The modelling of Surface-Water photoreactions made easier: introducing the concept of ‘equivalent monochromatic wavelengths’

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The recent development of successful model approaches that predict the photochemical behaviour of surface waters has greatly aided in the understanding of how water environments work and will likely work in the future, from a photochemical point of view. However, the inherent multi-wavelength (polychromatic) nature of environmental photochemistry causes the relevant mathematics to be quite complex, which discourages many scientists to carry out photochemical calculations. To greatly simplify model mathematics, this paper proposes a new approach that is based on a monochromatic approximation to the polychromatic problem, introducing the concept of what is here defined as equivalent monochromatic wavelengths (EMWs). The EMW is the single wavelength that reproduces the behaviour of the polychromatic system, using a monochromatic (Lambert-Beer based) equation. The EMW approach largely simplifies calculations, getting rid of integrals and allowing for much more straightforward and manageable equations to be obtained. In particular, this work shows that: (i) the EMW approach, although approximated, entails a negligible loss in accuracy compared to the exact polychromatic treatment of photochemical reactions; (ii) in the case of direct photolysis, the quantum yield is to be replaced by an apparent photon efficiency that is not bound to be < 1 (quantum yields can actually be > 1 for chain reactions and few other cases, but this is not the point here); (iii) the monochromatic Lambert-Beer equations work in most cases once the EMW is identified, with the present exception of sunlight absorption by chromophoric dissolved organic matter (CDOM). The latter spans a very wide wavelength range (from 300 to at least 600 nm), which makes a single-wavelength treatment more difficult. However, a relatively small modification to the monochromatic Lambert-Beer equation allows for successfully using the EMW approach, in the case of CDOM as well. The near-perfect coincidence between polychromatic and EMW-based predictions of photodegradation kinetics is here shown for the pollutants atrazine, bentazone, carbamazepine, diclofenac, diuron and ibuprofen. Extension to additional compounds requires translation of the traditional, polychromatic language into the EMW one. Hopefully, this contribution will introduce a new paradigm in the mathematical description of photochemical reactions in environmental waters. It could also become a new and simple way to treat multi-wavelength systems in general photochemistry studies, thereby completely changing the way multi-wavelength problems are dealt with.