Gibbs, adsorption equation chemical 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,... 

This is the Gibbs adsorption equation that relates y to the number of moles and the chemical potentials of the components in the interface. 

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... 

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]... 

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) ... 

Ta types of anions with chemical potentials /ii- Using Hansen s reference system and choosing the solvents a and P as reference substances, one can write down Hansen s rendition of Gibbs adsorption equation ... 

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. 

In this expression for the chemical potential, the first addendum, U(,(T), is a standard potential at a fixed pressure. The second addendum expresses the contribution from the fugacity of the pure component. The third addendum is due to mixing. The dependence/ (< >, T) may be found from the Gibbs adsorption equation (13b), where the integration is often carried out from zero pressure (and, correspondingly, the value of 0 is equal to zero). With such an expression for the chemical potentials of the adsorbed phase, equilibrium conditions (12), for the equilibrium with a nonideal gas phase, are reduced to the form... 

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 ... 

Equation (10.3) states that (given P, T) the boundary state, s, and the composition N) depend on juh the chemical potential of the components in the system, which has already been illustrated in Figure3-7. The 8( Ag) change in Ag2+(5S after the (/ - ) transformation at 176 °C indicates that point defects are adsorbed at the newly formed internal surfaces introduced into the crystal by this transformation, quite analogous to a Gibbs adsorption isotherm. For (isotropic) internal surfaces, the isotherm is... 

Fig. 2. Adsorption isotherms in all four model systems. F is the Gibbs excess adsoiption. The pressures corresponding to the three configurations shown in Figure 3 are marked with arrows. The pressure is plotted relative to the vapor pressure of the model fluid, as determined by independent Gibbs Ensemble Monte Carlo simulations. Chemical potentials were converted to pressures using a virial equation of state.

To calculate surface pressure-area isotherms, following the Gibbs convention, we used the adsorption equation relating surface tension to surface excess, T, and chemical potential, /a, in the form... 

Adsorption occurs because it lowers the free energy of the system. According to Gibbs, the chemical potential of the adsorbate is at equilibrium equal in the solution and at the surface. Fie further postulated an infinitely thin dividing plane between the two phases and then derived the equation... 

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. 

The decrease in the surface tension at constant adsorption, occurring in agreement with the Gibbs equation, is solely due to the increase in chemical potential of the adsorbed substance caused by the increased concentration of the latter in solution. As is commonly known, the increase in the chemical potential in a stable two-component system always corresponds to the concentration increase. For the present case it translates into the increase of surface concentration, and consequently, of the adsorption. Therefore, in the concentration region where the surface tension linearly depends on the log of concentration, a slow but finite, increase in adsorption not detected experimentally should occur. At the same time a sharp increase in the chemical potential of the surfactant molecules in the adsorption layer... 

During condensation with changing adsorption, the constant two-dimensional pressure corresponds to a constant value of chemical potential of the substance, in agreement with the Gibbs equation. This is similar to the three-dimensional case when the chemical potential is independent of the ratio of the liquid-vapor content. Thus, for soluble surfactants that undergo surface condensation, the surface condensation process should occur at some constant bulk concentration, cc, i.e. the condensation has the form of an abrupt change in adsorption from some Tc = 1/, vcNA to a value approximately equal to the limiting adsorption, Tmax. The dependence of adsorption on concentration is represented by curves 3 and 4 shown in Fig. 11-26. 

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. 

Of course, the problem here is that the standard term( )f(x) = - j,f/Z F, which expresses the standard chemical potential, varies in a stepwise manner aaoss the mixed-solvent layer, say, 1 nm thick, whereas the potential drop varies in a monotonic way in the absence of specific adsorption across the two back-to-back diffuse layers, say 10 nm thick. As a result. Equation 1.30 is a very rough approximation, as soon as the Gibbs energy of transfer of the ion is larger than 5 kJ-mol. ... 

---