Titration curve fitting acid-base curves

Potentiometric titration curves are used to determine the molecular weight and fQ or for weak acid or weak base analytes. The analysis is accomplished using a nonlinear least squares fit to the potentiometric curve. The appropriate master equation can be provided, or its derivation can be left as a challenge. 

To use this method, the sample is dissolved in a system containing two phases (e.g., water and octanol) such that the solution is at least about 5 x 10-4 M. The solution is acidified (or basified) and titrated with base (or acid) under controlled conditions. The shape of the ensuing titration curve is compared with the shape of a simulated curve, which is created in silico. The estimated p0Ka values (together with other variables used to construct the simulated curve such as substance concentration factor, CO2 content of the solution and acidity error) are allowed to vary systematically until the simulated curve fits as closely as possible to the experimental curve. The p0Ka values required to achieve the best fit are assumed to be the correct measured p0Ka values. This computerized calculation technique is called refinement , and is described elsewhere [14, 15]. 

Whatever the aim of a particular titration, the computation of the position of a chemical equilibrium for a set of initial conditions (e.g. total concentrations) and equilibrium constants, is the crucial part. The complexity ranges from simple 1 1 interactions to the analysis of solution equilibria between several components (usually Lewis acids and bases) to form any number of species (complexes). A titration is nothing but a preparation of a series of solutions with different total concentrations. This chapter covers all the requirements for the modelling of titrations of any complexity. Model-based analysis of titration curves is discussed in the next chapter. The equilibrium computations introduced here are the innermost functions required by the fitting algorithms. 

Figure 4-44. Fitted titration curve for the addition of base to a mixture of a diprotic and a strong acid.

For an indicator to be useful in detecting the end-point of an acid-base reaction, the pH range of the indicator must fit into the vertical part of the titration curve for that reaction. The vertical part of the titration curve will include the pH of the salt solution at the end-point. 

To extract acid dissociation constants from an acid-base titration curve, we can construct a difference plot, or Bjerrum plot, which is a graph of the mean fraction of bound protons, H, versus pH. This mean fraction can be measured from the quantities of reagents that were mixed and the measured pH. The theoretical shape of the difference plot is an expression in terms of fractional compositions. Use Excel SOLVER to vary equilibrium constants to obtain the best fit of the theoretical curve to the measured points. This process minimizes the sum of squares [nH(measured) -nH( theoretical) 2. 

In [351,2985-2990], the data points are too far from the PZC to make a reliable estimate. The PZC of hydrous zirconia at pH 2.9 reported in [2991] is probably due to occluded HNO,. The apparent PZC found for silica by titration in [2992] is probably due to occluded acid. The dependence of the apparent PZC on the pH of precipitation reported in [2993] is chiefly due to acid or base occluded in the precipitate. The charging curves in [2994] were set off to fit ion adsorption results. Most likely, surface charging of Y(OH)2 (NO lo 5 studied in [2995] was not the main process governing acid-base titrations of that compound. The equilibrium pH in water and 1 M KCl at low and unknown solid-to-liquid ratio reported in [172] and the equilibrium pH at low solid-to-liquid ratio reported in [175] are not... 

Fortunately, there is such a method, which is both simple and generally applicable, even to mixtures of polyprotic acids and bases. It is based on the fact that we have available a closed-form mathematical expression for the progress of the titration. We can simply compare the experimental data with an appropriate theoretical curve in which the unknown parameters (the sample concentration, and perhaps also the dissociation constant) are treated as variables. By trial and error we can then find values for those variables that will minimize the sum of the squares of the differences between the theoretical and the experimental curve. In other words, we use a least-squares criterion to fit a theoretical curve to the experimental data, using the entire data set. Here we will demonstrate this method for the same system that we have used so far the titration of a single monoprotic acid with a single, strong monoprotic base. 

The availability of a master equation for acid-base titrations, and of convenient non-linear least-squares curve-fitting methods such as incorporated in Excel s Solver, have made the determination of the unknown sample concentration(s) relatively easy a spreadsheet is all that is required for such an analysis. Of course, there is no guarantee that all component... 

One can of course fit experimental data to the entire, theoretical curve with a non-linear least-squares routine such as Solver. In this particular case, however, the direct, non-iterative method of using Gran plots provides a valid, simpler alternative. As illustrated below, such plots are quite linear, analogous to the Gran plots for the titration of strong acids and bases. 

Because of the close analogy between acid-base and redox behavior, it will come as no surprise that one can use redox titrations, and also simulate them on a spreadsheet. In fact, the expressions for redox progress curves are often even simpler than those for acid-base titrations, because they do not take the solvent into account. (Oxidation and reduction of the solvent are almost always kinetically controlled, and therefore do not fit the equilibrium description given here. In the examples given below, they need not be taken... 

Usually one titrates from acidic to alkaline in order to minimize the interference by carbon dioxide. It is important that the compound stays in solution in the course of the titration. Compound precipitation can lead to significant errors in the pKa determination. This problem is particularly acute when one works with a compound concentration close to its solubility limit. In this case partial precipitation of the substance induces a distortion in the titration curve, while a good fit to the fitting of the experimental can still be achieved. For bases, partial precipitation leads to an underestimation of the pKa. For acids, partial precipitation leads to an overestimation of the pKa. ° When the compound concentration is way above its solubility limit, a stronger distortion of the titration curve is observed and the experimental data no longer fit with the proposed model. Argon is used to minimize carbon dioxide uptake in the course of the titration. 

Figure 7a shows a acid-base turbidimetric titration curve for the PE carrageenan and Chy (a basic protein) in the presence of different NaCl concentrations [38]. The pHc was calculated as the intersection of the tangent at the inflection point with the plateau of the plot while the pHcj) was obtained by fitting the data to a 4-parameter sigmoid function. 

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