# Useful Spectrokinetic Methods for the Investigation of Photochromic and Thermo-Photochromic Spiropyrans

## Abstract

**:**

## 1. Introduction

## 2. Physical properties, reactivity and kinetic study of spiropyrans

#### Background

_{spiro}). The two possible enantiomers are separated by an energy barrier of approximately 86 kJ mol

^{-1}[48]. Depending on the molecular structure, the reactivity of spiropyran derivatives is believed to be governed by reversible thermal and/or photochemical mechanisms (Scheme 1). Hence, either or both A and B species can be thermally and photochemically reactive and the reaction sequence might include up to two photochemical and two thermal reaction steps.

_{spiro}–O bond. The activation energy of the thermal conversion ranges between 80 and 130 kJ mol

^{-1}[46].

^{1}SP*) within less than 200 fs of the excitation pulse [50]. The reaction stages subsequent to the initial bond cleavage have been extensively studied. They ultimately lead to eight possible “cis” and “trans” isomers of MC. The four cis conformations are much less stable than their trans counterparts (Scheme 3). Also, according to computational calculations, TTC is the most stable isomer for SP molecules bearing NO

_{2}groups [51,52]. Nonetheless, the overall thermal isomerization pathway, the major resonance forms and/or the most stable isomer remain subject to debate.

^{1}SP

^{*}, produced after light excitation, is dominated by either a singlet or a mixed singlet-triplet pathways depending on whether or not the SP molecule possesses a nitro substituent. The successive transformations leading to the ground state MC (

^{1}SP

^{* }→

^{1}MC

_{0}), which include cis and trans isomer intermediates (X), have various time scales ranging between tens of picoseconds to a few milliseconds [2,3,50,51,52] (Scheme 4).

^{1}MC

^{*}→ SP) has also been investigated by time resolved techniques [53,54]. The phototransformation of MC is believed to proceed via similar types of decay routes (i.e. which can involve a mixed singlet-triplet and/or a singlet pathway depending on whether a nitro substituent is present in the MC molecule). Here as well, the longest lifetime recorded for the intermediates does not exceed a few ms.

#### Considerations for continuous irradiation experiments

## 3. AB systems with similar kinetic behaviour to spiropyrans

**Figure 2.**A selection of photochromic and/or thermochromic AB systems possessing a kinetic behaviour similar to that of spiropyrans.

## 4. Typical kinetic mechanisms

_{4}, S

_{8}and S

_{10}), are included here to complete the series of kinetic elucidation methods for all possible cases of AB reactions.

## 5. Observables and Experimental Conditions

#### Photochemical Reactors

^{3}). Ususally, the reactor serves also as a sample holder for spectrophotometric measurements during reaction progress.

#### Irradiation Conditions

_{0}and ϕ, respectively) may vary with wavelength.

_{probe}) and lateral irradiation beam (l

_{irr}) correspond to the dimension of the photoreactor (1 cm for the classical spectrophotometric cuvettes). However, for top irradiations (Scheme 4b), l

_{irr}will be equal to the length of the liquid solution inside the photochemical reactor that is subjected to the irradiation (e.g. 0 cm < l

_{irr}< 4 cm, for a cuvette). In all cases, it is important to identify the (experimental) values of l

_{probe}and l

_{irr}as they are included in the formalisms of the kinetic elucidation methods (see next section).

_{irr}) is identical to that of an isosbestic point (or isosbestic region, see next section) whereas the latter means that λ

_{irr}is different from that of an isosbestic point. The elucidation methods developed hereafter may require either or both types of irradiations.

#### Observables

_{0}), the intensity of the irradiation beam (I

_{0}), the medium temperature, the optical path lengths l

_{probe}and l

_{irr}are all measurable quantities which can be defined before the start of the experiment. The accessible kinetic data collected on a given photochromic and/or thermochromic reactive system are represented by plots of the measurable variation of the absorbance (M) with reaction time (t). It is important to underline that spectrophotometric techniques of analysis allow only one absorbance (labelled M), monitored by the probing light at the condition of observation (i.e. where the optical path length is l

_{probe}), to be recorded. The absorbance of the medium in the condition of irradiation (along the optical path length l

_{irr}) is usually not directly accessible (Scheme 4), and hence it is not considered to be an observable.

