If we are titrating our analyte with volume V_{titrant} of the titrant solution, and we have V_{titrant} as a function of various known constants and [H^{+}], would the comprehensive method of finding first the [H+] at the equivalence point and then (by back-substitution) the V_{titrant} at which each equivalence point is reached be to take the derivative with respect to [H+] of V_{titrant}, and then set this as being equal to 0, and solve (with each real, positive solution corresponding to an equivalence point)? This seems feasible because the equivalence point is defined as the point at which rate of change of [H+] is greatest.
OK, so then the equivalence point(s) do not exactly coincide with the solutions, even from a theoretical point of view, there are better methods and this derivative is not the definitive solution for the equivalence point. From the level of detail my book goes into I get the picture there is no direct and definitive solution?
Solution to what? Determining the equivalence point is an experimental problem, not a theoretical one.
Ah ok ... and that could be the source of our inaccuracy when we try and calculate it with a purely theoretical method like the one I just said (or any purely theoretical method). When I said solution I meant the various values (all real positive roots) you get as results if you take the first derivative and then set this =0.
No. There is no problem with calculating equivalence points theoretically, as we simply apply the definition to calculate whatever we want. There is a problem with experimental detection of the equivalence points, as data collected during experiment is never perfect.
OK, that means for certain real-world applications we will need to fit data to an experimental plot. Let's say we already know all initial concentrations of all components in both titrant and analyte, along with all equilibrium constants being known, and the starting volume of the analyte. The determination of the equivalence points (V_{titrant} required and pH at each equivalence point), that should then be confirmed in experiment, is purely a theoretical problem, then, based on stoichiometry? In that case, if I have V_{titrant} as a function (in terms of these initial concentrations and equilibrium constants, all known) of [H^{+}], derived from theoretical simultaneous equations of the equilibria, and then take the derivative of this and solve as before, would my values match up directly with those calculated from the definition of equivalence point?
Hello, Could you show your math from post #7 ? Because I want to be sure I understand your question completely.
Well an issue is that you may not be familiar with this method as proposed by Robert de Levie in his Oxford Chemistry Primer 'Aqueous Acid-Base Equilibria and Titrations'. On the other hand you may be familiar with it - I have no clue how much of de Levie's system is common knowledge and how much was created by him. Basically our result is that, for any acid, we can write an "acid dissociation function" F_{a} which is a function in [H^{+}] and the acid dissociation constants (how to work out F_{a} is more complicated and I cannot explain it simply; simply assume the function is robust, if you haven't seen the method before) for each acid (or indeed acid, base or salt) in solution, and if we then multiply F_{a} (for that acid) by C_{a} for each acid, F_{a} by C_{s} for each salt, F_{a} by C_{b} for each base and then sum these all together, we will have a function in [H^{+}] and the acid dissociation constants, and when we then add K_{w}/[H^{+}]-[H^{+}] to this expression and set it to 0, we have an polynomial equation we can solve for [H^{+}]. With titrations, de Levie shortly shows how to express V_{titrant} similarly to how we just wrote the entire equation, i.e. V_{titrant} is a function of V_{analyte}, all starting concentrations, all equilibrium constants, and H^{+}. The issue with normal titrations as Borek said earlier is that we do not start by knowing all starting concentrations. Thus we must carry out the experiment to determine the concentration. However, let us say all starting concentrations are given, as well as V_{analyte} and all constants, so we can write V_{titrant} exclusively as a function of [H^{+}]. Would we get the exactly (mathematically) same results from differentiating this function with respect to [H^{+}] and setting equal to 0, as we would from using the definition of equivalence point to calculate what volume we need to reach each one and what [H^{+}] there will be at each one? Hope this has cleared up what my problem means. (In other words, I'm asking if the points of 0 gradient directly coincide, in theory, with the equivalence points, or if these are just approximations we take in experimental practice to make it easier to spot the equivalence points?)