Problem 198 - A Complete Resource Book in Chemistry for JEE Main

Question

In a fuel cell methanol is used as fuel and oxygen gas is used as an oxidizer. The reaction is \(\mathrm{CH}_{3} \mathrm{OH}(\mathrm{l})+3 / 2 \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{~g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{l})\) At \(298 \mathrm{~K}\) standard Gibb's energies of formation for \(\mathrm{CH}_{3} \mathrm{OH}\) (1), \(\mathrm{H}_{2} \mathrm{O}\) (l) and \(\mathrm{CO}_{2}(\mathrm{~g})\) are \(-166.2 .-237.2\) and \(-394.4 \mathrm{kJmol}^{-1}\) respectively. If standard enthalpy of combustion of methanol is \(-7269 \mathrm{~kJ} \mathrm{~mol}^{-1}\), efficiency of the fuel cell will be: [2009] (a) \(87 \%\) (b) \(90 \%\) (c) \(97 \%\) (d) \(80 \%\)

Step-by-Step Solution

Answer

The efficiency of the fuel cell is 97% (option c).

Step 1: Understand the Reaction

The reaction for the fuel cell is the conversion of methanol and oxygen into carbon dioxide and water. This transformation is represented by: \(\mathrm{CH}_{3} \mathrm{OH}(\mathrm{l})+\frac{3}{2} \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{~g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{l})\).

Step 2: Determine the Change in Gibbs Free Energy

To find the change in Gibbs Free Energy (\(\Delta G^0_{rxn}\)) for the reaction, use the standard Gibbs energies of formation. The formula is: \(\Delta G^0_{rxn} = \sum \Delta G^0_{products} - \sum \Delta G^0_{reactants}\). For our reaction: \(\Delta G^0_{rxn} = [(-394.4) + 2(-237.2)] - [(-166.2)] = -702.8 \text{ kJ/mol}\).

Step 3: Calculate the Maximum Work

The Gibbs Free Energy change (\(-702.8 \text{ kJ/mol}\)) represents the maximum reversible work that the fuel cell can do.

Step 4: Calculate the Efficiency of the Fuel Cell

Efficiency is calculated using the formula: \(\eta = \frac{\Delta G^0_{rxn}}{\Delta H_{combustion}} \times 100\%\). Substitute the values: \(\eta = \frac{-702.8}{-726.9} \times 100\% \approx 96.7\%\).

Step 5: Compare to Options

From the efficiency calculation, the obtained value is \(96.7\%\). Looking at the given options, \(96.7\%\) is closest to \(97\%\).

Key Concepts

Gibbs Free Energy

In the context of fuel cells, Gibbs Free Energy (93G) is crucial as it indicates the maximum usable work that can be extracted from a chemical reaction at constant temperature and pressure. For students learning about fuel cells, understanding Gibbs Free Energy is essential because it measures the spontaneity of a reaction and the potential energy that can be harnessed.

In a methanol fuel cell, we use the standard Gibbs energies of formation to determine the change in Gibbs Free Energy (94G^0_{rxn}). For our specific reaction, this is calculated using the formula:

• 94G^0_{rxn} = 94G^0_{products} - 94G^0_{reactants}

This approach involves plugging in the values of the Gibbs energies of formation for methanol, carbon dioxide, and water. By performing these calculations, we find that 94G^0_{rxn} is 702.8 kJ/mol. This value gives us the maximum amount of work, implying how much energy can be converted into electrical work from the total energy generated by the reaction. These calculations provide insights into how efficient the energy conversion process is in a fuel cell.

Standard Enthalpy of Combustion

The standard enthalpy of combustion (94H_{combustion}) represents the heat released when one mole of a compound completely reacts with oxygen under standard conditions. For methanol fuel cells, the standard enthalpy of combustion is 726.9 kJ/mol.

This value is significant because it shows the total energy change when methanol is combusted under ideal conditions. Unlike Gibbs Free Energy, the enthalpy of combustion accounts for all forms of energy released as heat without focusing on the work that can be extracted. It's crucial for students to understand that while enthalpy gives the overall energy picture, not all of it is convertible to work.

• It helps in calculating the theoretical maximum energy output.
• It's a benchmark for comparing different fuels' energy efficiencies.

Understanding the standard enthalpy of combustion makes it easier to appreciate the energetic profile of methanol and other fuels used in similar applications.

Efficiency Calculation

Efficiency in fuel cells is a measure of how well energy is being converted from chemical to electrical form. It's calculated using the formula:

• B7 = 94G^0_{rxn} / 94H_{combustion} 15 100%

In this exercise, by inserting the values into the efficiency formula, we get a result of approximately 96.7%, which is very close to 97%. This means that methanol fuel cells are quite efficient, as a large proportion of the chemical energy is being converted into electricity rather than being lost as heat.

Students can use this formula to determine and compare the efficiency of different fuel cells and energy processes. Efficiency calculation helps in making decisions on the best types of fuels to use in specific scenarios depending on energy needs and availability. It's an essential tool for evaluating the real-life performance of energy systems.

Methanol Fuel Cells

Methanol fuel cells operate by converting chemical energy contained in methanol directly into electrical energy through a chemical reaction with oxygen. This differs from traditional combustion processes as it directly transforms fuel energy, maximizing efficiency while reducing emissions.

Methanol is a practical choice for fuel cells due to its high energy density and liquid state at room temperature, making it easy to store and transport. Hence, it's commonly used in portable electronics and for stationary power generation.

• Methanol provides a cleaner alternative to fossil fuels.
• It helps in reducing the carbon footprint and is vital in green energy solutions.

Understanding how methanol fuel cells work empowers students to appreciate developments in sustainable energy and innovating technologies that address environmental concerns. Methanol fuel cells illustrate the transition from conventional energy systems to greener alternatives, highlighting the importance of renewable energy in our future.