Gibbs-Duhem relations

The Gibbs-Duhem relation gives the interrelationship of the chemical potentials of the components in a binary compound. 

A binary oxide has the general formula MO. Its equilibrium with its ions can be represented as  

In terms of the electrochemical potential, we can write Equation 18.68 for this system. 

is the number of charges on the ion i, e is the electronic charge, and p is the electric potential. The electrochemical potential of the oxide is constant. Therefore, 

2 Show that the Gibbs-Duhem relationship holds well. 

By means of the Gibbs-Duhem relation to check the mutual consistency of freezing point measurements and e.m.f. measurements on aqueous solutions of HCl. 

The freezing point measurements of Randall and Vanselow (J. Amer. Chem. Soc. 1924,46, 2418) have been quoted and analjrsed in problem 79. 

The freezing point measurements have been analji ed in problem 

79 and the osmotic coefficient 9 (on the molality scale) has been fitted by the formula 

We shall analyse the e.m.f. measurements in a manner precisely the same as in problem 81 and shall fit the activity coefficient y (on the molality scale) of HCl by the formula 

The thermodynamic state of a system is defined by its state variables (T,p,..la/s). Because the state variables are not independent of each other, specification of only a limited number, given by the Gibbs-Duhem relation, is sufficient to determine the thermodynamic state. 

The Gibbs-Duhem relation is obtained by equating the differential of a function of state (as defined by Equations 3.10 through 3.13) to the corresponding differential expressions (Equations 3.8 and 3.14 through 3.16). 

Equation 3.82 or 3.83 is the Gibbs-Duhem relation. It indicates the number of variables that can be independently varied for a system. At constant T and p, for a homogeneous system containing k components the chemical potentials of (k - 1) components can be chosen at will. 

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When both solutes are present in large amounts, i.e. greater than about 1 at. 7o of each, no simple theoretical treatment is available to predict their mutual effects on thermodynamic properties. In this case, recourse must be made to the various solutions of the ternary Gibbs-Duhem relation... 

The argument required here is, of course, the equivalent of the Gibbs-Duhem relation. 

The fundamental thermodynamic equation relating activity coefficients and composition is the Gibbs-Duhem relation which may be expressed as ... 

Taking the partial derivative with respect to the mole fractions in the micelle (x ) and using the Gibbs-Duhem relation to eliminate some of the resulting terms gives... 

It is known, however, from the Gibbs-Duhem relation that... 

For example, we may choose to as the average volume velocity, to = (c, V,) v,-. In more general terms, we may define to by X Pi vi > with Y Pi = 1 The s are weighting factors. If we formulate Eqn. (4.72) for two different reference velocities, to and to", and take into account the partial molar volumes (V,) which are not independent of each other (Gibbs-Duhem relation), we obtain after some algebraic rearrangements [H. Schmalzried (1981)] the quite general expression... 

For the external, observer, jA +/B = 0. From this condition and the Gibbs-Duhem relation, the local lattice velocity becomes... 

Assumption of local equilibrium permits the Gibbs-Duhem relation to be written... 

Under the assumption of local equilibrium, the Gibbs-Duhem relation applies, which places an additional constraint on chemical potential changes in Eq. 6.7 and implies that only IV — 1 of the m can vary independently ... 

In addition, because of the Gibbs-Duhem relation, c a dp a + cb d/xs = 0, the chemical potential gradients are interdependent ... 

Using all these variables the relations, which form the starting point for the further calculations, can be constructed. These relations are the energy density , the dissipation function R, the Gibbs-relation and the Gibbs-Duhem relation. To illustrate the idea of our model we split up e and R into several parts according to the different origin of the variables ... 

Equation (15) is called the general Gibbs—Duhem relation. Because it tells us that there is a relation between the partial molar quantities of a solution, we will learn how to use it to determine a Xt when all other X/ il have been determined. (In a two-component system, knowing Asolvent determines Asolute.) This type of relationship is required by the phase rule because, at constant T, P, and c components, a single-phase system has only c — 1 degrees of freedom. 

In a binary solution, the Gibbs-Duhem relation [Eq. (15)] determines the variation of a partial molar property of one component in terms of the variation of the partial molar quantity of the other component. This relation is useful for obtaining chemical potentials in binary solutions when only one of the components has a measurable vapor pressure. Applying Eq. (15) to chemical potentials in a binary solution,... 

Show that the chemical potentials for the ideal solution satisfy the Gibbs-Duhem relation [Eq. (29)]. 

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... 

The advantage of this choice of the X dependence for the correlation functions and the bridge function relies on the fact that the excess chemical potential, and the one-particle bridge function as well, can be determined unambiguously in terms of B(r) as soon as n and m are known. To address this problem, the authors proposed to determine the couple of parameters (n m) in using the Gibbs-Duhem relation. This amounts to obtaining values of n and m from Eq. (87), which is considered as supplementary thermodynamic consistency condition that have to be fulfiled. 

Gibbs s analytical proof of the adsorption formula is, mutatis mutandis, analogous to the analytical deduction of the Gibbs-Duhem relation both depend on the integration of the formula for the increment in energy, followed by differentiation and comparison of the result with the original formula. 

After applying the Gibbs-Duhem relations to the differential of this equation one is left with ... 

Equation (1.113) is called the Gibbs-Duhem relation, which becomes particularly useful at isobaric and isothermal conditions, and when the force and electrical work are neglected, we have... 

Using the molar-specific volume (v = I IN) and molar-specific entropy (5 = SIN), a simplified version of the Gibbs-Duhem relation results... 

For a binary vapor-liquid system, the Gibbs-Duhem relations are... 

Thermodynamic correction factor Y is defined using the Gibbs-Duhem relation... 

For a system in mechanical equilibrium in which the pressure gradient is balanced by the mass forces, the Gibbs-Duhem relation becomes... 

Prove that for the internal energy function the following Gibbs-Duhem relation must hold Ed(l/T) + Vd(P/T)... 

We now attend to the third term above on the right, beginning with the Gibbs-Duhem relation at constant T and P Equation (1.22.26) reads... 

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