I was wondering if the following is a legitimate proof of the statement above. Assume we have
HA -> A- + H+
where [HA] + [A-] is our buffer.
If we add a small amount of H+ one of two things will happen:
1) The H+ will not react with A- ...thereby raising the pH so we are done.
2) The H+ will react with A- ...converting to [HA]
looking at the HH equation we know that
pH = pKa + log(ratio)
since ratio_before != ratio_after
pH_after = pKa + log(ratio_after) and pH_before = pKa + log(ratio_before)
pH_after - log(ratio_after) = pH_before -log(ratio_before)
Now assume pH_after == pH_before.... we get -log(ratio_after) = -log(ratio_before) which is a contradiction of above, so pH_after != pH_before.
What are your thoughts?