Over a year ago, I wrote here about ice. It has 16 known forms with different crystal geometries. The most common form on Earth, hexagonal ice I, is a surprisingly subtle blend of order and randomness:

Liquid water is even more complicated. It’s mainly a bunch of molecules like this jostling around:

The two hydrogens are tightly attached to the oxygen. But accidents do happen. On average, for every 555 million molecules of water, one is split into a negatively charged OH⁻ and a positively charged H⁺. And this actually matters a lot, in chemistry. It’s the reason we say water has pH 7.

Why? By definition, pH 7 means that for every liter of water, there’s 10-7 moles of H⁺. That’s where the 7 comes from. But there’s 55.5 moles of water in every liter, at least when the water is cold so its density is almost 1 kilogram/liter. So, do the math and you see one that for 555 million molecules of water, there’s only one H⁺.

Acids have a lot more. For example, lemon juice has one H⁺ per 8800 water molecules.

But let’s think about this H⁺ thing. What is it, really? It’s a hydrogen atom missing its electron: a proton, all by itself!

But what happens when you’ve got a lone proton in water? It doesn’t just sit there. It quickly attaches to a water molecule, forming H₃O⁺. This is called a hydronium ion, and it looks like this:

But hydronium is still positively charged, so it will attract electrons in other water molecules! Different things can happen. Here you see a hydronium ion water molecule surrounded by three water molecules in a symmetrical way:

This is called an Eigen cation, with chemical formula H₉O₄⁺. I believe it’s named after the Nobel-prize-winning chemist Manfred Eigen—not his grandfather Günther, the mathematician of ‘eigenvector’ fame.

And here you see a hydronium ion at lower right, attracted to water molecule at left:

The is a Zundel cation, with chemical formula H₅O₂⁺. It’s named after Georg Zundel, the German expert on hydrogen bonds. The H⁺ in the middle looks more tightly connected to the water at right than the water at left. But it should be completely symmetrical—at least, that’s the theory of how a Zundel cation works.

But the Eigen and Zundel cations are still positively charged, so they attract more water molecules, making bigger and bigger structures. Nowadays chemists are studying these using computer simulations, and comparing the results to experiments. In 2010, Evgenii Stoyanov, Irina Stoyanova and Christopher Reed used infrared spectroscopy to argue that a lone proton often attaches itself to 6 water molecules, forming H⁺(H₂O)₆, or H₁₃O₆⁺, like this:

As you can see, this forms when each hydrogen in a Zundel cation attracts an extra water molecule.

Even this larger structure attracts more water molecules:

But the positive charge, they claim, stays roughly within the dotted line.

Wait. Didn’t I say the lone proton was right in the middle? Isn’t that what the picture shows—the H in the middle?

Well, the picture is a bit misleading! First, everything is wiggling around a lot. And second, quantum mechanics says we don’t know the position of that proton precisely! Instead, it’s a ‘probability cloud’ smeared over a large region, ending roughly at the dashed line. (You can’t say precisely where a cloud ends.)

It seems that something about these subtleties makes the distance between the two oxygen nuclei at the center is surprisingly large. In an ordinary water molecule, the distance between the hydrogen and oxygen is a bit less than 100 pm—that’s 100 picometers, or 100 × 10-12 meters, or one angstrom (Å) in chemist’s units:

In ordinary ice, there are also weaker bonds called hydrogen bonds that attach neighboring water molecules. These bonds are a bit longer, as shown in this picture by Stephen Lower, who also drew that great picture of ice:

But the distance between the two central oxygens in H₁₃O₆⁺ is about 2.57 angstroms, or twice 1.28:

Stoyanov, Stoyanova and Reed put the exclamation mark here. I guess the big distance came as a big surprise!

I should emphasize that this work is new and still controversial. There’s some evidence, which I don’t understand, that 20 is a ‘magic number': a lone proton is happiest when accompanied by 20 water molecules, forming H⁺(H₂O)₂₀. One possibility is that the proton is surrounded by a symmetrical cage of 20 water molecules shaped like a dodecahedron! But in 2005, a team of scientists did computer simulations and arrived at a different geometry, like this:

This is not symmetrical: there’s a Zundel cation highlighted at right, together with 20 water molecules.

Of course, in reality a number of different structures may predominate, in a rapidly changing and random way. Computer simulations should eventually let us figure this out. We’ve known the relevant laws of nature for over 80 years. But running them on a computer is not easy! Kieron Taylor did his PhD work on simulating water, and he wrote:

It’s a most vexatious substance to simulate in useful time scales. Including the proton exchange or even flexible multipoles requires immense computation.

