Why does the pH of a diluted solution change?

[Bohrok]

The first time I came across a chemistry problem about the pH of a solution that's been diluted, I thought there was an argument for a change and for no change. After doing some searching online, it seems that the pH does change since [H₃O⁺] changes, which I understand. However, using the Henderson–Hasselbalch equation equation both before and after dilution would give the same pH, right?

As an example, if I have x moles each of HA and A^- in 1 L solution, then$$pH = pK_a + \log\left(\frac{\frac{x\text{ mol }A^-}{1 L}}{\frac{x\text{ mol }HA}{1 L}}\right) = pK_a + \log(1) = pK_a$$Now if the solution were at 2 L to start with, then$$pH = pK_a + \log\left(\frac{\frac{x\text{ mol }A^-}{2 L}}{\frac{x\text{ mol }HA}{2 L}}\right) = pK_a + \log(1) = pK_a$$

How is this setup giving the wrong answer as opposed to recalculating [H₃O⁺] after the dilution?

 

[ChiralWaltz]

It looks like x moles/x moles=1 is the error. It should look more like X mol per L/Y mol per L or (.10 M/.12 M), when X=.10, Y=.12 and M=(mol/L).

Due to the common ion effect and Le Chatelier's principal, the weak electrolyte ionizes less than the strong electrolyte. Therefore the equilibrium shifts in favor of the strong electrolyte, creating an unequal molar concentration of acid and base (example .10 M and .12 M). Hopefully I got the question right.

 

[MrSid]

A hint; take a look at the derivation of the HH equation and the simplifying assumptions that are made...

On the surface definition of a buffer, the pH is largely defined by pKa and has a broad largely unchanged pH over a range defined by the buffer's capacity. Dilution of the buffer only changes the capacity but not the pH range to the first order approximation.

In reality, the quantities used are not the formal concentrations of the corresponding acid and conjugate, but their Activities which do have a bearing on changing the pH as one changes concentration.

Borek's pH calculator (chembuddy) uses Debye Huckel and Extended Debye Huckel Limiting Law to make better approaches to an iterative solving of the pH in buffers and titrations. He has a good explanation on the web site.