Equilibrium free energy relationships

Table 3.2. Composition-Equilibrium-Free-Energy Relationships ... 

Eq. 3 is the function developed by Henry Eyring to describe equilibrium activation free energy relationships [141 ]. 

In this chapter it is clearly impossible to do more than sample the extensive literature on the carbon acidity of sulfinyl and sulfonyl compounds, as it illuminates the electronic effects of these groups, particularly in connection with linear free-energy relationships. There are three main areas to cover first, as already indicated, equilibrium acidities (pKa values) second, the kinetics of ionization, usually studied through hydrogen isotopic exchange and finally, the kinetics of other reactions proceeding via carbanionic intermediates. 

The Hammett equation is a linear free energy relationship (LFER). This can be demonstrated as follows for the case of equilibrium constants (for rate constants a similar demonstration can be made with AG instead of AG). For each reaction, where X is any group,... 

Fig. 4. Linear free-energy relationship for the reaction, Co(NH3)50H2 " +X 5 Co(NH3)5X + H2O. Log k (rate coefficient) vs. log K (equilibrium constant).

Sander and Jencks introduced a linear free energy relationship for nucleophilic addition to carbonyls. The equilibrium nucleophilicity of a species HNu is given by... 

Before terminating our discussion of the Hammett equation, we should note that the existence of linear correlations of the type indicated by equation 7.4.20 implies a linear free energy relationship. The rate or equilibrium constants can be eliminated from this equation using equation 7.4.1 that is,... 

One of the most important theoretical developments of the last decade is due to Chris Jarzynski, who established a remarkably simple relationship between the equilibrium free energy difference and an ensemble of properly constructed irreversible... 

Equation (12) is a linear free-energy relationship, since activity coefficients/can be represented as AG° values. The reason for defining the slope parameter as in equation (12) (subscript e for equilibrium) is that a little rearranging of equations (11) and (12) leads to the easy-to-use Bunnett-Olsen equation for equilibria, equation (13) 30... 

The structure-reactivity relationship is a concept familiar to every organic chemist. As commonly used it refers to a linear free energy relationship, such as the Bronsted or Hammett equations, or some more general measure of the effect of changing substituent on the rate or equilibrium of a reaction. A substituent constant is conveniently defined as the effect of the substituent on the free-energy change for a control reaction. So the so-called structure-reactivity relationship is in fact usually a reactivity-reactivity relationship. 

The pyridinium chlorochromate (PCC) oxidations of pentaamine cobalt(III)-bound and unbound mandelic and lactic acids have been studied and found to proceed at similar rates.Free-energy relationships in the oxidation of aromatic anils by PCC have been studied. Solvent effects in the oxidation of methionine by PCC and pyridinium bromochromate (PBC) have been investigated the reaction leads to the formation of the corresponding sulfoxide and mechanisms have been proposed. The major product of the acid-catalysed oxidation of a range of diols by PBC is the hydroxyaldehyde. The reaction is first order with respect to the diol and exhibits a substantial primary kinetic isotope effect. Proposed acid-dependent and acid-independent mechanisms involve the rapid formation of a chromate ester in a pre-equilibrium step, followed by rate-determining hydride ion transfer via a cyclic intermediate. PBC oxidation of thio acids has been studied. ... 

Let us call the melt phase a and the solid phase with complete immiscibility of components y. P is constant and fluids are absent. The Gibbs free energy relationships at the various T for the two phases at equilibrium are those shown in figure 7.2, with T decreasing downward from Ty to Tg. The G-X relationships observed at the various T are then translated into a T-X stability diagram in the lower part of the figure. 

A plot of the logarithm of a rate constant (or an equilibrium constant) for one series of reactions versus the logarithm of the rate constant (or the equilibrium constant) for a related series of reactions. (Recall that at constant temperature and pressure the logarithm of an equilibrium constant is proportional to AG°, and the logarithm of a rate constant is proportional to AG ). An example of a linear free energy relationship is provided by the Hammett crp-equation. With equilibrium constants, this relationship is given by the expression ... 

Equilibrium Partition Constants and Standard Free Energy of Transfer Effect of Temperature on Equilibrium Partitioning Using Linear Free Energy Relationships (LFERs) to Predict and/or to Evaluate Partition Constants and/or Partition Coefficients Box 3.2 Partition Constants, Partition Coefficients, and Distribution Ratios -A Few Comments on Nomenclature Concluding Remarks... 

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