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How can we identify bases and acids by their chemical formulas?

Take for example this reaction. I know that HNO3 is an acid and KOH is a base, but if I didn't know that how would I be able to identify that using the chemical formulas

$\ce{HNO3(aq) + KOH(aq) \rightarrow KNO3(aq) + H2O(l)}$

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I'm going to assume you are in high school, thus talking about Brønsted acids/bases.

A Brønsted acid is a compound that can donate a proton $\ce{H^+}$ whereas the base can accept a proton. For some chemicals it can be easily seen that they can donate a proton, like in the example of $\ce{HNO3}$ you give. There is obviously a proton in there so it makes sense that it is an acid. For chemicals with $\ce{O^-}$ in their structure (like the $\ce{KOH}$ in your example which dissociates into $\ce{K^+}$ and $\ce{OH^-}$) it is also clear that they can accept a proton and thus are obviously a base.

Now the difficult part is when a molecule can do both: these are called amphiprotic substances, the most famous being water $\ce{H2O}$, but there are many more of these, think about $\ce{H2PO4^-}$, it both can accept and donate a proton. In these situations molecules can indeed be considered both a base and an acid and the way they behave depends on the other molecules they come into contact with. If the other chemical is a stronger acid, as measured by the acid dissociation constant then the current molecule will act as base and vice versa.

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Look at The Periodic Table. Metal (Group I and II) hydroxides are bases, non-metal "hydroxides" are acids (especially with high positive charges on the non-metal.) Then we have amphoteric hydroxides like zinc and aluminum that afford zincates and aluminates in very alkaline media. $\ce{Mn(OH)3}$ is a base but $\ce{HOMnO3}$ is a strong acid. Look at core oxidation state, charge polarization, and oxophilicity of that ion.

Arrhenius, Brønsted-Lowry, Lewis; hard, soft; solvo-cation, solvo-anion. How do you classify $\ce{Al(OMe3)}$ that is a Brønsted base for methoxide and a Lewis acid for aluminum(III)?

$\ce{NH4}{^+}$ $\ce{NH3}$ $\ce{NH2}{^-}$ $\ce{NH}{^{2-}}$ $\ce{N}{^{3-}}$. Ammonium is definitely the acid. Nitride is definitely the base. What are the species in-between?

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