_{irr}) and an observation (λ

_{obs}) wavelength as well as to l

_{probe}, l

_{irr}, I

_{0}and the medium temperature (T). Such curves are called kinetic traces. They are obtained either during the irradiation or the thermal relaxation of the reactive medium. The photochemical traces relate to the progress of both photochemical and thermal reactions of a given sequence while thermal curves are exclusively due to thermal processes (these traces are recorded in the dark, i.e. I

_{0 }= 0). Also, it is worth noting that traces for individual species are most often accessible if the reaction medium is observed at a wavelength (λ

_{obs}) where only one species absorbs (most likely in the visible region of the spectrum because the spectra of reactant and product often overlap throughout the UV region).

_{A}= ε

_{B}, over a spectral range).

#### The unknowns

_{AB}or ϕ

_{BA}) and the rate constants of the thermal reactions (k

_{AB}and k

_{BA}). The inaccessible spectroscopic data are the molar extinction coefficients of species A and B (ε

_{A}and ε

_{B}respectively). Quantum yields and molar extinction coefficients are wavelength dependent while rate constants are temperature dependent. Of course, the total number of unknowns varies with the sequence as indicated in Table 2. In this respect, it is relevant to consider that ε

_{A}is unknown for all thermally reactive initial species (A) because the progress of such reactions is immediate following the preparation of the stock solutions. (Reliable spectrophotometric measurements cannot be carried out promptly enough due to relatively fast rate constants, k

_{AB}[2,3]). In addition to the above unknowns, the species equilibrium concentrations (at the PSS, C

_{i}(PSS), and at the STE, C

_{i}(STE); with i = A or B) are not measurable in the type of experimental conditions previously set and therefore they must be accurately defined by the methods as well.

**Table 2.**Sets of unknowns specific to each kinetic case of Table 1.

Reaction scheme | Possible unknowns for the sequence |
---|---|

S_{11} | k_{AB}, k_{BA}, ϕ_{AB}, ϕ_{BA}, ε_{A} and ε_{B} |

## 6. Fundamental Kinetic Laws

_{4}, a

_{5}, a

_{19}, M, m

_{0}….) is similar to that used in the original papers [90,91,92,93,94,95]. A glossary of the symbols and labels adopted in this review is included in Section 13.

_{0}) are referred to an excitation and an observation wavelength (λ

_{exc}/λ

_{obs}), i.e. either irr or isos for the excitation of the photochemical reactions, and either obs, irr or an arbitrary λ, for the monitoring wavelength. The photokinetic factor (F) which is always defined by the irradiation features ( , , l

_{irr}...etc) can be time dependent F

^{irr}(t) or constant when measured at t = 0, F

^{irr}(t) = , or corresponding to either the PSS, F

^{irr}(PSS) = , or the STE, F

^{irr}(STE) = . The absorbance monitored in the observation conditions (l

_{probe}), is labelled throughout.

#### The Fundamental Kinetic Equations for AB Systems

_{irr}) is supposed to correspond to a spectral region where species A and B absorb different amounts of the incident light, and the concentration of the excited state remains negligible. C

_{A}(t) and C

_{B}(t) are the concentrations of species A and B, respectively. Eq.1 corresponds to the most inclusive reaction mechanism S

_{11}(in Table 1). Hence, individual differential equations for the remaining S

_{1}-S

_{10}sub-sequences can easily be inferred from Eq. 1.

^{irr}is the variable photokinetic factor, which is expressed as a function of the total absorbance at the irradiation wavelength (l

_{irr}),

^{isos}). These photokinetic factors are given by the following equation where only observables are used (with, in Eq. 3b, init = 0, STE or PSS),

_{irr}and observed at λ

_{obs}(with λ

_{irr}≠ λ

_{obs }), as

_{0}, as

_{0}= C

_{A}(t) + C

_{B}(t) = C

_{A}(0) + C

_{B}(0) = C

_{A}(STE) + C

_{B}(STE) = C

_{A}(PSS) + C

_{B}(PSS)

_{A}/dt = 0).

