Surface, equimolar

Figure 3.2 Variation of excess chemical potential of CHjF as a function of distance of the carbon atom from the liquid water-hexane interface at 310K (Pohorille and Wilson, 1996 Pratt and Pohorille, 2002). The hydrophobic contribution, obtained by eliminating electrostatic interactions, is the dot-dash curve and the electrostatic contribution is the dashed curve, lowest on the right. The water equimolar surface is at z = 0. The combination of these two contributions leads to interfacial activity for this simple solute.

As the interface is diffuse, h can be only defined as a nominal interface position, which may be identified with the location of a particular isodensity level. The most natural choice is the Gibbs equimolar surface, which satisfies the relation... 

Suppose now that the interface is weakly curved, so that isodensity levels do not coincide anymore with planes 2 = const. The nominal location of a curved diffuse interface (e.g. the Gibbs equimolar surface) can be used to describe it in the language of differential geometry commonly applied to sharp interfaces. Its spatial position can be defined in a most general way as a vector function X(4) of surface coordinates 4- A curved interface can be approximated locally by an ellipsoid with the half-axes equal to the principal curvature radii. If both radii far exceed the characteristic interface thickness, all isodensity levels are approximated by ellipsoidal segments equidistant from the interface. The density changes along the direction 2 normal to isodensity surfaces, and the... 

FIG. 2 Density profile in the interfacial region with the definition of the equimolar surface as the location where the excess density vanishes. 

Since the number density p is uniquely defined it follows from (2.41) that there is no arbitrariness about the adsorptions with respect to the equimolar surface, that is r ,p) = ir and F (p) = <. However the values of At) and A depend on the zeros of entropy and energy, so it follows from (2.42) and (2.43) that Fp( , and Fp( > are not uniquely defined. It is convenient to consider only the configurational part of entropy and energy. 

For a planar surface TJAp is (z -zj, die separation of the equimolar surface and the surfiuie of tension (2.48). For a curved suif ... 

The equation equivalent to (4.112) for d, the separation of the equimolar surface and the surface of tension defined via (2.89) and the pressure tensor of Irving and Kirkwood, is ... 

Consider a flat gas-liquid surface lying in the x, y-plane and take the origin of the coordinate system to lie on a convenient dividing surface, e.g. the equimolar surface, Fp = 0. Let u(r) be an external potential which deforms the flat surface into a spherical one with radius of curvature R (Fig. 4.2). The potential can be arbitrarily weak since we shall be interested only in the limiting behaviour R ->0. If Ap(t) is the change of density on applying the potential... 

We have seen in 4.8 the difficulty of reconciling the arbitrariness of the surface of tension, z defined medianically (or quasi-thermodynamically) by (2.89) and the planar limit of the surface of tension of a drop, introduced through the thermodynamic arguments of 2.4—arguments which fix its position with respect to the equimolar surface, R. One way of analysing problems of this kind is by exact calculations for a model system. The mean-field treatment of the penetrable-sphere model is not exact, but, as we have seen, it becomes so in the two limits of (1) infinite dimensionality at all temperatures, and (2) zero temperature for all dimensions. Here we examine the three-dimensional spherical drop (and bubble) in the mean-field approximation and show that the results resolve some of the difficulties of 4.8. 

We have therefore two independent equations for R the radius of the surface of tension, or, what is equivalent, for S, its separation from the equimolar surface. Figure 5.13 ows aja- as a function of Rj. The points are the values of c, calculated from (5.152) and the straight line has a slope -28, with S calculated by eliminating cr, between the two thermodynamic equations (5.153), viz.. 

It is noteworthy that tr, is greater than zero temperature since S can then be shown to be exactly Calculations for different sizes of drop and for different temperatures show that 8 is always negative and that it changes only slowly with these variables. Within the accuracy of the calculations, the apparent limit at T-0 and Re 0 agrees with the exact result of At T = 0 the mean-field approximation becomes exact, and there is no uncertainty in the calculation of o-, by... 

