MODIFIED HELGESON, KIRKHAM AND FLOWERS EQUATION OF STATE FOR STANDARD PARTIAL MOLAL PROPERTIES OF IONS

An equation of state for standard thermodynamic properties of aqueous species at P and T above the critical point of H₂O was developed by Helgeson and Kirkham (1976) and Helgeson, Kirkham and Flowers (1981) and later revised (revised HKF). The electrostatics in the revised HKF equations are based on Born charging from a vacuum to a dielectric medium. This produces a linear dependence of standard Gibbs energy on the inverse of the ion’s radius, 1/r_i. In order to fit the data available r_i of cations and anions are a function of P and T. Also r_i for cations and anions of the same crystallographic size are different which contravenes Born charging theory. Because of the quadrupole properties of H₂O it is proposed that in addition to a Born charging term the Gibbs energy of ion-water-dipole and ion-water-quadrupole interactions G^o_ion-q be explicitly considered:

G^o_ion-q = - n_iN^o|Z_i|e_oμ_H2O/(r_i + r_H2O)² + n_iN^oZ_ie_oq_H2O/2(r_i + r_H2O)³

In the equation n_i is the number of H₂O molecules in the primary hydration sphere of ion i, N^o and e^_o are Avogadro's number and the electron charge, Z_i gives the ion's charge, μ_H2O stands for the dipole moment of a water molecules, r_H2Odenotes the radius of a water molecule and q_H2O is its quadrupole moment. As the equation indicates negatively charged ions will have a greater stability than positive ions of the same charge and size as experimental heat of hydration experiments indicate. The electrostatic contribution to standard Gibbs energy is then

G^o_i = (N^o(Z_ie_o)²/2(r_i + 2r_H2O)(1/ε-1) - n_iN^o|Z_i|e_oμ_H2O/(r_i + r_H2O)² + n_iN^oZ_ie_oq_H2O/2(r_i + r_H2O)³

where ε is the dielectric constant of water. A term for induced dipoles which vary as 1/(r_i+r_H2O)⁴ could also be added which gives a "fit" equation in terms of 1/(r_i+r_H2O)^n_j , n_j = (1, 2...). Provisions for transition metal ions with unsymmetrical 3d orbitals might also be needed as they have larger heats of solvation than s-orbital ions of similar charge and size.