When discussing energy transformations on the molecular scale, we usually use electronvolts as the energy unit. This is handy because chemical bond energies are a few electronvolts in magnitude.

I had an idea to go through some thermodynamic tables and convert them into molecular-scale units (energy in $\mathrm{eV}$, entropy in bits, etc.) in order to get a feel for how chemistry works on the scale of single molecules. However, for talking about volume changes (that is, average volume change per molecule), I can't think of a convenient unit. All of the most commonly used volume units ($\mathrm{\mu L}$ etc.) are far too large, apart from Planck units, which are far too small. Even yoctolitres are a bit on the big side, and I've never heard of anyone tabulating values in $\mathrm{yL}$.

Of course it wouldn't be hard to define a convenient unit, for example by deriving it from the ideal gas law at a standard temperature and pressure. But it would be better to use a unit that people already recognise. So my question is: is there a unit of volume in common use that is of a convenient size for thinking about single molecules? That means, I guess, that it should be between around $10^{-31}$ to $10^{-27}\;\mathrm{m^3}$, and preferably towards the smaller end of that.

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    $\begingroup$ I guess you wouldn't just want to work with cubic nanometers? Or perhaps a cubic Angstrom? $\endgroup$ – tpg2114 May 20 '14 at 0:54
  • $\begingroup$ $nm^3$ are a bit big, but actually I guess cubic Angstroms are probably the obvious choice, I just hadn't thought of it. (But the most important thing is whether people actually use those units for this purpose already.) $\endgroup$ – Nathaniel May 20 '14 at 0:56
  • $\begingroup$ I suppose it's a matter of choice depending on the size of your molecule -- complex hydrocarbons may be on the cubic nanometer scale. But between the two, you have your $10^{-30}$ or your $10^{-27}$ as the case may be. I've seen crystal lattices described by either their Angstrom spacing and angles or the volume in Angstroms of the circumscribed sphere. $\endgroup$ – tpg2114 May 20 '14 at 1:00
  • $\begingroup$ It's for considering volume changes rather than absolute volumes. Typically, volume changes are rather smaller than the volume of an individual molecule, unless the reaction involves a phase change to or from the gas state, in which case it goes up to $k_BT/p$ per molecule ($\approx 4\times 10^{-26}\;\mathrm{m^3}$ at standard $T$ and $p$). From working through a few examples, though, it seemed like it would be more convenient to use a smaller unit. $\endgroup$ – Nathaniel May 20 '14 at 1:09
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    $\begingroup$ I still think cubic Angstroms may be the best bet, but I don't know what kinds of calculations you're dealing with. I don't mind using values from 1e-3 to 1e3 regularly so something like $0.001$ cubic Angstroms isn't unbearable for me. That's probably within the range of volume changes you would encounter maybe? $\endgroup$ – tpg2114 May 20 '14 at 1:55

There are atomic units, where

Length is: Bohr radius = 1

Mass is: mass electron = 1

Time is: Bohr period = 1

See above link for a full explanation of the system.


Crystallographers mostly talk about the spacings between lattice planes in Ångstroms, as tpg2114 says in a comment.

I have found myself tabulating number densities for different materials in atoms per barn-cm. This is because the transmission through an absorber with number density $n$, thickness $\ell$, and capture cross section $\sigma$ is $$ T = \exp -n\sigma\ell $$ and I typically have cross sections in barns and care about lengths in centimeters. A barn is an area unit equal to 100 fm2 = 10-28 m2 = 10-24 cm2. This is a very small unit of volume, but it's a very long and skinny volume.

I can't remember if I started doing that on my own or if I copied it from someone else, but it was damned useful for doing transmission calculations. Don't know how appropriate for thermodynamics, though. Interesting question!


A cubic nm contains 33 molecules of water. The smallest amount of H2O needed to make water, is six molecules, or 0.18 cu. nm.


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