Simplified methods of pH calculation of amphiprotic salt solution

As usual, the most general approach is unsuitable for calculations done by hand. For salts with one strong and one weak component the best way of pH calculation it to treat conjugate acid (or base) as the only source of H⁺ (or OH^-) ions, and do the calculations in the same way as it was described in weak/acid base section.

For example if we have a solution of a salt of a weak acid (with dissociation constant K_a) and a strong base, the reaction of hydrolysis is

and the equilibrium is described by the conjugate base dissociation constant

where

Starting from these equations we can calculate pOH and pH of the solution using method and assumptions shown for weak acid an base. Exactly the same approach can be used for salt of strong acid and weak base - just using the K_a constant for the weak base conjugate acid . If the acid (or base) is polyprotic we can use one of the methods described in the polyprotic simplified section.

However, what to do when we have a solution of a salt that contains both weak acid and weak base? In general we should use the full equation derived on the pH of the salt solution page. As it yields 4^th degree polynomial, its applicability is at least questionable. Luckily, if the solution is not too diluted we can easily derive much simpler formula.

Let's say we have a solution of an AB salt of a weak acid and a weak base of concentration C and dissociation constants K_a and K_b. Equations for K_a and K_b have their standard forms:

We will solve them for [H⁺] and [OH^-]:

Putting calculated concentrations into water ion product equation we get:

or

Now let's take a look at the charge balance equation:

As long as we deal with not very diluted solution, we can safely assume concentrations of both H⁺ and OH^- are much lower than concentrations of A^- and B⁺. That yields an important approximation:

From the mass balance we know that

combining it with the equation 13.10 we get another approximation:

Using 13.10 and 13.12 we can now rewrite equation 13.8 as

which is a very important result. Knowing ratio of concentrations of the acid and its conjugate base we can easily calculate H⁺ concentration, just by rearranging acid dissociation formula and plugging the ratio into:

Taking logs of both sides:

If we remember the relation between K_a and K_b we can rewrite this equation as

where pK_aB is acid dissociation constant of the base B conjugate acid. This result is interesting, as the equation is identical to that derived for the pH of solution of the amphiprotic salt.

How good is the equation 13.16? For ammonium acetate (CH₃COONH₄) pK_a=4.75 and pK_b=4.75 too. This salt is so symmetrical, that results of pH calculation just must be perfect:

Results of pH calculation of ammonium acetate

C (M)	pH calculator	pH eq. 13.16
1	7.00	7.00
10-8	7.00	7.00

And what about aniline formate? pK_a=3.745 and pK_b=9.4, so there is a substantial difference in the strengths.

Results of pH calculation of aniline formate

C (M)	pH calculator	pH eq. 13.16
1	4.17	4.17
0.1	4.17	4.17
0.01	4.18	4.17
0.001	4.24	4.17

The more diluted the solution, the worse the results, but if the solution is at least in 0.01M range both equations - 12.9 and 13.16 - give acceptable results.