Free energy relationship with equilibrium constant

Keeping in mind the relationship between standard free energy and the equilibrium constant, it is very easy to modify the Langmuir isotherm to take into account the variation of AG with 0. To do this,... 

Solvent effects are essentially free-energy correlations [57] and we shall omit in this chapter those of strongly acidic media as these are clearly not pertinent to biochemistry. The effect of variation of solvent on a rate or equilibrium constant may be treated in the same way as a substituent effect. We may apply the same criteria to elucidate mechanistic or rate-determining step changes to free-energy relationships with solvent effects. Let us consider the hydrolysis of 5-phenyl-oxazoline-2-ones... 

The relationship (Equation 4.3) that correlates energy change with equilibrium constant involves standard enthalpy (AH°), temperature (K), and standard entropy (AS°) and is called the Gibbs standard free energy (AG°) ... 

Values of A5, AfT and A5° are given and a value of less than unity for the gradient of the free energy relationship [log(rate constant) as ordinate versus log(equilibrium constant)] implies an mechanism. The planar complexes [NiX2L2] (X = Cl, Br, I, or NCS L = pyridine derivative with methyl groups in the 2-and 6-position) are uniquely inert to substitution and are not even attacked by strong mineral acids.This lack of reactivity is attributed to steric hindrance. 

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,... 

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 ... 

The surface complexation models quantify adsorption with experimentally determined equilibrium constants. Another, less widely used approach considers the relationship between the equilibrium constant for the adsorption reaction and the associated free energy change (James and Healy, 1972). Attempts have been made to determine the chemical contribution to the overall adsorption free energy by fitting adsorption isotherms to the experimental data values of -50, -33 and —45 kj mol were found for the change in chemical free energy associated with adsorption of Cr, Ni and Zn, respectively, on ferrihydrite (Crawford et al., 1993). Values ranging from -21 to 241 kJ mol were found for Ni on hematite the actual value depended upon the hydrolysis species that were assumed to exist (Fuerstenau and Osseo-Assare, 1987). 

Organic chemists have studied the influence of substituents on various reactions for the better part of a century. Linear free energy relationships have played an important role in this pursuit by correlating equilibrium and rate processes. One of the earliest examples is now known as the Hammett equation. It emerged from the observation that the acidities of benzoic acids correlated with the rates at which ethyl esters of benzoic acids hydrolyzed. The relationship was expressed as follows in which K represents an equilibrium constant and k is a rate constant. The proportionality constant, m, is the slope of the log-log data plot for the two processes. 

Because of the bulk of comparable material available, it has been possible to use half-wave potentials for some types of linear free energy relationships that have not been used in connection with rate and equilibrium constants. For example, it has been shown (7, 777) that the effects of substituents on quinone rings on their reactivity towards oxidation-reduction reactions, can be approximately expressed by Hammett substituent constants a. The susceptibility of the reactivity of a cyclic system to substitution in various positions can be expressed quantitatively (7). The numbers on formulae XIII—XV give the reaction constants Qn, r for the given position (values in brackets only very approximate) ... 

The quality of this linear free energy relationship allowed us to predict with confidence the equilibrium constant of a subcomponent substitution reaction... 

Linear free energy relationship (LFER) — For various series of similar chemical reactions it has been empirically found that linear relationships hold between the series of free energies (-> Gibbs energy) of activation AG and the series of the standard free energies of reactions AGf, i.e., between the series of log fc (k -rate constants) and log K (Kt - equilibrium constants) (z labels the compounds of a series). Such relations correlate the - kinetics and -> thermodynamics of these reactions, and thus they are of fundamental importance. The LFER s can be formulated with the so-called Leffler-Grunwald operator dR ... 

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