J . Phys. Chem. 1989, 93, 1386-1392
1386
Polar Solvent Contributions to Activation Parameters for Model Ionic Reactions Terumitsu Morita,+*$Branka M. Ladanyi,*,+and James T. Hynes*** Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523, and the Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309-021 5 (Received: June 21 1988) ~
-
Transition-state-theory (TST) transfer activation parameters for model ionic reactions A+ + B- products in model polar solvents (water and chloroform) are calculated via reference linearized hypernetted chain theory of intermolecular structure. Transfer activation free energies (AAG’), enthalpies (AAH*), entropies (AAS’), and activation volumes (AV) are studied as functions of ionic sizes and the location of the transition state. We find that the molecular theory transfer activation parameters often differ considerably from dielectric continuum theory predictions. For example, for certain A+ + B- reactions the molecular theory predicts orders of magnitude greater solvent-induced rate enhancement than the continuum theory. In addition, molecular AVt values, which reflect polar solvent electrostrictiveeffects, significantlyexceed continuum predictions. Nonetheless, qualitative similaritiesbetween the predictions of the two theories are often observed; an example is an approximate linear correlation between AAS’ and AV values. For weak electrostatic interactions, e.g., when reactant ions have low charge density, it is found that electrostatic and hard-sphere solvent structural contributionsto activation parameters can be comparable. It is also pointed out that significant pressure variation of AV should be accounted for in isolating possible dynamic solvent-induced breakdown of TST rate predictions.
I. Introduction It is well-documented and appreciated that the rates of many ionic chemical reactions are strongly influenced by solvent pol a r i t ~ . l -To ~ quote but one of many possible examples, the rate constant for the SN1 ionization of tert-butyl chloride increases by over 10 orders of magnitude in the solvent series benzene, acetone, water.2a Within the widely adopted framework of transition-state theory (TST) of reaction rates,s such effects are usually interpreted as electrostatic solvent effects on activation free energies. Other activation parameters are interpreted in this fashion as well.I-l2 For example, the activation volume AV, determined by the pressure variation of the rate constant, frequently is associated with so-called electrostriction effects, in which, for example, polar solvent molecules are “released” upon formation of a dipolar transition state from the ionic reactants.’-I2 Quite often the electrostatic conception of these solvent effects is expressed via explicit continuum dielectric models of chargesolvent interaction.l-l2 In such calculations, the solvent dielectric constant t is the sole solvent property that enters. Continuum model interpretations, while occasionally successful, more often fail in a significant fashion. For example, KohnstamIo details the failures of the continuum theory of electrostriction to account for the measured activation volumes of the ionic Menshutkin reaction between pyridine and methyl iodide. Such failures signal an important practical limitation of the continuum theory, given the widespread use of AV to probe reaction mechanisms and solvent effects thereon.l*2s’0Indeed, the dielectric continuum description of reaction activation parameters has a fundamentally highly suspect feature: The activation free energy and the activation enthalpy, entropy, and volume depend on the solvent only through the dielectric constant t and its temperature and pressure derivatives (cf. section 11). One would expect these activation parameters to depend instead on molecular level aspects of the interaction between the reaction system charges and solvent molecules. One response to this difficulty in a dielectric continuum description is the use of the solvent polarity scale^.^^^^ Such polarity scales, which in some sense reflect “local” polarity of the solvent in the neighborhood of solutes, are frequently successful in correlating rates. Another, more fundamental, response is the application of a molecular theory to the transition-state solvation problem. We follow this route here. Colorado State University. *University of Colorado. 8 Present address: Sendai College, 2526 Takicho, Sendai, Kagoshima, Japan 895-02.
