Question

The overall reaction and equilibrium constant value for a hydrogen-oxygen fuel cell at \(298 \mathrm{~K}\) is $$2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(I) \quad K=1.28 \times 10^{83}$$ a. Calculate \(\mathscr{C}^{\circ}\) and \(\Delta G^{\circ}\) at \(298 \mathrm{~K}\) for the fuel cell reaction. b. Predict the signs of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) for the fuel cell reaction. c. As temperature increases, does the maximum amount of work obtained from the fuel cell reaction increase, decrease, or remain the same? Explain.

Short answer

a. The standard cell potential, \(\mathscr{C}^{\circ}\), is approximately \(1.229 \ \mathrm{V}\), and the standard Gibbs free energy change, \(\Delta G^{\circ}\), is approximately \(-474920 \ \mathrm{J/mol}\). b. The signs of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) for the fuel cell reaction can be predicted as negative (exothermic) and positive, respectively. c. As the temperature increases, the maximum amount of work obtained from the fuel cell reaction decreases.

Step-by-step solution

a. Calculating standard cell potential and standard Gibbs free energy change

Using the Nernst equation, we can calculate the standard cell potential, \(\mathscr{C}^{\circ}\), from the equilibrium constant, \(K\), as follows: \[\mathscr{C}^{\circ} = \frac{RT}{nF} \ln K\] Where: - \(R\) is the gas constant, \(8.314 \mathrm{J} \cdot \mathrm{mol}^{-1} \cdot K^{-1}\) - \(T\) is the temperature in Kelvin, here \(298 \mathrm{K}\) - \(n\) is the number of electrons transferred, which is \(4\) for \(2 \mathrm{H}_{2} + \mathrm{O}_{2} \longrightarrow 2 \mathrm{H}_{2}\mathrm{O}\) - \(F\) is Faraday's constant, \(96485 \ \mathrm{C/mol}\) - \(K\) is the equilibrium constant, \(1.28 \times 10^{83}\) Plugging in the values, we get: \[\mathscr{C}^{\circ} = \frac{8.314 \times 298}{4 \times 96485} \ln (1.28 \times 10^{83}) \approx 1.229 \ \mathrm{V}\] Now, we can calculate the standard Gibbs free energy change, \(\Delta G^{\circ}\), using the cell potential and Faraday's constant: \[\Delta G^{\circ} = -nF\mathscr{C}^{\circ}\] \[\Delta G^{\circ} = -4 \times 96485 \times 1.229 \approx -474920 \ \mathrm{J/mol}\]

b. Predicting the signs of ΔH° and ΔS° for the fuel-cell reaction

Since the fuel cell reaction generates a positive cell potential and has a large positive equilibrium constant, we can deduce that the reaction is spontaneous and exergonic, meaning the standard enthalpy change, \(\Delta H^{\circ}\), should be negative (exothermic). The spontaneous nature of the reaction also indicates that the standard entropy change, \(\Delta S^{\circ}\), will likely be positive. An increase in entropy usually accompanies the conversion of reactant gases into fewer product molecules.

c. Effect of temperature on the maximum work obtained from the fuel-cell reaction

To determine how temperature affects the maximum work output of the fuel-cell reaction, we need to consider the following relationships: - \(\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}\) - \(W_{max} = -\Delta G\) As noted earlier, \(\Delta H^{\circ} < 0\) and \(\Delta S^{\circ} > 0\). As the temperature increases, the term \(T\Delta S^{\circ}\) increases, making the standard Gibbs free energy change, \(\Delta G^{\circ}\), less negative or closer to zero. As a result, the maximum work that can be obtained from the reaction, \(W_{max}\), decreases. Therefore, as the temperature increases, the maximum amount of work obtained from the fuel cell reaction decreases.

