What does mercury being liquid at room temperature have to do with Einstein's theory of relativity?

Image: Wikipedia Commons

One of the great moments in twentieth century science came when Paul Dirac married quantum mechanics with Einstein's Special Theory of Relativity to produce relativistic quantum mechanics. Dirac's theory did many things - predict electron spin and the positron, analyze atomic collisions, jump-start the revolution in quantum electrodynamics - but it also had very significant repercussions for chemistry. However these repercussions did not become known for another few decades because it turned out that for solving most problems in chemistry you could neglect relativistic effects. Figuring out chemical bonding, predicting the thermodynamic properties of molecules and rates of chemical reactions, understanding the molecular glue that holds proteins together; all these problems succumbed to calculation without chemists worrying about relativity.

All except one problem, that is. And it deals with a question that every child since antiquity has asked: Why is mercury liquid at room temperature? Mercury - the only metal with this property - has beguiled and fascinated men for centuries; a glittering substance that flows with studied gravity, supports the weight of coins, magically seems to dissolve other metals and resists all attempts to scoop it up. A substance that can aid health when calibrated inside a thermometer and can kill when it accumulates in living tissues. But the one quality of mercury that is apparent to everyone who has even the slightest acquaintance with it is its liquid nature.

Why is this so? It turns out that sometimes simple observations in science can have complicated although very interesting explanations, and this is one of those cases. Fortunately the crux of the matter is simple, and it has received its most complete and satisfying treatment in a recent paper published in the journal Angewandte Chemie. But first let's go back to the basics. Mercury is a metal, which means that it occupies the middle of the periodic table along with other metals like gold, zinc and cadmium. In fact it is in the same group as zinc and cadmium, and yet it couldn't be more different from them. Zinc and cadmium are not liquids at room temperature and they crystallize in a different form from mercury. In addition mercury is right next to gold, and yet their properties are utterly dissimilar.

Recall from college chemistry that atomic orbitals come in different flavors; s, p, d and f orbitals are distinguished by different quantum numbers and different "shapes". Metals are characterized by significantly occupied d orbitals. In addition, filled orbitals imply special stability. The singular fact that distinguishes mercury from its neighbors is that it has a filled outermost 6s atomic orbital. This means that the electrons in the orbital are happily paired up with each other and are reluctant to be shared among neighboring mercury atoms. Where the theory of relativity comes in is in accounting for subtle changes in the masses of the electrons in mercury and the atomic radii which nonetheless have profound effects on the physical properties of the metal.

According to special relativity, the apparent mass of an object increases as its velocity approaches the speed of light. From Niels Bohr's theory of atomic structure we know that the velocity of an electron is proportional to the atomic number of an element. For light elements like hydrogen (atomic number 1) the velocity is insignificant compared to the speed of light so relativity can be essentially ignored. But for the 1s electron of mercury (atomic number 80) this effect becomes significant; the electron approaches about 58% of the speed of light, and its mass increases to 1.23 times its rest mass. Relativity has kicked in. Since the radius of an electron orbit in the Bohr theory (orbital to be precise) goes inversely as the mass, this mass increase results in a 23% decrease in the orbital radius. This shrinkage makes a world of difference since it results in stronger attraction between the nucleus and the electrons, and this effect translates to the outermost 6s orbital as well as to other orbitals. The effect is compounded by the more diffuse d and f orbitals insufficiently shielding the s electrons. Combined with the filled nature of the 6s orbital, the relativistic shrinkage makes mercury very reluctant indeed to share its outermost electrons and form strong bonds with other mercury atoms.

The bonding between mercury atoms in small clusters thus mainly results from weak Van der Waals forces which arise from local charge fluctuations in neighboring atoms rather than the sharing of electrons. But all this was conjecture; someone had to do the rigorous calculations, treating every electron in the element relativistically and calculating the relevant properties. In this case the relevant property is the heat capacity of a substance which dramatically changes during a phase transition, say from solid to liquid. The question was simple; using the most state-of-the-art calculations, could you predict the temperature at which mercury melts as indicated by a sudden change in heat capacity? In a paper published in Angewandte Chemie this month, chemists from New Zealand, Germany and France have provided a result which is the most complete one to date. They actually simulated the melting of mercury using quantum molecular dynamics, solving the Schrodinger equation, calculating forces and velocities from quantum mechanics and allowing the atomic clusters to sample different geometric orientations randomly. They carried out the calculations first by excluding relativity and then by including it, and the results were unambiguous; when relativistic effects were taken into account, the melting point of mercury dropped from 355 kelvin to 250 kelvin, in excellent agreement with experiment and accompanied by a sudden change in the heat capacity.

The liquid nature of mercury is not the only thing that the special theory explains. It also explains why gold is yellow while silver is white. In this case, the splitting of orbitals and the lower energy of the 6s orbital results in gold absorbing blue light and emitting yellow and red. Since the 6s level is higher in silver, the energy required to excite an electron corresponds to the UV region instead of the visible region; consequently silver appears devoid of colors from the visible region of the spectrum.

I always feel a twang of pleasure when I come across studies like this. There are few things more satisfying than the successful application of our most cherished and accurate theories to explaining life's most humdrum and yet fascinating phenomena. That's what science is about.

References:

1. Evidence for Low-Temperature Melting of Mercury owing to Relativity; F. Calvo et al. Angew. Chem. Intl. Ed. Engl. 2013, 10.1002/anie.201302742

2. Why is Mercury Liquid? L. Norrby, J. Chem. Ed. 1991, p. 110.

3. Relativistic Effects in Chemistry, D. McKelvey, J. Chem. Ed. 1983, p. 112

This post was first published on the Scientific American Blog Network.

