No need for that. Sometimes we will titrate to the "first visible color change". As a rule of thumb we assume it means concentration of one form 10 times higher than concentration of the other,
Surely there is only one "end-point" of a titration (more only if the indicator is polyprotic)? A range may be more practicable, but as those links and others show there is just one end-point. I'm assuming this is indeed when [HInd]=[Ind-], so the colours balance perfectly, so the pH of this point can be predicted as being directly = to pK
Ind of the indicator (even the activities cancel out nicely)?
I think we can find the pH at the bottom of the range as easily as the pH at the top of the range. Call the pH at the bottom pH
min and the pH at the top pH
max (pH at the exact end-point continues to be pH
end).
If [HInd] is the form present in acidic excess and [Ind-] in alkali excess, a rearrangement of [HInd]/[Ind-]=10=10^(pKInd-pH) and 1/10=10^(pKInd-pH) shows that:
pH
min=pK
Ind-1 (the pH where [HInd] is 10 times more concentrated than [Ind-])
pH
max=pK
Ind+1 (the pH where [Ind-] is 10 times more concentrated than [HInd])
So as we see, this really isn't too different from knowing what the pH at the end-point is; the range is just extended by 1 pH unit on either side.
Side note: we can I think replace 10 with any other factor, let's call it n, and get similar equations. pH
min=pK
Ind-log
10(n) and pH
max=pK
Ind+log
10(n), and of course pH
end=pK
Ind.
Is this reasonable?
To some extent yes, but look above.
I see, so as long as the pH of the equivalence point is within pH
min and pH
max we'll be fine for the titration. If we want the perfectly neutral colour, though, we'll want pH
equivalence=pH
end=pK
Ind.
Your approach looks logical, but I have never seen these things done this way. Most likely because there are not many commercially available indicators with many color changes, spaced exactly as you want them.
I'm just worried about figuring out where the end-points should be ideally for now - the commercial reasoning is of course true. Perhaps this is why they are rarely used in experiments to find multiple equivalence points?
As far as I can see, the same rules that apply for the pH at one end-point in relation to the pK
Ind should apply for multiple ones, so the range we want will be the same (e.g. pH
max,3, i.e. max pH for the third end-point, can be found by pH
max,3=pK
Ind,3]+log
10(n). This is not directly related to equivalence points: all it shows is the maximum ranges within which we can watch the pH values change for each end-point, but if each equivalence point could ideally be situated within the range for one of the end-points we could track all of the equivalence points.
Why not? Plenty of examples of classic techniques that did use indicators in such cases.
Potentially because of the reason you just pointed out above - they cannot be spaced correctly for such an inconsistently changing set of equivalence points.