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

Expert verified
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

01

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.
02

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).
03

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.
04

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.
05

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.

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