_{B}(PSS) and C

_{B}(STE) can be worked out from Eqs. 7 and 8. By using the latter equations in the equation of , Eq. 4, we can write formula, as

#### Basic Kinetic Laws for AB Kinetics When F is Constant

_{A}and C

_{B}), for all AB(nk,nϕ) systems studied under isosbestic irradiations (with its intensity), whether they involve pure photochemical, pure thermal or a combination of photochemical and thermal reaction steps, have been defined [90] as:

_{i}coefficients are functions of the kinetic parameters [90]; they are defined for the photochemical reaction (where both the thermal and the photochemical processes operate, and indexed by the letter P, a

_{iP}) and the thermal reaction (indexed T, a

_{iT}, where only the thermal processes are active, i.e. after the irradiation is cut off and = 0). Hence, for an AB(2k,2ϕ) system as given by Sequence S

_{11}(Table 2), we obtain :

_{4T}= k

_{AB}a

_{5T}= −k

_{AB}

_{isos}in Eqs. 11 depends on the attributes of the isosbestic point ( , , ), the irradiation condition (l

_{irr }= l

_{isos}) and F

^{isos}, the photokinetic factor for the isosbestic irradiation (Eq. 3b).

_{i}(Eqs. 11 and 12) and the total initial concentration.

## 7. Data Simulation, Fitting of the Kinetic Traces and Testing the Methods

#### Data simulation

#### Fitting of the Kinetic Traces

#### Testing the Methods

_{P}and Tr

_{T}, respectively). Then, in order to achieve the conditions of a realistic experiment, Tr

_{P}and Tr

_{T}together with C

_{0}, I

_{0}, l

_{probe}and l

_{irr}are considered as observables of known values whereas set A will represent the experimentally inaccessible set of unknowns that the method must define (i.e. set A, including the values of ϕ, k and ε, will remain unknown during the rest of the treatment). If data are obtained by an isosbestic irradiation, the traces Tr

_{P}and Tr

_{T}are fitted to the appropriate model equations and their respective fitting coefficients determined. In the case of non-isosbestic irradiation, the equilibrium absorbances and/or the initial velocities are determined. The adequate spectrokinetic method is then applied and an elucidation solution is found. The latter elucidation solution, corresponding to a set of values of the unknown parameters, is called set B.

## 8. Distinguishability and Identifiability Problems

_{3}and S

_{7}which involve respectively two and three reaction steps, are non-distinguishable from one another if not solved analytically. Indeed, both are represented by the same model Eqs.18, i.e. their experimental traces can both be fitted by the same Eqs. 18 and in both cases the corresponding fitting coefficients are extracted (§ 7). Therefore, at such a simple analysis level (i.e. if the kinetic analysis is only limited to observing a good fit of the traces with Eqs. 18), the distinguishability problem will remain undoubtedly unsolved. The requirement for more powerful elucidation methods is then required.

_{11}is an obvious example for identifiability when only kinetic data is used (§ 9).

## 9. The Spectrokinetic Elucidation Methods

#### AB(1ϕ) reactions, S_{1}

_{1}-type reactions [93]. The closed-form integration of Eq. 5 is achieved with a variable separation method after proceeding with a change of variables. The irradiation and observation wavelengths (λ

_{irr}= λ

_{obs}) must be equal. The resulting kinetic “Log-exp” model is [93]:

**Figure 4.**Comparison between fitting simulated RK data to (a) monoexponential and (b) analytical model Eq.19a [reprinted from Ref. 93 with permission of the Journal of Photochemistry and Photobiology: A Chemistry].

**Figure 5.**Photokinetic traces in hexane solution (1.82 10

^{-5}M (ο) and 3.28 10

^{-5}M (□); 15°C) at two irradiation wavelengths in the visible. Experimental data (circles and squares) are readily fitted by the theoretical model, Eq.19a (lines) [reprinted from Ref. 93 with permission of the Journal of Photochemistry and Photobiology: A Chemistry].

