Question

A voltaic cell is based on \(\mathrm{Ag}^{+}(a q) / \mathrm{Ag}(s)\) and \(\mathrm{Fe}^{3+}(a q) /\) \(\mathrm{Fe}^{2+}(a q)\) half-cells. (a) What is the standard emf of the cell? (b) Which reaction occurs at the cathode and which at the anode of the cell? (c) Use \(S^{\circ}\) values in Appendix \(\mathrm{C}\) and the relationship between cell potential and free-energy change to predict whether the standard cell potential increases or decreases when the temperature is raised above \(25^{\circ} \mathrm{C}\) .

Short answer

(a) The standard emf of the cell is \(E_{cell}^° = 0.03 V\). (b) The cathode reaction is \(Ag^+(aq) + e^− \rightarrow Ag(s)\) and the anode reaction is \(Fe^{2+}(aq) \rightarrow Fe^{3+}(aq) + e^−\). (c) We cannot predict the effect of temperature on the standard cell potential without the provided entropy values.

Step-by-step solution

Find standard reduction potentials for each half-cell

Consult a table of standard reduction potentials to find the values for the half-cells given. The values for each half-cell are: \(Ag^+(aq) + e^− \rightarrow Ag(s) \) \(E^° = 0.80 V\) \( Fe^{3+}(aq) + e^− \rightarrow Fe^{2+}(aq) \) \(E^° = 0.77 V\)

Calculate the EMF of the voltaic cell

From the standard reduction potentials, we can calculate the EMF for the cell using the formula: \(E_{cell}^° = E_{cathode}^° - E_{anode}^°\) Since the EMF of the voltaic cell should be positive, we can choose the half-cell with the higher EMF as the cathode and the lower EMF as anode: \(E_{cell}^° = 0.80 V - 0.77 V = 0.03 V\)

Determine the reactions at cathode and anode

The half-cell with the higher reduction potential, Ag+/Ag, acts as the cathode, and its reaction will be the reduction: Cathode: \(Ag^+(aq) + e^− \rightarrow Ag(s)\) The half-cell with the lower reduction potential, Fe3+/Fe2+, acts as the anode, and its reaction will be the oxidation (reversed reduction reaction): Anode: \(Fe^{2+}(aq) \rightarrow Fe^{3+}(aq) + e^−\)

Predict the effect of temperature on standard cell potential

To predict if the standard cell potential increases or decreases when the temperature is raised above 25°C, we will use the relationship between cell potential and free-energy change, expressed by the equation: \(\Delta G^° = -nFE_{cell}^°\) Where \(\Delta G^°\) is the standard Gibbs free energy change, \(n\) is the number of electrons transferred, and \(F\) is the Faraday constant (96,485 C/mol). Additionally, we know the relationship between Gibbs free energy, enthalpy, and entropy: \(\Delta G^° = \Delta H^° - T \Delta S^°\) Using the given entropy values (\(S^°\)) for the reactions at the cathode and anode and comparing their sum to zero, we can determine if the standard cell potential increases or decreases when the temperature is raised above 25°C: - If \(\Delta S^° > 0\), the cell potential will increase as temperature increases. - If \(\Delta S^° < 0\), the cell potential will decrease as temperature increases. However, since the given problem does not provide the entropy values, we cannot predict the effect of temperature on the standard cell potential using this method.

Key concepts

Standard Reduction Potentials

Understanding standard reduction potentials is crucial for mastering the basics of electrochemistry. These potentials, denoted as \(E^{\text{\textdegree}}\), are a measure of the tendency of a chemical species to be reduced, and they are determined under standard conditions, which include a concentration of 1 M for aqueous species, a pressure of 1 bar for gases, and a temperature of 25℃. Standard reduction potentials are always given for half-reactions, which only represent either the reduction or oxidation process, not the full redox reaction that occurs in an electrochemical cell.

When it comes to voltaic or galvanic cells, where chemical energy is converted into electrical energy, these potentials help us predict which direction electrons will flow. A higher standard reduction potential indicates a greater likelihood that the species will gain electrons and be reduced. During the calculation of the electromotive force (EMF) of a cell, we use these potentials to identify the cathode (where reduction occurs) and the anode (where oxidation occurs). The species with the higher reduction potential becomes the cathode, and that with the lower potential, the anode. The difference in potentials between the cathode and anode gives us the EMF of the cell.

Electrochemical Cell Reactions

An electrochemical cell creates electrical current through chemical reactions, specifically redox reactions. In a voltaic cell, two different metals or metal ions are placed in separate solutions, each forming a half-cell that hosts one half of the redox reaction. One half-cell undergoes oxidation (loss of electrons), while the other undergoes reduction (gain of electrons). The electrode where oxidation occurs is called the anode, and the one where reduction occurs is the cathode.

Electrons flow from the anode to the cathode through an external circuit, producing an electric current. A salt bridge or porous partition is used to maintain the charge balance because as electrons leave one half-cell and enter the other, the solutions would otherwise become either too positive or too negative. The cell's EMF is a result of the potential difference between the two electrodes, driven by the difference in standard reduction potentials. The cell reactions are effectively harnessing the energy change associated with the movement of electrons from a high energy state to a lower one.

Gibbs Free Energy in Electrochemistry

The relationship between Gibbs free energy \(\Delta G^{\text{\textdegree}}\) and electrochemical cells is a key concept in understanding how cells function. Gibbs free energy refers to the maximum amount of work that a thermodynamic system can perform at constant temperature and pressure. In the context of electrochemistry, it is directly related to the cell potential and the capacity of the cell to do electrical work.

The equation \(\Delta G^{\text{\textdegree}} = -nFE_{cell}^{\text{\textdegree}}\) links Gibbs free energy change to the EMF of the cell. Here, \(n\) represents the number of moles of electrons exchanged in the redox reaction, and \(F\) is the Faraday constant, which is the charge of one mole of electrons. This equation shows that a spontaneous reaction (one that can occur without external energy) corresponds to a negative \(\Delta G^{\text{\textdegree}}\) and a positive \(E_{cell}^{\text{\textdegree}}\). Understanding this relationship helps us to calculate the thermodynamic feasibility of a reaction and predict how much electrical energy a reaction can produce.

Temperature Effects on Cell Potential

The standard cell potential is dependent on certain variables, including temperature. To understand the effect of temperature on cell potential, we look at the Gibbs free energy formula \(\Delta G^{\text{\textdegree}} = \Delta H^{\text{\textdegree}} - T\Delta S^{\text{\textdegree}}\), which relates enthalpy change (\(\Delta H^{\text{\textdegree}}\)) and entropy change (\(\Delta S^{\text{\textdegree}}\))—both of which can have temperature dependencies.

As temperature increases, the term \(T\Delta S^{\text{\textdegree}}\) becomes more significant. If the process involves an increase in entropy (\(\Delta S^{\text{\textdegree}} > 0\)), the free energy change will decrease more with increasing temperature, which can result in a higher cell potential. Conversely, if the process results in a decrease in entropy (\(\Delta S^{\text{\textdegree}} < 0\)), the cell potential may decrease with an increase in temperature. Understanding how temperature affects cell potential is important for assessing and predicting the performance of electrochemical cells under different thermal conditions.