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ION–SOLVENT INTERACTIONS 209

By similar reasoning,

By using Eqs. (A2.2.7) and (A2.2.8), Eq. (A2.2.2) becomes

Since zlr = cos and 2dq is the dipole moment

or the ion–dipole interaction energy is given by

APPENDIX 2.3. INTERACTION BETWEEN AN ION AND A WATER
QUADRUPOLE
Instead of presenting a sophisticated general treatment for ion–quadrupole inter-
actions, a particular case of these interactions will be worked out. The special case to
be worked out is that corresponding to the water molecule being oriented with respect
to a positive ion so that the interaction energy is a minimum.
In this orientation (see Fig. A2.3.1), the oxygen atom and a positive ion are on the
y axis, which bisects the H–O–H angle. Further, the positive ion, the oxygen atom,
and the two hydrogen atoms are all considered in the xy plane. The origin of the xy
coordinate system is located at the point Q, which is the center of the water molecule.
The ion is at a distance r from the origin.
The ion–quadrupole interaction energy is simply given by the charge on the
ion times the potential at the site of the ion due to the charges of the quadrupole,

The potential is the sum of the potentials due to the four charges and
in the quadrupole (1 and 2 are the positive charges at the hydrogen, and 3 and 4 are
the negative charges at the oxygen). That is,