^{3+}―hv→ Fe

^{2+}). So far, the experimental data of the ferrioxalate actinometery were usually analysed by Runge-Kutta or Simpson’s numerical integration methods [2,3].

#### AB(1k) Systems, S_{2}

_{19T}, which is either equal to (– k

_{AB}) for A→B transformations or (– k

_{BA}) for B→A reactions (Eq. 14b), is easily obtained from the curve fitting, the determination of the reaction rate constant is straightforward.

#### AB(1ϕ,1k) Systems, S_{3}

_{AB}and a rate constant k

_{BA}, respectively). The reaction progress, under irradiation, leads to a PSS from which the thermal relaxation takes place in the dark (Figure 7). When an isosbestic irradiation is used, the model equation for the photochemical reaction is [92]

_{19T}and .

**Figure 7.**Kinetic traces of the photocolouring and the thermal fading reactions of NO

_{2}-BIPS (Scheme 6) in ethyl acetate (plots represent Absorbance vs. Time (in s)). The experimental data (squares) are fitted by model Eqs. 21a and 20b (solid lines). T = 14˚C, C

_{0}= 5.83 10

^{-5}M, λ

_{isos}= 339 nm, λ

_{obs}= 580 nm [reprinted from Ref. 92 with permission of the International Journal of Chemical Kinetics].

_{19T}(which is equal to (– k

_{BA}) as described for data treatment corresponding to S

_{2}kinetics), and the absolute value of the quantum yield is obtained as

_{BA}) are known, the calculation of species concentrations at PSS is achieved on the basis of Eqs. 17c and 17d. This allows the determination of the exact value of the molar extinction coefficient of the transient species ( at the observation wavelength) from the equation and value of (Eq. 15, with init = PSS). Accordingly, the kinetic elucidation is completed.

**Figure 8.**Absorption spectra of species A (plain line) and B (dashed line) for Scheme 6. The spectrum of B has been reconstructed using data from the elucidation method and Eq.21c. (ε in L.mol

^{-1}.cm

^{-1}and λ in nm) [reprinted from Ref. 92 with permission of the International Journal of Chemical Kinetics].

#### AB(1k,1ϕ) Systems, S_{4}

_{3}because species B can be obtained in a stable form and hence its spectra can be recorded. When species B is the coloured form [78], species A can be regenerated by applying a non-isosbestic irradiation to B and the efficiency of the photoreaction at a given irradiation wavelength ( ) is determined as per S

_{1}. Once the phototransformation of B is complete (i.e. ) and while keeping the photoirradiation light on, the full spectrum of species A can be recorded. The thermal rate constant k

_{AB}is subsequently obtained by fitting the thermal relaxation trace of species A (recorded in the dark) to the model Eq. 20a (the same treatment used for S

_{2}applies). The photochemical efficiencies at wavelengths where both species absorb can be derived from Eq. 6 (where = k

_{BA}= 0).

_{3}(i.e. the elucidation method developed for the kinetic case S

_{3}applies except that the labelling of the equations must take into account that now the initial species is B).

_{3}) because the full spectra of both species are easily accessible.

#### AB(2k) Systems, S_{5}

_{2}kinetics and Table 2). Alternative physical data can be considered. In this instance, the determination of the equilibrium concentration of one or both species can be obtained, for example, by NMR. This allows the calculation of the equilibrium constant, K

_{e}= k

_{AB}/k

_{BA}(as equal to C

_{B}(STE)/C

_{A}(STE)). However, the determination of the individual rate constants is impossible if no complete kinetic traces (at least one corresponding to either A , B or A and B) are available. This issue is discussed further with relation to the kinetics of thermo-photochromes (S

_{11}).

**Figure 9.**Examples of conmplete kinetic traces for AB(2k). (a) Observation of individual traces of species A and B. (b) Observation of the temporal evolution of the total absorbance of the same medium as in (a).