0022-3654/89/2093-1386$01.50/0
In this paper we present a molecular level examination of the solvent contribution to a range of activation parameters for model activated bimolecular ionic reactions. We compare the results with continuum model predictions and find quite large deviations both qualitative and quantitative. Nonetheless, we find that there are often surprising qualitative similarities between molecular and continuum predictions. All our calculations employ the reference linearized hypernetted chain (RLHNC) integral equation theory for pure polar solvents and for ions dissolved in these solvents developed by Patey and c o - ~ o r k e r s . ~ ~In - ~this * approach, spherical reacting ions in a solvent consisting of hard spherical molecules with embedded point dipole and quadrupole moments are described. This is a reasonable approach for the two solvents, water and chloroform, examined in the present study. While other approaches are possible,19 a
( I ) Laidler, K. J. Chemical Kinetics, 3rd ed.; Harper and Row: New York, 1987. (2) (a) Reichardt, C. Soluent Effects in Organic Chemistry; Verlag Chemie: Weinheim, 1979; Pure Appl. Chem. 1982, 54, 1867. (b) Blandamer, M.; Burgess, J. Pure Appl. Chem. 1982, 51, 2087. (c) Dack, M. R. J. J . Chem. Educ. 1974, 51, 231. (3) Entelis, S. G.; Tiger, R. P. Reaction Kinetics in the Liquid Phase; Wiley: New York, 1976. (4) Isaacs, N. S. Liquid Phase High Pressure Chemistry; Wiley: New York, 1981. (5) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory ofRate Processes; McGraw-Hill: New York, 1941. (6) Pearson, R. G. J . Chem. Phys. 1952, 20, 1478. (7) Bell, R. P. J . Chem. SOC.1943, 629. (8) Laidler, K. J.; Chen, D. T. Y . Trans. Faraday Soc. 1958, 54, 1026; Can. J . Chem. 1959, 37, 599. (9) Schaieger, L. L.; Long, F. A. Adu. Phys. Org. Chem. 1963, I , 1. ( I O ) Kohnstam, G. Prog. React. Kiner. 1976, 5, 335. ( 1 1 ) LeNoble, W. J. Adu. Phys. Org. Chem. 1965, 5 , 207. Asano, T.; LeNoble, W. J. Chem. Reu. 1978, 7 8 , 407. (12) Hammann, S. D. Reu. Phys. Chem. Jpn. 1980, 50, 147. (13) Patev. G . N. Mol. Phys. 1977. 34, 427; 1978, 35, 1413. (14) Leveique, D.; Weis, J.J.; Patey, G. N. Phys. Lett. A 1978, 66A, 115; J . Chem. Phys. 1979, 72, 1887. (15) (a) Patey, G. N.; Levesque, D.; Weis, J. J. Mol. Phys. 1979, 38, 1635. (b) Patey, G . N.; Levesque, D.; Weis, J. J. Ibid. 1979, 38, 219. (16) Carnie, S. L.; Patey, G. N. Mol. Phys. 1982, 47, 1129. (17) Patey, G. N.; Carnie, S . L. J . Chem. Phys. 1983, 78, 5183. (18) Perkyns, J. S.; Fries, P. H.; Patey, G. N. Mol. Phys. 1986, 57, 529. (19) (a) Hirata, F.; Rossky, P. J.; Pettitt, B. M. J . Chem. Phys. 1983, 78, 4133. (b) Pettitt, B. M.; Rossky, P. J. J . Chem. Phys. 1986, 84, 5836. (c) Friedman, H. L.; Ramanathan, P. S . J . Phys. Chem. 1970, 74, 3756. (d) Ramanathan, P. S.; Friedman. H . L. J . Chem. Phys. 1971, 54, 1086.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 4, 1989 1387
Activation Parameters for Model Ionic Reactions feature of the R L H N C method important in the present investigation of activation parameters is its ability when applied to reasonably simple models (described within) to accurately reproduce the experimental dielectric properties of the pure solvents being modeled, starting from known gas-phase values of permanent and induced molecular moments. This ensures a measure of consistency in the comparison of calculated molecular theory and continuum theory predictions. In addition to their intrinsic interest in the context of the TST predicted rates, TST activation parameters are important to understand for a quite different reason. In recent years, the nonequilibrium dynamic-as opposed to static, equilibrium TSTaspects of polar solvents have been argued to play a key role in solution-phase ionic From an experimental viewpoint, these dynamic effects can only be sorted out when the undoubtedly large static TST effects are properly taken into a c c o ~ n t . ~In~ particular, ~*~ the potential use of pressure variation to examine dynamic solvent effects on ionic reaction rates requires special attention to the TST activation volume A P , and we give this parameter particular attention herein. The outline of the paper is as follows. In section 11, we describe the reaction model and the formulation for calculating reaction activation parameters. In section 111, the models for water and chloroform solvents are described together with calculational procedures required for the activation parameter evaluations. The results are described in section IV. while section V concludes. 11. Reaction Model and Formulation
