Question

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?

Short answer

The uncertainty in pH is approximately ±0.017. The uncertainty in H+ concentration is between \(1.11 \times 10^{-4}\text{ M}\) and \(1.49 \times 10^{-4}\text{ M}\). To achieve an uncertainty of ±0.02 pH unit, the potential must be measured with a precision of approximately ±0.00118 V.

Step-by-step solution

Write the given equation

The Nernst equation relating the measured potential with pH is given by: \[\mathscr{C}_{\text{meas}} = \mathscr{E}_{\text{ref}} + 0.05916 \cdot \text{pH}\]

Rearrange the equation to find pH

Let's rearrange the equation to find the expression for pH: \[\text{pH} = \frac{\mathscr{C}_{\text{meas}} - \mathscr{E}_{\text{ref}}}{0.05916}\]

Insert the given values and find pH

Assume that \(\mathscr{E}_{\text {ref}}=0.250 \text{ V}\) and \(\mathscr{C}_{\text {meas }}=0.480 \text{ V}\). Plug in these values to find the pH value: \[\text{pH} = \frac{0.480 - 0.250}{0.05916} \approx 3.89\]

Calculate the uncertainty in pH

The uncertainty in the measured potential is given, \(\pm 0.001 \text{ V}\). Use the same rearranged equation to find the uncertainty in pH: \[\Delta \text{pH} = \frac{\Delta \mathscr{C}_{\text{meas}}}{0.05916} = \frac{0.001}{0.05916} \approx \pm0.017\]

Calculate the uncertainty in H+ concentration

The relationship between pH and H+ concentration is: \[\text{pH} = -\log_{10}[\text{H}^{+}]\] Now, to find the uncertainty in H+ concentration, first find the H+ concentration without uncertainty: \[[\text{H}^{+}] = 10^{-\text{pH}} = 10^{-3.89} \approx 1.29 \times 10^{-4}\text{ M}\] Then, find the upper and lower limits of H+ concentration using uncertainty in pH: \[[\text{H}^{+}]_{\text{min}} = 10^{-(3.89+0.017)} \approx 1.11 \times 10^{-4}\text{ M}\] \[[\text{H}^{+}]_{\text{max}} = 10^{-(3.89-0.017)} \approx 1.49 \times 10^{-4}\text{ M}\] So, the uncertainty in H+ concentration is between \(1.11 \times 10^{-4}\text{ M}\) and \(1.49 \times 10^{-4}\text{ M}\). #b. Precision required in potential for the uncertainty in pH to be ±0.02 pH unit#

Use the relationship between potential uncertainty and pH uncertainty

To find the required precision in the potential measurement to achieve the desired uncertainty in pH, use the relationship obtained in step 4: \[\Delta \text{pH} = \frac{\Delta \mathscr{C}_{\text{meas}}}{0.05916}\] Let the required uncertainty in the potential be \(\Delta \mathscr{C}_{\text{req}}\).

Solve for the required potential uncertainty

Now, we need to find the potential uncertainty to maintain the pH uncertainty as \(\pm0.02\). \[\Delta \text{pH} = 0.02 = \frac{\Delta \mathscr{C}_{\text{req}}}{0.05916}\] Now, solve for \(\Delta \mathscr{C}_{\text{req}}\): \[\Delta \mathscr{C}_{\text{req}} = 0.05916 \times 0.02 \approx 0.00118 \text{ V}\] So, the potential must be measured with a precision of approximately \(\pm 0.00118\) V for the uncertainty in pH to be maintained at \(\pm 0.02\) pH unit.

Key concepts

Nernst Equation

The Nernst equation plays a crucial role in understanding how the measurement of pH works with a glass electrode. Essentially, it allows us to relate the potential difference we measure in a pH meter with the pH value of the solution. At a temperature of 25°C, the formula for a general pH meter is expressed as follows:

• \(\mathscr{C}_{\text{meas}} = \mathscr{E}_{\text{ref}} + 0.05916 \cdot \text{pH}\)

Here, \(\mathscr{C}_{\text{meas}}\) is the measured cell potential, \(\mathscr{E}_{\text{ref}}\) refers to the reference electrode potential and other potentials that are not dependent on the hydrogen ion concentration, and \(\text{pH}\) is the acidity or alkalinity of the solution.
Through this equation, changes in the measured potential \(\mathscr{C}_{\text{meas}}\) are linked directly to changes in the hydrogen ion concentration, which affects the pH.

Glass Electrode

A glass electrode is a type of sensor that is widely used for measuring the pH of a solution. It consists of a thin glass membrane, which is sensitive to hydrogen ions, making it essential for accurate pH measurements. When placed in a solution, the glass electrode generates an electric potential proportionate to the hydrogen ion concentration in the solution.

• The interior of the glass electrode is filled with a buffer solution of known pH and ionic strength.
• The outer surface is in contact with the test solution.

Because the glass is selectively permeable to hydrogen ions, it allows the concentration of these ions to determine the potential difference across the membrane. This potential difference is then measured and interpreted by a pH meter, which uses the Nernst equation to relate the potential to pH. The glass electrode is key because it determines the sensitivity of the pH measurement process.

Potential Uncertainty

Measuring potential with a pH meter comes with certain challenges, particularly when it comes to the potential uncertainty. This refers to the possible variation in the potential measurement, which directly affects the accuracy of the calculated pH value. In practice, small fluctuations in the potential can cause changes in the pH value, which is why precision is of utmost importance.
If the uncertainty in the measured potential is \(\pm 0.001 \text{ V}\), this can lead to a pH uncertainty of approximately \(\pm 0.017\).
To ensure that the pH determination is reliable, it’s essential to control the potential measurement within a strict range. For example, if you want the pH uncertainty to be within \(\pm 0.02\), the potential measurement should be accurate to about \(\pm 0.00118 \text{ V}\).

• High precision instruments are required for minimal potential uncertainty.
• Calibration and maintenance of pH meters help minimize potential uncertainty.

Hydrogen Ion Concentration

The concentration of hydrogen ions, represented as \([\text{H}^+]\), is fundamental to determining the pH of a solution. The relationship between hydrogen ion concentration and pH is logarithmic, which means small changes in \([\text{H}^+]\) are amplified in the pH scale.

• The formula \(\text{pH} = -\log_{10}[\text{H}^+]\) showcases this relationship.
• pH is inversely proportional to \([\text{H}^+]\); the higher the concentration, the lower the pH.

In terms of uncertainty, if there is a variance in the measured pH, this will directly translate to an uncertainty in the hydrogen ion concentration. Knowing the exact \([\text{H}^+]\) is crucial in many scientific and industrial applications, such as chemistry labs, environmental monitoring, and pharmaceuticals. Precisely determining this concentration can help ensure consistent and reliable results.