Zn(III)? Not so fast

Chemists love rogues, oddballs which seem to defy the rules and bond, react, and exist on their own terms. The rogues are valuable because they push the boundaries, teach us about new principles of structure and reactivity and challenge coveted preconceptions. One of the most striking rogues in the history of chemistry was the compound xenon hexafluoroplatinate which shattered belief in the non-reactivity of noble gases. Another immensely productive rogue was the first stable carbocation, a species that was considered too unstable to isolate until George Olah surmounted the energy barrier. Some rogues are literal rogues in the sense that they need to be incarcerated in order to prevent their unruly bonds from going haywire; witness the classic taming of cyclobutadiene by Donald Cram. There's no doubt about it; rogues' galleries are shining gems in the chemical establishment.

A couple of months ago it seemed that another minor rogue had made its appearance. Students around the world know that the most stable oxidation state of zinc is +2. The rationale is simple; unlike transition metals like iron and nickel, zinc contains a fully filled d orbital with an electronic configuration of d¹⁰s². It is therefore quite happy to lose its outermost 2s electrons and remain stable. Thus it came as a surprise when a paper detailing theoretical calculations on zinc compounds predicted the existence of a complex with Zn as Zn(III). That fact would have raised the eyebrows of a million freshmen memorizing transition metals trends for their final exams. The existence of the unusual zinc was based on quantum chemical calculations done by Puru Jena's group at VCU and their basic rationale was that a ligand that was oxidizing and electronegative enough would essentially force the metal to donate extra electrons. They seemed to find such a complex in Zn(AuF₆)₃. On the face of it that would make sense, but it's worth keeping in mind that nature loves filled orbitals; for instance, even a ligand as oxidizing and electronegative as CF3 does not force copper to adopt the +2 state (copper's most stable oxidation state being +1, resulting from its d¹⁰s¹ configuration).

Now another paper suggests, with ample evidence it seems, that the suspected Zn(III) rogue might be an upstanding chemical citizen after all. Sebastian Riedel's group in Berlin has examined the purported Zn(III) complex and found that not only does zinc adopt its good old +2 oxidation state in the compound, but that the compound would not be thermodynamically stable. They use a quantum chemical method more sophisticated than the previous one and include more relativistic effects. Where does relativity enter the picture, you ask. It turns out that for heavy metals, the speed (in a crude sense) of the d s-electrons can be fast enough to warrant relativistic calculations; indeed, relativistic quantum chemistry has shed light on many commonplace and yet unusual phenomena, like the color of gold and the liquid state of mercury at room temperature. Relativity is not the exclusive domain of physicists.

I am not enough of an expert in quantum chemistry, but it clearly seems from the latest paper that the more accurate calculations which account for finer details of electron correlation and relativistic effects indicate something striking; the purported complex with Zn in the +3 oxidation state would undergo an exothermic reaction. Put simply, it would not be stable and would decompose to a compound with good old Zn(II). For me, a more troubling fact was the structure of the compound published in the original paper. The structure was supposed to be a low-energy minimum but it has two fluorines practically colliding with each other in an unholy, 197 pm, sub-Van der Waals radius squeeze (illustrated above). The present calculations find another low-energy structure existing only 10 kJ/mol above the previous structure that is consistent with a Zn(II) state. Other calculations indicate that the fluorines in the previously proposed structure are best described as radical anions bridging two AuF₅ units.

There's other theoretical evidence in there too, along with citations of experimental facts that point to the recalcitrance of Zn to adopt a +3 state. The study seems to be as solid a piece of evidence against Zn(III) as can be obtained using current levels of theory. But the real reason I want to point out this paper is because it illustrates one of the key roles of theory and computation in chemistry; as an invitation to experiment. Just because the complex is thermodynamically unstable does not mean it cannot be isolated under any conditions. One of the great lessons of chemical science in the last forty years has been the fact that given the right conditions almost anything can be synthesized, stored and isolated (Cram's cyclobutadiene again being a case in point) and that the words stable and unstable are highly relative constructions created by an impoverished chemical lexicon.

This tussle between two predictions illustrates the function of theory in chemistry as the ultimate teaser; if I were an experimentalist I would be rubbing my hands in glee, sending my armies of postdocs and graduate students into the lab to try to synthesize Zn(AuF₆)₃. As Excimer commented on another site, "Enough theory! Someone make the damn thing already".
Image source: ACS publications.

The passing of Gossypium extraordinarium

Whenever I hear the name "cotton", the first thing that comes to my mind is "...and wilkinson". The second thing, apart from the cotton we all love to wear, is the dreaded group theory, for which Cotton's again was the pioneering text.

I could never really make sense of group theory. But today, the author of my (and millions of others') favourite advanced inorganic textbook has passed into history, and I regret his loss along with everyone else.

"Cotton and Wilkinson" has the same relation to its discipline as "Samuelson" and "The Feynman Lectures on Physics" have to theirs. Cotton was a truly prolific chemist. I am not qualified to judge whether his discovery of metal-metal quadruple bonds merits a Nobel or not, but I suspect that he would more than deserve a Coreyesque lifetime achievement prize. His output, honours, and contributions to research and teaching are immense. F. Albert Cotton, RA (Rest Assured) rather than RIP. Your legacy will live. My dog-eared copy of "C & W" and thousands of others bear witness to that.

P.S. Since we are on the topic, I have to say that my all-time favourite descriptive introductory inorganic text is J D Lee's "Concise Inorganic Chemistry", an affair that dates back to early college days. The book is so engaging and informative that it makes for great bedtime reading. For a more advanced treatment, I also love Huheey.