Gibbs adsorption equation, case

Ionic surfactants are electrolytes dissociated in water, forming an electrical double layer consisting of counterions and co-ions at the interface. The Gouy-Chapman theory is used to model the double layer. In conjunction with the Gibbs adsorption equation and the equations of state, the theory allows the surfactant adsorption and the related interfacial properties to be determined [9,10] (The Gibbs adsorption model is certainly simpler than the Butler-Lucassen-Reynders model for this case.). 

Surface Adsorption. From Fig.l and Fig.2 we can calculate the total surface adsorption (17 ) of the RDH-surfac-tant mixture by applying the Gibbs adsorption equation(7). In the case of a mixed aqueous solution with a constant ionic strength, the equation is written as... 

In line with the Gibbs adsorption equation (equation 3.33 in chapter 3), the presence of thermodynamically unfavourable interactions causes an increase in protein surface activity at the planar oil-water interface (or air-water interface). As illustrated in Figure 7.5 for the case of legumin adsorption at the n-decane-water interface (Antipova et al., 1997), there is observed to be an increase in the rate of protein adsorption, and also in the value of the steady-state interfacial pressure n. (For the definition of this latter quantity, the reader is referred to the footnote on p. 96.)... 

In this case (in the absence of a surface charge) the interface is neutral. The dependence of the interfacial tension y on the uni-univalent electrolyte concentration cE can be calculated using the Gibbs adsorption equation ... 

For a neutral gas, the adsorption (which is proportional to the free energy of the system, see Eq. (8)) can be calculated easily. For charged particles one should account in the Gibbs adsorption equation for the adsorption of all particles of the system (including those responsible for the charging of the surface [23]). Therefore, one should first identify the mechanism of formation of the double layer. In this case,... 

The Gibbs-Duhem equation is used in several cases in electrochemistry, e.g., in the derivation of - Gibbs adsorption equation or -> Gibbs-Lippmann equation since Eq. (4) can be extended by surface work ... 

The elaboration depends on details of the cell, the composition of the solution and the nature of the reference electrode, which can be a (non-reversible) calomel electrode, a reversible (normal) hydrogen electrode or an electrode that is reversible to the cation or anion In the solution. For the last two cases the cell tensions were called E and B respectively. For the Gibbs adsorption equations in. say KNO3, we derived (1.5.6.16) ... 

The comparison between Equations 5.42 and 5.43 shows that the Gibbs adsorption equation can be expressed either in terms of a, and a or in terms of o, E , and a. Note that Equations 5.42 and 5.44 are valid under equilibrium conditions, whereas Equation 5.43 can be used also for the description of dynamic surface tension (Section 5.2.2) in the case of surfactant adsorption under diffusion control, assuming local equilibrium between adsorptions E and subsurface concentrations of the respective species. 

As an example, let us consider again the special case of SDS + NaCl solution. In this case, the Gibbs adsorption equation takes the form ... 

When the point, C20, is on the linear portion of the curve (Figure 5-3), that is, if the surface is essentially saturated when the surface tension of the solvent has been reduced by 20 dyn/cm (20 mN m ), which is generally the case for most surfactants, then for the linear portion of the plot, the Gibbs adsorption equation becomes... 

There are two principal problems with penetration experiments the adsorption characteristics of the protein have to be understood, and the amount of protein that adsorbs to the interface when lipid is present has to be determined. Previously, most researchers used the change in film pressure (Atr) as a measure of the amount of protein that interacted with the lipid monolayer. However, this approach implicitly assumes that the adsorption of protein can be described by Gibbs adsorption equation, but as pointed out by Colacicco (6), this is invalid for proteins which adsorb irreversibly. Because the surface concentration of protein is unknown, radiolabeled proteins have been used (8, 9, 10). This work has been concerned exclusively with highly water-soluble proteins whose prime mode of interaction with monolayers (and bilayers) is electrostatic. In these cases a simple description of the packing in the mixed lipid-protein films was impossible (6). 

Role of Adsorbed Surfactant Layer. Foams, irrespective of the nature of liquid and gas involved, require a third component for stabilization of thin films (lamellae) of the liquid. In the familiar case of aqueous soap films, this third component is the soap, a surface-active chemical that adsorbs at the gas—liquid interface and lowers the surface tension of water. The two effects, adsorption at the liquid surface and the depression of surface tension, are intimately linked and occur concomitantly. The adsorption is defined as the excess moles of solute per unit area of the liquid surface. In a binary system, this surface excess can be directly related to the lowering of surface tension by Gibbs adsorption equation ... 

Just as it was interesting to verify experimentally the Gibbs adsorption equation in the case of solution-vapor interfaces [26], it should be worthwhile to verify such aspects of the Young and Dupre Equation 1 as are experimentally accessible. The approaches outlined above are not thermodynamic in the sense that particular molecular models are... 

