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Offline Big-Daddy

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End-Point and Indicators
« on: November 02, 2012, 11:07:15 AM »
Let us say the equilibria in question is:

HInd + H2O ⇌ H3O+ + Ind-

And as usual, H2O is taken as constant and ommitted from the discussion.

I take it the end-point of a titration is defined by the pH when [HInd]=[Ind-], i.e. when the colours of the two forms are in exact balance. In that case, at the end-point H3O+=KInd.

Is this accurate? And how can this approach be extended to indicators with multiple dissociations (I hear phenolphthalein is one such example, with 3 dissociations from KInd,1 (from H3Ind+ to H2Ind) to KInd,3 (from HInd- to Ind2-))? I'm looking to extend my understanding of textbook stuff here.

Offline Big-Daddy

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Re: End-Point and Indicators
« Reply #1 on: November 02, 2012, 11:12:51 AM »
Further, there is the problem of choosing indicators. Would it be accurate to say that, if I know (either from calculation or from a titration curve which makes it obvious) the pH at the equivalence point of a titration, I should choose the indicator with the closest pKInd available, and the closer my indicator's pKInd to the pH at the equivalence point, the closer the end-point of the titration will be to the equivalence point?

Again, I don't understand how to relate this to titrations with more than one end-point or equivalence point (though we can assume that I am given the pH values at all equivalence points in the solution).

Offline Borek

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Offline Big-Daddy

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Re: End-Point and Indicators
« Reply #3 on: November 02, 2012, 06:39:31 PM »
See if these don't help:

http://www.titrations.info/titration-end-point-indicators

http://www.titrations.info/titration-end-point-detection

http://www.titrations.info/acid-base-titration-indicators

http://www.titrations.info/acid-base-titration-end-point-detection

They do help, but I still need confirmation of my thoughts. Let me number my queries. The first two are simple "yes/no" answers, the next two will take a bit more thought (at least I don't know how they might work).

1. The end-point of a titration occurs when the concentration of the two forms of indicator are the same, i.e. [HInd]=[Ind-], and so KInd=[H3O+]end (just cancel down from the expression of KInd to see this). So the pH of the end-point can be found from the KInd: pH is the same as the pKInd.

2. Indicators are normally chosen to change colour at the equivalence point, correct? So we want an end-point as close as possible to the equivalence point. So if the pH of the equivalence point is known, and pHend-point=pKInd, we want the pKInd to be as close to the pH of the equivalence point as possible.

All I need for the above two is confirmation that I am correct.

The next two are merely extensions:

3. How do I find the pH of the 3 different end-points of an indicator which has three different dissociations? Possibly, I'm thinking, [H3O+]end-point-1=KInd,1, [H3O+]end-point-2=KInd,2, [H3O+]end-point-3=KInd,3, etc., so pHend1=pKInd1, pHend2=pKInd2, pHend3=pKInd3, etc., but I'm not sure.

4. How do I choose an indicator for its pKInd values if I know the pH at the multiple equivalence points of my titration? I'm thinking, if my understanding of part 3 is right, that I should get an indicator with pKInd values as close as possible to my pH values at the equivalence points (i.e. if pKInd,1 is very close to the one of the equivalence points, I will see the neutrality of colour between the two forms represented by KInd,1 very close to that equivalence point, at a pH defined by pKInd1 which as I have just pointed out is close to the equivalence point). And yes, I am aware that indicators are not normally used for weak acid-base reactions, let alone polyprotic acid-base reactions.

Is this accurate?

Offline Borek

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Re: End-Point and Indicators
« Reply #4 on: November 03, 2012, 03:17:20 PM »
1. The end-point of a titration occurs when the concentration of the two forms of indicator are the same, i.e. [HInd]=[Ind-]

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,

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2. Indicators are normally chosen to change colour at the equivalence point, correct? So we want an end-point as close as possible to the equivalence point. So if the pH of the equivalence point is known, and pHend-point=pKInd, we want the pKInd to be as close to the pH of the equivalence point as possible.

To some extent yes, but look above.

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How do I find the pH of the 3 different end-points of an indicator which has three different dissociations?

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.

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And yes, I am aware that indicators are not normally used for weak acid-base reactions, let alone polyprotic acid-base reactions.

Why not? Plenty of examples of classic techniques that did use indicators in such cases.
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Offline Big-Daddy

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Re: End-Point and Indicators
« Reply #5 on: November 03, 2012, 04:00:46 PM »

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 pKInd 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 pHmin and the pH at the top pHmax (pH at the exact end-point continues to be pHend).

