Question

Direct methanol fuel cells (DMFCs) have shown some promise as a viable option for providing "green" energy to small electrical devices. Calculate \(\mathscr{E}^{\circ}\) for the reaction that takes place in DMFCs: $$ \mathrm{CH}_{3} \mathrm{OH}(l)+3 / 2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(l) $$ Use values of \(\Delta G_{\mathrm{f}}^{\circ}\) from Appendix \(4 .\)

Short answer

The given reaction does not involve any transfer of electrons, and it is not a redox reaction. Therefore, we cannot directly calculate the standard cell potential for this reaction using the provided information.

Step-by-step solution

Write down the given values of ΔG°f from Appendix 4

The values of $$\Delta G_{\mathrm{f}}^{\circ}$$ are given for the individual compounds in the reaction. Let's write down the values for the reactants and products. $$ \mathrm{CH}_{3} \mathrm{OH}(l) \rightarrow \Delta G_{\mathrm{f}}^{\circ 1} \\ \frac{3}{2} \mathrm{O}_{2}(g) \rightarrow\Delta G_{\mathrm{f}}^{\circ 2} \\ \mathrm{CO}_{2}(g) \rightarrow\Delta G_{\mathrm{f}}^{\circ 3} \\ 2 \mathrm{H}_{2} \mathrm{O}(l) \rightarrow\Delta G_{\mathrm{f}}^{\circ 4} $$

Calculate the Gibbs free energy change ΔG° for the overall reaction

To calculate the Gibbs free energy change for the overall reaction, we can use the formula: $$ \Delta G_{\mathrm{reaction}}^{\circ} = \sum n_i \Delta G_{\mathrm{f}}^{\circ i}\ (\textnormal{products}) - \sum n_i \Delta G_{\mathrm{f}}^{\circ i}\ (\textnormal{reactants}) \\ $$ So, $$ \Delta G_{\mathrm{reaction}}^{\circ}=(1 \times \Delta G_{\mathrm{f}}^{\circ 3}+2 \times \Delta G_{\mathrm{f}}^{\circ 4})-(1 \times \Delta G_{\mathrm{f}}^{\circ 1}+\frac{3}{2} \times \Delta G_{\mathrm{f}}^{\circ 2}) $$ Plug in the given values from Appendix 4 and find the Gibbs free energy change for the overall reaction.

Determine the number of moles of electrons transferred in the reaction

For this step, we need to balance the given reaction and find the number of moles of electrons transferred. However, since there are no charges in the given reaction (i.e., no ions are formed or consumed), no electrons are transferred within this reaction. The reaction is already balanced.

Calculate the standard cell potential

Since we found out that no electrons are being exchanged in this redox reaction, we can't use the given relationship to determine the standard cell potential. In the provided reaction, there is no transfer of electrons, and the process does not happen in a typical electrochemical cell setup. Therefore, it is not possible to directly calculate the standard cell potential for this reaction using the provided information.

Key concepts

Gibbs Free Energy

Understanding Gibbs free energy is crucial when studying chemical reactions, as it offers insight into a reaction's spontaneity and the amount of work it can perform. Named after the scientist Josiah Willard Gibbs, it is represented by the symbol \( G \). The change in Gibbs free energy, symbolized as \( \Delta G \), is particularly key in predicting the direction of a chemical process. This value is calculated by subtracting the free energy of the reactants from that of the products.

In the context of direct methanol fuel cells (DMFCs), which are eco-friendly energy sources ideal for small devices, \( \Delta G \), indicates how much energy is available to do work when methanol and oxygen are converted into carbon dioxide and water. To calculate this for a specific reaction, we use the formula: \[ \Delta G_{\mathrm{reaction}}^{\circ} = \sum n_i \Delta G_{\mathrm{f}}^{\circ i} \ (\textnormal{products}) - \sum n_i \Delta G_{\mathrm{f}}^{\circ i} \ (\textnormal{reactants}) \]
Using values from thermodynamic tables, we can evaluate whether the DMFC reaction will occur spontaneously under standard conditions—typically a good indication of the cell's practicality for energy generation.

Electrochemistry

Electrochemistry is the branch of science that deals with the relationships between electricity and chemical reactions. It's particularly involved with the exchange of electrons between species, which is the foundation for understanding how batteries and fuel cells operate. In electrochemistry, we study the movement of ions in a solution and the transfer of electrons at the interface between an electrode and the solution.

For direct methanol fuel cells (DMFCs), electrochemistry helps us to comprehend how methanol functions as the fuel to produce electricity. The DMFC, however, presents a unique case within electrochemistry because the reaction \( \mathrm{CH}_{3} \mathrm{OH}(l) + \frac{3}{2} \mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g) + 2 \mathrm{H}_{2} \mathrm{O}(l) \) does not involve explicit electron transfer as seen in typical redox reactions. This means we can't directly calculate the number of moles of electrons transferred or apply the usual electrochemical techniques to determine the cell's standard potential, which leads us into the nuances of cell potentials and how they quantify a cell's energy-producing capability under standard conditions.

Standard Cell Potential

Standard cell potential, denoted as \( \mathscr{E}^{\circ} \), is a key parameter in electrochemistry that tells us the voltage that a cell can produce under standard conditions, which are 1 M solutions, 1 bar pressure, and a temperature of 25°C or 298 K for all the reactants and products. This voltage is fundamentally a measure of the driving force behind the electrochemical reaction and, hence, the cell's ability to do electrical work.

The standard cell potential is determined by considering the reduction potentials of the half-reactions that occur at the anode and cathode. This value provides a quantitative idea of the maximum electrical work that a fuel cell or battery can perform. In typical cases, the relationship between \( \Delta G \) and \( \mathscr{E}^{\circ} \) is given by the equation \[ \Delta G = -nFE^{\circ} \], where \( n \) is the number of moles of electrons transferred in the reaction, \( F \) is the Faraday constant, and \( E^{\circ} \) is the standard cell potential. However, without electron transfer, as in the case of DMFCs described in the provided exercise, this relation is not applicable. This demonstrates the complex nature of certain fuel cells and signifies a departure from conventional electrochemical cells.