Gibbs adsorption equation linear

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... 

According to the Gibbs adsorption equation, the linear plot of surface or interfacial tension vs logarithm of the bulk concentration must indicate that the surface saturation has reached the... 

The linear differential equation given by Eq. (8) uniquely characterizes the interfaces using the surface tension and the various excess quantities. Applying the Young-Schwartz theorem, the Gibbs adsorption equation for a solid/fluid interface s = S fl can be deduced directly ... 

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. 

Point defects segregate to planar defects just as they do to linear defects. Adsorption on surfaces and the related catalysis of reactions on surfaces are well-known phenomena. The amount of segregation is related to the reduction in surface tension y for the boundary by the Gibbs adsorption equation, which for a multicomponent system is... 

Equation 9 states that the surface excess of solute, F, is proportional to the concentration of solute, C, multipHed by the rate of change of surface tension, with respect to solute concentration, d /dC. The concentration of a surfactant ia a G—L iaterface can be calculated from the linear segment of a plot of surface tension versus concentration and similarly for the concentration ia an L—L iaterface from a plot of iaterfacial teasioa. la typical appHcatioas, the approximate form of the Gibbs equatioa was employed to calculate the area occupied by a series of sulfosucciaic ester molecules at the air—water iaterface (8) and the energies of adsorption at the air-water iaterface for a series of commercial aonionic surfactants (9). 

At low surfactant concentrations the linear a(c) dependence can indeed be experimentally verified. Thus, a simple model describing adsorption layers as ideal solutions allows one to integrate (within a certain approximation) the Gibbs equation. 

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... 

When one examines the shape of the surface tension-ln C curve for a surfactant, it can be seen that the curve becomes approximately horizontal at some concentration below the cmc. It can be shown that the effectiveness of the adsorption of a surfactant, Ao-omo, can be quantitatively related to the concentration of surfactant at which the Gibbs equation becomes linear, Q, the surface tension attained at Ci, o-i, and the cmc. The relationship has the general form... 

This is dealt with in detail in Chapter 3 and only a summary is given here. Adsorption of surfactants at the air/liquid or liquid/Hquid interface lowers the surface or interfacial tension y. Just before the c.m.c., the y — log[CsAA] curve is linear and above the c.m.c. y becomes virtually constant. From the slope of the linear portion of the y — log C curve one can obtain the amount of surfactant adsorption T (mol m ), usually referred to as the surface excess using the Gibbs equation [8], where R is the gas constant and T is the absolute temperature,... 

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