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Free Energy Relationships in Organic and Bio-organic Chemistry

By Andrew Williams

The Royal Society of Chemistry

Copyright © 2003 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85404-676-8

Contents

Chapter 1 Free Energy Relationships, 1,
Chapter 2 The Equations, 17,
Chapter 3 Effective Charge, 55,
Chapter 4 Multiple Pathways to the Reaction Centre, 75,
Chapter 5 Coupling Between Bonds in Transition Structures of Displacement Reactions, 107,
Chapter 6 Anomalies, Special Cases and Non-linearity, 129,
Chapter 7 Applications, 158,
Appendix 1 Equations and the More O’Ferrall–Jencks Diagram, 201,
Appendix 2 Answers to Problems, 210,
Appendix 3 Tables of Structure Parameters, 258,
Appendix 4 Some Linear Free Energy Equations, 282,
Subject Index, 290,

CHAPTER 1

Free Energy Relationships

There are many techniques of varying degrees of generality for the study of mechanisms and that of free energy relationships is the most readily applicable and general. Free energy relationships comprise the simplest and easiest of techniques to use but the results are probably the trickiest to interpret of all the mechanistic tools.

The quantitative study of free energy relationships was first introduced by Brønsted and Hammett in the early part of the twentieth century. As well as playing a major rôle in understanding and deducing mechanism, free energy relationships have been incorporated into software to provide computational methods for calculating physico-chemical parameters of molecules. Such techniques are of substantial technical importance in drug design where parameters such as solubility, pKa, etc. can be useful indicators of the biological activity of chemical structures of potentially therapeutic compounds. Free energy relationships may be employed in the design of synthetic routes.

The most recent undergraduate texts devoted to free energy relationships were published in 1973. Since that time the application of free energy relationships to the study of organic and bio-organic mechanisms has undergone substantial transformation and fierce critical appraisal. The slopes of linear free energy correlations of polar substituent effects are related directly to changes in electronic charge (defined as effective charge) at a reaction centre. Since charge difference is a function of bonding change, polar substituent effects are related to bonding in the transition structure relative to that in the reactant; it has to be recognised that this relationship is often not a simple one. Effective charge is measured by comparing the polar substituent effect with that of a standard equilibrium, such as the (heterolytic) dissociation of an acid, where the charge difference is arbitrarily defined. Comparison of the polar substituent effect with that of a standard dissociation is a special case of the similarity concept where the change in free energy in a reaction or process is compared with that of a similar standard process. This way of utilising the polar substituent effect provides a direct relationship between an empirical parameter (the slope of the free energy relationship) and an easily understood physical quantity, namely charge, developing in the reference process.

Polar substituent effects are observed when there is a change in the polarity of the substituent in the system. The observer’s tool (the substituent change) is part of the system being observed and will thus affect the results. The observational effect is inherent in all measurements (which must be made by an observer) but is usually negligible, for example in chemical experiments involving relatively non-invasive techniques, such as electromagnetic radiation, as the means of observation. In free energy relationships the effect could be serious unless it is recognised. Indeed, as we shall see in Chapters 5 and 6 the curvature sometimes observed in free energy correlations is a direct consequence of the connection between observational tool and observed reaction centre. The additional complication that changing the substituent could have a gross effect on the mechanism is a very useful tool as explained in Chapter 7.

We shall be discussing the following main attributes of the technique of free energy relationships throughout the text:

• Linear free energy relationships are empirical observations which can be derived when the shapes of the potential energy surfaces of a reaction are not substantially altered by varying the substituent.

• The slopes of linear free energy relationships for rate constants are related to transition structure.

• Transition structure is obtained from knowledge of the bond orders of the major bonding changes as well as of solvation.

• Changes in the shape of the energy surface of a reaction give rise to predictable changes in the slopes of free energy relationships.

• Polar substituent effects arise from changes in electronic charge in a reaction brought about by bond order and solvation changes.

1.1 MECHANISM AND STRUCTURE

The mechanism of a reaction is described as the structure and energy of a system of molecules in progress from reactants to products. This representation gives rise to an expectation that experimental methods should indicate the structures as shown in static diagrams; it obscures the fact that the experimental techniques refer to assemblies of structures each of which has rotational, vibrational and translational activity.

