Problem 103 - Chemistry Principles and Practice

Question

The acid-base titration curves discussed in Chapter 16 can be determined using a \(\mathrm{pH}\) meter, which measures the potential of a cell made up of a reference electrode and an indicating electrode that responds to the hydrogen ion concentration in solution. Assume that the cell potential, in volts, follows the equation \(E_{\text {cell }}=k-0.059 \mathrm{pH}\) The potential of the cell is \(135 \mathrm{mV}\) at the start of a titration of a \(0.032 \mathrm{MHCl}\) solution with \(\mathrm{NaOH}(\mathrm{aq}) . \mathrm{What}\) is the potential of this cell when the equivalence point is reached?

Step-by-Step Solution

Answer

The cell potential at the equivalence point is -0.278 volts.

Step 1: Understand the Given Information

We are given the equation \(E_{\text{cell}} = k - 0.059 \mathrm{pH}\) for the cell potential in volts. At the start, the potential \(E_{\text{cell}}\) is 135 mV, which is equivalent to 0.135 volts. We know we're dealing with a \(0.032 \text{M HCl}\) solution being titrated with \(\text{NaOH}\). Our task is to find the potential at the equivalence point.

Step 2: Identify Conditions at Equivalence Point

At the equivalence point in a titration of a strong acid with a strong base, the solution consists of mostly water and salt; the \(\mathrm{pH}\) should be neutral, or \(\mathrm{pH} = 7.\) Thus, substituting \(\mathrm{pH} = 7\) into the equation will give the cell potential at this point.

Step 3: Substitute pH into the Equation

Replace \(\mathrm{pH}\) with 7 in the given equation: \[E_{\text{cell}} = k - 0.059 \times 7.\]

Step 4: Simplify the Equation

Calculate the resulting expression:\[E_{\text{cell}} = k - 0.413.\]

Step 5: Solve for the New Cell Potential

Since we need the value of \(k\) to find the potential at \(\mathrm{pH}=7\), use the initial condition. At the start, the cell potential is 135 mV (0.135 volts). Assuming this is when \(\mathrm{pH}=0\):\[0.135 = k - 0.059 \times 0\]\[k = 0.135\] Thus, at equivalence:\[E_{\text{cell}} = 0.135 - 0.413 = -0.278\] volts.

Key Concepts

pH Meter

A pH meter is an essential tool used to measure the acidity or basicity of a solution. It functions by measuring the hydrogen ion concentration in that solution.
This is done through the use of two electrodes:

• A reference electrode, which provides a constant voltage against which measurements are compared.
• An indicating electrode that is sensitive to hydrogen ion activity and varies its voltage in relation to the solution's pH.

These electrodes create a complete electric circuit in the solution, with the resulting potential difference being recorded as the pH measurement.
By interpreting this electrical signal, the pH meter can calculate the pH value based on the cell potential equation, given as:\[E_{\text{cell}} = k - 0.059 \ \text{pH}\]Here, \(k\) is a constant that needs to be determined experimentally, involving the initial conditions of your setup.
This equation implies that the potential recorded by the pH meter changes linearly with pH, allowing us to predict changes in acidity or alkalinity as reactions progress during titrations.

Equivalence Point

The equivalence point in a titration is a critical moment, representing the exact point at which the amount of titrant equals the quantity of the substance in the solution being titrated.
This specifically means that all the acid has reacted with base, or vice versa, forming products typically comprising a neutral salt and water.At this point in a strong acid-strong base titration, as in the problem discussed, you will find:

• The solution's pH will be near neutral, often around pH 7.
• The solution will primarily consist of water and salt, with minimal amounts of excess reactant remaining.

In the exercise, by substituting \(\text{pH} = 7\) into the cell potential equation, we determine what the electric potential must be at this specified moment in the reaction.
This step is crucial because it ensures that chemical stoichiometry is balanced, and it allows for the accurate measurement of pH changes using the initial cell potential as a reference.

Cell Potential

Cell potential, a crucial concept in electrochemistry, is the measure of the voltage between two electrodes in an electrochemical cell.
This potential arises from the natural tendency of electrons to flow from areas of high to low concentration, corresponding to the differing reactivity of the substances.In our context:

• The cell potential equation \(E_{\text{cell}} = k - 0.059 \cdot \text{pH}\) describes the relationship between pH and voltage in a titration process.
• Initially, when you start with a known acid-base concentration, the cell potential can be measured (0.135 volts in this scenario).
• As the reaction proceeds, changes in pH cause the potential difference recorded by the electrodes to change, following the given equation.

By understanding and using the formula, one can determine the change in potential during the course of titration.
This change in potential can help identify the equivalence point, which we calculated to be -0.278 volts in this titration, helping visualize how reactions proceed to completion.