Intensive variables phase rule

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). 

This is Gibbs s phase rule. It specifies the number of independent intensive variables that can and must be fixed in order to estabbsh the intensive equibbrium state of a system and to render an equibbrium problem solvable. 

The intensive state of a PVT system is established when its temperature and pressure and the compositions of all phases are fixed. However, for equihbrium states these variables are not aU independent, and fixing a hmited number of them automaticaUy estabhshes the others. This number of independent variables is given by the phase rule, and is called the number of degrees of freedom of the system. It is the number of variables which may be arbitrarily specified and which must be so specified in order to fix the intensive state of a system at equihbrium. This number is the difference between the number of variables needed to characterize the system and the number of equations that may be written connecting these variables. 

For a system containing N chemical species distributed at equihbrium among 7C phases, the phase-rule variables are temperature and pressure, presumed uniform throughout the system, and N — mole fraciions in each phase. The number of these variables is 2 -t- (V — 1)7T. The masses of the phases are not phase-rule variables, because they have nothing to do with the intensive state of the system. 

Since the phase rule treats only the intensive state of a system, it apphes to both closed and open systems. Duhem s theorem, on the other hand, is a nJe relating to closed systems only For any closed system formed initially from given masses of preseribed ehemieal speeies, the equilibrium state is completely determined by any two propeities of the system, provided only that the two propeities are independently variable at the equilibrium state The meaning of eom-pletely determined is that both the intensive and extensive states of the system are fixed not only are T, P, and the phase compositions established, but so also are the masses of the phases. 

The general XT E problem involves a multicomponent system of N constituent species for which the independent variables are T, P, N — 1 liquid-phase mole fractions, and N — 1 vapor-phase mole fractions. (Note that Xi = 1 and y = 1, where x, and y, represent liquid and vapor mole fractions respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to estabhsh the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by siiTUiltaneous solution of the N equihbrium relations ... 

The term ff denotes the number of independent phase variables that should be specified in order to establish all of the intensive properties of each phase present. The phase variables refer to the intensive properties of the system such as temperature (T), pressure (P), composition of the mixture (e.g., mole fractions, x ), etc. As an example, consider the triple point of water at which all three phases—ice, liquid water, and water vapor—coexist in equilibrium. According to the phase rule,... 

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... 

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. 

If we now recall the phase rule, it is evident that, at the P-T conditions represented by point D in figure 2.5, slight variations in the P or T values will not induce any change in the structural state of the phase (there are one phase and one component the variance is 2). At point A in the same figure, any change in one of the two intensive variables will induce a phase transition. To maintain the coexistence of kyanite and andalusite, a dP increment consistent with the slope of the univariant curve (there are two phases and one component the variance... 

In terms of the phase rule, we must utilize Eq. (2.24) instead of (2.23), since we have only one noncompositional intensive variable, T (pressure is fixed), so A = 1. Application of Eq. (2.24) to points A and C in Figure 2.4 indicates that the liquid and solid regions (0 = 1 C = 2) correspond to F = 2, which means that both temperature and composition can be altered independently in these regions. Point B is in the liquid -b solid two-phase region (0 = 2 C = 2), so that F = 1. Temperature and composition are not independent in this region any change in temperature necessarily results in a... 

It should be emphasized that the criterion for macroscopic character is based on independent properties only. (The importance of properly enumerating the number of independent intensive properties will become apparent in the discussion of the Gibbs phase rule, Section 5.1). For example, from two independent extensive variables such as mass m and volume V, one can obviously form the ratio m/V (density p), which is neither extensive nor intensive, nor independent of m and V. (That density cannot fulfill the uniform value throughout criterion for intensive character will be apparent from consideration of any 2-phase system, where p certainly varies from one phase region to another.) Of course, for many thermodynamic purposes, we are free to choose a different set of independent properties (perhaps including, for example, p or other ratio-type properties), rather than the base set of intensive and extensive properties that are used to assess macroscopic character. But considerable conceptual and formal simplifications result from choosing properties of pure intensive (R() or extensive QQ character as independent arguments of thermodynamic state functions, and it is important to realize that this pure choice is always possible if (and only if) the system is macroscopic. 

Eq.(2.2-4) is the phase rule of Gibbs. According to this rule a state with II phases in a system with N components is frilly determined (all intensive thermodynamic properties can be calculated) if a number of F of the variables is chosen, provided that g of all phases as function of pressure, temperature and composition is known. 

Since both the oxide reactant and the spinel are ternary (nonstoichiometric) compounds when equilibrated with each other, at a given P and T, the boundaries (A, B)0/spinel and spinel/(B, A)203 are not invariant. They become invariant (and thus provide unique boundary conditions for the reaction) only if an additional intensive thermodynamic variable can be predetermined. This is a consequence of Gibbs phase rule. 

