Equation first Gibbs-Duhem

According to the first Gibbs-Duhem equation, free enthalpy of the entire solution is equal... 

It is strictly for convenience that certain conventions have been adopted in the choice of a standard-state fugacity. These conventions, in turn, result from two important considerations (a) the necessity for an unambiguous thermodynamic treatment of noncondensable components in liquid solutions, and (b) the relation between activity coefficients given by the Gibbs-Duhem equation. The first of these considerations leads to a normalization for activity coefficients for nonoondensable components which is different from that used for condensable components, and the second leads to the definition and use of adjusted or pressure-independent activity coefficients. These considerations and their consequences are discussed in the following paragraphs. 

Trustworthy thermodynamic data for metal solutions have been very scarce until recently,25 and even now they are accumulating only slowly because of the severe experimental difficulties associated with their measurement. Thermodynamic activities of the component of a metallic solution may be measured by high-temperature galvanic cells,44 by the measurement of the vapor pressure of the individual components, or by equilibration of the metal system with a mixture of gases able to interact with one of the components in the metal.26 Usually, the activity of only one of the components in a binary metallic solution can be directly measured the activity of the other is calculated via the Gibbs-Duhem equation if the activity of the first has been measured over a sufficiently extensive range of composition. 

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. 

We derived Equation (4.30) from first principles, using pure mathematics. An alternative approach is to prepare a similar equation algebraically. The result of the algebraic derivation is the Gibbs-Duhem equation ... 

We have already obtained the first Maxwell relation (Equation (4.37)) by comparing the Gibbs-Duhem equation with the total differential ... 

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... 

The change of the interfacial tension can be calculated with the help of the Gibbs-Duhem equation even when a potential is applied. In order to use the equation, we first need to find out which molecular species are present. Evidently, only those which are free to move are of interest. In the electrolyte we have the dissolved ions. In the metal the electrons can move and have to be considered. 

Finally, consider a two-phase, two-component system in which the two phases are separated by an adiabatic membrane that is permeable only to the first component. In this case we know that the temperatures of the two phases are not necessarily the same, and that the chemical potential of the second component is not the same in the two phases. The two Gibbs-Duhem equations for this system are... 

First we consider the binary systems when no inert gas is used. When only one of the components is volatile, the intensive variables of the system are the temperature, the pressure, and the mole fraction of one of the components in the liquid phase. When the temperature has been chosen, the pressure must be determined as a function of the mole fraction. When both components are volatile, the mole fraction of one of the components in the gas phase is an additional variable. At constant temperature the relation between two of the three variables Pu x1 and yt must be determined experimentally the values of the third variable might then be calculated by use of the Gibbs-Duhem equations. The particular equations for this case are... 

The three Gibbs-Duhem equations can also be used to determine the change in the mole fraction of one of the phases with temperature or with pressure. We choose here to determine dx i/dT. We first eliminate dfi2 from the three equations, to obtain... 

In the first example we assume that the species in the gas phase are A, B, and A2B, where A and B represent the same molecular entities as the components 1 and 2, respectively. The Gibbs-Duhem equations are... 

There are six variables and five equations, and the system is univariant. We wish to determine the change of the partial pressure of the species A2B with change of composition of the condensed phase that is, the derivative (din PAB/dx1)T- In the solution of the set of Gibbs-Duhem equations, we therefore must retain nAB and or expressions equivalent to these quantities. The solution can be obtained by first eliminating n2 and fiB from Equations (11.167) and (11.168) by use of Equations (11.169) and (11.171). The expressions... 

On eliminating the chemical potential of the first component between the two Gibbs-Duhem equations applicable to the two phases, we obtain... 

The first term on the right-hand side f(° is the standard chemical potential of component i, the second term is the ideal mixture contribution, and the third term is the nonideality caused by the hard-sphere interactions. The chemical potential of the hard-sphere droplet (component g) is obtained using the Gibbs—Duhem equation... 

Although the correlations provided by the Margules equations for the two sets of VLE data presented here are satisfactory, they are not perfect. The two possible reasons are, first, that the Margules equations are not precisely suited to the data set second, that the data themselves are systematically in error such that they do not conform to the requirements of the Gibbs/ Duhem equation. 

Now the Gibbs-Duhem equation must be satisfied (recall that the Gibbs-Duhem equation is a simple consequence of the fact that the function, or functional, that delivers the free energy is homogeneous to the first degree see Section V). For arbitrary 5rt/ resulting in some 6fi/, one must have... 

We shall examine first the case of isothermal equilibrium displacements. The three Gibbs-Duhem equations and (29.49) must be satisfied simultaneously at constant temperature (8T= 0)... 

Finally, it can be shown that the multicomponent competitive Langmuir isotherm (Eq. 4.5) does not satisfy the Gibbs-Duhem equation if the column saturation capacities are different for the components involved [13]. This profound inconsistency may explain in part why this model does not accoimt well for experimental results. There are two very different alternative approaches to the problem of competitive Langmuir isotherms when the saturation capacities for the two pure compounds are different. Before discussing this important problem and an interesting extension of the competitive Langmuir isotherm, we must first present the competitive bi-Langmuir isotherm model. 

The first term in the last equation vanishes because % is unchanged. The second and third terms add up to zero according to the Gibbs-Duhem equation. There is left only... 

These data can be studied in two ways. The first is to use the Gibbs-Duhem equation and numerical integration methods to calculate the vapor-phase mole fractions, as considered in Problem 10.2-6. A second method is to choose a liquid-phase activity coefficient model and determine the values of the parameters in the model that give the best fit of the experimental data. We have, from Eq. 10.2-2b, that at the jth experimental point... 

Differentiating both sides with respect to a and using the differential form of the First Law, dU=T dS-P,dV + p dN, one obtains the Gibbs-Duhem equation ... 

The starting point for the formalism is the Gibbs-Duhem equation, which we write first for a pure fluid [13] ... 

It can be seen from the Gibbs-Duhem equation that at constant temperature and volume the first two terms of the right-hand side of Equation 7 are equal and opposite, so that... 

We first prove that, if a solution is ideal with respect to r - 1 of its components, it is ideal with respect to component r. At constant temperature and pressure, the Gibbs-Duhem equation [Eq. (6-59)] becomes... 

Of practical importance are electrochemical systems with electrodes. Electrical and gravitational systems may be of a pseudo first order with respect to charge and mass additions, when only the contribution of the external field to the energy is considered. However, this does not mean that these electrical and gravitational contributions should be included into the Gibbs -Duhem equation [16]. Of course, it is a matter of definition of the thermodynamic variables, whether the thermodynamic similarity exists or not. 

The practical osmotic coefficient obtained directly from cryoscopic measurements gives a measure of the deviation from ideality for the solvent species. The corresponding measure of deviation from ideality for the solute species is the activity coefficient, y. y and (f> are related as a consequence of the Gibbs-Duhem equation. The following expression was first derived by Bjerrum ... 

The left-hand side of this equation must be identical to the first term of the right-hand side, according to equation (6-3). The so-called Gibbs-Duhem relationship thus states... 

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