Applications of the Gibbs adsorption equation

For a binary solution at constant temperature, Equation (7.20) may be expanded as 

In most instances the position of the Gibbs dividing surface may be chosen such that Fa = 0 and Equation (7.21) m be rearranged to express the surface excess of the solute as 

if increasing the chemical potential (i.e. the concentration) of component B decreases the interfacial energy that species will be adsorbed, whereas if doing this increases y it will be desorbed. 

The chemical potential of the solute may be expressed in terms of the activity as 

If the solution is dilute enough that the solute obeys Henry s law. Equation (1.54), then 

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In writing the present book our aim has been to give a critical exposition of the use of adsorption data for the evaluation of the surface area and the pore size distribution of finely divided and porous solids. The major part of the book is devoted to the Brunauer-Emmett-Teller (BET) method for the determination of specific surface, and the use of the Kelvin equation for the calculation of pore size distribution but due attention has also been given to other well known methods for the estimation of surface area from adsorption measurements, viz. those based on adsorption from solution, on heat of immersion, on chemisorption, and on the application of the Gibbs adsorption equation to gaseous adsorption. 

The role of the cosurfactant in reducing the interfacial tension can be understood from application of the Gibbs adsorption equation in the form (14). 

An important application of the Gibbs adsorption equation is to the calculation of the relative adsorption from measurements of the variation of surface tension with concentration ... 

Primarily, this approach was based on the formal analogy between a first order phase transition and the micellisation. When a new phase of a pure substance is formed the chemical potential of this substance and its concentration in the initial phase do not change with the total content of this substance in the system. A similar situation is observed above the CMC, where the adsorption and the surface tension become approximately constant. In reality variations of these properties are relatively small to be observed by conventional experimental methods. The application of the Gibbs adsorption equation shows that the constancy of the surfactant activity above the CMC follows from the constancy of the surfactant adsorption T2 [13]... 

An appendix to this paper discusses thermodynamic consistency tests for binary solute adsorption. Provided that the data are within the Henry s law region of the isotherm, it is shown that application of the Gibbs adsorption equation leads, for any closed path, to... 

Application of Gibbs Adsorption Equation. By substituting Cg = 1/A (where A is the area per molecule in the adsorbed monolayer), in the simple form of the Gibbs Adsorption Equation (Equation 1), we obtain... 

The approach of IAS of Myers and Prausnitz presented in Sections 5.3 and 5.4 is widely used to calculate the multicomponent adsorption isotherm for systems not deviated too far from ideality. For binary systems, the treatment of LeVan and Vermeulen presented below provides a useful solution for the adsorbed phase compositions when the pure component isotherms follow either Langmuir equation or Freundlich equation. These expressions are in the form of series, which converges rapidly. These arise as a result of the analytical expression of the spreading pressure in terms of the gaseous partial pressures and the application of the Gibbs isotherm equation. 

While the derivation of these quantities seems at first a bit of a mathematical card trick without a real application, interfacial scientists do utilize these equations on a day-to-day basis. The Gibbs adsorption equation that relates interfacial tension to the interfacial coverage is a perfect example of a thermodynamic relationship that can be obtained from... 

If one pictures the adsorbed monolayer as a two-dimensional imperfect gas, it seems reasonable to assume the applicability of a two-dimensional form of the van der Waals equation in which the gas pressure is replaced by the spreading pressure and the volume by the surface area. By combining this with the Gibbs adsorption equation - Equation (2.34) - de Boer (1968) obtained the equation... 

Benhamou and Guastalla (1960) were the first to question the assumption of irreversibility with an analysis of the adsorption of insulin, /3-lactoglobulin, and ribonuclease. They investigated whether the Gibbs adsorption equation was obeyed. This basic equation, applicable to reversible adsorption, is firmly based on thermodynamics and has been amply verified experimentally. It can be written in the simple form... 

Reversibility of Adsorption. Apparently, the data in Figure 10.13 imply that the Gibbs equation (10.2) does not hold for the protein. As we have seen, it is valid for the amphiphile. However, the slopes dll/d In c given in the figure differ only by a factor 2 between the two surfactants, whereas the values of Fm differ by two orders of magnitude. The explanation is not fully clear. Application of the Gibbs equation to polymers is anyway questionable, because it is generally not known what the relation is between concentration (c) and activity (a) of the surfactant. Moreover, proteins and other polymers are virtually always mixtures. 

The multicomponent adsorption isotherm for the component 1 is obtained by the application of the Gibbs equation ... 

As mentioned in Section 7.1, Langmuir s monolayers are not in chemical equilibrium with the solution, and, as a consequence, the Gibbs adsorption equation is not applicable to such monolayers. However, relations between n, T, and T are completely determined by the number of molecules, and the interactions between them, in the monolayer, irrespective of the way the monolayer has been formed. Equations of state are therefore identical for Gibbs and Langmuir s monolayers. 

Application of the extended set of tools allows isotherms of binary systems to be computed, on the basis of Eq. (8) and the Gibbs adsorption equation, without any direct adsorption experiment. 

Applications of the Gibbs equation (adsorption, monolayers, molecular weight of proteins)... 

What is the Gibbs adsorption equation Mention some of its practical applications. 

Equation 9 states that the surface excess of solute, T, is proportional to the concentration of solute, C, multiplied by the rate of change of surface tension, with respect to solute concentration, d m,/dCThe concentration of a surfactant in a G—L interface can be calculated from the linear segment of a plot of surface tension versus concentration and similarly for the concentration in an L—L interface from a plot of interfacial tension. In typical applications, the approximate form of the Gibbs equation was employed to calculate the area occupied by a series of sulfosuccinic ester molecules at the air—water interface (8) and the energies of adsorption at the air-water interface for a series of commercial nonionic surfactants (9). 

Adsorption can be measured by direct or indirect methods. Direct methods include surface microtome method [46], foam generation method [47] and radio-labelled surfactant adsorption method [48]. These direct methods have several disadvantages. Hence, the amount of surfactant adsorbed per unit area of interface (T) at surface saturation is mostly determined by indirect methods namely surface and interfacial tension measurements along with the application of Gibbs adsorption equations (see Section 2.2.3 and Figure 2.1). Surfactant structure, presence of electrolyte, nature of non-polar liquid and temperature significantly affect the T value. The T values and the area occupied per surfactant molecule at water-air and water-hydrocarbon interfaces for several anionic, cationic, non-ionic and amphoteric surfactants can be found in Chapter 2 of [2]. 

We have addressed the various adsorption isotherm equations derived from the Gibbs fundamental equation. Those equations (Volmer, Fowler-Guggenheim and Hill de Boer) are for monolayer coverage situation. The Gibbs equation, however, can be used to derive equations which are applicable in multilayer adsorption as well. Here we show such application to derive the Harkins-Jura equation for multilayer adsorption. Analogous to monolayer films on liquids, Harkins and Jura (1943) proposed the following equation of state ... 

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