As for thermal kinetics, in photokinetics, a rate law should be written for each species Y j (where 0 ≤ j ≤ 7 , with the reactant X = Y 0 and the photoproducts Y 1   t o   7 ) of the reaction mechanism (Scheme 1). The n Φ j reaction steps linking Y j can correspond to either forward or backward reactions with other Y j ′ species ( j ≠ j ′ ). Each rate law, for a given Y j ( r Y j λ i r r t ), can then take the following general formulation, irrespective of the actual reaction mechanism. r Y j λ i r r t = C Y j λ i r r t ′ = ∑ j ′ ≠ j n Φ j − Φ Y j   →   Y j ′ λ i r r   P a Y j λ i r r t + Φ Y j ′   → Y j λ i r r   P a Y j ′ λ i r r t . (1)

It is to be noted that it is unusual in kinetics that the concentration of species Y j at time t is written relative to the irradiation wavelength, as given in Eq. (1). This labelling is necessary simply because all the parameters on the right-hand side of Eq. (1) are dependent on λ i r r . This makes two traces of a species, obtained at two different wavelengths λ i r r 1 and λ i r r 2 (e.g., C Y j λ i r r 1 t and C Y j λ i r r 2 ), most certainly different from each other.

In Eq. (1), the light absorbed by species Y j or Y j ′ ( P a Y j λ i r r and P a Y j ′ λ i r r , with j ≠ j ′ ), constitutes fractions of the total absorbed light by the medium ( P a λ i r r ) at time t (expressed in e i n s t e i n   s − 1   d m − 3 ), according to the following expression: P a Y j   o r   j ′ λ i r r t = A Y j   o r   j ′ λ i r r t A t o t λ i r r t   P 0 λ i r r   1 − 1 0 − A t o t λ i r r t = A Y j   o r   j ′ λ i r r t   P 0 λ i r r   P K F t , (2) where P 0 λ i r r is the intensity of the incident radiation (see Section 3.9 for a detailed definition), the total absorbance ( A t o t λ i r r ) is a sum of the individual absorbances of the species ( A Y j λ i r r ) at λ i r r ( n s p , in Eq. (3), represents the number of all species Y j —reactant and products—in the medium at time t ), and P K F t is the dimensionless photokinetic factor. A t o t λ i r r t = ∑ j = 0 n s p A Y j λ i r r t . (3)

Obtaining analytical solutions of Eq. (1) rate-law equations, for a vast majority of cases, is simply impossible. This mathematical hurdle can be explained by the fact that these differential equations are non-linear due to the presence of the time-variable term 1 − 1 0 − A t o t λ i r r , which encompasses all absorbances of the species present in the reactive medium. This situation is expected to persist for the foreseeable future until a mathematical technique, capable of analytically solving these complicated differential equations, is devised.

The traces of a primary photoprocess subjected to monochromatic light were proven to obey a Φ -order equation that was analytically derived from the reaction’s rate law (Maafi and Brown, 2007). Previous work on other photomechanisms (Maafi and Maafi, 2013; Maafi and Maafi, 2014a; Maafi and Maafi, 2016), whose rate laws cannot be solved by closed-form integrations, has also shown that the traces of photoreactions, involving up to five photochemical steps, can be well described by specific semi-empirical formulae, which mathematically possess a Φ -order character, but under certain constraints on the total absorbance at the end of the reaction ( A t o t λ i r r ∞ ranging between 0.2 and 0.6). However, till now, no general formula has been proposed for the description of traces of photoreactions. A conjecture, based on the aforementioned results, might suggest that such a general equation would be of a Φ -order type.

Accordingly, the following explicit formula (Eq. 4) is proposed to describe the trace of species Y j ( C Y j λ i r r t , j = 0 for the reactant X ), irrespective of the mechanism undergone by the actual photoreaction, and the wavelength ( λ i r r ) of the non-isosbestic monochromatic irradiation light driving the reaction. The reaction mechanism may involve any number of photochemical reaction steps (e.g., within Scheme 1). The temporal variation in the concentration of a typical species Y j is given by Eq. 4 (where Log and e are, respectively, the base 10 logarithm and the exponential functions and i j is the number of mono- Φ -order terms under the sum). C Y j λ i r r t = C ∞ , j λ i r r + ∑ i = 1 i j ω i j λ i r r   Log 1 + c c j λ i r r e − k i j λ i r r   t . (4)

The expression for the reaction rate of species Y j is obtained by the differentiation of the corresponding Eq. 4 as C Y j λ i r r t ′ = r Y j λ i r r t = − ∑ i = 1 i j ω i j λ i r r   c c j λ i r r   k i j λ i r r   e − k i j λ i r r   t 1 + c c j λ i r r   e − k i j λ i r r   t   ln 10 , (5) from which we can derive the formula for Y j initial reaction rates (expressed in M   s − 1 ) as C Y j λ i r r t t = 0 ′ = r 0 Y j λ i r r = − 1 ln 10   ∑ i = 1 i j ω i j λ i r r   k i j λ i r r c c j λ i r r 1 + c c j λ i r r . (6)

The parameters ( C ∞ , j λ i r r , ω i j λ i r r , c c j λ i r r , and k i j λ i r r ) of Eq. 4 are specific to the irradiation wavelength ( λ i r r ), making C Y j λ i r r wavelength specific. The parameter C ∞ , j λ i r r , standing for the final concentration of species Y j , will be non-zero, but positive, if the species persists at the end of the reaction (typically for photostationary reactions and end products). The pre-exponential coefficient c c j λ i r r is considered here to be the same for all i j terms making up Eq. 4 of a given species Y j ( c c j λ i r r = c c λ i r r , a constraint that is verified practically). The pre-logarithmic parameter ω i j λ i r r , accepting positive and negative values, might be considered a weighing factor for the logarithm of the i t h kinetic regime of the considered species Y j . Parameter k i j λ i r r is the positive rate constant of the i t h kinetic regime (expressed in s − 1 ) and is proportional to the light intensity and the quantum yield of the i t h kinetic regime, but it may also depend on other attributes of the reactive species (these details are currently unknown).

