Mixtures Gibbs-Duhem equation

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

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 advantages of constant-pressure activity coefficients also become clear when we try to relate to one another the activity coefficients of all the components in a mixture through the Gibbs-Duhem equation (P6, P7). For variable-pressure activity coefficients at constant temperature we obtain... 

Equation (5.23) is known as the Gibbs-Duhem equation. It relates the partial molar properties of the components in a mixture. Equation (5.23) can be used to calculate one partial molar property from the other. For example, solving for dZ gives... 

In the case of ternary or higher-order mixtures, solution of the Gibbs-Duhem equation is again based on application of the properties of the exact differentials (Lewis and Randall, 1970) ... 

Let us add d i moles of C6H6 to the stoichiometric mixture. From the Gibbs-Duhem equation we can write... 

Chemical Potentials in Homogeneous Mixtures the Gibbs-Duhem Equation... 

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... 

In practice AG is known for a given T, p and x2 so that the other quantities based on the ratio a must be calculated from the excess functions for the mixture. Differentiation of eqn (26) with respect to yields, using the Gibbs-Duhem equation, In (fi/f2), and hence a. A second differentiation yields d In a/dxj. If eqn (27) is used to fit the Ge data then these quantities can be calculated from the /-coefficients. The arithmetic is tedious but a computer program can be used to advantage here (Blandamer et al., 1975b). Because the... 

Equation (169) must hold for arbitrary 5 / satisfying E8rij = 0, and clearly the only possibility is that/ (X/) = 1/X/, that is,fiX,) = InX/. The logarithmic form is the only one that satisfies the stated conditions, and hence the discrete equivalent of Eq. (21) simply follows from the definitions. Clearly, the argument can be generalized to a continuous description [one only needs to apply the Gibbs-Duhem equation in its continuous formulation to a 6 (a ) satisfying <8n(y)> = 0], and hence Eq. (21) is identified as simply the definition of a continuous ideal mixture. 

Frequently it is more convenient or only possible to measure activities or activity coefficients for a component that differs from the one in which the experimentalist is interested. In that case it is expedient to use the Gibbs-Duhem equation for a binary mixture. For a two-component system at constant T and P, we find n dii — —n2dfi2. On account of Eq. (3.6.2) this may be rewritten as... 

The last equation can be also derived starting from the Gibbs—Duhem equation for a ternary mixture (see Appendix 1). 

On the basis of the Gibbs—Duhem equations for a ternary mixture, Krichevsky and Sorina derived the following equation... 

This paper is devoted to the verification of the quality of experimental data regarding the solubility of sparingly soluble solids, such as drugs, environmentally important substances, etc. in mixed solvents. A thermodynamic consistency test based on the Gibbs-Duhem equation for ternary mixtures is suggested. This test has the form of an equation, which connects the solubilities of the solid, and the activity coefficients of the constituents of the solute-free mixed solvent in two mixed solvents of close compositions. 

Thermodynamic consistency tests are well known, and have been frequently used for vapour-liquid equilibrium data in binary mixtures (for reviews one can see Gmehling and Onken, 1977 Acree, 1984 Prausnitz et al., 1986). These tests are based on theGibbs-Duhem equation and allow one to grade the experimental data for vapor-liquid equilibrium in binary mixtures. A more difficult problem is the consistency of data regarding vapor-liquid equilibrium in ternary or multicomponent mixtures. However, several thermodynamic consistency tests, also based on the Gibbs-Duhem equation, were suggested for vapor-liquid equilibrium in ternary or multicomponent mixtures (Li and Lu, 1959 McDermott and Ellis, 1965). 

The isothermal-isobaric Gibbs-Duhem equation for an A-component mixture (A > 2) can be written as follows... 

Two limitations are involved in the derivation of the above equation (1) the compositions of mixed solvents (points c and d) should be close enough to each other for the trapezoidal mle used to integrate the Gibbs-Duhem equation to be valid, (2) the solubility of the solid should be low enough for the activity coefficients of the solvent and cosolvent to be taken equal to those in a solute-free binary solvent mixture. In addition, the fugacity of the solid phase in Eq. (4) should remain the same for all mixed solvent compositions considered. 

Eq. (8) allows one to derive a criterium for salting-in or salting-out for small cosolvent concentrations. Starting from the Gibbs-Duhem equation for a binary mixture... 

Gibbs-Duhem equation. (GDE). An exact thermodynamic relation that permits computation of the changes of chemical potential for one component of a uniform mixture over a range of compositions, provided the changes of potential for each of the other components have been measured over the same range. 

Not all the chemical potentials (and therefore, the activity coeflicients) in a mixture are independent of each other. They are all related to one another through the Gibbs-Duhem equation. To derive this equation, we start with tlie fundamental equation of thermodynamics for the Gibbs free energy, which can be written as... 

For 1 mole of a binary mixture, in general, by the Gibbs-Duhem equation at constant and p,... 

Activity coefficient is a function of the state of a mixture. An activity-coefficient equation is required to calculate the fugacities of real solutions. The interrelationship of the activity coefficients through the Gibbs-Duhem equation implies that the activity-coefficient equations of aU components are derivatives of a common thermodynamic function. Since the activity coefficient is an expression of the nonideal behavior of a component, a thermodynamic function is needed to express the nonideality of the total solution and then to obtain from it the activity-coefficient equation. 

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. 

Note that an ideal mixture identically satisfies the Gibbs-Duhem equation (Eq. 

Since both real and ideal mixtures satisfy the Gibbs-Duhem equation of Sec. 8.2,... 

Finally, it is useful to note that the Gibbs-Duhem equation can be used to get information about the activity coefficient of one component in a binary mixture if the concentration dependence of the activity coefficient of the other species is known. This is demonstrated in the next illustration. 

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