = f ( t d + c~[(df/dt)(dt/dc,)],,=,, f(td = Plo/V1, and (df/dt),,=o = (2t02 1)/9t02. Equation 1-5 thus reduces to
+
+
be considered.
molecules, whose radius in theory approaches infinity. Let p-5 denote the scalar product, and let (p-5)denote the statistical average, taken over all molecules of the given species. Then g is defined by
g = (pG-i)lw2
(1-1)
When there is more than one component,s it is instructive to introduce a separate factor g, for each component, as follows: gL = (P1*FL)/PL2
(1-2)
5, now denotes the vector sum of p,, the dipole vector of the ith molecule, and the net dipole moment of the surrounding sphere of molecules, which comprises molecules of all species. This formulation enables us to express the polarization of the liquid solution as a sum of additive terms for the individual components:2 f(€)
(t
- 1)(2t + 11/96 = CC,P,
4 pi = - .~rNo[a, + gL”’pL2/3kT] 3
+
+
+
+
(1-3a) (1-3b)
Here t denotes dielectric constant of the liquid solution, c, concentration, P,molar polarization, a, molecular polarizability, and +pL the molecular dipole moment of the ith species of molecules in the given solution. In general, +pl is somewhat greater than the intrinsic moment p L of the isolated molecule. The ratio, p , / p p L , may be calculated from Onsager’s reaction fieldg strictly only if all g,’s are unity, and even then may require ad hoc assumptions about molecular size and shape.l0 We decided to adopt the relation
h = p,/qp, = 1 - [(n2- I ) / ( n 2 2 ) ] 2(t - I)/(%
(1-6) It should be noted that Pl0denotes the molar polarization of the pure solvent, while Pzodenotes the molar polarization of the solute in an infinitely dilute solution. To solve for (dPJ dc&=O, we introduce (1-3b)and (1-4).The quantities ai and pi are constant, by definition, but gl, g2, and h ( n , t )are functions of c2. In writingll the final result (1-7),it is convenient to use the following parameters: clo = l/V1; cp = ( n 2- l ) / ( n 2 2) for the solution; q,= (no2- l)/(no2 2) for the pure ho = 1 solvent; R1 = 4nNoa1/3 = poV1; R2 = 4.1rNo(~2/3; 2cpo(to - 1 ) / ( 2 t o 1).In principle, R1 is the molar refraction of the solvent and R2 is the apparent molar refraction of the dilute solute. In practice we shall use molar refractions of the pure substances obtained at the sodium D line.
- [ k z 2 ( g 2- 1) + ~
1 (dgJdc2)l ~ ~
(1-7) 1
Note that the right-hand side of (1-7) separates terms without g factors, which can be obtained experimentally, from terms with g factors, which can be used to probe molecular interactions. Because of this, it is useful to introduce a parameter p2,app, which will be called apparent dipole moment.
Apparent Dipole Moment p2,app is
defined as follows:
2 p2,app
+ 1) (1-4)
which is one of the possibilities suggested by Kirkwood2 and has also been used by others.8-10 (In (1-4),12 denotes the refractive index of the solution ) Equation 1-4 can be derived by applying Onsager’s reaction field to a dipole imbedded in a polarizable cavity whose properties are those of the macroscopic solution.2 It has been shown in previous work,ll and will be shown in the present series, that eq 1-4 is reasonably accurate in practical applications. A valid relationship between PpL and pL is necessary in order to calculate g,, and thence to deduce structural information. However, the relationship should be based entirely on experimental data and should not require the making of arbitrary assumptions about microscopic or structural parameters. Equation 1-4 satisfies this criterion. The Journal of Physical Chemistry, Vol. 80. No. 27, 1976
(1-8) Note that p ~ , is ,a function ~ ~ solely of experimental quantities. On substituting in (1-7) and rearranging, we obtain p2,app2
= g z d + Ciopi2(dgl/dcz),,=o
(1-9)
The apparent dipole moment defined here should not be confused with the term “apparent dipole moment” as used in the older literature.12 The older term is simply the dipole moment measured in nonpolar solvents and calculated by Debye’s second method.13 The present usage is analogous to such familiar usage as “apparent molar volume” or “apparent solvation number”. In the absence of dipole correlation, gl =
~
293 1
Hydrogen Bonding in Polar Liquid Solutions
g2 1 and dg,/clcz = 0. I t can then be seen from (1-9) that, under such conditions, pz,appreduces to p2. When there is dipole correlation, pz,appmay deviate greatly from ~ 2 Indeed, . fi2,app2 may be a negative quantity!
