HYDROGEN BRIDGES I N ICE AND LIQUID WATER MAURICE L. HUGGINS Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland Received March 19, 1096
It was shown by W. H. Bragg (3) in 1922 that the x-ray diffraction data obtained from ice by Dennison (4) were in agreement with a structure in which each oxygen is equidistant from four others tetrahedrally arranged around it.’ This distribution of oxygens has since been verified by further work (1). Bragg proposed that, in agreement with the theory of “hydrogen bridges” (or “hydrogen bonds”) proposed independently by the writer ( 5 ) and by Latimer and Rodebush (9), there is a hydrogen midway between each pair of adjacent oxygen atoms. Although the hydrogens do not contribute appreciably to the x-ray scattering, this seemed to be the most reasonable distribution, considering the symmetry of the arrangement of oxygens and the probable forces between the atoms. That neighboring atoms in liquid water must be held together in very nearly the same way as in ice is indicated.by the slight change of density, dielectric constant (2, ll),and various other properties on melting, by the low heat of fusion compared with the heat of vaporization, and by x-ray data.obtained in this laboratory (8) and elsewhere (2) showing that on the average each oxygen in liquid water has about four neighboring oxygens a t a distance only slightly greater than the 0-0 distance in the solid. One is thus led to postulate hydrogen bridges throughout the liquid; many of these, however, being bent or stretched and some broken, a t any instant, as a result of the heat motions. The bending of many of these bridges when ice melts permits R slightly closer packing, hence a greater density. As the temperature rises, more and more of the bridges are broken and stretched, the average interoxygen distance becomes larger, and the density decreases. In several respects this picture has not been entirely satisfactory. The low conductivity of pure water compared with water containing hydrogen ions (and also negative ions, of course) seems quite inexplicable. Moreover the high dielectric constant in both the liquid and solid, down to temperatures considerably below zero (for low frequencies), seemed to be evidence for the existence of molecular units, some of which are quite free t o orient themselves in an electrostatic field. A tetrahedral arrangement of four oxygens around each oxygen was predicted by the writer on the basis of the theory of hydrogen bridges. 723
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MAURICE L. HUGGINS
Largely because of these two arguments the writer2 suggested that the hydrogen of each hydrogen bridge in solid and liquid HgO may be closer to one oxygen neighbor than to the other, in such a way that each oxygen keeps but two close hydrogen neighbors, all oxygens in ice and most of them in water also having two other hydrogen neighbors a t a somewhat greater distance. As Pauling (10) has recently shown, this picture makes possible a calculation of the entropy of ice more nearly in agreement with experiment than that calculated on the assumption of rotating molecules. The additional assumption that the potential energy hump in the center of each hydrogen bridge in the structure is small is sufficient to account for both the conductivity and dielectric constant behavior, as will now be shown. Although the shifting of a single hydrogen in a hydrogen bridge in ice or water from the vicinity of one oxygen to the vicinity of the other might be quite difficult, owing to the size of the energy hump, such a shift is much facilitated if a t the same time another hydrogen nucleus approaches the first oxygen (thereby making its effective charge less negative) and a hydrogen previously close to the second oxygen moves away from it (making its effective charge more negative). Jumps in unison of all of the hydrogens in a ring, in such a way that after the jumps, as before, each oxygen has but two close hydrogen neighbors, may therefore occur quite frequently, just as a result of the ordinary heat energy.
The presence of an external electrostatic field favors such jumping in all cases in which the resultant electric moment in the direction of the field is increased, hindering those jumps decreasing the electric moment in that direction. (For a 6-bridge ring the resultant moment change is
* I n a preliminary edition of his book on “Chemistry, the Science of Atoms and their Interactions,” in 1933. Essentially the same picture, as regards each atom’s immediate neighbors, has been proposed by Bernal and Fowler (2).
