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Quadrupole terms in the Maxwell equations: Born energy, partial molar volume and entropy of ions. Debye-Hückel theory in a quadrupolarizable medium

A new equation of state relating the macroscopic quadrupole moment density $Q$ to the gradient of the field $\nabla E$ in an isotropic fluid is derived: $Q = α_Q(\nabla E - U \nabla.E/3)$, where the quadrupolarizability $α_Q$ is proportional to the squared molecular quadrupole moment. Using this equation of state, a generalized expression for the Born energy of an ion dissolved in quadrupolar solvent is obtained. It turns out that the potential and the energy of a point charge in a quadrupolar medium are finite. From the obtained Born energy, the partial molar volume and the partial molar entropy of a dissolved ion follow. Both are compared to experimental data for a large number of simple ions in aqueous solutions. From the comparison the value of the quadrupolar length $L_Q$ is determined, $L_Q = \sqrt{α_Q/3ε}= 1-2 Å$. Further, the extended Debye-Hückel model is generalized to ions in a quadrupolar solvent. If quadrupole terms are allowed in the macroscopic Coulomb law, they result in suppression of the gradient of the electric field. In result, the electric double layer is slightly expanded. The activity coefficients obtained within this model involve three characteristic lengths: Debye length, ion radius and quadrupolar length $L_Q$. Comparison to experimental data shows that minimal distance between ions is equal to the sum of their bare ion radii; the concept for ion hydration as an obstacle for ions to come into contact is not needed for the understanding of the experimental data.