I have seen it thrown around a lot that the pH at which concentration of a conjugate base is at a maximum can be found by adding up the 2 pKa's whose reactions that base is involved in and dividing by 2. But I tried differentiating and this only appears to be the case for HA^{-} maximum concentration (reached when pH=pKa1+pKa2, for a diprotic acid). Any other case, and we reach a polynomial... so where does this simple method for finding the pH for maximum concentration come from? What approximation does it require?
I don't know where you saw what things thrown around, you would have to quote. You quoted results just for diprotic acids. In other cases you do have a higher polynomial. I am not aware of any particular simplification - even though the polynomials always do contain one zero coefficient. So nothing very nice there on the face of it, unless you can quote us something. Somewhat nice though, since you seem interested in these things, is that these maxima occur where a n-protic acid has bound overall 1, 2,... (n - 1) protons per acid molecule (moles/mole) - you might show this to yourself and us. It is not limited to proton binding of course but applies to any ligand that can multiply bind to another molecule.
Well I can quote a problem I recently did: "It is known that in the solution of citric acid H_{3}X, the maximum concentration of H_{2}X^{-} is at pH = 3.95; the maximum concentration of HX^{2-} is at pH = 5.57; the concentrations of H_{2}X^{-} and HX^{2-} are equal at pH = 4.76. Determine the acidity constants K_{a1}, K_{a2}, K_{a3} of citric acid." The method expected was to write (K_{a1}*K_{a2})^{1/2}=10^{-3.95}, (K_{a2}*K_{a3})^{1/2}=10^{-5.57}, K_{a2} = 10^{-4.76}. But surely neither of these two is justified, since we are dealing with a triprotic acid here? The results found were K_{a1} = 1.24 * 10^{-4}, K_{a2} = 1.14 * 10^{-5}, K_{a3} = 4.17 * 10^{-7}. Yet the result is highly accurate. e.g. for K_{a3} my exact calculation gave 4.2 * 10^{-7}. So the question returns: given that this seems to be a decent approximation often, how do we explain it, and under what conditions will it tend to work? I've done some investigating of this now. Any suggestions on what to interpret from this would be most welcome. What I've found is that (1) it can be algebraically shown and numerically checked that as the value of as the [H^{+}] for maximal [H_{2}X^{-}] grows larger, our approximation for K_{a3} grows more accurate; 2) the approximation consistently underestimates K_{a3}. The approximation is typically highly accurate. Can we use this to make any generalizations about this approximation? It does indeed look to me like the larger the K_{a1} value compared to K_{a2} and K_{a3}, the more we can ignore it in the calculation and treat the acid as if only K_{a2} and K_{a3} existed, but also that the approximation will be decent unless K_{a1} is exceedingly close. Can I ask whether you were taught this/found it somewhere, or worked it out yourself? If the former, I would love if you could suggest where I can find more tricky issues like this. It is indeed noticeable that when [H^{+}]=(K_{a(k)}*K_{a(k+1)})^{1/2}, the approximation discussed in this thread (the suggestion being that the maximum concentration of [H_{n-k}A] is reached at approximately this pH), as a result we always have [H_{n-k-1}A]=[H_{n-k+1}A] at such points - in fact this relies only on equilibrium constant expressions. But I don't see that this can be extended to tell us anything about any forms of the acid except [H_{n-k-1}A], [H_{n-k+1}A] and thus about the acid overall?
OK, they almost give you K_{a2} but please check as I get not far but significantly different from you , 1.74*10^{-5} . I don't know how you did your 'exact calculation' for K_{a3} so please give it. Now I see what you meant by your original question. I gather from your 'the method expected' you have been told somewhere that this is the method. You'd have to show the calculations you made in its justification if you want any comment on them. Please in future give all values as both K_{a} and pK_{a} otherwise a reader is having to do calculations just to try and follow what you are saying. You give two widely different values for K_{a2}. You give a K_{a1} which seems to correspond to a pH close to your maximum so I don't see how you can possibly have been using means. If you do not set out calculations then any errors, yours or mine, make what you did incomprehensible. Qualitatively in e.g. your first maximum you are more than 2 pH units away from pK_{a3}. Then the 3^{-} form should be less than 1% of all, so looks safe to ignore and the formulation they gave you right to a decent approximation. I had known about that little theorem but never had any use for it before now. You could I think use it in your exact calculations. I looked it up and the way it is presented in my book it is easier and you are far better off to derive it yourself! Which I did. Not difficult - try it for diprotic then triprotic acids. There might some small slips in calculations but you seem to be approaching this in a right way.
