Equilibrium constant Gibbs equation

Use the equilibrium constants in equations 7.3-2 and 7.3-3 to calculate the further transformed Gibbs energies of formation of the forms of the dimer of hemoglobin and of the pseudoisomer group at [02] = 5 xlO"6, 10"5, and 2 xlO"5M. Make a table with the last three columns and first four rows of Table 7.3. 

The standard Gibbs energy change, AG° (still often called standard free energy, but this term should be avoided), is related to the equilibrium constant by equation (6). 

It is this calorimetric form of this equation (and the more complex equations that can be developed and used to describe sequential, parallel and more complex reacting systems) that can be analysed to yield values for the target parameters, n, k, H, K, G, S and E. References 6 and 7 provide details on the derivation of calorimetric kinetic equations and describe how to manipulate calorimetric data to determine equilibrium constants, Gibbs functions, entropies and activation energies. The approach described shows that, under normal storage conditions ( room temperature i.e. 298 K and ambient humidity), it is possible, from only 50 h of power time (the calorimetric output is of W vs. t) data, through these techniques, to distinguish between a... 

A catalyst speeds up both the forward and the reverse reactions by the same amount. Therefore, the dynamic equilibrium is unaffected. The thermodynamic justification of this observation is based on the fact that the equilibrium constant depends only on the temperature and the value of AGr°. A standard Gibbs free energy of reaction depends only on the identities of the reactants and products and is independent of the rate of the reaction or the presence of any substances that do not appear in the overall chemical equation for the reaction. 

We saw in Section 9.3 that the standard reaction Gibbs free energy, AGr°, is related to the equilibrium constant of the reaction by AGr° = —RT In K. In this chapter, we have seen that the standard reaction Gibbs free energy is related to the standard emf of a galvanic cell by AGr° = —nFE°, with n a pure number. When we combine the two equations, we get... 

Relation between standard reaction Gibbs free energy and equilibrium constant van t Hoff equation ... 

If two reactions differ in maximum work by a certain amount 8wm (= -SAG ), it follows from the Brpnsted relation [when taking into account the Arrhenius equation and the known relation between the equilibrium constant and the Gibbs standard free energy of reaction, A m = exp(-AGm/J r)] that their activation energies will differ by a fraction of this work, with the opposite sign ... 

The Van t Hoff isotherm establishes the relationship between the standard free energy change and the equilibrium constant. It is of interest to know how the equilibrium constant of a reaction varies with temperature. The Varft Hoff isochore allows one to calculate the effect of temperature on the equilibrium constant. It can be readily obtained by combining the Gibbs-Helmholtz equation with the Varft Hoffisotherm. The relationship that is obtained is... 

As equation 2.4.8 indicates, the equilibrium constant for a reaction is determined by the temperature and the standard Gibbs free energy change (AG°) for the process. The latter quantity in turn depends on temperature, the definitions of the standard states of the various components, and the stoichiometric coefficients of these species. Consequently, in assigning a numerical value to an equilibrium constant, one must be careful to specify the three parameters mentioned above in order to give meaning to this value. Once one has thus specified the point of reference, this value may be used to calculate the equilibrium composition of the mixture in the manner described in Sections 2.6 to 2.9. 

An equilibrium constant is simply related to a standard Gibbs free energy change, as indicated by equation 2.4.7. 

The effect of pressure on chemical equilibria and rates of reactions can be described by the well-known equations resulting from the pressure dependence of the Gibbs enthalpy of reaction and activation, respectively, shown in Scheme 1. The volume of reaction (AV) corresponds to the difference between the partial molar volumes of reactants and products. Within the scope of transition state theory the volume of activation can be, accordingly, considered to be a measure of the partial molar volume of the transition state (TS) with respect to the partial molar volumes of the reactants. Volumes of reaction can be determined in three ways (a) from the pressure dependence of the equilibrium constant (from the plot of In K vs p) (b) from the measurement of partial molar volumes of all reactants and products derived from the densities, d, of the solution of each individual component measured at various concentrations, c, and extrapolation of the apparent molar volume 4>... 

