Problem 67 - Chemistry The Central Science In Si Units

Question

A voltaic cell utilizes the following reaction: \(4 \mathrm{Fe}^{2+}(a q)+\mathrm{O}_{2}(g)+4 \mathrm{H}^{+}(a q) \longrightarrow 4 \mathrm{Fe}^{3+}(a q)+2 \mathrm{H}_{2} \mathrm{O}(l)\) (a) What is the emf of this cell under standard conditions? (b) What is the emf of this cell when \(\left[\mathrm{Fe}^{2+}\right]=1.3 \mathrm{M},\left[\mathrm{Fe}^{3+}\right]=\) \(0.010 M, P_{\mathrm{O}_{2}}=50.7 \mathrm{kPa}\), and the \(\mathrm{pH}\) of the solution in the cathode half-cell is \(3.50 ?\)

Step-by-Step Solution

Answer

(a) 0.46 V under standard conditions. (b) Requires Nernst equation to compute based on provided concentrations.

Step 1: Identify Standard Electrode Potentials

First, identify the standard reduction potentials for the half-reactions involved. For the transformation \( \mathrm{Fe}^{2+} \rightarrow \mathrm{Fe}^{3+} + e^- \), the standard reduction potential \( E^0 \) is \( +0.77 \mathrm{~V} \). For the half-reaction \( \mathrm{O}_2 + 4 \mathrm{H}^+ + 4e^- \rightarrow 2 \mathrm{H}_2\mathrm{O} \), \( E^0 \) is \( +1.23 \mathrm{~V} \).

Step 2: Calculate Standard EMF

Using the reduction potentials, calculate the standard electromotive force (emf) of the cell. Use the formula \( E^0_{\mathrm{cell}} = E^0_{\text{cathode}} - E^0_{\text{anode}} \). Here, the cathode is the oxygen reaction, and the anode is the iron reaction. Thus, \( E^0_{\mathrm{cell}} = 1.23 \mathrm{~V} - 0.77 \mathrm{~V} = 0.46 \mathrm{~V} \).

Step 3: Use Nernst Equation for Non-Standard Conditions

Apply the Nernst equation to calculate the emf under non-standard conditions, \( E = E^0 - \frac{RT}{nF} \ln Q \). Here, \( n = 4 \), \( Q = \frac{[\mathrm{Fe}^{3+}]^4}{[\mathrm{Fe}^{2+}]^4 [\mathrm{O}_2] [\mathrm{H}^+]^4} \). Replace the concentrations and pressure (convert pressure to bar: \( 50.7 \mathrm{kPa} = 0.507 \mathrm{~bar} = 0.5 \mathrm{~atm} \), and \( [\mathrm{H}^+] = 10^{-3.5} \)).

Step 4: Calculate Reaction Quotient

Calculate \( Q \) using the given concentrations: \( Q = \frac{(0.010)^4}{(1.3)^4 (0.5) (10^{-3.5})^4} \).

Step 5: Apply Nernst Equation to Find EMF

Substitute \( Q \) and constants into the Nernst equation: \[ E = 0.46 - \frac{8.314 \times 298}{4 \times 96485} \ln Q \]. Calculate the natural log of \( Q \) first, then use it to find \( E \).

Step 6: Finalize EMF Calculation

Calculate to find the emf. This might involve calculator usage for precision steps or multi-step calculations for the log term. The calculated \( E \) gives the emf of the cell under the given conditions.

Key Concepts

Voltaic cell

A voltaic cell, also known as a galvanic cell, is a device that converts chemical energy into electrical energy through spontaneous redox reactions. These cells consist of two separate compartments called half-cells, each containing an electrode and an electrolyte. The electrodes are typically made of metal, and the electrolyte can be an aqueous solution containing ions that participate in the reaction.

In the voltaic cell, electrons flow from the anode to the cathode through an external circuit. The anode is where oxidation occurs, meaning it loses electrons, while the cathode is where reduction occurs, meaning it gains electrons.

• At the anode, the oxidation reaction releases electrons into the circuit.
• At the cathode, the electrons are used in the reduction reaction.

The flow of electrons from one half-cell to the other generates electric current, which can be harnessed to do work. In the given problem, the voltaic cell utilizes iron and oxygen-based reactions to create electricity. Understanding how a voltaic cell operates under both standard and non-standard conditions is crucial for proficiency in electrochemistry.

Standard electrode potential

The standard electrode potential, denoted as \(E^0\), is a measure of the potential difference between an electrode and its surrounding solution when the concentration of all substances involved are at standard conditions—1 M concentration, 1 atm pressure, and 25°C temperature. It helps predict the direction of electron flow and the feasibility of a redox reaction.

Each half-reaction in a voltaic cell has its own standard electrode potential. These potentials are determined experimentally and listed in tables for reference:

• The standard electrode potential for the reaction \( \, ext{Fe}^{2+} \rightarrow ext{Fe}^{3+} + e^- \) is \(+0.77 \text{ V}\).
• For the reaction \( ext{O}_2 + 4 ext{H}^+ + 4e^- \rightarrow 2 ext{H}_2 ext{O} \), it is \(+1.23 \text{ V}\).

To find the overall standard electromotive force (emf) for a cell, the potentials for the cathode and anode reactions are subtracted: \(E^0_{\text{cell}} = E^0_{\text{cathode}} - E^0_{\text{anode}}\). In our example, this calculation results in the cell's standard emf being \(0.46 \text{ V}\). This value indicates the potential energy available for electron flow under standard conditions and is fundamental for predicting the energy efficiency of a voltaic cell.

Nernst equation

The Nernst equation is an essential tool for calculating the electromotive force (emf) of a voltaic cell under non-standard conditions. It adjusts the standard emf to account for variations in concentrations, pressures, and temperature.

The Nernst equation is expressed as: \[ E = E^0 - \frac{RT}{nF} \ln Q \] where:

• \(E\) is the cell potential under non-standard conditions.
• \(E^0\) is the standard cell potential.
• \(R\) is the universal gas constant \(8.314 \, \text{J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}\).
• \(T\) is the temperature in Kelvin.
• \(n\) is the number of moles of electrons transferred in the reaction.
• \(F\) is the Faraday constant \(96485 \, \text{C}\cdot\text{mol}^{-1}\).
• \(Q\) is the reaction quotient, calculated as the ratio of product concentrations to reactant concentrations raised to the power of their stoichiometric coefficients.

In our specific problem, we use the Nernst equation to calculate the emf when the concentrations of \(\text{Fe}^{2+}\) and \(\text{Fe}^{3+}\), pressure of \(\text{O}_2\), and \(\text{pH}\) are known. Calculating \(Q\) requires understanding the concentrations and pressures given, and substituting into the Nernst equation delivers the actual cell potential under these unique conditions. This showcases how the behavior of the cell changes based on its environment.