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Table 9.2
Effect of solvent on the Menschutkin reaction
Solvent
Acetonitrile
Nitrobenzene
Acetone
Tetrahydrofurane
Benzene
⌬G‡ (kJ molϪ1)
⌬G0 (kJ molϪ1)
n‡a
98.5
100.0
100.4
105.2
110.7
Ϫ19.0
Ϫ14.6
Ϫ12.1
1.9
6.1
0.628
0.628
0.630
0.633
0.623
KT
0.40
0.30
0.48
0.54
0.10
a
Value which reproduces the activation energy using harmonic oscillators to describe the
C᎐I bond ( fCI ϭ 1.6ϫ103 kJ molϪ1 ÅϪ2 and lCI ϭ 2.207 Å) in the reactants and the NC bond
(fNC ϭ 2.95ϫ103 kJ molϪ1 ÅϪ2 and lNC ϭ 1.472 Å) in the products.
Figure 9.1 Variation of activation energy with reaction energy, for the Menschutkin reaction (data
from Table 9.2).
constant by a factor of 2.5ϫ104. However, as shown in Figure 9.1, this does not correspond
to a linear correlation between kinetic and thermodynamic parameters.
What seems to be happening is that, though the solvent does produce an effect on ⌬G°,
at the same time there is a change in the bond order of the transition state, n‡. According
to Figure 9.2, this can be correlated with the Kamlet–Taft solvent parameter KT, which is
an empirical measure of the capacity of the solvent to coordinate via the donation of a pair
of non-bonding electrons, i.e. the solvent acts as a Lewis base.
This effect of the solvent lone pair can be rationalised in terms of attraction by the positive part of the amine dipole, which leads to an increase in n‡, due to an increase in electron density in the N᎐C bond, as indicated diagrammatically in Figure 9.3.
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Figure 9.2 Correlation of transition state bond order (n‡) and the Kamlet–Taft KT parameter for the
Menschutkin reaction (data from Table 9.2). The intercept has a value n‡ ϭ 0.62.
δ
H
N
O
δ
C
I
H
O
R'
electron
donation
R'
electron
removal
Figure 9.3 Changes in electron density produced by electron donation and withdrawal by the solvent.
In addition to these limits of a solvent acting as an inert medium or as an active chemical or electronic species in the transition state, we will see a number of other effects of a
more physical nature can occur, as will be discussed in the following sections.
9.2
EFFECT OF DIFFUSION
With reactions in solution, even when the medium is chemically inert, the solvent will
exert an effect as a consequence of it being a condensed phase, with its molecules very
close together. As a consequence, the distribution of collisions is much more important
than their frequency, which is comparable, but generally 2–3 times greater in liquids
than in gases. As seen in a simulation below (Figure 9.4) due to Rabinowitch and
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9. Elementary Reactions in Solution
Figure 9.4 Contrast between collisions in the gas phase and encounters between the same two reactants in the solvent cage in a liquid.
Wood [3], while collisions between molecules in the gas phase are separated by reasonable time intervals; in liquids, collisions occur in groups of rapid sequences, known as
encounters.
In each encounter between two reactants, the molecules undergo at least four collisions,
until they manage to escape the cage of solvent molecules which surround them. However,
the frequency of these encounters is about 100 times lower than that for collisions of the
same molecules in the gas phase.
Based on this, a reaction in solution can be broken down into three steps: (1) diffusion
of reactants from bulk solution to the collision distance; (2) chemical reaction of the reacting species within the solvent cage; and (3) diffusion of the products from the solvent cage
to bulk solution. We can represent these processes in the following kinetic scheme:
A(s) + B(s)
kD
k− D
( A ⋅⋅⋅ B)(s)
sep
r
⎯⎯ ( P ⋅⋅⋅ Q )(s) ⎯⎯→ P(s) + Q(s)
→
k
k
(9.II)
The rate of reaction, measured by the decrease in concentration of species A, is given by
the expression:
−
d [A]
dt
= kD [ A ][ B] − k− D [ A ⋅⋅⋅ B]
(9.1)
and the formation of the encounter pair or complex, (AB)s, between reactants A and B is
d [ A ⋅⋅⋅ B]
dt
= kD [ A ][ B] − k− D [ A ⋅⋅⋅ B] − kr [ A ⋅⋅⋅ B]
(9.2)
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Assuming steady-state conditions for (AB)s, d[(AB)s]/dt ϭ 0, such that from the previous
equation
[A ⋅⋅⋅ B] =
kD [ A ][ B]
(9.3)
k − D + kr
Substitution for [(AB)s] into eq. (9.1) leads to the reaction rate
−
d [A]
dt
= kD [ A ][ B] − k− D kD
[A][B]
k − D + kr
(9.4)
which, on rearranging, gives
−
d [A]
dt
=
kr kD
[A][B]
k − D + kr
(9.5)
Eq. (9.5) shows two limiting conditions in the reaction kinetics, krkϪD and krkϪD. In
the first case, the reaction rate is given by the expression:
−
d [A]
dt
= kr
kD
[A][B]
k− D
(9.6)
This is the limit of reaction (or activation) control; the rate in solution is controlled by the
activation-controlled reaction between the reactants, kr. The second limit corresponds to
diffusional control, where the rate of reaction is controlled by the rate of diffusion of the
reactants, kD.