An assembly of reactant molecules is converted into product molecules through an assembly of transient structures which are not discrete molecules. Even though the structures making up the transition state assembly are not interconvertible within the period of their lifetimes (10-13 to 10-14 seconds) the assembly is postulated to have a normal thermodynamic distribution of energies. A definition of mechanism which can be fulfilled experimentally, at least in principle, is a description of any intermediates and the average transition structures intervening between these intermediates, reactants and products. This definition of mechanism is currently employed in mechanistic studies. More fundamental knowledge, such as that of the structure of the system between reactant and transition structure, is at present largely in the realm of theoretical chemistry.

Structure requires a description of the relative positions of nuclei and the electron density distribution in a system. The existence of zero-point vibrations indicates that even at absolute zero the exact positions of atoms in a molecule are uncertain. As the temperature is increased, higher quantum states are occupied for each degree of freedom so that fluctuations around the mean positions of atoms increase. In order to discuss the mechanism of any reaction, it is necessary to bear in mind precisely what the term structure means. A pure compound is usually visualised as an assembly of molecules each comprising atoms which have identical topology relative to their neighbours. The exact relative position of each atom is time-dependent, and a bond length, measured on a collection of molecules, is an average quantity. The two-dimensional static structures possessing bonds represented by lines (Lewis bonds) are hypothetical models. These structures, commonly designated Kekulé structures, are very convenient for visualising organic molecules as they are easily comprehensible and are easily represented graphically in two dimensions. Although they are often taken for reality (particularly in their three-dimensional form) they are simply representations of hypotheses which fit experimental knowledge of the compounds.

The reaction coordinate is a measure of the average structure taken by a single molecular system on passage from reactants to products. Structure usually equates with potential energy. The transition structure is not a molecular species and corresponds to the maximum of the Gibbs free energy in the system on progression from reactant to product state. The free energy is distributed amongst the available quantum states of the various degrees of freedom of the collection of pseudo-molecules of the transition state.

The structure of the molecular species in the transition state is thus an average but it should be stressed that this is no different in principle from that of a discrete molecule which is also an average.

1.1.1 Interconversion of States – Reaction and Encounter Complexes

In a bimolecular reaction in solution reactants diffuse through the assembly of solute and solvent molecules (Scheme 1) and collide to form an encounter complex within a solvent cage. Reaction is not possible until any necessary changes occur in the ionic atmosphere to form an active complex and in solvation (such as desolvation of lone pairs) to form a reaction complex in which bonding changes take place. The encounter complex remains essentially intact for the time period of several collisions because of the protecting effect of the solvent surrounding the molecules once they have collided. The products of the subsequent reaction could either return to reactants or diffuse into the bulk solvent.

A similar description applies to a unimolecular reaction except that formation of the transition structure is initiated by energy accumulation in the solvated reactant by collision.

Scheme 2 gives typical half-lives for reactant molecules destined to react. Many encounters do not lead to reaction and only a small fraction of the complexes will have the appropriate transition state solvation in place to form a reaction complex and for reaction to proceed.

Extracting conclusions from experimental results derived from reactions of collections of molecules requires the interpretation of measured global energy changes in these collections as a function of “average” structures. Free energy relationships provide an important technique for carrying out this interpretation. In general polar substituents will have their greatest effect on the process between reaction complexes (A in Scheme 2). Solvent change is largely effective on processes B and C but there is some effect on A due to solvent stabilisation of charge and dipole caused by bonding change.

1.2 UNIVERSAL MEASURE OF POLARITY

It is commonplace that some groups withdraw electrons from a reaction centre whereas others donate electrons. For example, a halo group is expected to attract electrons relative to hydrogen, and alkyl groups to donate electrons. Free energy relationships provide a means of quantifying polar effects. The concept of relative polarity is not simply that of an inductive effect (I) through σ-orbitals, because transmission of charge can occur through space as afield effect (F) or through a conjugated system of π-orbitals as a resonance effect (R). The extent to which these three transmission vectors contribute to the overall polarity of a substituent group is dependent on the system to which it is attached.

The original measure of substituent polarity was chosen to be the dissociation constant of a substituted benzoic acid in water (Equation 1). It is readily confirmed that the pKa<.sub> of the benzoic acid is raised by an electron donating substituent and lowered by an electron withdrawing substituent.

A polar substituent changes the energy of the transition state of a reaction by modifying the change in charge brought about by the bond changes (Hine). Thus substituents that withdraw electrons and disperse negative charge will tend to stabilise a structure if there is an increase in negative charge at the reaction centre (and vice versa). The polar substituent must be suitably placed in the molecule for the effect to be transmitted efficiently to the reaction centre through the bond framework, space or solvent.