In order to test the proposed equations (e.g., Eq. 4), a Φ -shaped mechanism is selected (Scheme 1, involving the reactant, X , seven photoproducts, Y , and 14 reaction steps). It is thought to be representative of a number of simple and complicated photoreactions encountered in photochemistry books and literature (Mauser et al., 1998; Crano and Guglielmetti, 2003; Tonnesen, 2004; Nakatani et al., 2016; Montalti et al., 2020; Scaiano, 2020; Pianowski, 2022). The number of sub-mechanisms that can be worked out from Scheme 1 exceeds 100 examplars (including the counting of the absorption of terminal species that might equal to zero).

The set of sub-mechanisms worked out from Scheme 1 includes, but is not limited to, those of the primary photoprocess with a transparent photoproduct (as for photochromic diarylethenes under visible light (Crano and Guglielmetti, 2003; Nakatani et al., 2016; Maafi and Alqarni, 2022; Pianowski, 2022)), the primary photoprocess with an absorbing photoproduct (as for nifedipine, nisoldipine, and dacarbazine drugs (Tonnesen, 2004; Maafi and Maafi, 2013; Maafi and Lee, 2015a)), the photoreversible reaction (as for stilbenoids (Crano and Guglielmetti, 2003; Maafi and Al-Qarni, 2019), anti-cancer drugs (Tonnesen, 2004; Maafi and Lee, 2015b), and dimethylhydropropenes (Ziani et al., 2021)), the consecutive double photoreversible system (as for fulgides in solution or in the solid state (Crano and Guglielmetti, 2003; Weerasekara et al., 2017; Kochman et al., 2022; Pianowski, 2022) and benzopyrans (Frigoli et al., 2020)), the multi-consecutive photoreaction (as for riboflavin (Maafi and Maafi, 2016)), and other more complex systems (Nakatani et al., 2016; Scaiano, 2020; Malato et al., 2021; Pianowski, 2022). The mechanisms and molecular systems reviewed previously have been extensively investigated, but their photokinetic studies tended to be either performed by using the equations of thermal kinetics (e.g., first-order exponential models), using various proposed (non-analytical) equations, or employing numerical integration. It is to be noted that no standard equation model has ever been provided in the aforementioned documentation and no integrated rate law has ever been analytically derived for photoreactions (including the simplest ones, such as the concurrent Y 3 ← X → Y 1 reaction).

From a kinetic view point, the rate of a thermal reaction of a species Y j does not depend on the properties (e.g., concentrations) of distant species from Y j (that are not generated by or generate themselves Y j ), whereas in photokinetics, reactions’ rates depend not only on the properties of the species involved in the considered reaction step, as in thermal kinetics, but also on those of all the absorbing species occurring in the reaction medium over time. In fact, the rate of a photochemical reaction depends explicitly on the absorbances of all the species with a non-zero absorbance at a given time t . This particular difference between photo- and thermal kinetics is expressed in the rate equation (Eqs 1, 2) by the total absorbance of the set of reactive species ( A t o t λ i r r t ) present in the medium at a given time. Such a subtle interdependence of the reaction species within the photokinetic rate law is responsible for the species’ competition for the incident light. Therefore, species that belong to completely separated reaction steps of the overall reaction mechanism still influence each other’s reactivities. In the same context, the kinetics of species Y j might also be dependent on the absorbance of one or more inert (non-reactive) species, present in the medium as spectators, when the latter absorb at the irradiation wavelength. This particular phenomenon is further described in Section 3.7.

Furthermore, since the coefficients of the rate-law equation correspond to the attributes of the species and light at a given wavelength, it is evident that the solution of that equation is also specific to that wavelength. Therefore, changing the irradiation wavelength will cause the trace to change accordingly. This imposes that all Eq. 4 coefficients are indexed by λ i r r (including the concentrations, C Y j λ i r r t ). In thermal kinetics, this would be equivalent to a change in temperature. Here, apart from a qualitative comparison (e.g., reaction speeds up between irradiations at λ i r r 1 and λ i r r 2 ), it is difficult to proceed with further quantitative analysis based on the comparison of the traces in the absence of an integrated rate law and the quantitative parameters it provides. For instance, the knowledge of the relative reaction velocity at a given time but at different wavelengths can help selecting the light wavelength range causing the species at hand to be most or least reactive depending on the purpose of the experimentalist/experiment. For some molecules (drugs, photochromes, etc.), it is it useful to specify the regions of light inducing most photoconversion, a situation which would be solved for photoreactions using Eq. 4, as will be shown in the following sections. One additional difference between thermal and photo-kinetics concerns an evidenced change in the photoreaction kinetic order. It was previously proven that the trimolecular cyclic photoreaction (Maafi and Brown, 2005b) and all its sub-mechanisms obeyed a first-order kinetics when irradiated with an isosbestic monochromatic beam ( λ i s o s coincides with an isosbestic point on overlayed successive absorption spectra of the medium through reaction time). It is reasonable to predict that any photoreaction will also be described by the first order kinetics whenever subjected to an isosbestic irradiation since the photokinetic factor [ P K F t , Eq. (2)], under these conditions, is independent of time (which converts the rate law into a first-order linear differential equation). However, the kinetics of any photoreaction, under non-isosbestic light, extracted from the Φ-shaped mechanism, is shown in the following sections to follow Φ -order kinetics. The conjecture is to observe similar conclusions for any photoreaction. Accordingly, a general statement can be derived from the aforementioned observations: the kinetic data of a photoreaction can either obey a first-order or a Φ -order kinetics, respectively, when driven by an isosbestic or a non-isosbestic irradiation light (when all other experimental and reaction attributes are kept the same). A true change in the kinetic order then specifically occurs for photoreactions (the equivalent for thermal reactions would perhaps correspond to a change in the kinetic order when temperature changes, which is not observed). Also, it is noticeable that the mathematical formulation of the rate law explicitly includes a rate-constant factor (k) for thermal reactions but not for photochemical reactions.