We will adopt a simple reaction model of two spherical ions of charges qA and q B approaching a transition-state separation r*. Here we are imagining that there is an intrinsic barrierZSto the reaction
+
-
AQA BQB
products
(2.2)
in which the molar transfer activation free energy (20) van der Zwan, G.;Hynes, J. T. J. Chem. Phys. 1982,76,2993; 1983, 78, 4174; Chem. Phys. Lett. 1983, 101, 367; Chem. Phys. 1984, 90, 21. (21) Bergsma, J. P.; Gertner, B. J.; Wilson, K. R.; Hynes, J. T. J. Chem. Phys. 1987, 86, 1356. Gertner, B. J.; Bergsma, J. P.; Wilson, K. R.; Lee, S.; Hynes, J. T.Ibid. 1987, 86, 1377. (22) Zichi, D.; Hynes, J. T. J. Chem. Phys. 1988, 88, 2513. (23) (a) Belch, A. C.; Berkowitz, M.; McCammon, J. A. J . A m . Chem. SOC.1986,108, 1755. (b) Karim, 0.A.; McCammon, J. A. Ibid. 1986,108, 1762. (c) Berkowitz, M. L.; Karim, 0. A.; McCammon, J. A.; Rossky, P. J. Chem. Phys. Lett. 1984, 105, 577. (d) Karim, 0. A,; McCammon, J. A. Chem. Phys. Lett. 1986, 132, 219. (24) Hicks, J. M.; Vandersall, M. T.; Sitzmann, E. V.; Eisenthal, K. B. Chem. Phys. Lett. 1987, 135, 413. (25) (a) Kessler, H.; Feigel, M. Acc. Chem. Res. 1982,15, 2. (b) Masnovi, J. M.; Kochi, J. K. J. A m . Chem. SOC.1985, 107, 7880. (c) Paradesi, C.; Bunnett, J. F. Ibid. 1985, 107, 8223.
(2.3)
is the difference between the activation free energies for the actual reaction and a corresponding reference reaction. One important choice for the reference reaction frequently adopted in the literature'~~ and the one used here is that of the hypothetical reaction corresponding to eq 2.1 for the uncharged reactants in the same solvent at the same T and p values and with the same transition-state location.26 In equilibrium statistical mechanical terms, the ratio of the rate constant k to its value krcfin the reference state is27-30
where wAB(r*)is the ion-ion potential of mean force evaluated at r* and wABref(r*) is the corresponding quantity in the reference state. A convenient separation of the potential of mean force is27 where uAB(r)is the direct interaction between the ions and YA&) is the solvent-dependent indirect correlation function. An analogous definition holds for the reference state: The important feature of these definitions is that the solventdependent quantities yAB(r) and yABref(r)are independent of the direct A-B interaction and can be evaluated without explicitly considering A and B to be reactive ~ p e c i e s . ~ ~ - ~ O We separate the direct ionic interaction uAB into the direct Coulomb part and the remainder u i e UAB(r)
(2.1)
which can arise from, for example, steric effects, a change in hybridization, of multiple-bond rearrangement during the reactive process for the molecular ionic species that AQ*and BQBmodel. This intrinsic barrier height will be modified by the solvent, and the latter effect represents the solvation activation free energy effect we wish to compute. In particular, we will focus on rate constant ratios appropriately defined such that the intrinsic barrier contributions cancel out, thereby isolating the solvent effects of interest here. We will select the transition-state location r* to correspond to contact or slight overlap of the reactant ions in the transition state. The spherical ion model is highly simplified and ignores, for example, orientational effects present in polyatomic reactant systems. It is however the same model used in most physical organic studies of transition-state solvent effects.'-'2 In the present work, it is the conventional continuum description of the solvent that is replaced by a molecular level treatment. According to the transition-state t h e ~ r y ,the ~ , ~reaction rate constant k may be related to a reference rate constant krcfvia k / k r e f= exp[-AAG*/RT]
AAG* = AG* - AGlrcf
=
U'AB(r)
+ qAqB/r
(2.7)
