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Can isotopes of any given element be represented using a completely different chemical symbol? What's the IUPAC's take on this?

Sure, ordinarily you would add the isotope's mass as a superscript to the element's symbol to differentiate it from other isotopes: For example, carbon-12 ($\ce{^{12}C}$) and carbon-14 ($\ce{^{14}C}$); however the base-symbol $\ce{C}$, for carbon, doesn't change.

But the isotopes of hydrogen don't seem to follow this strictly. Often, I see deuterium ($\ce{^{2}H}$) and tritium ($\ce{^{3}H}$) represented by $\ce{D}$ (I see this one in organic chem textbooks a lot) and $\ce{T}$ respectively. Does this "convention" fit in with IUPAC norms? If so, can isotopes of other elements be represented differently as well?

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IR-3.3.1 Isotopes of an element

The isotopes of an element all bear the same name (but see Section IR-3.3.2) and are designated by mass numbers (see Section IR-3.2). For example, the atom of atomic number 8 and mass number 18 is named oxygen-18 and has the symbol $\ce{^{18}_{}O}$.

IR-3.3.2 Isotopes of hydrogen

Hydrogen is an exception to the rule in Section IR-3.3.1 in that the three isotopes $\ce{^{1}_{}H}$, $\ce{^{2}_{}H}$ and $\ce{^{3}_{}H}$ can have the alternative names protium, deuterium and tritium, respectively. The symbols D and T may be used for deuterium and tritium but $\ce{^{2}_{}H}$ and $\ce{^{3}_{}H}$ are preferred because D and T can disturb the alphabetical ordering in formulae (see Section IR-4.5). The combination of a muon and an electron behaves like a light isotope of hydrogen and is named muonium, symbol $\ce{Mu}$.⁵ These names give rise to the names proton, deuteron, triton and muon for the cations $\ce{^{1}_{}H+}$, $\ce{^{2}_{}H+}$, $\ce{^{3}_{}H+}$ and $\ce{Mu+}$, respectively. Because the name proton is often used in contradictory senses, i.e. for isotopically pure $\ce{^{1}_{}H+}$ ions on the one hand, and for the naturally occurring undifferentiated isotope mixture on the other, it is recommended that the undifferentiated mixture be designated generally by the name hydron, derived from hydrogen.


Source:

N.G. Connelly, T. Damhus, R.M. Hartshorn, A.T. Hutton (eds) (2005). Nomenclature of Inorganic Chemistry (PDF). RSC–IUPAC. ISBN 0-85404-438-8.


Addendum:

The small subscript ⁵ present in the source is a reference to Names for Muonium and Hydrogen Atoms and Their Ions, W.H. Koppenol, Pure Appl. Chem., 73, 377–379 (2001) which can be viewed over at:

https://www.iupac.org/publications/pac/pdf/2001/pdf/7302x0377.pdf

$\cdots$ A particle consisting of a positive muon and an electron ($\pu{\mu^+ e^–}$) is named “muonium” and has the symbol $\ce{Mu}$. Examples: “muonium chloride,” $\ce{MuCl}$, is the equivalent of deuterium chloride $\cdots$

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    $\begingroup$ Additional historic information is contained in answers to chemistry.stackexchange.com/questions/35462/… $\endgroup$ – Jon Custer May 4 '17 at 16:13
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    $\begingroup$ Muons are negatively charged. Presumably you mean an antimuon? $\endgroup$ – Emilio Pisanty May 4 '17 at 19:53
  • $\begingroup$ It's supposed to be an antimuon, en.m.wikipedia.org/wiki/Muonium. @berry please check the original source. $\endgroup$ – Oscar Lanzi May 20 '17 at 19:45
  • $\begingroup$ @OscarLanzi Checked, the source reads muonium, also I've added the reference it was linked to. $\endgroup$ – Berry Holmes May 22 '17 at 5:29
  • $\begingroup$ @E.P. ^please see this addendum $\endgroup$ – Berry Holmes May 22 '17 at 5:30
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While it is read like "carbon 12", and "carbon 14", etc. the atomic notation of the nuclides is the other way around, i.e. the mass number precedes the element symbol. Hence it is not $\ce{C^{12}}$, but $\ce{^{12}C}$.

The special names and symbols seen for $\ce{^1H}, \ce{^2H}, \ce{^3H}$ (but not the other isotopes of neither hydrogen, nor the ones of other elements) follow a recommendation by IUPAC, published in Pure and Applied Chemistry in 1988 (doi 10.1351/pac198860071115, open access). It is still in power, as the corresponding recommendation by 2001 in the same journal shows (doi 10.1351/pac200173020377).

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    $\begingroup$ Preceding the symbol with the mass is a relatively modern invention. When I was involved in chemistry (be it 40 years ago) the historic positioning after the symbol was the rule and I see no need to change - it's change for change's sake. If it's good enough for Linus Pauling (College Chemistry, 1964 edition) then it's good enough for anyone. A similar situation occurred in electronics where educational institutions flip-flopped from conventional current flow to electron current & back again, causing confusion that still reigns. Planets asteroids & stars are named systematically - except a few $\endgroup$ – Magoo May 4 '17 at 17:19
  • $\begingroup$ @Magoo This is indeed a noteworthy hint. Hence I looked up page 63/700 in the archive (archive.org/stream/CollegeChemistry/…) in chapter "The structure of the Atomic Nuclei" and found it. Interesting reading, with figures clearer than in many contemporary (colourful) chemistry books. $\endgroup$ – Buttonwood May 4 '17 at 21:08
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    $\begingroup$ @Magoo - with many problems relating to ions and charge states being caused under that convention. I think various sub-fields in physics and chemistry arrived at slightly different conventions in the last century, and we might still be working out the bugs... $\endgroup$ – Jon Custer May 4 '17 at 22:55

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