For this case the Gibbs adsorption equation takes the form... 

The preceding discussion of the Gibbs adsorption equation was referenced to a fluid-fluid interface in which the surface excess, T, is calculated based on a measured quantity, a, the interfacial tension. For a sohd-fluid interface, the interfacial tension cannot be measured directly, but the surface excess concentration of the adsorbed species can be, so that the equation is equally useful. In the latter case. Equation (9.16) provides a method for determining the surface tension of the interface based on experimentally accessible data. 

First, for all reduced surfactants dissolved in solution at concentrations near their critical micelle concentrations (CMC), oxidation leads to an increase in the surface tension of the solution (Fig. 1). In the case of surfactants I and n, oxidation returns the surface tension of the solution to a value that is similar to the surfactant-free solution of electrolyte (approx. 72 mN/m). The excess surface concentration of surfactant, estimated using the Gibbs adsorption equation, decreases in the case of surfactant II from 10x10 to < 0.1 X10 mol/m upon oxidation. Clearly, oxidation drives the desorption of surfactant from the surface of the solution. The increase in surface tension of... 

In (2.12) n x) refers to the polymer segment concentration profile near a single wall whereas in (2.11) n x) is the profile between two walls. Expression (2.10) is the extension of the Gibbs adsorption equation for a single surface to the case of two surfaces at finite separation [6-8]. Integration of (2.10) gives... 

The Gibbs adsorption equation has been a subject of many investigations in the literature (Chattoraj and Birdi, 1984 Fainerman et al., 2002). Gibbs considered that the interfacial region is inhomogeneous and difficult to define, and he therefore also considered a more simplified case in which the interfacial region is assumed to be a mathematical plane. 

In the case of gas-crude oil systems, surface activity is defined by the Gibbs adsorption equation, which relates the adsorption (strictly the surface excess) of a solute, at the gas-crude oil interface to the derivative of the gas-crude oil surface... 

Even in the primitive versions of the van dcr Waals theory with m independent of p, that coefficient may still depend on the temperature T (or, equivalently, on the chemical potential fi) at which the phases are in equilibrium. While in Chapter 5 we shall see some examples, or limiting idealized cases, in which m is a fixed constant, independent of T, and is determined by the intermolecular forces alone, as in (1.38), it will, more generally, depend on T and in that event, as we shall see in 3.4, the connection between this theory and the Gibbs adsorption equation (2.31) is not entirely straightforward and requires discussion. 

Let us now consider fluid interfaces composed of chemical components which are soluble in (and equilibrated with) the adjacent bulk phases. If a new area is created at constant T, x p, H, and D, this does not correspond to any change of the physical state of the interface. In such a case, dtoVda = 0, and from Eq. (101), one realizes that y = to. Then, Eq. (101) reduces to a generalized form of the Gibbs adsorption equation cf. Eq. (1). It is now evident that the bending and torsion moments, B and , are connected with the curvature dependence of the interfacial tension. 

The surface pressuie/area equations that describe such films are the two-dim isional analogues of the well-known three-dimensional equations of state. As previously discussed, see also in Chapter 7 (Table 7.1), the ideal gas film equation of state (Equation 4.9) can be derived from the Gibbs adsorption equation, assuming that surface tension decreases linearly with concentration (Example 4.3). Such a dependency is a realistic picture for several (rather simple) cases, e.g. dilute surfactant or alcohol solutions and in general when the surface tension is not far away from the water surface tension. Liquid (and solid) films require more complex two-dimensional equations of state, e.g. van der Waals and Langmuir. Figures 4.5 and 4.6 present some surface pressure-area plots for liquid and solid films and discuss some of their important characteristics. 

Equilibrium molar surface concentrations per unit surface area (Gibbs adsorption equation surface concentration) are designated with the upper case Greek symbol, F. 

In the case of oxides, the Gibbs adsorption equation (8.12) gives the variation in interfacial tension of particles in the presence of a basic solution XOH (or an acidic solution HY) and an electrolyte XY [39-41] (... 

It is often useful (e.g. for dilute solutions) to express the adsorption of components with respect to a predominant component, e.g. the solvent. The component that prevails over m components is designated by the subscript 0 and the case of constant temperature and pressure is considered. In the bulk of the solution, the Gibbs-Duhem equation, , nt dpt = 0, is valid, so that... 

Adsorption at liquid surfaces can be monitored using the Gibbs adsorption isotherm since the surface energy, y, of a solution can be readily measured. However, for solid substrates, this is not the case, and the adsorption density has to be measured in some other manner. In the present case, the concentration of adsorbate in solution will be monitored. In place of the Gibbs equation, we can use a simple adsorption model based on the mass action approach. 

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