[magician4]

in my opinion, the "how to "depends...
... on what the original questioner really wants to see from you.


for all practical purposes, the usual approximation equations would give you excellent results already:

- in the beginning, pH = 14 - [ 0.5 (pKb - c₀(NH₃)]

- in the middle, excess H₂SO₄ (or HSO₄^- ... : I didn't calculate yet) would be the name of the game, with any NH₄⁺ formed being negligible  (as it is by far the weakest acid around)

- in the end , NH₄⁺ will dominate the game, as all further basic substances (SO₄^2- , NH₃) around will be negligible compared to this

the respective concentrations can be calculated easily from the n₀ values (more or less given) and the respective volumes at the given moment

for anything more refined you would need to work through balance equations and thatlike, yes


regards

Ingo

[Big-Daddy]

I tried using the balance equations and going for the approximation that [H⁺]=0 (the first approximation for most acid-base calculations) and thus leaving out the K_w expression, after which the solution came as a quadratic and still went to 10.588 for Solution 1. I just have a couple more questions:

1) Can we deduce whether Solutions 2 and 3 will be acidic, alkaline or neutral without much calculation? (No quadratics or higher-order, just a simple approach for estimation) And, for the method you present to do so, what cases will it be applicable in? (e.g. Kb being low for the weak acid and infinite for the strong acid, etc.)

2) What's a good general mathematical way to define the equivalence point? Because that appears to be the issue on which solving this problem turned as far as my knowledge goes. Here we used Σ_Bases (n₀(Base)*k(Base)) = Σ_Acids (n(Acid)*j(Acid)), where you have a set of k-protic bases being summed over on the LHS (initial number of moles * k for each base) and an analogous set of j-protic acids summed over on the RHS (initial number of moles * j for each base). I don't know if this is exact or not, but it seems to be good enough both for weak (NH₃) and strong (NaOH) cases.

My question then is, how do salts factor into the equivalence point? For instance, what is the equivalence point of a solution where I am titrating n₀(H₂CO₃) with a mixture of n₀(NH₄HCO₃) and n₀(K₂CO₃), i.e. an acid with a basic and an amphiprotic salt?

[magician4]

I'll start with # 2 :

question with "equivalent point" always is, than in not-so obvious cases you'll have to come up with a sensible definition of the very meaning all by yourself: "equivalent" to what exactly??

(and the answer has, of course, to be problem related)

..which gives you kind of a "responsible freedom", if you know what I mean

for a simple acid-base titration (or even a reverse titration, like in the original question) this usually is no problem, and the mathematical definition you proposed seems a good one to me.

when more complex situations are posed (like for example in the presence of buffers of some sort or another, or additional pH-active species), in my opinion the situation is too complex to have something like a meaningfull "equivalent point" at all - except you'll define one esp. for the setup given.

so, I think there's no general answer to the question how to handle those complicated cases: depending on the problem, you'll have to come up with a usefull definition of your own


ref #1:
yes, I think we can:
for solution #2 , the only resonable situation would be that it has still some acid capacity left - else additional KOH would be quite meaningless.
Furthermore, looking at the concentrations / volumes involved, there is strong evidence that hence the solution should be on the strong acidic side at this moment

for solution #3 , the substances  present (due to my proposal for "equivalence point") should be " dissolved K₂SO₄ plus (NH₄)₂SO4 " at approx. equal concentrations ("quick and dirty" calculation from the initial values given)
... which makes it a solution of ammonium, sulfate at approx. equal concentrations with respect to pH active species.
now, ammonium is by a factor of ~ 10³ the stronger acid than sulfate is basic, hence the total outcome should be on the acidic side.
being a weak acid nontheless, the result should be on the weakly acidic side, to be more specific.


regards

Ingo

[Big-Daddy]

I'll start with # 2 :

question with "equivalent point" always is, than in not-so obvious cases you'll have to come up with a sensible definition of the very meaning all by yourself: "equivalent" to what exactly??

(and the answer has, of course, to be problem related)

..which gives you kind of a "responsible freedom", if you know what I mean

for a simple acid-base titration (or even a reverse titration, like in the original question) this usually is no problem, and the mathematical definition you proposed seems a good one to me.

when more complex situations are posed (like for example in the presence of buffers of some sort or another, or additional pH-active species), in my opinion the situation is too complex to have something like a meaningfull "equivalent point" at all - except you'll define one esp. for the setup given.

so, I think there's no general answer to the question how to handle those complicated cases: depending on the problem, you'll have to come up with a usefull definition of your own

Hmm, so if the components are just acids and bases, it's ok to define the equivalence point as the point at which the formula I wrote above is true for the combined solution into which the titrant and analyte have been mixed up to that point.

