Gibbs adsorption equation surface concentration from

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 ... 

Hence the surface adsorption of surfactant 1 and 2, and their surface mole fractions can be obtained from the surface (interfacial) tension-concentration relationships (Fig.1 and fig.2) by applying the Gibbs adsorption equation. 

Takahashi et al.67) prepared ionene-tetrahydrofuran-ionene (ITI) triblock copolymers and investigated their surface activities. Surface tension-concentration curves for salt-free aqueous solutions of ITI showed that the critical micelle concentration (CMC) decreased with increasing mole fraction of tetrahydrofuran units in the copolymer. This behavior is due to an increase in hydrophobicity. The adsorbance and the thickness of the adsorbed layer for various ITI at the air-water interface were measured by ellipsometry. The adsorbance was also estimated from the Gibbs adsorption equation extended to aqueous polyelectrolyte solutions. The measured and calculated adsorbances were of the same order of magnitude. The thickness of the adsorbed layer was almost equal to the contour length of the ionene blocks. The intramolecular electrostatic repulsion between charged groups in the ionene blocks is probably responsible for the full extension of the... 

The surface excess concentration (T), which is the surface concentration of surfactant, can be determined by the representative Gibbs adsorption equation. The T can be obtained from the slope of a plot shown in Figure 2.1 (y versus log[C] at constant temperature). 

Once the additional interactions are known, the distribution of the concentration of ions near the interface can be obtained from eqs la and lb (for the appropriate boundary conditions), and consequently, the surface tension of electrolytes can be easily calculated, via the Gibbs adsorption equation. 

Equation (2.34) is often referred to as the Gibbs adsorption equation where the interdependence of r and p is given by the adsorption isotherm. TTie Gibbs adsorption equation is a surface equation of state which indicates that, for any equilibrium pressure and temperature, the spreading pressure II is dependent on the surface excess concentration r. The value of spreading pressure, for any surface excess concentration, may be calculated from the adsorption isotherm drawn with the coordinates n/p and p, by integration between the initial state (n = 0, p = 0) and an equilibrium state represented by one point on the isotherm. 

Figure 9.5. Gibbs adsorption equation. The surface excess F can be obtained (cf. equation 19) from a plot of surface tension 7 versus log activity (concentration) of adsorbate. The area occupied per molecule or ion adsorbed can be calculated.

Direct experimental verification of the Gibbs adsorption equation in aqueous solutions is difficult, because physical separation of the monomolecular layer at the water surface is required to compare the concentration differences between the surface layer and the bulk solution. Several attempts have been made on this subject from 1910 to the present day, and although an exact fit has never been obtained, the results show a good agreement with the theory. McBain and co-workers used a suitable microtome to cut off a thin layer of approximately 50-100 4m from the surface of phenol, p-toluidine etc. solutions and verified the Gibbs equation within experimental error in 1932. Later, isotopically labeled solute molecules were employed for this purpose. Beta-emitter molecules, such as 3H, 14C and 35S have also been used and the radioactivity close to the surface measured. Since electrons only travel a short distance, the recorded radioactivity comes from the interface or very near the interface. 

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]... 

The Gibbs adsorption equation allows for calculation of area per molecule from very simple measurements of surface (or interfacial) tension versus surfactant concentration in the solution. This calculation, in turn, enables one to study the relative area/molecule of a surfactant. Tighter molecular packing in the adsorbed film lowers the interfacial tension. 

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... 

This expression is called the Gibbs adsorption equation. The equation provides a mathematical basis for the generalization that if a substance tends to reduce surface tension it will collect in the surface phase while those that tend to increase surface tension will be retained in the bulk phase. We can also see from equation (13) that when the surface tension of a solution decreases with concentration, dy/dC, is negative, F, is positive and the concentration of the solute in the surface is greater than the bulk phase. The surface active agents exhibit this type of behaviour. On the other hand, when the surface tension of a solution increases with concentration, dy/dC, is positive, F, is negative and the body of the solution is richer in the solute than the surface. This is observed in solutions of many electrolytes. The above equation has also been experimentally verified by McBain and Humphrey. They employed an apparatus... 

Another interesting aspect to discuss is the calculation of the apparent area of the species adsorbed at the interface and its relationship with the molecular weight of said species. As is well known, Gibbs adsorption equation allows us calculate the adsorption T (the concentration of the species that is adsorbed at the surface) from measurements of tension as a function of the concentration of the surfactant in the solution (Hiemmenz, 1998). 

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. 

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 ... 

Hydrophilic surfactants adsorb best on aqueous phases, whereas hydrophobic surfactants adsorb best on lipophilic surfaces (oils). Data on adsorption at constant temperature are usually plotted as a function of the surfactant equilibrium concentration plots for solid substrates are termed Langmuir isotherms. From such isotherms the maximum surfactant concentration at the interface (Fmax) can be derived and the maximum area occupied by the surfactant at the interface ( max) can be calculated. In addition, the Gibbs adsorption equation can be extracted. 

Equations D3.5.30 and D3.5.32 are both very valuable. They state that the rate of adsorption can be obtained from plots of the interfacial tension versus either tA- (for t—>0) or lth (for the long-term solution f— >). With these two equations the tool to extract the adsorption rate from experimentally obtained surface tension-time curves is at hand. It should be noted that instead of the Gibbs model, one could use one of the previously mentioned adsorption isotherms such as the Langmuir adsorption isotherm to convert interfacial tension to interfacial coverage data. The adsorption isotherms may be obtained by fitting equilibrium surface tension data versus surfactant concentration. 

These insoluble monomolecular films, or monolayers, represent an extreme case in adsorption at liquid surfaces, as all the molecules in question are concentrated in one molecular layer at the interface. In this respect they lend themselves to direct study. In contrast to monolayers which are formed by adsorption from solution, the surface concentrations of insoluble films are known directly from the amount of material spread and the area of the surface, recourse to the Gibbs equation being unnecessary. 

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