How hydronium/hydroxyl ions move through strong acidic/basic solutions

One of water's various so called anomalous properties (e.g. the fact that the solid phase (ice) is less dense that the liquid phase (water)) is the fact that an excess proton in water, such as is produced when a strong acid dissociates, has an unusually high mobility or diffusion rate. The rate is much higher than can be explained by ordinary notions of diffusion. This anomalously high diffusion rate is responsible for the fact that strong acid solutions are good conductors of electricity - there is an abundance of highly mobile ions around. If ordinary notions of diffusion cannot explain this phenomenon, then what is the chemical process or mechanism by which excess protons in the form of hydronium ions move through aqueous solution?

In the early 19th century, the Dutch chemist de Grotthuss proposed that the hydrogen bonding in water is, at least in part, responsible for the anomalously high mobility of hydronium. He suggested that an excess proton moves from one water molecule to another by ``transferring'' through a hydrogen bond, thus detaching itself from one water molecule, leaving a lone pair (in the Lewis sense) and attaching itself to the lone pair on the water oxygen to which it is transferring. Furthermore, an excess proton could move through water by a chain of such reactions, one following the other. Thus, the hydronium moves through the hydrogen bond network in water by a series of proton transfer reactions. Thus, it appears as if a larger structure, the ${\rm H}_3{\rm O}^+$ ion, is, itself, diffusing through water by hopping along the hydrogen bond network:

Figure 1:

rather than ${\rm H}_3{\rm O}^+$ simply floating through the liquid and moving by an ordinary diffusive random walk process. This process is known as ``structural diffusion.''

Until very recently, however, it was not known why structural diffusion would occur at all - why does the proton hop through hydrogen bonds in the first place. Furthermore, no proposed mechanism or model could rationalize why the time for such a hopping event would be so short - it is approximately only 1.5 $\times 10^{-12}$ seconds. The first realistic elucidation of this process was first presented only three years ago.

To understand what drives the structural diffusion process, we need to consider how a hydronium ion fits into the hydrogen bond network of liquid water. The pictures shown below depict a first principles calculation of the dynamics of hydronium in water. The first is an actual configuration taken from the dynamics showing hydronium ion in a typical solvation state:

The next picture shows just the hydronium with its first and second water solvation shells around it, taken from the configuration in the preceding picture:

Notice that, while the hydronium only forms three hydrogen bonds to neighboring water molecules, each of those water molecule, in turn, is properly hydrogen bonded to four neighbors, as we would expect for a water molecule.

Given the three hydrogen bonds that hydronium can form, the three protons bonded to the central oxygen often migrate, one at a time, to the center of their respective hydrogen bonds, as if to ``probe'' possible pathways for a proton transfer reaction of the type depicted in the cartoon. However, these attempted transfers are rarely successful because if such a transfer were to occur, the newly formed hydronium would have four hydrogen bonds to neighboring water molecules, which is overcoordinated.

Something else, then, is driving structural diffusion. In fact, what drives the process is a manifestation of thermal motion. Thermal motion is often responsible for hydrogen breaking in water. Unlike in ice, where the hydrogen bond network is relatively static, in water, hydrogen bonds a constantly breaking and reforming. Hydrogen bond breaking might also be associated with the rotation of a water molecule. When one of the water molecules hydrogen bonded to the hydronium undergoes a hydrogen bond breaking event with one of its neighbors (besides the hydronium), then it is no longer fourfold coordinated, but is left with only three hydrogen bonds: one to the hydronium and two to other water molecules. In such an ``undercoordinated'' state, it looks like another hydronium only with one of its protons missing. Thus, it is in a perfect condition to accept one of the protons from the hydronium to which it is currently hydrogen bonded, and become the hydronium ion. This is depicted below, which is a configuration during proton transfer taken directly from the dynamics calculation.

The proton completes the process by making a full transfer to the undercoordinated water molecule, thus forming a new hydronium ion at a different oxygen site, as shown below.

Thus, the structure that has diffused is not just ${\rm H}_3{\rm O}^+$, but actually ${\rm H}_3{\rm O}^+$ with three hydrogen bonded neighbors, which constitutes a H$_9$O$_4^+$ complex. While the proton transfer is occurring, the proton spends enough time midway between the oxygens that an intermediate complex can be said to have formed, namely, an H$_5$O$_2^+$ complex, in which no definite hydronium ion can be said to exist, but rather something in between. Which of the three protons on the hydronium actually executes the jump is a matter of probability. There is an equal chance for each of the water molecules hydrogen bonded to the hydronium to become undercoordinated, thus promoting the transfer event. Thus, there is still a ``random walk'' aspect to structural diffusion, but it is highly directional, occurring only along one of the three hydrogen bonds.

This proposed mechanism has been subject to considerable scrutiny and has been found to be consistent with all known experimental data on proton mobility in water. For example, the rotation time of a water molecule in the liquid has been measured and found to be between 1.0 and 2.0 ps (1 ps = 10$^{-12}$ s). Given that molecular rotation can only occur by the breaking of a hydrogen bond, and that the time required to transfer a proton depends on the time to break a hydrogen bond, the predicted time for a transfer event is consistent with the time required to break a hydrogen bond. Also, the actual proton transfer rate, itself, has been measured and found to be approximately 1.5 ps. Other, more complex measurements, also yield results that are consistent with the proposed mechanism. Thus, there is substantial evidence suggesting that this mechanism is the correct one.

From this mechanism, it can be inferred that the picture of an excess proton in water as existing simply as an ${\rm H}_3{\rm O}^+$ ion is extremely crude. The diffusing structure is actually H$_9$O$_4^+$, which is constantly being interconverted to H$_5$O$_2^+$ by proton transfer events, and the structure, itself, it highly mobile. The picture is further complicated by the fact that the proton is a light particle that needs to be treated by quantum mechanics, in order to have an accurate description of it. You will learn next semester that quantum mechanical particles behave more like waves, which exist over large spatial regions, rather than as single points. Thus, a solvated proton can, in fact, be delocalized over several hydrogen bonds, as a result of the probabilistic nature of the transfer process. This further supports the notion that the solvated proton cannot be said to manifest itself only as the hydronium ion.

The situation for the hydroxyl ion is similar to hydronium but has a few new interesting features. There are essentially two types of solvation shells that can form around a hydroxyl ion. These are shown in the two configurations below, which, again, are taken directly from a dynamics calculation.

In one of these, the hydroxyl oxygen is coordinated by four water molecules, in which the four hydrogen bonds lie in a plane roughly perpendicular to the OH$^-$ bond axis. In the second, the hydroxyl oxygen is coordinated by three hydrogen bonds in a tetrahedral arrangement. Because of the 90 degree angle between the hydrogen bonds and the OH$^-$ bond axis, the first of these solvation structures does not allow proton transfer to the hydroxyl, since this would lead to a water molecule with a 90 degree internal angle (see figure below):

The second of the two solvation structures does allow for proton transfer. Thus, this structure is capable of undergoing structural diffusion, as discussed above. The process is similar to the hydronium case, although it does not depend as strongly on hydrogen bond breaking between first and second solvation shell members. The process is shown below, where a sequence of snapshots from the dynamics calculation is shown.

Thus, structural diffusion in a basic solution depends on the breaking of a hydrogen bonds, specifically a hydrogen bond between the hydroxyl and one of the four waters bonded directly to it in the first of the above solvation shells. This allows the inactive structure to devolve into the active structure (the one that permits proton transfer and structural diffusion). This is a relatively strong hydroggen bond, and for this reason, structural diffusion a basic solution is slower than it is in an acidic solution.