Temperature-Dependent Nonlinear Calibration of Glass pH Electrodes for Negative pH Applications

Glass pH electrodes are widely used to estimate proton activity (a _H⁺ ) in aqueous solutions. However, at high proton activity, conventional linear calibration using standard NIST/DIN buffers breaks down due to nonlinear electrode response and strong nonideality, leading to severe pH overestimation. To alleviate this limitation, this study develops and validates a nonlinear, temperature-corrected calibration protocol that extends the measurable and reliable range of pH using glass electrodes down to (pH = -log a _H⁺ ) = -5 (proton activity of 10⁵) over a temperature range between 5 and 60 °C. The calibration is done using NIST buffers (pH 1.68 to 10.01) and sulfuric acid standards (concentrations ranging between 0.16 mmol·L^-1 and 9.69 mol·L^-1 H₂SO₄). Proton activities are calculated using the Pitzer model with the MacInnes assumption implemented in PHREEQC. At a given temperature, the response curve between pH and the electrode electromotive force in the low to negative pH range is modeled best with a logistic equation. Incorporation of temperature-dependent parameters enables construction of continuous nonlinear calibration curves across the studied temperature range. Application of this calibration protocol reduces systematic pH overestimation by up to >3 pH units and yields accurate proton activity under strongly acidic conditions. Sensitivity analysis demonstrates that meaningful measurements are achievable down to approximately pH ≈ -5, below which signal-to-noise constraints dominate. This work establishes a transferable and practical methodology that extends the operational range of conventional glass electrodes by several orders of magnitude in proton activity, enabling more reliable pH measurements in extreme natural and industrial environments.