Question

Hydrogen gas has the potential for use as a clean fuel in reaction with oxygen. The relevant reaction is $$ 2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) $$ Consider two possible ways of utilizing this reaction as an electrical energy source: (i) Hydrogen and oxygen gases are combusted and used to drive a generator, much as coal is currently used in the electric power industry; (ii) hydrogen and oxygen gases are used to generate electricity directly by using fuel cells that operate at \(85^{\circ} \mathrm{C}\) . (a) Use data in Appendix C to calculate \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) for the reaction. We will assume that these values do not change appreciably with temperature. (b) Based on the values from part (a), what trend would you expect for the magnitude of \(\Delta G\) for the reaction as the temperature increases? (c) What is the significance of the change in the magnitude of \(\Delta G\) with temperature with respect to the utility of hydrogen as a fuel? (d) Based on the analysis here, would it be more efficient to use the combustion method or the fuel-cell method to generate electrical energy from hydrogen?

Short answer

The change in enthalpy (∆H) of the reaction is -571.66 kJ/mol, and the change in entropy (∆S) is -400.74 J/(mol ⋅ K). As the temperature increases, the change in Gibbs free energy (∆G) decreases in magnitude, implying it becomes less spontaneous. Consequently, the utility of hydrogen as a fuel decreases at higher temperatures. The fuel-cell method, operating at \(85^{\circ} \mathrm{C}\), would be more efficient in generating electricity from hydrogen and oxygen compared to the combustion method, as it maintains a more negative ∆G, improving the efficiency of energy conversion.

Step-by-step solution

Determine the change in enthalpy (∆H) of the reaction

To determine the change in enthalpy (∆H), we need to subtract the standard enthalpy of formation of the reactants from the standard enthalpy of formation of the products: \(\Delta H^{\circ} = \sum{Products}- \sum{Reactants}\) The relevant reaction is given by: \[2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{H}_{2}\mathrm{O}(l)\] Using the standard enthalpies of formation from Appendix C, we will have: \(∆H = 2 \times (-285.83) - (0)\) \(∆H = -571.66 \mathrm{kJ/mol}\) The change in enthalpy of the reaction is -571.66 kJ/mol.

Calculate the change in entropy (∆S) of the reaction

Similar to ∆H, to determine the change in entropy (∆S), we need to subtract the standard entropy of the reactants from the standard entropy of the products: \(\Delta S^{\circ} = \sum{Products}- \sum{Reactants}\) Considering the entropy values from Appendix C, we can determine the change in entropy: \(\Delta S = 2 \times (69.91) - (2 \times 130.68 + 205.2)\) \(\Delta S = -400.74 \mathrm{J/(mol \cdot K)}\) The change in entropy of the reaction is -400.74 J/(mol ⋅ K).

Analyze the trend of the change in Gibbs free energy (∆G) with temperature

To examine the trend of ∆G with temperature, we use the Gibbs-Helmholtz equation: \(\Delta G = \Delta H - T\Delta S\) Since ∆H is negative and ∆S is also negative, as the temperature increases, the term \(T\Delta S\) becomes more negative. Thus, ∆G will decrease in magnitude (i.e., become less negative) as the temperature increases from its standard value.

Discuss the significance of ∆G change with respect to hydrogen's utility as fuel

The change in Gibbs free energy with temperature has implications for the efficiency of the reaction as a fuel source. As the temperature increases, the reaction will become less spontaneous, meaning that it will be less favored at higher temperatures. Therefore, the utility of hydrogen as a fuel could potentially decrease at higher temperatures.

Decide the more efficient method between combustion and fuel-cell to generate electricity

Based on our analysis, using hydrogen and oxygen at a temperature of \(85^{\circ}\) C in a fuel cell, which directly produces electricity, would be more efficient than burning the gases at higher temperatures, such as in a combustion process. The lower operating temperature of the fuel cell would maintain a more negative ∆G, making the reaction more spontaneous, and thus improving the efficiency of energy conversion. In conclusion, the fuel-cell method would be more efficient in generating electricity from hydrogen and oxygen as compared to the combustion method.

Key concepts

Enthalpy Change

Enthalpy change, denoted by \( \Delta H \), reflects the heat absorbed or released during a chemical reaction. For hydrogen fuel cells, understanding \( \Delta H \) is crucial as it relates to the energy produced.

In the given reaction \( 2 \mathrm{H}_{2}(g) + \mathrm{O}_2(g) \rightarrow 2 \mathrm{H}_2\mathrm{O}(l) \), the enthalpy change is calculated using the standard enthalpies of formation for products and reactants. Here, since water forms from hydrogen and oxygen, and both are gases initially, the heat released (\( \Delta H \)) is found to be \(-571.66 \ \mathrm{kJ/mol}\).

This negative value indicates an exothermic reaction, meaning energy is released when hydrogen combines with oxygen to form water.

Entropy Change

Entropy change, \( \Delta S \), measures the disorder or randomness in a system. For chemical reactions, changes in entropy help understand how the energy is distributed.

In the hydrogen fuel cell reaction, the change in entropy is calculated by subtracting the total entropy of the reactants from that of the products. The calculated entropy change is \(-400.74 \ \mathrm{J/(mol \cdot K)}\).

This negative entropy change means that the disorder decreases when hydrogen and oxygen react to form water. Since gases have more entropy than liquids, this makes sense as both reactants are gases while the product is a liquid, leading to an overall decrease in disorder.

Gibbs Free Energy

Gibbs free energy, \( \Delta G \), determines the spontaneity of a reaction. It's a critical factor in assessing hydrogen's viability as an energy source.

The relation \( \Delta G = \Delta H - T\Delta S \) helps analyze how temperature affects \( \Delta G \). For hydrogen to be an efficient fuel, \( \Delta G \) should remain negative, ensuring the reaction is spontaneous.

In our case, both \( \Delta H \) and \( \Delta S \) are negative, so as temperature \( T \) rises, \( T\Delta S \) becomes more negative. Consequently, \( \Delta G \) becomes less negative, meaning it reduces in spontaneity. Thus, operating hydrogen fuel cells at moderate temperatures ensures they remain efficient and effective.

Energy Efficiency

Energy efficiency is vital in choosing between fuel cells and combustion for energy generation. Fuel cells, like those used with hydrogen, offer significant advantages.

Fuel cells convert chemical energy directly into electrical energy. This direct conversion generally leads to higher efficiency compared to combustion methods, which involves heat generation, losses, and mechanical energy conversion.

In a fuel cell using the hydrogen-oxygen reaction, less energy is lost as heat, preserving a more negative \( \Delta G \), which improves efficiency. Thus, for hydrogen as a clean fuel, using fuel cells is preferable, especially given their reduced operational temperatures and greater energy return.