Gibbs, adsorption equation thermodynamic potential

Information on the chemical potentials of components in a solution of biopolymers can serve as a guide to trends in surface activity of the biopolymers at fluid interfaces (air-water, oil-water). In the thermodynamic context we need look no further than the Gibbs adsorption equation,... 

Since l is a thermodynamic quantity, the most reliable procedures for its determination are based on a thermodynamic analysis of adsorption data, possibly at low coverages. Adsorption data to be analyzed by the Gibbs adsorption equation can be obtained by measuring the interfacial tension y, the charge density crM or the differential capacity C. Direct y measurements are equilibrium measurements that can only be carried out on mercury. Direct charge measurements are conveniently carried out by the potential-step chronocoulometric technique, which can be... 

The macroscopic description of the adsorption on electrodes is characterised by the development of models based on classical thermodynamics and the electrostatic theory. Within the frames of these theories we can distinguish two approaches. The first approach, originated from Frumkin s work on the parallel condensers (PC) model,attempts to determine the dependence of upon the applied potential E based on the Gibbs adsorption equation. From the relationship = g( ), the surface tension y and the differential capacity C can be obtained as a function of E by simple mathematical transformations and they can be further compared with experimental data. The second approach denoted as STE (simple thermodynamic-electrostatic approach) has been developed in our laboratory, and it is based on the determination of analytical expressions for the chemical potentials of the constituents of the adsorbed layer. If these expressions are known, the equilibrium properties of the adsorbed layer are derived from the equilibrium equations among the chemical potentials. Note that the relationship = g( ), between and , is also needed for this approach to express the equilibrium properties in terms of either or E. Flere, this relationship is determined by means of the Gauss theorem of electrostatics. 

A curious example is that of the distribution of benzene in water benzene will initially spread on water, then as the water becomes saturated with benzene, it will round up into lenses. Virtually all of the thermodynamics of a system will be affected by the presence of the surface. A system containing a surface may be considered as being made up of three parts two bulk phases and the interface separating them. Any extensive thermodynamic property will be apportioned among these parts. For example, in a two-phase multicomponent system, the extra amount of an i component that can be accom-mondated in the system due to the presence of the interface ( ) may be expressed as Qi Qii where is the total number of molecules of i in the whole system, Vj and Vjj are the volumes of phases I and II, respectively, and Q and Qn are the concentrations of i in phases I and II, respectively. The surface (excess) concentration of i is defined as Fj = A, where A is the surface area. At equilibrium, the chemical potential of any component is the same in each bulk phase and at the surface. The Gibbs adsorption equation, which is one of the most widely used expression in surface and colloid science is shown in Eq. (2) ... 

An alternative and insightful way to obtain the interaction potential is from the extended Gibbs adsorption equation [6-8]. The natural thermodynamic potential to describe the system depicted in Fig. 2.2 is the grand potential Q(r, V, p, h)... 

An alternative interpretation of the electrochemical double layer comes from a more thermodynamic approach. As an initial point, considering the Gibbs adsorption equation proved useful. This equation originally describes the dependence of the surface tension on the two-dimensional surface concentration (the surface excess F) of adsorbed particles as well as on their chemical potential p. The equation can be extended by introducing an electric term which considers the potential dependence of the surface tension. The Gibbs adsorption equation in its complete form is as follows ... 

They analyzed their results using a thermodynamic approach based on the Gibbs adsorption equation and the main conclusion of their work was that relative surface excesses of the ionic species were well described by the Gouy-Chapman theory. They adopted the MVN model of the ideally polarized interface stating that the compact layer is an ion-free layer consisting of laminated layers of water and nitrobenzene sandwiched between two diffuse layers. The potential difference across this inner layer was estimated to be about 20 mV at the PZC but was found to vary with the surface charge density. 

The above thermodynamic relationship, describing adsorption in a two-component system, was first derived by Gibbs and is known as the Gibbs equation [3]. It follows from the Gibbs equation that the excess of component within the interfacial layer determines how abrupt the decrease in the surface tension is with correspondingly increasing chemical potential of the adsorbed substance. 

Hall proposed an experimental procedure, in which changes in the area of the monolayer in equilibrium with the soluble surfactants should be performed to keep constant or control the chemical potential of the first component when the activity of the second component is varied. To summarise, the rigorous thermodynamic analysis of the penetration equilibrium, based on the Gibbs equation, can neither provide an equation of state of the monolayer, nor an adsorption isotherm for the soluble component. This analysis only enables one to formulate the conditions for a penetration experiment, which are, however, very difficult to implement. Therefore, to describe the thermodynamic behaviour of real mixed monolayers, at present one should use some approximate theoretical models. 

The most straightforward (and the most developed) approach to multicomponent adsorption is in further development of the thermodynamics of a surface phase, similar to the bulk-phase thermodynamics. In this way, the Gibbs surface thermodynamics should be completed by an equation of state or by an excess model for a proper thermodynamie potential. An extended review of the fundamentals and the history of the development of this approaeh may be found in Refs. 8, 9, and 78. The approach has become espeeially popular and widely used for praetieal modeling of multicomponent adsorption after the works of de Boer [79] and, especially, Myers and Prausnitz [80]. The latter authors made the natural step of introducing the activity coefficients y of the components in an adsorbed phase. In terms of these coefficients, the chemical potentials of the adsorbate may be expressed as... 

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