J Phys Chem 1987, 91, 1868-1873
1868
Boundary Continuity and Analytical Potentials in Continuum Solvent Models. Implications for the Born Model S. Ehrenson Department of Chemistry, Brookhaven National Laboratory, Upton, New York 1I973 (Received: September 2, 1986: In Final Form: December 8, 1986)
Some implications of imposing dielectric continuity at regional boundaries in continuous medium solvation models are considered. The familiar inverse radial exponential function of Block and Walker is examined and shown to be a member of a family of functions of similar properties. This family includes the Onsager or homogeneous medium function as a special member whose use produces the most common type of boundary discontinuity in continuum medium models. Application of various members of this family of functions to modify the Born model of ion solvation is described as are extensions to dual solvent region models. Near-homogeneous medium functions which simultaneously provide continuity and relatively simple potential expressions are outlined and discussed.
Some time ago, in an attempt to remedy the boundary discontinuity deficiency of the Onsager theory’ of gaseous and liquid-state dipole moments, Block and Walker2 introduced the following radial dependence to dielectric functions for liquid continua t(r) =
tg
exp(-K/r) = cB(’-‘/‘)
(1)
where tg is the bulk medium dielectric constant. They chose this function on the grounds that it provided an analytic solution to the differential equation for the radial potential, R, of an ideal dipole a t the center of a spherical cavity of radius a, i.e.
without requiring parameterization other than that imposed by the natural boundary conditions. The latter are, of course, that c(a) = 1, wherein lies the difference with the Onsager formulation, and €(a)= tg. Upon application of this modification, Block and Walker found improved correlation between the gas-phase dipole moments and liquid-phase dielectric constant properties of a variety of nonassociating molecules. Although not specifically attributed to such by these authors (they do, however, mention “local molecular interactions” ... resulting in ... “local order which generally, but not inevitably, have the effect of reducing permittivity”), the implication that their boundary continuity function might simulate the physical phenomenon of solvent dielectric saturation by solute-induced electric fields is, at least superficially, plausible. Since this time other uses of the formulation have been made3-6 and the implicit interpretation has gained credibility. The fact that improved results, to whatever degree, over those obtained with Onsager-type models will increase its’ future acceptance and lend credence to the electrical saturation rationale suggests that closer scrutiny might be profitable. The Block and Walker and Related Functions
The dielectric function, eq 1, may be readily shown to be the second in the series of simply related functions cn(r) =
tg
exp(-K,,/r‘)
z = l/n;
n = 0, 1, 2, 3, ...
(3)
( 1 ) Onsager, L. J . Am. Chem. SOC.1936, 58, 1486. (2) Block, H.; Walker, S. M. Chem. Phys. Lett. 1973, 19, 363. (3) (a) Abboud, J.-L.; Taft, R. W. J . Phys. Chem. 1979, 83, 412. (b) Kamlet, M. J.; Abboud, J.-L.; Taft, R. W. Prog. Phys. Org. Chem. 1980, 13,
485. (4) (a) Ehrenson, S. J . Comput. Chem. 1981,2,41. (b) J . Am. Chem. SOC. 1981, 103, 6036. (c) Ibid. 1982, 104, 4793. (5) Brady, J. E.; Carr, P. W. J . Phys. Chem. 1985, 89, 5759. (6) Abe, T. J . Phys. Chem. 1986, 90, 713.
