An upper bound on the rate coefficient

Reck and Prager [507] wrote the diffusion equation for fluorophor density, n, as 

As a steady state can be established and the effect of natural fluorescence decay can be removed with a transformation m = n exp f/r), so that the decay of exited fluorophors due to reaction with quenchers is represented by 

The density m(r) satisfies the same boundary conditions as it does in the Felderhof and Deutch [25] theory 

The average (macroscopic) density of excited fluorophors in a volume V is 

As the effect of the natural fluorescence decay has been removed from eqn. (263), the effective second-order quenching rate coefficient is k p). The rate of fluorescence quenching is kpM, where p is the quencher density. Because a steady state exists, this rate of quenching balances the 

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