3950
J . Phys. Chem. 1991, 95, 3950-3954
in the A N 0 basis set reduces the binding by 2.3 kcal/mol, a conservative upper bound for the effect of all higher correlation effects must be a further 2.0 kcal/mol reduction in binding energy. Thus if our A N 0 set is essentially complete and we have underestimated the correlation effect, the binding could be 2 kcal/mol smaller than what we compute. All other errors are expected to increase the binding energy. Thus we make our conservative estimate by adding 3 kcal/mol to the computed value, and assigning it a 5 kcal/mol uncertainty, giving a value of 35 f 5 kcal/mol. D. CH3CH2OH and CH3CH0. It was not possible to consider larger ligands using the correlated treatments, but we did consider CH3CH20H and CH’CHO at the TZ2P SCF level. (The geometry optimization and calculation of the zero-point energy for CH3CH20Hand Mg+-CH3CH20H are performed with a basis set where the two hydrogen polarization functions are replaced by one with an exponent of 1 .I 5; based on CH3CH20H this is expected to make essentially no difference in the reported values.) The Do values computed are 30.5 (H2O) < 34.2 (H2CO) < 35.0 (CH3OH) < 37.3 (CH’CH2OH) < 41.0 (CH3CHO) The calculations show that Mg+-H20 and Mg+-CH30H are computed to about the same accuracy at the SCF level and that correlation increases the binding by approximately 1 kcal/mol. Since the bonding is similar in Mg+-CH3CH20H to that in Mg+-H20 and Mg+-CH30H, we expect there will be a similar correlation contribution (1 kcal/mol). For Mg+-H2C0, however, correlation reduces the binding energy by about 2 kcal/mol. Since the bondings in Mg+-H2C0 and Mg+-CH3CH0 are similar, we expect a similar correlation contribution. Thus, at the SCF level, the Mg+ binding energies to H 2 C 0 and CH’CHO are expected to be about 3 kcal/mol too large relative to the SCF binding energies of Mg+ to H20, CH30H, and CH3CH20H. Therefore, the relative binding energies setting Mg+-CH30H to zero as in ref 3, are -4.5 (H20) < -3.8 (H2CO) < 0.0 (CH’OH) < 2.3 (CH’CHZOH) < 3.0 (CH3CHO) The computed results for Mg+-CH3CH20Hand Mg+-CH3CH0 are in reasonable agreement with the 1.77 f 0.1 and 2.28 f 0.1 kcal/mol deduced from experiment.’ As noted above, the relative
binding of Mg+-H20 and Mg+-CH30H is consistent with that deduced based on a comparison of the H+ and Mg+ affinities. Thus while the computed binding energies are very different from the experimental value, the relative binding energies seem to be quite good.
Conclusion As expected the ab initio calculations show the bonding in Mg+L to be electrostatic in origin. The best computed Mg+CH30H binding energy is 35.9 kcal/mol (which includes a -1.1 kcal/mol zero-point correction), and this value is almost certainly a lower bound. The computed binding energy for Mg+-H20 is 35.0 kcal/mol. Based on previous experience, these values are estimated to be about 5 kcal/mol too small leading to our predicted values of 41 f 5 and 36 f 5 kcal/mol for the binding energy of Mg+-CH30H and Mg+-H20, respectively. The difference between the binding energies is similar to that deduced based on a comparison of the Mg+ and H+affinities, but the theoretical values are approximately 20 kcal/mol smaller than the upper bounds determined by photodissociation experiments. The lowest absorptions are computed to occur at about 80 kcal/mol, which are 17 kcal/mol higher than that found in the experiments. In addition, this strong absorption is to an excited state that is bound with respect to Mg+(2P) and CH30H or H20. Our calculations demonstrate that the binding energies are much smaller than the upper bound determined from the photodissociation experiments. The calculations offer a possible explanation to what is observed in experiment. The ions are hotter than assumed such that excitation to the strongly allowed excited electronic state is occurring at 61 kcal/mol. This excited state is then dissociated with a second photon or be predissociated. It is hoped that additional experiments will be performed on these systems to help clarify the difference between theory and experiment. The binding of H 2 C 0 to Mg+ is predicted to be 35 f 5 kcal/mol. The relative binding energies for CH’OH, CH3CH20H,and CH3CH0 are in reasonably good agreement with experiment providing additional confidence in the accuracy of the reported results. Acknowledgment. We thank Ben Freiser for bringing this problem to our attention. We also thank Andy Komornicki for helpful discussions. Registry No. Mg,14581-92-1; CH,OH, 67-56-1; HzO, 7732-18-5; H2C0,50-00-0.
