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

Question

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?

Answer 1

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.