Photophysics and kinetics of naphthopyran derivatives, part 6: A fundamental system of equations to describe ABC(3k,6o) dynamics. A contribution to the elucidation of [2H]-naphthopyran kinetics

Article abstract of DOI:10.1002/kin.20306

A new spectrokinetic method free of any assumption is developed in this paper to investigate photochemical systems obeying ABC(3k,6o) kinetic mechanisms that can be characterized by up to 14 unknown parameters for a nonpreconceived mechanism. The mathematical framework of this method is derived as an application from an original formalism previously established to deal with the kinetic law descriptions relating to ABC reactive systems. The necessary number of linearly independent (but nonlinear) equations has been fully defined together with seven other supplementary equations useful as constraints during the iterative calculations. As a practical example, this spectrokinetic method has been used to investigate the dynamics of a new photochromic derivative belonging to the naphthopyran family (2,2-di(4-methoxyphenyl)-5-carbomethoxy-6- (4-methoxyphenyl)-9-methoxy-2H-naphtho[1,2-6]-pyran). The system of equations has been solved, and the problems of identifiability and distinguishability have been raised with relation to the degeneracy of the kinetic solution. To get more insight into the photochemical and thermal reaction steps involved, a progressive overall discussion leads to several quantitative conclusions whose background sets the dynamics for the present derivative to at least ABC(1k,4o) kinetics.

Full text of DOI:10.1002/kin.20306

Photophysics and Kinetics
of Naphthopyran
Derivatives, Part 6:
A Fundamental System of
Equations to Describe
ABC(3k,6φ) Dynamics.
A Contribution to the
Elucidation of [2H]-
Naphthopyran Kinetics
MOUNIR MAAFI,1 ROBERT G. BROWN2
1Leicester School of Pharmacy, De Montfort University, The Gateway, Leicester, LE1 9BH, UK
2Faculty of Health, Sport and Science, University of Glamorgan, Pontypridd, Cardiff CF37 1DL, UK
Received 6 September 2005; revised 10 July 2007; accepted 2 August 2007
DOI 10.1002/kin.20306
Published online in Wiley InterScience (www.interscience.wiley.com).
ABSTRACT: A new spectrokinetic method free of any assumption is developed in this paper
to investigate photochemical systems obeying ABC(3k,6φ) kinetic mechanisms that can be
characterized by up to 14 unknown parameters for a nonpreconceived mechanism. The math-
ematical framework of this method is derived as an application from an original formalism
previously established to deal with the kinetic law descriptions relating to ABC reactive sys-
tems. The necessary number of linearly independent (but nonlinear) equations has been fully
defined together with seven other supplementary equations useful as constraints during the
iterative calculations. As a practical example, this spectrokinetic method has been used to
Correspondence to: Mounir Maafi; e-mail: mmaafi@dmu.ac.uk.
2008 Wiley Periodicals, Inc.
c
ꢀ

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
269
investigate the dynamics of a new photochromic derivative belonging to the naph-
thopyran family (2,2-di(4-methoxyphenyl)-5-carbomethoxy-6-(4-methoxyphenyl)-9-methoxy-
2H-naphtho[1,2-b]-pyran). The system of equations has been solved, and the problems of
identifiability and distinguishability have been raised with relation to the degeneracy of the ki-
netic solution. To get more insight into the photochemical and thermal reaction steps involved,
a progressive overall discussion leads to several quantitative conclusions whosꢀe background
C
sets the dynamics for the present derivative to at least ABC(1k,4φ) kinetics.
2008 Wiley
Periodicals, Inc. Int J Chem Kinet 40: 268–281, 2008
INTRODUCTION
studied individually. Second, their electronic absorp-
tion spectra have similar features (often overlapping
entirely) and mix with the spectrum of the initial, ring-
closed, species over the whole ultraviolet region. As a
consequence, femtosecond–nanosecond time-resolved
studies are restricted to limited investigation options. In
the case of spectrokinetic approaches, in addition to the
above, another challenge has to be considered as well.
It is understood that an NPY system involving three
species is defined by up to 17 unknowns (potentially
including six photochemical yields, six thermal rate
constants, two extinction coefficients, and three con-
centrations of the species at the photostationary state
(PSS)). The total number of unknowns reduces to 14 if
we assume that not more than three thermal processes
are active in the dynamics, as is the case for nonther-
mochromic 2H- and 3H-NPY systems. The challenge
(so far not met) is to derive as many equations that
describe the dynamics as the number of the spectro-
scopic and kinetic unknown parameters that define the
reactive system so that these unknowns can (in theory)
be determined.
