Gibbs-Duhem equation homogeneous phase

In this section, we wish to derive the Gibbs-Duhem equation, the fundamental relationship between the allowed variations dRt of the intensive properties of a homogeneous (singlephase) system. Paradoxically, this relationship (which underlies the entire theory of phase equilibria to be developed in Chapter 7) is discovered by considering the fundamental nature of extensive properties Xu as well as the intrinsic scaling property of the fundamental equation U = U(S, V, n, n2,. .., nc) that derives from the extensive nature of U and its Gibbs-space arguments. 

The Gibbs-Duhem equation for a homogenous region in any phase is given by Equation (14.14). This equation, expressed in molar quantities, is... 

These two relationships are analogous to the Gibbs-Duhem equation for a homogeneous phase according to which... 

The relation (5.65) known as Gibbs-Duhem equation is valid for any thermodynamic property in a homogeneous phase. At T, P constant it becomes ... 

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. 

If in the course of a phase transition, there is a discontinuity in the entropy, other variables also show a discontinuity. For a homogeneous system, a phase, with one component, the Gibbs - Duhem equation holds ... 

This is essentially the equation widely used in conventional descriptions where the adsorption complexes AR and BR are simply A and B in the adsorbed state.By combination of these equations with a Gibbs-Duhem equation formulated for the sorption phase treated as a homogeneous solution, expressions are derived for -yar, 7br, In and In Kc for three cases (a) adsorption of a single substance (presumably from the vapour), (6) adsorption from a binary mixture in which a certain constant number of adsorption sites remain empty, and (c) adsorption from a binary mixture which fully saturates the surface (cr = 0). 

The only guidance comes from the Gibbs-Duhem equation (2 83). For a homogeneous phase of two components A and B this reads... 

The next step is to make clear the Gibbs adsorption amount, using the above relative adsorption. Recall (8.27), in which the interfacial tension is a function of i -H 2 independent variables. However, the Gibbs phase rule permits only i independent variables for two phases including i components. Therefore, the problem is how to reduce the number of intensive variables by two while keeping thermodynamical consistency. The Gibbs-Duhem equations for two homogeneous phases a and respectively, are... 

We have now reduced the degrees of freedom by one, employing (8.29) with respect to the volume. On the other hand, from (8.34) and (8.35), the Gibbs-Duhem equations per unit volume for homogeneous bulk phases become the following ... 

Figure 5.5. Homogeneous phase equilibrium in the system ice water -water vapour, shown schematically. At a given temperature T, the stable phase is the phase with the lowest free energy G. According to the Gibbs-Duhem equation, dG/dT)p = —S at constant pressure. As S s) < S ) < S(g), the increasing temperature will successively favour the states (s), ( ) and (g).

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