Interface surface excess quantities

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). 

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. 

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... 

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... 

It is clear from Equations 1.1 and 1.2 that surface excess quantities do take into account the variation of composition and propalies across an interfacial region of finite thickness. As we shall see shortly, they can be used to define interfacial tension. Moreover, since all surface excess properties are assigned to the reference surface S, the area and curvature of S can be identified as the corresponding properties of the interface and used, for example, to describe interfacial deformation. 

The meaning of the surface excess is illustrated in Fig. 1, in which the solid line represents the actual concentration profile of an adsorbate i, when the bulk concentration of i in the phase a (a = O or W) is c . The hatched area corresponds to be the surface excess of i, T,. This quantity depends on the location of the dividing surface. On the other hand, the experimentally accessible quantity should not depend on the location of the artificially introduced dividing surface. The relative surface excess, which is independent of the location of the dividing surface, is defined by relativizing it with respect to those of certain reference components. In oil water interfaces, the mutual solubility of solvents can be significant. The relative surface excess in Eq. (3) is then related to the surface excesses through... 

The surface concentrations T depend on the thickness of the interfacial region, and we would like to express them through quantities which are independent of it. This can be done for those species which occur both at the interface and in the solution. Usually one of the components of the solution, the solvent, has a much higher concentration then the others. We denote it by the index 0 , and introduce surface excesses with respect to the solvent in the following way In the bulk of the solution the Gibbs-Duhem equation (at constant T and p) is simply E Ni dfri = 0, or ... 

These excess quantities are independent of the thickness chosen for the interface as long as it incorporates the region where the concentrations are different from those in the bulk that is, it does not matter if one chooses too thick a region (see Problem 1). We cannot refer the surface concentrations of the metal particles M, Mz+, and e to the solution. Nevertheless we will drop the asterisk in their surface concentrations to simplify the writing we will eliminate these quantities later. We can now rewrite the Gibbs adsorption equation in the form ... 

It is easy to see that the surface excess is either a positive or a negative quantity, depending on whether the departure from the bulk concentration is positive or negative, i.e., on whether there is an accumulation or depletion of the particular species i in the interface. 

Often the surface excess of a particular species has been simply assumed to be the quantity of that species adsorbed on the surface of an electrode. To examine this point of view, consider the profile of the actual concentration. The interphase region can be said to begin from the point where the actual concentration departs from the bulk value. The amount of the species that can be said to have adsoibed per unit area of the interface is equal to the total amount of the species existing inside the interphase region divided by the area of the interface. In Fig. 6.49 the adsoibed material is indicated by the hatched area. 

What are the capabilities of this system Since the system consists of a polarizable interface coupled to a nonpolarizable interface, changes in the potential of the external source are almost equal to the changes of potential only at the polarizable interface, i.e., the changes in zl< ) across the mercuiy/solution interface are almost equal to changes in potential difference Vacross the terminals of the source. Hence, the system can be used to produce predetermined zl< ) changes at the mercuiy/solution interface (Section 6.3.11). Further, measurement of the surface tension of the mercuiy/solution interface is possible, and since this has been stated /Section 6.4.5) to be related to the surface excess, it becomes possible to measure this quantity for a given species in the interphase. In short, the system permits what are called electrocapillary measurements, i.e., the measurement of the surface tension of the... 

Thus, surface tension changes have been related to changes in the absolute potential differences across an electrode/electrolyte interface and to changes in the chemical potential of all the species, i.e., to changes in solution composition. Only one other quantity is missing, the surface excess. This can be easily introduced by recalling the definition of surface excess [Eq. (6.66)], i.e.,... 

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. 

The surface excess T is a positive quantity even when the interactions between surfactant and water become as favorable as those between surfactant and oil. In the latter case Ah0 = 0 and the concentrations in the two phases become equal. T does not vanish in that case, because the interactions between water and head group and between oil and hydrocarbon chain are stronger than those between water and hydrocarbon chain and oil and head group, respectively. It is, however, small because the interactions at the interface are not much stronger than those in the bulk. 

The surface tension depends on the potential (the excess charge on the surface) and the composition (chemical potentials of the species) of the contacting phases. For the relation between y and the potential see - Lipp-mann equation. For the composition dependence see -> Gibbs adsorption equation. Since in these equations y is considered being independent of A, they can be used only for fluids, e.g., liquid liquid such as liquid mercury electrolyte, interfaces. By measuring the surface tension of a mercury drop in contact with an electrolyte solution as a function of potential important quantities, such as surface charge density, surface excess of ions, differential capacitance (subentry of... 

