Chemistry - Do atoms form either a positive or a negative charge, but not both?
Solution 1:
Actually, in theory almost all of the elements can be found with both positive and negative oxidation numbers: it's just a matter of finding a system with the proper reagents and conditions to force it. If you isolate chemical species which have a very strong tendency of displaying some specific behaviour (accepting electrons, donating electrons, coordinating ions, releasing a leaving group, bonding to metals, releasing a proton, adopting a specific molecular geometry, or any other myriad properties), then you can often obtain strange results by clashing them with substances which also have the same tendency, but not quite as strong. This often causes the substance with the weaker behaviour to "operate in reverse".
Let me give a vivid and related example. As we all know, the alkali metals (group 1 elements) are exclusively present as cations with an oxidation number of +1, except in the pure metals, where it's zero, right? Well, here's something which might shatter your world: most of the alkali metals (with the exception of lithium, for now) also form alkalides, that is, stable salts containing discrete, clearly observed $\ce{Na^{-}}$, $\ce{K^{-}}$, $\ce{Rb^{-}}$ or $\ce{Cs^{-}}$ anions, with alkali metals displaying oxidation number -1.
How is this done? All you need is to find a neutral substance with a much stronger tendency to donate an electron than a neutral alkali metal atom (easier said than done). Since neutral alkali metals atoms form quite stable cations upon loss of an electron, this implies you need to search for a neutral substance capable of donating an electron and forming a cation with exceptional thermodynamic and/or kinetic stability. This can be achieved, for example, by the use of cryptands, which are cyclic molecules capable of coordinating very strongly to cations, strongly enough that they even coordinate alkali metal cations very well. The cryptand-coordinated cation is both thermodynamically and kinetically stable enough that alkalide anions, which would be extremely reactive otherwise, are not reactive enough in this case to immediately cause charge transfer and neutralize the negative charge.
Amusingly, it is actually possible to prepare a single compound which contains both alkali metal cations and anions, as exemplified by $\mathrm{[Na(2,2,2- crypt)]^{+} Na^{-}}$, which contains a sodium cation coordinated by a cryptand as a counterion to a natride/sodide ($\ce{Na^{-}}$) anion. One can imagine this compound being made by putting together a neutral sodium atom ($\ce{Na^{0}}$) and the neutral cryptand species $\mathrm{[Na(2,2,2- crypt)]^0}$. As Brian mentions in the comments, this latter species is actually an electride, which can be written as $\mathrm{[Na(2,2,2- crypt)]^{+}e^{-}}$ and thought of as a salt where the anion is a lone electron (!). Both the neutral sodium atom and the electride have a strong tendency to lose an electron in chemical reactions, but this tendency is much stronger for the electride. Thus, the electride ends up having its way, forcing its very loosely bound electron onto the neutral sodium atom, causing the neutral sodium atom to "operate in reverse" and accept an electron rather than donate it, resulting in the $\ce{Na^{-}}$ anion.
This table on Wikipedia is far more complete than most "common oxidation number" tables out there, and it lists many negative oxidation numbers for elements, including iron! For many of the transition metals, negative metal oxidation numbers are achievable by using the carbonyl ($\ce{CO}$) ligand, which removes electron density from the metal atom via backbonding. This stabilizes negative charges on the metal atom, once again allowing the resultant species to survive with the appropriate counterion.
Solution 2:
Nicolau Saker Neto already has an excellent answer, but he doesn't mention possibly the most common instance of an element being both a anion and a cation: Hydrogen
Hydrogen usually has a positive oxidation state, but in any of the metal hydrides, such as sodium hydride, it takes on a negative oxidation state.
Solution 3:
Unlike some, I wouldn't bet this is not common. Let's take a look at the biggest body in our system: the Sun. In its center, you can find for example some fluorine which is completely ionized. That's not really what we are used to with this electron lover.
We can also look for example at Titan's upper atmosphere. Here, interactions with the magnetic field of Saturn are responsible for the formation of species like $\mathrm O^+$, or almost every ionized molecule you can imagine with $\mathrm C$, $\mathrm N$, $\mathrm O$ and $\mathrm H$.
