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Chemistry Forums for Students => Undergraduate General Chemistry Forum => Topic started by: Big-Daddy on August 19, 2013, 04:48:59 PM
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Problem:
A 10.00 mL sample of an aqueous solution containing NH3 gas is added to 15.00 mL of 0.0100 M H2SO4. The resulting solution was titrated with 0.0200 M standard NaOH solution and the equivalence point was reached at 10.64 mL. Kb(NH3) = 1.8 · 10-5, Ka2(H2SO4) = 1.1 · 10-2.
Comments:
Never seen a back-titration involving equilibrium constants. Both guidance for this problem, and any links to more questions with back-titrations involving equilibrium constants (or general problems involving equilibrium constants where the initial conditions for each of the reactions are not always known, as in here), would be appreciated.
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Can you post the question that you are trying to answer? You only posted the setup of the problem and forgot to include the unknown that you need to find.
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My apologies. Here it is:
Calculate the pH of the solution at every stage of the titration. (Footnote: This means starting with the 10.00 mL sample (Solution 1), then the resulting solution after addition of sulphuric acid (Solution 2), then the solution when the equivalence point is reached after NaOH titration (Solution 3).)
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Ok, let's start: what species do you have after the titration with NaOH?
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NH4OH? We'll have Na+ ions, sulphate ions and NH4+ ions in solution.
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Sulphate ions? The second dissociation is not complete, so there is also hydrolysis.
Let's change the question a little: what is the equivalence point of the titration with NaOH?
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That is what we have to find (the pH at that equivalence point) as well as the pH at the two other main stages.
2.128 * 10-4 mol of NaOH were required to reach the equivalence point. This corresponds to an NaOH concentration of 5.9708 * 10-3 M in this solution. So I suppose we could say [Na+] = 5.9708 * 10-3 M. Then what?
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That is what we have to find (the pH at that equivalence point) as well as the pH at the two other main stages.
Actually I was thinking about a broader question like "When is the equivalence point reached in terms of reactions in the solution?" "What do we actually titrate up to the equivalence point?"
2.128 * 10-4 mol of NaOH were required to reach the equivalence point. This corresponds to an NaOH concentration of 5.9708 * 10-3 M in this solution. So I suppose we could say [Na+] = 5.9708 * 10-3 M. Then what?
This doesn't really help.
Again: what species do you have in solution after the titration with NaOH?
Sulphate ions? The second dissociation is not complete, so there is also hydrolysis.
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Actually I was thinking about a broader question like "When is the equivalence point reached in terms of reactions in the solution?" "What do we actually titrate up to the equivalence point?"
As far as I know, the mathematical definition of the equivalence point is as the point of inflexion on the titration curve. However, we aren't shown a titration curve here. So I'm unsure how to answer your question.
Perhaps you want me to say HSO4- (the prevalent species in acidic conditions) is titrated up to the equivalence point at which point [HSO4-] = [SO42-]? I'm not confident about defining the equivalence point by [HSO4-] = [SO42-] ...
Again: what species do you have in solution after the titration with NaOH?
After the titration I would suspect the solution to be more alkaline so sulphate ions should be the more prevalent? Before the titration the solution will be more acidic so HSO4- ions should be prevalent.
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http://www.titrations.info/titration-basic-terms
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So in the case of the m-protic acid titrated with the monoprotic base, what is the equivalence point: m*n0[Acid]=n0[Base] (where n0[Base] is the number of moles of the base added, and equivalent for the acid)? If not this, then I don't know what "stoichiometric amount of titrant to titrated substance" means.
And even if this is the equivalence point definition for a diprotic acid, how does it help, when we're considering a mixture of an acid (H2SO4) and a base (NH3)?
MrTeo's two major questions - "When is the equivalence point reached in terms of reactions in the solution?" "What do we actually titrate up to the equivalence point?" - would be nice to answer. But even after reading that page I do not understand equivalence point well enough to answer them. I assume it comes down to an issue of HSO4- and SO4-, or maybe NH4+, but as you can see my thinking is unclear here as I do not properly understand the situation. What happens when a mixture of NH3 and H2SO4 is titrated with NaOH?
