Phase Rule extensive

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. 

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). 

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. 

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. 

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. 

Duhem s theorem is another rule, similar to the phase rule, but less celebratec It applies to closed systems for which the extensive state as well as the intensiv state of the system is fixed. The state of such a system is said to be completel determined and is characterized not only by the 2 + (iV—l)ir intensive phase rule variables but also by the it extensive variables represented by the masse (or mole numbers) of the phases. Thus the total number of variables is... 

The two independent variables subject to specification may in general be either intensive or extensive. However, the number of independent intensive variables is given by the phase rule. Thus when F = 1, at least one of the two variables must be extensive, and when F = 0, both must be extensive. 

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. 

As a student and collaborator of Professor de Bonder, Professor R. Defay has contributed to the work of the Brussels School omthe Thermodynamics of chemical reactions. His own researches are chiefly related to azeotropism, to surface tension, to the extension of the phase rule, to capillary systems and to the study of surfaces not in adsorption equilibrium. 

The work of Gibbs was essentially theoretical and its full importance in physical chemistry was appreciated only after its wide applicability had been demonstrated by extensive experimental researches. Among these may be mentioned, for example, the work of Bakhuis Rooseboom which focused attention on the phase rule. At the same time Planck, van Laar, Duhem and van der Waals demonstrated clearly the importance of the concept of chemical potential and completed several aspects of Gibbs work. 

Variables of the kind with which the phase rule is concerned are called phase-rule variables, and they are intensive properties of the system. By this we mean properties that do not depend on the quantity of material present. If you think about the properties we have employed so fer in this book, you have the feeling that pressure and temperature are independent of the amount of material present. So is concentration, but what about volume The total volume of a system is called an extensive property because it does depend on how much material you have the specific volume, on the other hand, the cubic meter per kilogram, for example, is an intensive property because it is independent of the amount of material present. In Chap. 4 we take up additional intensive properties, such as internal energy and enthalpy. You should remember that the specific (per unit mass) values of these quantities are intensive properties the total quantities are extensive properties. 

For one phase, N p - 1 + 2 intensive variable (from the phase rule) plus one extensive variable are needed to completely specify a stream. 

Note too that by defining the system we mean defining the intensive parameters - the masses or volumes of the various phases or components are irrelevant in this traditional use of the Phase Rule. A gram of NaCl solution is no different from a kilogram of solution in this sense. We will consider what happens if extensive parameters are included in 3.7.1. 

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... 

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... 

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. 

For the subject of this book, Langmuir s (1933) extension of the phase rule for adsorption under equilibrium and non-equilibrium conditions is placed at the beginning of the treatment of surface thermodynamics. In the study of heterogeneous equilibrium by Gibbs methods, the term phase is used for a homogeneous part of a system without regarding for quantity or form. 

The state variables (intensive, i.e., independent of the amount of matter, and extensive that are proportional to it) are these quantities that describe a system, for example, by means of the equation of state. For the discussions of phase diagrams it is important to know how many of the state variables one may change without going through a phase transition. The total number of variables required to describe a system by Eq 2.14 is N + 2, where N is number of components and 2 stands, e.g., for V and T. For a closed system the number of intensive variables (also known as a degree of freedom), /, is given by the Gibbs the phase rule ... 

Note that in the above formulation both extensive and intensive variables are included, in contrast with the phase rule, where only intensive variables are considered Typical Ny variables are temperatures and pressures, partial flows, specific enthalpies, chemical potential or fugacities of components, heat duty and mechanical work, recycle ratios and split fractions. Be careful not to include in the list interdependent variables. 

For each single-phase stream containing C components, a complete specification of intensive variables consists of C -1 mole fractions (or other concentration variables) plus temperature and pressure. This follows from the phase rule, which states that, for a single-phase system, the intensive variables are specified byC- + 2= C+ l variables. To this number can be added the total flow rate, an extensive variable. Finally, although the missing mole fractions are often treated implicitly, it is preferable for completeness to include these missing mole fractions in the list of stream variables and then to include in the list of equations the mole fraction constraint... 

The number of components (C) will be taken to be 3 latex particles, free polymer and dispersion medium. The number of degrees of fre om (F) is 2, the latex particles concentration being taken as an extensive variable. The two intensive variables are the temperature and the concentration of free polymer, both of which can be varied independently. In accordance with the Phase Rule, the number of phases is given by P=C-F+2 = 3, one of which is the vapour phase. This means that two liquid phases are possible. These will be separated by a macroscopic liquid/liquid interface. Note that no information is obtained from this relationship as to the dispersed or flocculated nature of the latex particles in either liquid phase, nor to the respective concentrations of each component in the two phases. 

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