Nonequilibrium solvent polarization in kinetics of SN2 reactions

Article abstract of DOI:10.1002/kin.10105

The solvent effect on the experimental activation barriers for the reactions of methyl iodide with chloride and thiocyanate ions was analyzed according to the Marcus and Shaik theories, considering S_N2 mechanism in terms of a single electron shift. The linear increase in the solvent reorganization energy of the Marcus theory (after removing contributions from the specific solvation) with the solvent Pekar factor, describing the effect of the nonequilibrium solvent polarization, was observed for six aprotic solvents. The direct support of the title effect based on the Shaik theory was less evident; however, in general, the calculated activation barriers in 10 solvents change parallel with the experimental ones.

Full text of DOI:10.1002/kin.10105

Nonequilibrium Solvent
Polarization in Kinetics
of SN2 Reactions
J. S. JAWORSKI
Faculty of Chemistry, Warsaw University, Pasteura 1, 02-093 Warsaw, Poland
Received 13 June 2002; accepted 16 October 2002
DOI 10.1002/kin.10105
ABSTRACT: The solvent effect on the experimental activation barriers for the reactions of methyl
iodide with chloride and thiocyanate ions was analyzed according to the Marcus and Shaik the-
ories, considering SN2 mechanism in terms of a single electron shift. The linear increase in the
solvent reorganization energy of the Marcus theory (after removing contributions from the spe-
cific solvation) with the solvent Pekar factor, describing the effect of the nonequilibrium solvent
polarization, was observed for six aprotic solvents. The direct support of the title effect based
on the Shaik theory was less evident; however, in general, the calculated activation barriers
in 10 solvents change parallel with the experimental ones. ꢀ 2002 Wiley Periodicals, Inc. Int J
C
Chem Kinet 35: 61–66, 2003
solvent polarization in the transition state (TS). This
contribution results from a slow reorientation of sol-
vent molecules that cannot follow the instantaneous
position of the electronic charge in the TS and there-
fore they adjust to some averaged charge distribution
[8]. In the Marcus theory [8] this contribution is de-
scribed by the solvent reorganization energy λ0, which
depends (according to a simplified dielectric contin-
INTRODUCTION
The description of the SN2 mechanism as a single elec-
tron shift synchronized in a one-step reaction to the
bond interchange and solvent reorganization has been
repeatedly examined since the original proposition of
Shaik and Pross [1–7] based on the curve crossing or
state correlation diagrams.The unified theory for the
rates of the SN2 and electron transfer (ET) reactions
was also proposed by Marcus [8]. Both models were
successful in comparison with the experimental results
obtained in the gas phase and aqueous solutions [3,6–
8]; the first model was successful even in comparison
with the results obtained in DMF [3,6,7]. Considering
solvent effects on the SN2 rate constants, both models
take into account an important contribution to the acti-
vationfreeenergiesoriginatedfromthenonequilibrium
1
1
uum model) on the solvent Pekar factor γsolv
=
−
related to the static εs and optical εop electric εpopermiεts-
tivity (εop ≈ nD2). Contributions of λ0 to the activa-
tion barriers are well documented experimentally (by
the dependences of intrinsic activation barriers on the
Pekar factor) in the case of outer-sphere ET processes
and the bond cleavage of radical anions [9,10], which
can be described [11] as a concerted bond-breaking and
an intramolecular electron shift. On the other hand, the
effect of the nonequilibrium solvent polarization in the
TS for SN2 reactions was not directly demonstrated us-
ing experimental results in a number of solvents [1–8]
until our recent work [12]. It was shown [12] on the
basis of the Marcus theory [8] that the intrinsic activa-
tion free energy for the reaction between methyl iodide
Correspondence to: J. S. Jaworski; e-mail: jaworski@chem.uw.
edu.pl.
Contract grant sponsor: KBN.
Contract grant number: BST-761/8/2002.
2002 Wiley Periodicals, Inc.
c
ꢀ

62
JAWORSKI
and chloride ions in six aprotic solvents, after remov-
ing the contribution of the specific solvation, increases
linearly with the solvent Pekar factor. A similar anal-
ysis for the reaction of methyl iodide with thiocyanate
anions for which the rate constants were measured in
10 solvents (the greatest number in the Parker compi-
lation [13]) as well as a comparison with predictions of
the Shaik theory [6,7] for both reactions are presented
here.
