Equilibrium Gibbs phase rule

The general thermodynamic treatment of binary systems which involve the incorporation of an electroactive species into a solid alloy electrode under the assumption of complete equilibrium was presented by Weppner and Huggins [19-21], Under these conditions the Gibbs Phase Rule specifies that the electrochemical potential varies with composition in the single-phase regions of a binary phase diagram, and is composition-independent in two-phase regions if the temperature and total pressure are kept constant. 

Having phases together in equilibrium restricts the number of thermodynamic variables that can be varied independently and still maintain equilibrium. An expression known as the Gibbs phase rule relates the number of independent components C and number of phases P to the number of variables that can be changed independently. This number, known as the degrees of freedom f is equal to the number of independent variables present in the system minus the number of equations of constraint between the variables. 

The concept of chemical potentials, the equilibrium criterion involving chemical potentials, and the various relationships derived from it (including the Gibbs phase rule derived in Chapter 5) can be used to explain the effect of pressure and temperature on phase equilibria in both a qualitative and quantitive way. 

The international temperature scale is based upon the assignment of temperatures to a relatively small number of fixed points , conditions where three phases, or two phases at a specified pressure, are in equilibrium, and thus are required by the Gibbs phase rule to be at constant temperature. Different types of thermometers (for example, He vapor pressure thermometers, platinum resistance thermometers, platinum/rhodium thermocouples, blackbody radiators) and interpolation equations have been developed to reproduce temperatures between the fixed points and to generate temperature scales that are continuous through the intersections at the fixed points. 

The framework for constructing such multi-component equilibrium models is the Gibbs phase rule. This rule is valid for a system that has reached equilibrium and it states that... 

When treated by the modified Gibbs phase rule (Crisp, 1949 Defay, 1932), these results suggest that at equilibrium, the enantiomeric monolayer system... 

If solvent is added to either of the solid eutectics represented by e or e in Fig. 25a or b, the undissolved solid retains this composition while the saturated solution maintains the composition E or E, respectively. Again, Gibbs phase rule [145,146] can provide further insight into these systems. If the solid enantiomers are solvated, the compositions of the equilibrium solids are displaced symmetrically along the DS or LS axes to an extent determined by the stoichiometry of the solvates. Similarly, if the racemic compound is solvated, the stoichiometry of the equilibrium solid is displaced from R along the line RS to an extent determined by the stoichiometry of the solvate. 

The most broadly recognized theorem of chemical thermodynamics is probably the phase rule derived by Gibbs in 1875 (see Guggenheim, 1967 Denbigh, 1971). Gibbs phase rule defines the number of pieces of information needed to determine the state, but not the extent, of a chemical system at equilibrium. The result is the number of degrees of freedom Np possessed by the system. 

The relationship between the number of degrees of freedom, F, defined as the number of intensive parameters that can be changed without changing the number phases in equilibrium, and the number of phases, Ph, and components, C, in the system is expressed through Gibbs phase rule ... 

For a given set of constraints (for example temperature, pressure and overall composition), the algorithm identifies the phases present and the relative amounts of these phases, as well as the mole fraction of all the components in all phases. The global minimum evidently must obey the Gibbs phase rule, and not all phases need to be present at the global equilibrium. 

For three-component (C = 3) or ternary systems the Gibbs phase rule reads Ph + F = C + 2 = 5. In the simplest case the components of the system are three elements, but a ternary system may for example also have three oxides or fluorides as components. As a rule of thumb the number of independent components in a system can be determined by the number of elements in the system. If the oxidation state of all elements are equal in all phases, the number of components is reduced by 1. The Gibbs phase rule implies that five phases will coexist in invariant phase equilibria, four in univariant and three in divariant phase equilibria. With only a single phase present F = 4, and the equilibrium state of a ternary system can only be represented graphically by reducing the number of intensive variables. 

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressure. The Gibbs phase rule for a ternary system at isobaric conditions is Ph + F = C + 1=4, and there are four phases present in an invariant equilibrium, three in univariant equilibria and two in divariant phase fields. Finally, three dimensions are needed to describe the stability field for the single phases e.g. temperature and two compositional terms. It is most convenient to measure composition in terms of mole fractions also for ternary systems. The sum of the mole fractions is unity thus, in a ternary system A-B-C ... 

Let us now include an additional component to the Fe-0 system considered above, for instance S, which is of relevance for oxidation of FeS and for hot corrosion of Fe. In the Fe-S-0 system iron sulfides and sulfates must be taken into consideration in addition to the iron oxides and pure iron. The number of components C is now 3 and the Gibbs phase rule reads Ph + F = C + 2 = 5, and we may have a maximum of four condensed phases in equilibrium with the gas phase. A two-dimensional illustration of the heterogeneous phase equilibria between the pure condensed phases and the gas phase thus requires that we remove one degree of... 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

Gibbs phase rule phys chem A relationship used to determine the number of state variables F, usually chosen from among temperature, pressure, and species composition in each phase, which must be specified to fix the thermodynamic state of a system in equilibrium F = C - P - M+2, where C is the number of chemical species presented at equilibrium, P is the number of phases, and M is the number of independent chemical reactions. Also known as Gibbs rule phase rule. gibz faz, rijl I... 

In discussing chemical systems, one must be aware of the rules which determine the chemical species that are permitted to occur for a given set of conditions. The basic rule governing systems which are considered to be in thermodynamic equilibrium was first stated by J. Willard Gibbs as early as 1876. The Gibbs phase rule relates the physical state of a mixture with the chemical species of which it is composed and is given in its simplest form as... 

The identification of the superconducting phase YBagCug-O7 g provides an example in which knowledge of thermodynamics, i.e. the Gibbs phase rule and the theory of equilibrium phase diagrams coupled with X-ray diffraction techniques led to success. Further, the use of databases that can now be easily accessed and searched on-line provided leads to a preliminary structure determination. The procedures outlined here are among the basic approaches used in solid state chemistry research, but by no means are they the only ones. Clearly the results from other analytical techniques such as electron microscopy and diffraction, thermal... 

In carrying out the procedure for determining mechanisms that is presented here, one obtains a set of independent chemical reactions among the terminal species in addition to the set of reaction mechanisms. This set of reactions furnishes a fundamental basis for determination of the components to be employed in Gibbs phase rule, which forms the foundation of thermodynamic equilibrium theory. This is possible because the specification of possible elementary steps to be employed in a system presents a unique a priori resolution of the number of components in the Gibbs sense. 

According to the Gibbs phase rule (number of degrees of freedom = number of components - number of phases + 2 see Atkins, 1998), for a system containing a single chemical distributed between two phases at equilibrium, there is only one... 

Let us now discuss some details of practical relevance. From the Gibbs phase rule, it is evident that crystals consisting of only one component (A) become nonvariant by the predetermination of two thermodynamic variables, which for practical reasons are chosen to be Pand T. In these one-component systems, it is easy to recognize the (isobanc) concentration dependence of the point defects on temperature. From the definition of the vacancy chemical potential for sufficiently small vacancy mole fractions Nv, namely //v = /A (P, T) + RT- In Vv, together with the condition of equilibrium with the crystal s inerL surroundings (gas, vacuum), one directly finds... 

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. 

The phase coexistence of gels at the first-order transition is accompanied by a number of unusual features, of which a few will be mentioned below. First, the fact that the triphasic equilibrium persists over a wide temperature range is an apparent contradiction to the Gibbs phase rule. This rule predicts that the... 

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