0022-3654/87/2091-1868$01.50/0
all of which lead to analytic solutions of the differential eq 2.’ The first in the series is the Onsager function, a fact which will later be useful in the physical interpretation of previous and current results. All members of the series are uniquely defined by the boundary conditions at r = a and r = m (Le., with K, = alln In tB and when n = 0, in the limit of r a, vide infra). There are, however, negative aspects in such tight definition, especially if physical interpretations are sought. In particular, even for n = 1, the dielectric constant increases more slowly with r toward the bulk value limit ( e B ) than continuous field saturation appears to reFor larger n, the increase is even more gradual. It is true, of course, that continuous field saturation functions may not provide appropriate physical benchmarks over the entire solvent environment. On the grounds that distinction may be made between layers of solvent molecules on the basis of their ordering by solute, we may wish to retain structural features of continua only within shells representing these layers (Le., as averages over angle, and by single well-behaved functions within such shells). After all, it may be argued that this has already been done to the zeroth-order in defining the differences between the inside and outside of the original cavity (distinguishing solvent from solute, even though this is artificial, e.g., in the original Onsager and Block and Walker dipolar solvent applications). Following this reasoning suggests adoption of individual dielectric functions, each wellbehaved within its own region, and pieced to conform with the conditions of continuity at the boundaries, as in the Block and Walker and current expanded formulation^.^
-
Application to Modified Born Equation Calculations
Recent use of the Block and Walker (henceforth BW) function by Abe6 to modify the Born equationlo.” substantially improved (7) See,for example, Adams, E. P. “Smithsonian Mathematical Formulae and Tables of Elliptic Functions”; Smithsonian Miscellaneous Collections, 73, No. I , Publication 2672; Smithsonian Institution: Washington, 1947; Chapter VIII, eq 8.635, and the Appendix wherein an explicit solvable differential equation form of eq 2 and some solutions of particular interest are detailed. (8) Buckingham, A. D. J . Chem. Phys. 1956, 25, 428. Buckingham, A. D. Discuss. Faraday SOC.1957, 24, 151. Note that even faster approach to the bulk dielectric condition with distance is implied for ideal dipole-than ion-induced fields (proportional respectively to r4 and r4) in the Buckingham development. (9) These continuity conditions should not be confused with the conditions of equality of the potentials and of the normal components of the dielectric displacements at the boundaries, a, i s . , R+ = R. and t+(a)(dR+/dr), = c.(a)[dRJdr]a, which define the linear coefficients C, of the potentials. ( I O ) Born, M. 2.Phys. 1920, 1 , 45. ( 1 1 ) Others including Stiles and Laidler and Muirbead-Gould have also derived and used integral equations for Born charging energies where the dielectric constant of the solvent is radially dependent. (a) Stiles, P. J. Aust. J . Chem. 1980, 33, 1389. (b) Laidler, K. J.; Muirhead-Gould, J. S. Trans. Faraday Sot. 1967, 6j3953.
0 1987 American Chemical Society
Analytical Potentials in Continuum Solvent Models
The Journal of Physical Chemistry, Vol. 91, No. 7, 1987
TABLE I: Free Energies of Solvation for K+ Ion Computed from the Born and Single Solvent Region Model Modifications“ A4G8’(obsd - calcd) -ACSon =0 n=1 solvent tg (obsd) (Born) (BW Mod) n = 2 l,l-dichloroethane 1,2-dichloroethane tetrahydrofuran 1,2-dimethoxyethane ammonia acetone acetonitrile nitromethane N,N-dimethylformamide dimethyl sulfoxide N-methylformamide formamide 1 -propanol ethanol methanol water
9.9 10.2 7.4 7.3
73.2 74.2 73.3 73.1
33.4 32.9 29.4 29.3
-2.9 -3.4 -7.7 -7.8
-20.1 -20.7 -24.6 -24.6
16.9 20.5 36.0 37.0 36.7
79.1 76.7 77.6 78.9 79.5
32.8 36.7 38.2 36.6 36.0
-1.3 3.5 7.9 6.4 5.7
-18.7 -13.8 -9.0 -10.5 -11.2
46.7 182.4 109.5 20.4 24.3 32.6 78.4
80.5 81.9 81.8 77.1 78.0 79.9 80.8
35.5 36.3 36.4 36.1 36.2 35.4 35.4 (34.9)
6.4 13.2 11.2 2.9 3.8 4.6 8.8 (6.8)
-10.2 -1.8 -4.4 -14.4 -13.4 -12.4 -7.3 (15.0)
“All energies in kcal/mol. Parenthesized quantities at feet of col-
umns are the rms values.
