[magician4]

let's look at this for a moment from the more practical side, shall we?

"equivalent point" is nothing mother nature knows anything about nor cares for. With respect to physics, nothing special is happening there, and even biological systems pretty much don't give a damn about those , in general.
... except one biological system, and that is "humans", and amongst them mostly only those, who're in for chemical analysis.
WE want to know when something meaningfull we've done (for example: add an acid to a base) , and which we can measure in its extend, would be equivalent to something else we, however, can't see/measure/detect directly, and hence need to figure out indirectly.

That's what all this fuzz with "equivalent point" is all about, and why there at times might be several equivalent points (depending on your mode of detection) and all that.
It's us humans, looking out for some simple observables ( like: a huge jump in pH and thatlike) and our different opinions on which observables might be the most easiest ( best, sharpest, important, best suited to our machines at hand, ... you name it) to watch out for, who are responsible for this chaos of un-precision in terms of mathematical definitions.

lets, for example, take a look at the titration of phosphoric acid, waterbased(called "c₀" hereafter) , with (additional) NaOH staring "x" , and with pH ( solution) being the observable, i.e. "y"

the resulting graph should ( at a  fixed temperature and all that...) look like this:



(taken from: link )

now, everybody who even knows basics in analysis in math immediately will know, that for using calculus on this kind of problems you'd watch out for very special properties of this graph, like turning points ( i.e. y''=0 ), local maximum slope (i.e. y'''=0)
... even [itex] \lim_{x \to \infty} (y) [/itex] , which should tell you c₀(NaOH)   (in case you didn't know beforehand)

there you are: that's where all your differential equations come into play if you wanted to solve this properly. i.e extract c₀ from this graph

furthermore, you would immediately see that if you solved those equations, for example for those turning points c(acid)= c(corresp. base) is only approximately correct, and that, besides, pH=pKs isn't exactly true , either.

[irony on] take that, bugger, and stick'em pH-indicators somewhere the sun doesn't shine ... [/ irony off]

is it, really?
no, of course it is not.
Still our definition given earlier holds:

Σ_Bases (n₀(Base)*k(Base)) = Σ_Acids (n0(Acid)*j(Acid))

it's just the relation between those significant elements in the graph and our desired result that became a bit more complicated.
it's the function mother nature provides that doesn't make analysis of this problem easy, not our definition of what we want to know (even if there were several "ideal" equivalents to watch out for, i.e. n(NaOH) = n (H₃PO₄) ; n(NaOH) = n(H₂PO₄^-)= 2 n (H₃PO₄) and so on)

... and only if we try to simplify this function, i.e work out an equivalent function that in the desired area (i.e our ""equivalent point of choice") is next to identical in y ( i.e. neglect subterms of minor importance for y , like for example those depending on K_w or K_a of "far away" dissociation steps) will we get rid of those differential equations we else would need to recalculate the what we can measure to the what we want to know.

is this approach justified?
if we'd talk phosphoric acid: yes (as results usually still meet the traditional standards of precision required, i.e. +/- 0.1%), it's good enough for almost all relevant tasks we wish to perform (like, for example, prepare a buffer for an experiment)

with other acids (or , to be more general, "substances"), this might be different (as , for example, the K-values might be too close to "separate"), and you really might need to solve those differential equations to get somewhere at all.
but still, you'd be looking for the situation where Σ_Bases (n₀(Base)*k(Base)) = Σ_Acids (n0(Acid)*j(Acid)) would be true, not matter how complicated this might be to "extract" from the graph measured

regards

Ingo

[Big-Daddy]

Ah I see. So then I get the impression that the turning points will correspond to the points we would find if we experimentally took a set of readings of pH against V, fit them to a best-fit equation, and then differentiated. These tend to match up with the equivalence points we generally find, where "equivalence point" continues to be defined by our previous equation, Σ_Bases (n0(Base)*k(Base)) = Σ_Acids (n0(Acid)*j(Acid)).