Finally, the concentrations of the species involved in a photokinetic process must all belong to their respective calibration graphs (a redundant condition for thermal kinetics).

For generality purposes of the application of the model equation (Eq. 4), RK simulations were conducted on several sub-mechanisms derived from Scheme 1. Individual RK traces have been obtained for each species, involved in the sub-mechanism considered, and the fitting of each of these traces was performed with its corresponding Eq. 4.

The process of RK trace fitting was implemented according to a simple protocol. The first trace to be fitted was the reactant’s, followed by that of the nearest Y j species in the reaction mechanism, then the next, and so on. The fitting parameters k i j λ i r r obtained for an earlier species (e.g., k 10 λ i r r and k 20 λ i r r for X and j = 0 ) were kept the same in the subsequent equations, describing the trace of the next species linked to X , where the latter might involve new terms (e.g., k 11 λ i r r = k 10 λ i r r , k 21 λ i r r = k 20 λ i r r , and k 31 λ i r r for Y 1 ). Kinetically, this means that the trace of a given species in the reaction mechanism is necessarily dependent on the rate constants (both their number and values) of previous reaction steps generating that species in the reaction mechanism. Accordingly, the number i j of mono- Φ -order terms in Eq. 4 is minimal for the reactant and is expected to increase (or at least remains the same) for species occurring subsequently. The maximum number of terms in Eq. 4 that describe the trace of any single species cannot exceed the number of reaction steps occurring in the considered overall reaction mechanism ( i j ≤ n Φ ). The minimum number of these terms should, in principle, be equal to the number of reaction steps starting or ending at the considered species (i.e., n Φ j ). However, in some cases and depending on the kinetics, Eq. 4 can provide a good fit of the traces with less than n Φ j terms. Interestingly, it is predicted that most likely the same number of terms is found for each Eq. 4 of each species in a cyclic reaction mechanism involving three or more molecules (e.g., Figure 1).

Furthermore, it has been observed that this fitting protocol indicates that factor c c j λ i r r keeps the same value ( c c j λ i r r = c c λ i r r ) for all Eq. 4 of all the species belonging to a given reaction mechanism. C ∞ , j λ i r r takes the value of the final concentration of the considered species Y j , and ω i j λ i r r are different for different traces of the species of the same mechanism.

This coefficient, c c λ i r r , hence, plays the role of a coupling factor of the system of Eq. 4, describing the studied reaction. This feature will impose an additional constraint on the fitting of the photokinetic traces, which is expected to improve the reliability of the fitting parameters’ values.

It is possible to add a few more constraints to the fitting protocol, by considering Eq. 4 at the initial time ( t = 0 ) and the known initial concentration values of the species. C Y j λ i r r 0 = C ∞ , j λ i r r + ∑ i = 1 i j ω i j λ i r r   Log 1 + c c j λ i r r . (7)

In the present study, more than 200 RK traces were fitted according to the protocol described previously. These traces belonged to species involved in various sub-mechanisms derived from the Φ-shaped reaction (Scheme 1), including one to seven photoproducts and 1 to 14 reaction steps.

The individual species’ RK-generated traces were fit by adequate Eq. 4 of the reaction studied with various numbers of mono- Φ -order terms making up each of these equations. The protocol described previously for the treatment of the RK data with Eq. 4 worked extremely well for all traces. Excellent fittings of each individual trace (illustration in Figure 1) were obtained with correlation coefficient values, for RK-calculated vs. Eq. 4 data, not lower than 0.999, sums of squared errors (SSE) as low as 10⁻²², and root mean square errors (RMSE) of no higher than 10⁻⁹.

These results prove that the Φ -order kinetic behaviour is preserved for photochemical reactions, irrespective of the operating sub-mechanism that can be extracted from Scheme 1. This stands for a confirmation of our conjecture that the Φ -order equation is the seminal model for photoreactions despite the fact that it has only been analytically derived for the primary photoprocess whose photoproduct is transparent to the radiation used (Maafi and Brown, 2007). Such a situation might be an analogue to that observed for thermal reactions where the mono-exponential function, obeyed by the kinetics of the simplest unimolecular reaction, is the basis for the description of kinetic data belonging to any other, more complex thermal reaction mechanisms. Incidentally, this is somewhat supported by the typical Φ -order equations set out for C t by the semi-empirical method for several reactions (Maafi and Maafi, 2013; Maafi and Maafi, 2014a; Maafi and Maafi, 2014b; Maafi and Maafi, 2016).

The findings of the present work also allow us to put forward a conjecture stipulating that Φ -order kinetics (based on Eq. 4 template) maps out, in general, the behaviour of photoreaction kinetics (irrespective of the photomechanism considered, i.e., beyond Scheme 1).

Hence, the formulation of the integrated rate equation given by the proposed Eq. 4 becomes the unifying model in photokinetics. This represents the first time that a unique explicit description is proposed for the traces of photoreactions.

Conversely and even if some photokinetic traces may well be fitted by one or several mono-exponential functions, such an approach remains limited and cannot be generalised. It is unlikely to reach similar fitting results by using mono-exponential functions instead of Φ -order equations (of Eq. 4 template), keeping the kinetic meaning of the fitting results. It has already been shown that such an interchange of ( Φ -order/exponential) equations does not always work for the simplest primary photoreaction (Maafi and Brown, 2007). More importantly, this observation is fully justified, in principle, by the significant difference between the mathematical formulations of the rate laws describing photoreactions and thermal reactions. These differences in the differential equations lead to a predictable dissimilarity of the mathematical solutions that would be derived for each (e.g., first- and Φ -order kinetics).

