When the $pK_a$ scale is not applicable - when there is no proton/charged species transfer - what should be used to quantify the strength of an acid?
Answer
$$\ce{A~ + :\!\!B -> \ \ A\!\leftarrow:\!\!B}$$
$$K_f = \dfrac{\ce{[A\!\leftarrow:\!\!B]}}{\ce{[A][B]}}$$
See sections 4.1 and 4.2 in the following reference for more information:
http://goicoechea.chem.ox.ac.uk/teaching/abse/abse_lecture2.pdf
But $K_f$ is specific to an acid-base pair.
A more universal scale is formed by the Gutmann–Beckett method by observing the phosphorous NMR spectrum of triethylphosphine oxide as it interacts with a sample lewis acid.
An Acceptor Number (AN) scale is formed with AN=0 corresponding to triethylphosphine oxide in hexane (P chemical shift of 41), and AN = 2.21 (d-41), where d is the chemical shift of triethylphosphine oxide interacting with the sample. The greater AN is, the more acidic the compound.
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