Effects of Higher-order Electrostatic Terms

As shown in Chapter V. Pitzer included terms to account for the electrostatic effects of unaymmetrical mixing in his method. He stated (P5) that, while the effect on 1-2 mixing is small and may be safely omitted, it should be included when doing calculations for 3-1 and greater unsymmetrical types of mixing, 

Friedman (12), in 1962. proposed that there is a higher-order electrostatic effect, or limiting law, even when mixing ions of the same charge, Pitzer presented Friedmans equations for Gibbs excess free energy in a molality basis and omitted terms occurring due to differences in concentration, molality, etc, as these are dealt with by Pitzer s virial coefficients  

Ajjj - Debye-Hiickel coefficient for the osmotic coefficient 

Pitzer used the following expression to define the excess free energy  

The first terms on the right of equations (A.5.1) and (4.57) are expressions based on the Debye-Huckel limiting law. and Pitzer notes that the coefficients X correspond to the Jjj terms. 

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Pitzer, K. S., 1975, Thermodynamics of electrolytes, V, effects of higher order electrostatic terms. Journal of Solution Chemistry 4, 249-265. 

Pitzer, K. S. "Thermodynamics of Electrolytes. V. Effects of Higher-order Electrostatic Terms" J. Sol. Chem., 1975,... 

Thermodynamics of electrolytes. V. Effects of higher-order electrostatic terms. 

Pitzer, KS. (1975). Thermodynamics of electrolytes. V. Effects of higher-order electrostatic terms, J. Solution Chem, Vol. 4, No. 3, p>p. 249-265, ISSN 0095-9782 Pitzer, KS. (1977). Electrolytes theory-improvements since Debye-Hiickel, Account Chem. 

For an isolated system there are no external influences so the first term of Equation A4 is zero. The effective pair potential is a function of the pair separation = r, — r jl and is constructed in such a way as to include the true pair potential and average effects of higher order terms it often includes also electrostatic and dipolar effects. The total potential energy of the system is then a sum over all distinct pairs of particles... 

The observed elastic shear constant C44 of aluminum is 2,8 x 10" crg/cm, whereas the electrostatic contribution is 14.8 x 10" erg/cm (Harrison, 1966a, p, 179 the value there was based upon an effective charge 7.9 percent larger tlian the 3,0 appropriate here). The band-structure energy is an estimate of the difference, —12.0 x 10" crg/cm. Even if we sum all terms, the result is approximate because of the neglect ofterms of higher order than two and the use of an approximate pscudopoteiitial. We obtain the effect of the nearest lattice wave numbers here. 

The effective potential in this formulation V ff, is not the same as Vgff iii Eq. [45]. Whereas an effective polarization was treated in Eq. [45] by assigning fitted charges commensurate with an enhanced dipole moment, the charges in Eq. [46] should correspond to the gas phase moments. Although the self-consistent dipole polarization does select all N-body interactions arising from dipole polarization, the corrections to higher order electrostatic moments— especially the quadrupole moments— are not properly accounted for. Three-body corrections to dispersion and repulsive components of the potential still remain in the effective two-body V ff term. 

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