PHASE RULE. The phase rule, due to Gibbs, gives the number F of intensive variables which can be fixed arbitrary in a system in equilibrium. This number is also called the variance or the number of degrees of freedom of the system. Tt is given by... 

The variables (or rather, intensive variables ), are p (pressure), T (temperature), and the concentrations (e.g., mole fractions of the components) in each separate phase. P is the number of phases present at equilibrium, and C is the minimum number of components necessary to duplicate any system that represents the equilibrium in question. (The components may sometimes be chosen in several ways). Finally, F is the number of degrees of freedom or the number of independent variables. With P phases present, one can (within limits) assign values independently of F variables, but then all other variables are fixed by the conditions for equilibrium. [For example, one may apply the phase rule to the system CaO—CCL—H20, with one, two, three, four, or five phases, determine how many independent variables result, and decide what will be the most practical choice of variables. (Five phases might be CaC03(s), Ca(OH)2(s), ice, an aqueous solution, and a gas phase.)]... 

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. 

Consider the potential for hydrate formation from methane gas and free water. One question would be, At what temperature (T) will hydrates form for a given pressure (P) Before the calculation is done, one might wonder if there is a unique solution for the problem, or if the problem is under-specified (has an infinite number of solutions). Since the components are methane and water C = 2, and the phases are three (Lw-H-V) by the Gibbs Phase Rule, one intensive variable (F = 1), such as either T or P must be specified in order to obtain a unique solution for the formation of hydrates. 

Thus by specifying either T or P, there is a unique solution to determine the other intensive variables (such as the T or P that was not specified, as well as the other intensive variables listed above). Again, extensive variables such as the volumes or phase amounts are not considered in Gibbs Phase Rule. 

By Gibbs Phase Rule illustrated in this chapter s introduction, a second intensive variable is needed (in addition to either temperature or pressure) to specify the three-phase binary system with an inhibitor (F = 3 — 3 + 2). Typically, the concentration of the inhibitor in the free water phase is specified as the second intensive variable. Substances that have considerable solubility in the aqueous phase, such as alcohols, glycols, and salts, normally act as inhibitors to hydrate formation. The colligative mechanism of formation inhibition is aided by increased competition for water molecules by the dissolved inhibitor molecule or ion through hydrogen bonding for alcohols or glycols, or via Coulombic forces (for salt ions). 

This is the usual relation given for the phase rule. The difference V is called the number of variances or the number of degrees of freedom. (Note that some texts use F = C — P + 2, where F is the same as V.) These are the number of intensive variables to which values may be assigned arbitrarily but within the limits of the condition that the original number of P phases exist. It is evident from Equations (5.63) and (5.56) that the intensive variables are the temperature, pressure, and mole fractions or other composition variables in one or more phases. Note that the Gibbs phase rule applies to intensive variables and is not concerned with the amount of each component in each phase. 

The Gibbs phase rule provides the necessary information to determine when intensive variables may be used in place of extensive variables. We consider the extensive variables to be the entropy, the volume, and the mole numbers, and the intensive variables to be the temperature, the pressure, and the chemical potentials. Each of the intensive variables is a function of the extensive variables based on Equation (5.66). We may then write (on these equations and all similar ones we use n to denote all of the mole numbers)... 

The Gibbs-Duhem equation is the basis for the phase rule of Gibbs. According to the phase rule, the number of degrees of freedom F (independent intensive variables) for a system involving only PV work, but no chemical reactions, is given by... 

This shows that there are C + 2 intensive variables for a chemical reaction system at equilibrium, but only C + 1 of them are independent because of this relation between them in other words, for a one-phase system at chemical equilibrium the number F of degrees of freedom is given by F = C + 1. Since this is a one-phase system, it is evident that the phase rule for the reaction system is... 

The phase rule offers a simple means of determining the minimum number of intensive variables that have to be specified in order to unambiguously determine the thermodynamic state of the system. 

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. 

For c components in a phase, this equation has c + 2 terms, one more than given by the phase rule, because its integration requires the size of the phase as well as its intensive variables. The quantity (9X/dni)PTn is called the partial molar X with respect to i and given the symbol Xt ... 

As with all multiphase systems, the Gibbs phase rule provides a useful tool for determining the number of intensive variables (ones that do not depend on system mass) that can be fixed independently ... 

Nbf is the number of degrees of freedom, Nc is the number of components, and Np is the number of phases in the system. The number of degrees of freedom represents the number of independent variables that must be specified in order to fix the condition of the system. For example, the Gibbs phase rule specifies that a two-component, two-phase system has two degrees of freedom. If temperature and pressure are selected as the specified variables, then all other intensive variables—in particular, the composition of each of the two phases—are fixed, and solubility diagrams of the type shown for a hypothetical mixture of R and S in Fig. 1 can be constructed. 

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