Optimising nonlinear calibration of glass pH electrodes over the pH range -4.5 to 15.5 using improved statistical...

Accurate calibration of the widely used glass pH electrode is particularly challenging at high (pH < 1) and low (pH > 11) proton activities (a_H⁺), where the linear Nernstian relationship between the cell potential and log(a_H⁺) breaks down. An asymmetric double-logistic function, selected from seven nonlinear function candidates by the corrected Akaike Information Criterion (AICc) and RMSE, best described the electromotive force (EMF)-pH relationship across both the acid- and alkaline-error regions from pH -4.5 to 15.5 at 25 °C (R² = 1.00, RMSE = 2.3 mV). Reference pH was assigned from proton activity (Pitzer model, MacInnes convention) for five NIST/DIN buffers, ten H₂SO₄ standards (0.10-7.54 mol L^-1 H₂SO₄; pH -4.5 to 1) and nine NaOH standards (0.10-6.91 mol L^-1 NaOH; pH 12.9 to 15.5). Relative to conventional three-point calibration, the function reduced acid-error bias by up to 3.0 pH units at pH -4.5 and alkaline-error bias by up to 1.5 pH units at pH 15.5. All-subsets and leave-one-out Monte Carlo analyses further showed that five acidic and three alkaline standards, with the NIST/DIN-traceable buffers, reproduced full-dataset accuracy to within ±0.25 pH unit at the extremes. This nonlinear calibration model enables continuous calibration over nearly twenty pH units; when only one region is of interest, the model reduces to a four-parameter logistic function. The reduced standard set retains accuracy while reducing calibration time by at least 50 %.