#### Reversible Photochemical AB(2ϕ) and AB(2ϕ,1k) Systems, S_{6}, S_{7} and S_{8}

_{6}and S

_{7}are presented together in this section because they are solved by the same kinetic elucidation method [94]. We consider here the typical spectral features of photochromes where the spectra of the species overlap throughout the UV region while B has an individual absorption in the visible range.

_{6}) where each step is exclusively sensitive to either UV (e.g. ) or visible light (e.g. , Scheme 7). The second case is that of sequences S

_{6}and S

_{7}whose photoreactions are simultaneously initiated by UV light and are independent of the irradiation wavelength. In the third kinetics we will consider the preceding case but with forward photoreaction efficiencies dependent on irradiation. The much complex kinetics where both direct and reverse quantum yields are irradiation dependent is also briefly discussed.

^{isos}and , are calculated using Eqs. 3c and 3b, respectively.

#### Pure photochemical opposed reactions AB(2ϕ) which are responsive to different light ranges

#### AB(2ϕ) and AB(2ϕ,1k) systems where both photochemical reactions are responsive to the same excitation beam and their quantum yields are independent of the irradiation wavelength (e.g. UV).

_{6}) has demonstrated that the kinetic solution for this case can only be achieved provided that the ratio of the forward to reverse quantum yields, at any two non-isosbestic irradiations, is constant. This constant ratio hypothesis is a reasonable assumption according to Kasha’s rule. However, the mathematical formalism adopted for this method does not hold if an additional thermal reaction step is incorporated in the reaction scheme (S

_{7}). The new approach presented hereafter overcomes this problem and elucidates equally S

_{6}and S

_{7}kinetics.

_{irr}(Figure 10), are used for the treatments.

**Figure 10.**Examples of simulated (a) AB(1k,2ϕ) kinetic traces using RK integration for the photochemical reaction performed at two excitation conditions (here at exc/irr wavelength, ) and (b) AB(2 ϕ) photoreaction ( at the observation wavelength) [reprinted from Ref. 94 with permission of Photochemical and Photobiological Sciences].

_{BA}). It is a fitting parameter (–a

_{19T}) of Eq. 20b to the thermal relaxation trace. The mathematical formalism yields the analytical expression of the extinction coefficient of species B at the irradiation wavelength. The formula of is exclusively defined by experimentally available coefficients.

Initial parameters | ||||||

Figure # | C_{o}(M) | (M^{-1} cm^{-1}) | (M^{-1} cm^{-1}) | (Einst.s^{-1} dm^{-1}) | (Einstein.s^{-1} dm^{-1}) | k_{BA} (s^{-1}) |

10b | 8.5 10^{-6} | 18617 | 58690 | 6 10^{-6} | 2.0 10^{-6} | 0 |

Fitting results for the simulated kinetic traces and all experimentally accessible data | ||||||

Figure # | (0) ^{b} | ^{c} | ^{c} | γ_{isos} ^{c} | (PSS) | |

10b | 0.4988 | -0.0673 | -0.2651 | 0.2150 | 0.1285 | 1.99 |

^{a}Throughout l

_{irr}= l

_{probe}= 1 cm.

^{b}: .

^{c}: These parameters are given in s

^{-1}[reprinted from Ref. 94 with permission of Photochemical and Photobiological Sciences].

**Table 4.**Spectroscopic and kinetic values of the unknown parameters calculated by the spectrokinetic method (Eqs. 24) using data of Table 1.

Figure # | ϕ_{AB} | ϕ_{BA} | C_{A}(PSS) | ||

λ_{isos} ^{a} | λ_{irr} ^{a} | ||||

2 | 10359 | 0.76 | 0.47 | 3.25 10^{-6} | 8.36 10^{-7} |

^{a}: wavelength of the irradiation beam responsible for yielding C

_{A}(PSS) in solution [reprinted from Ref. 94 with permission of Photochemical and Photobiological Sciences].

#### AB(2ϕ) and AB(2ϕ,1k) systems where both photochemical reactions are responsive to the same excitation beam and their forward quantum yields are dependent on the irradiation wavelength.

