An alternative to using a visual indicator to locate a titration’s end point is to continuously monitor the titration’s progress using a sensor whose signal is a function of the analyte’s concentration. The result is a plot of the entire titration curve, which we can use to locate the end point with a minimal error.

Shown here are four examples of titration curves for the titration of 50.0 mL of 0.050 M CH_{3}COOH with 0.10 M NaOH in which we monitor the solution’s pH. The titration curve in (a) is the normal potentiometric titration curve obtained by monitoring pH as a function of the volume of NaOH added. We can enhance the equivalence point, which is useful when the change in pH at the equivalence point is small, by plotting (b) the first derivative of the titration curve or (c) the titration curve’s second derivative. Finally, we can seek a mathematical transformation of the titration curve that linearizes it. The titration curve in (d) transforms the data using the following equation

[H_{3}O^{+}] × *V*_{b} = *K*_{a}*V*_{eq} – *K*_{a}*V*_{b}

where *V*_{b} is the volume of titrant, *K*_{a} is the analyte’s weak acid dissociation constant, and *V*_{eq} is the volume of titrant at the equivalence point. For volumes of titrant before the equivalence point, a plot of [H_{3}O^{+}] × *V*_{b} versus *V*_{b} is a straight-line with an *x*-intercept of *V*_{eq} and a slope of –*K*_{a}. Figure 9.14d shows a typical result. This method of data analysis is called a Gran plot.