Use of the Gibbs-Duhem equations

A final example concerning phase equilibria in terms of species involves the use of Equation (8.102) or similar equations. We discuss only Equation (8.102) here, which is 

The use of the Gibbs-Duhem equations for phase equilibria when no chemical reactions occur in the system is discussed in Section 10.20. Useful expressions for the derivatives of one intensive variable with respect to 

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 

These equations are subject to the equilibrium conditions that hA2B hA hB = 0 hl = hA 

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The same type of polynomial formalism may also be applied to the partial molar enthalpy and entropy of the solute and converted into integral thermodynamic properties through use of the Gibbs-Duhem equation see Section 3.5. 

The activity of the water is derived from this expression by use of the Gibbs-Duhem equation. To utilize this equation, the interaction parameters fif ) and BH must be estimated for moleculemolecule, molecule-ion and ion-ion interactions. Again the method of Bromley was used for this purpose. Fugacity coefficienls for the vapor phase were determined by the method of Nakamura et al. (JO). 

The use of the Gibbs-Duhem equation to derive the limiting laws for coUigative properties is based on the work of W. Bloch. 

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. 

For the solvent, we must again make use of the Gibbs-Duhem equation to find a suitable function of the molarities of the solutes. We set... 

Three different uses of the Gibbs-Duhem equation associated with the integral method are discussed in this section (A) the calculation of the excess chemical potential of one component when that of the other component is known (B) the determination of the minimum number of intensive variables that must be measured in a study of isothermal vapor-liquid equilibria and the calculation of the values of other variables and (C) the study of the thermodynamic consistency of the data when the data are redundant. 

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... 

In a binary solution, the activity coefficient of component 2 can be determined from Eq. (11-155) by use of the Gibbs-Duhem equation. Since, in the applications of this method, the solution is dilute with respect to component 2 it will be most useful to determine y, the activity coefficient of component 2 with respect to the molality. At constant pressure, the Gibbs-Duhem equation may be written... 

This equation needs to be modified when only (pyX), (p,y)y or (x,y) is used to derive or when one component is involatile. In the former case, one of the error terms drops out but a further error will be introduced because an equation in some form, or an integration, or a differentiation, is necessary to reduce the data. When component B is involatile, is calculated directly from equation (2) using only (p, at) data since the vapour phase consists only of the volatile component and hence can only be calculated by making use of the Gibbs-Duhem equation. 

Applications of the Integrated Equations. The usefulness of the Gibbs-Duhem equation for establishing the thermodynamic consistency of, and for smoothing, data has been pointed out. The various integrated forms are probably most useful for extending limited data, sometimes from even single measurements, and it is these applications that are most important for present purposes. 

Activity coefficients for the CaMgSi206 component may be gotten from those of the CaAl2Si0g component by use of the Gibbs-Duhem equation ... 

To see the usefulness of the Gibbs-Duhem equation, lets examine the scenario where we wish to find the partial molar volume of species h in a binary solution when we know the partial molar volume of species a, Va, as a function of composition. If we apply Equation (6.19) to the property volume, we get ... 

Use of the Gibbs-Duhem Equation to Relate Partial Molar Properties... 

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