And the Gibbs-Duhem equation

B. Constant-Pressure Activity Coefficients and the Gibbs-Duhem Equation.. 158... 

Equation 8.5-3 indicates that the number of natural variables for the system is 6, D = 6. Thus the number D of natural variables is the same for G and G, as expected, since the Legendre transform interchanges conjugate variables. The criterion for equilibrium is dG 0 at constant T,P,ncAoi, ncA(3, /icC, and The Gibbs-Duhem equations are the same as equations 8.4-8 and 8.4-9, and so the number of independent intensive variables is not changed. Equation 8.5-3 yields the same membrane equations (8.4-13 and 8.4-14) derived in the preceding section. 

However, when M is given as a function of composition at constant P and the Gibbs/Duhem equation, Eq. (11.8), is... 

Using the forces and flows identified in Eq. (7.1), and the Gibbs-Duhem equation for an n-component system at constant temperature and pressure, we obtain... 

Henry s law applies to a species as it approaches infinite dilution in a binary solution, and the Gibbs/Duhem equation insures validity of the Lewis/Randall rule for the other species as it approaches purity. 

The entropy production term has been written as a sum of products of fluxes and forces. However, there are only q—1 independent mass fluxes js, and the Gibbs-Duhem equation (2.451) tells us that there are only q — 1 independent forces as well [5]. 

Thus, the interrelationships provided by Eqs. 8.2-8 through 8.2-15 are really restrictions on the mixture equation of state. As such, these equations are important in minimizing the amount of experimental data necessary in evaluating the thermodynamic, properties of mixtures, in simplifying the description of multicomponent systems, and in testing the consistency of certain types of experimental data (see Chapter 10). Later in this chapter we show how the equations of change for mixtures and the Gibbs-Duhem equations provide a basis for the experimental determination of partial molar properties. 

AH and AV are molar enthalpy and molar volume of mixing, respectively. The heat of mixing is negligible in most cases. In an isobaric system the second term vanishes, and the Gibbs-Duhem equation takes the form ... 

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 ... 

From these relations and the Gibbs-Duhem equation we also have... 

The densities in Eqns (3.1) and (3.2) are dependent on the locally well-defined temperature. Also, the classical thermodynamic equations such as the Gibbs and the Gibbs-Duhem equations... 

GENERAL PROPERTIES OF SOLUTIONS AND THE GIBBS-DUHEM EQUATION... 

From the internal energy density uix) and entropy density s(x), we obtain the local variables of (du/ds)yjj = T(x), -(duldV)s j = P, and (ds/dNi )u = -tx x)IT x). The densities in Eqs. (3.1) and (3.2) are dependent on the locally well-defined temperature. In a nonequilibrium system, therefore, the total entropy S is generally not a function of the total entropy U and the total volume V. Also, the classical thermodynamic equations such as the Gibbs and the Gibbs-Duhem equations... 

Classical thermodynamics based on the surface Laplace and the Gibbs-Duhem equations [43] have derived that Kq obeys the following relations [41, 44] ... 

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