Conditions of equilibrium for heterogenous systems

It is a general observation that any system that is not in equilibrium will approach equilibrium when left to itself. Such changes of state that take place in an isolated system do so by irreversible processes. However, when a change of state occurs in an isolated system by an irreversible process, the entropy change is always positive (i.e., the entropy increases). Consequently, as the system approaches equilibrium, the entropy increases and will continue to do so until it obtains the largest value consistent with the energy of the system. Thus, if the system is already at equilibrium, the entropy of the system can only decrease or remain unchanged for any possible variation as discussed in Section 5.1. 

We may obtain further insight into the conditions by considering the equation 

If the system is at equilibrium, it is incapable of doing work. If there is a possible variation of the system that would decrease the value of the energy at constant entropy, then the system would be capable of doing work. (The numerical work of dW t would be negative.) However, this is not possible and therefore the energy can only increase or remain constant. 

Equations (5.1) and (5.2) give the basic criteria of equilibrium, but it is advantageous to obtain more-useful conditions from them. We first consider an isolated heterogenous system composed of any number of phases and containing any number of components at equilibrium. These phases are in thermal contact with each other, and there are no walls separating them. 

Each phase may be considered as an open, nonisolated system. The variation of the energy of each phase is given by 

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The mathematical basis of classic thermodynamics was developed by J. Willard Gibbs in his essay [1], On the Equilibrium of Heterogeneous Substances, which builds on the earlier work of Kelvin, Clausius, and Helmholtz, among others. In particular, he derived the phase mle, which describes the conditions of equilibrium for multiphase, multicomponent systems, which are so important to the geologist and to the materials scientist. In this chapter, we will present a derivation of the phase rule and apply the result to several examples. 

Equation (6-36) is a consequence of the conditions of equilibrium for a heterogeneous system and will be discussed further in Chap. 9. 

The meaning and also the limitation of the term possible variations must be considered. For the purposes of discussion, we center our attention on Equation (5.2) and consider a heterogenous, multicomponent system. The independent variables that are used to define the state are the entropy, volume, and mole numbers (i.e., amount of substance or number of moles) of the components. The statements of the condition of equilibrium require these to be constant because of the isolation of the system. Possible variations are then the change of the entropy of two or more of the phases subject to the condition that the entropy of the whole system remains constant, the change of the volume of two or more phases subject to the condition that the volume of the whole system remains constant, or the transfer of matter from one phase to another subject to the condition that the mass of the whole system remains constant. Such variations are virtual or hypothetical,... 

Equation (12.2) is applicable to every phase in a heterogenous system. Because of the identity of this equation with Equation (4.12) with the exclusion of other work terms, the conditions of equilibrium must be the same as those developed in Chapter 5 in a heterogenous system without restrictions, the temperature of every phase must be the same, the pressure of every phase must be the same, and the chemical potential of a species must be the same in every phase in which the species exists. For phase equilibrium, then,... 

For heterogeneous systems, the set of reactions includes adsorption, dissociation, surface diffusion, desorption, and other processes. In this case, the rates of processes can differ by many orders of magnitude. In accordance with Eq. (30), the time step is determined by the fastest process in the system. This condition strongly restricts the maximum real time in the simulation and prevents modeling of rare processes. One of the ways of overcoming this problem can be to exclude all fast processes from the table of reactions and use equilibrium distributions for these processes. For example,... 

A great many electrolytes have only limited solubility, which can be very low. If a solid electrolyte is added to a pure solvent in an amount greater than corresponds to its solubility, a heterogeneous system is formed in which equilibrium is established between the electrolyte ions in solution and in the solid phase. At constant temperature, this equilibrium can be described by the thermodynamic condition for equality of the chemical potentials of ions in the liquid and solid phases (under these conditions, cations and anions enter and leave the solid phase simultaneously, fulfilling the electroneutrality condition). In the liquid phase, the chemical potential of the ion is a function of its activity, while it is constant in the solid phase. If the formula unit of the electrolyte considered consists of v+ cations and v anions, then... 

The equilibrium interfaces of fluid systems possess one variant chemical potential less than isolated bulk phases with the same number of components. This is due to the additional condition of heterogeneous equilibrium and follows from Gibbs phase rule. As a result, the equilibrium interface of a binary system is invariant at any given P and T, whereas the interface between the phases a and /3 of a ternary system is (mono-) variant. However, we will see later that for multiphase crystals with coherent boundaries, the situation is more complicated. 

There has recently been much activity in developing molecular spectroscopic probes of electrochemical interfaces, as for other types of heterogeneous systems. The ultimate objectives of these efforts include not only the characterization of adsorbate molecular structure interactions under equilibrium conditions, but also the extraction of mechanistic and kinetic information from spectral detection of reactive adsorbates. 

However, in heterogeneous catalysis, metals are usually deposited on nonconducting supports such as alumina or silica. For such conditions electrochemical techniques cannot be used and the potential of the metallic particles is controlled by means of a supplementary redox system [8, 33]. Each particle behaves like a microelectrode and assumes the reversible equilibrium potential of the supplementary redox system in use. For example, with a platinum catalyst deposited on silica in an aqueous solution and in the presence of hydrogen, each particle of platinum takes the reversible potential of the equilibrium 2H+ + 2e H2, given by Nemst s law as... 

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