See below. Note that K_{a2} = 10^{-4.76} = 1.74 * 10^{-5} to 3 sig figs, pKa=4.76 is known exactly as per the question. I don't want to show the entire differentiation process as it's quite long. I took the concentration of [H_{2}A^{-}] as a function of all Ka values and [H+] and differentiated with respect to [H+], set equal to 0 and got a polynomial for concentration of [H+] at which [H_{2}A^{-}] would be maximal. I did the same for [HA^{2-}]. This gives two equations, in which we have a total of 2 variables, Ka1 and Ka3, since Ka2 is known to be 10^{-4.76} (pKa=4.76) and the concentration of [H+] at which each of these two conjugate base forms is maximal is also known. I rearranged one equation for Ka1, substituted into the other and rearranged for Ka3. I can provide the final result if you'd like. As for "the method expected", I originally didn't know how to approach it because I knew that HA^{-} was the only form for which we have this simple exact result (and at the time the sheer length of differentiating them etc. as above made me think it probably wasn't what they wanted). I looked at the given solutions and they used this approximate method. Then I used the above differentiation method to calculate exact values and it seems that they are in pretty close agreement, which is what I want to find out about most. I haven't provided any exact calculation values for K_{a1}. The values that came as a result of the approximation discussed were given in my last post. But we hadn't calculated pKa3 yet at the time? Perhaps if we define symbolically the information given, it will make things easier to discuss. Let us say that [H+] at which concentration of [H_{2}A^{-}] is maximal is defined as [H+]_{k=1} and [H+] at which concentration of [HA^{2-}] is maximal is defined as [H+]_{k=2}. All we have to work with in the original question is Ka2, [H+]_{k=1}, [H+]_{k=2} (or if you prefer, pKa2, pH_{k=1}, pH_{k=2}). To get the rest, we have to choose our method - exact, or approximate, and if the latter is likely to give us a good result let's go for that! More broadly, how do we decide if the (Ka2*Ka3)^{1/2} formula (alternatively (1/2)(pKa2+pKa3) equation) for maximal concentration of the twice-dissociated form will give us a good, close result? Seems to require that Ka1 and I presume Ka4 (if the acid were 4+-protic), or pKa1 and pKa4, are not too close to Ka2 and Ka3 (pKa2 and pKa3). So then, if they are indeed not too close to Ka2 and Ka3 (pKa2 and pKa3), does it mean we are calling them negligible? In the problem, according to the approximate method (which is pretty close to the exact method) we got pKa1=3.91, pKa2=4.76, pKa3=6.38 (Ka1 = 1.24 * 10^{-4}, Ka2 = 1.14 * 10^{-5}, Ka3 = 4.17 * 10^{-7}). So it looks like the 'interfering' equilibrium constant has to be within a fraction of a pKa unit of the 'important' ones in order for the approximate method to give poor results? In other words, a very good approximation most of the time. I'll try to see if I can find some ways to prove the approximate equations from the exact procedure and get back to you. In the meanwhile any help you can give with qualitative interpretations of why the approximate method works will be most appreciated! I see. It seems like a slightly niche thing - my analytical chemistry textbook does not even broach the question ("what pH is the concentration of a given form maximal?") - so I was wondering if you knew a link or textbook I could look at which would contain more of these rarer issues. My "exact calculation" method is to differentiate the mole fraction of the form with respect to [H^{+}] and set equal to 0. For HA^{-} form from a diprotic acid, we get the neat (and exact) 1/2 (pK_{a1} + pK_{a2}) = pH equation (i.e. [H+]=(Ka1*Ka2)^{1/2}).
