In a galvanic cell, the cathode is an Ag+ | 1.00 M | Ag(s) half-cell. The anode is a standard hydrogen electrode immersed in a buffer solution containing 0.10 M benzoic acid (C6H5COOH) and 0.050 M sodium benzoate (C6H5COO-Na+). The measured cell voltage is 1.030 V. What is the pKa of benzoic acid?

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Identify the half-reactions for the galvanic cell. The cathode reaction is Ag+ + e- -> Ag(s), and the anode reaction is the standard hydrogen electrode (SHE), which is 2H+ + 2e- -> H2(g).

Use the Nernst equation to relate the cell potential to the concentrations of the reactants and products. The Nernst equation is E_cell = E°_cell - (RT/nF) * ln(Q), where Q is the reaction quotient.

Calculate the standard cell potential, E°_cell, using standard reduction potentials: E°_cell = E°_cathode - E°_anode. For Ag+/Ag, E°_cathode = +0.80 V, and for SHE, E°_anode = 0.00 V.

Determine the reaction quotient, Q, for the cell. Since the anode involves a buffer solution, use the Henderson-Hasselbalch equation: pH = pKa + log([A-]/[HA]), where [A-] is the concentration of the benzoate ion and [HA] is the concentration of benzoic acid.

Rearrange the Nernst equation to solve for pKa. Substitute the known values (E_cell, E°_cell, concentrations) into the Nernst equation and the Henderson-Hasselbalch equation to find pKa.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Galvanic Cell

A galvanic cell is an electrochemical cell that converts chemical energy into electrical energy through spontaneous redox reactions. It consists of two half-cells: an anode where oxidation occurs and a cathode where reduction takes place. The flow of electrons from the anode to the cathode generates an electric current, and the cell voltage can be measured to determine the driving force of the reaction.

Nernst Equation

The Nernst equation relates the cell potential to the concentrations of the reactants and products in a redox reaction. It is expressed as E = E° - (RT/nF) ln(Q), where E is the cell potential, E° is the standard cell potential, R is the gas constant, T is the temperature in Kelvin, n is the number of moles of electrons transferred, F is Faraday's constant, and Q is the reaction quotient. This equation is crucial for calculating the effect of concentration on cell voltage.

pKa and Buffer Solutions

The pKa is a measure of the strength of an acid in solution, defined as the negative logarithm of its acid dissociation constant (Ka). In buffer solutions, which resist changes in pH, the pKa indicates the pH at which the concentrations of the acid and its conjugate base are equal. For benzoic acid and sodium benzoate, the pKa can be determined using the Henderson-Hasselbalch equation, which relates pH, pKa, and the ratio of the concentrations of the acid and its conjugate base.

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