Question

A fuel cell designed to react grain alcohol with oxygen has the following net reaction: $$ \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)+3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g)+3 \mathrm{H}_{2} \mathrm{O}(l) $$ The maximum work that 1 mole of alcohol can do is \(1.32 \times\) \(10^{3} \mathrm{~kJ}\). What is the theoretical maximum voltage this cell can achieve at \(25^{\circ} \mathrm{C} ?\)

Short answer

The theoretical maximum voltage this fuel cell can achieve at 25°C is approximately 1.14 V.

Step-by-step solution

Identify relevant values and constants

We are given: - Maximum work for 1 mole alcohol: -1320 kJ/mol - Temperature = 25°C = 298.15 K - Faraday's constant (F) = 96485 C/mol

Calculate the number of moles of electrons transferred (n)

In order to find n, we need to balance the half-reactions: The oxidation half-reaction (alcohol to carbon dioxide): $$ C_2H_5OH \longrightarrow 2CO_2 + 6e^- $$ The reduction half-reaction (oxygen to water): $$ 3O_2 + 12e^- \longrightarrow 6H_2O $$ As we can see, the number of electrons transferred (n) is 12.

Convert ∆G to Joules

Given that the unit of ∆G (maximum work) is in kJ/mol, we need to convert it to Joules/mol in order to proceed. We can do this by multiplying it by 1000: ∆G = -1320 kJ/mol × 1000 J/kJ ∆G = -1320000 J/mol

Calculate the theoretical maximum voltage (E)

We can now use the relationship between Gibbs free energy change, Faraday's constant, the number of moles of electrons transferred, and the cell voltage: $$ ∆G = -nFE $$ Rearranging the equation to solve for E: $$ E= \frac{-∆G}{nF} $$ Substitute the values: $$ E = \frac{-(-1320000 \thinspace J/mol)}{12 \thinspace moles \times 96485 \thinspace C/mol} $$ Calculate the voltage: $$ E \approx 1.14 \thinspace V $$ So, the theoretical maximum voltage this cell can achieve at 25°C is approximately 1.14 V.

Key concepts

Understanding Electrochemistry

Electrochemistry is the branch of science that deals with the relationships between electricity and chemical reactions. Fuel cells are practical applications of electrochemistry, where chemical energy from a fuel is converted into electrical energy through a chemical reaction. This process occurs via two types of reactions: oxidation at the anode and reduction at the cathode, collectively known as redox reactions. In a fuel cell, the chemical reaction is continuously supplied with reactants, providing a steady flow of electricity, different from a traditional battery that stores energy and discharges.

The Role of Gibbs Free Energy

In electrochemistry, Gibbs free energy (\r\( \Delta G \) represents the maximum amount of work that can be performed by a chemical reaction at constant temperature and pressure. It is a measure of the usable energy that can be extracted from a process. When \r\( \Delta G \) is negative, the reaction occurs spontaneously, providing the driving force in galvanic cells. For the fuel cell in the exercise, the maximum work, or negative \r\( \Delta G \) value, determines the potential voltage achievable by the cell.

The Principle of Galvanic Cells

Galvanic cells, also known as voltaic cells, are devices that transform chemical energy into electrical energy by spontaneous redox reactions. They consist of two half-cells linked by a salt bridge, with each half-cell containing an electrode and an electrolyte. The electrons flow from the anode, undergoing oxidation, to the cathode, where reduction takes place. The flow of electrons through an external circuit is what we use as electricity. The maximum theoretical voltage of a galvanic cell is calculated using the Gibbs free energy change of the reaction.

Faraday's Constant in Calculations

Faraday's constant (F) is a fundamental value used in electrochemistry, representing the charge of one mole of electrons, approximately \r\( 96485 \, \text{C/mol} \) (Coulombs per mole). It's crucial when linking the macroscopic world of chemical reactions with the microscopic world of electron flow and charge. In the context of the exercise, Faraday's constant is used to relate the Gibbs free energy change to the charge transferred in moles of electrons, which then allows us to calculate the theoretical voltage of the cell.

Oxidation-Reduction Reactions Unpacked

Oxidation-reduction reactions, or redox reactions, are the foundation of electrochemical processes, including those in fuel cells. Oxidation involves the loss of electrons, while reduction involves the gain of electrons. In our fuel cell example, the grain alcohol (\r\( \text{C}_2\text{H}_5\text{OH} \) is oxidized to \r\( \text{CO}_2 \) and the \r\( \text{O}_2 \) is reduced to \r\( \text{H}_2\text{O} \) These two half-reactions occur in separate regions of the cell, allowing for the capture and use of the electrons that are transferred, which results in the creation of electrical current.