The Dielectric Continuum Assumption as a Description of Medium Effects in Binary Solvent Mixtures. I mpiications for Eiectron-Transfer Processes in Cosoivent Systems Orland W.Kolling Chemistry Department, Southwestern College, Winfield, Kansas 671 56 (Received: February 9, 1990)
Most physical models for solvent effects upon electron-transfer reactions represent the solvent role as a dielectric continuum and incorporate the bulk solvent properties of dielectric constant and/or index of refraction into their solvent-dependentkinetics and thermodynamic terms. Because such models may fail quantitatively when applied to cosolvent mixtures, two isodielectric pairs were used as initial test systems to compare trends in classical continuum functions with changing solvent composition. Particular functions considered included those of Marcus, Kirkwood, Block-Walker, and Brady-Carr; the cosolvent pairs were benzene-CCI, and nitrobenzene-acetonitrile. The conditions under which the mean-spherical approximation (MSA) model of solute solvation reduces to the simpler continuum models have been investigated.
Introduction Medium effects upon electron-transfer reactions between molecular and/or ionic species in solution can be quite pronounced and several theoretical models have been developed to interpret such solvent influences. Usually, the model for the process assumes that the solvent forms a dielectric continuum surrounding the 0022-365419 112095-3950$02.50/0
reactant and product species as well as any transition-state structures. The Marcus treatment is the most important example of this type of model and eq 1 gives the solvent reorganizational X, = (Ae)2(0.5/al
+ 0.S/a2 - 1 / R ) ( 1 / n 2 - 1 /es)
(1)
energy contribution (A,) to an intermolecular electron transfer 0 1991 American Chemical Society
Medium Effects upon Electron-Transfer Reactions Here, aI and a2 are reactant radii, R is their internuclear distance, n2 and e, are the optical and static dielectric constants, respectively, and Ae the magnitude of the charge being transferred. When extended to electron-transfer reactions in mixed solvents, the approximations of dielectric continuum models often prove to be incorrect as selective solvation and irregularities in cosolvent viscosities complicate the As an alternative, Lay2 has proposed that the Marcus reorganizational energy function be resolved into a linear sum of several very specific solvent contributions, including hydrogen bonding. The current report is focused upon the role of the medium itself in electron-transfer processes and is the outcome of a search for suitable binary solvent systems with which to test the various continuum models for medium effects. Initial criteria for selection of potential test solvent pairs were established in part from the Chastrette-Purcell nonhierarchical statistically based classification schemeas The specific criteria used herein were as follows: (a) initially exclude all hydrogen bonding solvents (classes HB and HBSA); (b) in order to minimize other specific solvation effects, exclude the stronger Lewis base solvents (class EPD); (c) examine solvents primarily in the FI principal component group where solvent polarizable characteristics dominate; and (d) include as possible model cosolvents those pairs within the same principal classes (Le. AD, AHD, ARA, ARP).5 The benzene-carbon tetrachloride system was the primary mixed solvent of choice since the two components are very nearly isodielectric with values of 2.274 and 2.228 (at 25 "C) for benzene and carbon tetrachloride, respectively. On the other hand, they differ substantially in their empirical dipolarities, i.e. 0.59 for benzene and 0.29 for CC14 on the Kamlet-Taft scale.6 A difference of this magnitude is comparable to that encountered with such dissimilar aprotic pairs as n-hexane and diethyl ether or acetone and dimethyl sulfoxide. Yet both benzene and carbon tetrachloride fall in the same solvent class in the Chastrette-Purcell general system for classification of pure solvent^.^