One way to simplify this problem is to propose a
mechanism involving a reduced number of photochem-
ical reaction steps. However, it is important to bear in
mind that as the true mechanism is not known, several
equally plausible sequences can usually be postulated
and it is difficult to argue rationally in favor of a partic-
ular one as the definitive true mechanism. In addition,
as we will see in this paper, it is not even possible to
be perfectly certain that the parameter values found for
a particular mechanism are unique, even though they
yield a good fit to the experimental data.
Naphthopyrans (whether 2H- or 3H-derivatives
(NPY); see Scheme 1) are frequent components of the
photochromic element employed in plastic ophthalmic
lenses [1,2]. Judging by the ongoing intensive research
effort, their impact can be foreseen in future techno-
logical applications in areas such as photoswitches,
recording media, domestic glass dying, and liquid
crystals [3,4].
Although these expectations have triggered a huge
interest in this family of photochromes, most of the
literature is involved with the preparation of new struc-
tures (see, e.g., [5]). There are only a few investigations
focusing on understanding and elucidating the mech-
anistic processes [4,6–12]. In this field, femto-, pico-,
and nanosecond studies represent the core of the in-
formation available that contributes to shed new light
on the mechanism of the 2H-NPY dynamics. Spec-
trokinetic and photokinetic studies of these molecules
are scarcer, and definitive procedures are yet to be pro-
posed and implemented to allow systematic investi-
gations. From this point of view, understanding and
unraveling the 2H-NPY mechanism offers a valuable
means of assisting the design of structures possessing
optimal features for specific applications.
The primary event of the photoreaction of a
2H-NPY is the ring opening of the pyran unit through
the cleavage of its intracyclic ether “C O” bond. This
process yields a number of geometric and rotation iso-
mers of the “open” form of the photochrome—the ex-
act number depending on the substitution pattern in the
molecule under study. Among these, only a few con-
formations are long lived enough to be observed over
time scales of seconds or greater (i.e., interesting from
both application and spectrokinetic aspects).
It is still unclear, however, as to the involvement
of these longer living species in the overall dynamics
when the reaction medium is submitted to continu-
ous irradiation. These difficulties arise from the con-
junction of two unfavorable factors. First, as has been
alluded to above, the observable species, even though
the most stable, cannot be physically isolated and
In our attempts to contribute to the study of ABC
systems in general and to those involving 2H- and 3H-
NPY in particular, we have developed a mathematical
formalism [13] that provides the kinetic laws for re-
active trimolecular systems of the ABC(nk,mφ) type
when exposed continuously to a monochromatic light
beam at a wavelength corresponding to an isosbestic
point. This formalism has been used as a theoretical
basis to develop several spectroscopic methods and
tools aimed at solving complex ABC kinetics [14–17].
International Journal of Chemical Kinetics DOI 10.1002/kin

270
MAAFI AND BROWN
Scheme 1 The three basic structural variations of the naphthopyran family.
Most of these studies have been applied to gain more
insight in the reactivity of 3H-NPY derivatives. In
this respect, one of our studies [16] has shown that
the reaction of 3,3-diphenyl-3H-naphtho[2,1-b]pyran
is governed by an ABC(1k,6φ) mechanism. The math-
ematical treatment employed for this study was based
on monoexponential kinetic traces and a system of 13
equations [16].
In the present paper, we propose a new and gen-
eral spectrokinetic method based on the mathematical
framework allowed by the formalism, capable of inves-
tigating an ABC(3k,6φ) system for the first time. It is
applied here to study the kinetics of a 2H-substituted
naphthopyran derivative 1 (see Scheme 2).
sition and treatment. The sample holder (designed for
a 1-cm cell) is equipped with a temperature control
Peltier system model Agilent 8453. The latter equip-
ment has a precision of 0.1◦C.
The irradiation equipment was manufactured by
Photon Technology International Corporation (Ford,
UK). The light source was an Ushio 1000 W xenon
arc-lamp located in a housing shell model A6000 and
powered by a model LPS-1200 power supply. The lamp
housing was connected to a model 101 monochroma-
tor, which is a special f /2.5 monochromator with a
1200-groove/300-nm blaze grating to allow irradia-
tion wavelength selection. The lamp was continuously
cooled by a flow of fresh water delivered by a Watson
peristaltic pump model 505S. The excitation beam was
guided through an optical fiber to impinge upon the top
of the sample cuvette. Hence, the directions of the ir-
radiation and analysis light beams were perpendicular
so as to avoid interference. A shutter situated between
the monochromator and the optical fiber allowed the
excitation light to be manually switched on and off.
An excitation wavelength of 381 nm corresponding to
a wavelength, where the range of absorption variation,
measured at 15◦C before irradiation and at the PSS,
causes less than 6% difference in the photokinetic fac-
tor values (F as given by Eq. (8); vide infra), i.e., F
can be approximated to a constant.
