Gibbs functions/rule

When a reversible transition from one monolayer phase to another can be observed in the 11/A isotherm (usually evidenced by a sharp discontinuity or plateau in the phase diagram), a two-dimensional version of the Gibbs phase rule (Gibbs, 1948) may be applied. The transition pressure for a phase change in one or both of the film components can be monitored as a function of film composition, with an ideally miscible system following the relation (12). A completely immiscible system will not follow this ideal law, but will... 

Fig. 8.1 Variation of the activity (logarithmic scale) or galvanic cell voltage as a function of the composition (schematic). The plateaux indicate multi-phase regions in which the activity is fixed according to Gibbs phase rule.

The well-known thermodynamic rule says that the two substances of different nature are miscible if the process brings about a gain in the value of the Gibbs function, AG, also called Gibbs energy or free enthalpy—that is, if AG > 0. The Gibbs function is connected with further basic thermodynamic quantities enthalpy and entropy, by the relation... 

This observation is the simplest case of the Gibbs phase rule (to be discussed in Section 7.1). It implies, for example, that pressure P = P(V, T) is uniquely specified when V and T are chosen, and similarly, that V = V(P, T) or T = T(P, V) are uniquely determined when the remaining two independent variables are specified. Such functional relationships between PVT properties are called equations of state. We can also include the quantity of gas (as measured, for example, in moles n) to express the equation of state more generally as... 

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 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 capabilities of MEIS and the models of kinetics and nonequilibrium thermodynamics were compared based on the theoretical analysis and concrete examples. The main MEIS advantage was shown to consist in simplicity of initial assumptions on the equilibrium of modeled processes, their possible description by using the autonomous differential equations and the monotonicity of characteristic thermodynamic functions. Simplicity of the assumptions and universality of the applied principles of equilibrium and extremality lead to the lack of need in special formalized descriptions that automatically satisfy the Gibbs phase rule, the Prigogine theorem, the Curie principle, and some other factors comparative simplicity of the applied mathematical apparatus (differential equations are replaced by algebraic and transcendent ones) and easiness of initial information preparation possibility of sufficiently complete consideration of specific features of the modeled phenomena. 

If a multiphase multicomponent system is to be at equilibrium (no change with time of the intensive variables) obviously temperature and pressure must be the same for all phases and also the chemical compositions (mole fractions of each constituent). In any given phase there are (C—1) independent mole fractions (their sum is unity by definition), so there are P.(C—1) composition variables involved and thus [P.(C—1) -1-2] intensive variables in total. But if chemical equUibrium in all phases simultaneously is to hold, the chemical potential of each constituent (a function of the composition) must be the same in each phase thus there are C.(P—1) independent constraints on the composition variables arising from the equilibrium condition (the chemical potential in one of the phases is used as the reference standard for the other phases). Thus F = [P.(C—1) -1-2] — [C.(P—1)] =C—P-F2. This is the famous Gibbs Phase Rule. 

After these introductory remarks, let us calculate the equilibrium concentrations of structural elements in a binary crystal MeX when the independent variables P, T, and the chemical potential pii of a component (Me or X) are given. From the Gibbs phase rule it is obvious that with P, r, and (or jUx) fixed, all independent variables are determined. That is, the concentrations of all defects can be calculated as functions of these independent variables. 

In Sec. 8.1.7, the Gibbs phase rule for a pure substance was written F = 3 — P. We now consider a system of more than one substance and more than one phase in an equilibrium state. The phase rule assumes the system is at thermal and mechanical equihbrium. We shall assume furthermore that in addition to the temperature and pressure, the only other state functions needed to describe the state are the amounts of the species in each phase this means for instance that surface effects are ignored. 

In eq 5.71, i) is a constant that depends on the particular equation of state used and Gm is an excess Gibbs function of mixing obtained from an activity coefficient model. Activity coefficients are usually obtained from measurements of (vapour-f liquid) equilibria at a pressure relatively low compared with the requirement of eq 5.67 for which p- ao the activity coefficients are tabulated, for example, those in the DECHEMA Chemistry Data Series. This distinction in pressure is particularly important because the excess molar Gibbs function of mixing, obtained from experiment and estimated from an equation of state, depends on pressure d(G /7 r)/d/)<0.002MPa for (methanol-f benzene) at a temperature of 373 K. Equation 5.71 does not satisfy the quadratic composition dependence required by the boundary condition of eq 5.3. However, equations 5.70 and 5.71 form the mixing rules that have been used to describe the (vapour + liquid) equilibria of non-ideal systems, such as (propanone + water), successfully in this particular case the three-parameter Non-Random Two Liquid (known by the acronym NRTL) activity-coefficient model was used for G and the value depends significantly on temperature to the extent that the model, while useful for correlation of data, cannot be used to extrapolate reliably to other temperatures. 

The second equation for the mixing rules is given by eq 5.69 in the absence of the divergence of the excess molar Gibbs function as the pressure tends to infinity it is unnecessary to impose the use of eq 5.69 and other choices, such as, b — ajRT could be used. 

The Wong-Sandler mixing rules extend the use of cubic equations of state to mixtures that were previously only correlated with activity-coefficient models. For many mixtures, the Gibbs-function model parameters in the equation of state could be taken to be independent of temperature, thereby allowing extrapolation of phase behaviour over wide ranges of temperature and pressure. For example, for (ethanol-h water) the activity-coefficient model reported in DECHEMA is at a pressure of 0.4 MPa and this model provides reasonable predictions of the phase boundaries at pressures up to 20 MPa. This means the method can be used with UNIversal Functional Activity Coefficient (known by the acronym UNIFAQ and other group-contribution methods to predict properties at elevated pressure. 

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