There are examples like these everywhere just in our Solar system.
Most chemistry classes seem to forget that our "standard room conditions" are far from being a standard in the whole universe. That's a shame because most of the "rules" we learn in high school (and some colleges) are based on the assumption that we are at those said standard conditions and stash the fact that the observed chemical behavior of elements is not necessarily even representative of what happen when you look at the whole universe.
Solution 4:
You are correct in saying that it is possible but not common. This comes about as a consequence of the effective nuclear charge on the atom or Z. eff, and the size of the atom. Z effective if you are not familiar is a concept usually touched upon in chem 2 as a periodic trend. It is a measure of the charge felt by the outer most electron. It is calculated by accounting for the electrons on the inner shells essentially canceling out or shielding the outer electron from feeling all of the charge from the positive nucleus. As you could expect this make the electron easier or harder to take away causing ionization depending on the Zeff. A small atomic radius causes the energy barrier to be higher to remove an electron, because the electron is closer to the nucleus. If the atom is larger it will be easier to remove the outer electron because it's further away from the nucleus. As you move along the periods of the periodic table from left to right and low to high, the effective nuclear charge increases. As you move along the periodic table from top down and right to left, the atomic radius increases. There are small changes in what you would expect to be a straight line trend, depending on what kind of orbital the electron "resides" in as well. The two concepts are shown here: Atomic radii and Zeff image.
From both of these concepts comes the reasoning:
Q: Why are some atoms harder to ionize to a positive charge?
A: Too high of an ionization energy required because of a lot of nucleus interaction with the outer electron.
Q: Why are some atoms not seen ionizing to a negative charge?
A: Electrons shield too much of the nucleus to allow for enough interaction to hold another electron.
Keep in mind that this holds for normal day to day reactions, and why some elements are found in the oxidation state they are on earth. Once you get into an extreme energy environments there are more possibilities.
Solution 5:
Your initial presumption is incorrect — both in the example it chose and in the logic it presents. Let’s tackle the logic first.
The most prevalent element that forms both a monoatomic cation and a monoatomic anion is hydrogen — although it is true that protons are hard to capture. Acidic compounds typically dissociate into $\ce{H+}$ and an anionic acid residue. And a wide range of relatively stable hydrides is known, including sodium hydride, tetrahydridoborates, aluminium hydride, lithium tetrahydridoaluminate and many more.
Carbon is another element that frequently forms anions and cations, depending on the mechanism. For example, $\mathrm{S_N1}$ reactions and $\mathrm{E1}$ eliminations both include a cationic carbenium intermediate. In a different cationic example, non-classical cations such as the norbonyl-cation, a carbonium ion, have been generated. Carbenium here refers to cations generated by removing a bond and carbonium to those generated by formally adding a bond onto a saturated carbon — in crystal structures, carbonium carbons seem bonded to five different residues.
Furthermore, it all boils down to how isolated the charge on a specific ion has to be. Frequently, if a carbenium ion is generated next to an oxygen, the more important resonance structure features a $\ce{C=O}$ double bond and the positive charge located on the oxygen, but that has less of a physical meaning than the carbenium cations noted above.
So what about iron? Well, you are correct that the $\mathrm{+II}$ and $\mathrm{+III}$ oxidation states along with the neutral $\pm 0$ are the most common. Another common and very stable iron compound is pentacarbonyliron $\ce{[Fe(CO)5]}$, which features iron(0). This can react with hydroxides in the following manner:
$$\ce{[Fe(CO)5] + OH- -> [Fe(CO)4(COOH)]- ->[][- CO2] [Fe(CO)4H]- ->[][- H+] [Fe(CO)4]^2-}$$
Since carbon monoxide is a neutral, two-electron donor, the double negative charge on the final complex formed by deprotonation of the intermediate hydridotetracarbonyliron(0) features an $\ce{Fe^2-}$ anion. Formally, the actual compound displaced is carbonate, not carbon dioxide.[1]
Reference:
[1]: W. Hieber, W. Beck, G. Braun, Angew. Chem. 1960, 72, 795. DOI: 10.1002/ange.19600722202.