Binary mixture grand-potential

The grand potential provides the standard state for the formation of adsorbed solutions from the pure components. Given the pressure (P), temperature (7), and mole fraction of component 1 in the gas phase (yj) for a binary mixture, three equations are solved simultaneously to establish the amounts adsorbed ( , n ) at the standard state ... 

As we are again interested in determining the phase behavior of the binary mixture in confinement and near solid interfaces, we are essentially confronted with the same problem already discussed in Section 4.5, namely finding minima of the grand potential for a given set of thermodynamic (T, /x) and model parameters [see Eqs. (4.125)]. To obtain expressions for u> that are tractable, at least numerically, we resort again to a mean-field approximation. That... 

We begin with the simplest case of a confined binary mixture, which is a symmetric mixture confined between diemically homogeneous, nonsclective planar substrates (slit-pore). Tlie grand-potential density governing the equilibrium properties of such a mixture is given by Eq. (D.29) for the special case Xb = X = 1 a = s- These equilibrium states are obtained tn... 

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. 

Of course, completely symmetric mixtures, where the chain lengths Na, Nb of both species A, B are strictly equal, Na = Nb, in practical reality are rather the exception than the rule. Although asymmetric binary fluids can be studied in a fully grand-canonical ensemble (where both chemical potentials, /J.A, /J-B, and the volume, V, of the system, and the temperature T are the independent control variables see, e.g., Mognetri et for a study of short alkanes dissolved in carbon... 

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