It would be very interesting if the computational complexity of water were higher, in some precise sense, than many other liquids. It’s weird in other ways. It takes a lot of energy to heat water, it expands when it freezes, and its molecules have a large ‘dipole moment’—meaning the electric charge is distributed in a very lopsided way, thanks to the ‘Mickey Mouse’ way the two H’s are attached to the O.

I’ve been talking about the fate of the H⁺ when a water molecule splits into H⁺ and OH⁻. I should add that in heavy water, H⁺ could be something other than a lone proton. It could be a deuteron: a proton and a neutron stuck together. Or it could be a triton: a proton and two neutrons. For this reason, while most chemists call H⁺ simply a ‘proton’, the pedantically precise ones call it a hydron, which covers all the possibilities!

But what about the OH⁻? This is called a hydroxide ion:

But this, too, attracts other water molecules. First it grabs one and forms a bihydroxide ion, which is a chain like this:


with chemical formula H₃O₂⁻. And then the bihydroxide ion attracts other water molecules, perhaps like this:

Again, this is a guess—and certainly a simplified picture of a dynamic, quantum-mechanical system.

References and digressions

For more, see:

• Evgenii S. Stoyanov, Irina V. Stoyanova, Christopher A. Reed, The unique nature of H⁺ in water, Chemical Science 2 (2011), 462–472.

Abstract: The H⁺(aq) ion in ionized strong aqueous acids is an unexpectedly unique H₁₃O₆⁺ entity, unlike those in gas phase H⁺(H₂O)n clusters or typical crystalline acid hydrates. IR spectroscopy indicates that the core structure has neither H₉O₄⁺ Eigen-like nor typical H₅O₂⁺ Zundel-like character. Rather, extensive delocalization of the positive charge leads to a H₁₃O₆⁺ ion having an unexpectedly long central OO separation of 2.57 Å and four conjugated OO separations of 2.7 Å. These dimensions are in conflict with the shorter OO separations found in structures calculated by theory. Ultrafast dynamic properties of the five H atoms involved in these H-bonds lead to a substantial collapse of normal IR vibrations and the appearance of a continuous broad absorption (cba) across the entire IR spectrum. This cba is distinguishable from the broad IR bands associated with typical low-barrier H-bonds. The solvation shell outside of the H₁₃O₆⁺ ion defines the boundary of positive charge delocalization. At low acid concentrations, the H₁₃O₆⁺ ion is a constituent part of an ion pair that has contact with the first hydration shell of the conjugate base anion. At higher concentrations, or with weaker acids, one or two H₂O molecules of H₁₃O₆⁺ cation are shared with the hydration shell of the anion. Even the strongest acids show evidence of ion pairing.

Unfortunately this paper is not free, and my university doesn’t even subscribe to this journal. But I just discovered that Evgenii Stoyanov and Irina Stoyanova are here at U. C. Riverside! So, I may ask them some questions.

This picture:

came from here:

• Srinivasan S. Iyengar, Matt K. Petersen, Tyler J. F. Day, Christian J. Burnham, Virginia E. Teige and Gregory A. Voth, The properties of ion-water clusters. I. The protonated 21-water cluster, J. Chem. Phys. 123 (2005), 084309.

Abstract. The ab initio atom-centered density-matrix propagation approach and the multistate empirical valence bond method have been employed to study the structure, dynamics, and rovibrational spectrum of a hydrated proton in the “magic” 21 water cluster. In addition to the conclusion that the hydrated proton tends to reside on the surface of the cluster, with the lone pair on the protonated oxygen pointing “outwards,” it is also found that dynamical effects play an important role in determining the vibrational properties of such clusters. This result is used to analyze and complement recent experimental and theoretical studies.

This paper is free online! We live in a semi-barbaric age where science is probing the finest details of matter, space and time—but many of the discoveries, paid for by taxes levied on the hard-working poor, are snatched, hidden, and sold by profiteers. Luckily, a revolution is afoot…

There are other things in ‘pure water’ beside what I’ve mentioned. For example, there are some lone electrons! Since these are light, quantum mechanics says their probability cloud spreads out to be quite big. This picture by Michael Tauber shows what you should imagine:

He says:

Schematic representation of molecules in the first and second coordination shells around the solvated electron. First shell molecules are shown hydrogen bonded to the electron. Hydrogen bonds between molecules of 1st and 2nd shells are disrupted.

