Phase rule variables

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 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. 

The equations that may be written connecting the phase-rule variables are ... 

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. 

The two degrees of freedom for this system may be satisfied by setting T and P, or T and t/j, or P and a-j, or Xi and i/i, and so on, at fixed values. Thus, for equilibrium at a particular T and P, this state (if possible at all) exists only at one liquid and one vapor composition. Once the two degrees of freedom are used up, no further specification is possible that would restrict the phase-rule variables. For example, one cannot m addition require that the system form an azeotrope (assuming this possible), for this requires Xi = i/i, an equation not taken into account in the derivation of the phase rule. Thus, the requirement that the system form an azeotrope imposes a special constraint and reduces the number of degrees of freedom to one. 

The intensive state of a system at equilibrium is established when its temperature, pressure, and the compositions of all phases are fixed. These are therefore phase-rule variables, but they are not all independent. The phase rule gives the number of variables from this set which must be arbitrarily specified to fix all remaining phase-rule variables. 

The phase-rule variables are intensive properties, which are independent of the extent of the system and of the individual phases. Thus the phase rule gives the same information for a large system as for a small one and for different relative amounts of the phases present. Moreover, the only compositions that are phase-rule variables are those of the individual phases. Overall or total compositions are not phase-rule variables when more than one phase is present. 

In a natural gasoline fractionation system there are usually six chemical species present in appreciable quantities methane, ethane, propane, isobutane, n-butane, and n-pentane. A mixture of these species is placed in a closed vessel from which all air has been removed. If the temperature and pressure are fixed so that both liquid and vapor phases exist at equilibrium, how many additional phase-rule variables must be chosen to fix the compositions of both phases ... 

The phase rule for nonreacting systems, presented without proof in Sec. 2.8 results from application of a rule of algebra. The number of phase-rule variable which must be arbitrarily specified in order to fix the intensive state of a syste at equilibrium, called the degrees of freedom F, is the difference between t total number of phase-rule variables and the number of independent equatio that can be written connecting these variables. 

The phase-equilibrium equations that may be written connecting the phase rule variables arc given by Eqs. (10.3) or Eqs. (11.29) ... 

The phase-rule variables are temperature, pressure, and the phase compositions. The composition variables are either the weight or mole fractions of the species in a phase, and they must sum to unity for each phase. Thus fixing the mole fraction of the water in the liquid phase automatically fixes the mole fraction of the alcohol. These two compositions cannot both be arbitrarily specified. 

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... 

When JV = 2, the phase rule becomes F = 4 - it. Since there must be at least one phase (it = 1), the maximum number of phase-rule variables which must be specified to fix the intensive state of the system is three namely, P, T, and one mole (or mass) fraction. All equilibrium states of the system can therefore be... 

Thus, one specifies either T or P and either the liquid-phase or the vapor-phase composition, fixing 1 + (N - 1) or N phase-rule variables, exactly the number required by the phase rule for vapor/liquid equilibrium. All of these calculations require iterative schemes because of the complex functionality implicit in Eqs. (12.1) and (12.2). In particular, we have the following functional relationships for low-pressure VLE ... 

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... 

Three species, but only two phases, are present in this case, solid NH4CI and a gas mixture of NH3 and HC1. In addition, there is a special constraint, because the requirement that the system be formed by the decomposition of NH4C1 means that the gas phase is equimolar in NH3 and HC1. Thus a special equation ytm, = yHci (=0.5), connecting the phase-rule variables can be written. Application of Eq. (15.34) gives... 

This result means that one is free to specify four phase-rule variables, for example, T, P, and two mole fractions, in an equilibrium mixture of these five chemical species, provided that nothing else is arbitrarily set. In other words, there can be no special constraints, such as the specification that the system be prepared from given amounts of CH4 and H20. This imposes special constraints through material balances that... 

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... 

The phase-equilibrium and ehemieal-reaetion-equilibrium equations are the only ones considered in the foregoing treatment as interrelating the phase-rule variables. However, in certain situations special constraints may be plaeed on tlie system that allow additional equations to be written over and above tliose eonsidered in the development of Eq. (13.36). If die number of equations resulting from speeial eonstraints is s, theiiEq. (13.36) must be modified to take accoimt of these 5 additional equations. The still more general fomi of the phase rule that results is ... 

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. 

How can we reconcile this apparent paradox with our previous statement Since the phase rule is concerned with intensive properties only, the following are phase-rule variables in the ideal gas law ... 

For a system containing N chemical species distributed at equilibrium among K phases, the phase rule variables are T and P, presumed uniform throughout the system, and N—1 mole fractions in each... 

The equilibrium equations that may be written express chemical potentials or fugacities as functions of T, P, and the phase compositions, the phase rule variables ... 

Example 3 Detv and Bubble Point Calculations As indicated by Example 2a, a binary system in vapor/liquid equilibrium has 2 degrees of freedom. Thus of the four phase rule variables T, P, x, and t/i, two must be fixed to allow calculation of the other two, regardless of the formulation of the equilibrium equations. Modified Raoults law [Eq. (4-307)] may therefore be applied to the calculation of any pair of phase rule variables, given the other two. 

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