The thermodynamic driving force was calculated
[8,12] as
ꢁ
◦
◦
ꢀG = FꢀE + DMeI − DMeSCN
(3)
◦
where ꢀE is the difference between formal poten-
tials for I/I− and SCN/SCN− redox couples, and DMeI
and DMeSCN are the energies of homolytic bond dis-
sociation in the compound indicated. Formal poten-
tials in different solvents were calculated from th−ose in
◦
aqueous solutions (E = 1.4 V vs. NHE for I/I and
−
◦
RESULTS AND DISCUSSION
E = 1.66 V vs. NHE for SCN/SCN [14,15]) and
the free energy of transfer of each anion from water
to a given solvent [16]; only for acetone (Ac) and N-
methylformamide (NMF), where free energies of trans-
fer are unknown, the proper values were estimated us-
Solvent Analysis in Terms
of the Marcus Theory [8]
A similar approach based on the Marcus theory [8] as
used previously [12] for reaction (1):
N
ing the dependence on the Dimroth and Reichardt ET
solventparameter[17]. Takingintoaccountthefactthat
solvent effects on DMeI and DMeCl are very small [12]
Cl− + MeI → MeCl + I−
(1)
the solvent-independent value of DMeI − DMeSCN
=
−38 kJ mol−1 was used in Eq. (3). The last value
was applied to reaction (2):
ꢁ
◦
was estimated for the aqueous solution using ꢀG
=
−13 kJ mol−1 as suggested by Pearson [15] on the
SCN− + MeI → MeSCN + I−
(2)
basis of the gas-phase dissociation energy DMeSCN
=
293 kJ mol−1
The values of ꢀG obtained in a number of solvents
.
The values of the activation barrier ꢀG‡ were cal-
ꢁ
◦
exp
culated from the experimental rate constants extracted
from the Parker compilation [13] assuming [12] the
ꢁ
◦
are shown in Table I. The solvent change of ꢀG is
only 3 kJ mol−1, which is very small as compared with
an analogous change (Table II) for reaction (1) [12].
Themainreasonofthissmallsolventchangeisasimilar
preexponential term to be equal to 1012 M−1 −1. The
s
values obtained are shown in Table I and it is evi-
dent that they are strongly solvent-dependent, changing
from 67 kJ mol−1 in N-methylpyrrolidinone (NMP) to
89 kJ mol−1 in water.
solvent effect on ꢀG tr of I− and SCN− ions, whereas
◦
this effect is much higher for Cl− ions. As a result, free
Table I Activation Free Energy and Its Components
According to Marcus Theory for the Reaction (2) of MeI
with SCN− in Different Solvents
Table II Activation Free Energy in Different Solvents
for Reactions Y− + MeI → MeY + I− Calculated
According to the Shaik Theory
ꢀG‡
−0.5ꢀG
w + λ0/4 λ0/4 + C
◦
Y− = Cl−
Y− = SCN−
exp
r
Solvent (kJ mol−1) (kJ mol−1) (kJ mol−1) (kJ mol−1
)
◦
−ꢀG
ꢀG‡
(kJ mol−1
ꢀG‡
calc
calc
H2O
88.5
87.3
84.5
82.2
79.9
79.3
74.8
73.1
73.1
67.4
6.5
5.5
6.3
6.0
5.5
5.3
5.0
6.0
6.5
6.0
65
63
61
58
55.5
55
50
49
50
44
37
42
39
38
42
42
38
39
38
34
Solventa ρb
(kJ mol−1
)
)
(kJ mol−1
)
MeOH
FAa
H2O
MeOH
FA
0.56
−3
0
3
124
120
106
–
119
117
103
106
108
113
103
108
100
110
NMFb
MeNO2
ACNc
DMF
ACd
0.55
0.47
0.49
NMF
–
MeNO2 0.51
13
21.5
22
20
27
28
105
104
94
ACN
DMF
AC
DMA
NMP
0.54
0.48
0.52
0.47
0.53
DMA
NMPe
99
90
98
a Formamide.
b N-Methylformamide.
c Acetonitrile.
a For solvent abbreviations see footnote to Table I.
b Solvent reorganization factor from Ref. [7] or calculated from
data in Ref. [21].
d Acetone.
e N-Methylpyrrolidinone.