EXPERIMENTAL
The naphthopyran derivative (1) used in this study (2,2-
di(4-methoxyphenyl)-5-carbomethoxy-6-(4-methoxy-
phenyl)-9-methoxy-2H-naphtho[1,2-b]-pyran)
is
a commercially available photochromic dye that
has been generously donated by PPG Industries
(Pittsburgh, PA, USA) (synthesis procedure as
previously described [18]).
The spectrophotometric-grade toluene solvent was
supplied by Aldrich (Gillingham, UK) and was used
without further purification.
The absorption spectra and kinetic profiles were
recorded on an Agilent 8453 diode array spectropho-
tometer equipped with a Chemstation for data acqui-
All measurements were carried out in a quartz cell.
The sample was maintained at a constant temperature
(of 15◦C) and was stirred continuously during the ex-
periment. Typically, an experiment was carried out
on a 2-mL volume of 7.52 × 10−5 M solution, mon-
itored at the wavelength of the open form isomer(s)
(λobs = 530 nm).
The experimental kinetic profiles were corrected for
the absorbance generated by irradiation with the spec-
trophotometer analyzing beam. Indeed, because the
reaction medium is continuously (at a sampling rate
of 5 s) analyzed by a wide wavelength range (190–
600 nm), a certain amount of photochemical reaction
is always present. Accordingly, the data acquisition be-
gan with a first (dark) period of 300 s in which only
the spectrometer analysis beam was allowed to reach
the sample. Then, the excitation beam from the lamp
Scheme 2 Structure of the 2,2-di(4-methoxyphenyl)-5-
carbomethoxy-6-(4-methoxyphenyl)-9-methoxy-2H-naph-
tho[1,2-b]-pyran derivative (1) used in this study.
International Journal of Chemical Kinetics DOI 10.1002/kin

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
271
was allowed to impinge upon the sample for a sec-
ond period of 400-s duration. At the end of each of
these time periods, the absorption versus time profiles
attained a plateau indicating that the photostationary
states were fully achieved. In a final stage of 400 s, the
irradiation light was switched off, and the data corre-
sponding to the thermal fading reaction accumulated.
These data have been corrected for the interference
(using the trend observed during the first period) due
to the diode array beam prior to analysis. (A typical
trace is shown as Fig. 2 later in this paper.)
Typically experimental errors were less than 5%.
A Levenberg–Marquardt iterative program and a
least-squares analysis tool within the Origin 6.0 soft-
ware package were used to run both nonlinear fitting
on the kinetic experimental data and linear correlation
between experimental and calculated results.
The set of equations describing the kinetics of the
system under consideration was input into Mathcad
(premium software version 2001i) and solved by the
find function command, which allows the solution of a
system of nonlinear equations. A set of parameter val-
ues, corresponding to the possible solutions of the non-
linear system of equations, were accordingly generated
(by varying the values given to initial parameters).
T, respectively. We also consider here the realistic sit-
uation of NPYs in which only one species (e.g., A)
is thermally stable, and the reaction starts out from an
initial concentration of that species (C0 = CA0 ).
In this situation, the kinetic traces are readily de-
scribed by the generic equation given by the following
expression:
ꢀ
ꢁ
mλ0 − M0λ − θ4λ · ϑ
ϑ − ϑ
−
+·t
Mλ(t) = −
· eϑ
−
+
+
ꢀ
ꢁ
mλ0 − M0λ − θ4λ · ϑ
−·t
+
· eϑ + θ4λ
(1)
ϑ − ϑ
−
+
In this formula, Mλ(t) represents the observed absorp-
tion intensity due to species B and C at different time
delays and at a given observation wavelength (λ). Be-
cause the same set of equations can be applied to the
photocoloring trace as well as the one corresponding
to the fading, all the coefficients are normally related
to the specific trace by indexing them with the letters
P or T (e.g., MPλ (t) and MTλ(t)). The remaining factors
of Eq. (1) are defined in [13] by the series of equations
given below.
The initial velocities recorded at the earliest stages
of each trace are expressed by
ꢀ
ꢁ
ꢀ
ꢁ
THE MATHEMATICAL FRAMEWORK
mλ0 = εBλ a4 + εCλ a7 · CA0 + εBλ a5 + εCλ a8 · CB0
ꢀ
ꢁ
+ εBλ a6 + εCλ a9 · CC0
(2)
As has been stated above, the dynamics of NPY sys-
tems can be characterized by at least 14 unknown
parameters if it is set to conform to an ABC(3k,6φ)
mechanism. So to tackle the kinetics of these pho-
tochromes, at least 14 nonlinearly dependent equa-
tions are required. To achieve this objective, we make
use of the flexibility of the general formalism to de-
velop a coherent spectrokinetic method for a general
ABC(3k,6φ) system. The full potential of the method
is attained when the reaction reaches a photostation-
ary state (PSS) within a reasonable period, the species
spectra intersect at an isosbestic point, and the pho-
tochemical buildup and the thermal relaxation traces
are collected at different wavelengths to that of the
isosbestic point (λIrr).∗
The cumulative absorption intensities at the beginning
of the trace (where z = 0 or PSS) are
Mzλ = εBλ · CBz + εCλ · CCz
(3)
The coefficient θ4λ represents the amount of absorption
recorded after a long-reaction period for a mechanism
that reaches equilibrium. Depending on the type of
the trace, it may correspond to the absorption intensity
at the photostationary state (i.e., w = PSS) or to the
absorbance reached at a long period of the fading reac-
tion (where w = ∞) if the latter generates species in
equilibrium.