I'll answer the easiest points now, may make additional ones later. Misunderstanding due to a typo IOW you solved two simultaneous linear equations in two unknowns. Why not? In particular did you have a zero coefficient in each polynomial? If not I think we need to look again. However the last result is telling me you did get a zero in the central term of the quadratic. The things that students tossed off routinely a few years back when it was called Question 6 of Exercise 1 of Chapter 3, Calculus tend to get considered, if not insoluble, an undertaking of the most extravagant labour when it is called Biophysics. My point was even the approximate value you gave seems grossly wrong, perhaps due to another typo. Your K_{a1} gives me pK_{a1} of 3.96, very close to and on the wrong side of your pH of maximum, 3.95. Instead the latter is supposed to be the mean of pK_{a1} and 4.76, from which I make pK_{a1} to be 3.14. If you pull your Profs up on tiny inaccuracies but make a huge one yourself they may take revenge! And I think you would do well (also for yourself later) if you set out all relevant data and results in a table (K's and pK's !) otherwise no one will understand this, nor you later. Very acute. It sounded better how I said it but I think we could have got from other stuff we had that it was a reasonable at least rough approximation. When you are 1 pH unit away from something its effect is 10% or less. The results fully confirm my intuition! waves hands Indeed in this field you may often find what may at first sound like circular arguments - but it is pretty indicative when the circles close. I thought you said you had done the first. I did the second in a previous post Yes it is slightly - but it sounds you are making it your niche. For most biochemistry and biophysics approximate solutions are sufficìent and the point, and exact ones useless and even irritating to hear about. In fact they are so thrown out that you'll likely forget them youself, so I recommend you to make good notes while this is fresh, including what I said about setting out and tabulating this problem. Where this could have application would be if you get into e.g. metal complexes of acids, amino acids, peptides etc. More mainstream would be in co-operative phenomena in proteins and other biomolecules. Probs are slightly different - your examples are called negative co-operativity - a proton dissociating makes it harder for the next one to do so - and so the K's are nicely separated. Instead the thing of most biophysical interst is the opposite, one thing making the next easier. That makes it almost impossible to determine the K's! But the practice you are giving yourself will stand you in good stead. Just remember experimetalists will be concerned with big effects they can measure, not tiny differences they can't. A mathematical/physical book about this is J. Wyman & S.Gill "Binding and Linkage" USB publ. and the result we mentioned is on p.72.
Yes I did. They were actually cubics since it's a triprotic acid, but each had one 0 coefficient. And of course I didn't have to solve the cubic equations as such because I wasn't solving for the leading term. Is there a book where this would be found as Question 6 of Exercise 1 of Chapter 3? I was quite happy to go for the differentiation. It just occurred to me that it couldn't possibly be expected for the problem - that's why the idea only came afterwards :P You're right, I think there is a typo in the solutions, because the approximate method gives me K_{a1} = 7.24 * 10^{-4}, pK_{a1} = 3.14. Thanks, I think I've understood this issue now. At pH when a certain form is maximal, so long as the equilibria that form is involved in have K values not too closely surrounded by the K values of adjacent equilibria, it can be assumed that that form and the two adjacent to it will take up a vast majority of the total concentration of the acid and can neglect the rest of the acid forms, and neglect the presence of all the other equilibria except the two which this form is involved in. This does indeed lead by the same procedure of differentiation to the approximate result we've been discussing. The approximate result can also be found by taking the appropriate limit on the exact expression. The limit taken is tantamount to saying that as K_{a1} → ∞ (pK_{a1} → -∞), the maxima for the twice-dissociated form converges to the value our approximation predicts. Since this is the same thing as we've said above - that the surrounding Ka values are far enough away from the too important ones that we can call them negligible - we haven't learnt anything new from this bit of maths. :P Thanks for all the advice. My main purpose in this investigation was less to show my determination to use exact solutions and more to try and find out why and when the approximation is likely to be good. Interestingly, with the exact approach I don't think it matters whether the dissociation constants get larger or smaller. The calculation relies on more pure maths starting from a mass balance and the equilibrium expressions, and that's it. But I see why the approximate approach would get screwed up if one thing makes the next easier - because at the maximum of a given form, it is helping to produce more and more of the next form onwards and so forth. We thus cannot assume that only the form in question and the two adjacent to it are present significantly at this maximum, because however much there is of these will promote even more of the later forms to be produced. (Maybe I'm explaining my intuition badly!) But I checked the approximate vs. exact calculations and think you're right to say the approximate method won't work there. Thanks again for the help.