In addition to the successful fitting of the traces of photoreactions by the unifying Eq. 4 model, this study has revealed an aspect that must be considered in any practical photokinetic investigation. Indeed, despite the constraints imposed on the fitting of the traces [as given by the coupling factor c c λ i r r and Eq. (7)], it has become evident that reaching a unique solution (a unique set of values for the fitting parameters c c λ i r r , ω i j λ i r r , and k i j λ i r r ) for a given reaction is not achievable. This translates the occurrence of an identifiability issue (Vajda and Rabitz, 1988; Maafi and Brown, 2005; Hattersley et al., 2011; Ovchinnikov et al., 2023), i.e., changing the initial values where the fitting calculation starts, yields a quite large number of different sets of fitting values of the parameters. Each of these sets has equally excellent fitting metrics of the RK traces, indicating that the Levenberg–Marquardt algorithm can converge to different minima. The reason behind the identifiability issue is a smaller number of linearly independent equations describing the reactive system compared to the number of fitting parameters. In addition, Eq. (7) is linearly independent from Eq. 4, but is non-linear.

This point is well illustrated in the case of the simplest photoreaction ( X → Y 1 , whose photoproduct is transparent to the irradiation light). Here, the trace is described by a mono- Φ -order equation. Nonetheless, it is not possible to single out the true solution of the system (defined by the true values of ω 10 λ i r r , c c λ i r r , and k 10 λ i r r describing the reaction), where successive fittings (with different initial values) yield several possible sets of fitting parameters’ values with excellent fit of the trace. For more extended reactions, the insolvability of the identifiability issue becomes obviously more acute as the ratio between parameters’ and equations’ numbers becomes even larger. Overall, the fundamental problem posed here emerges from the unavailability of ω i j λ i r r , c c λ i r r , and k i j λ i r r explicit expressions (which is itself due to the impossibility of an analytical integration of the system’s differential equations). This concept is easily verified for the case of the only reaction whose solution is analytically derived and the expressions of its parameters are known, namely, the primary photoprocess (Maafi and Brown, 2007). Indeed, among all possible sets of parameters of this reaction’s Eq. 4 (which is a mono- Φ -order equation model) that generate good fit of an RK trace, it is possible to single out the true solution set of fitting parameters’ values ( ω 10 λ i r r , c c λ i r r , and k 10 λ i r r ) based on the fact that the equations of the latter coefficients are known (Maafi and Brown, 2007).

In this context, it is important to stress that such an identifiability issue cannot be solved by considering a combination of traces obtained at “any” number of different irradiation wavelengths, as might have been previously proposed (Crano and Guglielmetti, 2003; Delbaere et al., 2011; Micheau et al., 2014). The reason for this inconsistency is a proportional increase of the unknown parameters with each additional Eq. 4 when increasing the number of traces at different λ i r r . The seemingly success of the aforementioned approach came at a cost: it must assume that the quantum yields of the individual reaction steps are necessarily invariants with λ i r r (i.e., each species has a unique quantum yield for all λ i r r , or Φ Y j λ i r r = Φ Y j ). The latter condition addresses a very particular situation, which, in any case, needs to be proven experimentally and not assumed (the description of the latter approach (Crano and Guglielmetti, 2003) has neither proven experimentally such an invariability of the quantum yield nor considered the necessity of recommending an experimental proof on the quantum yield invariability before envisaging its application).

In a previous study by our team (Maafi and Brown, 2008), detailed spectrokinetic methods have been proposed with the aim of solving the identifiability problem (i.e., leading to the true solutions) of photoreversible systems. They showed the limits of using several traces obtained for a reactive system exposed to different irradiation wavelengths.

For the temporal variation in the total absorbance of the reactive medium ( A t o t λ i r r / λ o b s t ) irradiated at a given λ i r r and monitored at λ o b s , where λ i r r and λ o b s may or may not be equal (i.e., ε Y j λ o b s and l o b s might be different from those measured at λ i r r ), Eq. 4 equivalent takes the following form: A t o t λ i r r / λ o b s t = A t o t λ i r r / λ o b s ∞ + ∑ i = 1 n Φ A ω i , A λ i r r   Log 1 + c c A λ i r r e − k i A λ i r r   t . (8)

Note that Eq. (8) has the template of Eq. 4 because the former is a linear combination of the latter for a given reactive system. Hence, the number of terms in Eq. (8) ( i A ) would, at most, be equal to the number of reaction steps occurring in the photoreaction ( n Φ ). For the trace of A t o t λ i r r / λ o b s t (Eq. (8)), A t o t λ i r r / λ o b s ∞ is the total absorbance of the reactive medium at the end of the reaction, the factor ω i , A λ i r r is equivalent to the ω i j λ i r r of the species individual traces in Eq. 4 multiplied by the respective constants ε Y j λ o b s   l o b s , the factor c c A λ i r r is also constant in all terms of Eq. (8), and the rate constants of the different reaction’s regimes, in Eqs. (4) and (8), are invariant, i.e., k i A λ i r r = k i j λ i r r .

It is important, however, to mention that the total absorbance trace varies according to the conditions in which the measurement is performed. A t o t λ i r r / λ o b s is depending on which of the following combinations is considered: i) λ i r r = λ o b s and l i r r = l o b s ; ii) λ i r r = λ o b s and l i r r ≠ l o b s ; iii) λ i r r ≠ λ o b s and l i r r = l o b s ; iv) λ i r r ≠ λ o b s and l i r r ≠ l o b s . Hence, the properties of A t o t λ i r r / λ o b s t traces, and consequently, the fitting parameters, except k i , A λ i r r , will change between situations (i–iv), for a given reactive system.

Overall, however, the fitting of the RK traces of given systems, with Eq. 4 models, will allow us to describe some general aspects of photokinetics under monochromatic light, as well as to construct a method to fully solve such a photokinetics even when Φ λ i r r is variable with λ i r r , as it will be discussed in the following sections.

Basically, there are two types of coefficients that fully define the kinetics of a photoreaction: extrinsic and intrinsic coefficients. The former, such as C X 0 , P 0 λ i r r , l i r r , and l o b s , are controlled by the experimentalist and represent the experimental conditions (assumed known). The latter ones are specific features of the different species involved in the photoreaction, i.e., ε Y j λ i r r , Φ Y j   →   Y j ′ λ i r r (with j ≠ j ′ ).

