The Extensive Phase Rule

Each of the Phase Rules above is used to define the equilibrium state , which means that they each relate the number of properties (understood to be intensive variables) of the system to the number of degrees of freedom. This defines the equilibrium state, but it does not define how much of the equilibrium state we have. The equilibrium state of f kg of water saturated with halite is the same whether we have f g or f kg of halite. But modeling programs commonly want to do more than to define the equilibrium state. They want to dissolve or precipitate phases during processes controlled by the modeler, and to keep track of the masses involved, so as to know when phases should appear or disappear. To do this, the mass of each phase is required, not just its presence or absence. Therefore, an additional piece of information is required for each phase present, or p quantities. Almost invariably, the mass of H2O is chosen as 1 kg, so that the concentration of basis species defines the mass of each.3 If solid or gas phases are specified, the mass is usually also specified. If we count these extra p pieces of data, the extensive Phase Rule becomes 

This relationship is also fairly intuitive. Look at it this way. The number of phases is always at least one (a system with no phases is not very interesting). To define a system having only an aqueous solution phase (p = 1), we must specify each of the solutes in the water, or b — 1 quantities. If there is one mineral in equilibrium with the water (p = 2), it controls one basis species, and so reduces b by one, and similarly for all p mineral or gas phases. This is Phase Rule (3.34). But defining the equilibrium state is not usually enough. We want also to know the mass of each phase, so we need p extra data, giving Phase Rule (3.35), which says that for any system we need b pieces of information. These b pieces of information are 

3The unit of concentration used in modeling calculations is invariably molality, or the moles of solute species per kilogram of pure water. Therefore, if the mass of water is fixed at 1 kg, the molality of a species automatically equals the number of moles of the species, which is readily convertible to grams. 

The term Extensive Phase Rule is our own terminology, and may prove confusing to geochemists more used to seeing it referred to as Duhem s Theorem. As expressed by Prigogine and Defay (1965), p. 188, Duhem s Theorem says 

Whatever the number of phases, of components or of chemical reactions, the equilibrium state of a closed system, for which we know the initial masses m. .. m°, is completely determined by two independent variables. 

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Mention may also be made of the extensive phase-rule type of studies based on the solubility isotherms of M(OR)m-M (OR) systems in the research school of Turova, as illustrated by Gibbs-Roseboom triangular plots of NaOMe-Fe(OMe)3-MeOH, Ba(OBu )2-Ti(OBu )4-Bu OH, Bi(OEt)3-WO(OEt)4-EtOH, Ba(OMe)2-Ta(OMe)5-MeOH, NaOMe-Al(OMe)3-MeOH, Ca(OEt)2-Ta(OEt)4-C6H6, and Al(OIV)3-Hf(OPr)4-Pr OH systems. 

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. 

The basic asymmetry between intensive and extensive vectors can also be recognized in the Gibbs phase rule. This establishes the dimensionality of Ms in terms of the number of independent intensities, as expressed in (11.9b) in terms of rank(M). An alternative extensity-based (or M-based) description necessarily diverges at points where M becomes singular, i.e., at critical limits, where dimensionality changes, as shown by (11.24). 

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 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. 

The first three are intensive variables. The fourth is an extensive variable that is not considered in the usual phase rule analysis. The fifth is neither an intensive nor an extensive variable but is a siugle degree of freedom that the designer uses in specifying how often a particular element is repeated in a unit. For example, a distillation column section is composed of a series of equilibrium stages, and when the designer specifies the number of stages that the section contains. 

Distinguish between intensive and extensive variables, giving examples of each. Use the Gibbs phase rule to determine the number of degrees of freedom for a multicomponent multiphase system at equilibrium, and state the meaning of the value you calculate in terms of the system s intensive variables. Specify a feasible set of intensive variables that will enable the remaining intensive variables to be calculated. 

Extensive Phase Rule (3.35) then says that even though we are invariant, that is, even though we have adequately described the equilibrium state, to make our description more useful we need / = b = 5 pieces of information either (with only water) the four concentrations and the mass of water, or (with water and three minerals) one concentration plus the masses of all four phases. 

