Ice I_h - Proton Disorder

Proton Disorder

The protons (hydrogen atoms) in the crystal lattice lie very nearly along the hydrogen bonds, and in such a way that each water molecule is preserved. This means that each oxygen atom in the lattice has two protons adjacent to it, at about 101 pm along the 275 pm length of the bond. The crystal lattice allows a substantial amount of disorder in the positions of the protons frozen into the structure as it cools to absolute zero. As a result, the crystal structure contains some residual entropy inherent to the lattice and determined by the number of possible configurations of proton positions which can be formed while still maintaining the requirement for each oxygen atom to have only two protons in closest proximity, and each H-bond joining two oxygen atoms having only one proton. This residual entropy S₀ is equal to 3.5 J mol−1 K−1.

There are various ways of approximating this number from first principles. Suppose there are a given number N of water molecules. The oxygen atoms form a bipartite lattice: they can be divided into two sets, with all the neighbors of an oxygen atom from one set lying in the other set. Focus attention on the oxygen atoms in one set: there are N/2 of them. Each has 4 hydrogen bonds, with two hydrogens close to it and two far away. This means there are

allowed configurations of hydrogens for this oxygen atom. Thus there are 6N/2 configurations that satisfy these N/2 atoms. But now consider the remaining N/2 oxygen atoms: in general they won't be satisfied (i.e., they won't have precisely two hydrogen atoms near them). For each of those, there are

possible placements of the hydrogen atoms along their hydrogen bonds, of which 6 are allowed. So, naively, we would expect the total number of configurations to be

Using Boltzmann's principle, we conclude that

where is the Boltzmann constant, which yields a value of 3.37 J mol−1 K−1, a value very close to the measured value. This estimate is 'naive' because it assumes the 6 out of 16 hydrogen configurations for oxygen atoms in the second set can be independently chosen, which is false. More complex methods can be employed to better approximate the exact number of possible configurations, and achieve results closer to measured values.

By contrast, the structure of ice II is very proton-ordered, which helps to explain the entropy change of 3.22 J/mol when the crystal structure changes to that of ice II. Also, ice XI, an orthorhombic, proton-ordered form of ice I_h, is considered the most stable form.