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slight; this is not so, however, for many larger, less symmetrical rings.) Qualitatively the effect is much the same as if the substance contained a small percentage of rotatable molecular dipoles, and the large dielectric constant in both solid and liquid is accounted for (cf. reference 2). At temperatures which are too low, the temperature energy is insufficient to make even these synchronized shifts possible, and the dielectric constant decreases to a very small value. The shifts in a ring do not occur instantaneously as soon m the field is applied, as the vibrations of the atoms must be suitably timed and must have sufficient energy. Because of this time lag the dielectric constant values are less for high frequencies than for low frequencies, at any given temperature. When the temperature energy available is sufficiently large, the synchronized jumps of the hydrogens in a ring become synchronized oscillations about their mean positions. The effect of the field on this portion of the structure then becomes practically negligible. This accounts for the diminution of dielectric constant as the temperature rises, at temperatures slightly below zero, for low frequencies. Each hydrogen nucleus in pure water may be considered, roughly, as bridging between an OH- ion and an H20 molecule. The bridge is obviously unsymmetrical, the hydrogen in question being held close to the OH- (so producing an H2O molecule) :
H
H
....,.6-H ........6-H
........
A jump of this hydrogen nucleus to the other end of the bridge, giving OHand H80+,practically never takes place (cf. figure 2) ; hence the extremely low conductivity of pure water. If an acid is dissolved in water the hydrogen nuclei produced by dissociation undoubtedly also serve to bridge between oxygens, each such nucleus connecting two HzO molecules. With less repulsion between the oxygens one would expect a shorter 0-0 distance than in the case just considered. The hydrogen would be expected to oscillate symmetrically about the midpoint. We may represent this situation thus:
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MAURICE L. HUGGINS
There is no a priori reason for eliminating t'he possibility of H7O8+and larger ions, containing two or more practically symmetrical bridges :
The larger such ions however, the less the stability of the symmetrical bridges; it seems likely that HsOz+ is the usual limit. The effective charge on each of the H5OZ+oxygens is less negative than that on an HzO oxygen. In view of this fact, we should expect such shifts as the following to occur readily:
In this process the charge shifts with but little actual motion of the atoms. In a potential gradient between two electrodes, shifts of this sort which move the positive charge toward the negative electrode are favored in preference t o shifts in the opposite direction. As these shifts would be very rapid this theory accounts for the high mobility of hydrogen ion in water. (The chain mechanism postulated here differs but little from that proposed by the writer (6) in 1931. With the symmetrical bridges assumed at that time however, there was no obvious reason for the low conductivity of pure water, as noted above.) In an OH- ion the effective charge on the oxygen atom-is more negative than on the oxygen atom in a water molecule, hence the 0-H distance in this ion is less than in water. In water solution such an ion, unlike H+, would probably not form aggregates such as (H-0-H-0-H)because the larger Coulomb repulsion between the oxygens and larger Coulomb attraction between the central hydrogen and each oxygen would tend to make the central hydrogen bridge an unsymmetrical one. The oxygen atom in an OH- ion would be expected to tend to surround itself with four unsymmetrical hydrogen bridges, like that in a water molecule or HS02+ion. Shifts such as
........H-0-H
.......0-H ........ +
........H-0 ........H-0-H .......
(2)
would be expected to occur, although not so readily, on account of the greater 0-0 repulsion, as shifts of the type of equation 1. It is thus
HYDROGEN BRIDGES IN ICE AND LIQUID WATER
727
