Surfaces excess volume

For the sake of completeness, Ua and Ha are separately defined above. Actually, when the enthalpy is defined in the standard way (H-U + pV), the Gibbs representation results in a useful simplification since the surface excess volume Va<=0 and we can write Ha = Ua (as in Sections 2.4.2 and 2.5.1). 

In Section I.D., it has been proven that the exact Gibbs equation (48) contains the surface excess volume, V°, defined by the relationship... 

Together with the function v P), is plotted the surface excess volume function V°(P) is calculated on the real supposition that in this range of pressure, V (P) decreases (see Fig. 4). In the left-hand side of Fig. 3, two linear functions V (P) are plotted ... 

A quite different means for the experimental determination of surface excess quantities is ellipsometry. The technique is discussed in Section IV-3D, and it is sufficient to note here that the method allows the calculation of the thickness of an adsorbed film from the ellipticity produced in light reflected from the film covered surface. If this thickness, t, is known, F may be calculated from the relationship F = t/V, where V is the molecular volume. This last may be estimated either from molecular models or from the bulk liquid density. 

Cf, C y, and Cq are the concentrations of the substance in question (which may be a colligend or a surfactant) in the feed stream, bottoms stream, and foamate (collapsed foam) respectively. G, F, and Q are the volumetric flow rates of gas, feed, and foamate respectively, is the surface excess in equilibrium with C y. S is the surface-to-volume ratio for a bubble. For a spherical bubble, S = 6/d, where d is the bubble diameter. For variation in bubble sizes, d should be taken as YLnid fLnidj, where n is the number of bubbles with diameter dj in a representative region of foam. 

Figures 3 and 4 give displacement isotherms of PVP adsorbed on pyrogenic silica. The polymer surface excess is plotted versus the volume fraction of displacer < >d. Results in Figure 3 are for water as the solvent and in Figure 4 for dioxane as the solvent. Various displacers were used, as indicated in the figures. One of the displacers was N-ethyl pyrrolidone (NEP), which can be considered as the monomer of PVP. From the isotherm in dioxane, it was found that In = 0.14. The aqueous system shows strong...

At(p = 70%, the relative surface excess is of the order of 0.1%. At this volume fraction, the surface stress has already reached its asymptotic value. Thus, the plastic strain of the surface is smaller than 0.1%. 

It should be evident from the foregoing discussion that the property defined to have zero surface excess may be chosen at will, the choice being governed by the experimental or mathematical features of the problem at hand. Choosing the surface excess number of moles of one component to be zero clearly simplifies Equation (44). The same simplification could have been accomplished by defining the mathematical surface so that Y2 would be zero, a choice that would obviously deemphasize the solute. If the total number of moles N, the total volume V, or the total weight W had been the property chosen to show a zero surface excess, then in each case both T, and Y2 (which would be identified as TN, rK, or T for these three conventions) would have nonzero values. Last, note that the surface excess is an algebraic... 

The Gibbs convention (Equation 17) states that we compare the real system with a fictitious one having the same total volume and total numbers of moles of all constituents as the real system. Under this convention the total moles of individual components 1 and 2 will differ between the real and the fictitious systems, but because the total of all moles of both components is the same, the surface excesses of each must sum to zero, and this is the meaning of Equation 20. 

It is a basic assumption that the preferential adsorption of ions of one sign is the sole source of the electric potential difference between the surface and the bulk solution. The counterions and similions not actually on the surface are assumed to be distributed according to the Boltzmann law, Equation 3, which with Equations 2a and 2b yields Equation 4. The infinite flat plate case is treated first. The volume element, dv, has been replaced by Adi and the exponentials have been combined to give the cosh u term. The factor 104 permits the area to be in square meters, consistent with the units of surface excess. 

In order to calculate the surface excess (which is a composite isotherm) from the derived individual isotherm for component two, Equation 6, a further assumption is necessary. That chosen is that the partial molecular volumes of the two components in the surface-containing region are the same as in the bulk solution, even though some of the ions are attached to the surface. The error introduced by this assumption leads to a volume of the surface region which might be somewhat too large, but the primary effect is in the second term in Equation 8b, which in these calculations is always less than 10 3 times the value of the first term. The error in the calculated surface coverage at saturation is probably less than 1%. 

For use in these equations, n2s is obtained from Equation 6. The volume of a sodium beta-naphthalenesulfonate molecule, calculated from bond lengths and appropriate van der Waals radii, is taken to be 330 A.3. An average molecular volume of water of 30 A.3 was calculated from the density of water at 25.0°C. Most of the numerical work was done on a Honeywell 800 digital computer. The symmetric surface excess and the surface charge densities were calculated over a wide range of surface potentials and concentrations. 

The equation also applies to the adsorption of a gas on a solid. At low gas pressures, p. the equilibrium pressure of the gas can be substituted for a, the activity of the solute. The amount of gas adsorbed v/V is equivalent to the surface excess T, where v is equal to the volume of gas adsorbed per gram of solid and V is the molar volume of the gas. The total free energy change at constant pressure is E[Pg.1582]

In TIRF protein adsorption experiments, it is desirable to correlate the intensity of excited fluorescence with excess protein concentration at the interface. Such an adsorbed layer is often in equilibrium with bulk-nonadsorbed protein molecules which are also situated inside the evanescent volume and thus contributing to the overall fluorescence. Various calibration schemes were proposed, using external nonadsorbing standards40,154 , internal standard in a form of protein solution together with a type of evanescent energy distribution calculation 154), and independent calibration of protein surface excess 155). Once the collected fluorescence intensity is correlated with the amount of adsorbed protein, TIRF can be applied in the study of various interactions between surface and protein. 

The competitive adsorption of a short symmetric PS-PI diblocks or a long asymmetric PS-PI diblock to the surface of a PS homopolymer was examined by Budkowski etal. (1995).They used nuclear reaction analysis (Section 1.4.18) with labelled diblocks to determine the concentration of deuterium atoms as a function of depth, and hence the volume fraction of labelled chains. It was thus found that the shorter diblock tends to adsorb preferentially to the interface. The surface excess of PS and its interfacial density were compared to a theory for bidisperse brushes, a generalization of the model due to Leibler (1988). Excellent quantitative agreement was found, with no adjustable parameters. 

FIGURE 3.6 Comparison of surface excess free energy (AGs) and volume excess free energy (AGv) as functions of cluster size. (Reproduced and modified from Larson, M.A., Garside, J., Chem. Eng. Sci., 41, 1285 (1986). With permission from Elsevier.)... 

An adsorption isotherm is a graph of the amount adsorbed versus the pressure of the vapor phase (or concentration in the case of adsorption from solution). The amounts adsorbed can be described by different variables. The first one is the surface excess I in mol/m2. We use the Gibbs convention (interfacial excess volume Va = 0). For a solid surface the Gibbs dividing plane is localized directly at the solid surface. Then we can convert the number of moles adsorbed Na to the surface excess by... 

The surface excess amount, or Gibbs adsorption (see Section 6.2.3), of a component i, that is, /if, is defined as the excess of the quantity of this component actually present in the system, in excess of that present in an ideal reference system of the same volume as the real system, and in which the bulk concentrations in the two phases stay uniform up to the GDS. Nevertheless, the discussion of this topic is difficult on the other hand for the purposes of this book, it is enough to describe the practical methodology, in which the amount of solute adsorbed from the liquid phase is calculated by subtracting the remaining concentration after adsorption from the concentration at the beginning of the adsorption process. 

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