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While learning about poly-atomic ions, our instructor stated that ammonium is one of the few poly-atomic cations. Are there other poly-atomic cations? And why are positively-charged poly-atomic ions "rarer" than negatively-charged poly-atomic ions?

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There is an infinite number of polyatomic cations, much like there is an infinite number of polyatomic anions. So your instructor is wrong there. But there is still some reason in the statement. Why? Polyatomic simple cations are rare (if you discount metal-ligand complexes).

Generating a simple polyatomic anion is easy: Take an element that is not fluorine, bond it to as many hydroxy groups as its valency is and deprotonate the hydrogens — voilà. You can then play around with oxidation states (i.e. add more oxygen) to generate even more anions, e.g. with $\ce{PO3^3-}$ and $\ce{PO4^3-}$.

This works because oxygen is a very electronegative element that, when bonded to a hydrogen, is strong enough to displace said hydrogen. For a polyatomic cation to form, we would need to reverse the idea: Take a stable, neutral molecule that has a lone pair and then add a hydrogen (one of the few rather electropositive elements that still has a tendency to form bonds).

And here are the problems. Most elements simply aren’t basic enough to cope with the additional hydrogen. In fact, basically any element that isn’t nitrogen will be too acidic in its protonated state. Also, you can’t really play with additional oxygens to create further oxidation states since increasing the oxidation state would destabilise the cation. Thus, simple polyatomic cations are few while simple polyatomic anions are plenty.

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  • $\begingroup$ That was a great exploration, I didn't know that electropositivity was a thing. Thanks! $\endgroup$ – user35425 Oct 14 '16 at 0:17
  • $\begingroup$ I think it could phrased better as inorganic polyatomic cations are rare. Every organic amine could form a cation of course. $\endgroup$ – MaxW Oct 14 '16 at 22:37
  • $\begingroup$ @MaxW Hence simple. Taking my arbitrary definition of simple, it would only include two elements, but any organic amine requires at least three elements (carbon, nitrogen and hydrogen). $\endgroup$ – Jan Oct 14 '16 at 22:48
  • $\begingroup$ Jan - understand. My point was just that "inorganic" is a fairly well defined term but "simple" is more fuzzy. $\endgroup$ – MaxW Oct 14 '16 at 22:52

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