Chapter 7. Relationships between Structure and Reactivity

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Owing to the exponential dependence of the rate upon energy, the rate problem reduces

mainly to the determination of the lowest energy barrier that has to be surmounted. The

ISM model, which was presented in the previous chapter, points, in general terms, to some

structural factors that control the barriers of chemical reactions and as a consequence the

rate constants. These relevant factors are (i) reaction energy, ⌬E0, ⌬H0 or ⌬G0; (ii) the

electrophilicity index of Parr, m, a measure of the electron inflow to the reactive bonds at

the transition state, also characterised as a transition-state bond order n‡; (iii) when the

potential energy curves for reactants and products can be represented adequately by harmonic oscillators, the relevant structural parameters are the force constants of reactive

bonds, fr and fp in reactants and products, respectively; and (iv) equilibrium bond-lengths

of reactive bonds, lr and lp in reactants and products, respectively.

Let us represent the course of reaction (7.I) by the intersection of potential energy

curves of the reactant BC and the product AB as

A + BC → AB + C



(7.I)



When the potential energy of the molecular species is represented by harmonic oscillators,

the common energy of the distended configurations of BC and AB at the transition state is

represented by the following expression:

1

1

2

fr x 2 = fp ( d − x ) + ΔE 0

2

2



(7.1)



where x is the bond extension of the reactant molecule BC to the transition state, and d represents the sum of the bond extensions of BC and AB on going to the transition state. In

eq. (7.1), dϪx represents the bond extension of the product molecule as illustrated in

Figure 7.1. This equation expresses reaction energy in terms of a thermodynamic quantity,

the internal energy, ⌬E0, rather than in terms of a difference in the potential energies of

products and reactants, ⌬V0.

Before pursuing further, a word of caution is necessary. Chemical thermodynamics

deals with macroscopic observations, but aims to build a bridge between macroscopic

variables and events at microscopic level by specifying the composition of the system in

molecular terms. Potential energy, V, is a microscopic variable defined for all configurations of the reacting system. Thus, V varies in a continuous manner along the progress variable and defines a reaction path. But thermodynamic energies are not continuous functions

along this path. They are only valid for stable or metastable states, i.e., for elementary

reactions they are only valid for reactants, products and the transition state. Whereas

eq. (7.1), defined in terms of ⌬V0, possesses a physical meaning for all the points

0 Յ x Յ d, the same mathematical expression defined in terms of ⌬E0, or in terms of any

other thermodynamic quantity, has no physical meaning for the points interpolated

between reactants, transition state and products.

The energy barrier for the prototype reaction, measured with respect to reactants, is

expressed by the equation

ΔE ‡ =



1

fr x 2

2



(7.2)



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Figure 7.1 Reaction coordinate of a bond-breaking–bond-forming reaction in terms of intersecting

harmonic oscillators. The reaction coordinate d represents the sum of the reactant and product bond

extensions, from equilibrium to transition sate configurations.



Under the particular condition of almost equal force constants for reactant and product,

f=fr=fp, eq. (7.1) becomes

1 2 1 2

1

fx = fd − fdx + fx 2 + ΔE 0

2

2

2



(7.3)



Solving this equation for the reactant distension, x, yields

x=



d ΔE 0

+

2

fd



(7.4)



This expression can also take the more convenient form



x=



1 ⎛ 2 ΔE 0 ⎞

d 1+

2 ⎜

fd 2 ⎟

⎝

⎠



(7.5)



If we make the substitution of x in the expression of the energy barrier, eq. (7.2), then we have

⎛

ΔE 0 ⎞

ΔE = ΔE ( 0) ⎜ 1 +

⎟

‡

⎝ 4 ΔE ( 0) ⎠

‡



‡



2



(7.6)



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7. Relationships between Structure and Reactivity



where ⌬E‡(0) represents the intrinsic barrier of reaction (7.I).

ΔE ‡ ( 0) =



1 2

fd

8



(7.7)



The intrinsic barrier represents the barrier of the reaction taking it as an isoenergetic

process, that is, with ⌬E0 = 0.

