Problem 90 - Chemistry The Science in Context

Question

A direct methanol fuel cell uses the oxidation of methanol by oxygen to generate electrical energy. The overall reaction, which is given below, has a \(\Delta G^{\circ}\) value of -702.4 kJ/mol of methanol oxidized. What is the standard cell potential for this fuel cell? $$\mathrm{CH}_{3} \mathrm{OH}(\ell)+\frac{3}{2} \mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(\ell)$$

Step-by-Step Solution

Answer

The standard cell potential for a direct methanol fuel cell is 0.606 V.

Step 1: Determine the balanced redox reaction for methanol oxidation

First, we need to determine the balanced redox reaction for methanol oxidation. Methanol (CH3OH) is oxidized to CO2, while O2 is reduced to H2O. We can start by writing the half-reactions: Oxidation half-reaction: $$\mathrm{CH}_{3}\mathrm{OH}\rightarrow\mathrm{CO}_{2}+e^{-}$$ Reduction half-reaction: $$\mathrm{O}_{2}+e^{-}\rightarrow\mathrm{H}_{2}\mathrm{O}$$ Now, we must balance the number of electrons in each half-reaction and then combine them to obtain the balanced overall redox reaction.

Step 2: Balance the electrons in each half-reaction and find the overall redox reaction

We first need to balance the number of electrons in both half-reactions so that the number of electrons lost in the oxidation reaction is equal to the number of electrons gained in the reduction reaction. Oxidation half-reaction (balanced): $$\mathrm{CH}_{3}\mathrm{OH}+3\mathrm{H}_{2}\mathrm{O}\rightarrow\mathrm{CO}_{2}+12\mathrm{H}^{+}+12e^{-}$$ Reduction half-reaction (balanced): $$3\mathrm{O}_{2}+12\mathrm{H}^{+}+12e^{-}\rightarrow6\mathrm{H}_{2}\mathrm{O}$$ Now, we can add the balanced half-reactions together to form the balanced overall redox reaction: $$\mathrm{CH}_{3}\mathrm{OH}+3\mathrm{H}_{2}\mathrm{O}+3\mathrm{O}_{2}\rightarrow\mathrm{CO}_{2}+6\mathrm{H}_{2}\mathrm{O}$$ This is the overall redox reaction for the direct methanol fuel cell.

Step 3: Determine the number of moles of electrons transferred in the reaction

From the balanced redox reaction, we can see that 12 moles of electrons are transferred for each mole of methanol oxidized. Therefore, n = 12.

Step 4: Calculate the standard cell potential (E°)

Now that we have the number of moles of electrons transferred (n), we can use the equation \(\Delta G^{\circ} = -nFE^{\circ}\) to find the standard cell potential (E°). First, convert kJ to J for ΔG°: ΔG° = -702.4 kJ/mol × 1,000 J/kJ = -702,400 J/mol Using ΔG°, n, and F, solve for E°: $$E^{\circ} = -\frac{\Delta G^{\circ}}{nF} = -\frac{-702,400\,\text{J/mol}}{12\,\text{mol e}^{-}\times 96,485\,\text{C/mol e}^{-}}$$ After solving for E°, we get: $$E^{\circ} = 0.606\,\text{V}$$ The standard cell potential for the direct methanol fuel cell is 0.606 V.

Key Concepts

Direct Methanol Fuel Cell

A direct methanol fuel cell (DMFC) is a type of renewable energy device that converts the chemical energy of methanol directly into electrical energy through electrochemical reactions. Methanol, a light, volatile, and flammable liquid, acts as the fuel, and it's combined with oxygen from the air to produce electricity, water, and carbon dioxide.

One of the primary benefits of DMFCs is their high energy density compared to other fuel cells, making them particularly well-suited for portable electronic devices. The technology behind DMFCs incorporates principles of redox reactions and electrochemistry. Understanding the significance of the standard cell potential in DMFCs helps us evaluate their efficiency and potential as sustainable power sources.

Redox Reactions

Redox reactions, short for reduction-oxidation reactions, are a fundamental type of chemical reaction where electrons are transferred between two species. In the context of electrochemistry, one species is oxidized (loses electrons) and another is reduced (gains electrons).

In a DMFC, the methanol oxidation and oxygen reduction are separate but interconnected redox reactions that occur at different electrodes. These reactions are crucial for the generation of electric current within the cell. The balanced overall redox reaction must account for all atoms and charges to accurately represent the process within the fuel cell, ensuring efficient energy conversion and the proper functioning of the device.

Gibbs Free Energy

Gibbs free energy, denoted by \(\Delta G\), is a thermodynamic quantity that measures the maximum amount of work that can be obtained from a chemical reaction at constant temperature and pressure. It is a predictor of the spontaneity of a chemical process; reactions with a negative \(\Delta G\) value are spontaneous, meaning they can occur without a continuous input of energy.

The direct methanol fuel cell's \(\Delta G^\circ\) value of -702.4 kJ/mol indicates that the reaction is spontaneous and capable of doing work, i.e., producing electricity. This relationship between \(\Delta G\) and electrical work links thermodynamics and electrochemistry and enables us to calculate the theoretical limit of the cell's electrical output.

Electrochemistry

Electrochemistry is the branch of chemistry that deals with the relationship between electricity and chemical reactions. It plays a critical role in a variety of technologies, including batteries, sensors, and fuel cells like the DMFC.

The standard cell potential (\(E^{\circ}\)) is a key concept in electrochemistry; it measures the voltage that a cell can produce under standard conditions (1 bar, 25°C, and 1 M concentrations), when no current is flowing. The formula \(\Delta G^{\circ} = -nFE^{\circ}\) directly connects Gibbs free energy (\(\Delta G^{\circ}\)) to the standard cell potential (\(E^{\circ}\)), with 'n' being the number of moles of electrons transferred and 'F' the Faraday constant (the charge of one mole of electrons).

In summary, by understanding the principles of electrochemistry, we can predict the behavior of electrochemical cells, calculate their theoretical potential, and optimize their performance for practical applications.