Derivation of the Nernst-to-Carnot conversion

This technical appendix provides the rigorous mathematical bridge between the electrochemical potential of a fuel cell and the thermal efficiency of a Carnot cycle. By tracking the entropy flux through the chemical species, we can prove that the "Voltage Discount" found at high temperatures is exactly proportional to the Carnot limit.

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Technical Appendix: A Formal Derivation of the Nernst-Carnot Equivalence

To demonstrate that the Nernst equation and the Carnot efficiency are different expressions of the same physical law, we define a closed-loop system where water is decomposed at a high temperature ( $T_H$ ) and recombined at a lower temperature ( $T_L$ ).

1. The Energy Balance of Water Decomposition

The total energy required to split a water molecule is the Enthalpy of Reaction ( $\Delta H$ ). In a thermochemical system, this energy is provided by two sources: electrical work ( $W$ ) and thermal energy ( $Q$ ). According to the Second Law:

$$\Delta G=\Delta H-T\Delta S$$

Where:

• $\Delta G$ (Gibbs Free Energy) is the maximum non-expansion work (electricity) required or produced.

• $T\Delta S$ is the heat exchange with the reservoir required to maintain isothermal conditions.

2. The High-Temperature Electrolysis (HTE) Stage

At $T_H=1500^{\circ }\text{C}$ ( $1773\text{ K}$ ), we provide a quantity of electrical work ( $W_{in}$ ) to the electrolyzer. To minimize the electrical input, we operate at the reversible limit where the cell absorbs heat ( $Q_{in}$ ) from the high-temperature source:

$$W_{in}=\Delta G(T_H)=\Delta H-T_H\Delta S$$

$$Q_{in}=T_H\Delta S$$

3. The Low-Temperature Fuel Cell (FC) Stage

The separated $H_2$ and $O_2$ are cooled (ideally via a perfect recuperator) and reacted in a fuel cell at $T_L=100^{\circ }\text{C}$ ( $373\text{ K}$ ). The electrical work produced ( $W_{out}$ ) is:

$$W_{out}=-\Delta G(T_L)=-(\Delta H-T_L\Delta S)$$

(Note: We assume $\Delta H$ and $\Delta S$ are approximately constant over this temperature range for the sake of the ideal derivation. In a real-world model, these would be integrated over $C_p$ .)

4. The Net Work and Efficiency

The Net Work ( $W_{net}$ ) produced by the entire chemical engine is the difference between the electrical energy generated at the cold end and consumed at the hot end:

$$W_{net}=W_{out}-W_{in}$$

$$W_{net}=(\Delta H-T_L\Delta S)-(\Delta H-T_H\Delta S)$$

$$W_{net}=(T_H-T_L)\Delta S$$

The Efficiency ( $\eta$ ) of the system is the Net Work divided by the external Heat Input ( $Q_{in}$ ) provided at $T_H$ :

$$\eta =\frac{W_{net}}{Q_{in}}=\frac{(T_H-T_L)\Delta S}{T_H\Delta S}$$

By canceling $\Delta S$ , we arrive at:

$$\eta =\frac{T_H-T_L}{T_H}$$

5. The Nernstian Perspective

To express this in Volts, we substitute the Nernst relationship ( $\Delta G=-nFE$ ) into the Net Work equation. Let $E_L$ be the potential at the fuel cell and $E_H$ be the potential at the electrolyzer:

$$nF(E_L-E_H)=(T_H-T_L)\Delta S$$

This shows that the Voltage Gap ( $E_L-E_H$ ) is the electrical representation of the temperature gradient. If you were to measure the voltage of a fuel cell at 100°C and an electrolyzer at 1500°C, the "missing voltage" ( $0.39\text{V}$ in our specific case) is the physical manifestation of the Carnot heat being converted into work.

6. Conclusion on System Boundaries

The derivation proves that a fuel cell does not "bypass" the Carnot limit. Instead, the fuel cell is merely the expansion stroke of a chemical heat engine.

• If you ignore the $H_2$ production, the fuel cell appears 100% efficient (relative to $\Delta G$ ).

• If you include the thermal $H_2$ production, the system boundary encompasses a classic heat engine where $H_{2}O$ is the working fluid and the "voltage discount" at $T_H$ is the entropy-driven mechanism of the Second Law.

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