Surface excess entropy

Harkins et al., 1940), where ne is the ESP, and Ae is the average area per molecule at the ESP as obtained from the 11/ 4 isotherm of the spread film. The temperature dependence of the ESP may then be used to calculate the excess surface entropies from (5) and enthalpies of spreading from (6). 

An alternative derivation of the same equations is possible. Imagine two equidistant planes separated at a distance h. The volume conhned between the two planes is thought to be filled with the bulk liquid phase (I). Taking surface excesses with respect to the bulk phases we can derive Equations 5.132 and 5.133 with s and F. being the excess surface entropy and adsorption ascribed to the surfaces of this liquid layer. A comparison between Equations 5.132 and 5.130 shows that there is one additional differential in Equation 5.132. It corresponds to one supplementary degree of freedom connected with the choice of the parameter h. To specify the model we need an additional equation to determine h. Eor example, let this equation be... 

If the excess components of y and dueto orientational entropy, are, respectively, 61 and i.e., excess surface entropy over that corresponding to a normal Eotvos constant [17]—Equation 1 becomes... 

Here A is the surface area eq. (11) is called the Gibbs-Helmholtz equation. Thus it is possible to estimate and the excess surface entropy per unit area, S IA, from a measurement of y T). 

T, may become positive or negative, depending on the particular interface in question. Other surface excess properties, such as the surface internal energy and surface entropy, are defined similarly ... 

For pure liquids the description becomes much simpler. We start by asking, how is the surface tension related to the surface excess quantities, in particular to the internal surface energy and the surface entropy ... 

If the temperature dependence of the surface tension is known, it is possible to obtain additional structured information by thermodynamic means. In principle, by differentiation of the surface tension with respect to the temperature we can obtain the surface excess entropy, which carries such information. However, the required analysis demcmds some scrutiny. Consider (4.2.71, which is rigorous (at given pressure) and contains S°, the excess entropy per unit cirea with reference to the surface entropy that a reference system with mole fractions (1 - x) and x would have up to the surface. From (4.2.7] we eliminated one of the chemiccd potentials using the Gibbs-Duhem rule as (1 - x)d/ij + xdp = 0. However, if the temperature becomes variable, the Gibbs-Duhem rule must be extended to... 

The derivative of the surface tension with respect to temperature at the interface between condensed phases in binary systems can be either positive, or negative, or even change its sign when the temperature changes, which makes it different from the vapor-liquid interface in a one-component system. Within a certain approximation one may assume that in binary systems, as in single-component ones, the value r = -do/dT is the excess of entropy within the discontinuity surface. Consequently, for the interface between condensed phases, the excess of entropy can not only be positive (as it was with singlecomponent systems), but also negative. This situation is especially typical for the interface between two mutually saturated liquid solutions. 

In systems with the upper critical temperature, rcu, the surface tension decreases with increasing temperature, and consequently the excess of entropy within the surface layer is positive (Fig. III-l, a). In systems with the lower critical temperature, TCL, (Fig. III-l, b) the increase in interfacial tension is observed above the point at which system separates into two phases the value of tj is hence negative. The latter may serve as evidence for the existence of strong coorientation between molecules within the interfacial layer, which is due to the presence of directed chemical bonding, such as hydrogen bonding. 

In systems having a closed region of phase separation (Fig. III-l, c) the temperature as a function of surface tension passes through a maximum the value of a approaches zero in the vicinity of both the lower and higher critical points. In such a case the excess of entropy within the interfacial layer is... 

The temperature coefficient of surface tension is plotted versus composition in Figure 8. One recognizes that the blend exhibits the temperature coefficient of PMA as long as PMA is in excess. At lower content of PMA, the coefficient steeply ascends to the temperature coefficient of PEO. This indicates that surface entropy of the blend is ruled by PMA in the range of low content of PEO. 

Here, T and are temperature and chemical potentials dU dS and dN i denote the excess surface internal energy, entropy, and number of molecules, respectively, of the ith component, belonging to the elementary parcel dA the symbol 8 denotes infinitesimal variation due to the occurrence of a thermodynamic process in the system. Then, one obtains [143]... 

Staverman uses Flory s model as a starting point, but introduces a form factor in the number of complexions, taking account of the fact that the molecules are only in contact at their surfaces. This enables him to construct a new expression for number of possibilities of introducing the molecule of polymer and the molecules of solvent. Thus, the excess molar entropy term of conformation is altered, and becomes ... 

Table 14 Equilibrium spreading pressures IF and surface excess free energies, entropies, and enthalpies of spreading for first and second eluting C-15 6,6 -A amide diacids. 

The expressions in the two parentheses can be identified as the surface excess moles and surface excess entropy defined by eqs. (6.2) and (6.5). Equation (6.12) thus reduces to... 

The exact position of the geometrical surface can be changed. When the location of the geometrical surface X is changed while the form or topography is left unaltered, the internal energy, entropy and excess moles of the interface vary. The thermodynamics of the interface thus depend on the location of the geometrical surface X. Still, eq. (6.13) will always be fulfilled. 

---