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[edit] Electrode calibration

Another use of the Gran plot is for electrode calibration.[1] The advantage of this method of electrode calibration is that it can be performed in the presence of a background medium of constant ionic strength, the same medium which may later be be used for the determination of equilibrium constants. The electrode is assumed to obey a modified form of the Nernst equation.

E=E^0+s \ln [H^+]\,

E is a measured electrode potential, E0 and s are calibration constants and [H + ] is the hydrogen ion concentration. The parameter s is ideally equal to RT/F (59.1 mV at 298 K) but in practice may not be exactly equal to the ideal value. Note that the electrode response is assumed to be linear in hydrogen ion concentration (not activity). This assumption is valid when the ionic strength is effectively constant. Rearrangement of this equation gives

[H^+]=10^{\frac{E^0-E}{s}}

The analytical concentration of the hydrogen ion, TH in a titration of a strong acid, concentration aH, with a strong base, concentration bH is given by

(T_H)_i=\frac{a_Hv_0-b_H v_i}{v_0+v_i}

By convention the concentration of hydroxide ions is given as minus the corresponding concentration of hydrogen ions. Note that for this application the concentrations of acid and base must both be known. The concentrations should be directly related to primary standards.

As described above, in acid solutions with a pH of 5 or less, the concentration of hydoxide ions is negligible compared to the concentration of hydrogen ions, so [H + ] = TH. Therefore

(v_0+v_i)10^{-\frac{E^0-E}{s}}=a_Hv_0-b_H v_i

and a plot of the function on the left-hand side, sometimes called the Gran function, against titre, vi should be a straight line cutting the x-axis at the (acid) equivalence point. The parameters E0 and s can be obtained by the method of least squares. By a similar argument, with alkaline solutions of pH 9 or above the left-hand side of the equation

(v_0+v_i)10^{-\frac{E-E^0}{s}pK_w}=a_Hv_0-b_H v_i

will yield a straight line with respect to titre, cutting tha axis at the (alkaline) equivalence point. The difference between the two equivalence points can be used to calculate a carbonate content value.