Explicit Expressions for the Excess Functions

The calculations necessary to obtain the explicit expressions for the excess functions are elementary but somewhat lengthy. We reproduce below as an example the calculation of the excess free energy. The other excess properties may be deduced exactly in the same manner. 

Let us separately consider these five contributions to the excess free energy. 

This can be calculated exactly as the first term. We obtain 

The evaluation of the third and fourth terms reqtdres the knowledge of VA — i AA. VB — VAA and vbb — vaa- These quantities can be deduced from the equation of state (10.3.1) and for the present purpose it is sufficient to retain the first order terms in 6, d and p. 

From this relation we can deduce va — vaa for b and va are related by (10.1.10). Remembering that here we have to keep only linear terms in fl, d and p, we have 

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Formnlae for the Excess Functions 185. 4. Explicit Expressions for the Excess... 

Primitive models have been very useful to resolve many of the fundamental questions related to ionic systems. The MSA in particular leads to relatively simple analytical expressions for the Helmholtz energy and pair distribution functions however, compared to experiment, a PM is limited in its ability to model electrolyte solutions at experimentally relevant conditions. Consider, for example, that an aqueous solution of NaCl of concentration 6 mol dm (a high concentration, close to the precipitation boundary for this solution) corresponds to a mole fraction of salt of just 0.1 i.e. such a solution is mostly water. Thus, we see that to estimate the density of such solutions accurately the solvent must be treated explicitly, and the same applies for many other thermodynamic properties, particularly those that are not excess properties. The success of the Triolo et approach can be attributed to the incor-... 

Equations 8 and 16 provide the basic thermodynamic equations which can predict the dependence on salinity, of the equilibrium radius r and the volume fraction at the transition between the region in which a microenulsion phase forms alone and that in which it coexists with an excess dispersed phase. Any addition to the system of excess dispersed phase having the same composition as the globules will change neither nor r in the microenulsion, as soon as the transition point is reached. To carry out such calculations, explicit expressions are needed for the interfacial tensionY as a function of the concentrations of surfactant and cosurfactant in the continuous phase, of salinity and radius r, as well as expressions for C and for the free energy Af. The interfacial tension depends on the radius for the following two reasons If the radius were increased at con-... 

The solute site charge fitting required in the site-site RISM-SCF treatment is eliminated for the ab initio MO method coupled with the 3D-RISM approach explicitly treating the solute electron distribution in the SCF loop. The effective potential of solvent acting on the solute electrons, F( ° )(r), is obtained by functional differentiation of the excess chemical potential of solvation with respect to the electron density distribution of the solute, Eq. (4.93). In the 3D-KH as well as 3D-HNC approximations (4.15) and (4.14) this leads to the solvent effective potential in the mean field form (4.101), expressed in terms of the pseudopotential of a solvent molecule acting on an external electron. It comprises partial contributions v r) centered on the interaction sites of the solvent molecule. The classical effective potential energy of the solute acting on solvent site 7,... 

Equation (15) can also be considered as a general form of adsorption (excess) isotherms applicable for liquid free surfaces. For example, let us suppose that the differential function of the measured relationship y versus a can be expressed in the following explicit form ... 

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