Equimolecular surface

Consider the boundary between an aqueous solution of a nonionic surfactant and a hydrophobic phase, air or oil. The dividing surface is usually chosen to be the equimolecular surface with respect to water, that is F , = 0. Then Equation 5.1 reduces to da = -F,r/pj, where the subscript 1 denotes the surfactant. Because the bulk surfactant concentration is usually not too high, we can use the expression for the chemical potential of a solute in an ideal solution Pj = + A rinc, where k... 

It is now evident that depending on the choice of the dividing surface position, the adsorption of component 1 can be either positive or negative (corresponding to a deficiency of a component within the discontinuity surface), or zero (note that the surface free energy is independent of the dividing surface position, see Chapter 1,1). The dividing surface, the position of which is chosen in such a way that T, = 0, is referred to as the equimolecular surface with respect to component 1 (i.e. the solvent). 

To illustrate the applicability of the above general expression, let us consider a specified system a microemulsion containing spherical droplets [D = 0, cf. Eq. (97)]. A typical microemulsion system contains the following components water (yv), oil (o), surfactant (s), cosurfactant (c), and neutral electrolyte (e) see Fig. 12. We choose the dividing surface to be the equimolecular surface with respect to water (i.e., F , = 0). Then, at constant temperature T and chemical potential of the oil phase, (jlq, and for quasistatic processes without shear deformation (7 = 8p = 0), Eq. (101) can be transformed to read [210]... 

The theoretical treatment which has been developed in Sections 10.2-10.4 relates to mass transfer within a single phase in which no discontinuities exist. In many important applications of mass transfer, however, material is transferred across a phase boundary. Thus, in distillation a vapour and liquid are brought into contact in the fractionating column and the more volatile material is transferred from the liquid to the vapour while the less volatile constituent is transferred in the opposite direction this is an example of equimolecular counterdiffusion. In gas absorption, the soluble gas diffuses to the surface, dissolves in the liquid, and then passes into the bulk of the liquid, and the carrier gas is not transferred. In both of these examples, one phase is a liquid and the other a gas. In liquid -liquid extraction however, a solute is transferred from one liquid solvent to another across a phase boundary, and in the dissolution of a crystal the solute is transferred from a solid to a liquid. 

For mass transfer to a surface, a similar relation to equation 12.117 can be derived for equimolecular counterdiffusion except that the Prandtl number is replaced by the Schmidt number. It follows that ... 

Obtain the Taylor-Prandtl modification of the Reynolds Analogy between momentum transfer and mass transfer (equimolecular counterdiffusion) for the turbulent flow of a fluid over a surface. Write down the corresponding analogy for heat transfer. State clearly the assumptions which are made. For turbulent flow over a surface, the film heat transfer coefficient for the fluid is found to be 4 kW/m2 K. What would the corresponding value of the mass transfer coefficient be. given the following physical properties ... 

Determination of adsorption of any component is made under the condition of a special location of the dividing surface, which corresponds to This surface is referred to as equimolecular for component i. 

The nature of the surface of alumina is important in its application as an adsorbent for chromatography. This subject is covered by special literature. Inorganic ions can be separated on alumina as well as on silica (241). Anions are adsorbed together with bivalent cations, but not together with univalent cations indicating, again, equimolecular... 

Strictly speaking, such position of the dividing surface differs from that of the equimolecular one (with respect to solvent), however, the difference between the two is too small to cause any significant influence on the results of the present treatment... 

According to the Gibbs phase rule, at constant temperature and volume the binary two-phase model system has only one degree of freedom, meaning that only one variable in eq. (II.4) is independent. It is possible then to replace partial derivatives by full ones. The treatment is simpler for a surface that is equimolecular with respect to solvent, for which T, = 0 the individual subscripts are no longer needed and can be omitted (i.e., p = p2 and T = T2), and eq. (II.4) becomes... 

The adsorption of the dye wool violet 5BN, ha ving the formula C41H45X8O (SOaNa), on the surface of an aged precipitate of lead chromate was studied, by colorimetric analj. sis of a solution of the dye before and after equilibration with precipitate, by Kolthoff and Eggertsen (J. Amer. Chem. Soc. 1940, 62, 2125). The adsorption reached a limiting value equivalent to 4.8 mg solid dye/g PbCrO pro ided that the solution also contained sufiftcient lead nitrate. Additional experiments using the radioactive lead isotope ThB as an indicator showed that equimolecular amounts of dye and lead ions were removed from solution. 

Let us consider the boundary between an aqueous solution of a nonionie surfaetant and the oil phase. We ehoose the dividing surface to be the equimolecular dividing surface with respect to water. The Gibbs adsorption equation then takes the form (6, 7) ... 

Equimolecular mixtures of H2SO4 and TFA supported on Si02 have also been claimed as suitable alkylation catalysts. Here, the catalyst is characterized by a specific surface area of up to 40 m /g, containing an equimolecular... 

The regioselective acylation of aromatic ethers with carboxylic acids (aromatic ether/carboxylic acid ratio =1) can be pafonned with an equimolecular mixture of trifluoroacetic anhydride adsorbed on the surface of alumina without any solvent [77]. The process can be applied, with nearly quantitative yields, to anisole and the three isomeric dimethoxybenzenes by using carboxylic acids. The authors outline that in the case of anisole, the acylation selectively occurs at the para position to the methoxy group. The reaction requires a large amount of alumina and trifluoroacetic anhydride, and consequently, it can only be appUed at the laboratory scale. The intervention of a mixed carboxyhc acidArifluoroacetic acid mixed anhydride intermediate is presumed. 

The summation in Equation 4.1 is carried out over aU components. UsnaUy an equimolecular dividing surface with respect to the solvent is introduced for which the adsorption of the solvent is set zero by definition [4,5]. Then the snmmation is carried ont over aU other components. Note that F, is an excess surface concentration with respect to the bulk F is positive for surfactants, which decreases o in accordance with Equation 4.1. On the contrary, F is negative for aqneous solutions of electrolytes, whose ions are repelled from the surface by the electrostatic image forces [5] consequently, the addition of electrolytes increases the surface tension of water [6]. For surfactant concentrations above the critical micellization concentration (CMC) = constant and, consequently, a = constant (see Equation 4.1). 

Raoult s Law. When one liquid is dissolved in another, the partial pressure of each is decreased. Assume two liquids, the molecules of which are the same size and which mix without the complicating effects of molecular association, chemical combination, and the like. In an equimolecular mixture of two such liquids, each unit of surface area of the liquid mixture will have in its surface half as many molecules of each component as exist in the liquid surface of that component in the pure state. Hence the escaping tendency or partial pressure of each component in the mixture will be half that of the same component in the pure state. Similarly, in a mixture containing 26 mol per cent of the first component and 75 mol per cent of the second, the first will exert a partial pressure 25 per cent of that of this component in the pure state. In more general terms, for any such mixture the partial pressure of any component will equal the vapor pressure of that component in the pure state times its mol fraction in the liquid mixture. This generalization is known as Raoult s law (Ref. 19). It is expressed in the relationship, = PaXa, where pa is the partial pressure of the component A in the solution, Xa is its mol fraction in the solution and... 

The second equation in Eq. (105) represents a form of the Laplace equation of capillarity. For the so-called surface of tension, P = 0 by definition [143,208] then, 7 = 0 and Eq. (105) coincides with Eq. (95). On the other hand, if the Gibbs dividing surface is defined as the equimolecular dividing surface (the surface for which the adsorption of solvent is equal to zero see Refs. 8 and 207), then B is not zero and the generalized Laplace equation should be used. It is interesting to note that for a flat intermolecular dividing surface, B... 

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