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:

pHmin=pKInd-1 (the pH where [HInd] is 10 times more concentrated than [Ind-])

pHmax=pKInd+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. pHmin=pKInd-log10(n) and pHmax=pKInd+log10(n), and of course pHend=pKInd.

Is this reasonable?

To some extent yes, but look above.

I see, so as long as the pH of the equivalence point is within pHmin and pHmax we'll be fine for the titration. If we want the perfectly neutral colour, though, we'll want pHequivalence=pHend=pKInd.

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 pKInd should apply for multiple ones, so the range we want will be the same (e.g. pHmax,3, i.e. max pH for the third end-point, can be found by pHmax,3=pKInd,3]+log10(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.

Offline Borek

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Re: End-Point and Indicators
« Reply #6 on: November 03, 2012, 04:47:38 PM »

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 pKInd of the indicator (even the activities cancel out nicely)?

Sorry, but I have no idea what you are talking about. Yes, there is one end-point. But if you have an indicator with pKa=8 and you want to titrate a strong acid to pH=7.0 it is perfectly correct to titrate till the first color change, which should be observable at pH 7.

Perhaps that's what you mean, as large part of your discussion circles about the idea, but it wasn't clear for me.

We can use any other factor than 10, but as I explained earlier, as a rule of thumb we assume color change becomes visible when ratio of concentrations of both forms is 10. My understanding is that - as approximate as this rule is - it follows observed lab reality.

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I'm just worried about figuring out where the end-points should be ideally for now

Answer is simple - end point should be where the equivalence point is. If there is anything to worry about it is the difference between equivalence and end point.

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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.

The idea that they are not used is FALSIFIED by the real methods that use indicators for weak acid-base reactions, and even for mixtures of strong and weak bases. So trying to find reasons why they are not used when they ARE used is somewhat nonsensical.
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Offline Big-Daddy

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Re: End-Point and Indicators
« Reply #7 on: November 04, 2012, 06:30:56 AM »
Sorry, but I have no idea what you are talking about. Yes, there is one end-point. But if you have an indicator with pKa=8 and you want to titrate a strong acid to pH=7.0 it is perfectly correct to titrate till the first color change, which should be observable at pH 7.

Perhaps that's what you mean, as large part of your discussion circles about the idea, but it wasn't clear for me.

We can use any other factor than 10, but as I explained earlier, as a rule of thumb we assume color change becomes visible when ratio of concentrations of both forms is 10. My understanding is that - as approximate as this rule is - it follows observed lab reality.

I was merely running some theoretical calculations, not asking about lab practice. I just wanted to make sure that pKInd does correlate with pH directly as I thought it might.

The use of factors other than n is not useful in determining minimum concentration for visibility as we can just assume this to be 10, but it is necessary to define the colour ratio we're looking for if the equivalence point is somewhere between pHmin and pHmax of the indicator. e.g. if pKInd=pHend=8 but the pH of the equivalence point is 8.3:

[HInd]/[Ind-]=10^(pKInd-pH) [this formula applies at any pH, so I can do the same thing at pH 6 and just get a very large HInd:Ind- ratio, 100:1)
[HInd]/[Ind-]=10^(8-8.3)=0.5012

So we're looking for the moment when Ind- concentration is approximately twice as strong as HInd, and then we know we have hit the equivalence point and can stop the titration.

The idea that they are not used is FALSIFIED by the real methods that use indicators for weak acid-base reactions, and even for mixtures of strong and weak bases. So trying to find reasons why they are not used when they ARE used is somewhat nonsensical.

Sorry, but I was under the impression NMR and pH meters were in far more common use in modern labs.

Offline Borek

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Re: End-Point and Indicators
« Reply #8 on: November 04, 2012, 07:29:46 AM »
I was merely running some theoretical calculations, not asking about lab practice. I just wanted to make sure that pKInd does correlate with pH directly as I thought it might.

You can't ignore lab practice here, that would be putting cart before the horse. There is such thing as a theoretical equivalence-point, but there is no such thing as a theoretical end point. End point is what you get in the lab practice using available indicators and selecting the one that fits best, not the one that has pKa equal to pH of equivalence point, as you don't have that many indicators to select from. There is a 99.9-100.1% rule that says you should select an indicator that changes color completely between equivalence point ± 0.1%, see below (drop size) for explanation why it makes sense. Besides, it is already discussed on one of the pages linked to earlier.

