Gibbs phase rule defined

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 ... 

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 ... 

For solid-liquid equilibrium in a quaternary system, the Gibbs phase rule allows four degrees of freedom. If T, P, xc, and xD (in which x is the mole fraction of component i in liquid solution) are specified, then xA, x, t/, and xAC (in which x is the mole fraction of component ij in solid solution) are determined, and the system is invariant. These variables are defined by the following equations ... 

An intensive variable [such as the temperature (T), pressure (P), or individual mole fractions of a single phase (xSi, Xu or y of the hydrate, liquid, or vapor phases, respectively)] is defined as a measured value that is independent of the phase amount. For example, T, P, xSi, xu y or density are intensive variables, while phase masses, volumes, or amounts are extensive variables, and thus not addressed by Gibbs Phase Rule. 

A phase is defined as the part of the system that has uniquely distinguishing properties from the other part of the system. That property can be, for example, density (e.g., water-ice-water vapor) or different crystallographic forms (e.g., a — Pd//3 - Pd). The coexistence and number of different phases p depends on the number of components c, and on external physical parameters called degrees of freedom /. These are most typically pressure and temperature. The governing relationship is the Gibbs phase rule. 

Many choices of independent variables such as the energy, volume, temperature, or pressure (and others still to be defined) may be used. However, only a certain number may be independent. For example, the pressure, volume, temperature, and amount of substance are all variables of a single-phase system. However, there is one equation expressing the value of one of these variables in terms of the other three, and consequently only three of the four variables are independent. Such an equation is called a condition equation. The general case involves the Gibbs phase rule, which is discussed in Chapter 5. 

The Gibbs phase rule allows /, the number of degrees of freedom of a system, to be determined. / is the number of intensive variables that can and must be specified to define the intensive state of a system at equilibrium. By intensive state is meant the properties of all phases in the system, but not the amounts of these phases. Phase equilibria are determined by chemical potentials, and chemical potentials are intensive properties, which are independent of the amount of the phase that is present. The overall concentration of a system consisting of several phases, however, is not a degree of freedom, because it depends on the amounts of the phases, as well as their concentration. In addition to the intensive variables, we are, in general, allowed to specify one extensive variable for each phase in the system, corresponding to the amount of that phase present. 

When multicomponent gas and liquid phases are in equilibrium, a limited number of intensive system variables may be specified arbitrarily (the number is given by the Gibbs phase rule), and the remaining variables can then be determined using equilibrium relationships for the distribution of components between the two phases. In this section we define several such relationships and illustrate how they are used in the solution of material balance problems. 

A simple classification scheme of solids is given in Fig. 7.1. In order to differentiate between the types of solids, we have to consider the Gibbs phase rule, which is discussed in any physical chemistry textbook. The basic question is whether the solid substance consists of only one chemical entity (component) or more than one. Usually the component is one molecular unit, with only covalent bonded atoms. However, a component can also consist of more constituents if their concentration cannot be varied independently. An example of this is a salt. The hydrochloride salt of a base must be regarded as a one-component system as long as the acid and the base are present in a stoichiometric ratio. A deficiency of hydrochloric acid results in a mixture of the salt and the free base, which behave as two completely different substances (i.e. two different systems). Polymorphic forms, the glassy state, or the melt of the base (or the salt) are considered as different phases within such a system (a phase is defined as the portion of a system that itself is homogeneous in composition but physically distinguishable from other phases). When the base (or salt) is dissolved in a solvent, a new system is obtained this is also tme when a solvent is part of the crystal lattice, as in the case of a solvate. Thus, each solvate represents a different multicomponent system of a compound, whereas, polymorphic forms are different phases. The variables in the solvate are the kind of solvate (hydrate. 

Duhem s theorem is applied to closed systems at equilibrium, when both intensive and extensive parameters are known, i.e., for systems of totally defined state. Both extensive and intensive parameters may be independent. However, their interrelation is defined by Gibbs phase rule. At C = 0 both parameters must be extensive, and at C = 1 at least one of them must be extensive. 

The answer is yes and we will digress a bit at this point to introduce these concepts as we did earlier in the chapter. The temperature and pressure conditions that govern physico-chemical behavior of liquids are defined in terms of thermodynamics. The Gibbs Phase Rule is a direct outcome of the physical chemistry of changes in the state of matter. The phase rule helps to interpret the physico-chemical behavior of solids, liquids, and gases within the framework of the kinetic-molecular theory of phase equilibria. 

This equation is called Gibbs phase rule. It defines the number of thermodynamic degrees of freedom /, i.e. the number of variables T, P, pi which can arbitrarily be varied without breaking the equilibrium among 7 phcises. When / = 0, the equilibrium is referred to eis non-variant, with / = 1 it is monovariant, when / = 2 it is bivariant, and with / > 3 it is polyvariant. 

As an illustration of the Gibbs phase rule, let us consider the equilibrium between solid, liquid and gas phases of a pure substances, i.e. one component. In this case we have C = 1 and P = 3, which gives / = 0. Hence for this equilibrium, there are no free intensive variables there is only one pressure and temperature at which they can coexist. This point is called the triple point (Fig. 7.1). At the triple point of H2O, T — 213A6K = 0.01 °C and p = 611 pa = 6.11 X 10 bar. (This unique coexistence between the three phases of water may be used in defining the Kelvin scale.)... 

Defining the critical point of a substance requires two degrees of freedom. (Those degrees of freedom are the critical temperature and the critical pressure.) Justify this fact in light of the Gibbs phase rule. 

Let us consider a binary solution that is composed of two liquid components that are not interacting chemically. If the volume of the liquid is equal to the size of the system, then we have only one phase and two components, so the Gibbs phase rule says that we have F=2 — 1+2 = 3 degrees of freedom. We can specify temperature, pressure, and mole fraction of one component to completely define our system. Recall from equation 3.24 that the mole fraction of a component equals the moles of some component i, divided by the total number of moles of all components in the system,... 

A single compound system exhibiting two solid states is univariant according to the Gibbs phase rule. Thus, at constant pressure, temperature is sufficient to define the solid state. The temperature at which both solid states (polymorphs) exist in equilibrium is called the transition temperature, T,. Essentially, there are two types of polymorphic transformations that can occur (a) reversible enan-tiotropic and (b) irreversible monotropic. The thermodynamic state of polymorphic transitions can be explained in terms of Gibbs free energy, G, which is given by the relation ... 

The above analysis is not intended to imply that the Kripke-Putnam theory of reference is free of any criticism. For example, two prominent philosophers of chemistry have objected to it on the basis that it relies too heavily upon micro-reduction. For example, van Brakel espouses a radical anti-reductionism in which he favors the manifest image over micro-reduction. His writings have included a thoroughgoing critique of Kripke s and Putnam s theory and a review of all other critiques that have been made in the context of chemical kinds (van Brakel 2000). Similarly, Needham takes an anti-reductionist approach and prefers to define substances through the Gibbs phase rule (Needham 2005). 

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