Structure of Water and Hydrophobic Bonding in Proteins. I. A Model for the Thermodynamic Properties of Liquid Water

The thermodynamic parameters of liquid water are derived by means of a statistical thermodynamic treatment, based on the ``flickering cluster'' model proposed by Frank and Wen. Various models proposed for the structure of liquid water are reviewed, and the advantages of the Frank—Wen model are pointed out. The hydrogen-bonded ice-like clusters of H2O molecules in equilibrium with non-hydrogen-bonded liquid are described quantitatively in terms of the molecular species participating in different numbers of hydrogen bonds in the clusters. Equations expressing the mole fractions of the various species in terms of the average cluster size are derived. The partition function derived for the description of liquid water is based on a distribution of the H2O molecules over five energy levels, corresponding to four, three, two, one, and no hydrogen bonds per molecule. The most probable values of the average cluster size, the mole fraction of non-hydrogen-bonded water, and the thermodynamic parameters are obtained from the partition function. The energy of the hydrogen bond and the molecular ``free volume'' of the unbonded molecules are introduced as adjustable parameters; they are shown to have physically reasonable magnitudes. The average cluster size ranges from 91 to 25 H2O molecules over the temperature range from 0° to 70°C, with the mole fraction of non-hydrogen-bonded molecules increasing from 0.24 to 0.39 over the same range of temperature. The calculated values of the free energy, enthalpy, and entropy of liquid water in this temperature range agree with experimental data to within an error of less than 3%. The calculated temperature dependence of cv is too large. The calculated results agree well with the radial distribution curve derived from x-ray diffraction data. The compressibility and the thermal expansion are considered in terms of two contributing forms in the liquid: the hydrogen-bonded clusters and the more closely packed molecules. The P—V—T data derived for the close-packed species from the model and from experimental data are shown to have physically reasonable magnitudes. Several other properties are discussed qualitatively on the basis of the model. The model can also be extended to provide an explanation of the properties of aqueous solutions of nonpolar substances.