Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2021 / Paper 2A / Section B / 22 / b

The phosphorescence quantum yield is the number of phosphorescence photons emitted per photon absorbed. In rate form,It is dimensionless. The denominator counts absorbed photons, not incident photons or the instantaneous triplet population.

Almost every absorbed photon generates a triplet, within the stated kinetic scheme. The branching fraction from the excited singlet isso $k_{\mathrm{ISC}}[S_1]\simeq I_{\mathrm{abs}}$ under steady illumination. This statement concerns the triplet formation yield; it does not mean that the triplet population or the phosphorescence yield is also one. The triplet can subsequently deactivate or be quenched.

The steady-state approximation for the excited singlet givesand that for the triplet givesCombining these with the quantum yield definition gives the more general expressionUnder the supplied limit $k_{\mathrm{ISC}}\gg k_F$, this becomesThe approximation is the near-unit triplet formation yield, not an assumption that phosphorescence is the only triplet loss process.

Without quencher, ${\varphi }_{P,0}=k_P/(k_P+k_D)$. Taking the ratio gives the Stern-Volmer equationThus the plot has unit intercept and slope $k_Q{\tau }_0$. A triplet formation yield less than one would cancel from this ratio if it remained independent of quencher concentration. The result assumes dynamic triplet quenching and no additional quencher-induced absorption or singlet loss.^[ref]

$k_Q=9.9\times 10^4\mathrm{d}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$. The unquenched triplet lifetime isSince the slope is $k_Q{\tau }_0$, the bimolecular rate constant isFor comparison, the unquenched phosphorescence yield in the unit-triplet-yield approximation is $180/198=0.909$.