# Are there separate quantity symbols for the pKa of a substance and the pKa of its conjugate acid?

For polyprotic acids, we distinguish the different deprotonation equilibria and their equilibrium constants by indices, e.g. phosphoric acid dissociation is described by $$K_1$$, $$K_2$$, and $$K_3$$. Considering the acid/base properties of water, there are two relevant equilibria (protonating or deprotonating water once) and equilibria with at least two other species ($$\ce{H4O^2+}$$ and $$\ce{O^2-})$$. Is there a systematic nomenclature for these equilibria, maybe something like $$K_1$$, $$K_2$$, and $$K_{-1}$$ and $$K_{-2}$$?

In tables (typically in organic chemistry), molecules are often listed alongside with the $$\mathrm{p}K_\mathrm{a}$$ values of the conjugate acid, e.g. comparing the basicity of amines and amides. There is potential confusion with the $$\mathrm{p}K_\mathrm{a}$$ values of the species listed (giving a measure of how acidic they are).

Is there are good way of symbolically distinguishing the $$\mathrm{p}K_\mathrm{a}$$ values of the substance itself and that of its conjugate acid?

• Isn't the simple answer that pKa should always refer to the subject compound, and any other usage (such as "the pKa of ammonia is 9.2") is incorrect even if the intended meaning is clear enough? Jan 23 at 17:15
• For an aqueous solution, may you point to evidence of $\ce{H4O^2+}$ and $\ce{O^2-}$? So far e.g., with magnesia ($\ce{MgO}$), teaching I encountered so far was it consists of $\ce{O^2-}$ ions, however upon contact with water, these would rapidly (kinetics) deprotonate the solvent to yield $\ce{OH^-}$ hydroxide as the far more stable (thermodynamics) form. Jan 23 at 18:11
• @Buttonwood I don't think twice-deprotonated or twice-protonated water exists in aqueous solution. Nevertheless, folks want to attach a pKa to reactions even when they aren't observed to show how they don't occur. Jan 23 at 20:41
• @Andrew That would be nice. However, sometimes the intent is to show that a substance will not act as a base by stating a very high pKa of the corresponding conjugage acid (which does not have a common name because it does not form readily). Jan 23 at 20:45

Clayden et al. [1, p. 174] suggest to use $$\mathrm{p}K_\mathrm{aH}$$ (however, they merely introduce the symbol once and never use it again in the book relying solely on $$\mathrm{p}K_\mathrm{a}$$):

The ‘$$\mathrm{p}K_\mathrm{a}\text{s}$$’ of bases

Chemists often say things like ‘the $$\mathrm{p}K_\mathrm{a}$$ of triethylamine is about 10.’ (It’s actually 11.0 but 10 is a good number to remember for typical amines). This may surprise you as triethylamine has no acidic hydrogens. What they mean is of course this: ‘the $$\mathrm{p}K_\mathrm{a}$$ of the conjugate acid of triethylamine is about 10.’ Another way to put this is to write ‘the $$\mathrm{p}K_\mathrm{aH}$$ of triethylamine is about 10.’ The subscript ‘$$\mathrm{aH}$$’ refers to the conjugate acid.

Note that older literature used $$\mathrm{p}K_\mathrm{b}$$ written for the equilibrium

$$\ce{B + H2O <=> BH+ + OH-}\quad K_\mathrm{b} = \frac{[\ce{BH+}][\ce{OH-}]}{[\ce{B}]}\tag{1}$$

instead of uniform equilibrium with conjugate acid

$$\ce{BH+ <=> B + H+}\quad K_\mathrm{a} = \frac{[\ce{B}][\ce{H+}]}{[\ce{BH+}]}\tag{2}$$

and suggesting to use a relation

$$\mathrm{p}K_\mathrm{a} + \mathrm{p}K_\mathrm{b} = \mathrm{p}K_\mathrm{w} = 14.00~(\text{at}~\pu{25 °C}),\tag{3}$$

which was not only inconvenient for comparison between two scales for acidity (for example, see What is the importance of using a pKa value instead of a pKb value when describing drug chemistry?), but was a source of errors when one didn't pay attention for what temperature the value was reported (i.e. at $$\pu{20 °C}$$ $$\mathrm{p}K_\mathrm{w}$$ is $$14.17,$$ not $$14.00).$$ There is also a conceptual flaw in using $$\mathrm{p}K_\mathrm{b}$$ [2, pp. 195–196]:

Today the use of $$K_\mathrm{b}$$ is avoided because it does not touch the heart of the matter, namely: an acid produces hydrogen ions and a base receives them. Thus both acids and bases are best related in terms of a single quantity, their affinity for the hydrogen ion. Such a relationship requires the use of the acidic constant $$(K_\mathrm{a})$$ for both acids and bases.

### References

1. Clayden, J.; Greeves, N.; Warren, S. G. Organic Chemistry, 2nd ed.; Oxford University Press: Oxford; New York, 2012. ISBN 978-0-19-927029-3.
2. Albert, A. The Determination of Ionization Constants: A Laboratory Manual, 3rd ed.; Springer Netherlands: Dordrecht, 1984. ISBN 978-94-009-5548-6.