Gibb relationship with equilibrium constant

Because it applies mostly to electrolytes, it is discussed in Chapter 15. Briefly, Helgeson models the behavior of solutes by developing equations for the standard state partial molar volume (Helgeson and Kirkham 1976) and standard state partial molar heat capacity (Helgeson et al. 1981) as a function of P and T, with adjustable constants such that they can be applied to a wide variety of solutes. If you know these quantities (V°, C°p), you can calculate the variation of the standard state Gibbs energy, and that leads through fundamental relationships to equilibrium constants, enthalpies, and entropies. 

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°) ... 

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... 

The steps for constructing and interpreting an isothermal, isobaric thermodynamic model for a natural water system are quite simple in principle. The components to be incorporated are identified, and the phases to be included are specified. The components and phases selected "model the real system and must be consistent with pertinent thermodynamic restraints—e.g., the Gibbs phase rule and identification of the maximum number of unknown activities with the number of independent relationships which describe the system (equilibrium constant for each reaction, stoichiometric conditions, electroneutrality condition in the solution phase). With the phase-composition requirements identified, and with adequate thermodynamic data (free energies, equilibrium con-... 

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 ... 

Equation (2-80) expresses the retention of an ionizable basic analyte as a function of pH and three different constants ionization constant adsorption constant of ionic form of the analyte (7 bh+) and adsorption constant of the neutral form of the analyte (T b)- These three constants describe three different equilibrium processes, and they have their own relationships with the system temperature and Gibbs free energy with respect to the particular analyte form. 

In principle all reactions are reversible just as reactants have a tendency to combine and form products, products have the tendency to recombine and form the initial reactants. At equilibrium, the forward rate is balanced by the reverse rate, all net conversion ceases, and the composition of the system becomes constant in time. Suppose we load a closed reactor with a mixture that contains arbitrary amounts of the reactant and product species, and initiate the reaction while maintaining constant temperature and pressure. If we monitor the progress to equilibrium through the extent of reaction, we will observe it to increase in the positive or negative direction, indicating that the reaction progresses in the forward or reverse direction, until equilibrium is reached. Since temperature and pressure are held constant, the equilibrium state corresponds to conditions that minimize the Gibbs free ener. This condition allows us to obtain precise mathematical relationships for the equilibrium constant of the reaction. As an example, consider the ammonia synthesis reaction. 

The equilibrium constant at 25 °C is calculated directly from tabulations of the Gibbs free energy of formation. Once this value is known, the equilibrium constant can be calculated at any other temperature. To obtain the equation that governs the variation of the equilibrium constant with temperature, the starting point is ea. 00.5). which provides the relationship between the Gibbs free energy, temperature, pressure, and composition ... 

The quantitative measure for the electron-donating ability of solvents is adequately given by reaction enthalpy e.g., for reaction (1.11.11) it is [D SbCl5]. Very diluted solutions are used in practice so that gas laws can be applied. Changes in the Gibbs energy (AG°) can be determined by spectroscopic and NMR measurements, thus enabling determination of the equilibrium constants of reactions with certain solvents. A linear relationship has... 

Keeping in mind the relationship between the standard Gibbs energy and the equilibrium constant, it is very easy to modify the Langmuir isotherm, taking into account the variation of AG g with 0. To do this, we replace AGj g in Eq. (11.4) by AG0 and combine with Eq. (11.10), to obtain... 

By starting with the relationship between standard Gibbs energy change and the equilibrium constant, the van t Hoff equation—relating the equilibrium constant and temperature—can be written (equation 13.25). With this equation, tabulated data at 25 °C can be used to determine equilibrium constants not just at 25 °C but at other temperatures as well. 

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. 

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. 

The Gibbs phase rule is the basis for organizing the models. In general, the number of independent variables (degrees of freedom) is equal to the number of variables minus the number of independent relationships. For each unique phase equilibria, we may write one independent relationship. In addition to this (with no other special stipulations), we may write one additional independent relationship to maintain electroneutrality. Table I summarizes the chemical constituents considered as variables in this study and by means of chemical reactions depicts independent relationships. (Throughout the paper, activity coefficients are calculated by the Debye-Hiickel relationship). Since there are no data available on pressure dependence, pressure is considered a constant at 1 atm. Sulfate and chloride are not considered variables because little specific data concerning their equilibria are available. Sulfate may be involved in a redox reaction with iron sulfides (e.g., hydrotroilite), and/or it may be in equilibrium with barite (BaS04) or some solid solution combinations. Chloride may reach no simple chemical equilibrium with respect to a phase. Therefore, these two ions are considered only to the... 

For a closed reaction system the determination of the compositions can be calculated with thermodynamic methods under some constraints. The constraints include mass conservation, constant temperature and constant total pressure. Under equilibrium conditions all systems obey the Gibbs phase rale, which relates the number of the species components (n) to the number of phases present (p) and the degree of freedom (/) together. The relationship is expressed by [3]... 

We study the consequences of applying the Gibbs-Duhem relation to a two-component system at constant T and P. This should make that relationship appear much less abstract it imposes important restrictions on components in equilibrium. We proceed as follows Divide the relation - - U2dfi2 = 0 by (ni + 2) to obtain x dfii + (1 x )dfi2 = 0. With dx2 = —dx we find that... 

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