In general, one of the metrics in kinetics is the reaction rate constants ( k ). However, this typical approach is compromised when the identifiability issue cannot be solved, as previously seen for the fitting of the photokinetic RK traces with Eq. 4. In order to circumvent such a hurdle, the use of the initial reaction rate of the reactant and/or those of the species emanating from the reactant would represent a more appropriate means to quantifying the variability of the reaction speed.

It is to be noted that this metric (e.g., R K : r 0 X λ i r r ) is the only accessible trace parameter by numerical integration (the rate constants of the reaction regimes are, in general, not calculated by NIMs and not possibly calculated by the experimentalist from those traces or NIM-fed parameters since their explicit formulae are not known). Therefore, since the initial species’ rates can readily be calculated directly by NIMs ( R K : r 0 λ i r r ) for all reactive species, they become good photokinetic metric tools, indicative of the reaction performance. The proportionality between r 0 X λ i r r and k i X λ i r r , as indicated by Eq. (6), validates the characterisation of the reactant’s photoreactivity by its initial rate.

Overall, the speed up (or the slowdown) of the reactant photoconversion is expected to be followed by the rest of the reaction species’ reactivities in relative proportions (for instance, a change in the rate when a value of an extrinsic coefficient is changed). Consequently, the variation in the reactant speed in one direction is indicative of the variation in the velocities of each of the subsequent species of the reaction in the same direction.

For the present study, another way is offered for the determination of the initial reaction rate, that is, by applying Eq. 4 to the RK trace of the reactant. The negative initial reactant rate [Eq. (9)], for the Φ -shaped reaction mechanism (Scheme 1) where X is depleted to form Y 1 and Y 3 , can be worked out from the following expression: F i t : r 0 X λ i r r = F i t : r 0 Y 1 λ i r r + F i t : r 0 Y 3 λ i r r = − 1 ln 10   c c λ i r r 1 + c c λ i r r ∑ i = 1 i 0 ω i 0 λ i r r   k i 0 λ i r r , (9)

where the individual, positive, and initial photoproduct rates ( F i t : r 0 Y 1 λ i r r and F i t : r 0 Y 3 λ i r r ) are obtained from their respective Eq. 4 as F i t : r 0 Y 1 λ i r r = − 1 ln 10   c c λ i r r 1 + c c λ i r r   ∑ i = 1 i 1 ω i 1 λ i r r   k i 1 λ i r r , (10) F i t : r 0 Y 3 λ i r r = − 1 ln 10   c c λ i r r 1 + c c λ i r r   ∑ i = 1 i 3 ω i 3 λ i r r   k i 3 λ i r r . (11)

It is important to notice that the values of F i t : r 0 λ i r r obtained in this way will not vary for the different sets of fitting parameters possibly generated for the RK traces (due to the occurrence of the identifiability issue), as long as each of the sets produces an excellent fit of the RK traces by the corresponding Eq. 4.

An additional way to confirm the validity of the aforementioned equations is to compare the values they generate with those obtained from the theoretical rate law of the reaction [Eq. (1)]. The values of T h e o : r 0 X λ i r r , T h e o : r 0 Y 1 λ i r r , and T h e o : r 0 Y 3 λ i r r can be obtained from the following equation: r 0 X λ i r r = − r 0 Y 1 λ i r r − r 0 Y 3 λ i r r = − Φ X → Y 1 λ i r r + Φ X → Y 3 λ i r r   P 0 λ i r r   1 − 1 0 − A X λ i r r 0 . (12)

The reliability of our methodology should be confirmed by the equality F i t : r 0 Y j λ i r r = T h e o : r 0 Y j λ i r r = R K : r 0 Y j λ i r r in all circumstances (Figure 2).

Accordingly, and despite the identifiability problem, it is possible to consistently evaluate the kinetics of a photoreaction, and therefore, r 0 λ i r r becomes one of the essential metric tools of photokinetics. In the case of a total absorbance trace, the initial velocity can be obtained from the following expression: F i t : r 0 A λ i r r = − 1 ln 10   c c A λ i r r 1 + c c A λ i r r   ∑ i = 1 i A ω i A λ i r r   k i A λ i r r = T h e o : r 0 A λ i r r = l i r r   o r   o b s ∑ j = 0 1,3 ε Y j λ i r r   o r   o b s   r 0 Y j λ i r r . (13)

It is to be noted that the expressions for the initial rates of the reactant and photoproducts ( X , Y 1 , and Y 3 ) cannot be obtained from the fitting of the total absorbance trace [second term of Eq. (13)] because the theoretical equation of the initial rate is a combination of terms as given by the last term of Eq. (13).

Nonetheless, r 0 A λ i r r is an as good metric for photokinetics as was F i t : r 0 X λ i r r , but with a considerable advantage, that is, of relieving the experimentalist from obtaining the individual species C Y j λ i r r t traces ( A t o t λ i r r / λ o b s t can be generated on a routine spectrophotometer, preferably with an in situ irradiation).

A spectator molecule (one of w molecules, SPM _r , with w ≥ r ≥ 1 ) is a species present in the reactive medium but does not interact with reactants and products of the photoreaction, has a constant concentration, is photo- and thermally inert, but absorbs at the irradiation wavelength. In thermal kinetics, a spectator of this type would have no effect on the reactivity of the species involved in the reaction mechanism. However, it is different in photokinetics. In the cases where a non-zero absorbance of the spectator molecule(s) at λ i r r , A S P M r λ i r r ≠ 0 (Eq. (14)), is considered, it must necessarily make part of the total absorbance of the medium (Eqs. 3, 14), since the fraction of the light absorbed by the spectators is lost for the reactive species. It is to be noted that no conditions on A S P M r λ i r r to belong to the linearity ranges of the S P M r calibration graph are necessary, since A S P M r λ i r r does not contribute to the mathematical formalism of the photokinetics other than by its actual constant value. A t o t λ i r r t = ∑ r = 1 w A S P M r λ i r r + ∑ j = 0 n s p A Y j λ i r r t . (14)

For this reason, an increase in the absorbance of the spectators is expected to result in a decrease of all species’ reactivities (i.e., rate constants and initial rates). Figure 3 depicts such a behaviour, where the rate of the reactant X and photoproduct, Y 7 , are reduced with an increase in the concentration of the spectator for the illustration mechanism shown in Figure 4 (a similar trend is also observed for the rest of the species rates, whose traces are not shown in Figure 3).