and select, as noted above, the reference state corresponding to uncharged particles A and B in the same solvent. Thus and the required mean potential difference for the rate constant ratio eq 2.4 is AwAB(r)
wAB(r) - wABrCf(r)= qAqB/r - kBT In bAB(r)/yABref(r)l(2.9)
containing the direct Coulomb interaction and indirect interactions. For ions at infinite dilution in a polar solvent, which is the system of interest here, yAB and yABrefarise from solvent-mediated A-B interactions. To evaluate these solvent-dependent terms, we will use the RLHNC theory for hard-sphere i ~ n s , ' ~according J~ to which yAB(r) is given by (2.10) Here yABHS(r)is the hard-sphere indirect correlation function, while AvAB originates from the charge-multipole interactions between the ions and the solvent molecules, and thus vanishes in the reference state. (Note that yABHs refers to interionic hardsphere interactions mediated by solvent molecules.) On combining eq 2.9 and 2.10 and noting that yABHS(r)= yABref(r),the mean potential difference in the R L H N C description is then (2.1 1) AwAB(r) = qAqB/r -k k B T A v A B ( r ) Note that AqAB, and thus AWAB,contain no direct hard-sphere (26) In a more detailed molecular treatment of the reaction system, one should take into account solvent shifts (cf. ref 22) in the transition-state location. (27) The equilibrium constant analogue of eq 2.4 was developed in: (a) Chandler, D.; Pratt, L. R. J. Chem. Phys. 1976, 65, 2925. (b) Pratt, L. R.; Chandler, D. Ibid. 1977, 66, 147. (28) The analogue of eq 2.4 for unimolecular isomerizations appears in: Chandler, D. J. Chem. Phys. 1978, 68, 2959. (29) Hynes, J. T.In The Theory of Chemical Reactions; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. 4, p 171. (30) Ladanyi, B. M.; Hynes, J. T. J. Am. Chem. SOC.1986, 108, 585.
1388 The Journal of Physical Chemistry, Vol. 93, No. 4, 1989
repulsion. Thus, overlapped configurations of ions A and B, which may occur at the transition state, are allowed. The function AvAB can be calculated by RLHNC theory, as described in section 111. With the formulation above, the molar transfer activation free energy in eq 2.2 can be calculated as AAG* = NAWAB(r*) = NqAqB/r*+ RTAr)AB(r*)
(2.12)
where N is Avogadro's number. (Note that no contribution of the intrinsic solvent-independent barrier enters either AAG* or k/kref'.) For comparison, the continuum or so-called primitive model prediction for AwAB(~*)is just the Coulomb interaction modified by the solvent dielectric constant AAG*, = NqAqB/€r*
(2.13)
as is w e l l - k n ~ w n . ~ * ~ J ~ The other transfer activation parameters follow from standard thermodynamic relations. Thus, the transfer activation entropy and enthalpy are
Morita et al. and d(3) = dr3 dQ3represents integration over all positions r3 of the center of mass and orientations Q3 of molecule 3. The RLHNC approximation is designed for systems with intermolecular potentials uaa(12) which can be separated into a shortranged, predominantly repulsive, central-force reference part (in the present case hard sphere) and a multipolar part
+
uaa(12) = uaaHS(r) Aua8(12)
(3.3)
where r = Irl - r21. The pair correlations ( X = h, c, 7) are similarly separated into reference and multipolar parts Xaa( 12) = XaaHS(r)+
A&&
12)
(3.4)
Very accurate methods exist for the evaluation of XasHS(r)for pure fluids31 and mixture^.^^^^^ The R L H N C theory provides an approximation to AXaa(12) by linearizing the hypernetted chain closure relation34 in powers of AXas( 1 2 )
AAS* = -(aAAG*/aT)p
AAH* = AAG*
+ TAAS*
(2.14)
where
+1
while the molar activation volume is
A P = (aAAG*/ap),
gaaHS(r)= hapHs(,) (2.15)
if r* is taken to be independent of p and T, as is usually assumed. AAS* and A v ' in the continuum model are determined by the T and p variation of the solvent dielectric c o n ~ t a n t : I ~ ~ J ~
(2.16) The activation entropy and volume factors have a particular interest, since by eq 2.12, 2.14, and 2.15, the RLHNC molecular values
(2.17) are determined by the solvent-mediated ion-ion interaction term RTAvAs(r*)and are independent of the direct Coulomb interaction between the ions when r* is independent of T and p . Thus, AAS* and A P are especially revealing probes of polar solvent effects on the rate process.