The Dielectric Continuum Assumption and Solvent Polarity Scales: Theoretical Considerations The multiple influences of the solvent upon rates for electron-transfer reactions are usually addressed through quantitative models in which the key solvent specification is given in terms of either (a) a bulk solvent physical property (Le. static dielectric constant, viscosity, or index of refraction), (b) an empirical reactivity parameter defined by comparison to one or more reference reactants or reactions, and (c) an empirical "solvent polarity" parameter based on a scale developed from a series of model solutes and having nearly universal and systematic correlations to some fundamental characteristics of the solvent (Le. solvatochromic shifts vs molecular dipole moments or electron pair donor-acceptor behavior). Theoretical approaches of the (a) and (c) categories are especially pertinent to the particular interpretive problems encountered with electron-transfer reactions in cosolvent systems. Classical Dielectric Continuum Models of Polar Solvent Effects in Single Solvents. Although the term "solvent polarity" is used qualitatively in several imprecise ways, a common definitional reference point is the early Onsager description for the mean electrostatic interaction at equilibrium between the central dipole of a reactant solute molecule and the dipole moments of all adjacent solvent molecules.' In the Onsager model, the force field experienced by the solute molecule is resolved into a cavity field component determined by the dimensions and shape of the solute molecule and a reaction field component (Le. solvent polarity) ( I ) Marcus, R. A. J . fhys. Chem. 1%3,6?, 853-857. (2) Lay, P. J . fhys. Chem. 1986, 90,878-885. (3) Lai, C.; Freeman, G. J . fhys. Chem. 1990.94, 302. (4) Blackbourn, R.; Hupp, J. lnorg. Chem. 1989, 28, 3786. ( 5 ) Chastrette, M.; Rajzmann, M.;Chanon, M.;Putcell, K. J . Am. Chem. SOC.1985, 107, 1-1 1. (6) Kamlet, M.; Abboud, J.; Abraham, M.; Taft, R. J . Org. Chem. 1983, 48, 2817. (7) Onsagcr. L. J . Am. Chem. Soc. 1936, 58, 1486-1493.
The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 3951 as an influence coming from the surrounding molecules of the solvent. The latter includes both an orientational effect upon the solute molecule and to a lesser extent a coupled inductive effect from the neighboring dipolar solvent molecules. For the solute solvent interaction energy, the Onsager version of a reaction field model can be stated in a simple generalized manner by eq 2, where
E,,, is a composite solvation energy for the dipolar solute having a permanent dipole moment p and a, the molecular radius of the solute. Formal expressions derived from the Onsager model for the free energy and entropy change accompanying solute solvation have been reported by Bishop.* In the original Onsager model, the role of the dielectric continuum from the surrounding solvent molecules was expressed by eq 3 in terms df the bulk dielectric constant (e); it differs from
A€)= 2(c - 1)/(2c
+ 1)
(3)
the Kirkwood continuum function by only the coefficient. Other more refined continuum functions forflc) have been proposed and among those stated in terms of bulk dielectric constant alone, the e(t) quantity of Abboud and Taft given by eq 4 is the most
=
3c In t t h e - c + l
- -6-
Ine
2
(4)
~ignificant.~In this instance the added theoretical feature of an exponential shift toward a condition of dielectric saturation across the solute cavity boundary is incorporated, and it is conceptually deduced from the study of Block and Walker.Io As was stated by Abboud and Taft? the e(c) function is built on an equilibrium model that assumes a localized solute dipole at the center of a spherical cavity. It does not include distortional inductive-orientational effects from highly polar solvent molecules within the surrounding reaction field. In order to overcome these limitations on the e(t) function, Brady and Carr" have proposed that a modified dielectric constant (ep) replace e with the stipulation that it include an inductive-orientationalcomponent originating from the solvent molecule and acting in opposition to the permanent dipolar influence from the dielectric constant alone. In eq 5 the tp=t-n2+1
(5)
