Gibbss Phase Rule

The Gibbs phase mle is very important in chemical and metallurgical engineering where there can be many phases in the solid state but seldom comes into play in synthetic chemistry. In fact, it is not easy to find a simple example in chemistry. Even so this topic is in our list of essential topics in 

Let there be C chemical species in each of P phases, then there will be P x C mol fractions needed to specify concentrations in each phase. To this we need to add temperature and pressure values so the sum of variables seems to be PC + 2, but these variables are not aU independent. 

In each phase (a, (3, y, 8.), the sum of the mole fractions is 1, so that reduces the degrees of fireedom by one for each phase or (—P) degrees of freedom. 

Then we note that the chemical potential of each component is equal within each phase P by virtue of the implicit equilibrium of the C components, so there is also one less degree of freedom relating the chemical potentials within each phase for a reduction of C P — 1) degrees of freedom. 

One laboratory experiment that illustrates the Gibbs phase rule is the equilibrium between SO2 and aniline. 

The Gibbs phase rule states the number of the degrees of freedom that results from the number of components and phases, coexisting in a system. 

The number 2 in the Eq. 8 arises from the two independent variables, pressure and temperature. Phases are limited, physically and chemically homogeneous, mechanically separable parts of a system. Components are defined as simple chemical entities or units that comprise the composition of a phase. 

In a system, where the number of phases and the number of components are equal, there are two degrees of freedom, meaning that two variables can be varied independently (e.g. temperature and pressure). If the number of the degrees of freedom is zero, then temperature and pressure are constant and the system is invariant. 

In a three-phase system including a solid and a liquid as well as a gas, the Gibbs phase rule is modified to  

N = number of possible equilibrium reactions (species, charge balance, stoichiometric relations)-P = number of phases 

Let us now consider a heterogeneous thermodynamic system at equilibrium. If there are O phases in the system, it can easily be seen that 0 — 1 equations of type 2.15 and 2.16 apply for each component in the system. Hence, if there are n components, the number of equations will be (0 - 1). Moreover, the following mass-balance equation holds for each phase  

The system variables are composed of n compositional terms plus ambient variables that are usually two in number temperature and pressure (hydrostatic and/ or lithostatic-isotropic pressure). The variance (F) of the system is readily obtained by subtracting the number of condition equations from the total number of variables (n -I- 2)  

When applying the Gibbs phase rule, it must be remembered that the choice of components is not arbitrary the number of components is the minimum number compatible with the compositional limits of the system. 

There are two phases in reaction and, apparently, three components, corresponding to atoms Ca, C, and O. However, the compositional limits of the system are such that 

Because conditions 2.40 and 2.41 hold for all phases in the system, the minimum number of components necessary to describe phase chemistry is 1. 

1 Binary phase diagrams from thermodynamics Gibbs phase rule 

In chemical thermodynamics the system is analyzed in terms of the potentials defining the system. In the present chapter the potentials of interest are T (thermal 

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The latter is used as a guideline to determine the relationship between the number of potentials that can be varied independently (the number of degrees of freedom, F) and the number of phases in equilibrium, Ph. Varied independently in this context means varied without changing the number of phases in equilibrium. 

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 pressures. Gibbs phase rule then becomes 

How many variables need to be specified in order to fix the state of a system In order to fix the state of a one-phase system, the composition of the phase must be specified as well as two additional 

For a system with tt phases, there are a total of (w + 1)% unknowns. However, not all of these are independent. The conditions for phase equilibrium (see Eqs. (3.8)) give us (w + 2)(7r — 1) equations that must be satisfied between each of tlie phases. The difference between the number of unknowns in the system and tlie number of constraints (or equations) is equal to the number of degrees of freedom f in tlie system. 

This is known as the Gibbs phase nile. It tells us the number of variables f that must be specified in order to fix the (intensive) state of tlie system. 

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2 Solidification of Pure Systems 11.2.1 Gibbs Phase Rule 

Typical melting curve showing the thermal arrest. 

In this chapter, the phase relations of those oxide components will be discussed that most commonly occur in ceramics, namely silica, alumina, calcia, magnesia, potassia, and iron oxide. The chapter is introduced by a cursory analysis of the anatomy and construction of phase diagrams based on Gibbs phase rule, progressing from one component to two components, and from three components to multicomponent systems. The thermodynamics of these phase assemblies will be described, some important phase diagrams discussed, and conclusions drawn. 