Chemical Model The definition of the correlation factor g, is so general that, in order to deduce specific structural information, one has to introduce additional assumptions. We shall assume that dipole correlation between solute and solvent is considerable only if there is molecular complex formation between the two kinds of molecules, and that the relative concentrations of complexed and uncomplexed species conform to the laws of mass action and thermodynamics. We shall call this assumption the chemical model. Thus the solute molecules may be divided into two groups: a fraction 1 - f remains uncomplexed, with correlation factor = 1;and a fraction f forms complexes with solvent molecules, with correlation factor gza. The overall average g2 is given by (1-10) Similarly, the solvent molecules may be divided into two groups: those that remain in the bulk solvent (correlation factor gll), and those that exist in solute-solvent complexes (correlation factor glz). Let c1 and c2 denote the concentrations of the formal components, regardless of complexing, and let m denote the average solvation number of the solventsolute complexes. Then the fraction of all solvent molecules combined in solvent-solute complexes is rnfczlc1, and the overall average g1 is given by g~ = (1- mfc2/cl)g11+ (mfcz/cdg12
(1-11)
In principle, ,411 and g12 are functions of cz. On expanding in Taylor's series about c z = 0 (i.e.,g11 = gllo + c z (dgll/dcz) . . .) and neglecting higher order terms in cz, we obtain
+
gl = g1l0 + (g1P - gllOj(mfcz/cd
+ cz(dglddcdc,=o
(1-12)
Finally, we take the derivative of (1-12),evaluate at cp = 0, and substitute in (1-9):
Equation 1-13neatly dissects p2,app2- pz2 into three additive contributions: pz2 with its associated change of correlation factor; p12 with the associated change of correlation factor as a fraction (fmczlcl) of the solvent molecules becomes associated with solute molecules; and a term comprising the solute-induced changes in dipole correlation in the bulk solvent. We shall call the third term the solute-induced medium effect (SIME). ( p ~- p z, 2 ) and ~ ~dddcz ~ are ~ both indications of the change of polarity. However, a positive pz,app2- p z 2 means that the overall interaction makes the system become more polar than it was before interaction, while a positive dddcz means that the system with solute is more polar than that without solute. To examine the structure of solvent-solute complexes, we need to know the interdependent parameters gz, g1Zo, f, and m. Equation 1-13 shows that these parameters are accessible only if the solute-induced medium effect (proportional to dgll/dcp) is known or can be predicted: the SIME is the key. We shall examine SIME's in hydroxylic solvents in parts 3 and 5.
References and Notes (1) (a) Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this work: (b) John Simon Guggenheim Fellow, 1975-1976. (2) (a) J. G. Kirkwood, J. Chem. Phys., 7, 91 1 (1939); (b) Trans. Faraday SOC., A42, 7 (1946). (3) (a) W. Dannhauser, J. Chem. Phys., 48, 1911 (1968); (b) G. P. Johari and W. Dannhauser, ibid., 48, 51 14 (1968). (4) G. Oster and J. G. Kirkwood, J. Chem. Phys., 11, 175 (1943). (5) C. P. Smyth, "Dielectric Behavior and Structure", McGraw-Hili, New York, N.Y., 1955. (6) J. A. Pople, Proc. R. SOC.London, Ser. A, 205, 163 (1951). (7) (a) G. H. Haggis, J. E. Hasted, and T. J. Buchanan, J. Chm. Phys., 20, 1452 (1952); (b) J. B. Hasted in "Dielectric and Related Molecular Processes", (A Specialist Periodical Report) Vol. 1, The Chemical Society, London, 1972, p 121 (8)(a) G. Oster, J. Am. Chem. SOC.,68,2036 (1946); (b) ibid., 66,948 (1944); (c) J. G. Kirkwood in "Proteins, Amino Acids, and Peptides", E. J. Cohn and J. T. Edsall, Ed., New York, 1943, Chapter 12. (9) L. Onsager, J. Am. Chem. SOC., 58, 1486 (1936). (IO) C. J. F. Bottcher. "Theory of Electric Polarisation", Elsevier, Amsterdam, 1952. ( I 1) E Grunwald and A. Effio. J. Solution Chem., 4, 373 (1973). This paper also describes a more detailed derivation of eq 1-7. (12) R. J. W. Le Fevre, "Dipole Moments", 3d ed, Methuen, London, 1953. (13) P. Debye, "Polar Molecules", Dover Publications, New York, N.Y., 1945.
The Journal of Physical Chemistry, Vol. 80, No. 27, 1976