Then the only remaining point of confusion for me is, how come there is only one equivalence point for our system I described in the OP (2 monoprotic bases and a diprotic acid), and only one equivalence point arising from this equation definition of equivalence point, whereas as you say there are often multiple turning points and multiple derivatives to be found?

[magician4]

Ah I see. (...)
These tend to match up (...)

yes, exactly, and you put it in nice words: they tend to, i.e. come close enough to, for our purposes

for example, the turning point at approx. 5 mL for all practical purposes is close enough to c(H₃PO₄) = c(H₂PO₄^-) (but not exactly identical in a mathematical sense)  to use the Henderson-Hasselbalch equation here (i.e. treat phosphoric acid like a one-protonic acid , in this very region of the graph)
... which of course is much more convenient than having to deal with the more correct, but very demanding differential equations that would give a correct picture of the situation, even in a mathematical sense

Then the only remaining point of confusion for me is,(...)

It's sulfuric acid there, isn't it? (the pKa ₁ of which is next to -3 , if memory serves)
so, even before you begin with your measurements / titration, everything concerning this first pKa already is over, simply just from pouring the sulfuric acid into water.
there is (for all practical purposes) no sulfuric acid , undissociated, in water, and the only equivalent point of use remaining (i.e. you can see, detect, that will show a sharp pH-shift, ...)  would be related to HSO₄^- / SO₄^2- (as it's the only game in town left)


regards

Ingo

[Big-Daddy]

... which of course is much more convenient than having to deal with the more correct, but very demanding differential equations that would give a correct picture of the situation, even in a mathematical sense

Differential equations? Shouldn't it simply be simultaneous equations for the equilibrium system?

there is (for all practical purposes) no sulfuric acid , undissociated, in water, and the only equivalent point of use remaining (i.e. you can see, detect, that will show a sharp pH-shift, ...)  would be related to HSO₄^- / SO₄^2- (as it's the only game in town left)

But this misses my question slightly. To me, Σ_Bases (n0(Base)*k(Base)) = Σ_Acids (n0(Acid)*j(Acid)) inherently suggests that only one equivalence point exists (because n₀(HSO4-) is not well-defined - all we can define is n₀(H2SO4), however much sulphuric we put in solution, and then note that most of the time n₀(HSO4-) is the same as this, but that need not be the case, for other acids or bases). What I really need I think is a primer in how to use this mathematical definition for various different scenarios, in light of the different equivalence points that could exist in a single system - i.e. how to allocate or specify this equation for each one of those equivalence points.

[magician4]

Differential equations?

from the graph, you can't gain nothin' but, don't you agree?

Shouldn't it simply be simultaneous equations for the equilibrium system?

we're interested in the total function denoting the graph, in a sense that thereby we'd like to identify each and every element of that term  (which each should be a function of c_add(NaOH)^order).
in those elements, all relevant constants will appear, amongst them K_a1, K_a1, K_a3, K_W, c₀(H₃PO₄).

you could achieve this by trying to collect as much differential equations as are available, solve them, and do a power series development on the initially resulting e-functions

working with simultaneous functions instead would require to know a lot of those constants beforehand, so you could set up a balance equation or something similar, for example  (so this would be kind of a semi-empirical approach, whereas the above would be ab initio)

To me, Σ_Bases (n0(Base)*k(Base)) = Σ_Acids (n0(Acid)*j(Acid)) inherently suggests that only one equivalence point exists

with all due respect, I don't agree: taking H₃PO₄ for example again, there would be at least 3 of those "equivalents"  , namely for j=1, 2, 3 , respectively)

(...)What I really need I think is a primer in how to use this mathematical definition for various different scenarios, in light of the different equivalence points that could exist in a single system - i.e. how to allocate or specify this equation for each one of those equivalence points.