111. Interaction Model and Calculation of Interionic Mean Potential Difference To calculate the interionic mean potential difference AwqB(r) for the reaction problem, we consider dilute solutions of ions, represented as charged hard spheres of charges qA and qB, in dipolar-quadrupolar-polarizable hard-sphere solvents. AWAB(r) is related to the ion-ion correlation function AvAB(r)by eq 2.12. We use the RLHNC appr~ximation'~ to obtain AvAB(r) in a manner now described. The RLHNC theory is based on the Ornstein-Zernike equation and an approximate closure relation, which relate the intermolecular pair correlation hap( 12) ( a , p = A, B, S ) and the direct correlation ca8( 12) functions of molecules or ions 1 and 2. For a dilute solution at solvent density ps, the Ornstein-Zernike equation takes the form
(3.6)
is the hard-sphere pair distribution function. The desired ion-ion correlation function AvAB is obtained from eq 3.1
and requires the knowledge of ion-solvent correlation functions AhAQ(12) and AcSB(12). These ionsolvent correlations are found from the solution of eq 3.1 and 3.5 and through eq 3.1 depend on the pure solvent direct correlation function css(12). Thus to find AqAB(r), we have to start with the evaluation of the intermolecular pair correlations for the pure solvent. We then consider the ion-solvent correlations. A . Solvent Model and Dielectric Properties. We model the solvent molecules as hard spheres of diameter us with dipoles, quadrupoles, and polarizabilities. The polarizability is taken into account by using the self-consistent mean-field (SCMF) app r ~ a c h ' in ~ ,conjunction ~~ with the RLHNC approximation.13J5J6 An effective dipole moment, me, for a polar polarizable fluid is found self-consistently by solving eq 3.1 and 3.5 for a polar nonpolarizable fluid with dipole moment me and by using the resulting pair correlations to evaluate the polarization energy. gsSHS(r), needed in eq 3.5, is calculated via the Verlet-Weis approxima tion. 31 We consider two solvents: a highly polar one, designed to be a model for water, and a weakly polar one, a model for chloroform. For both solvents the pair distributions are evaluated at a set of densities and temperatures ps and T . The pressure, p(p,,T) is obtained from the experimental equation of state data.36*37(This avoids certain difficulties with the RLHNC theory for predicting the p r e s s ~ r e . ' ~ ) Our model for water is that used by Carnie and Patey.I6 In this model a water molecule is assumed to be isotropically polarizable with polarizability a and to have a dipole moment fi and a tetrahedral quadrupole moment tensor
