Gibbs dividing plane

A careful observation of Equations 10.35 and 10.36 prompts some caution. In principle, the isotherm is expressed in terms of specific amounts adsorbed and These are however meaningless unless one is able to assess which part of the liquid is in the adsorption sphere (viz. the position of the Gibbs dividing plane). 

Figure D3.5.2 Definition of the Gibbs dividing plane based on the excess concentration of component A of the two phases a and p that are in direct contact with each other.

The interfacial coverage T was originally introduced as a simple means to define the position of the Gibbs dividing plane ... 

The right side of the equation does not depend on the position of the Gibbs dividing plane and thus, also, the left side is invariant. We divide this quantity by the surface area and obtain the invariant quantity... 

Example 3.1. To show how our choice of the position of the Gibbs dividing plane influences the surface excess, we consider an equimolar mixture of ethanol and water (p. 25 of Ref. [40]). If the position of the ideal interface is such that TH2o = 0, one finds experimentally that TEthanol = 9-5 x 10 7 mol/m2. If the interface is placed 1 nm outward, then we obtain YEthanol = -130 x 10 7 mol/m2. 

If the interface is chosen to be at a radius r, then the corresponding value for dV13/dA is r /2. The pressure difference T>f) — Pa can in principle be measured. This implies that pp pa 2-y/r and l,f) — Pa = Pf /r are both valid at the same time. This is only possible if, dependent on the radius, one accepts a different interfacial tension. Therefore we used 7 in the second equation. In the case of a curved surface, the interfacial tension depends on the location of the Gibbs dividing plane In the case of flat surfaces this problem does not occur. There, the pressure difference is zero and the surface tension is independent of the location of the ideal interface. 

One important relationship can be derived directly from Eq. (3.29). For pure liquids we choose the Gibbs dividing plane such that T = 0. Then the surface tension is equal to the free surface energy per unit area ... 

For a pure liquid the Gibbs dividing plane is conveniently positioned so that the surface excess is zero. Then the surface tension is equal to the surface free energy and the interfacial Gibbs free energy f[Pg.40]

An adsorption isotherm is a graph of the amount adsorbed versus the pressure of the vapor phase (or concentration in the case of adsorption from solution). The amounts adsorbed can be described by different variables. The first one is the surface excess I in mol/m2. We use the Gibbs convention (interfacial excess volume Va = 0). For a solid surface the Gibbs dividing plane is localized directly at the solid surface. Then we can convert the number of moles adsorbed Na to the surface excess by... 

A more comprehensive introduction is Ref. [399], We restrict ourselves to uncharged species and dilute solutions (not binary mixtures). The important subject of polymer adsorption is described in Ref. [400], Adsorption of surfactants is discussed in Ref. [401], Adsorption of ions and formation of surface charges was treated in Chapter 5. In dilute solutions there is no problem in positioning the Gibbs dividing plane, and the analytical surface access is equal to the thermodynamic one, as occurs in the Gibbs equation. For a thorough introduction into this important field of interface science see Ref. [8],... 

This expression describes the analyte retention in binary system using only the total volume of the liquid phase in the column, Vq, and total adsorbent surface area S as parameters and the derivative of the excess adsorption by the analyte equihbrium concentration. It is important to note that the position of Gibbs dividing plane in the system has not been defined yet. 

A division Into "adsorption from dilute solution" and "adsorption from binary (and multicomponent) mixtures covering the entire mole fraction scale" appears to be useful. For simplicity, we shall designate mixtures covering the entire mole fraction scale as binary mixtures, as opposed to dilute solutions. This distinction is a consequence of issues (1) - (3) above, and reflected in thermodynamic and statistical interpretations. For instance, in dilute solutions locating the Gibbs dividing plane is not a problem, but for a mixture in which one of the components cannot confidently be identified as the solvent, it is. 

Accounting for size differences can also be realized in terms of distribution functions, assuming certain interaction energies. Simply because of size differences between molecules preferential adsorption will take place, i.e. fractionation occurs near a phase boundary. In theories where molecular geometries are not constrained by a lattice, this distribution function is virtually determined by the repulsive part of the interaction. An example of this kind has been provided by Chan et al. who considered binary mixtures of adhesive hard spheres in the Percus-Yevick approximation. The theory incorporates a definition of the Gibbs dividing plane in terms of distribution functions. A more formal thermodynamic description for multicomponent mixtures has been given by Schlby and Ruckenstein ). 

The excess character of interfacial energy. The Gibbs dividing plane 2.11... 

In equations [2.2.9-12] the excesses U , S°, F , G° and n° are counted with respect to a reference system of homogeneous bulk phases whose volumes and/or amounts are defined by a suitably chosen dividing surface, the Gibbs dividing plane. As this plane is infinitely thin, V =0. For establishing relations between measurable quantities this formalism has no consequences. However, in model studies the finite thickness has to be considered, and this will often be done in this chapter. 

Using [2.2.9] the surface tension of a pure liquid can be split into its entropic and energetic contribution. For a pure liquid, where the Gibbs dividing plane is determined by setting n°= 0, introducing U° =U°/A and S° =S°/A as the interfacial excess energy and entropy p>er unit area, respectively, we have... 

Between [3.6.23a and b] there is a difference of principle, in that the Identity 7 = is operational, whereas in the first equality F° and the r s depend on the choice of the Gibbs dividing plane. Their sum, the grand potential per unit area, is independent of this choice. Equations [3.6.23a and b] do not help to measure interfacial tensions, but serve in Interpreting them. 

These equations show that it Is impossible to define and F independently. In dilute solutions, x 1, we can get away with defining, that is the surface excess of the solute (comjxrnent 2) with respect to the solvent (component 1) which is present in excess. This locates the Gibbs dividing plane (GDP). In sec. 4.3, where only dilute solutions will be considered, this reference is appropriate. However, in the present case the entire range of x from 0 to 1 has to be considered, so the distinction between solvent and solute becomes very impractical it makes no sense to define r as the excess of 2 with respect to 1 when the system contains negligible amounts of 1, not that there would be a way of measuring this quantity prof>erly. 

Figure 3.1 a. The variation of a molecular property, C with the distance, x in the interfacial transition layer between a and fi phases, b. The same plot after the mathematical Gibbs dividing plane is located at point xro. Axr is the thickness of the transition layer. 

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