r? term is the square of the index of refraction (or optical dielectric constant in the Marcus theory), and tp exhibits the requisite limiting behavior that tp N 1 at the lower t values for nonpolar liquids where the usual condition is that t H n2.11 Molecular Dynamics of Polar Solvent Effects in Single Solvents. It is also important to compare the classical continuum treatments of reactant solvation with the predictions from theoretical interpretations of the molecular dynamics of solute solvation. For the latter, the recent extensive work of Chandra et aI.I2J3is representative and especially useful in establishing the requisite conditions for the merger of the dynamic molecular models of solvation at an equilibrium state with continuum descriptions of that solvation equilibrium. The mean-spherical approximation (MSA)I3 model for obtaining the equilibrium solvation energy of the solute in a pure solvent begins with the equilibrium polarization function in eq 6, where is the permanent dipole moment for the solvent
(8) Bishop, W. H. J . fhys. Chem. 1979,83, 2338-2344. (9) Abboud, J.; Taft, R. J . fhys. Chem. 1979,83,412-419. (IO) Block. H.; Walker, S. Chcm. Phys. Lcrr. 1973. 19, 363-364. ( I 1 ) Brady, J.; Carr. P. J . fhys. Chem. 1982, 86, 3053-3057. (1 2) Chandra, A,; Bagchi. B. J . fhys. Chem. 1989, 93,6996-7005. (13) Bagchi, B.; Chandra, A.; Fleming, G. J . fhys. Chem. 1990, 94, 5 197-5200,
3952
The Journal of Physical Chemistry, Vol. 95, No. 10, 19#91
t
01 4
'
.
10 .
'
I
"
: "
RJ
Figure 1. Trends in the equilibrium solvation energy (E) with changing dielectric constant (e) of the solvent and with changing solute/solvent radius ratio ( R ) . Curves: B = Born continuum model; M = MSA dynamic solvation model; 6 = Block-Walker reaction field model. Dashed lines are linearized limits for the functions.
molecule, p the equilibrium solvent density function including the positional (r), orientational ( w ) , and time-dependent (t) vectors, and & ( w ) the unit vector with orientational w . Thus, the equilibrium strucrure of the polarized liquid surrounding the reactant solute can be determined from the changes in the equilibrium density function for the dipolar liquid solvent, which also depicts the dependence of the solvation energy (Eh) upon P(r,t).I2 For the objectives of the present investigation, the shift in the solute solvation energy as a function of both solvent dielectric constant t and the size ratio R comparing the molecular radii (r,lute/r,lv) are primary considerations. E,lV(MSA) = 4u3'2R( 3112 $ ) $ a , ( r ) dr
(7)
The variables 9 and aI in eq 7 refer to the solute charge and solute radius, respectively, with Emhbeing computed in units of -q2/2a. The overall trends in these variables are shown in Figure 1 which is based on the papers of Chandra et al.lzJ3 In each instance, the comparisons between the calculated MSA values for Ed and those for the representative continuum model of Born (Eb,,, = 1 - 1 /t) are depicted as well. Several relevant conclusions can be drawn from the graphic comparisons in Figure 1 between the molecular dynamic treatment (MSA) and continuum models for the solvation energy in a single electron-transfer process in pure solvents: 1. With respect to the influence from changes in the dielectric constant for an ideal polar solvent, the overall trends for the Born, MSA, and Block-Walker models are parallel in form and with a nonlinear increase in E,,, up to t 15. Although not shown in Figure 1, the Onsager function is nearly coincident with the Born curve. 2. At higher t values, both the Born and M S A descriptions become linear with only a small slope (-0.002), implying a nearly constant solvation energy or at least a simple proportionate increase in this t range. 3. The greatest difference between the Born and MSA models is seen for the influence of the solute/solvent radius ratio upon E& For the continuum model, the solvation energy is independent of R. However, the E,,,, function for the MSA model increases sharply with R up to R H 4,but beyond that it exhibits only a small and an approximately linear trend (slope -0.0075). Thus, for large spherical solute molecules undergoing electron transfer