In the past, Evans and Polanyi [2] have introduced the concepts of chemical driving

force and chemical inertia. By chemical inertia they meant the work that must be done to

produce reaction, partly in breaking BC bond and in placing the atom A sufficiently close

to B. In the present formulation this can be represented by the intrinsic barrier, ⌬E‡(0). By

chemical driving-force they meant the contribution that the energy of formation of the new

bond AB makes towards overcoming the inertia, which is represented by the reaction

energy ⌬E0. Eq. (7.6) reveals that under certain approximations, one can separate the thermodynamic contribution, ⌬E0, and the kinetic contribution, ⌬E‡(0), for the energy barrier

of the reaction ⌬E‡. Seminal ideas for such a separation go back to Jean-Auguste Muller

in 1886, but the correct quantitative formulation was proposed by Rudolph Marcus [3] in

the 1960s.

The sum of bond extensions to the transition state, d, was given by ISM, in the previous chapter as

⎡ aЈ ln ( 2) aЈ ⎛ ΔE 0 ⎞ 2 ⎤

⎥ lr + lp

d=⎢

+ ⎜

‡

2⎝ Λ ⎟ ⎥

⎠ ⎦

⎢ n

⎣



(



)



(7.8)



When the dynamic parameter ⌳ is, in absolute terms, much greater than the reaction

energy, ⌳ Խ⌬E0Խ, then d is independent of the reaction energy. Under such conditions d

has a constant value that is the sum of bond extensions of reactants and products at thermoneutrality, d(0)

d ( 0) =



aЈln ( 2)

n‡



(l + l )

r



(7.9)



p



with aЈ=0.156. Under this set of approximations, the separation between the intrinsic and

thermodynamic contributions for the energy barrier, ⌬E‡, is valid (see eq. (7.6)). Thus, the

intrinsic barrier can be expressed in terms of several molecular factors, specifically force

constant, transition-state bond order and the sum of the equilibrium bond lengths of reactant and product.



ΔE ‡ (0 ) =



1

8



⎡ 0.108

⎤

f ⎢ ‡ lr + lp ⎥

n

⎣

⎦



(



)



2



(7.10)



Eq. (7.6), known as the equation of Marcus, leads to a quadratic relationship between

the barrier of reaction and the reaction energy, which shows up also in the modified form



(



)



2



ΔE 0

ΔE 0

ΔE = ΔE ( 0) +

+

2

16 ΔE ‡ ( 0)

‡



‡



(7.11)



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Figure 7.2 Rate constants of proton transfers from enolate ions and water. The curve represents an application of the Marcus quadratic relationship, with ⌬G‡(0) = 57 kJ molϪ1 and kdif = 1011 molϪ1 dm3 secϪ1.



For reactions in solution, one tends to use Gibbs energy, G, instead of internal energy,

E. For this reason, eqs. (7.6) and (7.11) are called (Gibbs) free-energy relationships. In this

particular case such equations represent quadratic free-energy relationships (QFER).

Figure 7.2 illustrates a QFER for the proton-transfer reactions between enolate anions

and water molecules [4]. The experimental data can be reproduced by the equation of

Marcus with an intrinsic barrier of ⌬G‡(0) = 57 kJ molϪ1 and a pre-exponential factor of kdif =

1011 molϪ1 dm3 secϪ1. This is a good illustration of a family of reactions which have similar behaviour throughout a very extensive energy range, ⌬⌬G0 = 200 kJ molϪ1 and ⌬⌬G‡ =

85 kJ molϪ1 and covering reaction rates over 14 orders of magnitude. Changes in reactivity are controlled entirely by changes in ⌬G0. This means that the bonds to be broken and

formed, OH and CH, have the same force constants and bond lengths throughout the reaction series and that n‡ is also constant. This implies that the electrophilicity is quite invariant for the different enolates employed in this study.