This expectation has turned out to be true for all the cases investigated. It corroborates the experimental results obtained for the primary photoprocess ( X →   Y , ε Y = 0 ) (Maafi and Maafi, 2013; Maafi and Maafi, 2015a) and the photoreversible reaction (Maafi and Maafi, 2015b), where the effects were quantitatively demonstrated by plotting the rate constant versus the photokinetic factor. However, in the present study, the evidence is provided by a progressive decrease in the initial reactant rate with an increase in SPM concentration.

As shown in Figure 4, each data point corresponds to both R K : r 0 X λ i r r and F i t : r 0 X λ i r r , whereas the curve joining the data points corresponds to T h e o : r 0 X λ i r r as functions of A S P M r λ i r r .

Beyond an established effect of the spectator on photoreactivity, it is important to stress a practical aspect that might be found useful for some applications. In other words, the photoreactivity of a molecule can effectively and significantly be reduced or even virtually stopped by an adequate selection of the concentration of one or more convenient spectator molecules. The reduction in photoreactivity by the presence of a spectator molecule has widely been exploited in chromatic orthogonality by performing a particular reaction within multi-component mixtures, though this use of SPM was treated only through a qualitative approach that has not been mathematically formalised (Bochet, 2006; Hansen et al., 2015).

The photokinetics of reactions must be dependent on the initial concentration of the starting reactant owing to the rate law [Eq. (1)], and more specifically, its photokinetic factor [Eq. (2)], being a function of concentration. Theoretically, the Φ -order kinetic pattern of the traces should be preserved when changing C X λ i r r 0 , but the rate of the reaction should be affected by the value of C X λ i r r 0 (all remaining extrinsic factors being the same), together with the initial reactant rate (as our metric tool in this work). It is also predictable that such an effect of C X λ i r r 0 on the rate of the reactant would be passed on to the photokinetics of the photoproducts whose r i j λ i r r would be proportionally changed according to their particular kinetic properties (since their rate laws depend on the concentrations of all species including that of X ). In this configuration, an increase in − r 0 X λ i r r is expected for an increased C X λ i r r 0 (i.e., a higher C X λ i r r 0 leads to a higher A t o t λ i r r 0 = A X λ i r r 0 and, hence, higher values of [ 1 − 1 0 − A X λ i r r 0 ), Eq. (1)]. At the same time, increasing C X λ i r r 0 will cause a reduction in the rate constants ( k i j λ i r r ) due to smaller values of P K F (Eq. (2), when A t o t λ i r r t increases) as was suggested by the semi-empirical method (Maafi and Maafi, 2013; Maafi and Maafi, 2014a; Maafi and Maafi, 2014b; Maafi and Maafi, 2015a).

A confirmation of C X λ i r r 0 effects on r 0 X λ i r r was investigated by performing RK calculations. The absolute values of R K : r 0 X λ i r r for a given reaction increase with an increase in the values of C X λ i r r 0 , tending asymptotically towards a limit (the latter should be reached at a total absorption of the light). When the RK traces were fitted by the appropriate Eqs 4 or 8, a gradual reduction in k i j λ i r r values was recorded with an increase in the values of C X λ i r r 0 (Figure 5). These trends were independent of the actual mechanism of the photoreaction (an example is provided in Figure 5). Such general results on the effect of C X λ i r r 0 set out a fundamental photokinetic property, which does not seem to have been reported, thus far, in the literature as a general feature, i.e., irrespective of the mechanism of the reaction at hand. The auto-photostabilisation of the reaction with increasing C X λ i r r 0 has previously been experimentally evidenced for two reactions (using semi-empirical equations), namely, the primary photoprocess with an ε Y 1 λ i r r ≠ 0 (Maafi and Maafi, 2013; Maafi and Maafi, 2014a) and the photoreversible reaction (Maafi and Maafi, 2015a), whose k 10 λ i r r linearly correlated with P K F 0 . Incidentally, it has been observed that the ratios of pairs of species’ final concentrations (end products or photostationary species) were independent of the values of C X λ i r r 0 .

The results obtained for R K : r 0 X λ i r r have served to the endorsement of both F i t : r 0 X λ i r r and T h e o : r 0 X λ i r r results (Figure 5), hence confirming the usefulness and reliability of the proposed model equation (Eq. 4).

Furthermore, since l i r r is imbedded within the absorption formula [always multiplying the concentration in Eq. (1)], its variation will cause similar photokinetic effects to those described previously for a change in C X λ i r r 0 . This was observed by monitoring R K : r 0 X λ i r r and F i t : r 0 X λ i r r , indicating that Eq. 4 applies irrespective of the reactor’s length.

It is to be noted that the effect of varying C X λ i r r 0 and/or l i r r is equivalent to when a similar change in the ε λ i r r values occurs, since the former quantities always multiply the latter in Eq. (1) (within the expressions of absorbances). Therefore, investigating a change in C X λ i r r 0 both informs about the trend of varying l i r r and equates to studying a class of a reactive system, whose reactant and photoproduct absorption coefficients ( ε λ i r r ) were proportionally changed. This might be of interest when designing a photoreactive system for particular applications.

For the purpose of describing how the radiation intensity within the reactor, P 0 λ i r r , might affect the photoreaction reactivity, it is useful to define what this quantity stands for.

As P 0 λ i r r is certainly related to the energy supplied by the light source, the latter is a good starting point.