Kolling in common polar solvents, the distinction between the Born and M S A interpretations of the R dependence fades at the higher R values. 4. Although not included in Figure 1, the t dependence of the relaxation time for solvent reorganization about the solute species was found by Chandra et a1.I2 to merge toward the classical continuum description (zero slope) a t higher values of R. Therefore, it is clear that the major differences in polar solvent effects predicted by the MSA dynamic and the classical continuum models for electron transfer become important in lower dielectric constant media (e < 5 ) and for relatively small reactant molecules (R I! 1-3). These distinctions are well illustrated by the recent successful extension of a linear solvation energy description to the representation of the role of the solvent upon the forward electron transfer by the zinc( 11) porphyrin-amidoquinone (ZnTPPQ) m~lecule.'~ In this instance, the solute (ZnTPPQ) is hemispherical with a calculated radius of about 7.5 A, based on available related structural data.I5 For common polar solvents, the ZnTPPQ/solvent R values fall in the computed range of 4.1->11, namely in that region where predictions of the M S A and continuum models become largely indistinguishable. Empirical Scales for Solvent Polarity. A number of empirical multiple parameter linear equations have been developed to correlate a particular solvent-dependent aspect of a reaction or equilibrium with the solvation properties of the medium for that process. Among those linear functions, the Kamlet-Taft 'linear solvation energy relationship" as a free energy function has been the most completely investigated6 and the usual general statement of that relationship is given in eq 8 with respect to the primary
XYZ = XYZo + S ( U *
+ db) + UCY+ bp + hbH + et
(8)
variable XYZ (i.e. In k or In Kq). The most fundamental parameter in eq 8 is u* (a dipolarity parameter), which represents rather accurately the strength of dipole-dipole interactions originating with the solvent and correlates well with molecular dipole moments for pure solvents. Likewise, physical models supporting a molecular interpretation for the u* scale have been developed and analyzed by Abboud and Taft16J7as well as by Brady and Carr" using the reaction field theories. The additional molecular solvent characteristics included in eq 8 are the secondary db term for induced dipolar contributions (related tofln2) functions), the a value for hydrogen bond donor acidity, the /3 value for hydrogen bond acceptor basicity, and the Hildbrand parameter aH associated with solvent cavity creation by the solute.6 The final term measures electron pair donor strength and is significant only for very strong Lewis base solvents. The empirical weighting coefficients s, a, b, h, and e, obviously determine the relative magnitudes for each of the contributing terms in eq 8; for influences from nonpolar to very polar solvents, the s value is the greatest determinant regulating the value of XYZ.
Experimental Section Materials. Spectral grade benzene and carbon tetrachloride were dried over anhydrous calcium sulfate for one week, and this was followed by fractional distillation through a Raschig column. The center 65% cut was retained and the solvent purity was verified from reference data for normal boiling point and refractive index.'* Distilled solvents were passed through a chromatographic column of alumina just prior to their use. Water content determined by Karl Fischer titration did not exceed 0.001%. The purification procedures for the solvatochromic indicators have been reported p r e v i ~ u s l y . ' ~ ~ ~ (14) Kolling, 0. W.J . Phys. Chem. 1991. 95. 192-193. (1 5 ) Spaulding, L.; Eller, P.; Bertrand, J.; Felton, R. J . Am. Chem. Soc. 1974, 96,982-987. (16) Taft, R.; Abboud, J.; Kamlet, M.J . Am. Chem. Soc. 1981, 103, 1080-1086. (17) Abboud, J.; Guiheneuf, G.; Essfar, M.;Taft, R. W.; Kamlet, M.J. J . Phys. Chem. 1984,88,4414-4420. ( I 8) Riddick, J.; Bunger, W. Organic Sohents; Wiley-lnterscience: New York, 1970; pp 606-61 1 and 773-775.
