Surface excess quantities

With the preceding introduction to the handling of surface excess quantities, we now proceed to the derivation of the third fundamental equation of surface chemistry (the Laplace and Kelvin equations, Eqs. II-7 and III-18, are the other two), known as the Gibbs equation. 

The most widely used experimental method for determining surface excess quantities at the liquid-vapor interface makes use of radioactive tracers. The solute to be studied is labeled with a radioisotope that emits weak beta radiation, such as H, C, or One places a detector close to the surface of the solution and measures the intensity of beta radiation. Since the penetration range of such beta emitters is small (a ut 30 mg/cm for C, with most of the adsorption occurring in the first two-tenths of the range), the measured radioactivity corresponds to the surface region plus only a thin layer of solution (about 0.06 mm for C and even less for H). 

A quite different means for the experimental determination of surface excess quantities is ellipsometry. The technique is discussed in Section IV-3D, and it is sufficient to note here that the method allows the calculation of the thickness of an adsorbed film from the ellipticity produced in light reflected from the film covered surface. If this thickness, t, is known, F may be calculated from the relationship F = t/V, where V is the molecular volume. This last may be estimated either from molecular models or from the bulk liquid density. 

It was noted in connection with Eq. III-56 that molecular dynamics calculations can be made for a liquid mixture of rare gas-like atoms to obtain surface tension versus composition. The same calculation also gives the variation of density for each species across the interface [88], as illustrated in Fig. Ill-13b. The density profiles allow a calculation, of course, of the surface excess quantities. 

It has been pointed out [138] that algebraically equivalent expressions can be derived without invoking a surface solution model. Instead, surface excess as defined by the procedure of Gibbs is used, the dividing surface always being located so that the sum of the surface excess quantities equals a given constant value. This last is conveniently taken to be the maximum value of F. A somewhat related treatment was made by Handa and Mukeijee for the surface tension of mixtures of fluorocarbons and hydrocarbons [139]. 

The treatments that are concerned in more detail with the nature of the adsorbed layer make use of the general thermodynamic framework of the derivation of the Gibbs equation (Section III-5B) but differ in the handling of the electrochemical potential and the surface excess of the ionic species [114-117]. The derivation given here is after that of Grahame and Whitney [117]. Equation III-76 gives the combined first- and second-law statements for the surface excess quantities... 

This shows that the surface excess quantity on the left-hand side is proportional to the change in surface tension with concentration of the solute (d Y/d(fii(Csurlacejctlvesubs tance)- A plot of In (Cj) versus y gives a slope equal to... 

We return to further simplifications of this equation after a brief discussion of how to define the position of a surface and how to define surface excess quantities. 

This measured concentration change is usually converted into a surface excess quantity, analogous to that usually calculable from surface tension data for adsorption at the liquid/vapor interface. In the case of... 

For pure liquids the description becomes much simpler. We start by asking, how is the surface tension related to the surface excess quantities, in particular to the internal surface energy and the surface entropy ... 

Gibbs-Lippmann equation — The relationship between the - interfacial tension (y) and the surface excess quantities was derived by -> Gibbs, J. W. At constant temperature (T) and external pressure (/ )... 

The Gibbs representation provides a simple, clear-cut mode of accounting for the transfer of adsorptive associated with the adsorption phenomenon. The same representation is used to define surface excess quantities assumed to be associated with the GDS for any other thermodynamic quantity related with adsorption. In this way, surface excess energy (U°), entropy (Sa) and Helmholtz energy (Fa) are easily defined (Everett, 1972) as ... 

By adopting the usual conventions of chemical thermodynamics, we are able to derive from the surface excess chemical potential pa a number of useful surface excess quantities. Our purpose here is to draw attention to the difference between the molar and the differential surface excess quantities. 

We now return to the definition of the surface excess chemical potential fta given by Equation (2.19) where the partial differentiation of the surface excess Helmholtz energy, Fa, with respect to the surface excess amount, rf, is carried out so that the variables T and A remain constant. This partial derivative is generally referred to as a differential quantity (Hill, 1949 Everett, 1950). Also, for any surface excess thermodynamic quantity Xa, there is a corresponding differential surface excess quantity xa. (According to the mathematical convention, the upper point is used to indicate that we are taking the derivative.) So we may write ... 

The differential quantities of adsorption , are the differences between the differential surface excess quantities and the same molar quantity, as in Equation (2.46) or... 

In Chapter 2 we have introduced a number of thermodynamic surface excess quantities (Equations (2.11)—(2.14)) in the case of a simple gas adsorption system involving a single adsorptive. These quantities were expressed as a function of the surface excess amount, na. In the case of the process of immersion of a solid in a pure liquid, the same surface excess quantities can still be defined and it is useful to express them as a function of the surface area. Thus ... 

By combining Equations (5.51), (5.54) and (5.55), we obtain the relationship between the relative and reduced surface excess quantities ... 

The main result of the thermodynamic treatment in Section 8.6 is that the interphase between a 3D Me-S bulk alloy phase and an electrolyte phase can be described by relative specific surface excess quantities, q, Tand Fi with f = IC and X, analogous to the interphase concept of an ideally polarizable substrate S in contact with the electrolyte phase (Section 8.2). 

According to Gibbs we obtain surface excess quantities as the basic for adsorption isotherms and surface rheology. For flat or only slightly curved surfaces one can define the surface excess quantity convenient for discussion of a particular problem. It is assiuned that this freedom... 

Surface Excess Isotherm A function relating, at constant temperature and pressure, the relative adsorption, reduced adsorption, or similar surface excess quantity to the concentration of component in the equilibrium bulk phase. 

The reason for defining the reference system is that the properties of the interface are governed by excesses and deficiencies in the concentrations of components that is, we are concerned with differences between the quantities of various species in the actual interfacial region, with respect to the quantities we would expect if the existence of the interface did not perturb the pure phases, a and j8. These differences are called surface excess quantities. For example, the surface excess in the number of moles of any species, such as potassium ions or electrons, would be... 

As mentioned, it is extremely difficult to measure a surface excess quantity directly. Such measurements have been made at solid-liquid interfaces, but... 

Mote than a century ago, Gibbs (1878) introduced surface excess quantities as a first step toward resolving this problem. The basic idea is to choose a reference surface S somewhere in the interfadal region. This surface is everywhere perpendicular to the local density or concentration gradient. Consider a property such as internal energy in the region between surfaces and of Figure 1.2 whieh are parallel to S but are located in the respective bulk phases. Because the... 

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