Chemically equilibrium potentials

Section 8 now combines all the material on electrolytes, electromotive force, and chemical equilibrium, some of which had formerly been included in the old Analytical Chemistry section of earlier editions. Material on the half-wave potentials of inorganic and organic materials has been thoroughly revised. The tabulation of the potentials of the elements and their compounds reflects recent lUPAC (1985) recommendations. 

Table 1.7 shows typical half reactions for the oxidation of a metal M in aqueous solutions with the formation of aquo cations, solid hydroxides or aquo anions. The equilibrium potential for each half reaction can be evaluated from the chemical potentials of the species involved see Appendix 20.2) and it should be noted that there is no difference thermodynamically between equations 2(a) and 2(b) nor between 3(a) and 3(b) when account is taken of the chemical potentials of the different species involved. 

In the section on chemical equilibrium in gases we introduced a magnitude called the molecular chemical potential of a component ... 

This does not imply that all components will have the same chemical potential at equilibrium, only that the chemical potential for each component is the same in all phases. xThe independent components are those that are not related through chemical equilibrium reactions. See Section 1.2c for a discussion of components in a system. 

For t vo systems in chemical equilibrium we can calculate the equilibrium constant from the ratio of partition functions by requiring the chemical potentials of the t vo systems to be equal. 

Electrochemical reactions differ fundamentally from chemical reactions in that the kinetic parameters are not constant (i.e., they are not rate constants ) but depend on the electrode potential. In the typical case this dependence is described by Eq. (6.33). This dependence has an important consequence At given arbitrary values of the concentrations d c, an equilibrium potential Eq exists in the case of electrochemical reactions which is the potential at which substances A and D are in equilibrium with each other. At this point (Eq) the intermediate B is in common equilibrium with substances A and D. For this equilibrium concentration we obtain from Eqs. (13.9) and (13.11),... 

This last equation contains the two essential activation terms met in electrocatalysis an exponential function of the electrode potential E and an exponential function of the chemical activation energy AGj (defined as the activation energy at the standard equilibrium potential). By modifying the nature and structure of the electrode material (the catalyst), one may decrease AGq, thus increasing jo, as a result of the catalytic properties of the electrode. This leads to an increase in the reaction rate j. 

In the two bulk phases the potential of mean force is constant, but it may vary near the interface. The difference in the bulk values of the chemical part is the free energy of transfer of the ion, which in our model is —2mu (we assume u < 0). Let us consider the situation in which the ion-transfer reaction is in equilibrium, and the concentration of the transferring ion is the same in both phases the system is then at the standard equilibrium potential 0oo- In Ihis case the potential of mean force is the same in the bulk of both phases the chemical and the electrostatic parts must balance ... 

A third possibility is that mineral ions leak out of tissue in the presence of phenolic acids, not because membrane permeability is altered, but rather because the driving force that maintains high ion concentrations in cells (i.e. PD) is dissipated by the chemicals. Without an electrical potential, ions would distribute solely according to their chemical concentrations. Thus, most ions would leak out of cells to reach chemical equilibrium with the external environment. 

In order to obtain a definite breakthrough of current across an electrode, a potential in excess of its equilibrium potential must be applied any such excess potential is called an overpotential. If it concerns an ideal polarizable electrode, i.e., an electrode whose surface acts as an ideal catalyst in the electrolytic process, then the overpotential can be considered merely as a diffusion overpotential (nD) and yields (cf., Section 3.1) a real diffusion current. Often, however, the electrode surface is not ideal, which means that the purely chemical reaction concerned has a free enthalpy barrier especially at low current density, where the ion diffusion control of the electrolytic conversion becomes less pronounced, the thermal activation energy (AG°) plays an appreciable role, so that, once the activated complex is reached at the maximum of the enthalpy barrier, only a fraction a (the transfer coefficient) of the electrical energy difference nF(E ml - E ) = nFtjt is used for conversion. 

However, the value of the equilibrium electrode potential is often not well defined (e.g. when the electrode reaction produces an intermediate that undergoes a subsequent chemical reaction yielding one or more final products). Often, an equilibrium potential is not established at all, so that the calculated equilibrium values must often be used. 

Recently, Muha (83) has found that the concentration of cation radicals is a rather complex function of the half-wave potential the concentration goes through a maximum at a half-wave potential of about 0.7 V. The results were obtained for an amorphous silica-alumina catalyst where the steric problem would not be significant. To explain the observed dependence, the presence of dipositive ions and carbonium ions along with a distribution in the oxidizing strengths of the surface electrophilic sites must be taken into account. The interaction between the different species present is explained by assuming that a chemical equilibrium exists on the surface. 