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in my opinion, the only reasonable definition of "equivalent point" here would be, that n0(NaOH) + n0 (NH3) = 2 n0(H2SO4)
from this, I would start my calculation, regarding H2SO4 (undissociated) and OH- (unreacted) as inexistant
hence, equilibrium NH3 / NH4+ and SO42- / HSO4- would be the pH-determinating particles
regards
Ingo
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in my opinion, the only reasonable definition of "equivalent point" here would be, that n0(NaOH) + n0 (NH3) = 2 n0(H2SO4)
Hm. I would actually say (I'm thinking, for example, of a titration followed with a pH-meter) that the equivalence point is when all the HSO4- has been titrated by NaOH. There you should "see" (actually this is quite a nasty mixture I'd say, so I'm not sure whether this inflection point will be very clear on the diagram) the second equivalence point of the sulphuric acid titration. But I suppose it's mainly a terminology issue. [Edit: Ok, I think we're saying exactly the same thing ;D]
from this, I would start my calculation, regarding H2SO4 (undissociated) and OH- (unreacted) as inexistant
hence, equilibrium NH3 / NH4+ and SO42- / HSO4- would be the pH-determinating particles
Exactly. That's what I wanted to hear from BD. At the equivalence point the problem is similar to the one of the pH of a salt with hydrolysis of the anion and the cation at the same time (which btw under certain approximations that I should check has a result independent of concentration, but I don't know if this applies here).
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At the equivalence point the problem is similar to the one of the pH of a salt with hydrolysis of the anion and the cation at the same time (which btw under certain approximations that I should check has a result independent of concentration
http://www.chembuddy.com/?left=pH-calculation&right=pH-salt-simplified
And let's not forget about
http://www.titrations.info/titration-equivalence-point-calculation
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Ok from magician4's equation for the equivalence point I gather that both NH3 and NaOH are to be seen as bases and n0(NaOH) reaches the point where the equation is satisfied, at the equivalence point. n0(NaOH) for Solution 3, at the equivalence point, is given. n0(H2SO4) (for Solutions 2 & 3) is given. n0(NH3) is then calculated from the other two, for all 3 Solutions.
So now we just set up mass balance, charge balance and equilibrium equations for each of the three situations and solve them for [H+]? Well that's the exact (and un-insightful) method anyway. I tried this for the first solution, Solution 1, with only n0(NH3) present. It worked. This leads me to think it would work for the others as well.
So what method might we follow which would enable us to solve it for Solution 1, at least, without needing to handle a cubic or higher order equation? :D I'm talking about approximations
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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 (SO42- , 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
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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 Kw 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?
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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
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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.
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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
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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?
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(...) 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.
:rarrow: 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 (http://en.wikipedia.org/wiki/Ferroin), methylene blue (http://en.wikipedia.org/wiki/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
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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).
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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
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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.
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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
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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 pKa or pKb?
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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
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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) Kw 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?
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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?
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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:
(https://www.chemicalforums.com/proxy.php?request=http%3A%2F%2Fwww.jagemann-net.de%2Fchemie%2Fchemie13lk%2Fmassenwirkungsgesetz%2Ffiles%2Ftitration_h3po4.gif&hash=6ceb62a39729f49089d99cf55c991c073a7337ac)
(taken from: link (http://www.jagemann-net.de/chemie/chemie13lk/massenwirkungsgesetz/massenwirkungsgesetz.php) )
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 Kw or Ka 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
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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?
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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- / SO42- (as it's the only game in town left)
regards
Ingo
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... 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- / SO42- (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.
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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 cadd(NaOH)order).
in those elements, all relevant constants will appear, amongst them Ka1, Ka1, Ka3, KW, 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 :
(https://www.bio.cmu.edu/courses/03231/DryLab/pH/HEPES.gif)
(from: link (http://www.chemicalforums.com/index.php?topic=70449.0))
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 cadd, 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:
(https://www.chemicalforums.com/proxy.php?request=http%3A%2F%2Fwww.haverford.edu%2Fchem%2FScarrow%2FGenChem%2Facidbase%2Frealtitrations%2FAcetic_LP.gif&hash=bd3e5d48aaf080d1678b534737de03a2e3e03cdd)
(from: link (http://www.haverford.edu/chem/Scarrow/GenChem/acidbase/realtitrations/table.html))
... 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 cadd"
... 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. cadd, for pH titrations in general
regards
Ingo
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Well, I'm getting the impression that, when we have a set of data points of pH against Vtitrant, 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+?
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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.