−
d [A]
dt
= kD [ A ][ B]
(9.7)
An example of the transition of a process controlled by chemical reaction to one controlled by diffusion is found in the electron transfer between excited aromatic hydrocarbons (D*), and various nitriles A in heptane
D∗ + A → D + + A −
(9.III)
The aromatic hydrocarbons are stronger reducing agents when electronically excited than
in the ground state, and can donate an electron to a species A. The rate of this electron
transfer depends on the ease with which A is reduced, and the reaction follows a linear
free-energy relationship. When ⌬G0 is sufficiently small, such that krkϪD, then the reaction is diffusion controlled, with a rate constant which is constant and is independent of
the reaction energy. These effects are indicated in Figure 9.5.
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9. Elementary Reactions in Solution
Figure 9.5 The change from diffusional control to reaction control in electron transfer between electronically excited aromatic hydrocarbons and nitriles in heptane at room temperature. The dashed
line is diffusion rate constant calculated with eq. (9.23) using the viscosity of heptane. (Courtesy
Carlos Serpa.)
Another example, also involving a photochemical reaction in solution, is the decomposition of molecular iodine and the reverse process of recombination of iodine atoms within
the solvent cage.
I2
hν
kabs
k− D
( I : I)(s)
r
⎯k⎯ 2 I •
→
(9.IV)
This is a primary recombination process, which can be distinguished from secondary
recombination reactions, which occur after the two atoms have separated.
The yield of formation of atomic iodine is twice that of disappearance of molecular
iodine. Under steady-state conditions for (I:I)s, the reaction rate takes the form
v=
kabs kr
[I2 ]
k − D + kr
(9.8)
The yield of I2 loss is the ratio of the rate of formation of 2I to the total rate of dissociation of (I:I)s
φ2 I = −φI2 =
kr
k− D + kr
(9.9)
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Table 9.3
Yield of decomposition of molecular iodine in a photo-induced reaction
Ϫφ I2
Solvent
Hexane
Carbon tetrachloride
Hexachlorobuta-1,3-diene
s
0.5
0.11
0.0042
m
4.0
9.5
Ͼ30
B
B
B
r
B
B
A
B
B
B
B
Figure 9.6 Molecules A in solution surrounded by B.
When the process is diffusion controlled, this yield depends on the solvent viscosity, s,
and, as we will see shortly, the diffusion constants present an inverse relation with s. Data
for this reaction are given in Table 9.3. This cage effect becomes more significant when
the solvents have a high viscosity, and in this case the highest primary recombination
yields are observed. This cage effect is known as the Franck–Rabinowitch effect.
To obtain a more detailed understanding of the kinetics of these reactions, we will now
see how we can estimate the diffusion constants of reactants.
9.3
DIFFUSION CONSTANTS
We will start by considering an uncharged molecule A surrounded by a statistically symmetrical distribution of B molecules (Figure 9.6). The flow, J, of B molecules in the direction towards A can be considered to be the flow of B molecules which cross a sphere
centred on A, with radius r.