the latters’ ability to correlate the free energies of solution of singly charged ions, particularly if attention was restricted to nonpolar solvents. Including polar solvents resulted in less satisfactory agreement with experiment, an observation which appears consistent with previous analysis involving saturation effects. Thus, while the BW function may represent the near-cavity boundary electrical properties of solvent better than the Born (Onsager) function, it seems unable to discriminate the neighboring and distant solvent properly when the bulk dielectric is high. This distinction among solvents based on their polarities is not lost, it should be noted, when correction of the computed electrostatic energies by the several kcal/mol required to form the ionic cavity in the particular solvent is made.12 The corrected values are, in fact, closer to the experiniental solvation energies for the K+ ion, but still range from 10%too small for the least polar solvents to 10% too large for the most polar and have an average error about equal to the total spread of observed energies. See Table I, column for the n = 1 function. In contrast to the Born equation, AGSo = e2LZ2(l/tB- 1)/2a, which under modification by the BW function, n = 1, becomes
1869
TABLE 11: Solvation Free Energies for K+ Using Dual Solvent Region Models, Inner Shell Homoeeneousasb ~~
n, = n2 = 0
solvent
rC
1,l-DCE 2.600 1,2-DCE 2.546 THF 2.569 I,2-DME 2.789 1.796 NH3 acetone 2.485 2.222 CH,CN CH3N02 2.238 DMF 2.524 Me2S0 2.456 NMF 2.307 formamide 2.023 1-PrOH 2.498 EtOH 2.300 2.037 MeOH 1.553 H20 %undary
outer region homogeneous 44GS0 R 0.5 0.91 -0.1 0.91 -0.7 0.88 -1.4 0.88 1.2 0.92 0.0 0.95 1.4 0.97 -0.4 0.97 -2.6 0.97 -3.3 0.98 -2.8 0.99 -0.6 0.99 -0.6 0.95 -0.1 0.95 0.0 0.96 2.7 0.98 (1.57) (0.95) 2.0, t g
n, = 0; n2 = 1 outer region BW 44GS0 R 1.3 0.5 0.3 0.1 -0.2 0.6 1.6 -0.1 -1.7 -2.3 -1.2
0.0 -0.1 0.0 -0.4 1.5 (1.03) 2.35
n, = 0: n, = 2 J
44G,O
0.79 1.8 0.79 1.0 0.76 1.1 0.78 1.0 0.76 -0.8 0.82 0.6 0.83 1.4 0.83 -0.4 0.85 -1.7 0.85 -2.4 0.88 -1.2 0.86 -0.4 0.82 0.0 0.82 -0.1 0.81 -0.8 0.81 0.5 (0.82) (1.12) 2.50
R 0.75 0.74 0.73 0.74 0.70 0.77 0.77 0.78 0.80 0.80 0.83 0.79 0.77 0.76 0.75 0.74 (0.77)
Footnote a of Table I pertains. b R is the fractional energy contribution of the inner shell, see text. ‘The solvent (and hence the inner shell) “thickness”, in A, see text.
Based on this and other considerations, no single function bulk dielectric modification of the type presented, including those with fractional n, appears capable of matching the surprisingly good correlations of solvation free energies reported in ref 13 with a dual homogeneous solvent region model.
Dual Solvent Region Born Type Models If distinct regions of the solvent characterized by different spatial dielectric dependences are recognized, the free energy of solvation within the Born (spherical field) framework may be written generally as
-
-
if the n = 2 function is used, the expression for free energy of solution, AGs0, is (1
- t g + t g In
tg)
- 1
where e2L/2 is 166 kcal/mol, and for the K+ ion, Z = 1 and a = 1.33 A. The results obtained with this more slowly varying modification function are also presented in Table I; they are almost always, except those for the most polar solvents, in substantially poorer agreement with experiment than those obtained with the BW function. It is furthermore apparent from this table that proceeding from the n = 1 to 2 functions has about half the effect on the free energies computed than did the n = 0 to 1 modification. Higher-n functions are expected to provide even poorer agreement with the experimental results. K+ in the solvents he considered, and for some reason neglected by Abe,6 are sufficiently large to affect quantitative comparisons of model and experimental results. They may be estimated by the method of Abraham and Liszi;” their paper is adopted as well, following Abe, as the source of all other experimental free energy and solvent property data henceforth employed. ( 1 2) Such corrections, which amount to 5-7 kcal/mol for