Phase diagrams are chemographic representations of the famous Gibbs phase rule (Gibbs, 1876 Gibbs, 1878) that relates the number of phases P, the number of components C, and the number of degrees of freedom F of a (closed) system by the simple equation 

The components C are defined as those simple oxides that combine to constitute the phases P present at equihbrium that is, those portions of a system that are physically homogeneous and mechanically separable. The number of degrees of freedom, F, is the number of variables that can be altered without changing the number of phases present The number 2 in Eq. (3.1) is appUcable when 

1) Equilibrium is defined as that state of any reversible system when no useful energy passes from or into the system. Then, the driving force is zero for a transformation of one phase into a different phase, ffowever, it is sometimes difficult to determine experimentally whether equilibrium has 

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The state postulate refers to the entire system. A related concept is used to determine the number of independent, intensive properties needed to constrain the properties in a given phase, which is referred to as the degrees of freedom, As we will later verify (see Example 6.17), the Gibbs phase rule says that is given by  

For a system that contains a pure substance— that is, one component—Equation (1.12) reduces to  

The number of phases influences which properties are independent. The determination of which two properties we can choose to constrain the system according to the state postulate depends on the number of phases that are present. First, consider a system with only one phase present (77 = l). In this case the state postulate and the Gibbs phase rule are equivalent. Equation (1.13) tells us we need two independent properties to constrain the phase and, thus, the system. Specification of any two intensive properties—such as pressure, P, and temperature, T—constrains all the other properties in the system. The 

We next wish to examine how to constrain the state of systems with more than one phase present. If we have a pure substance with two phases present, the phase rule says we need just one property in each phase to constrain the values of all the other properties for that phase. However, the properties temperature and pressure present a special case, since they are equal in both phases. Most other properties are different between phases.Thus, if we know either T or P of the system, we constrain the properties in each of the phases. 

To illustrate this concept, consider a pure system of boiling water where we have both a liquid and a vapor phase. In this text, we use water to indicate the chemical species H2O in any phase solid, liquid, or gas. The phase rule tells us that for the liquid phase of water, we need only one property to constrain the state of the phase. If we know the system pressure, P, all the other properties (T, o , u ,. . . ) of the liquid are constrained. The subscript T refers to the liquid phase. It is omitted on T since the temperatures of both the liquid and vapor phases are equal. For example, for a pressure of 1 atm, the temperature is 100 [°C]. We can also determine that the volume of the liquid is 1.04 X 10 [m /kg], the internal energy is 418.94 [kj/kg], and so on. The system pressure of 1 atm also constrains the properties of the vapor phase. The temperature remains the same as for the liquid, 100 [°C] however, the values for the volume of the vapor (l.63[m /kg]), the internal energy (2,506.5 [kJ/kg]), and so on are different from those of the liquid. 

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Gibbsitic [14762-49-3] Gibbs-Kelvin equation Gibbs phase rule Gibbs s phase rule Gibbs s theorem Gibbs-Thomson equation... 

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. 

It was shown some time ago that one can also use a similar thermodynamic approach to explain and/or predict the composition dependence of the potential of electrodes in ternary systems [22-25], This followed from the development of the analysis methodology for the determination of the stability windows of electrolyte phases in ternary systems [26]. In these cases, one uses isothermal sections of ternary phase diagrams, the so-called Gibbs triangles, upon which to plot compositions. In ternary systems, the Gibbs Phase Rule tells us... 

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. 

While the Gibbs phase rule provides for a qualitative explanation, we can apply the Clapeyron equation, derived earlier [equation (5.71)], in conjunction with studying the temperature and pressure dependences of the chemical potential, to explain quantitatively some of the features of the one-component phase diagram. 

We will be looking at first-order phase transitions in a mixture so that the Clapeyron equation, as well as the Gibbs phase rule, apply. We will describe mostly binary systems so that C = 2 and the phase rule becomes... 

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 a reversible transition from one monolayer phase to another can be observed in the 11/A isotherm (usually evidenced by a sharp discontinuity or plateau in the phase diagram), a two-dimensional version of the Gibbs phase rule (Gibbs, 1948) may be applied. The transition pressure for a phase change in one or both of the film components can be monitored as a function of film composition, with an ideally miscible system following the relation (12). A completely immiscible system will not follow this ideal law, but will... 

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

A basic exposition of Gibbs phase rule is essential for understanding phase solubility analysis, and detailed presentations of theory are available [41,42]. In a system where none of the chemical species interact with each other, the number of independently variable factors (i.e., the number of degrees of freedom, F) in the system is given by... 

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 application of n additional thermodynamic potentials (of electric, magnetic or other origin) implies that the Gibbs phase rule must be rewritten to take these new potentials into account ... 

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