20 Responses to Water

  1. Greg Egan says:

    … not his grandfather Günther, the mathematician of ‘eigenvector’ fame

    This is an urban math. The truth is that ‘eigenvectors’ are named after my own great-great-great-grandfather, Phineas Egan, the Leopold Bloom Professor of Mathematics at University College Dublin from 1805 to 1865. German mathematicians later corrupted the spelling.

    • aneumaier says:

      Jeff Miller, the author of the authoritative

      Earliest Known Uses of Some of the Words of Mathematics

      writes in the entry


      ”The eigen terms are associated with David Hilbert (1862-1943), though he may have been following such constructions as Eigentöne in acoustics (cf. H. L. F. Helmholtz Lehre von den tonempfindungen). Eigenfunktion and Eigenwert appear in the first of Hilbert’s communications on integral equations “Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen”, Nachrichten von d. Königl. Ges. d. Wissensch. zu Göttingen (Math.-physik. Kl.) (1904) p. 49.-91. (The communications from 1904-1910 were collected as Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen 1912). In Whittaker & Watson’s Course of Modern Analysis “Eigenfunktion” is translated as “autofunction.” Eigenvektor appears in the exposition of the finite-dimensional case in Courant & Hilbert Methoden der Mathematischen Physik (1924).”

  2. John Baez says:

    A conversation with Viktor Bautista i Roca got me to do the calculation more precisely. The density of water changes with temperature, but more importantly, so does its pH.

    In reality the pH of water is 7 only at 25 °C. As it gets hotter, the pH drops because the water jostles around more and there are more free H⁺ ions around. By 100 °C its pH is just 6.14.

    So, at 25 °C there are 10-7 moles of H+ per liter of water, while at 100 °C there are 10-6.14 = 7.24 × 10-7.

    The density of water also changes with temperature. At 25 °C it’s 0.997047 kilograms per liter, while at 100 °C, right before it boils, it’s 0.958366.

    A mole of water is 18.02 grams, so are 55.49 moles of water per kilogram. So, at 25 °C there are 55.33 moles of water per liter, but at 100 °C there are just 53.18.

    So, at 25 °C there are 10-7 moles of H+ per liter of water, or 55.33 × 107 molecules of water for each H+ ion. That’s 553.3 million molecules of water for each H+ ion.

    But at 100 °C there are 7.24 × 10-7 moles of H+ per liter of water, or 53.18 / (7.24 × 10-7) molecules of water for each H+ ion. That’s 73.4 million molecules of water for each H+ ion!

    • John Baez says:

      Just to exorcise this obsession, I’ll also work it out at 0°.

      Here the pH is 7.47, so there are only 10-7.47 = 3.39 × 10-8 moles of H+ per liter of water.

      Here the density of water is 0.999842 kilograms per liter. so there are 55.49 moles of water per liter.

      So, there are 55.49 / (3.39 × 10-8) molecules of water for each H+ ion. That’s 1640 million molecules of water for each H+ ion!

      So it varies quite a lot with temperature.

  3. domenico says:

    All is interesting,
    I am thinking that the cations can be the cause of sonoluminescence: if it is so, then the emission spectrum can give information on the mass of the emitter, that can be the proton (a quantum transition of a proton), or the cation or plasmons.
    If it is possible to measure the ultrafast emission spectrum, and measure the change with the frequency of the ultrasound, than is it possible to evaluate the mass of the emitter?
    If the cations are common in water, then this must happen in other asymmetric molecule, so that ammonia, sulfure dioxide can give the same structures, and the emission spectrum can give information on the emitter (comparing the spectra of different liquid: hydrogen sulfide is the right choice, but it is dangerous).

    • domenico says:

      I am thinking that an amplification of the emission can be obtained using stimulated emission (like in the laser): if the ultrasound produce sonoluminescence (light emission with sound, instead of electromagnetic pumping energy), then two reflective mirror can produce stimulated emission, the emitter population can be deduced from the energy of the laser emission (and the frequency of the laser); so that if the length of the cavity can be changed (to obtain different frequencies), then some population number can be obtained. A comparison with the concentration of the cations, and higher level structure, can give the emitters type.

  4. maochoad says:

    Reblogged this on Cuaderno de trabajo and commented:
    An interesting summary, by John Baez, of experimental findings and theoretical conjectures around the hydronium chemical structure.

