TBR Titration Question

[justadream]

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TBR GenChem I page 324 #33

The question is somewhat complicated to explain without copying the entire passage + question + answer explanation so only people with access to the TBR GC book can probably help.

TBR says that the concentrations of H3CCO2H between the two points in question differ by a factor of 2. Then, TBR concludes that that means that the pH of the two solutions differs by -log(sqrt2).

How do you suddenly get a square root in there?

 

[The Brown Knight]

I am familiar with the trick they explained in the chapter but haven't done any passages here, but this would be my guess:
the shortcut eqn in the chapter states: pH = (1/2)pKa - (1/2)log[HA]
so at point A we'd have: pH(A) = (1/2)pKa - (1/2)log[x]
at point D, we have the same acid but is diluted by a factor of 2 (since you add 25 mL of titrant); hence, its concentration is [x/2]
pH(D) = (1/2)pKa - (1/2)log[x/2]
So:
pH(A) - pH(D) = -(1/2)log[x] + (1/2)log[x/2] ---(using log rules)---> (1/2)log(1/2) = -(1/2)log2 = -log (sqrt 2)