NONEQUILIBRIUM SOLVENT POLARIZATION IN KINETICS OF SN2 REACTIONS
63
energies of activation for reaction (2) do not depend on
the thermodynamic driving force ꢀG .
Assumingthesameapproximationfortheresonance
energy of interactions of the states as used previously
[12], the activation free energy according to the Marcus
model [8] can be described by Eq. (4):
the mean quadratic deviation from the correlation line;
95%errorsofregressioncoefficientsaregiveninparen-
theses. A similar dependence was found previously
[12] for the reaction with Cl− ions and was discussed
as the manifestation of specific solvation effects on the
work term (which involves a partial desolvation of an-
ions) and/or on the solvent reorganization energy (the
noncontinuum effect not included in the Pekar factor
[12]). In order to remove this specific solvent effect,
the term 28 ETN, i.e., the slope from Eq. (5), was sub-
tracted from the sum wr + 0.25λ0 giving 0.25λ0 + C,
where C is a solvent-independent constant. The values
of 0.25λ0 + C are shown in Table I. They depend on
the Pekar factor for six aprotic solvents (Fig. 2) giv-
ing the following correlation with the intercept equal
to zero:
ꢁ
◦
ꢀG‡ = wr + 0.25λ0 + 0.06(DMeI + DMeSCN
)
ꢁ
◦
+ 0.5ꢀG
(4)
Assuming further that the solvent effect on
0.06(DMeI + DMeSCN) is negligibly small, as was
found for the reaction with Cl− ions [12], the com-
ponent wr + 0.25λ0 (Table I) was calculated for a
constant value of 0.06(DMeI + DMeSCN) = 29.9 kJ
mol−1; the last value was estimated for the aqueous
solution.
0.25λ0 + C = 81( 3)γsolv
(6)
n = 6, r = 0.920, F = 27.4, δ = 1.05
It is evident that the sum of the work term and the
solvent reorganization energy wr + 0.25λ0 gives the
main contribution to the solvent effect on the activa-
tion free energy. There is no clear dependence of the
sumwr + 0.25λ0 onthesolventPekarfactor γsolv;how-
ever, separate correlations for the groups of protic and
aprotic solvents can be observed, as shown in Fig. 1.
On the other hand, the acceptable correlation was
A similar correlation found [12] for reaction (1) is also
shown in Fig. 2. If the null intercept in Eq. (6) indicates
that the specific solvation of negative charge in the TS
and connected with the work term were completely
removed, then the constant C = 0 and the obtained
values are equal to 0.25λ0 and can be described by the
classic Marcus expression [8]:
N
found with the ET solvent parameter:
N
wr + 0.25λ0 = 28( 9)ET + 39( 5)
n = 10, r = 0.937, F = 51.1, δ = 2.3
0.25λ0 = (e2/4)(1/2a1 + 1/2a2 − 1/R)
(5)
× (1/εop − 1/εs)
(7)
where n is the number of solvents, r the correlation
coefficient, F the value of the Snedecor test, and δ
Figure 2 Relationships between the solvent reorganization
energy (without the contribution from the specific solvation)
and the solvent Pekar factor for reactions of MeI with SCN−
(circles) and Cl− (squares, data from Ref. [12]). The cor-
relation lines for aprotic solvents (solid points) are shown.
Solvents: 1 – NMP, 2 – DMF, 3 – DMA, 4 – AC, 5 – MeNO2,
6 – ACN.
Figure 1 Dependence of the sum of the work term and the
solvent reorganization energy for the reaction of MeI with
SCN− on the solvent Pekar factor. Solvent abbreviations as
in Table I.

