# What is the smallest molar volume?

I wondered how small a volume a mole of material could occupy, so I started with carbon, which would need 12 grams. That's 60 carats, and there happens to be a famous 60 carat diamond. If my density/atomic mass calculations are correct, the substances with the smallest molar volume would be nickel, carbon, beryllium, and boron, with boron being the smallest. But I had trouble finding images of large-enough beryllium or boron crystals.

Are there any compounds of sufficient density and low enough atomic mass to beat boron? Can a maximal density boron crystal be made that large?

What substance has the smallest molar volume?

EDIT -- The Noor-ul-Ain diamond is closer to being 12 grams.

• Apparently you are insisting on only single crystals? A mole is just a particular (large) number. – Jon Custer Feb 21 '19 at 17:06
• A small number of crystals would be fine, so long as it made for a good image. – Ed Pegg Feb 21 '19 at 17:08
• If you're asking about molar volume in standard conditions, you should call it. – Mithoron Feb 21 '19 at 17:30
• You're really asking about the smallest molar volume, not the smallest moles per se. – MaxW Feb 21 '19 at 20:30
• As pointed else where, diamond has the smallest molar volume ($\pu {3.42 cm^3/mol}$). The other 4 of 5 smallest molar volumes are: Boron ($\pu {4.39 cm^3/mol}$) < Beryllium ($\pu {4.85 cm^3/mol}$) < Carbon ($\pu {5.29 cm^3/mol}$) < Nickel ($\pu {6.59 cm^3/mol}$) < Cobalt ($\pu {6.67 cm^3/mol}$), according to Elements' Handbook. Iron and Copper are not far behind with molar volumes of $\pu {7.09 cm^3/mol}$ and $\pu {7.11 cm^3/mol}$, respectively. – Mathew Mahindaratne Feb 21 '19 at 22:09

Boron is a covalent solid with high melting point, like diamond (though not quite), and hence its crystals are hard to make. Unlike diamond crystals, they are not nice and probably wouldn't make a great display.

The table on http://periodictable.com/Properties/A/MolarVolume.v.log.html seems to corroborate your findings about boron molar volume being the smallest among all elements. Pity it's wrong, and so are you. (Or rather, it is technically right, but in a way that conveys a wrong impression.) Some elements just tend to have multiple polymorphs (called allotropes in this case), and carbon is one of them. All data in the standard tables are for the standard polymorph, which is graphite. But diamond at $$3.5\;\ce{g/cm^3}$$ is much denser, and decisively beats boron in the contest for the smallest molar volume.

Sometimes it takes walking around the world to realize that the aim of your quest has been in your pocket all along. The picture of the "smallest mole" is the one you brought here.

So it goes.

A mole of neutrons in a neutron star would take up about $$10^{-20}$$ m$$^3$$. And in a black hole, they would be even smaller.

• For completeness, metallic hydrogen is predicted to have a metastable state that would exist at standard temp and pressure (after forming at very high pressure) and would occupy a smaller volume per mole than diamond. – Andrew Feb 22 '19 at 12:55
• @Andrew: At STP? Really? That's the first I've heard of such a claim, and Googling it turns up this question on physics.SE with a rather skeptical answer. – Ilmari Karonen Feb 22 '19 at 13:52
• Not that a black hole or neutron star is STP, either. While Ice VII might beat diamond (on an atomic density basis) on Earth by being in a high pressure matrix (see another answer), diamond is probably the best that's strictly at STP. – Oscar Lanzi Feb 25 '19 at 15:37

If you allow a mole of atoms, then some compounds come to the fore. Like water.

Ordinarily, liquid water occupies $$6.0\text{ cm}^3/\text{mol atoms}$$. Freezing this to ordinary ice (Ice $$\text{ I}_h$$) increases this volume slightly as water expands upon freezing. But there are high pressure ice phases that are denser and thus give diamond a run for its money ... or maybe more.

Ice $$\text{ VII}$$ has been found on Earth as inclusions in diamonds. According to Wikipedia this phase has a density of $$1.65\text{ g/cm}^3$$, which translates to about $$3.6\text{ cm}^3/\text{mol atoms}$$. But that is just at the minimum pressure for this phase, $$2.5\text{ GPa}$$. At higher pressures, which can be maintained internally within the diamond lattice, this phase is fairly compressible because the hydrogen bonds can be squeezed towards a symmetric bonding arrangement (at which point we would have Ice $$\text{ X}$$). So the densest arrangement of atoms naturally occurring on Earth might be not diamond per se, but Ice $$\text{ VII}$$ included within it.

To address you concern about boron, there is a cubic diamond form of boron nitride $$\ce{c-BN}$$, ICSD #182731 , posseses $$V_\mathrm{cell} = \pu{7.99 Å3}$$, $$Z = 2$$ and molar volume

$$V_\mathrm{m} = \frac{N_\mathrm{A}V_\mathrm{cell}}{Z} = \frac{\pu{6.022e23 mol-1}\cdot\pu{7.99 Å3}}{2} \approx \pu{2.406e-6 m3 mol-1}$$

which is about $$30\%$$ less than diamond. The only drawback is that this form of boron nitride is predicted to be stable above $$\pu{11 Mbar}$$. Figure 1. Unit cell of $$\ce{c-BN}$$. Color code: $$\color{#FFB5B5}{\Large\bullet}~\ce{B}$$; $$\color{#3050F8}{\Large\bullet}~\ce{N}$$.

### References

1. Qiu, S. L.; Marcus, P. M. Structure and Stability under Pressure of Cubic and Hexagonal Diamond Crystals of C, BN and Si from First Principles. Journal of Physics: Condensed Matter 2011, 23 (21), 215501. https://doi.org/10.1088/0953-8984/23/21/215501.
• The "cubic" structure to which you refer looks tetragonal to me, unless the given cell is contained in a larger cubic structure because $c/a=\sqrt{2}$. Also in a compound, is the molar volume based on atoms or "molecules"? – Oscar Lanzi Feb 23 '19 at 10:33
• The cubic diamond phase stems, it seems, from the arrangement of B- and N-networks, separately resembling the diamond phase; and the elongation is caused by the shift between the both. And yes, you are right, to me it also looks like a tetragonal crystal system, although it's been deposited as triclinic for some reason. The molar volume is determined for the formula unit. – andselisk Feb 23 '19 at 11:20
• The abstract in the reference tells me that the dense phase os not the usual cubic boron nitride. It's a collapsed phase that forms from the c-BN phase at 11 Mbar (and from cubic diamond at 13 Mbar). We have to check the actual cell structure for the condensed phase. I suspect the actual unit cell has only one formula unit so it's half as dense as the claim. – Oscar Lanzi Feb 23 '19 at 16:00
• @OscarLanzi That's why I uploaded the init cell content. There are $(1 + 8\cdot\frac{1}{8}) = 2$ B atoms and $4\cdot\frac{1}{2} = 2$ N atoms, which makes up to formula unit BN and $Z = 2$. As for "collapsed" phase, I wasn't able to find the reported crystal structure, and the search among boron nitrides in ICSD reveals structures with higher molar volumes. – andselisk Feb 23 '19 at 22:04