Gran Plot - Electrode Calibration

Electrode Calibration

Note that the above analysis requires prior knowledge of and .

If a pH electrode is not well calibrated, an offset correction can be computed in situ from the acid-side Gran slope:

For a titration of acid by base, the acid-side slope can serve to compute using a known value of or using the value given by the equivalence volume. can then be computed from the base-side slope.
For a titration of base by acid, as illustrated in the sample plots, the acid-side slope is similarly used to compute and the base-side slope is used to compute using a known value of or using the value given by the acid-side equivalence volume.

In the sample data illustrated in Figure 1, this offset correction was not insignificant, at -0.054 pH units.

The value of, however, may deviate from its theoretical value and can only be assessed by a proper calibration of the electrode. Calibration of an electrode is often performed using buffers of known pH, or by performing a titration of strong acid with strong base. In that case, a constant ionic strength can be maintained, and is known at all titration points if both and are known (and should be directly related to primary standards). For instance, Martell and Motekaitis (1992) calculated the pH value expected at the start of the titration, having earlier titrated the acid and base solutions against primary standards, then adjusted the pH electrode reading accordingly, but this does not afford a slope correction if one is needed.

Based on earlier work by McBryde (1969), Gans and O'Sullivan (2000) describe an iterative approach to arrive at both and values in the relation, from a titration of strong acid by strong base:

is first estimated from the acidic data according to Rossotti and Rossotti (1965), and is initially taken to have its theoretical value;
modified Gran function plots are drawn, using vs. on the acidic side of equivalence and vs. on the alkaline side, and the equivalence volumes and are computed therefrom, as before;
as before, the difference in the equivalence volumes is used to compute the carbonate content but also to calculate an 'effective base concentration' for the alkaline side of equivalence;
approximate values are computed on the acid side as and on the alkaline side as ;
the initial definition is rewritten as, and the data are plotted against, using those values corresponding to pH values in the 2.5-4.5 and 10.7-11.5 ranges (the linear response range for a glass electrode that avoids variation of junction potentials and/or alkaline error at the pH extrema, and that additionally avoids measurement uncertainties near the equivalence point as well as computational errors from the neglect of on the acid side and the neglect of the carbonate/bicarbonate equilibrium on the alkaline side); a linear least-squares treatment provides as slope and as intercept;
steps 2 and 3 are repeated with the new and values for greater precision in the equivalence volumes and the CO₂ content.

The procedure could in principle be modified for titrations of base by acid. A computer program named GLEE (for GLass Electrode Evaluation) implements this approach on titrations of acid by base for electrode calibration. This program additionally can compute (by a separate, non-linear least-squares process) a 'correction' for the base concentration. An advantage of this method of electrode calibration is that it can be performed in the same medium of constant ionic strength which may later be used for the determination of equilibrium constants.
Note that the regular Gran functions will provide the required equivalence volumes and, as is initially set at its theoretical value, the initial estimate for in step 1 can be had from the slope of the regular acid-side Gran function as detailed earlier. Note too that this procedure computes the CO₂ content and can indeed be combined with a complete standardization of the base, using the definition of to compute . Finally, the usable pH range could be extended by solving the quadratic for .