But you're saying the equivalence point has no meaning for cases involving salts rather than acids and bases? What if there is an inflection point(s) on the titration curve - if that exists, then it means there still is an equivalence point, no? (Even if they are not entirely, exactly, identical.) Would you propose a definition for the equivalence point in the case of the carbonate titration I suggested, or should we say that the equivalence point for a more complicated titration like this is meaningless?

ref #1:
yes, I think we can:
for solution #2 , the only resonable situation would be that it has still some acid capacity left - else additional KOH would be quite meaningless.
Furthermore, looking at the concentrations / volumes involved, there is strong evidence that hence the solution should be on the strong acidic side at this moment

Thanks, this one should have been obvious really - 1.5*10^-4 on the acid side vs. 8.72*10^-5 on the basic side, and on top of that the acid is dissociating completely once and then some more on top. And as if we actually needed to look at the calculations ... there wouldn't be an equivalence point with NaOH if Solution 2 was alkaline (as then we're saying n₀[NH3]>2n₀[H2SO4]).

... which makes it a solution of ammonium, sulfate at approx. equal concentrations with respect to pH active species.

How did you arrive at that? Even approximately?

now, ammonium is by a factor of ~ 10³ the stronger acid than sulfate is basic, hence the total outcome should be on the acidic side.

Logic here makes sense.

[magician4]

But you're saying the equivalence point has no meaning for cases involving salts rather than acids and bases? (...)

depending on the very problem you have to solve, it could have different meanings
in analytics, you want to measures something.
with complicated situations, you'll have to think what signal (usually a sharp pH shift for acid -base titrations, but inflection points might be helpfull, too, though they're more difficult to measure, yes) might be related to your very substance, and what (additional) acid / base capacities due to additional substances could be involved (and how to measure those): you might have several significant "jumps" in such a situation, and every jump with a different meaning = every jump "equivalent" to something different happening.

... and that's where a very rigid definition of THE equivalent point might become meaningless

How did you arrive at that? Even approximately?

2/3 of the sulfuric acid is left for consumption by KOH obviously
this leaves 1/3 that had been consumed initially by NH₃, resulting in 2/3 ammonium
so, by the end of the day we'll have 1 sulfate and 0.66 ammonium, ballpark
... and I called this "approx 1:1" (good enough for the pH - estimation I gave afterwards)


regards

Ingo

[Big-Daddy]

depending on the very problem you have to solve, it could have different meanings
in analytics, you want to measures something.
with complicated situations, you'll have to think what signal (usually a sharp pH shift for acid -base titrations, but inflection points might be helpfull, too, though they're more difficult to measure, yes) might be related to your very substance, and what (additional) acid / base capacities due to additional substances could be involved (and how to measure those): you might have several significant "jumps" in such a situation, and every jump with a different meaning = every jump "equivalent" to something different happening.

... and that's where a very rigid definition of THE equivalent point might become meaningless

I looked up Wikipedia for "Equivalence point" and they said that "In some cases there are multiple equivalence points, which are multiples of the first equivalence point, such as in the titration of a diprotic acid." Not only does this not fit with my definition of equivalence point so far proposed, but it also appears not to fit with this problem at all - they said "the equivalence point", despite the fact that there are 2 bases and a diprotic acid so there should have been at least 2 equivalence points. What's going on?

And is the equivalence point defined for titrations other than those of acid-base mixtures? (Such as redox titrations or complexometric titrations) How should I try to define it then?

so, by the end of the day we'll have 1 sulfate and 0.66 ammonium, ballpark

How come? You've said 2/3 of the original sulphuric acid is left, and since 1/3 of the sulphuric acid has been consumed, 2/3 of the original sulphuric acid is the equivalent to ammonium. But that's after the sulphuric acid addition, i.e. in Solution 2. What happens next when the NaOH is added?

[magician4]

(...) What's going on?

that's Wikipedia for you...
though more often than not a valuable source of first information, they at times also fail if you had to look deeper into a problem, and are a bit** if it came to amendments.
  I don't take them as "the holy writ"

And is the equivalence point defined for titrations other than those of acid-base mixtures? (Such as redox titrations or complexometric titrations) How should I try to define it then?

yes, there are equivalent points for other titrations like redox ,too, as as there are indicators (ferroin, methylene blue..)

and , as before, the "definition" again depends on your analysis / understanding of the problem, maybe even your choice if there were several options available.

How come? You've said 2/3 of the original sulphuric acid is left, and since 1/3 of the sulphuric acid has been consumed, 2/3 of the original sulphuric acid is the equivalent to ammonium. But that's after the sulphuric acid addition, i.e. in Solution 2. What happens next when the NaOH is added?

if my initial analysis of the situation was correct , it depends: you might wish to titrate to the point where all the sulfuric acid / HSO₄^- is consumed, but the ammonium is still left untouched: that was the point I was referring to
OR
you might wish to titrate to the point, where also the NH₄⁺ initially generated is transformed to NH₃ .
nevertheless, I think this point is of no special use for the measurement, as by then the result would simply equal the total amount of sulfuric acid initially present (which is already known from starting conditions), and only might be usefull as kind of a "backup" to the above approach [the consumption of KOH here, calculated from endpoint "a" ,  once more should equal the results calculated from approach "a")


regards

Ingo

[Big-Daddy]

if my initial analysis of the situation was correct , it depends: you might wish to titrate to the point where all the sulfuric acid / HSO₄^- is consumed, but the ammonium is still left untouched: that was the point I was referring to
OR
you might wish to titrate to the point, where also the NH₄⁺ initially generated is transformed to NH₃ .
nevertheless, I think this point is of no special use for the measurement, as by then the result would simply equal the total amount of sulfuric acid initially present (which is already known from starting conditions), and only might be usefull as kind of a "backup" to the above approach [the consumption of KOH here, calculated from endpoint "a" ,  once more should equal the results calculated from approach "a")

At our equivalence point, how should we know whether the solution is acidic, neutral or alkaline? It's difficult to tell whether HSO₄^- has been entirely reacted or not. That itself depends on whether the solution is acidic or alkaline (the more acidic, clearly the more HSO₄^- remains).

