Photostationary State Lifetime Averaging
1. System and Setup
UV irradiation of species A drives a photoreaction that converts a fraction of A into a second, spectroscopically distinct species B. The reverse process (“glow” — thermal or light-driven back-conversion) regenerates A. At photostationary state, the sample is a static mixture of A and B in fixed proportions, and both species fluoresce (with hν₁ ≈ hν₂, so the two emissions are not spectrally resolved).
Figure 1. UV light converts A into B; the reverse pathway regenerates A. Each species has its own intrinsic fluorescence lifetime, τ₁ (A) and τ₂ (B). UV light diminishes the concentration of A.
Defining the photostationary population ratio:
K = (1 − α) / α ⇒ α = 1 / (1 + K)
where α is the fraction of the ensemble remaining as A and (1 − α) is the fraction converted to B. Before irradiation (K = 0), the lifetime measured is simply that of pure A:
initial lifetime = τ₁
2. Observed Lifetime of the Mixture
After UV exposure, the emission is a mixture of A- and B-type decay (hν₁ ≈ hν₂, so they cannot be separated spectrally). Treating the ensemble decay as a population-weighted average lifetime:
τobs = α·τ₁ + (1 − α)·τ₂
As irradiation proceeds, K changes (more A is converted to B), so α is not fixed. Substituting α = 1/(1+K):
τobs = τ₁/(1+K) + K·τ₂/(1+K)
which combines into the single working expression:
τobs = (τ₁ + K·τ₂) / (1 + K)
3. Limiting Cases
This expression correctly reproduces every physically sensible limit:
| Condition | Result |
|---|---|
| K → 0 (no conversion — pure A) | τobs → τ₁ |
| τ₁ = τ₂ = τ (species indistinguishable kinetically) | τobs = τ·(1+K)/(1+K) = τ |
| τ₁ ≫ τ₂, K small (τ₁ ≫ Kτ₂) | τobs ≈ τ₁ |
| τ₁ ≫ τ₂, K large, with τ₁ ≈ Kτ₂ | τobs = 2τ₁ / (1+K) |
| Kτ₂ ≫ τ₁ (K → ∞, essentially full conversion to B) | τobs → τ₂ |
4. Caveat: What This Average Actually Represents
The expression τobs = ατ₁ + (1−α)τ₂ treats the decay as if it collapses into a single effective exponential with a population-weighted lifetime. Strictly, if A and B decay independently with genuinely different rate constants, the true time-resolved signal is bi-exponential:
I(t) = α·e⁻ᵗ/ᵗ¹ + (1−α)·e⁻ᵗ/ᵗ²
and this only reduces to a clean single exponential with lifetime ατ₁ + (1−α)τ₂ in special limits (e.g. τ₁ ≈ τ₂), or when the quantity being extracted is specifically the amplitude-weighted mean lifetime from a fit to the decay, rather than a genuine single-exponential fit.
Since hν₁ ≈ hν₂ here, the two emissions cannot be spectrally separated, so using an averaged/effective τobs as an approximation is reasonable — but it should be stated explicitly as an approximation to an unresolved bi-exponential decay, not derived as a first-principles single-exponential law. The rigorous approach is to fit the measured decay to a bi-exponential model
Posted Sep 18, 2026, 8:39 AM UTC