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The use of factors other than n is not useful in determining minimum concentration for visibility as we can just assume this to be 10, but it is necessary to define the colour ratio we're looking for if the equivalence point is somewhere between pHmin and pHmax of the indicator. e.g. if pKInd=pHend=8 but the pH of the equivalence point is 8.3:

[HInd]/[Ind-]=10^(pKInd-pH) [this formula applies at any pH, so I can do the same thing at pH 6 and just get a very large HInd:Ind- ratio, 100:1)
[HInd]/[Ind-]=10^(8-8.3)=0.5012

So we're looking for the moment when Ind- concentration is approximately twice as strong as HInd, and then we know we have hit the equivalence point and can stop the titration.

There is no way for you to be able to titrate to the moment when these concentrations are 2:1. First, you don't have eye sensitive enough to color. Second, the smallest drop of a titrant that you can add is about 1/20 mL, and close to the equivalence point this single drop can yield pH change well above 2 pH units.

I have a feeling you are trying to reinvent the wheel. Rules that I am talking about are based on lab practice and experience of generations of chemists, really no need to try to outwit them.

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Sorry, but I was under the impression NMR and pH meters were in far more common use in modern labs.

Depends on your definition of a modern lab. Nobody is going to use NMR for titration. pH meter, conductometer - sure. Some spectroscopic methods - perhaps. Thermometric methods can be used as well. But the hardware is expensive, and it is cost efficient only when used often enough. There is nothing unusual to use classic titrimetric methods for cases when you have to do some single analysis, or even if the analysis is repeated - but just once a week. Judging from the questions people post one the forum now and then, similar things are happening all around the globe, both in USA and Bangladesh.
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Offline Big-Daddy

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Re: End-Point and Indicators
« Reply #9 on: November 04, 2012, 09:02:59 AM »
You can't ignore lab practice here, that would be putting cart before the horse. There is such thing as a theoretical equivalence-point, but there is no such thing as a theoretical end point. End point is what you get in the lab practice using available indicators and selecting the one that fits best, not the one that has pKa equal to pH of equivalence point, as you don't have that many indicators to select from. There is a 99.9-100.1% rule that says you should select an indicator that changes color completely between equivalence point ± 0.1%, see below (drop size) for explanation why it makes sense. Besides, it is already discussed on one of the pages linked to earlier.

I see, OK - so what is the precise definition of end-point, just so we're both on the same page? If not when [HInd]=[Ind-], then when?

I'm not sure what you mean by the 99.9%-100.1% rule; by changes colour completely I'm guessing you mean from [HInd]=10[Ind-] to [HInd]=(1/10)[Ind-] (assuming you're adding base). I have no idea what "equivalence point ± 0.1%" means.

There is no way for you to be able to titrate to the moment when these concentrations are 2:1. First, you don't have eye sensitive enough to color. Second, the smallest drop of a titrant that you can add is about 1/20 mL, and close to the equivalence point this single drop can yield pH change well above 2 pH units.

I have a feeling you are trying to reinvent the wheel. Rules that I am talking about are based on lab practice and experience of generations of chemists, really no need to try to outwit them.

Haha don't worry I'm not trying to outwit them, just coming to a theoretical understanding of the topics at hand. I see your argument is true.

Offline Big-Daddy

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Re: End-Point and Indicators
« Reply #10 on: November 04, 2012, 10:19:39 AM »
x

By the way, you mentioned theoretical equivalence point. How does one find the theoretical equivalence point? Simply take the second derivative of the titration curve equation (listed on one of the pages on titrations.info) and set it to 0? Or is there some other method?

Robert de Levie mentions a "non-linear least squares best fit" method as being the best. Have you ever heard of this? Is it more accurate than the second derivative? If not, what are its advantages? As far as I can see, the second derivative should be definitive in finding the equivalence point (and if there is more than 1 the derivative will simply yield more than one volume as a result).

Offline Borek

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Re: End-Point and Indicators
« Reply #11 on: November 04, 2012, 05:37:34 PM »
I see, OK - so what is the precise definition of end-point, just so we're both on the same page? If not when [HInd]=[Ind-], then when?

End point is where you finish the titration thinking you have reached the equivalence point. Equivalence point is where the stoichiometric amount of the titrant was added - so while the equivalence point has a clear, theoretical value, end point is just a measurement result.