Verlet, L.; Weis, J. J. Phys. Reu. A 1972, 2, 939. Grundke, E. W.; Henderson, D. Mol. Phys. 1972, 24, 269. Pratt, L. R.; Hsu, C. S.; Chandler, D. J . Chem. Phys. 1978,68,4203. Friedman, H. L. A Course in Smtisfical Mechanics; Prentice-Hall: Englewood Cliffs, NJ, 1985; pp 195-197. (35) (a) Wertheim, M. S. Mol. Phys. 1973,25, 211; 1973.26, 1425; 1977, 33, 95; 1977, 34, 1109. (b) Herye, J. S.; Stell, G. J . Chem. Phys. 1980, 73, 461. (c) Pratt, L. R. Mol. Phys. 1908, 40, 347. (36) Water: A Comprehensioe Treatise; Franks, F., Ed.; Plenum: New York, 1972; Vol. I . (37) Schroeder, J . ; Schiemann, V. H.; Jonas, J. Mol. Phys. 1977,34, 1501. (31) (32) (33) (34)
The Journal of Physical Chemistry, Vol. 93, No. 4, 1989 1389
Activation Parameters for Model Ionic Reactions TABLE I: Solvent Molecular Properties
molecule us,A H20 2.8 CHC13 4.7685'
jt,
D
QT,L,D.A
Q
2.5 2.39
1.855 1.04
or
A' Me?D
1.444 6.71, 9.39
2.6 1.6
OValue at 30 OC. Value at 60 OC was computed from u,(T) = A - (0.80 X lo-' A)(T/K - 300) (see also eq 3.7). bmcvalues depend slightly on T and p . The following data give the range of values of mein the p , T range of interest. For H20: m,(25 OC, 1 bar) = 2.61 D,m,(25 "C, 3000 bar) = 2.65 D, m,(60 "C, 1 bar) = 2.63 D. For CHC13: m,(30 OC, 1 bar) = 1.60 D,m,(30 OC, 4500 bar) = 1.65 D, m,(60 OC, 1 bar) = 1.63 D. 4.771 1
TABLE 11: Calculated and Experimental Solvent Dielectric Prowrties
(a In C/aq,, 10-3
e
solvent H2W CHCl,*
calcd
exptl
calcd
79.9 4.20
78.3 4.64
-4.12 -3.28
K-1
exptl -4.55 -3.55
(a In +wT,
1 .o
1 0-5 bar-'
calcd
exptl
7.3 11.4
5.2 12.6
"At 25 OC and 1 bar. b A t 30 OC and 1 bar. where Pi and 8, are unit vectors along two mutually perpendicular directions orthogonal to the unit vector fii along the dipole direction of molecule i. Qr corresponds to a tetrahedral spatial arrangement of two pairs of positive and negative charges of equal magnitude. The parameters a,p, and QTand the hard-sphere diameter usare given in Table I. a, p, and QT correspond to experimental value^,^*^^^ and usis a reasonable estimate of the size of a water molecule. The required correlations c,, and h,, are obtained as described in ref 16. Chloroform is modeled as a molecule with axially symmetric anisotropic polarizability
a(Qi) = aiIfiifii + a , ( l - fiifii)
(3.9)
Figure 1. Mean potential difference AwAB, eq 2.9, divided by k~Tversus separation r scaled by the solvent diameter us, for oppositely charged equal size (aA = uB) ions in water: (---) continuum theory, (-) RLHNC molecular theory.