The labeling system used for the general formalism
[13] will be adopted here including the labeling of the
various coefficients, characterizing the photocoloring
and the thermal fading processes by the letters P and
θ4λ,w = εBλ · CBw + εCλ · CCw
(4)
The four exponential coefficients (ϑ and ϑ T ) cor-
respond to the overall kinetic rate constants that are
given by the following general analytical expressions:
P
∗
The differential equations describing the reactive system can-
not be integrated in closed form when the irradiation of the reactive
medium is carried out at wavelength where two or all species absorb
differently (nonisosbestic irradiation) because the photokinetic fac-
tor is then time dependent [13]. The only known kinetic model for
such conditions is that of the simplest AB(1φ) photoreaction [19].
ꢃ
ꢂ
ꢄ
ϑ
= 0.5 · a19
a129 + 4 · a20
(5)
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272
MAAFI AND BROWN
The a1–a9 coefficients used in the preceding equa-
tions represent the specific rate constants of the various
species’ reactions. They are defined and labeled ac-
cording to the most general ABC(6k,6φ) mechanism,
as it has been given in [13].
In the subsequent analysis, we will work with the
nine ai coefficients as the system variables instead of
the absolute ki and φi quantities. This eases the mathe-
matical manipulation of the equations and establishes a
general method that can handle various kinetic mecha-
nisms (as different sets of ki and φi can be equal to zero
for different specific sequences). The set of Eq. (6) cor-
responding to the mechanism depicted in Scheme 3, for
example, is obtained by replacing all the missing ki by
zero. For the ABC(3k,6φ) systems, this formal strategy
does not alter the basic information of the system (and
hence the number of unknowns).
So far, we have identified a number of equations rel-
evant to the analysis of the kinetics. Considering the set
of Eqs. (1)–(12) for the photocoloring reaction (where
all the quantities are indexed P), we find seven linearly
independent equations that describe the reactive sys-
tem including Eqs. (2), (9), and (10). Note that Eqs. (4),
(10b), and (10c) are linearly dependent (vide infra), so
that only two of them can be considered for the fun-
damental system of equations to be used to solve the
kinetic system. Hence, at this stage seven more equa-
tions are required to reach the necessary quorum.
The first two extra equations derive from the photo-
stationary state condition. This condition assumes that
the variation of the species concentrations is negligible.
However, because these are linearly dependent,
a1 = −kAB − φAB · γ − kAC − φAC · γ
a2 = +kBA + φBA · γ
a3 = +kCA + φCA · γ
a4 = +kAB + φAB · γ
a5 = −kBA − φBA · γ − kBC − φBC · γ
a6 = +kCB + φCB · γ
a7 = +kAC + φAC · γ
a8 = +kBC + φBC · γ
a9 = −kCB − φCB · γ − kCA − φCA · γ
(6a)
(6b)
(6c)
(6d)
(6e)
(6f)
(6g)
(6h)
(6i)
where ki are the thermal rate constants, φi the quantum
yields of the elementary photochemical reactions, and
γ a constant factor
γ = εisos · l · I0 · F
(7)
dependent on the extinction coefficient of the species at
the isosbestic point (εisos), the optical path length (l),
the incident light intensity (I0), and the photokinetic
factor (F), which is considered constant throughout
the reaction, as discussed in the experimental section
(and where Atot is the total absorbance of the medium
at the irradiation wavelength).
dCAPSS
dt
dCBPSS
dt
dCCPSS
dt
=
+
(11)
only two may be selected. From the basic system of
differential equations describing the trimolecular sys-
tem [13], we write those corresponding to species B
and C as equal to zero at the PSS:
1 − 10−A
tot
F =
(8)
Atot
The ai coefficients (Eq. (6)) are related by three linear
equations, as
a4P · CAPSS + a5P · CBPSS + a6P · CCPSS = 0 (12a)
a7P · CAPSS + a8P · CBPSS + a9P · CCPSS = 0 (12b)
a1 = −a4 − a7
a5 = −a2 − a8
a9 = −a3 − a6
(9a)
(9b)
(9c)
The concentrations of the species at the PSS have
been previously defined as exclusive functions of the
a1–a9 coefficients and the initial concentration
(Eq. (13)). They represent three additional basic equa-
tions of the dynamics.