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9. Elementary Reactions in Solution
This flow is defined in terms of Fick’s first law of diffusion as
J≡
dn
dc
= − DA
dt
dx
(9.10)
where dn represents the number of moles of substance which diffuses across a surface of
area A, D the diffusion coefficient and dc/dx the concentration gradient. For the moment
we will consider that molecule A is stationary. The negative sign is due to diffusion being
in the direction of lower concentration. The surface of the sphere centred on A, radius r is
A ϭ 4 r2, such that from eq. (9.10), we have
J≡
dnB
dc
= 4 r 2 DB B
dt
dr
(9.11)
Here the concentration gradient, dcB/dr, takes the increase of the radius as reference, such
that the negative sign disappears. Thus, the concentration gradient becomes
dcB
JB
=
dr
4 r 2 DB
(9.12)
Integrating this equation with the limits of concentration of B at the distance r from A, c BO ,
and the concentration of B in the bulk solution (rϭ∞), represented as [B]
∫
[B]
cB 0
dcB (r ) =
JB
4 DB
∫
∞
r
1
dr
r2
(9.13)
gives
cB
[B]
cB0
∞
JB 1
4 DB r r
(9.14)
JB
+ [ B]
4 DBr
(9.15)
=−
which leads to
cB0 = −
This result is valid for the conditions where A is stationary. However, as in practice this species
also diffuses, the equation should be modified to include the two diffusion coefficients
cB0 = −
4
JB
+ [ B]
DB + DA ) r
(
(9.16)
When the reaction is totally controlled by diffusion at the microscopic level, B reacts very
rapidly with A when they are separated by a critical distance, dAB. Therefore, when r ϭ dAB
the concentration c BO ϭ 0. This is because there is reaction between A and B on every
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Figure 9.7 Concentration B as a function of distance from A for a reaction completely controlled by
diffusion, following eq. (9.16).
encounter. Therefore, from eq. (9.16), we obtain
JB = 4
( DB + DA ) dAB [B]
(9.17)
The graph of concentration of B as a function of distance is shown in Figure 9.7.
The rate of reaction involving all molecules of A becomes
v=4
( DB + DA ) dAB [ A ][ B]
(9.18)
and the diffusion-controlled rate constant is
kD = 4
( DB + DA ) dAB
(9.19)
The Système International (SI) units for DA and DB are m2 secϪ1, such that kD will have
the units m3 secϪ1. Normally in chemical kinetics we use molar quantities and litres, which
requires multiplying kD/m3 secϪ1 by the Avogadro number NA and by 103 to give
kD = 4 103 N A ( DB + DA ) dAB
dm 3 mol −1 sec −1
(9.20)
When the diffusion coefficients are not known, we can determine them from the
Stokes–Einstein equation
D=
k BT
6 sr
(9.21)
where s is the solvent viscosity. The Stokes–Einstein equation is valid when the solute
molecules are much bigger than those of the solvent. The sum of the diffusion coefficients
is approximately
DA + DB =
k BT
6 s
⎛1 1⎞
⎜ + ⎟
⎝ rA rB ⎠
(9.22)
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9. Elementary Reactions in Solution
And, taking dAB ϭ rAϩrB, substitution of eq. (9.22) into eq. (9.20) leads to the diffusioncontrolled rate constant
2 k T ( rA + rB )
kD = B
rA rB
3 s
2
(9.23)
If the molecules have the same size (i.e. rAϭrB), we obtain
kD =
8 k BT
3 s
(9.24)
The rate constant k᎐D is the reciprocal of the time in which A and B remain as nearest
neighbours, which, in the absence of any specific interactions between A and B will be
k− D =
6 ( DA + DB )
(9.25)
2
dAB
When the reactions are controlled by diffusion, the rate will be proportional to the reciprocal of the solvent viscosity, as shown by eqs. (9.7) and (9.24), and Table 9.4. The typical activation energy of diffusion in solution is fairly small, ca. 15 kJ molϪ1. Figure 9.8
shows an Arrhenius plot of one of the systems in Figure 9.5 under the diffusion-controlled
regime, but measured in isopropyl ether. The experimental activation energy for this system in this solvent is 11 kJ molϪ1, and is entirely due to the temperature dependence of the
viscosity.
Table 9.4
Diffusion controlled rate constants calculated from eq. (9.24) and viscosities for some common
solvents at 25 °C
Solvent
Carbon dioxide
Diethyl ether
Hexane
Acetone
Toluene
Decane
Water
Ethanol
Propan-2-ol
Ethylene glycol
Glycerol
a
Viscositya ( ) (mPa sec)
kdiffb (MϪ1secϪ1)
0.0577c
0.222
0.294
0.316
0.558
0.861
0.890
1.083
2.044
19.9d
945
1.2 ϫ 1011
3.0 ϫ 1010
2.2 ϫ 1010
2.1 ϫ 1010
1.2 ϫ 1010
7.7 ϫ 109
7.4 ϫ 109
6.1 ϫ 109
3.2 ϫ 109
3.3 ϫ 108
7.0 ϫ 106
From Handbook of Chemistry and Physics. 59th Edn. (ed. RC Weast), CRC Press, West Palm Beach, FA, 1979.