7.2



LINEAR FREE-ENERGY RELATIONSHIPS (LFER)



Eq. (7.11) can be further simplified if the intrinsic barrier is much higher than the range of

reaction energies, such that 16(⌬E‡(0)) (⌬E0)2. Inserting this condition into eq. (7.11)

leads to



ΔE ‡ Ϸ ΔE ‡ ( 0) +



ΔE 0

2



(7.12)



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This expression reveals the existence of a linear relation between energy barrier and reaction energy. When expressed in terms of Gibbs energy, this represents a linear free-energy

relationship (LFER).

The equation of Marcus was developed in 1956 and ISM in 1986. Predictions on the

effects of substituents on the rate constants and the equilibrium constants are not due to

such developments, but to a rich history of empiricism. Nevertheless, many of those historical paths can now be encompassed by the Marcus equation and ISM.

7.2.1



Brönsted equation



In the acid–base reaction (7.II), it appeared reasonable to Brönsted and Pederson that if the

rates k at which proton is removed by a particular basis BϪ were compared for various

acids HA

HA + B− → A− + BH



(7.II)



then the base might remove the proton more rapidly from the stronger acids. In fact, relationships between rate of an acid–base reaction and equilibrium have been observed in

many cases and frequently obey an equation known as the Brönsted catalysis law

k = GK



(7.13)



where G and are empirical constants, k the rate for reaction (7.II) and K the acid dissociation equilibrium constant. In logarithmic form, the Brönsted equation can be written as

log k = log K + constant



(7.14)



The value of can be estimated from a plot of the second-order catalytic constant log k for

the reaction catalysed by the acid HA, against the pKHA. Equivalent procedures can also

be performed for base catalysed reactions. Figure 7.3 illustrates the application of

Brönsted relationship for the proton-transfer reactions [5].

XC6 H 4 CH 2 CH(COCH 3 )CO 2 C2 H 5 + RCO − → XC6 H 4 CH 2 C− (COCH 3 )CO 2 C2 H 5

2

+ RCO 2 H



(7.III)



In one set of the reactions, the substituent X of the carbon acid was varied and the same

base, CH3COOϪ, was used; this allows one to estimate the Brönsted constant for the acid

catalysis, . In another set of experiments the base was varied, but the acid was kept constant; this allows the estimation of the Brönsted parameter for base catalysis.

Taking into consideration the relations between rate constants k and ⌬G‡ and equilibrium constants K and ⌬G0,

k=



⎛ ΔG ‡ ⎞

k BT

exp ⎜ −

h

⎝ RT ⎟

⎠



⎛ ΔG 0 ⎞

K = exp ⎜ −

⎝ RT ⎟

⎠



(7.15)



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Figure 7.3 Correlation between the ionisation constants of carboxylic acids RCOOH or carbon

acids R3CH, and the second-order rate constant of the reactions in mechanism (7.III).



the Brönsted equation takes the form

ΔG ‡ = ΔG 0 + constant



(7.16)



showing that, at a constant temperature, this equation reflects a LFER.

Following a suggestion of Leffler [6], chemists have tried for a number of years to use

the Brönsted coefficient, = * ΔG ‡ * ΔG 0 , as a measure of the position of the transition state (TS) along the reaction coordinate, since usually 0Ͻ Ͻ1 , with = 0 for reactants, and = 1 for products. As a consequence, = 0.5 would imply that the transition

state is exactly half way between reactants and products.

As illustrated in Figure 7.4 for a thermoneutral reaction, is not always a measure of the

position of TS. When fr = fp one has for this kind of reaction = 0.5, reflecting correctly

the position of the transition state. But if fr Ͼ fp the transition state is before the measure

provided by ; if fr Ͻ fp the transition state is ahead of the measure of .

The Marcus equation, now expressed in terms of Gibbs energy, can be written as



(



) (



)



*ΔG ‡ 1 1 ΔG 0

= +

*ΔG 0 2 8 ΔG ‡ ( 0)



(7.17)



This equation does not give full support to the empirical equation of Brönsted, because the

relation is a quadratic one, while that of Brönsted is linear, such that under the relevant

approximations would be constant and always equal to 0.5. ISM, which is more general

than the equation of Marcus, accommodates LFER with values of

0.5 (0 Ͻ Ͻ1)

when the force constants do not have a common value, fr fp. In empirical terms there are

cases with Ͼ1 and Ͻ0, but these are anomalous situations which will be discussed later



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Figure 7.4 Effect of the force constants of the reactive bonds on the position of the transition state

for an isothermal reaction.



on in the Chapter 13, devoted to proton-transfer reactions. Identical considerations are

valid for the coefficients.

