Dividing plane

As in Section III-2A, it is convenient to suppose the two bulk phases, a and /3, to be uniform up to an arbitrary dividing plane S, as illustrated in Fig. Ill-10. We restrict ourselves to plane surfaces so that C and C2 are zero, and the condition of equilibrium does not impose any particular location for S. As before, one computes the various extensive quantities on this basis and compares them with the values for the system as a whole. Any excess or deficiency is then attributed to the surface region. 

If one now makes a second arbitrary choice for the dividing plane, namely. S and distance x, it must follow that... 

If the interaction between atoms that are not nearest neighbors is neglected, then the ratios B/A are each equal to the ratio of the number of nearest neighbors to a surface atom (across the dividing plane) to the number of nearest neighbors for an interior atom. The calculation then reduces to that given by Eq. Ill-15. 

Surface excesses are usually referred to the unit surface area of the dividing plane S (surface excess densities). 

Depending on the context, we sometimes prefer the term interphase over interface because the latter refers to an infinitely sharp dividing plane between two phases. Organisms generally form boundary layers, e.g. the cell wall, that are characterised by a gradual transition from the biological phase to the medium phase, and if we discuss the volume properties of such layers the term interphase is more appropriate. 

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 Surface concentration defined in this manner depends upon the location of the dividing plane y. This can be seen by considering another dividing plane at position y". Now the volume assigned to phase I is increased by (y" — y )As while the volume of phase II decreases by the same amount. Substitution into equation (12.55) shows that the surface concentration, now given by T", differs from T by the amount... 

Thus, r,- depends upon the location of the dividing plane unless ch,/ = cit,-. 

In considering the thermodynamics properties associated with a surface, it is convenient to choose a position for the dividing plane that makes the surface concentration zero. For a single component (pure substance), it is possible to do this. From equation (12.55), we can show that this occurs when the two areas in Figure 12.5b are equal. This will be our choice for the phase boundary for a pure substance. With two or more components, in general it is not possible to make more than one T, equal to zero. In this case, we usually choose the boundary to make Tj the surface concentration of the solvent zero. With this choice, T,- for a solute will not be zero unless cu = c j. 

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. 

For the case when component 1 is a solvent in which all other components are dissolved and thus have a much lower concentration than component 1, we choose the position of the dividing plane such that TJ = 0 and from Eq. (3.11) we get... 

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 ... 

To apply the thermodynamic formalism to surfaces, Gibbs defined the ideal dividing plane which is infinitely thin. Excess quantities are defined with respect to a particular position of the dividing plane. The most important quantity is the interfacial excess which describes the amount of substance enriched or depleted at an interface. 

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],... 

We now describe how the periodic orbit theory of Section 3.6, which relates the energy levels with the poles of the spectral function g(E), can be extended to two dimensions. For simplicity we shall illustrate this extension by the simplest model, in which the total PES is constructed of two paraboloids that cross at some dividing plane. Each paraboloid is characterized by two eigenfrequencies, [Pg.113]

For perpendicular (N, J, transmission coefficient t and the reflection coefficient r, are given by... 

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. 

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