Medium Effects upon Electron-Transfer Reactions TABLE I: Specific Linear Solvation Energy Relationships (LSER) nt 25 O C
solvatochromic indicator function Calibration Functions for Pure Solvents ( n = 15) , ,X = 4 9 . 0 ( ~ *- 0.156) + 554 Phenol Blue Nile Blue A oxazone A, 41 .O(r* - 0.256) + 507 1 -methyl-4-acetylpyridinyl radical" AH(acetyl) = 1.90(s* - 0.146)
+ 1.46
EmDirical Functions for Cosolvent Benzene-CCL- (.n = 1 1 ) ,A, = 49.0[** - ( 0 . 0 7 5 X c ~ Phenol Biue (6cCl4= 0.5; dCsHs = 1 .O)
+ 0.1SX,)] + 554
Nile Blue A oxazone (?ice,, = 0; , , X = 41.0(;* - (0.25X,) + 6C6H6 = 1.0) 507 I-methyl-4-acetylpyridinylradical" AH(acetyl)= 1 . 0 9 [ ~ *(O.O70X~~~, + 0.14Xb,)] + (bee,, = 0.5; 6CsHs = 1.0) 1.46
ESR probe. Instrumental Procedures. Measured dielectric constants at 25.0 OC for benzenecarbon tetrachloride mixtures were derived from changes in dielectric cell capacitance by following procedures already described.22 Literature valuesI8 for the pure solvents were used for calibration of the cell. Dipolarity values by the "solvatochromic comparison method" were obtained by following the procedures of Kamlet, Abboud, and TaftSz3 The two solvatochromic indicators selected for this purpose were Phenol Blue (PB) and Nile Blue A oxazone (NBAO), since these are capable of yielding some of the more precise r* values.20Jl Calibration functions on select pure solvents are given in Table 1. Methods for ESR spectral measurements on solvent mixtures using the 1-methyl-4-acetylpyridinyl probe have been reported in detail?4 However, in order to increase the precision the number of recorded scans was increased to 20 and the number of samples for each solvent mixture was expanded to 10.
Results and Discussion If the solvent pair is one in which the types and magnitudes of the solvent-solvent molecular interactions in the mixture are indistinguishable from those within the pure solvents and hydrogen bonding is absent, then the composite polarity of the mixture would be expected to conform to a simple summation from the proportionate contributions of the components of that mixture. For components I and 2 having mole fractions XI and X2, the linear combination in eq 9 should express the weighted composite polarity
of the mixture. Suppanu substituted the Onsager and Debye functions in dielectric constant (e) f o r m , ) descriptors of polarity; however, in the present study several other functions in both index of refraction (n) and static dielectric constant were considered. Even slight deviations from linearity in eq 9 such as A&,,,) are discernible and will identify those nonideal solvent pairs characterized by stronger dipole l-dipole 2 associations and/or cosolvent complexation.25 Although a fortuitous confirmation of eq 9 is possible through hidden arithmetic compensations from within pI and p2, it is clear that no binary solvent system for which A h , , , ) # 0 can be validly interpreted as a true continuum. The particular functions for may be placed into two separate groups: those expressed in a single polarity variable, e or t?, and those containing both t and n2 in reaction field functions in which the orientational and electronic polarizational components (p,)
mi)
(19) Kolling, 0.;Goodnight, J. Anal. Chcm. 1973, 45, 160-162. (20) Kolling, 0. W. Anal. Chem. 1976,48, 884-885. (21) Kolling, 0. W. Anal. Chem. 1981, 53, 54-56. (22) Kolling, 0. W. Trans. Kans. Acad. Sei. 1979. 82, 218-224. (23) Kamlet, M.; Abboud, J.; Taft, R. J . Am. Chem. Soc. 1977, 99, 6027-6037. (24) Kolling, 0. W. Anal. Chem. 1983,55. 143-144. (25) Suppan, P. J . Chem. SOC.,Faraday Trans. I , 1987, 83, 495-509.
The Journal of Physical Chemistry, Vol. 95, NO. 10, 1991 3953 TABLE II: Summary of Spectral Data at 25 OC for Probes in Benzene-CQ Mixtures a d Derived r* Values XCCI, A,,(PB)O A,(NBAO)' AH(acetyUb #(mean) 0.592 1.960 575.0 521.3 0.000 0.561 1.935 0.100 574.2 520.9 0.531 I .908 0.200 573.1 520.6 0.503 1.883 520.4 0.300 572.2 0.47 1 1.857 520.2 0.400 57 1 .O 0.442 1.830 570.0 520.0 0.500 0.412 1.805 519.8 0.600 569.2 0.384 1.778 568.2 519.7 0.700 0.353 1.752 567.1 519.4 0.800 0.324 1.726 0.900 566.0 519.3 0.293 1.701 565.2 518.9 1.ooo &0.004c *0.003C *0.2c *0.3c
"Wavelength in nm for the absorption maximum of the solvatochromic indicator. bHyperfine splitting constant in G. Uncertainty (std dev). TABLE HI: Nonspectral Cosolvent Parameters at 25.0 OC for the Benzene-CCL Solvent Svstem ~~
X C C ~ em" 0.000 2.274c 0.100 2.269 0.200 2.264 0.300 2.260 0.400 2.255 0,500 2.251
.Db
1.4980 1.4941 1.4903 1.4857 1.4822 1.4773
XCC, 0.600 0.700 0.800 0.900 1.000
em" nob 2.246 1.4739 2.242 1.4698 2.237 1.4655 2.233 1.4621 2.22gC 1.4578 10.003d &0.0002d
"Dielectric constant of the mixture. blndex of refraction for the mixture. CStandard values used for calibration. duncertainty (std dev).
