# Amount of substance expressed in yoctomole units

I stumbled upon a couple of papers where amounts were expressed in yoctomoles ([ymol], $\pu{10^{-24} mol}$) and I find it somewhat bothersome as 1 ymol would correspond to about 60% of atom/molecule/ion etc., which doesn't make much sense to me:

$$N_\mathrm{A} \times \pu{10^{-24} mol} = \pu{6.022 \times 10^{23} mol-1} \times \pu{10^{-24} mol} \approx \pu{0.6}\,\text{(elementary entities)}$$

Simply put, I'd say that using zeptomoles ([zmol], $\pu{10^{-21} mol}$) is already a stretch as one of the ideas behind a mole concept is to allow to handle large numbers of entities more conveniently, and at this scale there is no reason not to use direct quantities of molecules/atoms/ions.

Should the yoctomoles units really be used, and if so, then how to rationalize fractional elementary entities?

• I don't follow your conversion, but agree that a yoctomole would correspond to 0.6 molecules. Using the unit seems fine to me. – MaxW Oct 23 '17 at 3:46
• When it comes to homeopathic dilutions even this unit seems a little bit coarse-grained. Oscillococcinum e.g. has one part duck offal in $10^{400}$ parts water. – aventurin Oct 23 '17 at 21:53

Well, the closer you get to atomic detection limits, the less sense bulk units make, that is correct.

However, coming from lab scale units it is much easier to scale down directly with SI prefixes — the exact reason what they were designed for. Therefore, it makes sense that these papers might say ‘Our predecessors could only detect $\pu{10zmol}$, we have lowered the detection limit by a factor of $100$ to $\pu{0.1zmol}= \pu{100ymol}$.’ And that is basically what they say. I note that both abstracts immediately place a molecule count behind the amount in ymol, so they are well aware of the scale of their unit with respect to molecules.

Once you get to the full molecular scale, you will be dealing with a type of quantisation this way or that; whether you are reporting energy, charge, mass or amount. In none of the other cases is anything special described for atomic scales so why should we make an exception for moles? Just make sure that the value itself corresponds to a whole number of molecules within the boundaries of rounding errors.

An addition to the self-contained Jan's answer: an existence of yoctomole can be justified by using it as the systematic unit for the absolute detection limit of $$N_\mathrm{A}^{-1}\approx\pu{1.66 ymol}$$ below which a molecule seize to exist. The first instrumental method to allow for such high sensitivity was, as it appears, capillary electrophoresis with laser-induced fluorescence detection which approached yoctomolar limit of detection during 1980s (and the yocto- prefix officially adopted in 1991).

I cannot help not to quote a part of Castagnola's brief paper [1] mainly because it shares concerns raised in OP and because of the illustration (addressing aventurin's comment on homeopathy):

While we can count a dozen eggs one egg at a time, the magnitude of Avogadro’s number normally rules out direct counting of objects. … As far as egg counting goes, it is better and simpler to say, ‘Please, give me four eggs’, than to ask for ‘one third of a dozen eggs’. It should surely therefore be better to discuss sensitivity levels in term of hundreds of molecules than in terms of a ‘zeptomole sensitivity level’. … Moreover, it is obvious that the manipulation of a twentieth of a dozen inevitably leads to broken eggs and, therefore, researchers can only introduce the yoctomole limit if they are planning to expand their culinary skill towards the preparation of a sub-molecular omelette!

### References

1. Castagnola, M. Sensitive to the Yoctomole Limit. Trends in Biochemical Sciences 1998, 23 (8), 283. DOI: 10.1016/S0968-0004(98)01251-1.

It isn't just yoctomoles. Synthesis attempts for super heavy elements involve cross-section for nuclear reactions involving small fractions of the cross-sectional area associated with one proton or neutron. Compare the cross-secions in this passage from Wikipedia with roughly 0.01 barn, the cross-section typically rendered for a single nucleon:

In August–October 2011, a different team at the GSI using the TASCA facility tried a new, even more asymmetrical reaction: …

$$\ce{^{249}_{98}Cf + ^{50}_{22}Ti → ^{299}_{120}Ubn^* → no atoms}$$

Because of its asymmetry, … the reaction between $$\ce{^{249}Cf}$$ and $$\ce{^{50}Ti}$$ was predicted to be the most favorable practical reaction for synthesizing unbinilium, although it is also somewhat cold, and is further away from the neutron shell closure at $$N = 184$$ than any of the other three reactions attempted. No unbinilium atoms were identified, implying a limiting cross section of 200 fb. … Jens Volker Kratz predicted the actual maximum cross section for producing unbinilium by any of the four reactions $$\ce{^{238}U + ^{64}Ni}$$, $$\ce{^{244}Pu + ^{58}Fe}$$, $$\ce{^{248}Cm + ^{54}Cr}$$, or $$\ce{^{249}Cf + ^{50}Ti}$$ to be around 0.1 fb; … in comparison, the world record for the smallest cross section of a successful reaction was 30 fb for the reaction $$\ce{^{209}Bi(^{70}Zn,n)^{278}Nh}$$, … and Kratz predicted a maximum cross section of 20 fb for producing ununennium. … If these predictions are accurate, then synthesizing ununennium would be at the limits of current technology, and synthesizing unbinilium would require new methods.

The sub-particle cross-section comes from incorporating the low reaction probability into the calculation.