Brinkley (1947) published the first algorithm to solve numerically for the equilibrium state of a multicomponent system. His method, intended for a desk calculator, was soon applied on digital computers. The method was based on evaluating equations for equilibrium constants, which, of course, are the mathematical expression of the minimum point in Gibbs free energy for a reaction. 

The equilibrium state is generated by minimizing the Gibbs free energy of the system at a given temperature and pressure. In [57], the method is described as the modified equilibrium constant approach. The reaction products are obtained from a data base that contains information on the enthalpy of formation, the heat capacity, the specific enthalpy, the specific entropy, and the specific volume of substances. The desired gaseous equation of state can be chosen. The conditions of the decomposition reaction are chosen by defining the value of a pair of variables (e.g., p and T, V and T). The requirements for input are ... 

The following equation relates the Gibbs free energy to an equilibrium constant ... 

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). 

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... 

Note that Q, when at equilibrium, becomes K. This equation gives us a way to calculate the equilibrium constant, K, from a knowledge of the standard Gibbs free energy of the reaction and the temperature. 

Perhaps the two most important outcomes of the first and the second laws of thermodynamics for chemistry are representedby equation 2.54, which relates the standard Gibbs energy (ArG°) with the equilibrium constant (K) of a chemical reaction at a given temperature, and by equation 2.55, which relates ArG° with the standard reaction enthalpy (A rH°) and the standard reaction entropy (ArA°). 

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... 

AG° is the standard Gibbs free energy for the reaction at 298°K R is the ideal gas constant and T is 298°K. Since the actual temperature of most slurries or solutions in flue gas scrubbing applications usually does not exceed 50°C, the value of the equilibrium constant can be determined at some temperature other than 298°K by using the van t Hoff equation... 

As stated, the most commonly used procedure for temperature and composition calculations is the versatile computer program of Gordon and McBride [4], who use the minimization of the Gibbs free energy technique and a descent Newton-Raphson method to solve the equations iteratively. A similar method for solving the equations when equilibrium constants are used is shown in Ref. [7],... 

Now that we have considered the calculation of entropy from thermal data, we can obtain values of the change in the Gibbs function for chemical reactions from thermal data alone as well as from equilibrium data. From this function, we can calculate equilibrium constants, as in Equations (10.22) and (10.90.). We shall also consider the results of statistical thermodynamic calculations, although the theory is beyond the scope of this work. We restrict our discussion to the Gibbs function since most chemical reactions are carried out at constant temperature and pressure. 

The condition of Equation (13.7) can be met only if p,j = p,n, which is the condition of transfer equilibrium between phases. Or, to put the argument differently, if the chemical potentials (escaping tendencies) of a substance in two phases differ, spontaneous transfer will occur from the phase of higher chemical potential to the phase of lower chemical potential, with a decrease in the Gibbs function of the system, until the chemical potentials are equal (see Section 10.5). For each component present in aU p phases, (p 1) equations of the form of Equation (13.7) provide constraints at transfer equilibrium. Furthermore, an equation of the form of Equation (13.7) can be written for each one of the C components in the system in transfer equUibrium between any two phases. Thus, C(p — 1) independent relationships among the chemical potentials can be written. As chemical potentials are functions of the mole fractions at constant temperamre and pressure, C(p — 1) relationships exist among the mole fractions. If we sum the independent relationships for temperature. 

It is evident from equation 5.204 that the intrinsic significance of equation 5.206 is closely connected with the choice of standard state of reference. If the adopted standard state is that of the pure component at the P and T of interest, then AG%i is the Gibbs free energy of reaction between pure components at the P and T of interest. Deriving in P the equilibrium constant, we obtain... 

While the examples chosen here concern only dissociation of protons, the Hammett equation has a much broader application. Equilibria for other types of reactions can be treated. Furthermore, since rates of reactions are related to Gibbs energies of activation, many rate constants can be correlated. For these purposes the Hammett equation can be written in the more general form in which k may be either an equilibrium constant or a rate constant.13 The subscript / denotes the reaction under consideration and i the substituent influencing the reaction. 

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