question
When computing the Gibbs energy change of a reaction using Legendre-transformed $\Delta_fG'$ values for a given $\mathrm{pH}$, the formula of the $\Delta_rG'$ of the reaction is:
$\Delta_rG' = \Delta_rG'^\circ + RT \ln Q'$
How should $Q'$ be expressed in practice? Should it include $\ce{H^+}$? Based on the reasoning below, I think it shouldn't, but I would like to have confirmation.
example
Lets consider the following reaction;
$\ce{A^- + H^+ <=> AH}$
In conditions of constant temperature and pressure, the Gibbs energy change of the reaction can be formulated as
$dG_{T,P} = \mu_{\ce{A^-}} dn_{\ce{A^-}} + \mu_{\ce{H^+}} dn_{H^+} + \mu_{\ce{AH}} dn_{\ce{AH}}$
If we suppose that the reaction occurs in a buffered medium such that the activity of $\ce{H^+}$ is constant through the advancement of the reaction, a correct way to represent the system is as an open system exchanging $\{\ce{H^+}\}$ with its surrounding (c.f., Ref.1):
$dG_{T,P} = \mu_{\ce{A^-}sys} dn_{\ce{A^-}sys} + \mu_{\ce{H^+}sys} dn_{\ce{H^+}sys} + \mu_{\ce{AH}sys} dn_{\ce{AH}sys} + \mu_{\ce{H^+}buf} dn_{\ce{H^+}buf}$
where $sys$ denote the chemical system and $buf$ the buffer, which acts as the "surrounding" in the sense of the 2nd law of thermodynamics.
A commonly used way to enforce constant $\mathrm{pH}$ in this system is to apply a Legendre transform on $dG$ such that
$dG'_{T,P} = dG_{T,P} - d(n_{\ce{H^+}sys} \mu_{\ce{H^+}sys}) = \mu_{\ce{A^-}sys} dn_{\ce{A^-}sys} + \mu_{\ce{AH}sys} dn_{\ce{AH}sys} + \mu_{\ce{H^+}buf} dn_{\ce{H^+}buf} - n_{\ce{H^+}sys} d\mu_{\ce{H^+}sys}$
since $\mathrm{pH}$ is constant, $d\mu_{\ce{H^+}sys} = 0$ so
$dG'_{T,P,\mathrm{pH}} = \mu_{\ce{A^-}sys} dn_{\ce{A^-}sys} + \mu_{\ce{AH}sys} dn_{\ce{AH}sys} + \mu_{\ce{H^+}buf} dn_{\ce{H^+}buf}$
We can develop this formula by expressing $\mu_i = \mu_i^\circ + RT \nu_i \ln \frac{\{i\}}{c^\circ}$, with $\nu_i$ being the stoichiometric coefficient for chemical species $i$, $\{i\}$ being its activity and $c^\circ$ being its reference activity in standard state (i.e. $1$) to get to this familiar expression:
$dG'_{T,P,\mathrm{pH}} = \Delta_r \mu^\circ + RT \ln \frac{\{\ce{AH}\}}{\{\ce{A^-}\}} + RT \ln \frac{\{\ce{H^+}buf\}}{c^\circ}$
$RT \ln \frac{\{\ce{H^+}buf\}}{c^\circ}$ being constant, what is called $\Delta_r G'^\circ$ should be $\Delta_r \mu^\circ + RT \ln \frac{\{\ce{H^+}buf\}}{c^\circ}$, which means $Q' = \frac{\{\ce{AH}\}}{\{\ce{A^-}\}}$. The contribution of $\ce{H^+}$ is then not present in $Q'$ as it is in $\Delta_r G'^\circ$ already. Is that right?
Reference:
- Lionel M. Raff, William R. Cannon, "On the Reunification of Chemical and Biochemical Thermodynamics: A Simple Example for Classroom Use," Journal of Chemical Education 2019, 92(2), 274–284 (https://doi.org/10.1021/acs.jchemed.8b00795).