reasonable that the mobility of the hydroxide ion is greater than that of other ions which cannot exchange atoms with the solvent in this way, and still is less than that of hydrogen ion. In attempting to make the treatment of hydrogen bridges in ice and water more quantitative, we shall first consider a hypothetical bridge in which the energy of interaction between neighboring hydrogen and oxygen atoms is the same as in the OH molecule in its normal state. (Many of the calculations have also been carried out using the constants for a higher energy state, with results only slightly different from those to be described.) For this interaction we assume a modified Morse function, of the type recently applied successfully by the writer (7) to the calculation of molecular constants for diatomic molecules. This is of the form
u (in 10-12 ergs)
= e-dr-nd
- Cle-’(-rd
(3)
in which a, a’, r12 a n d 4 ’ are constants determined from band spectrum data. I n this case a = 6.0, a’ = 1.29, r12 = 1.125 (assumed to be the same as calculated for the Qf state) and C’ = 11.83. The equilibrium distance re is 0.969. For the method of calculating the other constants from a, r12, re and the vibration frequency w t , the writer’s first paper (7) on diatomic molecules may be consulted. Taking the oxygen to oxygen distance as 2.75 A.U. (its value in ice), the potential energy, for a hydrogen moving along the 0-0 centerline, changes as shown in the top curve of figure 1. It is seen that there are two minima, about 1.10 A.U. from the oxygens, with an energy hump between them of about 0.15 X 10-l2 erg. It may be noted that $ kT,the average molecular energy per degree of freedom, is only about 0.02 x lO-l* erg at room temperature. If the 0-0 distance is taken as 2.70 A.U., the experimental value for hydrogen bridges in HsBOs, AlHO2 and some other compounds, the size of the central hump is very much diminished and the shorter 0-H distance is increased to 1.13 A.U. For a distance between oxygens of 2.55 A.U., such as observed in NaHC08 and KHtPOa, there is but one minimum, at the center. From these results one must conclude that, if our assumption regarding the variation of the 0-H interaction energy with distance is even approximately correct, the hydrogen bridges in ice and liquid water cannot be symmetrical, with the hydrogens oscillating about points midway between the oxygens. Each hydrogen must remain considerably closer to one oxygen neighbor than to the other. It is obvious that the energy is lowest-the stability greatest-if each oxygen in ice or liquid water has two close hydrogen neighbors. The effective charge on the oxygen of an H30+ ion is less negative and that on the oxygen of an OH- ion is more negative than that on an oxygen of an
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MAURICE L. HUGGINS
HzO molecule. The attraction between the oxygen of an H30+ion and each of its hydrogens is therefore less than that between an HzO oxygen and each of its hydrogens; likewise the attraction between the two atoms of an OH- ion is greater. Calculation shows that a very slight difference in effective charge on the two oxygens joined by a hydrogen bridge suffices to eliminate the double hump in the potential energy curve, leaving only a
FIQ.1. Potential energy curves for hydrogen bridges with various fixed oxygenoxygen distances, assuming the 0-H interactions t o vary with distance as in the OH molecule.
single minimum on the side of the oxygen with the more negative effective charge. Adding to the 0-H interaction energy represented by the uppermost curve of figure 1 a term ae2/d,-o, to take account of a decreased Coulomb attraction between the hydrogen atom and one of the oxygens, unsymmetrical single-minimum curves (see the two lowest curves of figure 2) are obtained for all values of 01 greater than 1/20 (for do-o = 2.75 A.U.). If a t the same time another term of the same form but opposite sign to
HYDROGEN BRIDGES IN ICE AND LIQUID WATER
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represent an increased attraction between the hydrogen and 0 2 is included, the central hump is eliminated with still smaller values of a. One must conclude then that a shift of one hydrogen of an HzO molecule to a neighboring molecule to give HaO+ and OH- ions could not occur. If one decreases the effective negative charge equally on both oxygens of a hydrogen bridge, the central hump tends to disappear, both because of a
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FIQ.2. Potential energy curves for hydrogen bridges having do, fixed a t 2.75 A.U. To the 0-H interaction energy as in the OH molecule a Coulomb term ae2/do,-* is added for each of the two lowest curves. For the two others terms of this form are added for both 01-H and Os-H interactions.