are separated. Though not comprehensive in scope, the following are representative functions selected for testing the data for the isodielectric binary solvent systems. The notation is adapted from that used by Brady and Carr.11v26 Single-variable continuum functions: Kirkwood-Onsager-Debye [K(e)] (10) f i e ) (e - 1)/(2t + 1) Lorenz-Lorentz [L(n2)]
f i n 2 ) (n2 - 1)/(2n2 + 1) (11)
Block-Walker (Abboud-Taft) l6 [e( e)] 3e In t - -6- 2 fie) = cine-c+l Ine Two-variable continuum functions: 1 1 Marcus1 [M(e,n2)] fie,n2) --
;;;
Dipolar component [Lo(c,n2)]
f(c,n2)
0
t
2(c - n2) 2c + 1
Dipolar permittivity28 [c,m] 4tn2 - 5 e - n2 1 (17) fie,n2) 4cn2 - 4( + 8 n 2 + 1 Equation 12 has been restated by Brady and CamZdin the same algebraic form except that the static dielectric constant is replaced by e ,defined in eq 15. The relative merit of the various forms for tkejlp,) functions has been thoroughly discussed in the papers of Brady and Carr11*26+28 and will not be reviewed here since the central objective of this investigation was to establish the pattern
+
(26) Brady, J.; Carr, P.J. Phys. Chcm. 1984, 88, 5796-5799. (27) Bekarek, V. J. Phys. Chem. 1981,85, 722-123. (28) Brady. J.; Carr, P.J . Phys. Chcm. 1985, 89, 5759-5766.
Kolling
3954 The Journal of Physical Chemistry, Vol. 95, No. 10, 199‘1 TABLE I V Derived Linear Functions for Continuum Models Representing Cosolvent Systems fin) m 6 SD’
Benzene-CCI,* c
n fP
%fl
K(c) L(n2) K(t)L(n*) 4(c.n2) M(t,n2)
e(€)
-0.0460 -0.0402 0.0730 0.0240 -0.00450 -0.01 24 -0.00381 0.0269 0.0156 -0.004 IO
2.274 1.4980 1.030 1.009 0.2296 0.2267 0.05205 0.01082 0.00585 0.1271
(K, *O.OOI f0.0002 *0.001 *0.0006 *0.0001 *0.0001 *O.ooOo2
*0.0005 *0.0002 *0.0012
Nitrobenzene-AcetonitrileC c
K(d e(t)
1.19 0.00070 0.0029
34.82 0.4788 0.4398
onstrated. The assumed proportionate connection expressed by eq 9 can be restated as e, = (e2 - cl)& + el. Then, the statement in eq 18 follows where subscript m refers to the mixture. For the
*0.02 *0.0001 *0.0001
aStandard deviation ( n = 1 I ) forf(Pm)alsd. bfcp,) = mXcc14+ 6.