Thompson and Gubbins have studied diatomic molecules, using a site-site Lennard-Jones potential with two sites on each molecule. If these sites are dose together, as in a potential chosen to represent N2, then there is no discernible orientation in the surface layer. If they are well-separated, as in a potential with a separation of 0-608d chosen to represent Q2, then at low temperatures those molecules just below the equimolar surface tend to be aligned perpendicular to the surface, and... 

It was pointed out that a bimolecular reaction can be accelerated by a catalyst just from a concentration effect. As an illustrative calculation, assume that A and B react in the gas phase with 1 1 stoichiometry and according to a bimolecular rate law, with the second-order rate constant k equal to 10 1 mol" see" at 0°C. Now, assuming that an equimolar mixture of the gases is condensed to a liquid film on a catalyst surface and the rate constant in the condensed liquid solution is taken to be the same as for the gas phase reaction, calculate the ratio of half times for reaction in the gas phase and on the catalyst surface at 0°C. Assume further that the density of the liquid phase is 1000 times that of the gas phase. 

Let us thus consider a model in which the association energy depth changes when two reacting particles are approaching the surface see Refs. 86,90. If in the vicinity of the surface the binding energy is lower than it is far from the surface, the probability of the chemical reaction to occur in the surface zone decreases. Similarly to the previous case, we consider an equimolar mixture of associating hard spheres of equal diameters. The interaction between the species a and (3 is assumed in the form... 

The synthesis and surface-active properties of higher hydroxyalkanediphos-phonates are discussed in Ref. 67. Phosphorus-containing betaines as hydrolytically stable surfactants, free from alkali salt impurities, were prepared by a reaction of amidoamines and equimolar amounts of phosphonate esters with 1.5-2 eq of formaldehyde at 60-140°C in a polar solvent [72]. 

For forced-convection studies, the cathodic reaction of copper deposition has been largely supplanted by the cathodic reduction of ferricyanide at a nickel or platinum surface. An alkaline-supported equimolar mixture of ferri- and ferrocyanide is normally used. If the anolyte and the catholyte in the electrochemical cell are not separated by a diaphragm, oxidation of ferrocyanide at the anode compensates for cathodic depletion of ferricyanide.3... 

In this equation the entire exterior surface of the catalyst is assumed to be uniformly accessible. Because equimolar counterdiffusion takes place for stoichiometry of the form of equation 12.4.18, there is no net molar transport normal to the surface. Hence there is no convective transport contribution to equation 12.4.21. Let us now consider two limiting conditions for steady-state operation. First, suppose that the intrinsic reaction as modified by intraparticle diffusion effects is extremely rapid. In this case PA ES will approach zero, and equation 12.4.21 indicates that the observed rate per unit mass of catalyst becomes... 

Goldberg (2002) found no evidence of any competition in sorption of arsenate and arsenite on Al or Fe-oxides and montmorillonite, but only a small and apparent competitive effect of equimolar arsenate on arsenite sorption on kaolinite and illite. The minor competitive effect in this study was due to the small concentrations of arsenic which is very low for saturation site. Competition for sorption sites is evident by increasing the surface coverage of the sorbents. Arsenate prevents arsenite sorption on metal oxides when the surfaces of the sorbents are saturated by the anions (Jain and Loeppert 2000 Violante and Pigna 2002). 

Scheme V. Representation of the catalytic p-type Si photocathode for Ht evolution prepared by derivatizing the surface first with Reagent III followed by deposition of approximately an equimolar amount of Pd(0) by electrochemical deposition. The Auger/depth profile analysis for Pd, Si, C, and O is typical of such interfaces (49) for coverages of approximately 10 8 mol PQ2 /cm2.

First, consider the gradient of cA. Since A is consumed by reaction inside the particle, there is a spontaneous tendency for A to move from the bulk gas (cAg) to the interior of the particle, first by mass transfer to the exterior surface (cAj) across a supposed film, and then by some mode of diffusion (Section 8.5.3) through the pore structure of the particle. If the surface reaction is irreversible, all A that enters the particle is reacted within the particle and none leaves the particle as A instead, there is a counterdiffusion of product (for simplicity, we normally assume equimolar counterdiffusion). The concentration, cA,at any point is the gas-phase concentration at that point, and not the surface concentration. 

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