Key concepts

Gibbs Free Energy

Gibbs free energy (\( \Delta G^{\circ} \)) is a crucial concept in the realm of thermodynamics, particularly when assessing chemical reactions like that in a hydrogen-oxygen fuel cell. It gives us insight into the spontaneity of a process. A negative \( \Delta G^{\circ} \) value indicates a spontaneous reaction, meaning it can occur without any external input of energy. This occurs when the energy from the reactants is transformed into products.
In a hydrogen-oxygen fuel cell, the Gibbs free energy change is calculated using the equation \( \Delta G^{\circ} = -nF\mathscr{C}^{\circ} \), where \( n \) is the number of electrons transferred, \( F \) is Faraday's constant, and \( \mathscr{C}^{\circ} \) is the standard cell potential. For this particular reaction, we find \( \Delta G^{\circ} \approx -474920 \ \mathrm{J/mol} \), highlighting that the reaction is indeed spontaneous and can proceed under standard conditions. If \( \Delta G^{\circ} \) is less negative, the system is closer to equilibrium, indicating a decrease in the maximum obtainable work.

Nernst Equation

The Nernst equation is a key tool for examining electrochemical reactions, like those in fuel cells. It helps us determine the cell potential under non-standard conditions. The equation is:\[\mathscr{C} = \mathscr{C}^{\circ} - \frac{RT}{nF} \ln Q\]where \( \mathscr{C} \) is the cell potential, \( \mathscr{C}^{\circ} \) is the standard cell potential, \( R \) is the gas constant, \( T \) is temperature in Kelvin, \( n \) is the number of moles of electrons exchanged, \( F \) is Faraday's constant, and \( Q \) is the reaction quotient.
In the exercise, the Nernst equation allows us to calculate the standard cell potential from the equilibrium constant, leading to a calculated cell potential of approximately \( 1.229 \ \mathrm{V} \). The Nernst equation illustrates how slight changes in concentration can impact the voltage output of a fuel cell, an essential consideration for practical applications.

Standard Cell Potential

The standard cell potential, \( \mathscr{C}^{\circ} \), is a vital aspect of understanding the efficiency of a fuel cell. It signifies the voltage or electrical potential difference between the cathode and anode when reactants and products are at standard conditions. Determining the standard cell potential involves using the Nernst equation and the equilibrium constant:\[\mathscr{C}^{\circ} = \frac{RT}{nF} \ln K\]This parameter is a measure of the maximum potential difference that can be developed by the fuel cell under standard conditions. In the hydrogen-oxygen fuel cell discussed, the calculated standard cell potential is approximately \( 1.229 \ \mathrm{V} \). This high potential reflects the favorable nature of the reaction and the efficient conversion of chemical energy to electrical energy.

Enthalpy Change

Enthalpy change (\( \Delta H^{\circ} \)) signifies the heat changes in a reaction, an essential factor for thermodynamics. In the context of a hydrogen-oxygen fuel cell, calculating this change helps us understand the energy transformations during the reaction process.
Given the high equilibrium constant and exothermic nature of the reaction, \( \Delta H^{\circ} \) for the hydrogen-oxygen fuel cell is negative, indicating that energy is released to the surroundings. Reactions that emit heat are termed exothermic, and they often overlap with spontaneous processes because they favor energy dispersion, increasing the universe's entropy.

Entropy Change

Entropy change (\( \Delta S^{\circ} \)) is a measure of randomness or disorder in a system, offering insight into the feasibility of a process. For fuel cells, understanding \( \Delta S^{\circ} \) is pivotal in assessing how the disorder changes from reactants to products.
In our hydrogen-oxygen fuel cell, \( \Delta S^{\circ} \) is considered positive due to the conversion from gaseous reactants to liquid water, reducing the number of gaseous molecules. This decrease in gas molecules results in a positive entropy change, often correlating with an increase in randomness and contributing to system spontaneity. Consequently, the positive \( \Delta S^{\circ} \) supports the spontaneous and exergonic nature of the reaction. Additionally, a larger positive entropy change can temper the Gibbs energy change as temperature increases, decreasing the maximum obtainable work.