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I'm reading this paper, which states that the chemical potential $\mu$ is determined by the steady-state balance of up and down transitions in a fluorescent molecule. I am happy with this interpretation. But what I don't know how to verify is the next statement:

"For a high-quantum-efficiency fluorescent molecule illuminated at the earth surface, $\mu$ is usually $0.6$ or $0.7$ times $h\nu$ at the centre of the absorption band".

How would I show that $\mu$ is usually $0.6$ or $0.7$ times $h\nu$ at the centre of the absorption band?

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    $\begingroup$ This would seem to be due to the $T\Delta S$ caused by the Stokes shift that leads to heat not photons, but it's not clear why it has these values. You will have to test some values of frequency shift. $\endgroup$
    – porphyrin
    Commented Jun 22, 2020 at 11:10
  • $\begingroup$ Apologies, I'm not quite sure I understand. So, a portion of the absorption leads to heat - but surely at high efficiency, this would be diminished? $\endgroup$
    – Tomi
    Commented Jun 22, 2020 at 19:19
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    $\begingroup$ There can still be a wavelength shift between absorption and emission even if the efficiency is high. $v=0\to 1$ is different in wavelength $v=1\to 0$ due to relaxation in excited state vs ground state due to differing interactions with solvent caused by changed electron distribution. The mean absorption and mean emission will always have a shift . $\endgroup$
    – porphyrin
    Commented Jun 23, 2020 at 6:40

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