‘.IIJ”) = mxCH,CN + b.
for the representativef(p,) quantities with respect to changing solvent composition. An initial test on the experimental Kamlet-Taft A* values in Table I1 demonstrated that the ( A * + d6) variable conforms to eq 9 within the precision of the data. The specific equations were first deduced in terms of the XYZ (eq 8) response of the probe (Le. ,A, or A H ) and these are given in Table I. However, since the relative precision in the A* values is significantly less than that for the measured t and n data, this test cannot be considered to be definitive. For the benzene-carbon tetrachloride system, the trend in the c and n data in Table 111 with respect to cosolvent composition (Xcc14)is a linear one within the precision of the experimental values. This linearity carries over to all of thef(p,) functions in both the single- and two-variable relationships. The specific coefficients and intercepts for each of these continuum functions are listed now in Table IV and are derived from the usual least-squares treatment of the data. It will be noted that those instances showing negative slopes (i.e. K ( t ) , L(n% K(c)L(nZ),and O(t)) correspond to the condition of decreasing polarizabilitydipolarity of the medium in going from benzene-rich to CC14-rich mixtures. This behavior parallels the trend in the A* values for these nearly isodielectric mixtures, while being opposite that shown by the Marcus function as well as by ep. Also, the small values of m for the parametric variables, K(c)L($) and e(€), suggest that there can be considerable risk in making inferences about “ideal” vs “nonideal” isodielectric cosolvents based solely upon these more complex functions applied to narrow t intervals. By use of synthetic data in which e and/or d deviates only slightly from linearity in X,,it can be shown that the algebraic form of the O(t) and K(c)L(d)functions can minimize the nonlinear contributions from the primary variables (c and n). The circumstances for the empirical linear trends to occur between the Kirkwood and similar functions vs solvent mole fraction in nearly isodielectric binary systems can be easily dem-
+ 1)/(1
- 2K,) =
(€2
- tl)X2 + € 1
(18)
nearly isodielectric pairs, the calculated values of K , are small (0.46-0.48)and nearly constant with the effect of keeping the denominator (1 - 2Km)relatively constant. With such a restriction the empirical proportionality K , = X2 follows. With respect to other bulk solvent properties for mixed solvents, further support for the fit of the continuum model to benzene-CCI‘ can be found from its very small excess volume of mixing (AV@ maximum of 0.06 mL/mol) and its only slightly positive value in A*GE for activation of viscous The nearly isodielectric binary solvent, nitrobenzene-acetonitrile, was selected as a second test pair, although it was in the more intermediate dielectric constant range. The interval included here is 34.8-36.0(at 25 “C) and in terms of both the MSA and continuum models for solute solvation this range is well within the area where Ed” is essentially independent of dielectric constant (see Figure 1). On the other hand, just as is true for the benzene-CCl, case, the pure components (nitrobenzene and acetonitrile) differ substantially in their A* values (1.01 and 0.75, respectively) and in their molecular dipole moments. Initial tests on the single-variable comparisons of the dielectric constant, the Kirkwood-Onsager, or the e(€) functions with shifting mole fraction of the solvent demonstrated the linearity in conformity with eq 9. Specific slope-intercept values of the correlations for the nitrobenzeneacetonitrile cosolvent system are given in Table IV. Finally, Blackbourn and Hupp4 have advocated the use of the entropy changes accompanying electron-transfer reactions in binary solvents as probes for evidence of selective solvation of reactants and products. Their choice for the electrochemical reaction medium was the acetonitrile-dimethyl sulfoxide system. Although acetonitrile has long been used as a solvent for electrochemical studies, there is ample evidence that acetonitrilecosolvent pairs are usually not ‘ideal mixtures” in terms of”) vs X2 trends; the case of AN-DMSO exhibits such representative behavior.)O The significant positive value of the dielectric excess function (At,,) for AN-DMSO in acetonitrile-rich mixtures would suggest that the composite ASo, values‘ for redox couples in this cosolvent may well contain contributions that should be attributed in part to structure-breaking and reorganizational processes within the cosolvent itself as well as those to be assigned to unsymmetrical selective solvation of reactant species. In this regard, the acetonitrile-dimethylformamide and acetonitriledimethylacetamide binary systems more closely approach the requirements for the dielectric continuum models than does the AN-DMSO pair.’O Registry No. CCI,, 56-23-5; benzene, 71-43-2; nitrobenzene, 98-95-3; acetamido. 7065-76-1. (29) Heric, E.;Brewer, J. J . Chem. Eng. Dora 1967, 12, 574-583. (30) Kolling, 0.W.Anal. Chem. 1987,59,674-677.