_{AB}is irradiation dependent while ϕ

_{BA}is constant with irradiation wavelength (Eq. 25a), i.e. α ≠ 1 (Eq. 22).

_{i}take the formulae below.

_{isos}= 317 nm and λ

_{irr}= 345 nm (Figure 11), yielded the kinetic solution given in Table 5.

**Figure 11.**Experimental traces of DAE (5.6 10

^{-5}M) in hexane at 15 °C. The kinetic traces are labelled by the wavelengths at which the irradiation and the observation were carried out (λ

_{exc}/ λ

_{obs}). Two of the curves (317/317 and 317/345, in circles) have been performed using the isosbestic irradiation and the third resulted from a non-isosbestic irradiation (345/345, in squares). The line linking the points of trace 317/345 represents the best fit to the monoexponential model, Eq.21a. The line shown on the trace 345/345 has been calculated by a RK integration using the data supplied by the elucidation method (Table 5) [reprinted from Ref. 94 with permission of Photochemical and Photobiological Sciences].

_{3}, and, S

_{6}and S

_{7}with ϕ ≠ f(λ

_{exc}), do not yield consistent results indicating that the system under consideration does not obey AB(1ϕ,1k) or AB(2ϕ,1k) kinetics with irradiation independent quantum yields. Furthermore, the use of the second root obtained from Eq. 26a ( = 2094 M

^{-1}cm

^{-1}, since Δ > 0, is given in Table 5 as ) in Eqs. 26b-26d generates a negative value for ϕ

_{BA}and values over unity for and .

**Table 5.**Calculated and experimental spectroscopic and kinetic values for the unknown parameters of the system.

Parameters | ^{a} | ϕ_{BA} | ^{b} | ^{b} | ^{b} | ||

Results of the Elucidation Method | 8887.8 | 0.23 | 0.419 | 0.095 | 1.6 10^{-5} | 3.4 10^{-5} |

^{a}: units: mol

^{-1}·L.cm

^{-1}.

^{b}: The concentration of the initial species at the PSS (in mol·L

^{-1}); where the irradiation wavelength used for the experiment is also indicated.

_{BA}is zero for one solution) the distinguishability problems arise because of the occurrence of two options. In this case, the complete solution would require further information to discern the true solution.

#### AB(2ϕ) and AB(2ϕ,1k) systems where both photochemical reactions are responsive to the same excitation beam and both their forward and reverse quantum yields are dependent on the irradiation wavelength.

_{exc}) and = g(λ

_{exc}), an analytical (assumption-free) method based on the use of pure kinetic data is yet to be developed. The difficulty in finding such a useful kinetic elucidation method stems from the fact that the number of unknowns is higher than that of the linearly independent equations that describe the reactive system. Progress on a better physical description of the kinetics by the integration of the fundamental differential equation of such kinetics would certainly represent a hope to complete this kinetic elucidation story.

_{A}(PSS) (Eqs. 8 and 17c, respectively for non-isosbestic and isosbestic irradiations).

#### AB(2ϕ,1k) systems, S_{8}

_{7}and the corresponding elucidation method applies.

#### Thermophotochromic AB (1ϕ,2k) systems, S_{9} and S_{10}

_{8}and S

_{9}are characterised by opposing thermal reaction steps and a single photochemical step responsible for the transformation of either A into B (S

_{8}) or B into A (S

_{9}). The total number and the identity of the five unknown parameters is the same for both kinetics (Table 2) except that each reaction sequence involves a different quantum yield (ϕ

_{BA}replaces ϕ

_{AB}for S

_{9}). Since the lifetimes of the thermal reactions are relatively short for photochromes, recording individual spectra of initial species or the early stages of the kinetic traces by spectrophotometry is not possible. Therefore, for such kinetics the photochemical reaction is realised after the thermal equilibrium (STE) has been reached. In these circumstances, the two reaction sequences do not differ on principle in such a way that the kinetic elucidation method developed for S

_{8}[95] hereafter can similarly be used for the treatment of S

_{9}provided that the appropriate modifications due to the new reaction sequence are taken into account in the labelling of the equations.