water-like solvent." Solvent molecules are treated as dipolarquadrupolar hard spheres and the ions as charged hard spheres. Solvent-solvent polarization is taken into account by using the effective solvent dipole moment me instead of the "bare" dipole moment p in the ion-solvent potential. In the range of thermodynamic states we consider, meis nearly constant, but considerably larger than p: for water me(25 OC, 1 bar) = 1.41 D and for chloroform m,(30 OC, 1 bar) = 1.54 D. Ionsolvent polarization is neglected. Thus, the ion-solvent potential is
= '/zQ~(3fiiQi- 1)
(3.10)
appropriate for a symmetric top molecule. The diameter is taken to be linearly dependent on temperature u,(T)
= 4.7711 A - ( T - 300 K)(0.80
X
A/K)
(3.11)
du,/dT is taken from ref 37, and ~ ~ ( 3 K) 0 0is slightly smaller than the estimated hydrodynamic diameter.37 The values of the potential parameters are given in Table I. all, aL,and p are the experimental values of these quantities,& while QLwas estimated by using the known value of p and assuming that the charges are located on CI and H atoms. The SCMF theory for anisotropically polarizable polar molecules with linear quadrupole moments is given in ref 41 and the R L H N C theory for hard spheres with dipoles and linear quadrupoles in ref 15a. We combine these to find the pair correlations for chloroform. The RLHNC approximation along with the SCMF procedure leads to a very good description of dielectric properties of polar fluids of molecules with appreciable quadrupole moments and nearly spherical cores.16,18In the present two cases, the calculated dielectric constant and its temperature and pressure derivatives ~ ,shown ~ ~ in Table are in very good agreement with e ~ p e r i m e n t ,as 11. B. Ion-Solvent Distribution Functions. The ion-solvent pair correlation functions hi, and cis ( i = A, B) are calculated following the procedure developed by Patey and Carnie for ions in a (38) Murphy, W. F. J . Chem. Phys. 1977, 67, 5877. (39) Verhoeven, .I.Dynamus, ; A. J . Chem. Phys. 1970, 52, 3222. (40) Bottcher, C. J . F.; Bordewijk, P. Theory of Electric Polarization; Elsevier: Amsterdam, 1978; Vol. 2, Tables 42 and 43. (41) Perkyns, J . S.; Kusalik, P. G.; Patey, G. N. Chem. Phys. Lett. 1986, 129. 258.
+
+
ui,(12) = uiFS(r) ~ ~ : ~ ( 1 2 )ui,'Q(12)
(3.12)
where
a dipole moment p along the 3-fold symmetry axis tii, and a linear quadrupole moment tensor QL(Qi)
3.0
2.0 r/g,
ulsHS(r)=
0
for r < uis for r 2 uis
(3.13)
= (u, + us)/2 and U, is the diameter of the ion of type i. u,,CD(12) is the charge-dipole potential u,,
u , : ~ (12) = - ( q , m , / r z ) i ~ f i z
(3.14)
and u,,@( 12) is the charge-quadrupole potential. For tetrahedral quadrupoles - (i42)2]
(3.15)
u,,CQ(12) = q,QL/(2r3)[3(i42)2- 11
(3.16)
u,,CQ(l2) =
(qjQT/rJ)[(i42)2
and for linear quadrupoles
The hard-sphere pair distribution functions g,,HS(r),needed in eq 3.5, were evaluated by using the Gr~ndke-Henderson~~ and Pratt et al.33generalizations to mixtures of the Verlet-Weis approximati~n.~' With the procedure described above, the required mean potential difference AwAB(r), eq 2.11, for the rate problem was calculated for the model water and chloroform solvents. The results are described next.
IV. Results A . H20Solvent. The calculated potential of mean force AwAa(r) = wAB(r) - WABref(r),eq 2.18, for univalent ions A+ and B- with the same diameter (2.8 A) as that adopted for H 2 0 is displayed in Figure 1 . Outside the overlap region, the general features of AwAB are similar to those of wAB for a pair of nonreactive ion^.^^,^^*^^^,^^^^ The oscillations in AwAB, including a solvent-separated ion pair well and an enhanced well depth near contact,43reflect the solvent structure in the vicinity of the ions. (42) Kusalik, P. G.;Patey, G. N. J . Chem. Phys. 1983, 79, 4468.
Morita et al.
1390 The Journal of Physical Chemistry, Vol. 93, No. 4, 1989 TABLE 111: Rate Constant Ratios for Ionic Reactions in H20 at 298 K and 1 baP qA
e e e llle
qe
UA/Q,
Fe re
"All the results are for
14.71 127.16 18.86 11.18 f6.79
0.8
aA = UB.
1 H,O, p = 1 bar; a)
'01
-'OI
~~~/~r"'~~/~~/~'C'~~l
.o
1 0.8 1.o 1.o
0.8 1.0 1.0
Ffl/le
In
r'/aA
1 .o 1.o
Te
1
ui
u., rt
4 -:----I
TAAS*