The a19–a21 coefficients defining the overall kinetic
rate constants (Eq. (5)) are analytically expressed by
the following formulae:
a3P · a8P − a2P · a9P
CAPSS
CBPSS
CCPSS
=
=
=
· C0
· C0
· C0
(13a)
(13b)
(13c)
a19 = a1 + a5 + a9
a20 = a7 · (a3 − a2) + a8 · (a1 − a3)
(10a)
(10b)
ϑ+P · ϑ
−P
a3P · a4P − a1P · a6P
+ a9 · (a2 − a1)
ϑ+P · ϑ
−P
ꢀ
a21 = (a3 · a4 − a1 · a6) · εBλ
a2P · a7P − a1P · a8P
+P · ϑ
ꢁ
+ (a2 · a7 − a1 · a8) · εCλ · C0 (10c)
ϑ
−P
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PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
273
Scheme 3 Photochemical and thermal reactions of 1 (A) and its postulated open forms B and C. The mechanism encompasses
all nine possible reaction steps, which are labeled according to the notation used in [13].
ꢅ
ꢆ
εBλ
εCλ
εCλ
εBλ
εBλ
εCλ
Two more equations are supplied by expressions found
in the intermediary stage of the general formalism
development. These relate to the formula for the ab-
sorbance of species B at the photostationary state:
a1λ1 = a6
+ a9 − a5 + a8
− a6
− a9
1
·
(16b)
ꢀελ · εCλ
ꢅ
ꢆ
C0
ꢀελ
CAPSS
ꢀελ
MPλSS
εCλ
εBλ
εBλ
εCλ
1
ꢀελ
APBSS,λ
=
−
−
(14)
a1λ2
=
a5 + a8
− a6
− a9
·
· C0
εCλ · ꢀελ
(16c)
with
The final series of equations is obtained from Eqs. (1)–
(12) for the thermal reaction (i.e., the fading process).
For this process, the coefficients are indexed by the let-
ter “T” and all the quantum yields in a1T–a9T (Eq. (6))
are equal to zero. In total, we can identify five equa-
tions to characterize the fading, i.e., Eqs. (2), (3), and
(10). However, if the fading does not lead to a ther-
mal equilibrium (this cannot be postulated for NPYs)
the coefficient a21T (Eq. (10c)) is equal to zero. Fur-
thermore, we cannot consider the initial absorbance
of the thermal reaction (Eq. (4)) because it is redun-
dant and equal to the absorbance recorded at the PSS
(MPλSS = M0λ,T ). Thus, only three equations are effec-
tively useful (Eqs. (2), (10a), and (10b)).
εCλ − εBλ
εCλ · εBλ
ꢀελ =
(15)
and the derivative of the cumulative absorbance at the
PSS (which may be equated to zero):
a1λ0 · CAPSS + a1λ1 · MPλSS + a1λ2 = 0
(16)
Here the a10–a12 coefficients are defined by the expres-
sions below [13], but are not considered as independent
variables themselves.
a1λ0 = εBλ · a4 + εCλ · a7
Accordingly, the fundamental system of lin-
early independent equations that describes fully any
ABC(3k,6φ) dynamics is given by the 14 equations
defined above and derived from the photocoloring (P)
ꢅ
ꢆ
εCλ
εBλ
εCλ
1
ꢀελ
− a5 + a8 − a6 − a9
(16a)
εBλ
International Journal of Chemical Kinetics DOI 10.1002/kin

274
MAAFI AND BROWN
and the fading (T) reactions. These are Eq. (2) (T,P),
Eq. (10a) (T,P), Eq. (10b) (T,P), Eq. (10c) (P), Eq. (12)
(2 × P), Eq. (13) (3 × P), Eq. (14) (P), and Eq. (16) (P).
It is important to note that if the method has been
successful in describing the kinetic system by achiev-
ing the required number of linearly independent equa-
tions to match that of the unknown parameters in-
volved, the latter system of equations still holds a
challenge to a complete kinetic elucidation due to the
nonlinear character of the equations.
equations) that have to be fulfilled by the solutions.
