Problem 30 - Chemistry and Chemical Reactivity

Question

One half-cell in a voltaic cell is constructed from an iron electrode in an \(\mathrm{Fe}\left(\mathrm{NO}_{3}\right)_{2}\) solution of unknown concentration. The other half-cell is a standard hydrogen electrode. A potential of \(0.49 \mathrm{V}\) is measured for this cell. Use this information to calculate the concentration of \(\mathrm{Fe}^{2+}(\mathrm{aq})\)

Step-by-Step Solution

Answer

The concentration of Fe\(^{2+}\) is approximately 0.0093 M.

Step 1: Understand the electrochemical cell setup

We have a voltaic cell with two half-cells. One is a standard hydrogen electrode (SHE) with a potential of 0 V, and the other is an iron half-cell with an unknown concentration of Fe\(^{2+}\). The cell potential is measured as 0.49 V.

Step 2: Identify the standard reduction potential

The standard reduction potential for the iron reaction \( ext{Fe}^{2+} + 2e^- \rightarrow ext{Fe(s)} \) is \(-0.44 \text{ V}\).

Step 3: Use Nernst equation

The Nernst equation is a crucial tool in calculating the concentration of ions in an electrochemical cell not at standard conditions:\[E = E^0 - \frac{RT}{nF} \ln Q\]Where:- \( E \) is the measured cell potential (0.49 V),- \( E^0 \) is the standard cell potential, \( E^0 = E^0_{ ext{cathode}} - E^0_{ ext{anode}} = 0 - (-0.44) = 0.44 \text{ V} \),- \( R \) is the gas constant \( 8.314 \text{ J/K mol} \),- \( T \) is the temperature in Kelvin (assumed to be 298 K),- \( n \) is the number of electrons transferred (2 for the Fe half-reaction),- \( F \) is Faraday's constant \( 96485 \text{ C/mol} \),- \( Q \) is the reaction quotient.

Step 4: Calculate Q using the Nernst equation

Rearranging the Nernst equation to solve for \( Q \): \[Q = e^{\frac{nF(E^0 - E)}{RT}}\]Substitute the values:\[Q = e^{\left(\frac{2 \times 96485 \times (0.44 - 0.49)}{8.314 \times 298}\right)}\]

Step 5: Calculate the concentration of Fe\(^{2+}\)

After substituting the values and calculating, we find that:\[Q = [\text{Fe}^{2+}] \approx 0.0093\]Thus, the concentration of Fe\(^{2+}\) is approximately 0.0093 M.

Key Concepts

Voltaic Cell

A voltaic cell, also known as a galvanic cell, is a type of electrochemical cell that converts chemical energy into electrical energy through spontaneous redox reactions. This cell consists of two half-cells, each containing an electrode and an electrolyte. These half-cells are connected by a wire and a salt bridge or a porous partition. In the given exercise, the voltaic cell comprises an iron half-cell and a standard hydrogen electrode. The iron half-cell is where the reduction or oxidation of iron ions occurs, while the standard hydrogen electrode (SHE) serves as a reference with a known potential. The cell generates electrical energy as electrons flow from the anode to the cathode through the external circuit. This electron flow results from the oxidation of iron ions in one half-cell and reduction in another, leading to the generation of a measurable cell potential.

Standard Hydrogen Electrode

The Standard Hydrogen Electrode (SHE) is a crucial reference point in electrochemistry. It is used to measure the electrode potentials of other half-cells. The SHE is assigned a potential of 0 V under standard conditions. It consists of a platinum electrode in contact with 1 M H extsuperscript{+} ions and close to pure hydrogen gas at 1 atm pressure. In our voltaic cell, the SHE acts as the reference half-cell. Because its potential is set to 0 V, any measured cell potential can be attributed to the potential of the other half-cell. In this case, when paired with the iron half-cell, the iron's half-cell potential can be determined relative to the hydrogen electrode. This comparison allows the calculation of unknown concentrations in the iron cell through the overall cell voltage observed.

Nernst Equation

The Nernst Equation is an essential tool for determining the cell potential in non-standard conditions. It shows how the electromotive force (EMF) of a cell adjusts based on ion concentrations within the reactants and products of a chemical reaction. The equation is written as:\[ E = E^0 - \frac{RT}{nF} \ln Q \]Where:- \( E \) is the measured cell potential- \( E^0 \) is the standard cell potential- \( R \) is the universal gas constant (8.314 J/K mol)- \( T \) is the absolute temperature (in Kelvin)- \( n \) is the number of moles of electrons transferred- \( F \) is Faraday's constant (96485 C/mol)- \( Q \) is the reaction quotient, expressing the ratio of product concentrations over reactant concentrations.In the practice exercise, the Nernst equation is pivotal to finding the concentration of \( \mathrm{Fe}^{2+} \) ions by rearranging the equation to solve for \( Q \), and consequently calculating the concentrations inside the cell.

Iron Electrode

In this voltaic cell, the iron electrode is a key component involved in the redox reaction. Typically, the iron half-cell is written as \( \mathrm{Fe}^{2+} + 2e^- \rightarrow \mathrm{Fe(s)} \), indicating that iron ions are gaining electrons (reduction) or solid iron is losing electrons (oxidation).The potential for this half-reaction is determined in relation to the SHE, which is significant in understanding the total cell potential. Knowing the standard reduction potential of iron, \( -0.44 \) V, helps when applying the Nernst Equation to account for non-standard conditions. This potential value is used in calculating the overall cell potential and determining the behavior of the iron ions in the solution.Through this information, we can further analyze the concentration and potential changes in the voltaic cell environment.

Concentration Calculation

Concentration calculation in electrochemical cells involves understanding the relationship between the electrode potential and ionic concentrations of the reactants/products. The Nernst Equation aids in this by linking the measured cell potential with the unknown concentration of reactants.By rearranging the Nernst Equation:\[ Q = e^{\frac{nF(E^0 - E)}{RT}} \]We find the reaction quotient \( Q \), which, in a simple one-species scenario, might reflect the ion concentration directly. In this exercise, the value of \( Q \) corresponds to the concentration of \( \mathrm{Fe}^{2+} \) ions. Using the given values (e.g., \( E = 0.49 \) V, \( E^0 = 0.44 \) V, and others), we determine the \( \mathrm{Fe}^{2+} \) concentration to be 0.0093 M, showcasing the impact of electrochemical principles on concentration determination.