Problem 112 - Chemistry: The Central Science

Question

In a galvanic cell the cathode is an \(\mathrm{Ag}^{+}(1.00 \mathrm{M}) / \mathrm{Ag}^{(s)}\) halfcell. The anode is a standard hydrogen electrode immersed in a buffer solution containing \(0.10 \mathrm{M}\) benzoic acid ( \(\left.\mathrm{C}_{6} \mathrm{H}_{3} \mathrm{COOH}\right)\) and \(0.050 \mathrm{M}\) sodium benzoate \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COON} \mathrm{Na}^{+}\right)\). The measured cell voltage is \(1.030 \mathrm{~V}\). What is the \(\mathrm{p} K_{4}\) of benzoic acid?

Step-by-Step Solution

Answer

The pKa of benzoic acid is 4.61, which we found by using the given cell voltage (1.030 V) in the Nernst equation to find the cell potential (E = 0.235 V), and then using the Henderson-Hasselbalch equation with the given concentrations of benzoic acid (0.10 M) and its conjugate base (0.050 M).

Step 1: Write the Nernst Equation

The Nernst equation relates the cell potential, E°, and the reaction quotient, Q, as follows: \[E = E° - \frac{RT}{nF} \ln Q\] Here, E - cell potential E° - standard cell potential R - gas constant (8.314 J/(mol·K)) T - temperature (25°C or 298 K) n - number of electrons transferred in the redox reaction F - Faraday's constant (96485 C/mol) Q - reaction quotient

Step 2: Find the Standard Cell Potential

The standard cell potential, E°, can be found from the given cell voltage (1.030 V) and the hydrogen electrode potential (0 V): \[E°_{Ag^+/Ag} = E_{cell} - E°_{H^+/H_2} = 1.030 \mathrm{V} - 0 \mathrm{V} = 1.030 \mathrm{V}\]

Step 3: Identify the reaction and write the balanced equation

The overall reaction in the galvanic cell is as follows: \[2H^+ (aq) + 2e^- + Ag^+ (aq) \rightarrow H_2 (g) + Ag (s)\] The number of electrons transferred, n, in this redox reaction is 2.

Step 4: Write the simplified Nernst equation

First, we express the reaction quotient, Q, in terms of concentration of benzoic acid and its conjugate base: \[Q = \frac{[Ag^+][H^+]^2}{[H^+]^2} = [Ag^+]\] Then, we can rewrite the Nernst equation: \[E = E° - \frac{RT}{nF} \ln Q = E° - \frac{RT}{nF} \ln [Ag^+]\] Since we already know E, E°, n, and [Ag^+], we can now solve for the cell potential, E.

Step 5: Solve the Nernst equation for the cell potential

Plug the known values into the Nernst equation and solve for E: \[E = 1.030 \mathrm{V} - \frac{(8.314 \mathrm{J/(mol·K)}) (298 \mathrm{K})}{(2)(96485 \mathrm{C/mol})} \ln (1.00 \mathrm{M})\] E = 0.235 V

Step 6: Use the Henderson-Hasselbalch equation to find the pKa of benzoic acid

The Henderson-Hasselbalch equation is as follows: \[\mathrm{pH} = \mathrm{p}K_a + \log \frac{[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}^-]}{[\mathrm{C}_{6} \mathrm{H}_{3} \mathrm{COOH}]}\] We already know the concentrations of benzoic acid (0.10 M) and its conjugate base (0.050 M) in the buffer solution. The cell potential E = RT/nF, so to find the pH, we can solve for [H+] concentration: \[\mathrm{pH} = -\log \left[\frac{RT}{(2F)E} \right] = -\log \left[\frac{(8.314 \mathrm{J/(mol·K)}) (298\mathrm{K})}{(2)(96485 \mathrm{C/mol})(0.235 \mathrm{V})}\right] = 3.31\] Now, we can find the pKa: \[\mathrm{p}K_a = \mathrm{pH} - \log \frac{[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}^-]}{[\mathrm{C}_{6} \mathrm{H}_{3} \mathrm{COOH}]} = 3.31 - \log \frac{0.050}{0.10} = 3.31 + 0.301 = 4.61\] Therefore, the pKa of benzoic acid is 4.61.

Key Concepts

Electrochemistry

Electrochemistry is the branch of chemistry that explores the relationship between electrical energy and chemical changes. It primarily focuses on reactions where oxidation and reduction take place, known as redox reactions. In electrochemistry, these reactions occur with the transfer of electrons. A typical way to study these reactions is through galvanic cells or electrolytic cells. Electrochemistry plays a vital role in various applications including batteries, electroplating, and electrolysis.

• Oxidation: This involves the loss of electrons from a substance.
• Reduction: This involves the gain of electrons by a substance.

A crucial aspect of electrochemistry is understanding how these electron transfers can generate electric currents, which is a fundamental concept in designing devices like batteries.
The Nernst Equation is a central concept in electrochemistry. It helps calculate the potential of an electrochemical cell under any conditions. Understanding this equation is crucial for studying how different concentrations affect cell potential.

Galvanic Cells

A Galvanic cell, also known as a voltaic cell, is a device that converts chemical energy into electrical energy through redox reactions. It consists of two different metals connected by an external circuit, with electrolytes facilitating the movement of ions.
Each metal, submerged in its respective electrolyte, serves as an electrode. One acts as the anode, where oxidation occurs, and the other as the cathode, where reduction happens.

• Anode: Site where oxidation takes place, generating electrons.
• Cathode: Site where reduction takes place, accepting electrons.

In galvanic cells, electrons flow from the anode to the cathode through an external circuit, while ions travel through a salt bridge within the solution to maintain charge neutrality.
Using the Nernst equation, it's possible to calculate the cell's potential under non-standard conditions. This can be essential in determining the efficiency and feasibility of a given cell reaction. The relationship between the concentration of ions involved in the reactions and the cell voltage can provide insights into optimizing the cell's design and function.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch Equation is a simple formula used to relate the pH of a solution to the pKa (acid dissociation constant) and the concentration ratio of the conjugate base to the acid. It's often used in buffer solution calculations, which are crucial in maintaining a stable pH in chemical and biological systems.
The equation is given by:\[\text{pH} = \text{p}K_a + \log \frac{[\text{A}^-]}{[\text{HA}]}\]

• [A⁻]: Concentration of the conjugate base.
• [HA]: Concentration of the acid.

By applying the Henderson-Hasselbalch equation, one can determine the pKa if the pH and the concentrations of the acid and base are known. This is particularly useful when dealing with buffer solutions, which in practice helps maintain the pH during reactions.
In electrochemistry exercises, such as those involving galvanic cells, the equation aids in the interrelation of chemical equilibria and electrochemical reactions, tying together the chemistry of acids and bases with redox systems. This is evident in the use of this equation to deduce the pKa of benzoic acid from the pH, facilitating a deeper understanding of the behavior of acids in electrochemical contexts.