64
JAWORSKI
wherea1 anda2 representtheradiiofspheresofanelec-
tron localization and R is the distance between them in
the TS, and the term in the last parentheses is the Pekar
factor. The comparison of Eqs. (6) and (7) indicates that
the same slope (equal to 82) as the experimental one
(equal to 81) can be obtained assuming a1 = 350 pm as
for the solvated iodide ion [12] but a2 = 213 pm as for
bare thiocyanate ion [18] and R = a1 + a2 + 2r = 717
pm (with r = 77 pm corresponding to the covalent ra-
dius of the carbon atom). Thus, the contribution to the
intrinsic activation barrier from the nonequilibrium sol-
vent polarization of the TS looks reasonable. Moreover,
it can be found that the solvent reorganization energy
0.25λ0 describing the effect of the nonequilibrium sol-
vent polarization is high for aprotic solvents and repre-
sents 46% (in NMP) to 50% (in DMF) of the intrinsic
activation barrier. The last effect is a little lower for
protic solvents (with the lowest value equal to 39%
in water).
It is evident from Fig. 2 that the point for water de-
viates completely from the correlation line given by
Eq. (6) in a similar manner as was previously ob-
served for reaction (1) between MeI and Cl− ions
[12]. However, for other protic solvents the points
are more close to the correlation line, giving, for
n = 9, r = 0.890 but with a high scatter of points
(δ = 1.13). Taking into account precision of the es-
timated data, this means that in general for all solvents,
with the exception of water, the nonequilibrium po-
larization effect in the TS seems to play an impor-
tant role and the Marcus model [8] can reasonably
describe it.
approximated [6,7] by
‡
◦
ꢀG ≈ [ fR + fP + (1 − fR − fP)ꢀG /GR]
◦
× GRGP/(GR + GP − ꢀG ) − B
(8)
Energy gaps for the vertical electron transfers in so-
lution can be approximated [3,7] using the gas-phase
energy gaps and solvation free energies of both anions
−
SY: and SX: and radical anion S(R-·)
(IY: − ARX)s ≈ (IY: − ARX)g + (1 + ρ)SY:
−
− (1 − ρ)S(R-·) ≈ (IY: − ARX g
)
+ SY: − SX: + ρ(SY: + SX:)
(IX: − ARY)s ≈ (IX: − ARY)g + SX: − SY:
+ρ(SY: + SX:)
(9)
(10)
where the solvent reorganization factor ρ = (εs −
nD2)/[nD2(εs − 1)] [3,6,7] is a function of the static
electric permittivity and the refractive index.
It is more convenient to analyze the solvent ef-
fect on the total energy barrier according to the above
equations than to separate the intrinsic barrier as in
the Marcus model. For the reaction between chloride
ions and MeI (reaction (1)) the gas-phase energy gaps
[7] are equal to (ICl: − AMeI)g = 389.1 kJ mol−1 and
(II: − AMeCl)g = 422.6 kJ mol−1, the slope factors in
all aprotic solvents were assumed to be the same as
in DMF [7] ( fR = 0.243, fP = 0.242) and in all pro-
tic solvents to be the same as in water ( fR = 0.239,
fP = 0.243). The solvation free energies were calcu-
lated using the free energy of transfer of a given anion
from water to a given solvent [16] and absolute stan-
dard molar Gibbs free energies of hydration equal to
SCl: = 347 kJ mol−1 and SI: = 283 kJ mol−1 [19].
Thereactionwiththiocyanateions(reaction(2))was
not considered previously [3,6,7]. The gas-phase ion-
ization potential of the nucleophile ISCN:(g) = 224 kJ
mol−1 was estimated from the correlation (r = 0.995)
of IY:(g) for five nucleophiles (H2N•, H2P•, HO•, HS•,
and CN• taken from Table 4.1 in Ref. [7]) with the
electron affinity of corresponding radicals [20], and
the vertical gas-phase electron affinity of MeSCN was
calculated from the bond dissociation energies of the
molecule and the radical anion using the following
equation [3,7]:
Solvent Analysis in Terms of the Shaik
Theory [3,6,7]
The activation barrier for the nonidentity SN2 reaction
was modeled [1–7] in terms of the deformations re-
quired to allow resonance between ground and charge
transfer states of reactants, i.e., as a result of the avoided
crossing between those states. Thus, the activation bar-
rier depends on at least five parameters: two vertical
ET energy gaps GR = IY: − ARX and GP = IX: − ARY
(where I and A are the vertical ionization potential and
the electron affinity, respectively, of indicated species
and Y: and X: denote the nucleophile and the leaving
group, respectively), two slope factors approximated
by the total delocalization indices of the charge trans-
•
•
−
•
ARY = AY· − DR-Y + DR-·Y
(11)
fer states fR ≈ wR:(Y /(R– X) ) and fP ≈ wR:(X /(R–·
Y)−), and the avoided crossing interaction B (i.e., a
measure of the resonance energy in the TS) taken usu-
where AY = 224 kJ mol−1, DR-Y = 293 kJ mol−1 (as
estimated in the previous part), and DR-·Y = −86 kJ
mol−1;thelastvaluewasestimatedfromthecorrelation
•
ally [3,6,7] as a constant equal to ca. 58.6 kJ mol−1
.
