The measurement of pH using a glass electrode obeys the...

The measurement of pH using a glass electrode obeys the Nernst equation. The typical response of a pH meter at $25.00^{\circ} \mathrm{C}$ is given by the equation $$\mathscr{C}_{\text {meas }}=\mathscr{E}_{\text {ref }}+0.05916 \mathrm{pH}$$ where $\mathscr{E}_{\text {ref }}$ contains the potential of the reference electrode and all other potentials that arise in the cell that are not related to the hydrogen ion concentration. Assume that $\mathscr{E}_{\text {ref }}=0.250 \mathrm{~V}$ and that $\mathscr{C}_{\text {tme\pi }}=0.480 \mathrm{~V}$ a. What is the uncertainty in the values of $\mathrm{pH}$ and $\left[\mathrm{H}^{+}\right]$ if the uncertainty in the measured potential is $\pm 1 \mathrm{mV}(\pm 0.001 \mathrm{~V})$ ? b. To what precision must the potential be measured for the uncertainty in $\mathrm{pH}$ to be $\pm 0.02 \mathrm{pH}$ unit?

The measurement of pH using a glass electrode obeys the Nernst equation. The typical response of a pH meter at $25.00^{\circ} \mathrm{C}$ is given by the equation
$$\mathscr{C}_{\text {meas }}=\mathscr{E}_{\text {ref }}+0.05916 \mathrm{pH}$$
where $\mathscr{E}_{\text {ref }}$ contains the potential of the reference electrode and all other potentials that arise in the cell that are not related to the hydrogen ion concentration. Assume that $\mathscr{E}_{\text {ref }}=0.250 \mathrm{~V}$ and that $\mathscr{C}_{\text {tme\pi }}=0.480 \mathrm{~V}$
a. What is the uncertainty in the values of $\mathrm{pH}$ and $\left[\mathrm{H}^{+}\right]$ if the uncertainty in the measured potential is $\pm 1 \mathrm{mV}(\pm 0.001 \mathrm{~V})$ ?
b. To what precision must the potential be measured for the uncertainty in $\mathrm{pH}$ to be $\pm 0.02 \mathrm{pH}$ unit?

Key Concepts

Uncertainty Propagation

Uncertainty propagation refers to the process of determining how errors in measurement, such as a potential reading, affect the calculated results like pH and hydrogen ion concentration. This concept is essential for understanding and quantifying the reliability and limitations of experimental data by translating a known measurement error into uncertainties in derived quantities.

Measurement Precision

Measurement precision in the context of electrochemical sensors is about determining the required accuracy of potential measurement to ensure that resulting quantities, such as pH, meet specific uncertainty criteria. This involves understanding the relationship between the sensitivity of the sensor, as defined by the Nernst equation, and the acceptable limits of error in the measured parameters.

Nernst Equation

The Nernst equation is a fundamental relationship in electrochemistry that relates the potential of an electrode to the concentration of reactive species, temperature, and fundamental constants. It is particularly important in the context of pH measurement because it explains how variations in hydrogen ion concentration can be directly translated into changes in electrode potential.

pH Measurement

pH measurement using a glass electrode leverages the sensitivity of a specialized membrane to hydrogen ion concentrations. By measuring the potential difference between a glass electrode and a reference electrode and applying the Nernst equation, one can accurately determine the pH of a solution, which is critical in many chemical and biological applications.

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