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Part 2: Preliminaries

2.1. Motivating the Mathematical Model

The data gathered when performing an acid-base titration are the pairs of values

(total volume of base added to solution, resulting pH of solution)

To better view the relationship, we usually graph these data points, thus producing a plot of volume of base added versus pH of the solution. A typical titration graph is shown in Figure 2, repeated here.


Figure 2.

When calculating the unknown concentration of acid using the standard method illustrated in the HCl/NaOH example above, the only information from the data that is actually used is the shape of the curve near the equivalence point so that the location of the equivalence point can be determined. The rest of the data, which may contain additional information about the relationship, are usually unused.

In practice the location of the equivalence point is found by identifying the approximate location of the point of most rapid increase, essentially by inspection. Often, first and second derivative data are calculated numerically from the original data and their plots are used to help locate the equivalence point. However, the accuracy of any such estimate is always limited by the distance between two consecutive data points (i.e., the size of the drops of base added to the solution), since the true equivalence point could occur anywhere between two data points.

So, in this project, we will investigate the following question:

In an attempt to answer this question, we will construct a mathematical model that will describe the relationship between the volume of base added and the resulting pH of the solution. Real titration data are provided, and the initial concentration of the acid can be determined by fitting the model to these data -- thus making use of most of the data available rather than just those data values near the equivalence point.

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