Determination of thermodynamic equilibrium constants

Kt may therefore be determined from measured values of over a range of ionic strength values and extrapolating the K versus y/fi plot to y/n = 0. This is a general technique for the determination of all types of thermodynamic equilibrium constants, e.g., solubility, stability and acid dissociation constants. 

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R. P. Sperline, and H. Freiser, Spectrophotometric Determination of pH and Its Applications to Determination of Thermodynamic Equilibrium Constants, Anal. Chem. 1992,64, 2720. 

Different di- and trihydroxy bile salts and drugs that differ in lipophilicity, basicity, and structure have been compared in order to examine solvatochromic equilibria. The principle exploited in the determination of thermodynamic equilibrium constants is the indirect measurement of the capacity factor affected by the tenside concentration (in this case, the bile acid concentration). A pronounced shift in the migration times and thus effective ionic mobility... 

Zils R (2000) Direct determination of the equilibrium constant and thermodynamic parameters in the reaction of pentadienyl radicals with O2. CR Acad Sci Ser lie Chim 3 667-674 Zils R, Inomata S, Imamura A, Washida N (2001) Determination of the equilibrium constant and thermodynamic parameters for the reaction of pentadienyl radicals with 02.J Phys Chem 105 1277-1282... 

When the pH is specified, each biochemical half reaction makes an independent contribution to the apparent equilibrium constant K for the reaction written in terms of reactants rather than species. The studies of electochemical cells have played an important role in the development of biochemical thermodynamics, as indicated by the outstanding studies by W. Mansfield Clarke (1). The main source of tables of ° values for biochemical half reactions has been those of Segel (2). Although standard apparent reduction potentials ° can be measured for some half reactions of biochemical interest, their direct determination is usually not feasible because of the lack of reversibility of the electrode reactions. However, standard apparent reduction potentials can be calculated from for oxidoreductase reactions. Goldberg and coworkers (3) have compiled and evaluated the experimental determinations of apparent equilibrium constants and standard transformed enthalpies of oxidoreductase reactions, and their tables have made it possible to calculate ° values for about 60 half reactions as functions of pH and ionic strength at 298.15 K (4-8). 

Evans and Fitch102 developed an electrochemical method to determine the thermodynamic equilibrium constant K. The value obtained by this technique (1.9 xlO-3 at 298 K) for bianthrone is in excellent agreement with spectro-photometric measurements.97 This method allows the estimation of the equilibrium constant of 1,1 "-disubstituted bianthrones, even if the substitution prevents any observable thermochromic behavior. 

Values of thermodynamic equilibrium constant for individual reactions under standard conditions may be found either in reference literature or may be determined by calculations. In reference literature they can be either thermodynamic or concentration. They are often expressed as pK = -log K. 

At constant temperature all terms to the right of the equal mark are constant and equal to the logarithm of thermodynamical equilibrium constants. If we determine the chemical potential of a nonpolar component in different media relative one and the same its standard state, i.e., if then these equilibrium constants are equal to 1. 

The amplitudes of chemical relaxation processes are determined by the equilibrium concentrations (and strictly speaking, associated activity coefficients) and by thermodynamic variables appropriate for the particular perturbation method used. Thus, for example, an analysis of the amplitudes of relaxation processes associated with temperature jump measurements can lead to determination of the equilibrium constants and enthalpies associated with the mechanism under study. As might be anticipated from our previous discussion, the relaxation amplitudes are determined by normal mode thermodynamic variables which are linear combinations of the thermodynamic variables associated with the individual steps in the mechanism. The formal analysis of relaxation amplitudes has been developed in considerable detail [2, 5,7],... 

A special problem arises if the molecular property measured in the course of the determination of the equilibrium constant is also dependent on the temperature. For example, the temperature dependence of the absorbance may render impossible the use of spectrophotometry in the determination of thermodynamic data from the equilibrium constants relating to various temperatures. In such cases AH may be obtained by means of a separate calorimetric measurement, and the entropy term is found in the usual manner from this AH value and from AG calculated from the equilibrium constants, relating to a single temperature ... 

Determination of the equilibrium constants on the basis of spectral data and statistical thermodynamics... 

The fourth and final chapter in this voliune is given over to the determination, both experimentally and by computation, of the values of the parameters associated with chemical reactions. Thermochemistry for enthalpy, the determination of the entropies, specific heat capacities and Gibbs energy values ultimately lead to the determination of the equilibrium constants. Analysis of the different thermodynamic tables and methods for estimating unknown values enable us to proceed to the practical application and finally computation of the equilibria by the equilibrium constant method and minimization of Gibbs energy. 

For the thermodynamic characterization of enzyme stability, the most critical step is the determination of the equilibrium constant of denat-uration. The equilibrium constant can be calculated from knowledge of the relative proportions of native and denatured enzymes at a particular temperature. The equilibrium constant can thus be calculated as... 

