Problem 113 - 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 \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}\right)\) 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 \(\mathrm{p} K_{a}\) of benzoic acid?

Step-by-Step Solution

Answer

The pKa of benzoic acid is 4.16.

Step 1: Write down the half-cell reactions and the Nernst equation

The two half-cell reactions for this galvanic cell are: Cathode: \(Ag^{+}(aq) + e^{-} \rightarrow Ag(s)\) (Reduction) Anode: \(2H^{+}(aq) + 2e^{-} \rightarrow H_{2}(g)\) (Oxidation) The overall cell reaction is: \(2Ag^{+}(aq) + H_{2}(g) \rightarrow 2Ag(s) + 2H^{+}(aq)\) The Nernst equation relates the cell potential, concentrations of the reactants and products, and the standard cell potential. The equation is: \(E_{cell}=E^{\circ}_{cell} - \frac{RT}{nF} \ln Q\) Where: - \(E_{cell}\) is the measured cell potential, - \(E^{\circ}_{cell}\) is the standard cell potential, - \(R\) is the gas constant (8.314 J/mol K), - \(T\) is the temperature (in Kelvin), - \(n\) is the number of electrons transferred in the overall cell reaction, - \(F\) is the Faraday's constant (96,485 C/mol), and - \(Q\) is the reaction quotient.

Step 2: Calculate the standard cell potential

The standard cell potential, \(E^{\circ}_{cell}\), for this galvanic cell is the difference between the cathode reduction potential and the anode reduction potential. We know that: \(E^{\circ}_{Ag^{+}/Ag} = 0.799 V\) \(E^{\circ}_{H^{+}/H_{2}} = 0 V\) Therefore, the standard cell potential is: \(E^{\circ}_{cell} = E^{\circ}_{Ag^{+}/Ag} - E^{\circ}_{H^{+}/H_{2}} = 0.799 V - 0 V = 0.799 V\)

Step 3: Calculate the reaction quotient

The reaction quotient, \(Q\), for this galvanic cell is the ratio of the product of the concentrations of the products and the product of the concentrations of the reactants, each raised to their stoichiometric coefficients. Since we are given the concentrations of benzoic acid (0.10 M) and sodium benzoate (0.050 M), we can write the reaction quotient as: \(Q = \frac{[H^{+}]^{2}}{[Ag^{+}]^{2}}\)

Step 4: Calculate the concentration of hydrogen ions

We can now solve the Nernst equation for the concentration of hydrogen ions, [H+], by substituting the given values and solving for [H+]. 1.030 V = 0.799 V - (2.303 RT/nF) log(\(Q\)) We know that n = 2, T = 298 K, R = 8.314 J/mol K, and F = 96485 C/mol. Therefore, 1.030 V = 0.799 V - [2.303 (8.314 J/mol K)(298 K)]/[(2)(96485 C/mol)] log(\([H^{+}]^{2}/[Ag^{+}]^{2}\)) From this, we find that: [H+] = 1.14 × 10^-5 M

Step 5: Calculate the pKa of benzoic acid

Now that we have the concentration of hydrogen ions, we can use the Henderson-Hasselbalch equation to calculate the pKa of benzoic acid: \(pH = pK_{a} + \log \frac{[A^{-}]}{[HA]}\) We know that pH = -log([H+]), [A-] is the concentration of sodium benzoate (0.050 M), and [HA] is the concentration of benzoic acid (0.10 M). Therefore, pH = -log(1.14 × 10^-5) = 4.94 Substituting the values into the Henderson-Hasselbalch equation: 4.94 = pKa + log(\(0.050/0.10\)) From this, we find that: pKa = 4.94 + log(0.5) = 4.16 Hence, the pKa of benzoic acid is 4.16.

Key Concepts

Nernst equation

The Nernst equation is a powerful tool used to determine the cell potential of an electrochemical cell under non-standard conditions. It allows us to account for changes in concentrations or partial pressures of the reactants and products involved in a chemical reaction. This equation is central in studying galvanic cells like the one presented in the original exercise.

The standard form of the Nernst equation is: \[ E_{cell} = E^{\circ}_{cell} - \frac{RT}{nF} \ln Q \]Here, \( E_{cell} \) is the actual cell potential we measure, and \( E^{\circ}_{cell} \) is the standard cell potential, which is the potential under standard conditions where all solutes are 1 M and gases are at 1 atm.

• \( R \) is the universal gas constant \((8.314 \text{ J/mol K})\),
• \( T \) is the temperature in Kelvin,
• \( n \) is the number of electrons transferred in the reaction, and
• \( F \) is the Faraday constant \((96485 \text{ C/mol})\).

\( Q \) is the reaction quotient, calculated as the ratio of the concentrations of products to reactants, each raised to their stoichiometric coefficients. Through the Nernst equation, we can understand how the cell potential varies with the concentration of different species in the cell. In this exercise, the Nernst equation helps us find the hydrogen ion concentration, which is crucial for calculating the pKa of benzoic acid.

standard cell potential

The standard cell potential, denoted as \( E^{\circ}_{cell} \), is a measure of the potential difference between two half-cells of a galvanic cell under standard conditions. This potential is crucial for predicting the spontaneity of a redox reaction.

In the exercise, we calculate the standard cell potential by using the standard reduction potentials of the respective half-cells involved in the galvanic cell: \( Ag^{+}/Ag \) for the cathode and \( H^{+}/H_{2} \) for the anode.

The difference between these potentials gives us the standard cell potential: \[ E^{\circ}_{cell} = E^{\circ}_{Ag^{+}/Ag} - E^{\circ}_{H^{+}/H_{2}} \] Plugging in the given values: \[ E^{\circ}_{cell} = 0.799 \text{ V} - 0 \text{ V} = 0.799 \text{ V} \]A positive \( E^{\circ}_{cell} \) signifies that the cell reaction is spontaneous under standard conditions, meaning the cell can produce electricity from chemical energy. Understanding this concept is essential for analyzing the overall efficiency and potential output of galvanic cells.

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a useful formula in chemistry for estimating the pH of a buffer solution. It relates the pH to the acidity constant (pKa) and the concentrations of the acidic and basic components of the buffer. This is especially helpful in calculating the pKa of weak acids like benzoic acid.

The equation is written as: \[ pH = pK_{a} + \log \frac{[A^{-}]}{[HA]} \]where:

• \( pH \) is the measure of acidity of the solution,
• \( [A^{-}] \) is the concentration of the conjugate base (in this case, sodium benzoate), and
• \( [HA] \) is the concentration of the weak acid (benzoic acid).

In the provided exercise, once we calculate the pH using the formula \( pH = -\log([H^+]) \), we substitute all the values into the Henderson-Hasselbalch equation to solve for \( pKa \). This equation elegantly ties together the composition of the buffer with its acid-base properties, providing a simplified approach to finding unknown variables in buffer solutions.