Freedoms phase rule

Degrees of Freedom, phase rule or, charge to batch still, mols... 

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. 

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. 

For a system such as discussed here, the Gibb s Phase Rule [59] applies and establishes the degrees of freedom for control and operation of the system at equilibrium. The number of independent variables that can be defined for a system are ... 

This is a consequence of the phase rule there are two components in three phases, hence the number of degrees of freedom is 2, so that when the temperature and pressure are fixed, the composition of each layer is also defined. 

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 system, therefore, is at equilibrium at a given temperature when the partial pressure of carbon dioxide present has the required fixed value. This result is confirmed by experiment which shows that there is a certain fixed dissociation pressure of carbon dioxide for each temperature. The same conclusion can be deduced from the application of phase rule. In this case, there are two components occurring in three phases hence F=2-3 + 2 = l, or the system has one degree of freedom. It may thus legitimately be concluded that the assumption made in applying the law of mass action to a heterogeneous system is justified, and hence that in such systems the active mass of a solid is constant. 

The phase rule as has been pointed out in the preceding paragraph deals with the behavior of heterogeneous systems at equilibria. It essentially includes three special terms. These are (i) number of phases in the system (P) (ii) the number of components for the system (C) and (iii) the number of degrees of freedom available to the system (F). A system for the present purpose could be any substance or combination of substances, which is set apart from its surroundings or other substances, such that its equilibrium state may be studied. The simplest way to express the rule in the form of an equation combining the three terms is as follows ... 

The coverage thus far has provided an account of the usefulness of phase rule to classify equilibria and to establish the number of independent variables or degrees or of freedom available in a specific situation. In the following paragraphs the equilibria used in mass transfer are analyzed in terms of phase rule in the case of leaching, drying and crystallisation. 

The difference between the number of variables involved in a design and the number of design relationships has been called the number of degrees of freedom similar to the use of the term in the phase rule. The number of variables in the system is analogous to the number of variables in a set of simultaneous equations, and the number of relationships analogous to the number of equations. The difference between the number of variables and equations is called the variance of the set of equations. 

A basic exposition of Gibbs phase rule is essential for understanding phase solubility analysis, and detailed presentations of theory are available [41,42]. In a system where none of the chemical species interact with each other, the number of independently variable factors (i.e., the number of degrees of freedom, F) in the system is given by... 

The most broadly recognized theorem of chemical thermodynamics is probably the phase rule derived by Gibbs in 1875 (see Guggenheim, 1967 Denbigh, 1971). Gibbs phase rule defines the number of pieces of information needed to determine the state, but not the extent, of a chemical system at equilibrium. The result is the number of degrees of freedom Np possessed by the system. 

The phase rule (Eqn. 3.52), then, predicts that our system has N = Ni degrees of freedom. In other words, given a constraint on the concentration or activity of each basis species, we could determine the system s equilibrium state. To constrain the governing equations, however, we need Nc pieces of information, somewhat more than the degrees of freedom predicted by the phase rule. 

If there is no liquid phase present, then from the phase rule there will be two degrees of freedom. Thus both the total pressure and the operating temperature can be fixed independently, and Pb = P — Pa (which must not exceed the vapour pressure of pure water if no liquid phase is to appear). 

The relationship between the number of degrees of freedom, F, defined as the number of intensive parameters that can be changed without changing the number phases in equilibrium, and the number of phases, Ph, and components, C, in the system is expressed through Gibbs phase rule ... 

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressures. Gibbs phase rule then becomes... 

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressure. The Gibbs phase rule for a ternary system at isobaric conditions is Ph + F = C + 1=4, and there are four phases present in an invariant equilibrium, three in univariant equilibria and two in divariant phase fields. Finally, three dimensions are needed to describe the stability field for the single phases e.g. temperature and two compositional terms. It is most convenient to measure composition in terms of mole fractions also for ternary systems. The sum of the mole fractions is unity thus, in a ternary system A-B-C ... 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

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... 

The phase rule is often used in the form t = c - p + 2 to ascertain the number of degrees of freedom of a system even when the concentration units in the aqueous phase are molal (m) or molar. This is not correct because the phase rule is derived 1n terms of mole fractions (X). Thus, an additional quantity, the total number of moles, is required to convert X into m. This is illustrated by equations below which we shall find useful later on. 

The phase rule is expressed in terms of p, the number of phases in the system C, the number of components and F, the number of degrees of freedom or the variance of the system. 

The Phase Rule. The number of degrees of freedom is the difference between the number of variables needed to describe the system and the number of independent relationships or constraints among those variables ... 

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