In this group of seven equations, we find Eqs. (4) and
(9), the mass balance equation (17), and two equations
endowed with the same meaning but expressed with
different sets of coefficients. They define the concen-
tration of species A at the PSS, one (Eq. (18)) is equiv-
alent to the forms of Eqs. (12), and the other (Eq. (19))
is found in the body of the bulk development of the
formalism [13] and will not be demonstrated further
here:
It is well known that a set of n linear and linearly
independent equations can readily be solved to give a
unique solution, represented by a set of values for the
n unknowns. For a kinetic system, this unique solu-
tion represents the true kinetic solution. However, the
situation is different for a set of n nonlinear and lin-
early independent equations. Indeed, more than one set
of values (for the n unknowns) can be found to mathe-
matically represent an exact solution to the set of equa-
tions. Furthermore, depending on the size and/or the
complexity of the latter set of equations, the total num-
ber of solutions can exceed the number of equations or
even be infinite. Since each set of values found for the
n unknowns is a solution to the system of equations,
the solution is said to be degenerate. In this respect, in
kinetics it is most important not to confuse any “plau-
sible” unique solution amongst the degenerate solution
set, which can be worked out based on sound assump-
tions, with the true solution because degeneracy means
that a similar probability of occurrence is allocated to
all options represented by the degenerate solutions.
Accordingly, the prospect of finding several sets of
parameter values as solutions of our system of equa-
tions is significantly increased. Despite the fact that all
of them are exact solutions, they are genuinely differ-
ent and most dramatically they are indiscernible, since
there are no effective means to allow us to eliminate
particular solutions and to single out the unique and
true kinetic solution. In general terms, this outcome
relates to the issues of identifiability and distinguisha-
bility [20–23], which we have discussed previously in
relation to NPY kinetics [14,16,17].†
C0 = CAPSS + CBPSS + CCPSS
(17)
a1P · CAPSS + a2P · CBPSS + a3P · CCPSS = 0 (18)
a1λ3 · CAPSS + a1λ4 · MPλSS + a1λ5 = 0
(19)
where the a13–a15 coefficients are defined by the fol-
lowing expressions:
ꢅ
ꢆ
a2P
εBλ
a3P
εCλ
1
ꢀελ
a1λ3 = a1P
−
−
(19a)
(19b)
(19c)
ꢅ
ꢆ
a3P
εCλ
a2P
εBλ
a3P
1
a1λ4
a1λ5
=
=
−
−
ꢆ
εCλ ꢀελ · εCλ
ꢅ
a2P
εBλ
a3P C0
εCλ
−
ꢀελ
APPLICATION TO 2H-NPY KINETICS
Absorption Spectra
The absorption spectrum of the closed form of 1 is
characterized in toluene by a strong absorption band at
280 nm and a second band with vibrational structure
between 300 and 350 nm. The bands are attributed to
π, π∗ transitions [7], and the last band envelop ends
at the lower limit of the visible (400 nm; Fig. 1). A
broad absorption band spanning the spectral region
between 400 and 650 nm appears after the solution
containing the closed form has been exposed to UV
irradiation. Absorption maxima are situated at 530 and
437 nm with a shoulder at 420 nm. These new spectral
features are due to the open forms that emerge from the
photochemical cleavage of the intracyclic C O bond
(Scheme 3).
The strong donor effects of the multiple p-methoxy-
phenyl groups cause a significant bathochromic shift
of the maximum of the absorption band of open
forms (λmax = 530 nm) of compound 1 relative to the
unsubstituted 2,2-diphenyl-2H-naphtho[1,2-b]-pyran
(λmax = 467 nm in acetonitrile [24] and 480 nm in
ethanol [7]). Also, the absorption maximum of com-
pound 1 is red-shifted by about 40–80 nm compared
In view of this complication, it is advisable to use as
much information and/or equations as possible to char-
acterize the dynamic system, to reduce the number of
the resulting solutions. With this aim, the following ex-
tra equations were used in the solving process for the
system, mainly as constraints (and not as fundamental
†
Note that in the much simpler cases where the three species’
individual Ci (t) experimental traces are available, we have found
(Maafi, M., 2006, unpublished data) that it is possible to analytically
solve up to ABC(6k,6φ) kinetics by using our formalism [13] (i.e.,
distinghuishability and identifiability problems are readily solved,
and the true kinetic solution is obtained).
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PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
275
Figure 2 Photocoloring and fading kinetic traces of 1,
7.52 × 10−5 M solution in toluene at 15◦C (λirr = 381 nm
and λ = 530 nm). Experimental data ( ) and fitting curves
◦
obs
using Eq. 1 (—). [Color figure can be viewed in the online
issue, which is available at www.interscience.wiley.com.]
Figure 1 Absorption spectra of 1 before and after
irradiation. [Color figure can be viewed in the online issue,
which is available at www.interscience.wiley.com.]
utes. These features represent a reliable indication of a
trimolecular mechanism [14].
The results corresponding to the fading trace
(Table I) strongly suggest that species C is virtually
thermally stable since one of its overall rate constants
(ϑ−T) can be approximated to zero. This indicates that
the trans–cis isomerization (C → B) is very slow, i.e.,
we can set kCB = 0 in the mechanism in Scheme 3.
