# Obtaining activity coefficients of conjugate acids of some common carboxylic acid molecules

Recently, I have been wondering about justifying the notion on how some common acids (e.g. carboxylic acids) are themselves weaker bases than water to the point that we don't usually consider them to act as base. To begin, we use the formula for $\mathrm{p}K_\mathrm{b}$ and consider for example acetic acid (written in organic chemistry convention, with $\ce{OAc^-}$ being the acetate group $\ce{CH_3COO^-}$ for unfamiliar readers)

$$\mathrm{p}K_\mathrm{b} = -\log \left(\frac{a_{\ce{H2OAc+}}a_{\ce{OH-}}}{a_{\ce{HOAc}}}\right)$$

where everything is measured at room temperature, 1 atm.

As usual, if the solution is dilute enough, we can approximate the activity with concentration.

However, the species $\ce{H2OAc+}$ seemed to be not very popular as NIST have few data on its thermodynamic parameters. While it is in principle possible to perform an experiment to measure $a_{\ce{H2OAc+}}$, if it is indeed very weak as a base, then the measurement result will only be reflecting the acetate ionisation equilibrium and masked any tiny result of $a_{\ce{H2OAc+}}$ if any.

How does one obtain the necessary thermodynamic parameters for a potentially highly unstable protonated species in aqueous medium in order to work out the $\mathrm{p}K_\mathrm{b}$ of the common carboxylic acids?

\begin{align} I &= \frac{1}{2}\sum_{{\rm i}=1}^{n} c_{\rm i}z_{\rm i}^{2} \\ \log{\gamma} &= -0.509 . ( z_+^m . |z_-|^n )^{\frac{1}{m+n}}. \sqrt{I} \quad (*)\\ \end{align}
$$c_i$$ is the ion concentration.
$$z_i$$ is the ion charge.
$$I$$ is the Ionic strength.
$$\gamma$$ is the activity ciefficient by the Debye-Hückel equation.
(*) - See the link for more info and more complex formulas for the medium $$I$$.