where d(r/a) 4r/a)(r/a)*
(7)
and 1, and u, are the lower and upper radial bounds, relative to cavity size, of this region of solvent within which the dielectric function t,(r/a) holds. For dual (solvent) region models, there are two finite boundaries of importance, Le., I,, at the cavity interface with nearest-neighbor solvent, and ul = I,, where the two solvent regions meet, and one u2, at infinity. In the Abraham and Liszi approach,13 two dielectric regions are employed, both of the Onsager (homogeneous medium) type. The inner, of thickness corresponding to the “thickness” of the solvent molecule (as defined by molar volumes within the Stern-Eyring pre~cription’~), is arbitrarily assigned a dielectric constant of 2. The outer, which occupies all the rest of solvent space, is supposed to have the bulk dielectric for this liquid. The single strong dielectric discontinuity of the original Born equation is replaced in this model by two. One minor discontinuity exists at the cavity-nearest-neighbor interface, and a second, generally strong discontinuity is found where the two solvent regions meet. The K+ ion results of ref 13 with this model are essentially reproduced in the second numerical column of Table II.15 They (13) Abraham, M. H.; Liszi, J. J . Chem. SOC.,Faraday Trans. 1 1978, 1604. (14) Stern, A. E.; Eyring, H. J . Chem. Phys. 1937, 5, 113. (1 5 ) Differences of no more than 0.2 kcal/mol from the ref 13 values are sometimes found, apparently due to different choices of “similar” solvents for estimation of the cavity formation corrections mentioned earlier,’* and/or roundoff.
1870
Ehrenson
The Journal of Physical Chemistry, Vol. 91, No. 7 , 1987 ,
.
35t I * e t Ill
I
'
I I' I t ' (IX
I
Ii I '
1'1
1
I I I I
I I I I
0 -I7 : H
-I4
c
II
1
t 11
5 I L
W
I
0
,
I
3
I
t
- 2 1 ,
,
I
,
,
,
V
I
-3 -4
*
,
;
1 5
80
75 -4G:
(kcal/mol)
Figure 1. AAG,O, observed - calculated, for , ' K as functions of the free energies of solvation AGSo for a variety of solvents and the radial dielectric function employed. (a) Single region functions with the arrows going from the n = 0 to 1 and to 2 type function results (see eq 3). (b) Optimized two region functions, n, = 0, with arrows indicating n2 = 0 to 1 to 2 function results and circles marking indistinguishably small AAG,O shifts upon these function changes. (c) Two region functions, n, = 1 and n2 = 1 to 2 change.
are displayed as observed minus calculated free energy differences and are accompanied by estimates of the relative contributions to the calculated AG,' values of the inner-shell and outer-solvent regions. The latter are presented as the ratios R = I,/(Zl + I*). Perhaps more striking than how well the model seems able to correlate the solvation energies is how much of the total contribution comes from the nearest-neighbor shell (>90% for all but the two least polar solvents). The relative constancy of this contribution is likewise remarkable. The standard deviation about the root mean square value of R is C0.04. Some reflection on the intrinsic simplicity of the homogeneous dual medium model and aided by examination of the first numerical column of Table I1 can help to rationalize these latter results. It is easily ascertained that IIin this model is (1 - I / ( 1 + p ) ) / 2 and I , is 1/( 1 + p)/cB, where p is the thickness of the solvent shell, r , normalized to the ionic radius, here 1.33 A.
IO
(r/o) Figure 2. Optimum fitting two-region dielectric distance profiles for H 2 0 and THF. The n, = 0, n2 = 1 profiles are easily distinguished by their homogeneous inner regions. The n , = 1, n2 = 1 curves are shown by dashed lines, the n , = 1, n2 = 2 curves by full lines. Crosses indicate solvent boundary conditions.