  5. John Baez says:

    There’s also a bit of hydrogen peroxide in ordinary water, but I’m finding it hard to read how much there is in equilibrium. I guess I’ll have to calculate it starting from first principles (the enthalpies of various substances). Hydrogen peroxide is sufficiently stable that water from different sources contains significantly different amounts of the stuff.

  6. Berényi Péter says:

    Water is just amazing. See Life Depends upon Two Kinds of Water by Philippa Wiggins.

  7. nad says:

    This paper is free online! We live in a semi-barbaric age where science is probing the finest details of matter, space and time—but many of the discoveries, paid for by taxes levied on the hard-working poor, are snatched, hidden, and sold by profiteers. Luckily, a revolution is afoot…

    Interesting question in this context is how much of science research is still being paid for by taxes.

    • nad says:

      I mean there nowadays seem to exist new methods for more or less cost-effective scientific reasoning, like on this page you can make 100 euros by solving these:

      Problem 2.3 (Martin-Löf random sequences and convergence)
      100 Euro are offered for the construction of a universal semimeasure with posterior convergence individually for all Martin-Löf random sequences. Universal may be defined in either of the following ways:
      (a) dominating all enumerable semimeasures Eq.(2.27),
      (b) being Solomonoff’s M for some universal Turing machine U Eq.(2.21),
      (c) being Levin’s mixture ξU for some U with general weights Eq.(2.26).

      and there are of course also corporations etc. which may be interested in funding research which is dedicated to finding new solutions for their evergrowing quests….

      • John Baez says:

        Nobody is going to support themselves doing scientific research if they only earn 100 euros for solving a problem of that difficulty! You can make money more quickly scrubbing floors.

        I bet most published scientific research is at least partially supported by government grants. (A lot of privately supported research doesn’t get published.)

  8. Simon Burton says:

    I keep seeing these beautiful photos of ice crystals on my facebook. Does anyone know what is the state of the art in understanding the structure of these ? For example why is there so much detail in the six-fold symmetry?


    • John Baez says:

      The growth of snowflakes depends heavily on the temperature and humidity of the air, and as they fall or get blown back up through different air conditions, they grow in different ways at different times, leading to complex and individual patterns.

      I’m not an expert on how this works, but I’ve posted a couple of articles about them on Google+, which I’ll append below. You definitely want to check out Kenneth Libbrecht’s website for more information!

      You may also enjoy my blog article about ice, even though that’s not about snowflakes.



      Snowflakes are amazingly diverse! The first scientific classification of them goes back to Ukichiro Nakaya. Trained as a nuclear physicist, in 1932. Nakaya was appointed to a professorship in Hokkaido, in the north of Japan. There were no facilities to do nuclear research, so Nakaya turned his attention to snowflakes! Besides studying natural snow, he created artificial snowflakes, and figured out which kinds form in which conditions. In 1954 he summarized his work in a book, Snow Crystals: Natural and Artificial. This is his classification.

      For more on the history of snowflakes, and amazing photos of them, see Kenneth Libbrecht’s page:

      • Kenneth Libbrecht, Early snow crystal observations.


      Triangular snowflakes! They’re one of the minor mysteries of science. They’re uncommon, but pretty easy to find if you look. How do they form? Apparently not just by chance! This paper claims that if you start with a snowflake that’s a slightly irregular hexagon, as it falls through the air it will tend to become triangular in shape:

      • K. G. Libbrecht and H. M. Arnold, Aerodynamic stability and the growth of triangular snow crystals, http://arxiv.org/abs/0911.4267.

      A perfectly hexagonal snowflake falls horizontally through still air. But suppose one of its six edges grows out a bit more than the rest, by some random fluctuation. Then this edge becomes shorter – yes, think about it, or look at the picture in the paper! This makes the snowflake unbalanced: the opposite edge tilts down. It and its two neighboring edges then start to grow faster, because they have the first chance to meet the incoming moist air. This makes the snowflake become a bit triangular – like the one in the upper left picture here.

      But more research is called for! The authors write:

      “Unfortunately, producing a more complete, quantitative aerodynamic model will be difficult. Researchers have only recently developed viable numerical techniques for modeling the diffusion-limited growth of faceted crystals, even for relatively simple physical cases. Adding aerodynamic instabilities and their resulting growth changes in full 3D will likely be a considerable challenge. Nevertheless, even our simple qualitative model makes a number of testable predictions.”

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