The incident radiation of the monochromatic light beam, which can be measured by a physical actinometer as a spectral irradiance of the light source, E s p − i r r , is expressed in W   c m − 2   n m − 1 or J   s − 1   c m − 2   n m − 1 (i.e., energy ( J ) per unit time, t ( s ), per unit area [1 c m 2 ), and per wavelength, λ i r r ( n m )]. Since the radiation must be monochromatic for the kinetic model considered in the present work, the measured spectral irradiation corresponds to a single wavelength. This allows to ignore the unit, per wavelength ( n m − 1 ), in the dimension of the spectral irradiation. The unit J / s corresponds to an energy flux.

In experimental photochemistry, the number of photons is preferred to the energy of the beam since the former allows an adequate quantitative metric because the reaction occurs between this number of photons and the number of photoactive species in the medium. This number of photons is determined as follows.

The energy ( E h ν λ i r r in J ) carried by one photon of the radiation is a function of the wavelength λ i r r . E h ν λ i r r = h   c λ i r r , (15) where h is Planck’s constant expressed in J   s ( h = 6.62608   1 0 − 34   J   s ), c is the velocity of light given in m / s ( 299792458   m   s − 1 ), and λ i r r is the wavelength in m .

For an e i n s t e i n (a mole) of photons of a given wavelength (or Avogadro’s number of photons per mole, N a , N a = 6.02214   10 23 , in e i n s t e i n − 1 ), carrying a total energy E m o l   h ν λ i r r , the latter equation becomes E m o l   h ν λ i r r = N a h   c λ i r r . (16)

Applying this formula to the physically measured energy of the radiation, provided by the spectral irradiance, E s p − i r r , gives the number of photons (number of einstein or moles of photons) carried by the radiation per units of time and area. This quantity represents the incident photon flux of the monochromatic light source through a unit area, f l x h ν λ i r r . f l x h ν λ i r r = E s p − i r r E m o l   h ν λ i r r = E s p − i r r   λ i r r N a h   c . (17)

The dimension analysis of Eq. (17), f l x h ν λ i r r = E s p − i r r   λ i r r / N a h   c = J   s − 1   c m − 2   m / e i n s t e i n − 1   J   s   m   s − 1 = e i n s t e i n   s − 1   c m − 2 , confirms a flux dimension ( e i n s t e i n   s − 1 ) per unit area.

Therefore, in order to express the number of photons entering the reactor per s for a given experimental setup, with the specific irradiated area of the sample, it is necessary to multiply the incident flux per unit surface f l x h ν λ i r r , by the actual irradiated area of the sample, S i r r (in c m 2 ). f l x h ν , S i r r λ i r r = f l x h ν λ i r r   S i r r . (18)

Finally, f l x h ν , S i r r λ i r r is to be reported to the actual volume ( V i r r in d m 3 ) of the sample subjected to irradiation. The derived quantity, representing the P 0 λ i r r in Eq. (2), is expressed as P 0 λ i r r = f l x h ν λ i r r   S i r r V i r r , (19) whose dimension analysis gives P 0 λ i r r = e i n s t e i n   s − 1   c m − 2   c m 2 / d m 3 = e i n s t e i n   s − 1   d m − 3 . P 0 λ i r r is the number of photons entering the reactor per second, per irradiated area and volume of the investigated sample.

The dimension of P 0 λ i r r is adequate for a rate law describing a photochemical reaction [Eq. (1)]. The dimension characterising the left-hand side term of that differential equation, corresponding to a differentiation of the concentration with time, as m o l / L   s or m o l   d m − 3   s − 1 , and the right-hand term of the equation has the unit of P 0 λ i r r (both quantum yields, Φ λ i r r , and factors multiplying P 0 λ i r r [Eq. (2)] are dimensionless). Therefore, the same unit is found for either side of the differential equation (the similarity claim is consistent, since for photons count, m o l is equivalent to e i n s t e i n ).

The IUPAC (Braslavsky, 2007) defines a quantity q n , p , λ 0 as the radiation spectral photon flux on amount basis, with a dimension of e i n s t e i n   s − 1   n m − 1 or e i n s t e i n   s − 1 for a monochromatic light. In the expression given therein for the rate law, the spectral photon flux is divided by the volume irradiated ( q n , p , λ 0 / V , in e i n s t e i n   s − 1   d m − 3 ). The latter quantity has the same dimension as P 0 λ i r r [Eq. (19)]. However, it is to be noted that Eq. (19) takes into account the actual irradiated area of the sample (the two equations will be equal if S i r r = 1   c m 2 ).

Therefore, the value of P 0 λ i r r can change due to a variation in either the incident flux of the light source or lamp ( f l x h ν λ i r r ), the irradiated area ( S i r r ), or the volume ( V i r r ) of the sample exposed to the light. In this context, it is obviously important to ensure a vigorous stirring of the reactive medium, throughout irradiation, for consistency with the value of V i r r in Eq. (19). It is to be noted also that the ratio V i r r / S i r r , in Eq. (19), is not necessarily equal to the optical path length of the light beam inside the sample l i r r (it would only be the case if the whole volume is exposed to the light through one of its full sides, but in general, V i r r / S i r r ≥ l i r r ).

This, then, commends that the three properties are reported in the investigation in order to complete the adequate photokinetic analysis. Unfortunately, in many published studies, these quantities are missing. This might explain, at least in part, the difficulties in comparing the photokinetic results that were obtained in different laboratories.

Practically, the changes in f l x h ν λ i r r , S i r r , and V i r r can be controlled by the experimentalist by, respectively, varying the electric potential of the lamp, the blinded surface, or volume irradiated of the sample (these are part of the extrinsic parameters).

The qualitative effect of increasing P 0 λ i r r is an increase of the photoreaction reactivity. The quantitative assessment of such an effect is provided by proportional changes in the values of k i j λ i r r (as proven in the previous work (Maafi and Maafi, 2013; Maafi and Maafi, 2014a; Maafi and Maafi, 2014b; Maafi and Lee, 2015a; Maafi and Maafi, 2015a; Maafi and Lee, 2015b; Maafi and Maafi, 2016; Maafi and Al-Qarni, 2019)). In addition, an increase in P 0 λ i r r leads to a reduction in the overall time necessary for the reaction to reach completion and, hence, a reduction in t 1 / 2 . Furthermore, a variation in P 0 λ i r r causes a proportional change in the value of r 0 λ i r r .