Using viscosities in mPa sec, it is necessary to multiply by 106 to get values in MϪ1 secϪ1.
c
At 35 °C and 10 MPa, in the supercritical phase, from http://webbook.nist.gov/.
d
At 20 °C.
b
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Figure 9.8 Temperature dependence of electron transfer between electronically excited naphthalene
and fumaronitrile in isopropyl ether. The line is the diffusion rate constant calculated at the different
temperatures. (Courtesy Paulo Gomes.)
Although from eq. (9.23) the diffusion-controlled constant, kD, depends on the size of
molecules, according to
( rA + rB )
rA rB
2
= 2+
rA rB
+
rB rA
(9.26)
With the normal range of sizes of molecules of interest, this only shows a modest change with
rA/rB. For example, the ratio of eq. (9.26) will be 4.5 for rA/rB ϭ 2 and 4.0 with rA/rB ϭ 1.
However, dramatic effects can be seen when one of the species is a polymer, and the other is
a small molecule. For example, with the case given in Chapter 3 of triplet energy transfer in
benzene from biphenyl to the polymer MEH–PPV, of molecular weight 1.5ϫ106, the experimental rate constant is 1.97ϫ1011 dm3 molϪ1 secϪ1 [4], which can be compared with the normal diffusion controlled rate constant in benzene at 25 °C (1.1ϫ1010 dm3 molϪ1 secϪ1) [5].
Using eq. (9.23) and reasonable estimates for the polymer and biphenyl size, we can calculate
a value kD ϭ 1.2ϫ1011 dm3 molϪ1 secϪ1 for this system.
For reactions between species of similar size in water at 25 °C, eq. (9.24) leads to a value
of the diffusion-controlled rate constant for neutral species of 7ϫ109 dm3 molϪ1 secϪ1. This
is probably an underestimate, and in Table 9.5, some experimental rate constants are given
for diffusion-controlled reactions in water.
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9. Elementary Reactions in Solution
Table 9.5
Experimental rate constants for some typical diffusion controlled reactions in
aqueous solutionsa
k (109 MϪ1secϪ1)
Reaction
H · ϩ H· → H 2
H· ϩ ·OH → H2O
·
OH ϩ ·OH → H2O2
·
OH ϩ OHϪ → O·Ϫ ϩ H2O
H· ϩ O2 → HO2·
H· ϩ Fe(CN)63Ϫ → Hϩ ϩ Fe(CN)64Ϫ
C6H6 (benzene) ϩ ·OH → C6H6OH·
7.75
7.0
5.5
13
21
6.3
7.8
a
Data selected from ref. [6].
Table 9.6
Rate constants for reactions of hydrated electrons (eaqϪ) with charged and
uncharged species in aqueous solutiona
k (1010 MϪ1secϪ1)
Reaction
eaqϪ ϩ H· → H2 ϩ OHϪ
eaqϪ ϩ eaqϪ → H2 ϩ 2OHϪ
eaqϪ ϩ OH· → OHϪ
eaqϪ ϩ Hϩ → H·
eaqϪ ϩ O2 → O2·Ϫ
eaqϪ ϩ O2·Ϫ → O22Ϫ
eaqϪ ϩ Agϩ → Ag(0)
eaqϪ ϩ Cu2ϩ → Cuϩ
eaqϪ ϩ UO22ϩ → UO2ϩ
eaqϪ ϩ Co(NH3)63ϩ → Co(NH3)62ϩ
eaqϪ ϩ Au(CN)2Ϫ → Au(0) ϩ 2CNϪ
eaqϪ ϩ Al(OH)4Ϫ → Al(OH)42Ϫ
eaqϪ ϩ Fe(CN)63Ϫ → Fe(CN)64Ϫ
eaqϪ ϩ C6H6 (benzene) → C6H7·
eaqϪ ϩ coenzyme B12 → reduced form
2.5
0.55
3.0
2.3
1.9
1.3
3.7
3.3
1.7
8.8
0.35
0.00055
0.31
0.0009
3.2
a
Data selected from ref. [6].
For reactions between species of charge Z1e and Z2 e, eq. (9.24) needs to be modified to
allow for the electrostatic attraction or repulsion between ions
kD =
8 k BT
3 s exp (
) −1
(9.27)
with
=
Z A Z B e2
4 kBTdAB
(9.28)