Free energy relationships, classes equations

In a broad sense, one may include the Free-Wilson equation within the class of linear free energy relationships (LFER). It is also subjected to the assumption of additivity of the contributions to the biological activity by substituent groups at different substitution sites. The assumption requires, for example, that there is no hydrogen bonding interaction between the various substitution groups. 

Class I free energy relationships compare a rate constant with the equilibrium constant of the same process. The Bronsted 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. 

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 -> B -t- A)... 

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

Taft s polar substituent constant, a, applicable to aliphatic systems is based on the notion that free energy relationships can exist between rate processes as well as between rate and equilibrium processes (Equations 3-5, Chapter 1). A rate process can thus be employed as a reference reaction. A simple example of such a Class II rate-rate free energy relationship is shown in Figure 3 where the reaction of phenoxide ion with 4-nitrophenyl-substituted benzoate esters is compared with the corresponding reaction of hydroxide ion, which therefore becomes the reference free energy change. 

The Bronsted equation is a Class I free energy relationship and this may be shown by considering as an example the acid-catalysed dehydration of acetaldehyde hydrate (Equation 30). This reaction also provides a good example of an acid-catalysed reaction following a Bronsted equation (Figure 7). 

These arguments do not invalidate the immense array of linear free energy relationships gathered over the past 70 years or so. Class II systems such as Hammett or Taft relationships can be excluded from these considerations because the two variables are independent. The following arguments can be employed to demonstrate experimentally that a Class I correlation such as a Bronsted or a Leffler Equation does not arise from a statistical artifact in a system under investigation. Rearranging the Leffler Equation (Equation 27) yields Equation (29). 

The similarity coefficient 6z in the free energy equation (Equation 2) compares the change in the process under investigation (the unknown system) with that for a standard or reference (the known) system. The coefficient measures the similarity between the two systems. In the case of Class I relationships the similarity coefficient measures the extent to which the transition state resembles the products compared with the reactants. The similarity coefficient for Class II systems measures the extent to which conversion to the transition state resembles the conversion of reactants into products in the standard or reference system. 

Provided there is no resonance transmission in the effect of substituents on the dissociation of acetic acids (as would be expected) then the linear equation indicates that the transmission of the effect of the meta substituent does not involve resonance either. The correlation shows that provides a useful secondary definition of Oi. This relationship is a Class II free energy correlation between the dissociation of substituted acetic acids and weZa-substituted benzoic acids. 

Here is a useful leisure time exercise for a very attentive reader. The purpose is to understand the connection between virial expansion (8.7) and the well known van der Waals equation of state (i.e., the relationship between volume, pressure, and temperature) for an ordinary imperfect gas. You may have studied van der Waals equation in general physics and/or general chemistry class, it reads p- -a/V )(V — b) = NksT. Say, the volume is V, and the number of molecules in the gas is N. Then n = N/V. You can work out the pressure by differentiation p = — (9F/9V), where free energy F is defined by formula (7.19), F = U — TS = t/ + / , the internal energy U is given by (8.7), and... 

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