3 December 1956
To: Dr. Oleg Jardetsky
From: Linus Pauling
Subject: The Structure of Water
Your interest in the configuration of the water molecules about an ion in an ionic solution has led me to consider the question of the structure of water, and this consideration has resulted in the ideas described in the following paragraphs.
The calculations were made with my 5-inch slide rule, and will have to be checked.
Water has a definite density, 1.00 g/cm.4, essentially constant over the temperature range 0°C to 100°C, and over a moderate range of pressures. The density is, of course, determined by the interatomic distances and the mode of packing.
I think that it is likely that there is predominantly one structure in water, and we may accordingly search for a structure that leads to the correct value of the density of water.
There is evidence that, whereas the maximum number of hydrogen bonds are formed between water molecules in ice, somewhat less than the maximum number, approximately three-quarters perhaps, are present in liquid water at 0°C, and a still smaller number of higher temperatures. We may assume that the O-H•••O hydrogen-bond distance in water is 2.76 Å, the value for ice. I do not think that this value can be as much as 1% wrong. In addition, we may assume that the intermolecular distance for non-bonding contacts is somewhat greater than 2.76 Å.
The ice structures, similar to the structures of cristabolite and tridmite, lead to a value for the density about 10% too small.
The density of quartz is 10% greater than the density of cristabolite or that of tridymite, and accordingly the predicted density for water with the structure of quartz is about right. It is presumably for this reason that Bernal and Fowler suggested that water has a quartz-like structure. However, there is no reason to accept this suggestion, because it fails to account satisfactorily for the properties of water, in particular for its mobility. A quartz-like structure should be as difficult of deformation as a cristabolite-like structure or a tridymite-like structure, and if the interatomic interruptions were such as to stabilize a quartz-like structure (presumably very small, crystallite soft quartz with a quartz-like structure, free to roll over one another), rather than a similar structure based on the cristabolite or tridymite structure, for liquid water, then the same interatomic interactions ought to stabilize the quartz-like structures for ice. I think that for these reasons the quartz-like structure is not acceptable. I may mention that the stability of this structure for quartz itself is probably associated with the fact that the bond-angle on oxygen is about 135°. In ice or water we want the hydrogen bonds to be straight, or nearly straight.
Another more or less reasonable structure that may be considered is related to that of chlorine hydrate, the hydrates of the noble gases, methane, etc. These hydrates occur with either one of two structures. One of them was determined by Dr. Marsh and me, in 1952. You can get a reprint from Mrs. Wulf. For chlorine hydrate, 6Cl2•46H2O, the value of the cube edge is 11.82 Å, corresponding to a hydrogen-bond distance of about 2.75 Å. In this structure there are two pentagonal dodecahedra of water molecules, in a body-centered arrangement. This accounts for forty water molecules. Each of the pentagonal dodecahedra forms eight hydrogen bonds with the eight surrounding similar complexes, and has accordingly twelve oxygen atoms left to form hydrogen bonds. Six additional water molecules are introduced in such a way as to satisfy the remaining hydrogen-bond-forming powers of the two dodecahedra, giving a total of 46H2O in the molecule, within the cube, with each molecule forming four hydrogen bonds, essentially in tetrahedral directions. There are eight cavities in the unit cube, three in the centers of the two dodecahedra, with distance 3.08 Å to the twenty surrounding oxygen atoms, and six in positions where they are surrounded by twenty-four water molecules. In chlorine hydrate these six larger vacancies are occupied by Cl2 molecules, and the smaller ones are unoccupied, or perhaps occupied by oxygen molecules or nitrogen molecules. In methane hydrate and the hydrates of the noble gases all eight cavities are occupied: methane hydrate has the formula HCH4•46H2O.
Let us consider a crystal 8H2O•46H2O, with A0 = 11.82 Å. In this crystal forty-six of the water molecules form all four hydrogen bonds possible for them, and the other eight water molecules form no hydrogen bonds; hence 81% of the maximum number of hydrogen bonds are formed.
