Phase rule counting variables

How many variables are required to specify a process stream completely To characterize a process streani we can select among temperature, pressure, total flow, overall mole fractions, phase fractions, mole fractions, total enthalpy, phase enthalpies, and so on. How many variables are needed to fix completely the state of the stream if we assume that phase and chemical equilibrium exist in the stream A review of the phase rule shows that for a single phase, the degrees of freedom associated with a stream are equal toC— 1+2 = C+1 intensive variables. A little thought will indicate that in the absence of reaction, you must in addition add the total flow rate of a stream to the count to specify the stream completely. The variables often associated with a stream are / , T, the total molar (mass) flow, and C — 1 overall mole (mass) fractions for the components. An equivalent set would be p, T, -and-the-individual component molar flow rates.-You cannot specify-G-eompositionsy and the molar flow rate as one composition is redundant. 

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... 

For a system of only one component it is possible to derive, as was done in Table 12.1, the consequences of the phase rule quite easily. The equilibria are readily represented by lines and their intersections in a two-dimensional diagram of the type we have used in this chapter. It hardly seems necessary to have the phase rule for such a situation. However, if the system has two components, then three variables are required and the phase diagram consists of surfaces and their intersections in three dimensions. If three components are present, surfaces in a fourdimensional space are required. Visualization of the entire situation is difficult in three dimensions, impossible for four or more dimensions. Yet the phase rule, with exquisite simplicity, expresses the limitations that are placed on the intersections of the surfaces in these multidimensional spaces. For this reason, the Gibbs phase rule is counted among the truly great generalizations of physical science. 

The phase rule relates only to intensive variables it makes no statements about the relative amounts of individual phases. It makes no statements about the values of the intensive variables, but only uses them to count phases. 

The phase rule is derived by counting the variables we can specify for an individual phase, and then subtracting the number of relations that must exist between those... 

The number of independent chemical reactions we need to specify can be obtained by using the Gibbs phase rule for reacting systems. The phase rule for reacting systems is obtained by counting the total number of variables in the system and making sure we have the same number of independent equations. It is accomphshed in much the same way we accounted for variables for nonreacting systems in Example 6.17. 

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