Since the initial condition for the thermal reaction
implies that all three species are present (the concen-
trations of the closed forms, B and C, are likely to
be nonzero at the PSS), the apparent monoexponential
trend of the fading trace may account for two mech-
anistic options. One assumes that B undergoes a di-
vergent reaction (kBA and kBC both nonzero, kCB = 0),
and the other only takes into account the recyclization
of B (kBC = kCB = 0). From a theoretical viewpoint, the
difference between these two possibilities can be trans-
lated from the formalism [13] into Eq. (20) where the
absorbance recorded after a long-fading period (A∞C ) is
either equal to or greater than the absorbance of species
C at the PSS (ACPSS). If kBA and kBC are both nonzero,
then AC∞ > APCSS, whereas AC∞ = ACPSS if kBC = 0.
to the absorption maxima of several substituted 2H-
naphthopyrans recorded in various solvents, includ-
ing halogen-substituted structures with one, two, or
three identical or different F, Cl, Br atoms [24],
derivatives incorporating methyl-groups (Lapachenol)
(λmax = 452 nm in acetonitrile/water) [25,26], methyl-
and/or carbomethoxy-groups (λmax = 450–470 nm in
ethanol) [7], and that modified by the presence of an
adamantylidene group [27].
However, the spectrum of the open form(s) of 1
is blue-shifted compared to those corresponding to
rigid structures containing a benzene linked to the 2H-
structure through a cyclopentene ring (λmax ∼ 560 nm
in ethanol) [7]. These results indicate that the deep pink
color of the open forms of 1 may contribute to expand-
ing the application capabilities of these photochromes
as dyes.
Analysis of the Traces
The experimental data corresponding to the kinetic
traces recorded at 530 nm are given in Fig. 2. The
model equation (Eq. (1)) has been used to fit the pho-
tochemical (θ4 = 0) and the thermal (θ4 = 0) evolution
of the cumulative absorbance of the open forms with
reaction time (respectively, for the sample exposed to
UV light and in the dark). The fitting results reported
in Table I are characterized by a good accuracy and by
correlation coefficients over 0.99 for the experimental
versus calculated data of both traces.
kBC
A∞C = ACPSS + CBPSS · εC530
·
(20)
kBA + kBC
The 19F-NMR experimental evidence reported
by Arnall-Culliford et al. [25] for a 2,2-bis(4-
fluorophenyl)-2H-benzo[h]-chromene proved that the
thermal cis–trans isomerization (B → C) does not oc-
cur for this compound. In general, all literature reports
dealing with such photochromes [4,7,12, 26–28] have
ignored, de facto, this reaction step in the mechanism.
In the present study, we assume that the results ob-
tained for the 2,2-bis(4-fluorophenyl)-2H-benzo[h]-
chromene derivative are plausible and can be adopted
As is usually observed for such compounds, the pho-
tocoloring trace obeys a biexponential model (Eq. (1))
with fast and slow kinetic regimes (ϑ+P/ϑ = 6.35),
−P
and a photostationary state reached within a few min-
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MAAFI AND BROWN
Table I Fitting Results and the Defining Coefficient Values for the Traces of 1 in Toluene (15◦C)
Photocoloring tracea
Coefficientsb
Valuec
M0λ,P
0
mλ0,P
ϑ
ϑ
MPλSS
0.1838
+P
−P
0.00483
−0.0277
−0.00436
Fading tracea
d
Coefficientsb
Valuesc
M0λ,T
0.1838
M0λ,T
−0.00317
ϑ
ϑ
A∞C,T
+T
−T
−0.01858
0.00001
0.0143
a Experimental traces are fit by model Eq. (1), with correlation coefficients exceeding 0.99.
b The coefficient meaning and expressions are defined in the text, with indices P and T indicating, respectively, the data from the photocoloring
and the thermal fading.
c Values are obtained at 15◦C and at a monitoring wavelength λobs = 530 nm.
d This value is provided by the kinetic trace from the fading reaction (Fig. 2).
the minimum values of the extinction coefficients εBλ
and εCλ (2000 dm3 mol−1 cm−1), and the accuracy index
on the predicted set of results (convergence tolerance
<10−8), in carrying out the calculations. The results
obtained from the simultaneous resolution of the sys-
tem of equations (Appendix) are reported in Table II.
for the thermal reaction of our compound 1. Ac-
cordingly, the overall mechanism of 1 reduces to an
ABC(1k,6φ), where the fading is represented by a sin-
gle reaction step (kBA in Scheme 3). This reduces the
total number of the unknown parameters of the system
to 12.
