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I am reading an article where they describe azaindoles (bicyclic pyridine/pyrrole rings) as being basic with $\mathrm{p}K_\mathrm{a} \approx 4.6.$

I suppose at physiological $\mathrm{pH}$ and after rearranging the Henderson–Hasselbalch equation

$$\mathrm{pH} = \mathrm{p}K_\mathrm{a} + \log\frac{[\ce{A-}]}{[\ce{HA}]}\tag{1}$$ $$\mathrm{pH} - \mathrm{p}K_\mathrm{a} = \log\frac{[\ce{A-}]}{[\ce{HA}]}\tag{2}$$ $$10^{\mathrm{pH} - \mathrm{p}K_\mathrm{a}} = \frac{[\ce{A-}]}{[\ce{HA}]}\tag{3}$$ $$10^{7.4 - 4.6} > 1,\tag{4}$$

the numerator is greater than denominator, i.e. more product than reactant and it is a strong acid. Does this mean the compound would be negatively charged at physiological $\mathrm{pH}$?

What exactly makes this a base? Or are they referring to the lone pair on nitrogen acting as a Lewis base?

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The formula you have used implies the acid is neutral, like e.g. acetate/acetic acid conjugate pair. But it is not always the case, like for ionic acids like $\ce{NH4+}$ or $\ce{H2PO4−}$.

By other words the symbolic $\ce{HA}$/$\ce{A-}$ pair suggests $\ce{A-}$ has 1 negative charge, while more generally it can have any charge.

In the question context, we can apply the concept of Broensted-Lawry (B-L) acids/bases:

  • An acid is a substance able to release a proton ( in water a hydrated H+ ion ).
  • A base is a substance able to capture a proton.

A generalized B-L neutralization reaction is :

$$\ce{HA^{n} + B^{m} <=> A^{n-1} + BH^{m+1}}$$

where $m$ and $n$ are integer numbers indicating the charge. They can be positive, zero or negative.

Typically for $\ce{N}$-based bases, acids are positive ions and bases neutral. Like

$$\ce{H3O+(aq) + NH3(aq) <=> H2O(l) + NH4+(aq) }$$ or $$\ce{H3O+(aq) + Py(aq) <=> H2O(l) + PyH+(aq) }$$ where Py = pyridine

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