The amounts n and na are extensive quantities, which depend on the extent of the interface. The related surface excess concentration , r, is an intensive quantity, which is defined as... 

The basic information in the study of sorption processes is the quantity of substances on the interfaces. In order to measure the sorbed quantity accurately, very sensitive analytical methods have to be applied because the typical amount of particles (atoms, ions, and molecules) on the interfaces is about I0-5 mol/m2. In the case of monolayer sorption, the sorbed quantity is within this range. As the sorbed quantity is defined as the difference between quantities of a given substance in the solution and/or in the solid before and after sorption processes (surface excess concentration, Chapter 1, Section 1.3.1), all methods suitable for the analysis of solid and liquid phases can be applied here, too. These methods have been discussed in Sections 4.1 and 4.2. In addition, radioisotopic tracer method can also be applied for the accurate measurement of the sorbed quantities. On the basis of the radiation properties of the available isotopes, gamma and beta spectroscopy can be used as an analytical method. Alpha spectroscopy may also be used, if needed however, it necessitates more complicated techniques and sample preparation due to the significant absorption of alpha radiation. The sensitivity of radioisotopic labeling depends on the half-life of the isotopes. With isotopes having medium half-time (days-years), 10 14-10-10 mol can be measured easily. 

For easy reference the expressions for the various surface excess functions are collected in appendix 2. These thermodynamic amd statlstlccd equations constitute the framework onto which further elaborations are anchored. All thermodynamic interfacied excess functions, except 12 , depend on the choice of the dividing plane. Of course, none of the measurable quantities are sensitive to this choice. Charged interfaces are also covered, provided the charge is caused by the preferential adsorption of one of the ionic species. Then, the components i refer to electro-... 

Some authors call this tensor the surface (or interfacial) tension tensor. The interfacial tensor is an excess quantity, (hence the superscript o) and acts in two dimensions (its SI units are N m as compared with N m for bulk stresses). Equation [3.6.14) applies to an isolated Interface. In reality isolation is of course impossible the interface is in contact and at mechanical equilibrium with the bulk. Otherwise the interface would accelerate, slow down, or display shear with respect to the adjacent bulk. An alternative way of formulation would be to retain the bulk tensor [3.6. Ij of which five components are zero in the interface. 

In the field of adsorption from solution, many discussions and reviews were published about the measurement of the adsorbed amount and the presentation of the corresponding data [14, 45—47]. Adsorption isotherms are the first step of any adsorption study. They are generally determined from the variation of macroscopic quantities which are rigorously measurable far away from the surface (e.g., the concentration of one species, the pressure, and the molar fraction). It is then only possible to compare two states with or without adsorption. The adsorption data are derived from the difference between these two states, which means that only excess quantities are measurable. Adsorption results in the formation of a concentration profile near an interface. Simple representations are often used for this profile, but the real profile is an oscillating function of the distance from the surface [15, 16]. Without adsorption, the concentration should be constant up to the soHd surface. Adsorption modifies the concentration profile of each component as well as the total concentration profile. It must be noted also that when the liquid is a pure component its concentration profile, i.e., its density, is also modified. Experimentally, the concentration can be measured at a large distance from the surface. The surface excess of component i is the... 

The experimentally controlled variables which are explicit in equation (10.9.12) are the activity of the adsorbate and the temperature T. The experimentally observed quantity is the surface excess Fa. The dependence of the adsorption on the electrical state of the interface is expressed through the local effective field E perpendicular to the interface and the average dipole moments of the adsorbate (/7a) and water molecules p ) in the same direction. The contribution of the last term is much larger under most circumstance for adsorption at the electrode solution interface than at the solution air interface. As a result, further treatment of the two problems is quite different. 

There are generally two approaches for treating surfactant adsorption at the A/L and L/L interfaces. The first approach, adopted by Gibbs, treats adsorption as an equilibrium phenomenon whereby the Second Law of Thermodynamics may be applied using surface quantities. The second approach, referred to as the equation of state approach, treats the surfactant film as a two-dimensional layer with a surface pressure jt that may be related to the surface excess F (the amount of surfactant adsorbed per unit area). These two approaches are summarised below. 

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