Gibbs-Duhem equations chemical equilibrium

Chapter 4 presents the Third Law, demonstrates its usefulness in generating absolute entropies, and describes its implications and limitations in real systems. Chapter 5 develops the concept of the chemical potential and its importance as a criterion for equilibrium. Partial molar properties are defined and described, and their relationship through the Gibbs-Duhem equation is presented. 

As usual, our goal is to find the minimum of G( ) in order to determine the equilibrium position ( = eq) of the chemical reaction at constant T and P. From (6.10c) [cf. the Gibbs -Duhem equation (6.36a)], the differential dG under these conditions is simply... 

As a first example, consider a pure liquid in equilibrium with its vapor. Because I wish to focus attention on the liquid/gas interface to the exclusion of adsorption effects at solid boundaries, I shall suppose the containing vessel to be chemically inert. The Gibbs-Duhem equation for the system is then... 

This is the Gibbs equation, which is particularly important for understanding phase equilibria. A related expression, called the Gibbs-Duhem equation, states that at equilibrium the change of chemical potential of one component results in the change of the chemical potentials of all other components... 

If one or more chemical reactions are at equilibrium within the system, we can still set up the set of Gibbs-Duhem equations in terms of the components. On the other hand, we can write them in terms of the species present in each phase. In this case the mole numbers of the species are not all independent, but are subject to the condition of mass balance and to the condition that , vtpt must be equal to zero for each independent chemical reaction. When these conditions are substituted into the Gibbs-Duhem equations in terms of species, the resultant equations are the Gibbs-Duhem equations in terms of components. Again, from a study of such sets of equations we can easily determine the number of degrees of freedom and can determine the mathematical relationships between these degrees of freedom. 

The common characteristics of phase transitions are that the Gibbs energy is continuous. Although the conditions of equilibrium and the continuity of the Gibbs energy demand that the chemical potential must be the same in the two phases at a transition point, the molar entropies and the molar volumes are not. If, then, we have two such phases in equilibrium, we have a set of two Gibbs-Duhem equations, the solution of which gives the Clapeyron equation (Eq. (5.73))... 

Gibbs-Duhem Equation and the Phase Rule at Chemical Equilibrium... 

GIBBS-DUHEM EQUATION AND THE PHASE RULE AT CHEMICAL EQUILIBRIUM... 

This is the Gibbs-Duhem equation, which relates the variation in temperature, pressure, and chemical potentials of the C components in the solution. Of these C + 2 variables, only C + 1 can vary independently. The Gibbs-Duhem equation has many applications, one of which is providing the basis for developing phase equilibrium relationships. 

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. 

Summary. Section 2.4 illustrates the extension of rational thermodynamics methodology on mixtures with chemical reaction(s) using a very simple model of two-component uniform mixture. The composition variable(s) enters the constitutive equations, cf. (2.76)-(2.79). In a uniform mixture, the classical chemical thermodynamics was obtained, i.e., its validity also in nonequilibrium covered by this model was demonstrated, cf. e.g., (2.82), (2.83), (2.85), (2.87), (2.88). Traditional quantities known from the equilibrium chemical thermodynamics may be thus introduced and used out of equilibrium—affinity by (2.89), chemical potential by (2.93) or (2.100), or Gibbs energy by (2.97). Gibbs and Gibbs-Duhem equations also remain valid,... 

These and all previous results of thermodynamic mixture which also fulfil Gibbs-Duhem equations (4.263) show the complete agreement with the classical thermodynamic of mixtures but moreover all these relations are valid much more generally. Namely, they are valid in this material model—linear fluid mixture—in all processes whether equilibrium or not. Linear irreversible thermodynamics [1-4], which studies the same model, postulates this agreement as the principle of local equilibrium. Here in rational thermodynamics, this property is proved in this special model and it cannot be expected to be valid in a more general model. We stress the difference in the cases when (4.184) is not valid—e.g. in a chemically reacting mixture out of equilibrium—the thermodynamic pressures P, Pa need not be the same as the measured pressure (as e.g. X =i Pa) therefore applications of these thermodynamic... 

The origin of the chemical signal can be expressed in thermodynamic terms. At equilibrium the number of moles n of all species and their chemical potentials // in a phase (e.g. in a chemically selective layer) are related through the Gibbs-Duhem equation which says that if a new species enters the organic layer the chemical potentials of all species in that layer must change. These include the change of the electrochemical potential of the electron - the Fermi level and therefore the electron work function. 

Equilibrium is characterized by the equality of the chemical potential. Nonequilibrium is therefore induced by the gradient of the chemical potential. In the thermodynamics of irreversible processes, chemical potential gradient is the fundamentally correct driving force for diffusion. According to the Gibbs-Duhem equation (2.3-5), we have ... 

Chemical thermodynamics was developed by Pierre Maurice Martin Duhem (Paris, lo June i86i-Cabrespine, 14 September 1916), professor of theoretical physics in Bordeaux, who published on the equations for heats of solution and dilution which had been deduced by Kirchhoff, on the liquefaction of gaseous mixtures, eutectic and transition points for binary mixtures which can form mixed crystals, and a long series of papers on false equilibrium of doubtful value. He published some books on thermodynamics and later on the history of science. An important general thermodynamic equation (Gibbs-Duhem equation) was deduced independently by Gibbs and Duhem. ... 

We further impose an experimentally accessible condition that the clathrate hydrate is in equilibrium with a fluid mixture of guest and water. This is realized by requiring that the chemical potentials of water and guest in the fluid phase are equal to those in the clathrate and the pressure in the fluid is equal to p in Eq. (7). Then, A fig in Eq. (8) is replaced in terms of dp and dJfrom the Gibbs-Duhem equations as... 

For the composition of a system to vary, component particles should be exchanged with its surrounding, and the exchange is driven by a difference in component chemical potentials across the boundary. Once component chemical potential distributions are rendered uniform, particle exchange ceases and then the system is said to be in external equilibrium. For the case of binary MOi+g, the number of composition variables is only one (8), and only one of the two component chemical potentials or activities um and aai can be varied independently at given temperature T and pressure P due to the Gibbs-Duhem equation ... 

The Gibbs-Duhem equation shows that as the cocenn-tration of one species of a mixture approaches zero, its chemical potential approaches minus infinity. This makes the chemical potential (partial molar Gibbs energy) an inconvenient working property for equilibrium calculations. For this reason we use the fugacity (Chapter 7) instead. 

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