decrease in the 0-0 distance (figure 1) and because of the decreased Coulomb attraction between the hydrogen and each oxygen (see the two upper curves of figure 2). This furnishes the justification for the assumption that the hydrogen ion consists of a relatively strong, symmetrical bridge joining two HzO groups. Reversing the argument, a hydrogen bridge connecting two OH- ions would be weaker, longer, and more un-
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MAURICE L. HUGGINS
symmetrical than the bridges in pure water (connecting, in effect, OHions and HzO molecules). As will be shown elsewhere, the F-F distance in the F H F bridge can be calculated with reasonable accuracy, using constants obtained entirely from other sources. To a large extent this distance depends on the interpenetration repulsion between the two fluorine atoms, assumed to be representable by an exponential term, e-@(r-rlz). Because of inaccurate knomledge of the best value of r12to use for the repulsion between two oxygen atoms, of the effective charges on the oxygen and hydrogen atoms, and of the magnitudes of the van der Waals constants, the corresponding calculation for OH0 bridges cannot a t present be carried out satisfactorily. I t is instructive however to calculate the value of r12 which is necessary to give equilibrium for different 0-0 distances, neglecting the van der Waals terms (which are small) and the Coulomb terms (which do not change rapidly with the distance). Taking a = 4.78 as for neon and fluorine, one obtains i-12 = 2.80 for do, = 2.75 A.U. and do,, = 1.10 A.U. For 0-0 and 01-H distances of 2.70 A.U. and 1.13 A.U., respectively, roo = 2.78 A.U. For do, = 2.55 A.U., with the hydrogen in the middle, roo = 2.70 A.U. It is seen that a small decrease in the “basic radius” ( T , = yoO/2) suffices to change an unsymmetrical bridge (two minima in the potential curve) into a symmetrical one (only one minimum, in the middle). This “radius” is a measure of the extension in space of the outermost electrons in the atom in the direction of the other oxygen. Its value depends on the tightness with which these electrons are held by the kernel and on what is on the other side-on the polarizability of the atom and on the magnitudes and directions of the forces tending to polarize it. Increased polarization resulting in a decreased basic radius for the oxygen at each end of a hydrogen bridge-and so a decreased 0-0 distance-may result from an increase in the number of close hydrogens or from their replacement by atoms of a sufficiently electronegative element such as carbon or phosphorus. In either case a single minimum potential curve is favored. This line of reasoning furnishes an additional argument for the structure
(H-0-H-0-H 7 H)+ with a strong central bridge, for the hydrogen ion in water solut’ion. It also accounts for the small 0-0 distances observed in sodium bicarbonate and potassium dihydrogen phosphate. SUMMARY
Evidence for the existence of hydrogen bridges in liquid and solid water and against their being symmetrical is reviewed. With sufficiently small
731
HYDROGEN BRIDQEB IN ICE AND LIQUID WATER
energy barriers in the center of each OH0 bridge, synchronized jumps over them of all of the hydrogens in a ring of hydrogen bridges are possible. The resulting partial freedom of orientation of the dipoles accounts satisfactorily for the dielectric constant data for ice and liquid water. Reasons are given for considering hydrogen ion in water to be (HzOHOHz)+, consisting of a relatively strong, symmetrical hydrogen bridge connecting two HzO groups. By very slight contemporaneous shifts of two hydrogen nuclei, such an ion can, in effect, add OH2 at one end and lose HzO at the other, thus shifting the position in the solution of the excess positive charge. This picture thus accounts for the high mobility of the hydrogen ion in water solution. Potential energy curves for OH0 bridges are calculated for different 0-0 distances, taking the 0-H interactions as in the OH molecule. For the smaller distances there is a single potential minimum; for the larger distances, such w observed in liquid and solid water, it is double. The central hump is eliminated if the Coulomb attraction between the hydrogen and one or both of the oxygens is slightly greater than in the OH molecule. The hydroxide ion is probably O h - rather than (HOH0H)-, the greater repulsion between the oxygen atoms making a symmetrical bridge less likely than in the cwe of the positive ion. The shift H-0-H . , 0-H to H-0 . . . . H-0-H should occur, but less readily than the corresponding shift in the case of hydrogen ion, hence the mobility should be less, as observed.
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REFERENCES
(1) BARNES,W. H.: Proc. Roy. SOC.London l M A , 670 (1929). (2) BERNAL, J. D.,AND FOWLER, R. H.: J. Chem. Physics 1, 615 (1933). (3) BRAQQ, W.H.: Proc. Phys. SOC.34, 98 (1922). (4)DENNISON, D.M.: Phys. Rev. 17, 20 (1921). (5) HUQQINS, M.L.: Undergraduate thesis (1919). (6) HUQQINS, M.L.: J. Am. Chem. SOC.63, 3190 (1931). (7) HUQQINS, M.L.: J. Chem. Physics 3, 473 (1935); also another article in press. (8) KATZOFF, S.: J. Chem. Physics 2, 841 (1934). (9) LATIMER, W. M.,AND RODEBUSH, W. H.: J. Am. Chem. SOC.42, 1419 (1920). (10) PAULINQ, L.: J. Am. Chem. SOC.67, 2680 (1935). (11) SMYTR,C. P., AND HITCHCOCK, C. 8.: J. Am. Chem. SOC.64, 4631 (1932).