_{1}). As previously stated, this first trace in generally not experimentally accessible and its specific equations are not considered in the elucidation formalism. The second kinetic trace starting from STE, corresponds to the photochemical transformation due to an isosbestic irradiation (labelled P for photochromism). When light is switched off, the system relaxes thermally from PSS to STE during a last stage called thermochromism 2 (Th

_{2}).

**Figure 12.**Typical kinetic traces depicting the three successive phases (Th1 (ο), P (▪) and Th2 (+)) of a thermophoto-reactive system (with M

_{tot}the total absorbance of the medium at the observation wavelength; irr. on and irr. off represent respectively the start and end of the isosbestic irradiation) [reprinted from Ref. 95 with permission of the International Journal of Chemical Kinetics].

_{B}(STE) in order to determine the value of [2,3], is not valid.

_{AB}, the absorbance of species B at the thermal equilibrium ( ) is used. (Note that in the visible region, = 0).

_{BA}= −a

_{19T}− k

_{AB}

#### Thermophotochromic AB (2ϕ, 2k) systems, S_{11}

_{5}), be obtained by a physical method (e.g. NMR, realised at the same temperature and the same solvent as that of the kinetic study). The kinetic data required are represented by a set of one thermal and one photochemical trace (realised under an isosbestic irradiation). In these conditions, the system will formally behave similarly to Figure 12.

_{19T}(Eq.14b) will allow the extraction of the absolute values for the individual rate constants of the thermal reactions (k

_{AB}and k

_{BA}). Also, the value of the sum of quantum yields is obtained as follows:

## 10. Determination of reaction quantum yields at any irradiation wavelength

_{isos}or at both λ

_{isos}and λ

_{irr}). This is true for the solved kinetics involving photoreactions (S

_{1}, S

_{3}, S

_{4}, S

_{6}-S

_{10}).

_{1}(Eq. 19), the determination of the values of the quantum yields at different irradiations must be carried out after the kinetic elucidation is completed in the conditions set in Section 9. In these circumstances, the spectra of both species as well as the thermal rate constants are all available. This is important because if the reaction, subjected to the new non-isosbestic irradiation wavelength (labelled irr

_{x}), is performed with all experimental conditions similar to those that served for the kinetic elucidation (of Section 9) then the photochemical quantum yields values are the only unknowns of the new kinetics. Accordingly, the absolute value of the forward quantum yield ( ) can be derived by several ways; one of which being the use of the initial velocity Eq. 6, as:

_{6}–S

_{8}and S

_{10}), the value of the reverse quantum yield ( ) can be worked out from the equation of the total absorbance at the PSS (Eq. 9). Hence, individual dependence of the efficiencies ( and ) on the irradiation wavelength can easily be studied.

## 11. Applicability of the kinetic elucidation methods to non-chromic systems

## 12. Conclusions

## 13. Glossary

#### Labelling

AB (nk,m) | Bimolecular kinetic system |

A | Species A (initial reactant) |

B | Species B (thermo- or photoproduct) |

AB | Transformation of species A into B |

BA | Transformation of species B into A |

λ | An arbitrary wavelength |

obs | Observation wavelength (_{obs}) |

irr | Irradiation wavelength (_{irr}) |

isos | Isosbestic-point wavelength (_{isos}) |

0 | Index or argument corresponding to time zero of the reaction |

PSS | Index or argument relative to the reaction at the Photo-Stationary State |

STE | Index or argument relative to the reaction at the State of thermal Equilibrium |

init | Index or argument equal to 0, STE or PSS |

P | Photochemical process |

T | Purely thermal process |

Ω | Index standing for either P or T letters |

f(λ_{exc}) | A function dependent on the excitation wavelength |

g(λ_{exc}) | A function dependent on the excitation wavelength |

#### Irradiation and observation conditions

λ_{exc} | An arbitrary wavelength used to irradiate the sample; it includes _{isos} which is the wavelength of an isosbestic point and _{irr} (≠ _{isos}) an irradiation wavelength where (and both ≠ 0). |

