Gibbs-Duhem theorem

Derive Eq. (2.107) using this relationship and the Gibbs-Duhem theorem,... 

The derivation of the phase rule is based upon an elementary theorem of algebra. This theorem states that the number of variables to which arbitrary values can be assigned for any set of variables related by a set of simultaneous, independent equations is equal to the difference between the number of variables and the number of equations. Consider a heterogenous system having P phases and composed of C components. We have one Gibbs-Duhem equation of each phase, so we have the set of equations... 

Thus, another approach consists in selecting some boundary conditions and properties. It is obvious that all exact correlation functions must satisfy and incorporate them in the closure expressions at the outset, so that the resulting correlations and properties are consistent with these criteria. These criteria have to include the class of Zero-Separation Theorems (ZSTs) [71,72] on the cavity function v(r), the indirect correlation function y(r) and the bridge function B(r) at zero separation (r = 0). As will be seen, this concept is necessary to treat various problems for open systems, such as phase equilibria. For example, the calculation of the excess chemical potential fi(iex is much more difficult to achieve than the calculation of usual thermodynamic properties since one of the constraints it has to satisfy is the Gibbs-Duhem relation... 

Galvanostatic transients, 66, 357, 359, 394 Gas-diflusion electrodes, 484 Gauss theorem. 192, 339 Gibbs adsorption isotherm, 228 Gibbs-Duhem equation, 235... 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

Since Zi is homogeneous in the zeroth degree in ni, n2,..., etc., Euler s theorem (Chapter 2) gives the Gibbs-Duhem equation ... 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

Intercepts and Common Tangents to Agmixing ts. Composition in Binary Mixtures. Euler s integral theorem and the Gibbs-Duhem equation provide the tools to obtain expressions for Agmixing and (9 Agmixing/9y2)r,/) in binary mixtures. This information allows one to evaluate the tangent at any mixture composition via the point-slope formula. For example, if i i = and p,2 = M2 when the mole fraction of component 2 is y, then equations (29-73) and (29-76) yield ... 

Euler s theorem and the Gibbs-Duhem relation are useful in making calculations. 

Section 4.6 Euler s Theorem and the Gibbs-Duhem Relation... 

Spear F. S. (1988) The Gibbs method and Duhem s theorem the quantitative relationships among P, T, chemical potential, phase composition and reaction progress in igneous and metamorphic systems. Contrib. Mineral. Petrol 99, 249-256. 

In the last chapter we established two powerful general theorems, those of Gibbs and Duhem, relating to heterogeneous systems. We shall now consider in more detail the quantitative behaviour of some simple systems, beginning with a study of the phase changes of a pure substance. A study of more complex heterogeneous systems will follow in later chapters after we have discussed the thermodynamic conditions of stability. 

The above equations enable us to derive two important theorems first enunciated by Gibbs, and later rediscovered by Konovalow and Duhem. 

Duhem s theorem is applied to closed systems at equilibrium, when both intensive and extensive parameters are known, i.e., for systems of totally defined state. Both extensive and intensive parameters may be independent. However, their interrelation is defined by Gibbs phase rule. At C = 0 both parameters must be extensive, and at C = 1 at least one of them must be extensive. 

We start in 9.1 by giving prescriptions for determining the number of properties needed to identify the thermodynamic state in multicomponent mixtures. Those prescriptions include Duhem s theorem and the Gibbs phase rule as special cases. The required number of properties determines the dimensionality of the state diagram needed to represent phase behavior. Then in 9.2 we summarize some features of pure-component diagrams that have not been discussed in earlier chapters. 

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