Chapter 9. Elementary Reactions in Solution

Ch009.qxd



12/22/2006



10:31 AM



Page 224



224



9. Elementary Reactions in Solution



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.



Ch009.qxd



12/22/2006



9.2



10:31 AM



Page 225



Effect of Diffusion



225



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



Ch009.qxd



12/22/2006



10:31 AM



Page 226



226



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)



Ch009.qxd



12/22/2006



9.2



10:31 AM



Page 227



Effect of Diffusion



227



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.



Ch009.qxd



12/22/2006



10:31 AM



Page 228



228



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)



Ch009.qxd



12/22/2006



9.3



10:31 AM



Page 229



Diffusion Constants



229



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.



Ch009.qxd



12/22/2006



10:31 AM



Page 230



230



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



Ch009.qxd



12/22/2006



9.3



10:31 AM



Page 231



Diffusion Constants



231



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)



Ch009.qxd



12/22/2006



10:31 AM



Page 232



232



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



Ch009.qxd



12/22/2006



9.3



10:31 AM



Page 233



Diffusion Constants



233



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.



Ch009.qxd



12/22/2006



10:31 AM



Page 234



234



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)