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. 

Write down the phase rule, define each parameter in the phase rule, and be able to apply the phase rule to determine the degrees of freedom, or the number of components, or the phases that exist in a system. 

At this point the system has throe phases (CUSO4 CuS04,Hj0 HjO vapour) and the number of components is two (anhydrous salt water). Hence by the phase rule, F + F = C + 2, t.e., 3+F = 2 + 2, or F=l. The system is consequently univariant, in other words, only one variable, e.g., temperature, need be fixed to define the system completely the pressure of water vapour in equilibrium with CUSO4 and CuS04,Hj0 should be constant at constant temperature. 

For a PVnr system of uniform T and P containing N species and 7T phases at thermodynamic equiUbrium, the intensive state of the system is fully deterrnined by the values of T, P, and the (N — 1) independent mole fractions for each of the equiUbrium phases. The total number of these variables is then 2 + 7t N — 1). The independent equations defining or constraining the equiUbrium state are of three types equations 218 or 219 of phase-equiUbrium, N 7t — 1) in number equation 245 of chemical reaction equiUbrium, r in number and equations of special constraint, s in number. The total number of these equations is A(7t — 1) + r -H 5. The number of equations of reaction equiUbrium r is the number of independent chemical reactions, and may be deterrnined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

For a system such as discussed here, the Gibb s Phase Rule [59] applies and establishes the degrees of freedom for control and operation of the system at equilibrium. The number of independent variables that can be defined for a system are ... 

Step 2. Determine the vapor pressure in the evaporator. According to the phase rule, for a mixture of two components (propane and butane) it is necessary to establish two variables of the liquid-vapor system in the evaporator to completely define the system and fix the value of all other variables. The assumed liquid mol fraction and a temperature of 0°F is known. The... 

This is a consequence of the phase rule there are two components in three phases, hence the number of degrees of freedom is 2, so that when the temperature and pressure are fixed, the composition of each layer is also defined. 

Regardless of their possible metallic properties, metal-rich Zintl system or phases are defined here as cation-rich compounds exhibiting anionic moieties of metal or metalloid elements whose structures can be generally understood by applying the classical or modern electron counting rules for molecules. 

To this point, the acceptance rules have been defined for a simulation, in which the total number of molecules in the system, temperature and volume are constant. For pure component systems, the phase rule requires that only one intensive variable (in this case the system temperature) can be independently specified when two phases... 

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 ... 

A phase is defined as a state of matter that is uniform throughout in terms of its chemical composition and physical state in other words, a phase may be considered a pure substance or a mixture of pure substances wherein intensive properties do not vary with position. Accordingly, a gaseous mixture is a single phase, and a mixture of completely miscible liquids yields a single hquid phase in contrast, a mixture of several solids remains as a system with multiple solid phases. A phase rule therefore states that, if a limited number of macroscopic properties is known, it is possible to predict additional properties. 

The degrees of freedom F in the phase rule refer to the number of externally controllable conditions of the system which must be specified to define uniquely the state of the system at equilibrium. In chemical systems the controllable variables are the temperature, pressure, and the proportions of the components of the system. The degree of freedom has a direct parallel in algebra where the "phase rule" is... 

Before moving on, it is wise here to note two important limitations of the phase rule. The criteria for components only prescribe that they be able to represent each phase in the system. The phase rule says nothing about how these components may combine to give other species and, thus, does not define the number or nature of other species in the system. That is, given the components CaO and C02, the phase rule cannot predict the existence of the intermediate compound CaCOs. 

The normal freezing point of the liquid under pressure is given by Tp, and OS is the melting curve of the substance, i.e. the locus of the points defining the co-existence of solid and liquid. If we measure the freezing point of a liquid in a closed system, the Phase Rule tells us that since at that temperature all three phases will be in equilibrium, F=0, and we obtain the... 

The relation between the non-stoichiometry and the equilibrium oxygen pressure mentioned in Section 1.1 can be deduced from the phase rule. For the purpose of the derivation of the phase rule, we shall review fundamental thermodynamics. Gibbs free energy G is defined by the relation... 

A phase rule study of the system Pb0-As205-H20 has been made over the acid range at 25° C. and the conditions for the existence of the two acid lead arsenates defined.7... 

Gas chromatography involves chemical equilibria between phases to bring about a particular separation. Thus, a brief discussion of phase equilibria is pertinent at this point. Phase equilibria separations can be understood with the use of the second law of thermodynamics. The phase rule states that if we have a system of C components which are distributed between. P phases, the composition of each of these phases will be completely defined by C-l concentration terms. Thus, to have the compositions of P phases defined it is necessary to have P(C-l) concentration terms. The temperature and pressure also are variables and are the same for all the phases. Assuming no other forces influence the equilibria it follows that. 

FREEDOM (Degrees oD. I. The number of variables whieh must be fixed before Ihe state of a system may be defined according to the phase rule. See also Phase Rule. The relationship between the number of degrees... 

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. 

After the conditions of equilibrium have been determined, we can derive the phase rule and determine the number and type of variables that are necessary to define completely the state of a system. The concepts developed in this chapter are illustrated by means of graphical representation of the thermodynamic functions. 

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