Theoretical calculation of the water ion product K_W

A dielectric solvation model is applied to the prediction of the equilibrium ionization of liquid water over a wide range of density and temperature with the objective of calibrating that model for the study of ionization in water of organic acids, {\it e.g.\/}, proteins and nucleic acids. The model includes an approximate description of the polarizability of the dissociating water molecule. The calculated pK$_W$ are very sensitive to the value of the radii that parameterize the model. The radii required for the spherical molecular volumes of the water molecule in order to fit the experimental ion product are presented and discussed. These radii are larger than those commonly used. They decrease with increasing density as would be guessed but the rate of decrease is slight. They increase with increasing temperature, a variation opposite to what would be guessed if radii were strictly viewed as a distance of closest approach. The molecular theoretical principles that might provide an explanation of the thermodynamic state dependence of these radii are discussed.