# should charge balance be based on activities?

We have a fluid with $\pu{0.01 M}~\ce{HCl}$ and $\pu{0.02 M}~\ce{K2HPO4}$, so if we assume activity coefficients are $1$ throughout we can find pH as $6.800012$ and confirm that at that pH there is approximately zero balance ($6.104839\!\times\!10^{-13}$) between positive and negative charges calculated as

$$\mathrm{ K^+ +H^+ – OH^- - Cl^- -}\ce{H2PO4-}\mathrm{ - 2\times} \ce{HPO4^2-} \mathrm{\ -\ 3\ \times}\ce{PO4^3-}$$

Now we can find the ionic strength and via Debye-Hückel/Davies find activity coefficients for absolute charge 1-3. This allows correction of the thermodynamic dissociation coefficients for phosphoric acid (and water dissociation) so we can correct the concentrations and find a new pH to minimize charge balance. After $4$ iterations we have pH 6.436864 at a charge balance of $1.429412\!\times\!10^{-15}$. This procedure allows reproduction of the results in a recent paper (Schell et al J Chem Eng Data 2018; 63: 2151-2161) with correlation $>0.9999$.

Now the question: these results requires the charge balance being formulated strictly in concentrations multi-plied by the relevant charges. I simply wonder why we should not for the charge balance equation multiply the found activity coefficients on the charged species so making the balance to minimize a function of $f_3\times 3\times [\ce{PO4}]$ and so forth with $f_3$ being the trivalent activity coefficient from Davies?

• It doesn't seem strange at all to me. You are doing a balance of charges, and the charges don't know what activity is. – Chet Miller Sep 9 '18 at 11:56
• Sticking with concentrations certainly gives a perfect fit compared to other methods, but all the literature on single ion activity coefficients for strong ions made me wonder if the charge should not be discounted. I am happy if that is not the case. – user37217 Sep 9 '18 at 17:37
• Would there be any online literature Chester Miller could cite to make it obvious that charges are oblivious to activities? – user37217 Sep 10 '18 at 6:09
• I don't know. But check out Handbook of Aqueous Electrolyte Thermodynamics – Chet Miller Sep 10 '18 at 12:48
• I would have corrected the typesetting, but I do not understand the formula at all. 2*K+H – OH- Cl -H2PO4 - 2*HPO4 - 3*PO4 What exactly are you adding or subtracting there? It is certainly not the elements/ions, but some kind of property of them. Please try to be more precise about this. I still don't understand the question and believe it should remain closed until it can be clarified. – Martin - マーチン Sep 12 '18 at 15:44