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I'm not sure what you mean by the 99.9%-100.1% rule; by changes colour completely I'm guessing you mean from [HInd]=10[Ind-] to [HInd]=(1/10)[Ind-] (assuming you're adding base). I have no idea what "equivalence point ± 0.1%" means.

See above - equivalence point is when you added 100% of the titrant (exactly stoichiometric amount). As I wrote earlier, these things are described on titrations.info.

By the way, you mentioned theoretical equivalence point. How does one find the theoretical equivalence point? Simply take the second derivative of the titration curve equation (listed on one of the pages on titrations.info) and set it to 0? Or is there some other method?

Equivalence point is not related to the titration curve, just to the stoichiometry of the reaction. While it can be calculated from the titration curve, it doesn't make sense to me - you calculate titration curve from the stoichiometry, so doing additional step to find out when the stoichiometry is correct sounds circular.

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Robert de Levie mentions a "non-linear least squares best fit" method as being the best. Have you ever heard of this? Is it more accurate than the second derivative? If not, what are its advantages? As far as I can see, the second derivative should be definitive in finding the equivalence point (and if there is more than 1 the derivative will simply yield more than one volume as a result).

I suppose he means detection of the equivalence point (note that to be nitpickingly precise we should call the result of this detection the end point, not the equivalence point), that's not the same as calculating a theoretical equivalence point.
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Offline Big-Daddy

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Re: End-Point and Indicators
« Reply #12 on: November 05, 2012, 12:55:37 PM »
I see, OK - so what is the precise definition of end-point, just so we're both on the same page? If not when [HInd]=[Ind-], then when?

End point is where you finish the titration thinking you have reached the equivalence point. Equivalence point is where the stoichiometric amount of the titrant was added - so while the equivalence point has a clear, theoretical value, end point is just a measurement result.

Quote
I'm not sure what you mean by the 99.9%-100.1% rule; by changes colour completely I'm guessing you mean from [HInd]=10[Ind-] to [HInd]=(1/10)[Ind-] (assuming you're adding base). I have no idea what "equivalence point ± 0.1%" means.

See above - equivalence point is when you added 100% of the titrant (exactly stoichiometric amount). As I wrote earlier, these things are described on titrations.info.

By the way, you mentioned theoretical equivalence point. How does one find the theoretical equivalence point? Simply take the second derivative of the titration curve equation (listed on one of the pages on titrations.info) and set it to 0? Or is there some other method?

Equivalence point is not related to the titration curve, just to the stoichiometry of the reaction. While it can be calculated from the titration curve, it doesn't make sense to me - you calculate titration curve from the stoichiometry, so doing additional step to find out when the stoichiometry is correct sounds circular.

Quote
Robert de Levie mentions a "non-linear least squares best fit" method as being the best. Have you ever heard of this? Is it more accurate than the second derivative? If not, what are its advantages? As far as I can see, the second derivative should be definitive in finding the equivalence point (and if there is more than 1 the derivative will simply yield more than one volume as a result).

I suppose he means detection of the equivalence point (note that to be nitpickingly precise we should call the result of this detection the end point, not the equivalence point), that's not the same as calculating a theoretical equivalence point.

de Levie offers rather complicated calculations to find the equivalence point theoretically from the gradient of the theoretical titration curve (as, for a given Va, Ca, Vb and Cb, along with all needed equilibrium constants, you can work out the theoretical pH at any point). What is the general method for finding the pH and Vtitrant at the theoretical equivalence point for any titration? (Web pages that explain this would be nice if you know of any) I can imagine simply having the graph of Vtitrant (y-axis) over pH (x-axis) and then taking the second derivative of the known equation for Vtitrant (this is linked on one of the titrations.info pages you pointed me to). (Vtitrant is the volume of titrant added at any one point in the titration) This will be some function in [H3O+] (see the titrations.info page), which I can then set to 0 and solve to find the [H3O+] values at the equivalence point (easy to convert to pH), and then I can plug that value of [H3O+] back into the original Vtitrant equation for my titration and get the Vtitrant needed to reach the equivalence point. Or is there some other superior method of calculating the theoretical equivalence point for a titration?

Offline Borek

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Re: End-Point and Indicators
« Reply #13 on: November 05, 2012, 05:43:07 PM »
de Levie offers rather complicated calculations to find the equivalence point theoretically from the gradient of the theoretical titration curve (as, for a given Va, Ca, Vb and Cb, along with all needed equilibrium constants, you can work out the theoretical pH at any point).