[magician4]

It's difficult to tell whether HSO4- has been entirely reacted or not.

look at the pKa - values involved: pKa II (H₂SO₄) = 1.9 , pKa(NH₄⁺) = 9.25

that huge a difference , the respective signals in titration should clearly separate

At our equivalence point, how should we know whether the solution is acidic, neutral or alkaline?

as I said, if the very situation that in my opinion is usefull to look at for solving this problem indeed  is the one I've described (and I don't see any alternative for the time being), then we would have sulfate, ammonium in approx. equal concentrations, and the solution hence should be on the weak acidic side

regards

Ingo

[Big-Daddy]

we would have sulfate, ammonium in approx. equal concentrations,

We started with 1.5*10^-4 mol of sulphuric acid and 8.72*10^-5 mol of ammonia. So if all the sulphuric acid has been converted to sulphate, and the ammonia to ammonium, by the addition of NaOH and the sulphuric acid respectively, we should have around 1.5*10^-4 mol of sulphate and 8.72*10^-5 mol of ammonium. Is this is the "approximately equal concentration"? Then because sulphate hydrolyses much less than ammonium, the acidic action outweighs the basic action and the solution is weakly acidic.

[magician4]

Is this is the "approximately equal concentration"?

yes, and I have to admit that I was a bit generous with rounding, and didn't look into the fines of the calculation too deep
this is more next to 2:1 than 1:1, agreed

... which is , as we're agreed also, of no importance for our ballpark estimation of the pH

and yes, this:

Then because sulphate hydrolyses much less than ammonium, the acidic action outweighs the basic action and the solution is weakly acidic.

is how I would go on from there


regards

Ingo

[Big-Daddy]

Thanks for your help. Can I just ask one more question here: for systems with various equilibria going on, besides the well-known approximations (e.g. neglect Kw in a system which is significantly acidic or alkaline), what should be the general quantitative approach to approximations for equilibrium systems? e.g. How should I tell which equilibria I can safely discard (pretending they, and any species present in them alone, do not occur) so as to get a system of equilibrium equations that can be solved much more easily? For example, I think the rule of thumb for acid-base calculations is a 3-unit jump in pK_a or pK_b?

[magician4]

to put it like this: in my opinion, the successfully analysis of complex systems and what to take into account for, what to neglect, pretty much comes close to an art (which usually also gets better with respective practice)

there's a multitude of aspects that might lead to one decision or another (with the 3 dimensions jump in K values being one of them), and I think that it would need a pretty huge catalogue if you wanted to write all of them down

but maybe someone else knows about a good comprehensive paper /article / book  where at least the gist of it is summarized...

regards

Ingo

[Big-Daddy]

Ok, so it's a topic in itself. Maybe it would be good for me to get down a couple of rules of thumb rather than try to understand every possibility. Perhaps I could start, and you might add a couple you feel are pertinent or very important?

a) K_w can be neglected for a solution known to be significantly acidic or alkaline (even NH3 fits the bill, hence why we can get the quadratic for the problem of the OP rather than a full cubic). The higher the value of Ka or Kb for your acid or base respectively, the more valid this approximation.

b) In a set of multiple successive equilibria (e.g. complexation or ligand exchange, or acid dissociation or base association, etc.), if a later K is roughly 3 orders of magnitude or more smaller than the one immediately before it, then that later K as well as all those following it can be neglected.

I'm tempted to generalize and say equilibria with much smaller K than the other(s) can usually be discarded. Would you agree? If so, what circumstances should I watch out for where doing this would be a bad idea?

[Big-Daddy]

I have a separate question which has risen to my attention.

Earlier in this thread we used the definition of equivalence point governed by the rule Σ_Bases (n₀(Base)*k(Base)) = Σ_Acids (n₀(Acid)*j(Acid)), where k(Base) and j(Acid) are the maximum number of protonations or dissociations the acid or base can undergo in total from the initial form. (So for EDTA, j(Acid)=4, despite the possibility of super-protonated forms.) We agreed that equivalence point then is poorly defined in cases where we have salts in the titration as well as acids and bases.

But now that I've read up some more on the subject, I'd like to reconcile this definition with a few issues I have found. Firstly, I heard that there would be multiple equivalence points for a system like this (2 bases and a diprotic acid) - and that the accurate way of finding them is via use of first or second differentials, rather than simply equating of the initial numbers of moles as I have suggested? So which definition is exact? And if the definition I used above is exact, then why the need for differentiation, when we could just solve the equations from scratch after noting the equivalence of initial number of moles as in my definition, without needing calculus?