The amount of material and its flow rate are not controlled by the Gibbs phase rule. The phase rule refers to intensive variables such as pressure, tenperature, or mole fraction, which do not depend on the total amount of material present. The extensive variables, such as number of moles, flow rate, and volume, do depend on the amount of material and are not included in the degrees of freedom. Thus a mixture in equilibrium must follow Table 2-1 whether there are 0.1,1.0,10,100, or 1,000 moles present. 

The state variables are those intensive or extensive quantities that describe a system, for example, by means of the equation of state. The total number of variables required to describe a system with i number of components is i+1 (cf. Eq. 2.11, i accounts for the i composition variables, and 2 accounts for, e.g., P and T). For the discussions of phase diagrams, it is important to know how many of the state variables can be varied without going through a phase transition. For a closed system with i number of components and II number of phases, the number of intensive variables (cf. thermodynamic degrees of freedom, /) is given by the Gibbs phase rule ... 

Let us compare the Gibbs phase rule and the Duhem equation the Gibbs phase rule specifies the total number of independent intensive variables regardless of the extensive variables in the system, while the Duhem equation specifies the total number of independent variables, intensive or extensive, in a closed system. 

The Gibbs phase rule gives the number of independent intensive variables in a simple system that can have several phases and several components. The equilibrium thermodynamic state of a one-phase simple system with c components is specified by the values of c -F 2 thermodynamic variables, at least one of which must be an extensive variable. All other equilibrium variables are dependent variables. The intensive state is the state of the system so far as only intensive variables are concerned. Changing the size of the system without changing any intensive variables does not affect the intensive state. Intensive variables cannot depend on extensive variables, so specification of the... 

Historically, one of the main contributions of J. W. Gibbs to the development of thermodynamics was his extension of G = H TS to open systems. This is an important consideration for onstream processes encountered by chemical engineers. We have already introduced the concept of the chemical potential, p., = (Gilni), in two previous applications in this chapter first in the treatment of gas species p(T, F) = p° - - RT In P and then again in the discussion of the Gibbs phase rule. So far the treatments referred to closed systems and it seemed that p is just... 

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. 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

The orbital mixing theory was developed by Inagaki and Fukui [1] to predict the direction of nonequivalent orbital extension of plane-asymmetric olefins and to understand the n facial selectivity. The orbital mixing rules were successfully apphed to understand diverse chemical phenomena [2] and to design n facial selective Diels-Alder reactions [28-34], The applications to the n facial selectivities of Diels-Alder reactions are reviewed by Ishida and Inagaki elesewhere in this volume. Ohwada [26, 27, 35, 36] proposed that the orbital phase relation between the reaction sites and the groups in their environment could control the n facial selectivities and review the orbital phase environments and the selectivities elsewhere in this volume. Here, we review applications of the orbital mixing rules to the n facial selectivities of reactions other than the Diels-Alder reactions. 

The orbital mixing rule demonstrates that the direction of the FMO extension is controlled by the relative energies of the Jt-HOMO (ej and the n-orbital of X (8 ). In the case of 5-acetoxy- and 5-chloro-l,3-cyclopentadienes, the jt-HOMO lies higher than n (e > ej. In this case, the ji-HOMO mainly contributes to the HOMO of the whole molecule by an out-of-phase combination with the low-lying n. The mixing of a-orbital takes place so as to be out-of-phase with the mediated orbital n. The HOMO at Cl and C4 extends more and rotates inwardly at the syn face with... 

In the general case a complex behaviour may be expected for the extension of the terminal solid solutions which, for a pair of metals Mb M2, also depends on the stoichiometry and stability of the M (or, respectively, M2) richest phase. However a certain regularity of the dependence of the mutual solid solubility on the position of the metals involved in the Periodic Table may be observed. This can be related to the so-called Hume-Rothery rules (1931) ... 

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