Grand canonical equilibrium unifies quantum electrodynamics and statistical mechanics in wet electrolytic environments at room temperature
Abstract
Quantum rate (QR) theory has previously shown that the charge-relaxation resistance of ferrocene redox monolayers is locked at Rq = h/2e2 ≈ 12.9 kΩ at room temperature, and that electron-transfer rates are accessible from equilibrium quantum capacitance without kinetic fitting. What has remained without a first-principles thermodynamic explanation is why a room-temperature ionic bath sustains rather than destroys quantum coherence. Here, we provide that mechanism. By applying the fluctuation-dissipation theorem to the grand canonical equilibrium of an electrochemical junction, we show that minimisation of the grand canonical potential (ΔΩ = 0) defines an isoscopic condition-a thermodynamic attractor in which the classical ionic reorganisation energy (e2/Ce) and the quantum charging energy (e2/Cq) are exactly equal and opposite, compelling the electrolyte to mirror rather than oppose the quantum state. A direct experimental test of this mechanism is that the charge-relaxation resistance must be universally locked at Rq = h/2e2 regardless of the solvent. We verify this prediction in ferrocene redox monolayers across three chemically distinct electrolytic environments (acetonitrile/water, acetonitrile, and dichloromethane) with a relative error of less than 7%. Furthermore, the isoscopic condition yields a parameter-free prediction of the Butler-Volmer standard rate constant, νμ = 8kBT/Nh, which agrees quantitatively with an independent Laviron analysis. Together with prior QR…
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