The measurement of \(\mathrm{pH}\) 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 $$ \mathscr{E}_{\text {meas }}=\mathscr{E}_{\text {ref }}+0.05916 \mathrm{pH} $$ 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 \(\mathscr{E}_{\mathrm{ref}}=0.250 \mathrm{~V}\) and that \(\mathscr{B}_{\text {meas }}=0.480 \mathrm{~V}\). 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?

To calculate the uncertainty in pH, we'll first need to find the actual pH value using the provided equation. Given: \(\mathscr{E}_{meas} = 0.480 V\) \(\mathscr{E}_{ref} = 0.250 V\) Equation: \(\mathscr{E}_{meas} = \mathscr{E}_{ref} + 0.05916 \cdot \text{pH}\) First, find the actual pH value by rearranging the equation and plugging in the given values: \[ \text{pH} = \frac{\mathscr{E}_{meas} - \mathscr{E}_{ref}}{0.05916} \] \[ \text{pH} = \frac{0.480 - 0.250}{0.05916} \approx 3.89 \] Now, we will calculate the uncertainty in pH due to the uncertainty in the measured potential. The uncertainty in measured potential is ±0.001 V. Using the same equation, we find the pH value of both the upper limit (0.481 V) and lower limit (0.479 V) of the measured potential: Upper limit: \[ \text{pH} = \frac{0.481 - 0.250}{0.05916} \approx 3.90 \] Lower limit: \[ \text{pH} = \frac{0.479 - 0.250}{0.05916} \approx 3.88 \] The uncertainty in the pH value is ±0.01. Next, we calculate the uncertainty in the hydrogen ion concentration, [H⁺]. To find [H⁺], we can use the equation: \[ [\text{H}^{+}] = 10^{-\text{pH}} \] Using the pH values we found earlier, we will calculate the [H⁺] values at the upper and lower limits: Upper limit: \[ [\text{H}^{+}] = 10^{-3.88} \approx 1.32 \times 10^{-4} \] Lower limit: \[ [\text{H}^{+}] = 10^{-3.90} \approx 1.26 \times 10^{-4} \] The uncertainty in [H⁺] is approximately ± 0.03 x 10⁻⁴.

To determine the necessary precision in potential measurement to achieve the desired pH uncertainty, we need to manipulate the Nernst equation and use the desired pH uncertainty value. The desired pH uncertainty is ±0.02 pH unit. Revisiting the Nernst equation, we can rearrange it to find the required uncertainty in potential measurement: \[ \Delta \mathscr{E}_{meas} = 0.05916 \Delta \text{pH} \] Plugging in the desired pH uncertainty value: \[ \Delta \mathscr{E}_{meas} = 0.05916 \cdot 0.02 \] \[ \Delta \mathscr{E}_{meas} \approx 0.001183 \, \text{V} \] Or: \[ \Delta \mathscr{E}_{meas} \approx 1.183 \, \text{mV} \] So to have an uncertainty of ±0.02 pH unit, the potential must be measured with a precision of about ±1.183 mV.