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Is the Avogadro constant equal to one?

Bane asked and provoked a discussion:

I was tasked with creating a presentation on Avogadro's work, and this is the first time I actually got introduced to the 'mole' and to 'Avogadro's constant'. And, to be honest, it doesn't make any mathematical sense to me.

\[ \eq{ 1\,\,{\rm mole} &= 6.022 \times 10^{23} \\
\text{Avogadro's constant} &= 6.022 \times 10^{23} \, {\rm mole}^{-1} } \]

This [w]hole field seems very redundant. There are four names for the same thing! Since when is a number considered to be a measurement unit anyway?!

LM answers:

Yes, Avogadro's constant is a redundant artifact from the era in the history of chemistry in which people didn't know how many atoms there were in a macroscopic amount of a material and it is indeed legitimate to set Avogadro's constant equal to one and abandon the awkward obsolete unit "mole" along the way. This $$$N_A=1$$$ is equivalent to\[

1\,\,{\rm mole} = 6.023\times 10^{23}\, \text{molecules or atoms}

\] and the text "molecules or atoms" is usually omitted because they're formally dimensionless quantities and one doesn't earn much by considering "one molecule" to be a unit (because its number is integer and everyone may easily agree about the size of the unit). We may use the displayed formula above to replace "mole" (or its power) in any equation by the particular constant (or its power) in the same way as we may replace the word "dozen" by 12 everywhere (hat tip: Mark Eichenlaub). We can only do so today because we know how many atoms there are in macroscopic objects; people haven't had this knowledge from the beginning which made the usage of a special unit "mole" justified. But today, the particular magnitude of "one mole" is an obsolete artifact of social conventions that may be eliminated from science.

Setting $$$N_A=1$$$ is spiritually the same as the choice of natural units which have $$$c=1$$$ (helpful in relativity), $$$\hbar=1$$$ (helpful in any quantum theory), $$$G=1$$$ or $$$8\pi G=1$$$ (helpful in general relativity or quantum gravity), $$$k=1$$$ (helpful in discussions of thermodynamics and statistical physics: entropy may be converted to information and temperature may be converted to energy), $$$\mu_0=4\pi$$$ (vacuum permeability, a similar choice was done by Gauss in his CGSM units and with some powers of ten, it was inherited by the SI system as well: $$$4\pi$$$ is there because people didn't use the rationalized formulae yet) and others. See this article for the treatment of all these universal constants and the possible elimination of the independent units:
Let's fix the value of Planck's constant
In every single case in this list, the right comment is that people used to use different units for quantities that were the same or convertible from a deeper physical viewpoint. (Heat and energy were another example that was unified before the 20th century began. Joule discovered the heat/energy equivalence which is why we usually don't use calories for heat anymore; we use joules both for heat and energy to celebrate him and the conversion factor that used to be a complicated number is one.) In particular, they were counting the number of molecules not in "units" but in "moles" where one mole turned out to be a very particular large number of molecules.

Setting the most universal constants to one requires one to use "coherent units" for previously independent physical quantities but it's worth doing so because the fundamental equations simplify: the universal constants may be dropped. It's still true that if you use a general unit such as "one mole" for the amount (which is useful e.g. because you often want the number of moles to be a reasonable number comparable to one, while the number of molecules is unreasonably large), you have to use a complicated numerical value of $$$N_A$$$.

There is a long discussion under my answer at the Physics Stack Exchange.

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reader Harry Dale Huffman said...

It is not called "the Avogadro constant", it is called "Avogadro's number" (that much is scientific--one may say, social--convention), and it is of course not one: The reciprocal of one is one, while the reciprocal of Avogadro's number is the Atomic Mass Unit, AMU (= 1.66 E-24 grams).

reader Luboš Motl said...

Dear Harry, you're totally mistaken about the conventions and, more importantly, about the whole point of this article.

Be sure that what I called the Avogadro constant is called the Avogadro constant (click and read) and it may be set to one because it contains "inverse mole", a new unit according to the SI system.

Historically, Avogadro's number was the dimensionless numerical constant only which is obviously not equal to one and cannot be set to one. But the Avogadro constant is dimensionful and it can be set to one.

You're also sloppy when writing that Avogadro's number is reciprocal to the atomic mass unit. Only the numerical values match in the SI system, in other systems, they wouldn't. And it's because the units don't match. The atomic mass unit contains the units of "grams" so it can obviously be reciprocal neither to Avogadro's number nor the Avogadro constant.

You're just completely misunderstanding the concept of a unit and the important fact that a unit that is chosen by someone isn't naturally or objectively "1" in any sense which is why it is so important to write the unit explicitly. When some laws of physics are understood at the fundamental level, one may choose natural units that everyone may agree upon and that contain no social baggage.