Photochemical characterization of rate laws, rate constants and photonicities in optically–dense, multiphoton–reactive systems

[Display omitted] •Averaged intensities account for Beer’s Law–induced decreases in intensity.•Averaged intensities facilitate analysis of optically–dense, multiphoton reactive systems.•Incident intensities yield unreliable rate constants.•Averaged intensities yield reliable rate constants and photonicities.•Averaged intensities apply well to multiphoton synthesis and photodynamic therapy. A novel method for characterizing the rate constants k and photonicities n of multiphoton–induced reactions in optically dense media is presented. Conventionally, k and n are calculated by irradiating samples of identical absorbance with different incident initial intensities I0,0 and characterizing the rates of change in absorbance, with incident rate constants k0,0 being proportional to I0,0n for n–photon processes. In our method,I0,0n is replaced by a longitudinally–and–temporally–averaged nth–order intensityIx,tn¯=1-10-nA¯2.303nA¯I0,0n which accounts for the exponential decrease in intensity across the optical pathlength due to the average absorbance A¯ over an irradiation interval of duration t. We demonstrate the utility of Ix,tn¯ by contrasting longitudinally–and–temporally–averaged rate constants kx,t and photonicities nx,t with incident rate constants k0,0 and photonicities n0,0 for the 532 nm, two–photon induced photodegradation of β–carotene in CCl4 solvent. Because the actinic intensity declines more rapidly with x in high (A ∼ 1.0) than low (A ∼ 0.2) absorbance samples, the incident rates R0,0=k0,0I0,0n are faster in the low absorbance samples: i.e., k0,0(A532 = 0.2) > k0,0(A532 = 1.0). In significant contrast, we find that kx,t(A532 = 1.0) ∼ kx,t(A532 = 0.2). indicating that kx,t provides better estimates of the rate constant than k0,0. As with kx,t, we find that the photonicities are minimally affected by A532; i.e., n0,0 ∼ nx,t for both low and high absorbance samples. Hence, the dependence of the rate on the actinic absorbance is incorporated principally into the incident rate constants k0,0; kx,t, n0,0 and nx,t are largely unaffected by A532. We detail potential applications of our method to both multiphoton–induced chemical synthesis and multiphoton–induced photodynamic therapy scenarios, as high absorbance samples are advantageous for both of these applications.