Some injected wastes are persistent health hazards that need to be isolated from the biosphere indefinitely. For this reason, and because of the environmental and operational problems posed by loss of permeability or formation caving, well operators seek to avoid deterioration of the formation accepting the wastes and its confining layers. When wastes are injected, they are commonly far from chemical equilibrium with the minerals in the formation and, therefore, can be expected to react extensively with them (Boulding, 1990). The potential for subsurface damage by chemical reaction, nonetheless, has seldom been considered in the design of injection wells. 

In general, the first derivative of the Gibbs energy is sufficient to determine the conditions of equilibrium. To examine the stability of a chemical equilibrium, such as the one described above, higher order derivatives of G are needed. We will see in the following that the Gibbs energy versus the potential variable must be upwards convex for a stable equilibrium. Unstable equilibria, on the other hand, are... 

Note that the chemical potential of a given component at the interface is equal to that in the two adjacent phases. This is important since this implies that adsorption can be treated as a chemical equilibrium, as we will discuss in Section 6.3. 

The exp-6 potential has also proved successful in modeling chemical equilibrium at the high pressures and temperatures characteristic of detonation. However, to calibrate the parameters for such models, it is necessary to have experimental data for product molecules and mixtures of molecular species at high temperature and pressure. Static compression and sound-speed measurements provide important data for these models. 

Exp-6 potential models can be validated through several independent means. Fried and Howard33 have considered the shock Hugoniots of liquids and solids in the decomposition regime where thermochemical equilibrium is established. As an example of a typical thermochemical implementation, consider the Cheetah thermochemical code.32 Cheetah is used to predict detonation performance for solid and liquid explosives. Cheetah solves thermodynamic equations between product species to find chemical equilibrium for a given pressure and temperature. From these properties and elementary detonation theory, the detonation velocity and other performance indicators are computed. 

However, a more realistic model for the phase transition between baryonic and quark phase inside the star is the Glendenning construction [16], which determines the range of baryon density where both phases coexist. The essential point of this procedure is that both the hadron and the quark phase are allowed to be separately charged, still preserving the total charge neutrality. This implies that neutron star matter can be treated as a two-component system, and therefore can be parametrized by two chemical potentials like electron and baryon chemical potentials [if. and iin. The pressure is the same in the two phases to ensure mechanical stability, while the chemical potentials of the different species are related to each other satisfying chemical and beta stability. The Gibbs condition for mechanical and chemical equilibrium at zero temperature between both phases reads... 

Let us start by giving a brief introduction into the general method of constructing mixed phases by imposing the Gibbs conditions of equilibrium [23, 18]. From the physical point of view, the Gibbs conditions enforce the mechanical as well as chemical equilibrium between different components of a mixed phase. This is achieved by requiring that the pressure of different components inside the mixed phase are equal, and that the chemical potentials (p and ne) are the same across the whole mixed phase. For example, in relation... 

We consider stellar matter in the quark core of compact stars consisting of electrons in chemical equilibrium with u and d quarks. Hence Pd = pu + Pe, where pe is the electron chemical potential. The thermodynamic potential of such matter is... 

The second integral in Equation 17 is the chemical contribution due to chemical reactions of the potential determining ions with the surface groups. This term may be recast as follows. Chemical equilibrium between potential determining ions bound on the surface and those in the solution adjacent to the surface during the changing process means that the chemical part of the chemical potentials are equal, i.e. 

The movement of solute across a semipermeable membrane depends upon the chemical concentration gradient and the electrical gradient. Movement occurs down the concentration gradient until a significant opposing electrical potential has developed. This prevents further movement of ions and the Gibbs-Donnan equilibrium is reached. This is electrochemical equilibrium and the potential difference across the cell is the equilibrium potential. It can be calculated using the Nemst equation. 

The chapter starts with a brief review of thermodynamic principles as they apply to the concept of the chemical equilibrium. That section is followed by a short review of the use of statistical thermodynamics for the numerical calculation of thermodynamic equilibrium constants in terms of the chemical potential (often designated as (i). Lastly, this statistical mechanical development is applied to the calculation of isotope effects on equilibrium constants, and then extended to treat kinetic isotope effects using the transition state model. These applications will concentrate on equilibrium constants in the ideal gas phase with the molecules considered in the rigid rotor, harmonic oscillator approximation. 

The two partial derivatives in Equation 4.34, which are equal to each other, are very important in the thermodynamics of chemical equilibrium, and are referred to as the chemical potential, p, . 

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