I'd like to return your statement to you, and kindly would ask you what, where and how you'd like to identify the story behind this system if you were asked for c₀, and didn't know nothin else about this system :

(from: link)

to me, the turning point would be of highest interest here, and I would try to figure out his relation to a situation where a certain amount of c₀ equals a certain amount of c_add, as per our definition

this choice, of course, would be strongly influenced by the fact, that there's nothing much else of use around here I could hang my hat on.
if, on the other hand, I had the titration of acetic acid with NaOH, I would try to reference the steepest part of the function instead of the turning point, i.e take my pick from at least two different options to define an "equivalent" from:

(from: link)

... and with H₃PO₄, I'd have a multitude of potential reference points I could try to relate to my "here, c₀ is equivalent so so and so much c_add"

... and I don't see how you'd potentially could have a general mathematical "primer", with such a multitude of situation related behaviours of the pH graph vs. c_add, for pH titrations in general


regards

Ingo

[Big-Daddy]

Well, I'm getting the impression that, when we have a set of data points of pH against V_titrant, yes, of course, we need to simply find the function (via differential equations) and then differentiate for the turning points.

But let's say that, as in the initial example of this thread, we already have a lot of information about the system - enough to define the equivalence points by Σ_Bases (n0(Base)*k(Base)) = Σ_Acids (n0(Acid)*j(Acid)), since these n₀ values are directly provided for 2 of the 3 components and "the" equivalence point is given. So now the main thing to concern ourselves with is how to use this equation to search for multiple equivalence points, as many systems do possess them - for which you just now suggested varying the values of j(Acid) for each acid and (I would guess) k(Base) for each base, from 1 (?) up to the maximum number of protons the acid can lose or base can gain.* If we use j(Acid)=1 then we are looking for the point when the undissociated form of the acid is equivalent in concentration to the once-dissociated form; j(Acid)=2 looks for the point when the once-dissociated form is equivalent to the twice-dissociated form; so forth. If the nth and mth dissociations were strong, we could not bother looking for the equivalence points where j(Acid)=n or j(Acid)=m; if the first 4 dissociations were considered strong (for example), we would start our equivalence point searching at n=5 (for the point where the 4th-dissociated form is equivalent to the 5th-dissociated form in concentration).

For a system with multiple components, we have to vary j and k for each acid and base from 1 to their maximum, and try every combination of these, and that will find us all the equivalence points; e.g. let's say H3PO4 and H2SO4 are both in solution and we're titrating it with NaOH and NH3, then we'd use (j(H2SO4)=2,j(H3PO4)=1),(j(H2SO4)=2,j(H3PO4)=2),(j(H2SO4)=2,j(H3PO4)=3) to find each of the three equivalence points (because the first sulphuric acid dissociation is strong we start at the first weak one, the second, so j(H2SO4)=2, and because there is no third we also stop at the second; meanwhile, for phosphoric acid, all dissociations are weak so we must cover them all).

All of this assumes that we do have detailed knowledge about the components, not just a set of pH vs V values, in which case we would have to try and find the function. Am I right with everything so far?

If I'm right - I would much appreciate if you could confirm - then I think this is all clear to me (for the time being!). In the problem in the OP, they could say that "the equivalence point" was found at a certain volume because there is only one equivalence point for H2SO4 if we assume that the first dissociation is strong (in reality, this holds unless the conditions are extraordinarily acidic), and with the first dissociation being strong we start at j(Acid)=2 leaving only one definition of equivalence point for this particular system. But more equivalence points could be defined if there were other components with (one component having) multiple weak dissociations or associations; if, for instance, they had used phosphoric instead of sulphuric acid, the question would have been invalid and could not have been asked without specifying which equivalence point was reached after a certain volume of NH3, e.g. "the equivalence point of H2PO4-/HPO42- was reached at 10.64 mL" in which case we would use the definition where j(Acid)=2 as this refers to the second dissociation of phosphoric acid.

*Side-question: how would vary j(Acid) to find each equivalence point for something like EDTA, which starts as H₄Y but whose deprotonations begin from H₆Y²⁺?

[Big-Daddy]

Could you please give that last post a look over? It would be a great help to me and if my understanding is correct then I don't think I will have to ask any more questions.