The volume of the unit is calculated to be 1000 cm.4 per mole, and the mass 980 grams. The predicted density is accordingly 0.98 g/cm.4, in satisfactory agreement with experiment.
However, this structure, like the quartz-like or cristabolite structure, is not satisfactory, because it is rigid, and does not provide an explanation of the mobility of liquid water. We need a structure in which there are hydrogen-bonded complexes that are capable of rolling over one another easily, without the rupture of a great number of hydrogen bonds.
Now let us assume that water has as its principle structural feature a complex of twenty-one water molecules, twenty of them forming a hydrogen-bonded pentagonal dodecahedron and the twenty-first, forming no hydrogen bonds, occupying the center of the dodecahedron. These complexes could form hydrogen bonds with one another, to some extent, as, for example, in the crystal of chlorine hydrate. The distance from the center of the dodecahedron to the corner is 3.87 Å, and half of the hydrogen-bond distance is 1.38 Å, so that the effective radius of the complex is 5.25 Å. Two such complexes would be expected to remain with their centers about 10.5 Å apart. We may expect these complexes to roll over one another, breaking hydrogen bonds and forming new ones, and the liquid might be described as similar to liquid metal, with the (H2O)21 complexes taking the places of the metal atoms. One complex might instantaneously be surrounded by eight others, as in the body-centered arrangement. There would then be small cavities, which could be occupied by additional water molecules forming hydrogen bonds with the large complexes. In this region of the liquid the structure would resemble that described above, and the density would be 0.98. Another possibility is that the arrangement be a face-centered arrangement, with each complex surrounded by twelve complexes. The edge of the cubic unit containing four of these complexes would then be about 14.8 Å. If there were also in this unit twelve additional water molecules (a structure of this sort has been described by Rodebush and a co-worker some time ago) the volume of the unit would be 19.60 cm.3 and the calculated density 100 g/cm.4
I think that this is the best description that can now be given of liquid water. I think that it would be worth while to check up on the X-ray diffraction liquid, and perhaps to discuss in greater detail the question of how the interstitial water molecules may be introduced.
One possibility that we might consider is that somewhat larger complexes are formed by condensation of the polyhedra. For example, a complex of thirty-seven water molecules can be formed by fusing two of the twenty-one H2O complexes together in such a way that they have a pentagonal face in common. A tetrahedral complex containing about sixty-five water molecules can be made by allowing each of four dodecahedral complexes to share a pentagonal face with each of three others. I have made a calculation that shows that the density is increased somewhat by this condensation process. If the condensation could be carried out completely, so that the structure consisted of dodecahedra filling space, the density would be 1.11; however, such a structure cannot be built.
The solution of methane, for example, in liquid water might occur by the displacement of the central water molecule of one of these complexes by a methane molecule. If this were so, possibly the properties of the solution would show a difference in concentration dependency when the concentration reached that of one methane molecule to about twenty-six water molecules. Perhaps, however, the other cavities in the liquid could also be occupied by methane molecules, and the change in concentration dependence would not make itself evident.
I have had a discussion with Professor Schomaker, in which another point has been brought out. This relates to the stability of the pentagonal dodecahedral complex of twenty-one water molecules and other larger complexes formed by condensation, relative to complexes with the tridymite or cristabolite structure.
The point is that a larger fraction of possible hydrogen bonds is formed in the dodecahedral complexes than in any other complexes, so far as I have been able to discover.
For example, the pentagonal dodecahedron has twenty unformed half-hydrogen bonds sticking out, for twenty-one water molecules. The double dodecahedron has thirty for thirty-seven water molecules and the quadruple has forty for sixty-one. The ratios are 0.96, 0.82, and 0.66, respectively. A three-layer tridymite complex with thirty-nine water molecules has forty-one half-hydrogen bonds, with the ratio 1.02. This is about comparable to the double dodecahedron, and we see that the ratio is about 20% smaller for the double dodecahedron than for the tridymite complex.
I think that we might formulate the mathematical question, what complexes can be formed can be formed that minimize the number of unformed hydrogen bonds sticking out form the surface, for a given number of the water molecules in the complex.
Linus Pauling:LL
December
3, 1956
