I am slightly baffled by a seemingly simple situation: I want to calculate the free energy of association between a molecule $\ce{A}$ and the solvent $\ce{B}$ using quantum methods (e.g. DFT).
From what I've understood, the basic way to do this in most situation is to optimize the geometry of each compound, then perform frequency calculations to obtain free energy corrections to the SCF. Let the SCF plus the corrections for specie $\ce{X}$ be $\ce{G_X}$. This yields the free energy of each compound at 1 atm, which can be converted into the free energy at 1 M via a factor of 1.89 kcal/mol. Thus, for the complexation $\ce{A + B \rightleftharpoons AB}$, the free energy of complexation would be
\begin{align} \Delta G_{comp}^0 &= (G_{AB}^0 + \pu{1.89 kcal/mol}) - (G_{A}^0 + \pu{1.89 kcal/mol}) - (G_{B}^0 + \pu{1.89 kcal/mol}) \\ &= G_{AB}^0 - G_{A}^0 - G_{B}^0 - \pu{1.89 kcal/mol} \end{align}
However, in the case of association between molecule $\ce{A}$ with concentration $\ce{C_A}$ and solvent $\ce{B}$ with concentration $\ce{C_B}$, how could one obtain the free energy of complexation?
There is the equation $\Delta G = \Delta G^0 - RT \ln\left(\frac{C_{AB}}{C_A*C_B}\right)$, but the "initial" concentration of the adduct is null. There seems to be a $RT \ln(C_B)$ term floating around in the literature, but I'm not too sure where it's coming from and how to handle this situation rigorously.