The determination of the values of P 0 λ i r r is essential for quantitative photochemistry and fundamental in converting a reactive system into an actinometer. The value of P 0 λ i r r is obtained from Eq. (20) [which is a rearranged Eq. 12]. Hence, photokinetic data are used to perform actinometry, and this methodology is dubbed here: kinactinometry. P 0 λ i r r = − T h e o : r 0 X λ i r r   o r   F i t : r 0 X λ i r r Φ X → Y 1 λ i r r + Φ X → Y 3 λ i r r   1 − 1 0 − A X λ i r r 0 . (20)

Actinometry is the measure of the incident light flux per unit area and volume ( P 0 λ i r r ) at λ i r r , entering a given reactor (a slab-shaped reactor for our study), and a chemical actinometer is a standardised reactive system that can deliver P 0 λ i r r . The standardisation can be achieved kinetically by quantifying the linearly correlated variation of P 0 λ i r r with the rate constant of a given reaction step ( k i j λ i r r ) of the global photoreaction, and/or the initial reactant rate ( r 0 X λ i r r or r 0 A λ i r r , the metric in this study). Such a linear correlation is an advantage provided by a description of the photokinetic traces with equations of the type of Eq. 4. The correlation between the number of photons entering the reactor, P 0 λ i r r , and r 0 Y j λ i r r follows from the relationship between P 0 λ i r r and the number of photons absorbed by the considered species, P a Y j λ i r r .

Technically, when varying P 0 λ i r r , the model equations (Eq. 4) of the different species of the reaction share the same factors C ∞ , j λ i r r , ω i j λ i r r , and c c λ i r r since all are independent of P 0 λ i r r . The final concentrations (at t = ∞ ) of the reactive species ( C ∞ , j λ i r r ) are invariant with P 0 λ i r r , but they are reached faster (shorter t 1 / 2 ) for higher values of P 0 λ i r r . Hence, changing P 0 λ i r r will induce a proportional change in the k i j λ i r r factors, as the only changing fitting parameter, which adds robustness of the methodology and, overall, eases the fitting process.

Since the aforementioned behaviour is expected for each of r 0 Y j λ i r r factors (e.g., for X , Y 1 , and Y 3 ), then each is a good metric tool for actinometry (with each r 0 Y 0,1   o r   3 λ i r r factor having a different proportionality to P 0 λ i r r ). A feature that can be beneficial experimentally as it means that a single species’ trace (e.g., the reactant’s) would be sufficient for achieving an actinometric measurement. In addition, it is important to use the total absorbance trace to demonstrate its usefulness for actinometry. It turns out that the fitting parameters, k i j λ i r r factors and r 0 A λ i r r , of the total absorbance traces with increasing P 0 λ i r r values also deliver linear correlations (Figure 6). This is an additional confirmation of the validity of the model, but, as importantly, it proves the usefulness of a simplified way to experimentally achieve actinometry. Indeed, all is required is to fit the A t o t λ i r r t trace with Eq. 4. This kinactinometric approach has never been reported in the literature.

The excellent linear correlation of P 0 λ i r r with R K : r 0 X λ i r r and T h e o : r 0 X λ i r r stands for a confirmation of that proportionality based on RK calculations. The similar correspondence found with F i t : r 0 X λ i r r and F i t : r 0 A λ i r r is not only another validation of the proposed Eq. 4 but also a simple means to perform actinometry and recruit new actinometers. For a given experiment, the value P 0 λ i r r is worked out from the specific equation (e.g., examples given in Figure 6) corresponding to the quantity measured from the fitting equation.

Experimentally, one needs to determine the value of P 0 λ i r r . Reference and precise values of P 0 λ i r r , at given wavelengths, can also be determined by using the set of reliable actinometers that were previously proposed by our team on the basis of semi-empirical equations (i.e., Eq. 4 type (Maafi and Lee, 2015a; Maafi and Maafi, 2015b; Maafi and Maafi, 2013; Maafi and Maafi, 2014a; Maafi and Maafi, 2014b, Maafi and Maafi, 2015a, Maafi and Maafi, 2016; Maafi and Al-Qarni,2019)).

The large spectral region coverage by these actinometers (ranging between 250 and 580 nm, Figure 7) is useful for virtually any organic photoactive molecules. Because the implementation of these actinometers is very easy, requiring straightforward kinactinometric methods, they are strong alternatives to the ferrioxalate actinometer whose experimental setup is much more demanding. The list of these actinometers (Figure 7) can be extended by new (cheaper, water soluble, nanocarrier, etc.) candidates using the kinactinometric method proposed in the present work.

The quantum yield of a reaction step ( Φ Y j → Y j ′ λ i r r ) is an intrinsic feature of the reagent Y j for the reaction step Y j →   Y j ′ at the irradiation wavelength considered. The monochromatic light is a requirement for the determination of the absolute values of the quantum yield (Braslavsky, 2007). Traditionally, Φ Y j → Y j ′ λ i r r was held as the most important characteristic of a photoreaction (Crano and Guglielmetti, 2003; Tonnesen, 2004; Pianowski, 2022). However, photokinetic analyses have shown that Φ Y j → Y j ′ λ i r r is a very important factor in defining the reactivity of a photospecies, nonetheless, in only a partial capacity, as it is one of a set of other elements [Eq. (1)]. The explicit expression of the primary photoprocess rate constant, which has been analytically derived (Maafi and Brown, 2007), indicates that the photochemical quantum yield is, at least, one of the four factors defining the reaction rate constant ( k X λ i r r = Φ X λ i r r   ε X λ i r r   P 0 λ i r r   l i r r   ln 10). A similar conclusion has been reached from the expressions of rate constants of the unimolecular (Maafi and Maafi, 2013), reversible (Maafi and Maafi, 2014a), and multiconsecutive (Maafi and Maafi, 2016) photoreactions (where the k i j λ i r r formulae have been obtained by a semi-empirical method).