In the free energy notation the full activation barrier is

NONEQUILIBRIUM SOLVENT POLARIZATION IN KINETICS OF SN2 REACTIONS
65
(all values in kcal mol−1):
DR-·Y = −0.31( 0.03)AY• − 3.9( 1.9)
(as shown in Fig. 3):
ꢀG‡
= 0.7( 0.2)ꢀG‡exp + 49( 12)
calc
(13)
n = 8, r = 0.959, F = 68.6, δ = 3.2
n = 9, r = 0.993, F = 532, δ = 0.74 (12)
For reaction (2), the observed scatter of points is sub-
stantially greater with the same most deviating points
forFAandNMP, butasimilargeneraltrendisobserved.
found for nine leaving groups (F, Cl, Br, I, HO,
MeO, HS, MeS, H2N) using data from Table 4.3 in
Ref. [7]; finally, AMeSCN(g) = −155 kJ mol−1. The
value of fP = 0.311 in water (and other protic solvents)
was estimated from the suggested [6] dependence on
1/(AY• − A )2 with A = constant, using fP values
It should be stressed here that the ꢀG‡
values
calc
were calculated against the free energy of ion–dipole
complexes of reactants but experimental barriers in-
clude the formation of these complexes. The last effect
was discussed [7] in terms of a part of solvation free
energy of an anion SY: but the addition of SY: as the
second explanatory parameter to Eq. (13) is statisti-
cally not significant. Moreover, it is evident from Fig. 3
that the calculated values for both reactions are much
higher than the experimental ones. The same tendency
was found [7] but not explained for a number of SN2
reactions in aqueous solutions where the calculated ac-
tivation barriers were 16–32% higher than experimen-
tal ones and in DMF where they were 32–40% higher.
In contrast to that, the calculated and the experimental
values in the gas phase were similar [7]. The possible
explanation of higher activation barriers in solutions,
as suggested by a referee, can be caused by neglecting
the intermolecular term in the simplified calculation of
the reorganization factor ρ, which was originally esti-
mated [3,7] on the basis of the solvation of individual
ions (e.g. nucleophile) instead of a molecular complex
(i.e. the transition state).
In conclusion, the plots shown in Fig. 3 indicate that
solvent effects on the activation barriers for reactions
of interest calculated using the Shaik theory in general
reflect the observed trends but the direct support of a
fundamental role of the nonequilibrium solvent polar-
ization in TS is difficult to obtain. On the other hand,
using the Marcus theory the direct support is evident at
leastforaproticsolvents;however, inaqueoussolutions
some additional effects should be taken into account.
•
•
R
R
[6,7] for PhS, HS, HO, and CN and their oxidation po-
tentials in water [15] as the approximation of solution
AY values. A slightly higher value of fP = 0.313 was
•
assumed for aprotic solvents [6,7]. The free energy of
hydration of SCN− ion is SSCN: = 287 kJ mol−1 [19]
and in other solvents the corresponding values were
obtained from the free energy of transfer of that anion
[16].
Activation free energies for reactions (1) and (2) cal-
culated according to Eqs. (8)–(10) are given in Table II.
◦
The free energy ꢀG of reaction (1), based on the liter-
ature data [12,22], is also given therein; it can be added
◦
that ꢀG does not include free energy of the forma-
tion of ion-dipole complexes of reactants and prod-
ꢁ
◦
ucts, which is contrary to ꢀG from Marcus theory,
but in Eq. (3) the work terms were also neglected. For
the reaction with chloride ions (reaction (1)) the com-
parison of the calculated and experimental activation
barriers, shown in Fig. 3, indicates a rather weak de-
pendence; excluding the most deviating point for for-
mamide (FA) it gives the following correlation line
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66
JAWORSKI
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