7.2.2



Ϫ

Ϫ

BellϪEvansϪPolanyi equation



In the mid-1930s of the twentieth century, Bell, Evans and Polanyi correlated energies of

activation, Ea, for several reactions in the vapour phase with heats of reaction, ⌬H0,

according to the following expression:

Ea = ΔH 0 + constant



(7.18)



which is also a LFER. The additive constant in this relationship will be taken to assume,

according to Rudolph Marcus, the presence of an intrinsic barrier.

Both free-energy relationships expressed by eqs. (7.6) and (7.18) would imply that reactions which have a strong driving force thermodynamically will also proceed rapidly.

However, if we consider this more deeply we will see that such an implication does not

hold generally. For example, the energetically favourable oxidation of hydrocarbons in the

presence of air may not take place for years, whereas the energetically unfavourable hydration of carbon dioxide takes place in seconds. For elementary reactions, the correct interpretation rests on the values of the intrinsic barriers.

7.2.3



Hammett and Taft relationships



In January 1937, Hammett published a paper on the “Effects of structure upon the reactions of benzene derivatives”. Let us consider the dissociation of benzoic acid

C6 H 5 − COOH → C6 H 5 − COO − + H + ,



K 0 = 6.27 × 10 −5



(7.IV)



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Hammett realised that the addition of a substituent in the aromatic ring has a quantitative effect on the dissociation constant K. For para-nitrobenzoic acid the dissociation

constant

O2 N − C6 H 5 − COOH → O2 N − C6 H 5 − COO − + H + ,



K 0 = 37.1 × 10 −5



(7.V)



increases, indicating an increment in the stabilisation of the negative charge generated by

dissociation. If the substituent is an ethyl group in para position, the opposite effect is

observed with respect to the unsubstituted reaction

C2 H 5 − C6 H 5 − COOH → C2 H 5 − C6 H 5 − COO − + H + ,



K 0 = 4.47 × 10−5



(7.VI)



indicating that the ethyl group destabilises the negative charge generated in dissociation.

Hammett found, for example, that the nitro group has a stabilising effect on other dissociation reactions such as those of phenylacetic acid. He then proposed a quantitative

relationship to account for such findings

log K = log K 0 +



(7.19)



The same kind of expression is also valid for the rate constants, k

log k = log k 0 +



(7.20)



K0 and k0 denotes the corresponding constants for the “parent” or “unsubstituted” compound.

The substituent constant is a measure of the electronic effect of replacing H by a given

substituent (in the para or meta position) and is, in principle, independent of the nature of

reaction. The reaction constant depends on the nature of the reaction, including conditions such as solvent and temperature; is a measure of the susceptibility of the reaction

to the electronic effects of substituents. The reference reaction is the ionisation of unsubstituted benzoic acid in water at 25 °C, with = 1.

Thus it is clear that just a few values of ␴ and can summarise a large body of equilibrium and rate measurements, and can help to predict rate coefficients and equilibrium

constants for reactions which have not yet been studied. The substituent constant ␴ is a

positive or negative number (Table 7.1, Figure 7.5). For example, p = 0.71 for ϪNO2

indicates that this group has an electron-removing effect higher than H-atom, whereas

p= Ϫ0.66 for ϪNH2 indicates that this group has an electron-donating effect stronger than

H-atom. Reactions with Ͼ 0 are facilitated by electron-removing substituents.

Although the Hammett relationship works well for meta- and para-substituted aromatic

compound, it frequently fails for either ortho- aromatic (due to steric effects) or aliphatic

compounds, and other scales have been proposed, such as that of Taft.

log k − log k0 = * *



(7.21)



Figure 7.6 compares the performance of the equations of Hammett and Taft with respect

to the hydrolysis of ketals that proceed according to mechanism (7.VII) [7].



