Free energy linear changes

Following the method established originally by Rehbinder, let us relate the surface energy to the internal pressure. The latter is the other quantity used to characterize the intermolecular interactions. To make things simpler, let us assume that the liquid phase is non-volatile (/ " / ), and that the free energy density,/ changes linearly from the bulk value f to some value/, within the entire discontinuity surface of thickness 8 = 8 (see Fig. 1-7). Let us also treat the surface tension a as the work, required to bring molecules contained in the volume of 1 m2 x 8 m = 8 m3 (i.e. 8 n molecules, provided that n is a number of molecules per 1 m3) from the bulk to the surface. Such treatment allows us to write... 

Table 3. Conformational free energy simulation of linear DPDPE. Changes in free energy and its components. Units kcal/mol...

The Cyc conformer represents the structure adopted by the linear peptide prior to disulfide bond formation, while the two /3-turns are representative stable structures of linear DPDPE. The free energy differences of 4.0 kcal/mol between pc and Cyc, and 6.3 kcal/mol between pE and Cyc, reflect the cost of pre-organizing the linear peptide into a conformation conducive for disulfide bond formation. Such a conformational change is a pre-requisite for the chemical reaction of S-S bond formation to proceed. 

The temperature is expressed ia degrees Celsius. The empirical equation for the Gibbs free energy change was found to be linear with temperature for AG° ia kJ/mol, Tia Kelvin. 

Neither the principles of thermodynamics nor theories of reaction rates require that there should be such linear relationships. There are, in fact, numerous reaction series that fail to show such correlations. Some insight into the origin of die correlation can be gained by considering the relationship between the correlation equation and the free-energy changes involved in the two processes. The line in Fig 4.2 defines an equation in which m is the slope of the line ... 

Since AG and AG are combinations of enthalpy and entropy terms, a linear free-energy relationship between two reaction series can result from one of three circumstances (1) AH is constant and the AS terms are proportional for the series, (2) AS is constant and the AH terms are proportional, or (3) AH and AS are linearly related. Dissection of the free-energy changes into enthalpy and entropy components has often shown the third case to be true. °... 

We consider the effect of the structural change on the free energy of the transition state. Leffler - postulated that changes in G can be represented as a linear combination of changes in the free energies of reactants and products,... 

As seen earlier, these logarithmic terms are linearly related to free energy changes ... 

Not all free energy relationships (Hammett plots) are linear. Aside from occasional scatter or unexplained deviations from an otherwise linear plot, it sometimes happens that the plot is smooth but nonlinear. Two causes have been identified, and many examples are known. One cause is a change in a rate-determining... 

Finally, two sets of physical properties have been correlated by the Hammett equation. Sharpe and Walker have shown that changes in dipole moment are approximately linearly correlated with ct-values, and Snyder has recently correlated the free energies of adsorption of a series of substituted pyridines with u-values. All the reaction constants for the series discussed are summarized in Table V. 

A treatment partially based on the Bunnett-Olsen one is that of Bagno, Scorrano, and More OTerrall, which formulates medium effects (changes in acidity of solvent) on acid-base equilibria. An appropriate equilibrium is chosen as reference, and the acidity dependence of other reaetions compared with it, by use of the linear free energy equation... 

Another simple approach assumes temperature-dependent AH and AS and a nonlinear dependence of log k on T (123, 124, 130). When this dependence is assumed in a particular form, a linear relation between AH and AS can arise for a given temperature interval. This condition is met, for example, when ACp = aT" (124, 213). Further theoretical derivatives of general validity have also been attempted besides the early work (20, 29-32), particularly the treatment of Riietschi (96) in the framework of statistical mechanics and of Thorn (125) in thermodynamics are to be mentioned. All of the too general derivations in their utmost consequences predict isokinetic behavior for any reaction series, and this prediction is clearly at variance with the facts. Only Riietschi s theory makes allowance for nonisokinetic behavior (96), and Thorn first attempted to define the reaction series in terms of monotonicity of AS and AH (125, 209). It follows further from pure thermodynamics that a qualitative compensation effect (not exactly a linear dependence) is to be expected either for constant volume or for constant pressure parameters in all cases, when the free energy changes only slightly (214). The reaction series would thus be defined by small differences in reactivity. However, any more definite prediction, whether the isokinetic relationship will hold or not, seems not to be feasible at present. 

The necessity of the statistical approach has to be stressed once more. Any statement in this topic has a definitely statistical character and is valid only with a certain probability and in certain range of validity, limited as to the structural conditions and as to the temperature region. In fact, all chemical conceptions can break dovra when the temperature is changed too much. The isokinetic relationship, when significantly proved, can help in defining the term reaction series it can be considered a necessary but not sufficient condition of a common reaction mechanism and in any case is a necessary presumption for any linear free energy relationship. Hence, it does not at all detract from kinetic measurements at different temperatures on the contrary, it gives them still more importance. 

---