Have a look at the general buffer derivation of the Henderson-Hasselbalch formula from the mass balances and charge balances. The exact formula I wrote is the one you will arrive at.[itex][H^+]+\frac{K_w}{H^+}[/itex] is an extremely useful factor in this type of chemistry for exact calculation. (See "Aqueous Acid-Base Equilibria and Titrations", Robert de Levie, Oxford Chemistry Press, for some more detail.)

Will try if I will be able to find the book, in my position it can be rather difficult.

Besides, I have a feeling we are comparing apples and oranges here. It could be that de Levie uses the term "HH equation" in a non standard way. Typically HH equation is just the one we got by rearranging dissociation constant definition, it is not something derived using mass and charge balance.

The OP has multiple salts, of polyprotic acids, together and thus that equation up there will not even come close to an exact calculation! The real deal is much more complicated.

Actually as long as we are close enough to pKa2 (say any pH between 5 and 9) we don't have to worry - both other dissociations steps are far enough to not matter.

For an exact calculation you would probably need to treat the dissociation steps separately. Obviously let's remember that there's no such thing as an **exact** calculation of pH because finally the best we have for ionic activity corrections is the approximate Davies equation.

Yes, but it depends on what you mean by "exact". Exact in general means solving 5th degree polynomial for phosphoric acid alone, without adding base. That's not a thing to be done manually, numerical approach is the only viable one - and then it is better to use ready software, and to not solve each case separately.

Also note "exact" equation that you listed is for sure not exact in this context, it is obvious it relies on some simplifying assumptions as well. Easy to check that it is not self consistent, let's take an equimolar solution of acetic acid and acetate:

C

_{a} = 5×10

^{-4} M

C

_{s} = 5×10

^{-4} M

K

_{a} = 10

^{-4.757}Using full blown approach to the equilibrium one can find exact pH of the solution to be 4.79, or [H

^{+}] = 1.639×10

^{-5} M (calculated using BATE, ionic strength ignored, but it is low enough to not substantially change the final result; equilibrium concentrations of [HAcetate]=4.836×10

^{-4} M, [Acetate

^{-}]=5.164×10

^{-4} M, you can easily check that these numbers perfectly fit initial conditions, mass & charge balances and K

_{a}). Let's insert these values into the "exact HH" equation:

[tex]\begin{align}

[H^+] &= K_a \frac {C_a - [H^+] + \frac {K_w}{[H^+]}}{C_s + [H^+] - \frac {K_w}{[H^+]}}\\

&= 10^{-4.757} \frac {5\times 10^{-4} + 1.639\times 10^{-5} - \frac {10^{-14}}{1.639\times 10^{-5}}}{5\times 10^{-4} - 1.639\times 10^{-5} + \frac {10^{-14}}{1.639\times 10^{-5}}}\\

&=8.86\times 10^{-6}\end{align}[/tex]

So we have 1.639×10

^{-5} = 8.86×10

^{-6} which is obviously not true - and it means the equation is inconsistent with the numbers given. As these numbers do describe correctly the solution, equation must be wrong.

Obviously let's remember that there's no such thing as an **exact** calculation of pH because finally the best we have for ionic activity corrections is the approximate Davies equation.

Once again, it depends. Davies equation is an approximation used for solutions with ionic strengths above 0.1, as long as the ionic strength is below 0.1 Debye-Huckel theory (and Debye-Huckel equation, especially in the rarely used version taking ionic radii into account) gives pretty good results.