Relations involving partial molar quantities

Here we derive several useful relations involving partial molar quantities in a single-phase system that is a mixture. The independent variables are T, p, and the amount ni of each constituent species i. 

From Eqs. 9.2.26 and 9.2.27, the Gibbs-Duhem equation applied to the chemical potentials can be written in the equivalent forms 

These equations show that the chemical potentials of different species cannot be varied independently at constant T and p. 

A more general version of the Gibbs-Duhem equation, without the restriction of constant T and p, is 

This version is derived by comparing the expression for dG given by Eq. 9.2.34 with the differential dG= l i -I- nt djjii obtained from the additivity rule G= iXifii. 

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Under these conditions 8HA and 8HB are constants, so that (3.123) immediately reduces to (3.121). Further aspects of such additivity relations involving partial molar quantities will be explored in Section 6.2. 

The general relations in Sec. 9.2.4 involving partial molar quantities may be turned into relations involving partial specific quantities by replacing amounts by masses, mole fractions by mass fractions, and partial molar quantities by partial specific quantities. Using volume as an example, we can write an additivity relation V = and Gibbs-... 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

We can obtain experimental values of such partial molar quantities of an uncharged species as Vi, Cpj, and Si. It is not possible, however, to evaluate the partial molar quantities Ui, Hi, Ai, and G, because these quantities involve the internal energy brought into the system by the species, and we cannot evaluate the absolute value of internal energy (Sec. 2.6.2). For example, while we can evaluate the difference Hi — H from calorimetric measurements of enthalpies of mixing, we cannot evaluate the partial molar enthalpy Hi itself. We can, however, include such quantities as Hi in useful theoretical relations. 

The Kirkwood—Buff (KB) theory of solution (often called fluctuation theory) employs the grand canonical ensemble to relate macroscopic properties, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volnmes, to microscopic properties in the form of spatial integrals involving the radial distribution function. This theory allows one to obtain information regarding some microscopic characteristics of mnlti-component mixtures from measurable macroscopic thermodynamic quantities. However, despite its attractiveness, the KB theory was rarely used in the first three decades after its publication for two main reasons (1) the lack of precise data (in particular regarding the composition dependence of the chemical potentials) and (2) the difficulty to interpret the results obtained. Only after Ben-Naim indicated how to calculate numerically the Kirkwood—Buff integrals (KBIs) for binary systems was this theory used more frequently. 

Up to this point, we were interested in the difference Gbs Gas only. However, the Kirkwood-Buff theory allows us to express both Gbs and Gas in terms of measurable quantities. Again, the algebra involved is quite lengthy. We therefore present the final result only. First we express the partial molar volume of S in the limit of very dilute solution in terms of the Kirkwood-Buff integrals. This relation is... 

Figure 21e shows the quantity Anics/AE(oj), obtained by eliminating the anion contribution, which means that this partial electro-gravimetric transfer function is only related to the contribution of the cation and solvent and remains in the third quadrant. By taking Wa = 35.5 gmoP, i.e. the molar mass of CP ions, the low-frequency loop disappears. This demonstrates that CP is the anion involved in the charge compensation and that it has the lowest time constant. 

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