Consequently, R = 1/(1 + (2/p)/tB), and since tB>> 2 and p is usually 2, the generalizations are understandable. One should, by the same token, recognize the importance of choice of shell dielectric constant on the free energies computed, and why there would be little accompanying change in R. If the shell dielectric constant in this model is optimized, the result is 2.02 with rms for AAG,' dropping insignificantly to 1S O kcal/mol. Similar results have been obtained for Na+ ion as well, where the optimum e , value is 2.03, again statistically an insignificant improvement over the assumed value. Subsequent investigation should reveal whether this (Le., e l = 2) is in general a good characterization of the inner shell of solvent. The problem of important boundary discontinuity has, however, only been shifted from the cavitysolvent interface in the original Born model to between the inner- and outer-solvent regions here. If, alternatively, a nonhomogeneous medium model for the outer solvation region is adopted, with the constraint that its' dielectric constant match that of the inner shell at their common boundary, this inconsistency is easily removed. Moreover, if functions of
-
Analytical Potentials in Continuum Solvent Models
The Journal of Physical Chemistry, Vol. 91, No. 7, 1987 1871
---Ii-
I
(r/o)
Pigure 3. Single region dielectric-distance profiles for H20with n values of 0.0, 0.02, 0.1, 1, and 2, respectively, characterizing the plots from left to right.
the type discussed earlier, Le., the BW or higher-n value types, are appropriate, the criterion of analytic solvability of the radial potential equation within each region of space can also be satisfied. Table 11, last two column pairs, contains some fitting results for such two region models. In the first of these, the BW function is combined with the homogeneous shell whose optimized el value is 2.35 and whose thickness is that of the particular solvent molecule ( r of Table 11). The fit of the K+ free energies by this model is judged to be significantly better (to >99.9% level) than for the two homogeneous regions model. The relative contributions of the inner shell are also found to be significantly reduced in this modification. When the n = 2 function is employed to describe the radial dependence of the outer region dielectric, insignificantly poorer fitting (confidence level a) everywhere but at the boundary and hence the expansion may be limited to the first order in x, Le., X = x everywhere but at the boundary. All terms in X , independent of the magnitude of x , vanish with vanishing 6 to yield the correct Onsager limit for the potential. The corresponding cavity (E,) and reaction ( E R )fields for this model in the limit of small 6 are
(21) Cf. Courant, R. Differential and Integral Calculus, Blackie: London, 1937; Vol. I, pp 33, 52 and Appendix to Chapter 3.
J. Phys. Chem. 1987, 91, 1873-1877
1873
6) homogeneous in the region r = a to
m
('46) where X , is the value of X at the cavity boundary, where r = a . It is of some interest to note that in the limits of cB = 1 and 03, the 6-coefficients for the cavity field are zero and 2, respectively, and for the reaction field, 3 and zero. For the solvent range of usual interest, say, eB from 10 to 80, the former coefficient ranges from 0.8 to 1.4 and the latter from 1.8 to 0.9. What has been obtained are nonanalytic, albeit tractable, expressions for the potential and fields of probable interest corresponding to a dielectric function, eq A3, which is continuous at the cavity boundary, and has as its' recognizable limit the Onsager-Born form. To complete the development as regards application to free energies of solvation, where the solvent is (in the limit of vanishing
The energies may be obtained from tabulations of X , quantities or from the sum which is relatively rapidly converging for all media of interest. E.g., for eB = 120, only one sum term is required for better than 1 part in 1000 accuracy for AAG,' with 6 = 0.001 and five terms if 6 = 0.01; for cB = 10 to the same accuracy, one and three terms, respectively, are required with these &values. Similar but slightly more complicated expressions can also be derived for multiregion solvents with each or some regions approaching homogeneity in the limit of vanishing 6.
Kinetics of the Reactions of Unsaturated Hydrocarbon Free Radicals (Vinyl, Allyl, and Propargyl) with Molecular Chlorine Raimo S. Timonen: John J. Russell, Dariusz Sarzyiiski, and David Gutman* Department of Chemistry, Illinois Institute of Technology, Chicago, Illinois 60616 (Received: May 12, 1986; In Final Form: October 3, 1986)
The kinetics of the reactions of three unsaturated free radicals (vinyl, allyl, and propargyl) with molecular chlorine have been studied by using a tubular reactor coupled to a photoionization mass spectrometer. The radicals were homogeneously generated by the pulsed photolysis of precursor molecules at 193 nm. The subsequent decays of the radical concentrations were monitored in real-time experiments as a function of CI2 concentration to obtain the rate constants of these CI atom metathesis reactions. Rate constants were measured as a function of temperature to obtain Arrhenius parameters. The following rate constant expressions were obtained (units of the preexponential factor are cm3 molecule-' s-l, and activation energies are in kJ mol-'; the temperature range covered in each study is also indicated): C2H, + C12 (10-'1.06exp(2/RT), 298-435 K); C3H, C12 exp(-28/RT), 525-693 K); C3H5+ Cl, (10-'o.81exp(-l8/RT), 487-693 K). The factors governing the reactivity of polyatomic free radicals in these and other CI atom transfer reactions with molecular chlorine are reviewed.
