Gibbs Duhem equation defined

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. 

The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

An alternative method of integrating the Gibbs-Duhem equation was developed by Darken and Gurry [10]. In order to calculate the integral more accurately, a new function, a, defined as... 

In Chapters 16 and 17, we developed procedures for defining standard states for nonelectrolyte solutes and for determining the numeric values of the corresponding activities and activity coefficients from experimental measurements. The activity of the solute is defined by Equation (16.1) and by either Equation (16.3) or Equation (16.4) for the hypothetical unit mole fraction standard state (X2° = 1) or the hypothetical 1-molal standard state (m = 1), respectively. The activity of the solute is obtained from the activity of the solvent by use of the Gibbs-Duhem equation, as in Section 17.5. When the solute activity is plotted against the appropriate composition variable, the portion of the resulting curve in the dilute region in which the solute follows Henry s law is extrapolated to X2 = 1 or (m2/m°) = 1 to find the standard state. 

Equation (1.16) is one example of the Gibbs-Duhem equation, which is one of the most useful formulae in thermodynamics. Thus, the partial molal quantity defined as the quantity satisfying the additive property is easily understandable. 

Thus the Gibbs-Duhem equation represents one of the 2D thermodynamic potentials that can be defined for a system, but this thermodynamic potential is equal to zero. It should be emphasized that there is a single Gibbs-Duhem equation for a one-phase system and that it can be derived from U, H, A, or G. 

These four Legendre transforms introduce the chemical potential as a natural variable. The last thermodynamic potential U T, P, /<] defined in equation 2.6-6 is equal to zero because it is the complete Legendre transform for the system, and this Legendre transform leads to the Gibbs-Duhem equation for the system. 

If the 5rif are of the form Xidn-j-, the Gibbs-Duhem equation would be trivially satisfied since dixj = 0 for all fs. Now suppose the Stij sum to some nonzero dtiy. One can define new 6n/ s as the original ones minus jc/6/It. and these would result in the same 6ju/ S, but their sum would be zero. It follows that one only needs to make sure that the Gibbs-Duhem equation is satisfied for all sets of drij that satisfy Ebtii = 0. If Eq. (166) is to hold, one has... 

The molarity of components 2, 3 and 4, defined by Hill to be Ni/Ni, can be obtained by applying the Gibbs-Duhem equation... 

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... 

The thermodynamic theory of the ideally polarised electrode has been extensively reviewed in the past few decades [1-5], and the relationship with the ideally non-polarisable interface has been derived in an elegant treatment by Parsons [6]. The starting point in all derivations is the Gibbs-Duhem equation which defines the relationship between the extensive thermodynamic variables. For a bulk phase this has the form ... 

The temperature on the coexistence curves corresponds to the temperature at which the transition from one phase to another takes place at a given pressure. Thus, if we obtain an explicit relation between the pressure and the temperature that defines the coexistence curve, we can know how the boiling point or freezing point changes with pressure. Using the condition for equilibrium (7.1.2), we can arrive at a more explicit expression for the coexistence curve. Let us consider two phases denoted by 1 and 2. Using the Gibbs-Duhem equation, dyL = -Sjn dT + Vm dp, one can derive a differential relation between p and Tof the system as follows. From (7.1.3) it is clear that for a component k, d i[ = dpi- Therefore we have the equality... 

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... 

This is the Gibbs-Duhem equation of the ensemble. When the extensive variables per small system are defined by X = X,/N, (5.8) and (5.9) are rewritten as... 

Define a partial molar property and describe its role in determining the properties of mixtures. Calculate the value of a partial molar property for a species in a mixture from analytical and graphical methods. Apply the Gibbs-Duhem equation to relate the partial molar properties of different species. 

Additionally, the pure species property, ki, is defined as the value of that property of species i as it exists as a pure species at the same T and P of the mixture. Values of a partial molar property for a species in a mixture can be calculated from an analytical expression by applying Equation (6.15) and by graphical methods, as illustrated in Figure 6.13. In the case of infinite dilution, species i becomes so dilute that a molecule of species i will not have any like species with which it interacts rather, it will interact only with unlike species. Additionally, partial molar properties of different species in a mixture can be related to one another by the Gibbs-Duhem equation ... 

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). 

They define interfacial energy starting from the equation of Gibbs Duhem, which is then written for any transformation, as... 

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