Transformed chemical potential of formation: something weird in Alberty 2003

I am trying to wrap my head around transformed chemical potentials, as described in Thermodynamics of biochemical reactions (R. Alberty, 2003), however, two equations seem to be contradictory in this book.

Lets consider a chemical species $$AH_n$$, which has n protons. The formation reaction of that species would then be $$\ce{A^{n-} +nH+<=> AH_n}$$

According to equation 4.4-6 of Alberty 2003, $$\Delta_f G$$; the Gibbs energy of formation for $$\ce{AH_n}$$ (under constant temperature and pressure) would be

$$\Delta_f G = \Delta_f G^0 + RT \ln⁡ \{\ce{AH_n}\} \tag{4.4-6}$$

Where $$\Delta_f G^0$$ is the standard Gibbs energy of formation for $$\ce{AH_n}$$ and $$\{\ce{AH_n}\}$$ the activity of $$\ce{AH_n}$$.

According to equation 4.4-8 Alberty 2003, $$\Delta_f G'$$; the Legendre-transformed Gibbs energy of formation for $$\ce{AH_n}$$ (under constant temperature and pressure) would be

$$\Delta_f G' = \Delta_f G'^0 + RT \ln⁡ \{\ce{AH_n}\}\tag{4.4-8}$$

Where $$\Delta_f G'^0$$ is the Legendre-transformed standard Gibbs energy of formation of $$\ce{AH_n}$$, defined in equation 4.4-9 of Alberty's book as

$$\Delta_f G'^0 = \Delta_f G^0 - n(\Delta_f G^0_{\ce{H^+}} + RT \ln⁡ {10}^{-\mathrm{pH}})\tag{4.4-9}$$

that is

\begin{align}\Delta_f G'^0 &= \Delta_f G^0 - n(\Delta_f G^0_{\ce{H+}} - RT \ln⁡ (10) \cdot \mathrm{pH}) \\ \Delta_f G'^0 &= \Delta_f G^0 - n \Delta_f G^0_{\ce{H+}} + n \cdot RT \ln⁡ (10) \cdot \mathrm{pH} \end{align}

At an ionic strength of 0; $$\Delta_f G^0_{H^+} = 0$$. So,

$$\Delta_f G'^0 = \Delta_f G^0 + n \cdot RT \ln⁡ (10) \cdot \mathrm{pH}$$

This development seems contradicted by what I can see in equation 4.3-9 of the same book; it states that the Legendre-transformed chemical potential of a pseudo isomer "1" is

$$\mu_1'^0 = \mu_1^0 - N_{H,1} \cdot RT \ln⁡(10) \cdot \mathrm{pH} \tag{4.3-9}$$

where $$\mu_1^0$$ is the standard chemical potential of pseudo-isomer "$$1$$" and $$N_{H,1}$$ is the number of hydrogen atoms in the pseudo-isomer "1", which is equivalent to the variable $$n$$ used above. I suppose that in conditions of constant temperature and pressure, chemical potentials are equivalent to Gibbs energies of formation. While this formula is given for a pseudo isomer, it should also be true out of this context, since it has to be true for a group consisting of a single pseudo isomer. What I do not understand is why the sign of the $$- N_{\ce{H},1} \cdot RT \ln⁡(10) \cdot \mathrm{pH}$$ term is negative given the development of equation 4.4-9. Can someone tell me how those two equations (4.3-9 and 4.4-9) can be true at the same time?

• What is equation 4.4-9? Commented Sep 23, 2020 at 17:49
• What is the meaning of $\Delta_f G$, $\Delta_f G'$, $\Delta_f G^0$, $\Delta_f G'^0$, And is ln$⁡{AH_n}$ equal to ln$⁡[AH_n]$ ? Commented Sep 23, 2020 at 18:51
• I edited my question to clarify this Commented Sep 24, 2020 at 10:02
• $- n(\Delta_f G^0_{\ce{H^+}} + RT \ln⁡ {10}^{-\mathrm{pH}})=- n\Delta_f G^0_{\ce{H^+}} - nRT \ln⁡ [\ce{H^+}])$. It therefore would seem to me that $\Delta_f G'^0$ is the standard free energy formation of $AH_n$ minus the standard free energy of formation of n times $H^+$ (plus an extra term ...) Commented Sep 24, 2020 at 13:49
• The extra term appears to come from the expression for the reaction quotient. There appears to be terms for $A^{n-}$ missing. Otherwise the expressions are pretty normal. Commented Sep 24, 2020 at 13:55