Surface excess, definition equation

Equation (3) has the same form as one of Gibbs s fundamental equations for a homogeneous phase, and owing to this formal similarity the term surface phase is often used. It must be remembered, however, that the surface phase is not physically of the same definiteness as an ordinary phase, with a precise location in space neither do the quantities c , if, mf refer to the total amounts of energy, entropy, or material components present in the surface region as it exists physioally they are surface excesses , or the amounts by which the actual system exceeds the idealized system in these quantities. Care must be taken not to confuse the exact mathematical expression, surface phase , with the physical concept of the surface layer or surface film. 

We now return to the definition of the surface excess chemical potential fta given by Equation (2.19) where the partial differentiation of the surface excess Helmholtz energy, Fa, with respect to the surface excess amount, rf, is carried out so that the variables T and A remain constant. This partial derivative is generally referred to as a differential quantity (Hill, 1949 Everett, 1950). Also, for any surface excess thermodynamic quantity Xa, there is a corresponding differential surface excess quantity xa. (According to the mathematical convention, the upper point is used to indicate that we are taking the derivative.) So we may write ... 

Substituting definitions (4.2.3a and b) of the surface excesses in the Gibbs equation yields... 

Now, since the surface excess, [J) = Nf/ASNA] by definition from Equation (223), then Equation (442) becomes... 

In most cases of interest, the surface excess mass E is small, so that the acceleration and body force terms may be neglected. Then Equation 1.40 simplifies to two conditions. One of them, V y = 0, requires that interfacial tension be uniform. The other is the Young-Laplace equation (Equation 1.22), which was obtained previously from thermodynamics for situations where body force and acceleration terms were unimportant. That the same equation (Equation 1.22) results from independent thermodynamic and mechanical derivations implies that interfacial tension must have the same value whether it is defined as in Equation 1.9 from energy considerations or as in Equation 1.39 from force considraations. Simply put, the force and energy definitions of interfacial tension are eqnivalrait, a conclusion emphasized in the work of Buff (1956). 

Notice that since A/j. is negative then by definition IT is positive.) The excess surface energy,, is obtained from the product of the surface excess, r, and the change in chemical potential, provided the surface is flat. Using the above equation then... 

The Gibbs equation expresses the equilibrium between the surfactant molecules at the surface or interface and those in the bulk solution. It is a particularly useful equation since it provides a means by which the amount of surfactant adsorbed per unit area of the surface, the surface excess , may be calculated. The direct measurement of the surface excess provides almost insuperable experimental problems and hence the Gibbs equation is widely used as an alternative method of determining this quantity. In the derivation of this equation a definite boundary between the bulk of the solution and the interfacial layer is imagined (see Fig. 1.3). The real system containing the interfacial layer is then compared to this reference system in which it is assumed that the properties of the two bulk phases remain unchanged up to the dividing surface. 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

The excesses of the system can be represented by three-parameter quantities. For the existence of the equation of definition, the value of the factors can be chosen arbitrarily. The choice = 0 is permitted. When the values of the factors (v = l v = l / = 0) are chosen, the excess adsorption of a multicomponent system is equal to the adsorption capacity of the surface layer, i.e. to the real amount in the layer (Guggenheim excess if the factors are previously given, the relative Gibbs excess or the Findenegg excess, etc., can also be defined). In an inverse procedure the value of belonging to a given excess can be obtained. Thus, relationships of quantities that are inaccessible in the traditional set of tools can be derived. 

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). 

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