Most experimental techniques measure differences; a change in the rate constant of a reaction brought about by a change of substituent represents the difference in the effect on reactants compared with that on the transition state. An increase in rate constant could result from the reactant becoming less stable, a more stable transition state or a combination of these effects. The observed effect is due to the difference between the effects on the reactant ([partial derivative]Gr) and transition ([partial derivative]G[double dagger]) states as illustrated in Scheme 3.

1.3 CLASSES OF FREE ENERGY RELATIONSHIP

A free energy relationship is defined by Equation (2) where the parameter a is called the similarity coefficient.

ΔG = a ΔGs + b (2)

The term ΔG is the free energy of a process such as a rate or equilibrium and ΔGs is the free energy of a standard process, often an equilibrium, which could be the process under investigation or some other standard reaction.

Equation (2) holds for many processes over relatively large changes in free energy and is usually written in terms more directly related to those which have been experimentally determined such as the logarithms (to base 10) of rate (k) or equilibrium (K) constants (Equations 3–5).

logK = a1logKs + b1 (3)

logK = a2logKs + b2 (4)

logK = a3logKs + b3 (5)

Class I free energy relationships compare a rate constant with the equilibrium constant of the same process. The Brønsted and Leffler equations (see Chapter 2) are examples of this class of free energy correlation.

The rate or equilibrium constant in Class II is related to the rate or equilibrium constant of an unconnected but (often) similar process. Class II free energy relationships are in general more common than those of Class I because equilibrium constants are more difficult to measure than rate constants (except in certain cases such as dissociation constants). The Hammett equation is the best-known Class II free energy relationship.

1.4 ORIGIN OF FREE ENERGY RELATIONSHIPS

The existence of Class I free energy relationships can be deduced from the energy profile of a reaction. Figure 1 illustrates what happens to the profile as a substituent is changed. We shall assume that the reaction (Equation 6) consists of a single bond fission (A-B [right arrow] B + A)

A-B [right arrow] A-C + B (6)

and a single bond formation (C + A [right arrow] A-C) related to inter-atom distance by Morse curves (see Appendix 1, Section A 1.1). The intersecting point (Figure 1) represents the transition structure where cross-over from one system to the other results in a smooth junction between the two processes (as shown). For small changes in overall energy the variation in ΔEa is linearly related to the change in ΔEο by Equation (7) (see Appendix 1, Section A 1.2, for derivation).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Over the small range of ΔEο the entropic component of the free energy can be assumed not to change so that [partial derivative]ΔG and [partial derivative]ΔG[double dagger] are proportional to [partial derivative]ΔE and [partial derivative]ΔE[double dagger] are respectively and Equation (8) follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Integration and rearrangement of this equation gives rise to the Class I free energy relationship (Equation 9).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

This equation fails if the slopes m1 and m2 change owing to large changes in ΔGο. The Class I relationship can alternatively be derived from the Marcus-type function (Chapter 6).

In the meantime it is necessary to define what is meant by the term reaction coordinate. The reaction coordinate for a dissociation process of a diatomic species is simply the distance between the atoms as the reaction proceeds. The majority of reactions are not of diatomic molecules and there is usually more than one bond undergoing a major bonding change, so that the reaction coordinate can no longer be regarded as a simple distance between atoms in a two-dimensional representation. For the purposes of Figure 1, the reaction coordinate has the dimension of interatomic distance for each component process, even though the identity of the major bonding change alters as the system progresses from reactants to products.

The origins of the Class II free energy relationships are not so easy to visualise as those of Class I in terms of a molecular model, because there is no direct atomic interaction between the systems giving rise to the two free energy changes (as in Figure 1). In this case it is necessary to invoke chemical intuition to indicate that substituent effects on chemically similar processes will be related to each other. No such assumption is necessary in the Class I case.

Leffler and Grunwald devised a treatment which derives both Class I and Class II free energy relationships and is more general than that using crossed energy surfaces. The equation, [partial derivative] ΔGR = a.[partial derivative]GRs relating the free energy of reaction of the reactant XR with that of the standard reaction XRs can be derived using the assumption that free energies of substances are additive functions. Let us consider a transformation (Equation 10) of a system (XR) comprising a single interaction between variable, non-reacting, substituent (X) and the reacting group (R) and product group (P).

(Continues…)Excerpted from Free Energy Relationships in Organic and Bio-organic Chemistry by Andrew Williams. Copyright © 2003 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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