Phase rule independent equations

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). 

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. 

The total number of independent equations is therefore (tt — )N + r In their fundamental forms these equations relate chemical potentials, which are functions of temperature, pressure, and composition, the phase-rule variables. Since the degrees of freedom of the system F is the difference between the number of variables and the number of equations. 

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 proof of the phase rule is actually implicit in the derivation of the governing equations (Eqns. 3.32-3.35), and is not repeated here. It is interesting, nonetheless, to compare this well-known result with the governing equations, if only to demonstrate that we have reduced the problem to the minimum number of independent variables. 

Equation (6.42) introduces a new independent variable of the system the mean curvature c = (c1 +C2). This variable must be taken into account in the Gibbs phase rule, which now reads F + Ph = C + 2 + 1. The number of degrees of freedom (F) of a two-phase system (Ph = 2) with a curved interface is given by... 

Equation (2.23) is a very important result. It is known as the Gibbs Phase Rule, or simply the phase rule, and relates the number of components and phases to the number of degrees of freedom in a system. It is a more specific case of the general case for N independent, noncompositional variables... 

This observation is the simplest case of the Gibbs phase rule (to be discussed in Section 7.1). It implies, for example, that pressure P = P(V, T) is uniquely specified when V and T are chosen, and similarly, that V = V(P, T) or T = T(P, V) are uniquely determined when the remaining two independent variables are specified. Such functional relationships between PVT properties are called equations of state. We can also include the quantity of gas (as measured, for example, in moles n) to express the equation of state more generally as... 

Let the number of basic overall equations be Q it is evident that Q< P. Let us denote the number of substances participating in the reaction as M and the number of independent components, in the sense this notion is used in the Gibbs phase rule, as C then... 

Because the values of Gm depend on P and T, this expression can be regarded as an equation relating P and T. That is, we have an equation relating two unknowns (P and T) so if P is set, T must have a corresponding value. Only one of the variables, P or T, is independent, so f = 1, as in the phase rule. Now consider three phases a, (3, and y in equilibrium. There are now three equations that have to be satisfied ... 

In a part of this system which has been studied by Hemley (11), four phases can exist at equilibrium aqueous solution, solid quartz, solid kaolinite (Al1 Si2Or)(OH)4), and potassium mica (KAl tSi 4Oio(OH)2). The variables are p, T, and the various concentrations [K+], [IF], [CT], [Al]a(J, [Si(OH)4].i(, etc. If we apply the phase rule (Equation 1) to equilibria of the four phases mentioned, we find F = 5 -f 2 — 4 = 3. The most practical choice of independent variables would seem to be p, T, and [CT]. These are easy to control, and CT is the one ion that must remain in the aqueous phase since there is no place for it in the solid phases. The phase rule now states that after the values for these... 

The Gibbs phase rule relates the number of equations that are required to describe a system at equilibrium to the number of variables necessary to describe the system. The number of degrees of freedom is the number of variables that can be changed independently without affecting the number of phases in equilibrium. It is the difference between the number of variables and the number of equations describing equilibrium. 

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 derivation of the phase rule is based upon an elementary theorem of algebra. This theorem states that the number of variables to which arbitrary values can be assigned for any set of variables related by a set of simultaneous, independent equations is equal to the difference between the number of variables and the number of equations. Consider a heterogenous system having P phases and composed of C components. We have one Gibbs-Duhem equation of each phase, so we have the set of equations... 

The Gibbs-Duhem equation is applicable to each phase in any heterogenous system. Thus, if the system has P phases, the P equations of Gibbs-Duhem form a set of simultaneous, independent equations in terms of the temperature, the pressure, and the chemical potentials. The number of degrees of freedom available for the particular systems, no matter how complicated, can be determined by the same methods used to derive the phase rule. However, in addition, a large amount of information can be obtained by the solution of the set of simultaneous equations. 

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... 

N02 + fH20 - NH3 + 1 02 Equations (6) and (7) represent a set of independent reactions for which r = 2. Other equivalent sets of two reactions may be obtained by different combination procedures. By the phase rule,... 

Solution According to the phase rule (see Sec. 15.8), the system has two degrees of freedom. Specification of both the temperature and the pressure leaves no other degrees of freedom, and fixes the intensive state of the system, independent of the initial amounts of reactants. Therefore, material-balance equations do not enter into the solution of this problem, and we can make no use of equations that relate compositions to the reaction coordinate. Instead, phase equilibrium relations must... 

It must be modified for application to systems in which chemical reactions occur. The phase-rule variables are the same in either case, namely, temperature, pressure, and N - 1 mole fractions in each phase. The tothl number of these variables is 2 + (N - 1)(7r). The same phase-equilibrium equations apply as before, and they number (it - 1)(N). However, Eq. (15.8) provides for each independent reaction an additional relation that must be satisfied at equilibrium. Since the Hi s are functions of temperature, pressure, and the phase compositions, Eq. (15.8) represents a relation connecting the phase-rule variables. If there are r independent chemical reactions at equilibrium within the system, then there is a total of (it - 1)(N) + r independent equations relating the phase-rule variables. Taking the difference between the number of variables and the number of equations, we obtain... 

For a nonreacting equilibrium system with n species and p phases, the number of independent phase equilibrium equations is (p - 1 )n. The number of phase-rule variables is 2 + (n — 1 )p, consisting of intensive variables of temperature pressure and (n — 1) compositions for each phase. The difference between the phase-rule variables and the number of independent phase equilibrium equations is the degrees of freedom of the system, F... 

Equation (2.1.8) specifies the famous phase rule of Gibbs (1875—1878). Knowing the number of components and phases in a given system, and assuming that T and P for the system as a whole are uniformly adjustable, Eq. (2.1.8) indicates how many state variables may be independently adjusted without altering the number of phases of the system. The ramifications of the phase rule will be discussed in Section 2.3. 

The framework within which condensed phases are studied is provided by the phase rule, which gives the number of independent thermodynamic variables required to determine completely the state of the system. This number, called the number of degrees of freedom or the variance of the system, will be denoted by F. The number of phases present (that is, the number of homogeneous, physically distinct parts) will be denoted by P, and the number of independently variable chemical constituents will be called C. By independently variable constituents, we mean those whose concentrations are not determined by the concentrations of other constituents through chemical-equilibrium equations or other subsidiary conditions. The phase rule states that... 

If r independent reactions occur among the system components and the reactions proceed to equilibrium, then the right-hand side of this equation must be reduced by r. [Note Perry s Chemical Engineers Handbook (see footnote 1), p. 4-24, presents a proof of the phase rule and outlines a method for determining how many independent reactions may occur among the components of a system.]... 

When we have reduced the representing of a problem by an equation to be no more than an algebraic expression, the first point we have to examine is the following How many distinct quantities are there in the equations of the problem And the examination of this point is immediately followed by the study of this other point Among these distinct quantities, how many independent relations does algebra furnish In making this double enumeration for the problems of chemical mechanics, J. Willard Gibbs was led to the propositions whose ensemble constitutes the phase rule. 

The phase rule for nonreacting systems, presented without proof in Sec. 2.7, results from application of a mle of algebra. Thus, the number of variables that may be independently fixed in a system at equilibrium is the difference between the total number of variables that characterize the intensive state of the system and the number of independent equations that can be written connecting the variables. 

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