Question

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 \(\mathscr{E}_{\text { meas }}=0.480 \mathrm{V}\) a. What is the uncertainty in the values of \(\mathrm{pH}\) and \(\left[\mathrm{H}^{+}\right]\) if the nncertainty in the measured potential is \(+1 \mathrm{mV}\) \(( \pm 0.001 \mathrm{V}) ?\) 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

For part a), the uncertainty in pH is approximately \(±0.02\) and the uncertainty in H+ concentration is approximately \(±1.50 \times 10^{-5} M\). For part b), the potential must be measured with a precision of approximately \(±0.00118 V\) for the uncertainty in pH to be \(±0.02\) pH unit.

Step-by-step solution

Find pH value using the given equation

First, we need to find the value of pH using given values of \(\mathscr{E}_{meas}\) and \(\mathscr{E}_{ref}\). Using the given equation, we can find the pH value as follows: \[ \mathrm{pH} = \frac{\mathscr{E}_{meas} - \mathscr{E}_{ref}}{0.05916} \]

Calculate the pH using given values

Now, let's plug the given values of \(\mathscr{E}_{meas}\) and \(\mathscr{E}_{ref}\) into the equation: \[ \mathrm{pH} = \frac{0.480 V - 0.250 V}{0.05916} \approx 3.89 \]

Calculate the uncertainty in pH

The uncertainty in pH can be calculated by taking the derivative of the equation for pH with respect to the measured potential and multiplying it by the uncertainty in the measured potential: \[ \Delta \mathrm{pH} = \frac{d(\mathrm{pH})}{d(\mathscr{E}_{meas})} \times \Delta \mathscr{E}_{meas} \] Since \(\frac{d(\mathrm{pH})}{d(\mathscr{E}_{meas})} = \frac{1}{0.05916}\), we can plug in the given uncertainty of the measured potential: \[ \Delta \mathrm{pH} = \frac{1}{0.05916} \times 0.001 V \approx 0.02 \]

Calculate the uncertainty in H+ concentration

To find the uncertainty in H+ concentration, first, use the relationship between pH and H+ concentration: \[ [\mathrm{H}^{+}] = 10^{-\mathrm{pH}} \] Now, we can find the uncertainty in H+ concentration by taking the derivative of this equation with respect to pH and multiplying it by the uncertainty in pH: \[ \Delta [\mathrm{H}^{+}] = \frac{d([\mathrm{H}^{+}])}{d(\mathrm{pH})} \times \Delta \mathrm{pH} \] Since \(\frac{d([\mathrm{H}^{+}])}{d(\mathrm{pH})} = -10^{-\mathrm{pH}} \mathrm{ln}(10)\), we can plug in the calculated pH and uncertainty in pH: \[ \Delta [\mathrm{H}^{+}] = -10^{-3.89} \times 2.3026 \times 0.02 \approx 1.50 \times 10^{-5} \mathrm{M} \]

Find precision required for given uncertainty in pH (part b)

Now in part b, we need to determine the required precision for an uncertainty of \(±0.02\) pH unit. Using the relationship we found in step 3: \[ \Delta \mathrm{pH} = \frac{1}{0.05916} \times \Delta \mathscr{E}_{meas} \] Isolate the uncertainty in the measured potential and plug in the given uncertainty in pH: \[ \Delta \mathscr{E}_{meas} = 0.05916 \times 0.02 \approx 0.00118 V \] #Conclusion#: For part a), the uncertainty in pH is approximately \(±0.02\) and the uncertainty in H+ concentration is approximately \(±1.50 \times 10^{-5} \mathrm{M}\). For part b), the potential must be measured with a precision of approximately \(±0.00118 V\) for the uncertainty in pH to be \(±0.02\) pH unit.

Key concepts

Nernst Equation

Understanding the Nernst equation is crucial for grasping how a pH meter functions. This fundamental scientific equation describes the relationship between ion concentration and the electric potential of an electrochemical cell. In simplified terms, the Nernst equation allows us to calculate the voltage generated by a cell due to ion exchange. It takes into account the temperatures and concentration differences of ions on either side of a membrane.

In the context of pH measurement with a glass electrode, the Nernst equation reflects how the potential difference is influenced by the concentration of hydrogen ions. The equation given for the pH meter at 25°C is:\[ \mathscr{E}_{\text{meas}} = \mathscr{E}_{\text{ref}} + 0.05916 \times \text{pH} \]

Here, \( \mathscr{E}_{\text{ref}} \) is a constant related to the reference electrode, while 0.05916 is a constant derived from the Nernst equation, representing the temperature dependence at standard conditions.

Glass Electrode

The glass electrode is a key component in pH measurement, used due to its sensitivity to hydrogen ion concentration. Unlike metal electrodes, a glass electrode relies on ion-exchange properties of glass. When immersed in a solution, the electrode interacts at the surface, allowing H+ ions to move across its membrane.

A typical glass electrode setup includes two key parts:

• A thin glass bulb housing the measuring electrode
• A reference electrode to stabilize readings

The thin glass bulb is selectively permeable, meaning it responds effectively to changes in hydrogen ion concentration. This permeability causes a potential difference, which is what the pH meter reads.

Because the surface of the glass electrode gets coated with ions, it is important to periodically recalibrate it using standard solutions. This ensures accurate measurements.

Uncertainty Analysis

Performing uncertainty analysis in pH measurement is essential to understand how precise the measurement is. Uncertainty quantifies how much the measured value of pH or hydrogen ion concentration could deviate from the actual value.

To analyze uncertainty in this context, the derivative of the pH equation with respect to the measured potential is used. This quantifies how sensitive pH is to changes in measured potential. The formula is given by:\[ \Delta \text{pH} = \frac{1}{0.05916} \, \times \, \Delta \mathscr{E}_{\text{meas}} \]

Calculating this helps in understanding how variations in voltage readings due to intrinsic or instrumental factors affect the pH value. For high degrees of accuracy, it's crucial to minimize these uncertainties as much as possible.

Hydrogen Ion Concentration

Hydrogen ion concentration is directly associated with the pH of a solution. Understanding its role is vital in chemistry as it determines a solution's acidity or basicity. pH is a measure of the hydrogen ion concentration; the lower the pH, the higher the concentration of \( \mathrm{H^+} \) ions.

The equation connecting pH and hydrogen ion concentration is:\[ [\mathrm{H}^+] = 10^{-\mathrm{pH}} \]

This equation implies that as the pH decreases by one unit, the hydrogen ion concentration increases tenfold. This logarithmic relationship is why small changes in pH can correspond to large changes in ion concentration.

Thus, accurately determining hydrogen ion concentration from pH is fundamental in fields like biochemistry, environmental science, and medicine, affecting decisions in enzymatic activity, water quality, and drug development.