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Table 7.1

Hammett (

Substituents

᎐NH2

᎐NMe2

᎐OH

᎐OMe

᎐OAc

᎐SMe

᎐H

᎐Me

᎐Et

᎐n-Pr

᎐t-Bu

᎐C6H5

᎐COMe

᎐F

᎐Cl

᎐Br

᎐I

᎐CN

᎐NO2

᎐CF3



m,



p)



and Taft ( *) coefficients of common substituents in organic molecules

m



p



Ϫ0.16

Ϫ0.10

0.02

0.12

0.39

0.14

0

Ϫ0.06



Ϫ0.66

Ϫ0.32

Ϫ0.22

Ϫ0.27

0.31

0.06

0

Ϫ0.14



Ϫ0.09

0.05

0.36

0.34

0.37

0.37

0.34

0.62

0.71

0.46



Ϫ0.15

0.05

0.47

0.15

0.24

0.26

0.28

0.71

0.78

0.53



*(CH2Y)



0.555

0.52



Ϫ0.10

Ϫ0.10

Ϫ0.13

Ϫ0.165

0.215

0.60

1.1

1.05

1.00

0.85

1.30

1.40

0.92



Figure 7.5 Application of the Hammett equation to the reactions following mechanism (7.III).



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Figure 7.6 Application of Hammett and Taft equations to the rate of the hydrolysis of ketals that

proceed according to mechanism (7.VII).



+



H

R



R



OC2H5

+



C

H3C



OC2H5

C



+ BH



H3C



OC2H5



+ B

OC2H5



slow

+

R



R

C



H3C



O



+ BH+



H2O

fast



C

H3C



OC2H5



+ C2H5OH



(7.VII)



These empirical correlations call for an explanation, by which we mean a theoretical

basis, however profound or modest. From eq. (7.20) we can write

ΔG ‡0 = ΔG ‡0 − 2.3 RT

0



(7.22)



where Δ G ‡0 corresponds to the Gibbs activation energy of the reference reaction. For

0

other homologous reactions, with the same set of substituents, one has

ΔG ‡0 Ј = ΔG ‡0 Ј − 2.3 RT Ј

0



(7.23)



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Those two equations can be written in a different form

ΔG ‡0



=



ΔG ‡0

0



− 2.3 RT



ΔG ‡0 Ј ΔG ‡0 Ј

0

=

− 2.3 RT

Ј

Ј



(7.24)



and if one subtracts the other, we obtain

ΔG ‡0



−



ΔG ‡0 Ј ΔG ‡0 ΔG ‡0 Ј

0

0

=

−

Ј

Ј



(7.25)



We can therefore write a linear equation between the free energies of activation of one set

of reactions and those of a corresponding set

ΔG ‡0 −



Ј



ΔG ‡0 Ј = constant



(7.26)



where the coefficient / Ј has the same value for all reactions in the set. Hammett equations are thus another kind of LFER.

The equations of Hammett and Taft apply reasonably well to Gibbs free energies. One

might anticipate that this will also be valid for changes in enthalpies and entropies.

ΔG ‡ = ΔH ‡ − T ΔS ‡



(7.27)



However, this is not the case. The changes in ⌬H‡ and in ⌬S‡ show a considerable scatter

when compared with ⌬G‡, and there is a compensating effect between ⌬H‡ and ⌬S‡. A substituent group that induces a strong interaction of the solute with solvent molecules and

decreases ⌬H‡, also decreases ⌬S‡. Figure 7.7 illustrates this effect of compensation in the

alkaline hydrolysis of ethyl benzoate in water/alcohol and water/dioxane mixtures [8],

which can be translated by the linear relationship

ΔH ‡ =



iso



ΔS ‡



(7.28)



when a parameter of the reaction (solvent, substituent, etc.) is changed,

ΔG ‡ = ΔH ‡ − T ΔS ‡ = (



iso



− T ) ΔS ‡



(7.29)



The parameter iso represents the isokinetic temperature, the real or virtual temperature for

which all members of the series have the same rate constant (Figure 7.8). Equivalent considerations are also valid for equilibrium constants.