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Justification for using effective radiation temperature Ty in Carnot efficiency and Boltzmann factor

Establish a rigorous justification for substituting the wavelength-dependent, non-equilibrium effective radiation temperature Ty(1) into the Carnot efficiency formula nmax = 1 − Tout/Ty(1) and into the Boltzmann factor exp(−ΔE/kBTy(2)) when analyzing photosynthetic light-powered systems, or determine the precise conditions under which these substitutions are valid, given that Ty(1) does not correspond to an equilibrium temperature.

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Background

The paper reviews the historical use of an effective radiation temperature (Ty) derived from diluted solar blackbody radiation to analyze theoretical efficiencies in photosynthetic systems. Traditionally, this Ty has been inserted directly into Carnot efficiency and Boltzmann factor expressions to estimate ideal performance and concentration ratios.

The author points out that Ty(1) is wavelength-dependent and represents a non-equilibrium quantity rather than a true thermodynamic temperature. As such, applying Ty(1) in equilibrium-based formulae (Carnot efficiency and Boltzmann factor) lacks a clear theoretical foundation, raising unresolved questions about validity and applicable conditions. The paper proposes a unified energy–entropy flow framework as an alternative, but explicitly notes that these historical questions remain unresolved in the literature.

References

Although this effective temperature has been widely used in the physicochemical analysis of the theoretical efficiency of photosynthesis, certain questions and critical discussions remain unresolved [17-21, 26-33]. These questions [21,24,27,33] stem mainly from the fact that this temperature Ty (1), which is not in equilibrium (i.e., not a true temperature), is automatically applied to Carnot efficiency and Boltzmann factor formulae, as nmax = 1 -Tout/Ty(1) and exp(-AE/kBTy(2)), respectively, without proper justification.

— Generalized formulation for ideal light-powered systems through energy and entropy flow analysis Part 2: Beyond the first-order evaluation under realistic conditions (2502.06797 - Yabuki, 28 Jan 2025) in Section 1 (Introduction)