+
Introduction The gas-phase reactions of hydrocarbon free radicals (R) with molecular chlorine are part of a well-known two-step free-radical chain which substitutes chlorine for hydrogen in the starting material:
+ R H -.HCl+ R R + CI, RCI + CI
CI
(A)
-
(B)
The differences in reactivity among these C1 atom transfer reactions appear to be associated in part with the formation of polar transition states whose stabilities are affected by the presence of electron-donating or -withdrawing groups attached to the carbon radical center, a situation also found in prior investigations of related reactions.I6-l8 (1) Huyser, E. Free Radical Chain Reactions; Wiley-Interscience: New York, 1970; Chapter 5. (2) Potsma. M. L. In Free Radicals: Kochi. J. K.. Ed.: Wilev-Interscience: New York, 1973; Vol. 2, pp 159-229.' (3) Benson, S. W. The Foundarions of Chemical Kinetics; McGraw-Hill: New York, 1960; Chapter 13. (4) "Chemical Kinetics and Photochemical Data for Use in Stratospheric Modeling. Evaluation Number 7"; JPL Publ. 1985, No. 85-37. (5) Rowland, F. S.; Molina, M. J. Rev. Geophys. Space Phys. 1975, 13, ,
Both steps are exothermic, have little or no activation energy, and constitute an efficient cyclic process for the production of chlorine-containing hydrocarbons. Although there is today a considerable body of knowledge regarding the kinetics and dynamics of reaction A,e11 very little is known about the kinetics of reaction B. In order to gain a more quantitative understanding of gas-phase chlorination processes, we are currently studying the kinetics of R C12 reactions. To date, this investigation has included studies of reaction B involving alkyl radicals (R = CH,, C2H5,i-C3H7, and t-C4H9)IZand halogen-substituted methyl radicals (R = CF3, CF2CI, CFCI2, CCI3).l3 Rate constants were measured as a function of temperature, and where possible, mechanisms were confirmed. Only one of these reactions (CH, Cl,) had been the subject of any prior direct i n v e ~ t i g a t i o n . ~ ~ . ' ~
+
+
'Permanent address: Department of Physical Chemistry, University of Helsinki, Helsinki, Finland.
0022-3654/87/2091-1873$01.50/0
I
1.
(6) Lewis, R. S.;Sander, S. P.; Wagner, S.; Watson, R. T. J . Phys. Chem. 1980, 84, 2009-2015. (7) Brunning, J.; Stief, L. J. J . Chem. Phys. 1985, 83, 1005-1009. (8) Manning. R. G.; Kurvlo, M. J. J . Phvs. Chem. 1977, 81. 291-296. (9) Buss, R-J.; Coggiola,-J. J.; Lee, Y. T:Faraday Discuss. Chem. SOC. 1979, 67, 162. (10) Moss, M. G.; Ensiminger, M. D.; Stewart, G. M.; Mordaunt, D.; McDonald, J. D. J. Chem. Phys. 1980, 73, 1256-1264. (1 1) Durana, J. F.; McDonald, J. D. J . Chem. Phys. 1976,64,2518-2527. (12) Timonen, R. S.; Gutman, D. J . Phys. Chem. 1986, 90, 2987-2991. (13) Timonen, R. S.; Russell, J. J.; Gutman, D. Int. J . Chem. Kiner. 1986,
____
I R_ I, 191-1_21-14 _ _ ..
(14) McFadden, D. L.; McCollough, E. A,; Kalos, F.; Ross, J. J . Chem. P h p . 1973, 59, 121-30. (15) Kovalenko, L. J.; Leone, S. J . Chem. Phys. 1984, 80, 3656-67. (16) Russell, G. A. In Free Radicals; Kochi, J. K., Ed.; Wiley-Interscience: New York, 1973; Vol. 1, p 275.
0 1987 American Chemical Society