λ_{obs} | An arbitrary observation wavelength (which might be equal to _{irr}) |

Incident light intensity of the excitation beam at λ_{ιρρ}(${\text{I}}_{0}^{\text{irr}}$) or _{isos} (${\text{I}}_{0}^{\text{isos}}$) | |

l_{probe} | Optical path length of the spectrophotometer (probing) light inside the sample |

l_{irr} | Optical path length of the irradiation light inside the sample |

#### Concentrations

C_{i}(t) | Concentration of species i = A or B at time t |

C_{i}(PSS) | Concentration of species i = A or B at time PSS |

C_{i}(STE) | Concentration of species i = A or B at time STE |

C_{0} | Total concentration of the species in the reactive medium |

#### Kinetic parameters

Generic quantum yield of the transformation j when the reaction medium is subjected to an excitation beam whose wavelength is λ_{εξχ} (= λ_{ισοσ} or λ_{irr}). | |

Quantum yield of the transformation of A into B at the excitation wavelength λ_{exc} | |

Quantum yield of the transformation of B into A at the excitation wavelength λ_{exc} | |

k_{BA} | First-order rate constant of the thermal transformation of B into A or A into B |

F^{irr}(t) | Photokinetic factor at the irradiation wavelength and time t |

F^{isos} | (or F^{isos}(t) = F^{isos}) Constant photokinetic factor at λ_{isos} |

(or F^{irr}(0)) Constant photokinetic factor at λ_{irr} and t = 0 | |

(or F^{irr}(PSS)) Constant photokinetic factor at λ_{irr} at PSS | |

(or F^{irr}(STE)) Constant photokinetic factor at λ_{irr} at STE | |

Initial velocity of reaction Ω when irradiation has been carried out using λ_{exc} (= isos or irr) and observed at λ_{obs} (= irr) | |

a_{19Ω} | Overall first-order reaction rate for the process Ω (λ_{exc} = λ_{isos}) |

a_{4Ω} | First-order direct reaction rate for the process Ω (λ_{exc} = λ_{isos}) |

a_{5Ω} | First-order reverse reaction rate for the process Ω (λ_{exc} = λ_{isos}) |

γ_{isos} | Constant factor that multiplies quantum yields when λ_{exc} = λ_{isos} |

#### Spectroscopic parameters

Extinction coefficient of species i (A or B) at λ (= isos, irr or obs) | |

The total absorbance of the medium measured at various excitation/observation wavelengths’ combinations (e.g. , and respectively at isos/isos, isos/irr and irr/irr) | |

Particular constant values of the total absorbance measured at init = 0, PSS, STE |

#### Acronyms

HPLC | High Performance Chromatography |

MC | Merocyanine molecule |

NMR | Nuclear Magnetic Resonance |

RK | Runge-Kutta numerical integration method |

SP | Spiropyran molecule |

TTC | Trans-Trans-Cis isomer of spiropyran (the same meaning for T and C applies for TTT, CTT and CTC) |

UV | Ultraviolet spectrophotometry or range |

V_{is} | Spectrophotometry or spectral range relating to the visible range of the electromagnetic spectrum |

## Acknowledgements

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- Sample Availability: Samples are not available.

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Maafi, M.
Useful Spectrokinetic Methods for the Investigation of Photochromic and Thermo-Photochromic Spiropyrans. *Molecules* **2008**, *13*, 2260-2302.
https://doi.org/10.3390/molecules13092260

**AMA Style**

Maafi M.
Useful Spectrokinetic Methods for the Investigation of Photochromic and Thermo-Photochromic Spiropyrans. *Molecules*. 2008; 13(9):2260-2302.
https://doi.org/10.3390/molecules13092260

**Chicago/Turabian Style**

Maafi, Mounir.
2008. "Useful Spectrokinetic Methods for the Investigation of Photochromic and Thermo-Photochromic Spiropyrans" *Molecules* 13, no. 9: 2260-2302.
https://doi.org/10.3390/molecules13092260