There is something wrong here. I don't know de Levie book, so I have no idea what, but either you are misunderstanding what he is doing, or his wording is lousy - or there is something else that is wrong. I wrote it several times already but looks like it still didn't sink - equivalence point is defined by the stoichiometry of the reaction. You add stoichiometric amount of titrant to the titrated substance and that's the equivalence point. As you know amounts of substances mixed you know the composition of the solution at the equivalence point, so you can calculate everything just from the solution composition. Calculating titration curve to find where is the equivalence point to calculate back the equivalence point makes no sense, as calculation of pH (or whatever value describes the titration curve) at equivalence point is already part of the titration curve calculation.

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What is the general method for finding the pH and Vtitrant at the theoretical equivalence point for any titration? (Web pages that explain this would be nice if you know of any)

If you will browse titrations.info you will find this information.

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Or is there some other superior method of calculating the theoretical equivalence point for a titration?

I have a feeling you are asking questions, ignoring answers or just skimming them, and reinventing the wheel again and again. These are quotes from my previous post:

Equivalence point is where the stoichiometric amount of the titrant was added

Quote
equivalence point is when you added 100% of the titrant (exactly stoichiometric amount). As I wrote earlier, these things are described on titrations.info.

Quote
Equivalence point is not related to the titration curve, just to the stoichiometry of the reaction.

After reading this you still ask how to use titration curve to calculate equivalence point? Sigh. Makes me think I am talking to a wall.
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Offline Big-Daddy

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Re: End-Point and Indicators
« Reply #14 on: November 05, 2012, 06:26:10 PM »
There is something wrong here. I don't know de Levie book, so I have no idea what, but either you are misunderstanding what he is doing, or his wording is lousy - or there is something else that is wrong. I wrote it several times already but looks like it still didn't sink - equivalence point is defined by the stoichiometry of the reaction. You add stoichiometric amount of titrant to the titrated substance and that's the equivalence point. As you know amounts of substances mixed you know the composition of the solution at the equivalence point, so you can calculate everything just from the solution composition. Calculating titration curve to find where is the equivalence point to calculate back the equivalence point makes no sense, as calculation of pH (or whatever value describes the titration curve) at equivalence point is already part of the titration curve calculation.

I have read what you are saying before but it doesn't seem to me that this solves any problems. That is like saying that acid and base reactions involve stoichiometry - of course they do, but the main complexity is in dealing with the dissociation constants. That is again the problem here: simply knowing the concentrations and volumes is only enough if all of my dissociations are taken as complete, which is not the case in a general method. That is why saying that stoichiometry is equivalent is a good definition on paper but when it comes to calculating the theoretical equivalence point doesn't help me much.


After reading this you still ask how to use titration curve to calculate equivalence point? Sigh. Makes me think I am talking to a wall.

Like I said earlier, I have absolutely no idea of how to go from the statement "equivalence point is defined by the stoichiometry of the reaction" to calculating the equivalence point because the amount of ions produced is governed by dissociation constants.

I have looked on the page titrations.info has on calculating the equivalence point for titrations, and they do not contain information enough to solve even for weak acids and bases, much less for polyprotics or mixtures as a general method must do. Here is the page in question: http://www.titrations.info/acid-base-titration-equivalence-point-calculation (there is a solution for weak acids/bases but it is inexact even at that level and so is not the sort of approach I'm looking for).

If it is true that the equivalence point is solely to do with reaction stoichiometry, then you should be able to help me calculate it mathematically. My current problem is that Vtitrant is not known at the equivalence point - if we did know it, working out the pH at that point would be simple (just run a calculation with the same equation, using Vtitrant for the equivalence point as the Vb).

Let us take as our starting point the main pH equation used by the BATE pH calculator since I think you are probably familiar with this already. Link: http://www.chembuddy.com/?left=BATE&right=pH_calculation

And let's say I have a mixture of a polyprotic acid that I'm titrating with a polyprotic base. How do I find expression(s) for the equivalence point(s)? (To my knowledge titrations of polyprotics tend to have more than one point of inflection, i.e. more than one equivalence point.)

I.E. Let us find the pH at the equivalence point(s) first, by getting some form of expression for [H+]. The question now is - how do we use that equation to input the correct value of Ca and Cb at the equivalence point? This you will have to tell me, unless you mean to say that the equivalence point is defined as the point where Ca=Cb. But if this is the case (and even if it isn't, depending on how you propose to find the equivalence point), how does that method account for the existence of more than one equivalence point in the titration? (Which we know there are for polyprotic mixtures.)

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