Question

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

Short answer

The pKa of benzoic acid is approximately 4.2.

Step-by-step solution

Identify the half-reactions

The half-reactions of the given galvanic cell are: \[\textrm{Cathode: } Ag^+ + e^- \rightarrow Ag(s)\] \[\textrm{Anode: } 2H^+ + 2e^- \rightarrow H_2(g)\]

Write the overall cell reaction

Combine the half-reactions to get the overall cell reaction: \[Ag^+ + H_2(g) \rightarrow Ag(s) + 2H^+\]

Find the standard cell potential

Standard cell potential, \(E^0_{cell}\), can be found by using the known standard reduction potentials: \[-E^0(H^+/H_2) = -0.000\mathrm{V}\] \[E^0(Ag^+/Ag) = 0.800\mathrm{V}\] Adding these two half-cell potentials, we get: \[E^0_{cell} = E^0(Ag^+/Ag) - E^0(H^+/H_2) = 0.800\mathrm{V} - (-0.000\mathrm{V}) = 0.800\mathrm{V}\]

Application of the Nernst equation

The Nernst equation relates cell potential (\(E_{cell}\)), standard cell potential (\(E^0_{cell}\)), and concentration of species involved in the reaction at equilibrium: \[E_{cell} = E^0_{cell} - \frac{RT}{nF} \ln{Q}\] In this galvanic cell, n=2 moles of electrons are transferred in the balanced cell reaction, and we are given the cell voltage: \(E_{cell} = 1.030\mathrm{V}\). Now we need to calculate the reaction quotient, Q, using the concentration of benzoic acid (HA) and sodium benzoate (A^-) given.

Calculate Q

The reaction quotient, Q, for the buffer solution containing benzoic acid and sodium benzoate is given by: \[Q = \frac{[H^+]^2}{[Ag^+][P_{H_2}]}\] Since we are not given the partial pressure of \(H_2\), we can assume it to be 1 atm. From the definition of Ka: \[[H^+]^2 = K_{a} [HA] [A^-]\] Subtitute the values of [HA] and [A^-]: \[Q = \frac{K_{a} (0.10) (0.050)}{(1.00)(1)}\]

Solve the Nernst equation for Ka

Substitute the values in the Nernst equation and solve for Ka: \[1.030\mathrm{V} = 0.800\mathrm{V} - \frac{(8.314 \mathrm{J/mol\cdot K})(298\mathrm{K})}{(2)(96485 \mathrm{C/mol})} \ln{\left(\frac{K_a(0.10)(0.050)}{1}\right)}\] Rearrange and solve for Ka: \[K_a = 6.30 \times 10^{-5}\]

Calculate pKa

Now, we can find the pKa of benzoic acid as follows: \[pK_a = -\log {K_a}\] \[pK_a = -\log (6.30 \times 10^{-5}) = 4.2\] So, the pKa of benzoic acid is approximately 4.2.

Key concepts

Standard Hydrogen Electrode

The Standard Hydrogen Electrode (SHE) is a reference electrode frequently used in electrochemical measurements. It consists of a platinum electrode coated with platinum black, in contact with a solution of 1 M hydrogen ion (H⁺) and bathed in hydrogen gas at 1 atmosphere pressure and 25°C (298K). The SHE serves as a benchmark for measuring the standard reduction potentials of other half-cells, assigning it a potential of exactly 0.000 V. Understanding the SHE is vital when discussing standard cell potentials, as it provides a comparison point for determining the tendencies of other electrodes to gain or lose electrons.
In the given galvanic cell problem, the anode is a SHE, providing a necessary baseline for the calculations required to determine the unknown pKa value of benzoic acid. By convention, the potentials of all other half-cells are measured relative to this fundamental standard.

Nernst Equation

The Nernst equation is a fundamental relation used in electrochemistry to calculate the electrical potential of a cell under any conditions, not just standard conditions. It accounts for temperature, the number of moles of electrons transferred in the redox reaction (n), the universal gas constant (R), the Faraday constant (F), and the reaction quotient (Q). The equation is given by:
\[E_{cell} = E^0_{cell} - \frac{RT}{nF} \ln{Q}\]
Where, \(E_{cell}\) is the cell potential under non-standard conditions, \(E^0_{cell}\) is the standard cell potential, and Q is the reaction quotient representing the ratio of the activities or concentrations of the chemical species involved in the reaction. In the textbook solution, the Nernst equation plays a crucial role in determining the pKa value of benzoic acid based on the measured cell voltage.

Reaction Quotient

The reaction quotient, Q, is used in the context of the Nernst equation to express the ratio of the concentrations of products to reactants for a reaction at any given moment, regardless of whether or not equilibrium has been reached. The equilibrium constant (K) and Q share the same expression, but while K is calculated at equilibrium, Q can be used for any set of concentration conditions.
Typically, Q is written as:
\[Q = \frac{\text{products}}{\text{reactants}}\]
For redox reactions occurring in a galvanic cell, Q will often have the form:
\[Q = \frac{[\text{oxidized species}]}{[\text{reduced species}]}\]
In the exercise provided, Q is calculated for the buffer solution of benzoic acid and sodium benzoate in the galvanic cell. This calculation is fundamental for solving the Nernst equation and ultimately finding the desired pKa value.

pKa Calculation

The pKa value of an acid is a measure of its acidity, defined as the negative base-10 logarithm of its acid dissociation constant (Ka). The lower the pKa value, the stronger the acid because it means the acidic proton is released more easily. The connection between Ka and pKa is expressed as:
\[pK_a = -\log(K_a)\]
pKa is valuable in understanding acid-base properties of molecules in various chemical and biological contexts. By converting the Ka value to a logarithmic scale, pKa provides a convenient and interpretable value. In our exercise, the pKa calculation is the final step that requires the prior determination of Ka via the Nernst equation and the given cell voltage.

Standard Reduction Potential

The standard reduction potential, often denoted by \(E^0\), represents the tendency of a chemical species to gain electrons and thereby be reduced. It is measured under standard conditions (1M concentration for solutions, 1 atm pressure for gases, and 25°C temperature). The standard reduction potential is critical in determining the direction and spontaneity of redox reactions.
Every half-cell reaction has a characteristic standard reduction potential, and by comparing the potentials for the anode and cathode, one can calculate the cell's standard potential \(E^0_{cell}\) as follows:
\[E^0_{cell} = E^0_{\text{cathode}} - E^0_{\text{anode}}\]
In the exercise, we calculated the standard cell potential of the galvanic cell by using the known standard reduction potentials for both the silver and hydrogen electrodes. The standard reduction potential is an integral component that leads to the completion of the Nernst equation and the determination of the pKa of benzoic acid.