Several features of equation 6.50 deserve mention. First, as the ionic strength approaches zero, the activity coefficient approaches a value of one. Thus, in a solution where the ionic strength is zero, an ion s activity and concentration are identical. We can take advantage of this fact to determine a reaction s thermodynamic equilibrium constant. The equilibrium constant based on concentrations is measured for several increasingly smaller ionic strengths and the results extrapolated... 

The mean chemical shifts of A- unsubstituted pyrazoles have been used to determine the tautomeric equilibrium constant, but the method often leads to erroneous conclusions (76AHC(Sl)l) unless the equilibrium has been slowed down sufficiently to observe the signals of individual tautomers (Section 4.04.1.5.1). When acetone is used as solvent it is necessary to bear in mind the possibility (depending on the acidity of the pyrazole and the temperature) of observing the signals of the 1 1 adduct (55) whose formation is thermodynamically favoured by lowering the solution temperature (79MI40407). A similar phenomenon is observed when SO2 is used as solvent. 

Besides electronic effects, structure sensitivity phenomena can be understood on the basis of geometric effects. The shape of (metal) nanoparticles is determined by the minimization of the particles free surface energy. According to Wulffs law, this requirement is met if (on condition of thermodynamic equilibrium) for all surfaces that delimit the (crystalline) particle, the ratio between their corresponding energies cr, and their distance to the particle center hi is constant [153]. In (non-model) catalysts, the particles real structure however is furthermore determined by the interaction with the support [154] and by the formation of defects for which Figure 14 shows an example. 

For four cations (349b-e), the thermodynamic parameters of the equilibrium (Table 3.21) and the exponents n in the equation of the equilibrium constant were determined. 

This permits provisional calculation of the compositional dependence of the equilibrium constant and determination of provisional values of the solid phase activity coefficients (discussed below). The equilibrium constant and activity coefficients are termed provisional because it is not possible to determine if stoichiometric saturation has been established without independent knowledge of the compositional dependence of the equilibrium constant, such as would be provided from independent thermodynamic measurements. Using the provisional activity coefficient data we may compare the observed solid solution-aqueous solution compositions with those calculated at equilibrium. Agreement between the calculated and observed values confirms, within the experimental data uncertainties, the establishment of equilibrium. The true solid solution thermodynamic properties are then defined to be equal to the provisional values. 

By examining the compositional dependence of the equilibrium constant, the provisional thermodynamic properties of the solid solutions can be determined. Activity coefficients for solid phase components may be derived from an application of the Gibbs-Duhem equation to the measured compositional dependence of the equilibrium constant in binary solid solutions (10). 

By examining the compositional dependence of the equilibrium constant, the thermodynamic properties of the solid solution can be determined if the final solution is either at equilibrium or stoichiometric saturation. That is, the provisional activities and activity coefficients will be valid if either equilibrium or stoichiometric saturation is attained in the solubility data. 

Most thermodynamic data for solid solutions derived from relatively low-temperature solubility (equilibration) studies have depended on the assumption that equilibrium was experimentally established. Thorstenson and Plummer (10) pointed out that if the experimental data are at equilibrium they are also at stoichiometric saturation. Therefore, through an application of the Gibbs-Duhem equation to the compositional dependence of the equilibrium constant, it is possible to determine independently if equilibrium has been established. No other compositional property of experimental solid solution-aqueous solution equilibria provides an independent test for equilibrium. If equilibrium is demonstrated, the thermodynamic properties of the solid solution are also... 

Values of the equilibrium constants at 298°K can also be calculated from tabulated thermodynamic properties. The standard Gibbs free energy of the reaction at 298°K is first calculated, and the equilibrium constant at 298°K is then determined from the equation... 

Consequently, it is seen, from the measurement of the overall reaction rate and the steady-state approximation, that values of the rate constants of the intermediate radical reactions can be determined without any measurement of radical concentrations. Values k exp and xp evolve from the experimental measurements and the form of Eq. (2.31). Since (ki/k5) is the inverse of the equilibrium constant for Br2 dissociation and this value is known from thermodynamics, k2 can be found from xp. The value of k4 is found from k2 and the equilibrium constant that represents reactions (2.2) and (2.4), as written in the H2 Br2 reaction scheme. From the experimental value of k CX(l and the calculated value of k4, the value k3 can be determined. 

AH values for various monomers. The AS values fall in a narrower range of values. The methods of evaluating AH and AS have been reviewed [Dainton and Ivin, 1950, 1958], These include direct calorimetric measurements of AH for the polymerization, determination by the difference between the heats of combustion of monomer and polymer, and measurements of the equilibrium constant for the polymerization. The overall thermodynamics of the polymerization of alkenes is quite favorable. The value of AG given by... 

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