Parameterization
DISCUSSION
By setting APCSS = AC∞ as has been discussed earlier, we
find that the coefficient a19T (Eq. (10a)) is equivalent
Time-resolved and NMR studies have shown that for
the 2H-benzopyrans and 2H-naphthopyrans the ring-
opening reaction is the prelude to the formation of sev-
eral intermediate species as the open forms can adopt
a number of different conformations dependent on the
orientation of the three exocyclic (two double and one
single) bonds. However, it has been strongly suggested
that many of these are short living species (less than a
nanosecond) and in general only two of these structures
have lifetimes longer than a millisecond. We attribute
to the cis–trans isomer the highest thermal rate constant
because it requires a single bond rotation to achieve re-
cyclization (species B in Scheme 2). The trans–trans
isomer (C) is now considered thermally inert during the
course of the photoreaction (500 s; Fig. 2), as is shown
in Scheme 3 (with kBC = kCB = 0). Since our experi-
mental setup allows us to collect data in time intervals
not less than 0.5 s, we can only observe the kinetics
due to these three species (A, B, C). And hence, the dy-
namics of 1 can be considered as a trimolecular system
that obeys an overall assumed ABC(1k,6φ) kinetics.
The fading reaction is characterized by rate constant
values comparable to that of other similar 2H-NPY
ranging between 1 × 10−3 and 2 × 10−2 s−1 depend-
ing on structure, solvent, and temperature. In general
terms, these values are of the same order of magnitude
as those of the unsubstituted 2,2-diphenyl-2H-NPY. It
seems that the variation of the rate constant values is
more sensitive to geometry and steric hindrance than
to the negative value of the thermal rate constant kBA
,
a20T (Eq. (10b)) is equal to zero and the numerical value
of the absorbance of species B at the PSS (Eq. (14))
is defined experimentally by using the expression of
the cumulative absorbance (Eq. (4)) and APCSS. This
also implies that one parameter of the system has been
identified, meaning that only 11 equations are now
required to fully describe the reactive system under
study.
Taking into account the discussion above, it is pos-
sible to define the set of equations that will be used for
the parameterization. We give in the appendix the series
of fundamental as well as the supplementary equations
employed in the present study to describe and to solve
the dynamics of 1. As can be seen, many equations
have been attributed a numerical value by using those
supplied by the fitting results as given in Table I. For
the a19–a21 coefficients, the values are determined on
the basis of the following expressions that we have set
previously [15]:
a19 = ϑ + ϑ
(21)
(22)
(23)
+
−
a20 = −ϑ · ϑ
+
−
a21P = −a20P · MPλSS
Beside the supplementary equations (Appendix), we
use constraints on the sign of the a1–a9 coefficients,
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PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
277
Table II Six Predicted Sets of Kinetic and Spectroscopic Parameter Values for 1 Photochemical Reaction in Toluene
at 15◦Ca
Result Sets
S
a
S
S
S
S
S
f
b
c
d
e
Kinetic Datab
ABC(1k,6φ)
ABC(1k,5φ)
ABC(1k,4φ)
φ
φ
φ
φ
φ
φ
γ
γ
γ
γ
γ
γ
0.00505
0.00240
0.00171
0.00176
0.00238
0.00708
0.00203
0.00432
0.00225
0.00268
0.00118
0.00306
0.00785
0.00089
0.00038
0.00580
0.00139
0.00205
0
0.00424
0
4.63 × 104
8.04 × 103
6.98 × 10−5
3.66 × 10−6
1.78 × 10−6
0.00832
0.00079
0
BA
AB
BC
CB
CA
AC
0
0.00157
0.00250
0.00029
3.11 × 104
3.03 × 103
6.50 × 10−5
5.45 × 10−6
4.71 × 10−6
0
0
0.00428
0.00006
0.00429
0.00007
7.99 × 104
1.28 × 104
7.20 × 10−5
2.12 × 10−6
1.12 × 10−6
0.00017
0
εB530
2.65 × 104
2.67 × 103
6.34 × 10−5
6.40 × 10−6
5.35 × 10−6
2.86 × 104
3.82 × 103
6.55 × 10−5
5.93 × 10−6
3.75 × 10−6
7.04 × 104
1.14 × 104
7.16 × 10−5
2.40 × 10−6
1.25 × 10−6
εC530
CAPSS
CBPSS
CCPSS
a The value of the rate constant (kBA) of the single thermal reaction step is given by −ϑ from Table I.
+T
b The product φi · γ in s−1; the extinction coefficients of species B and C at the observation wavelength (530 nm) in dm3 mol−1 cm−1; and
the species concentrations at the PSS in mol dm−3
.
to any electronic effects introduced by the substitutent
groups. As a matter of fact, it has been difficult to cor-
relate such variation to either the electronic properties
of the substitution groups or the other spectroscopic or
kinetic features of series of these compounds.
ruled out. Consequently, to give an account of these
observations, the two metastable open forms of 3,3-
diphenyl-3H-naphtho[2,1-b]pyran that are produced
either have a common excited state intermediate pre-
cursor or thermally interconvert between one another.
Conversely, the photoreactivity of the open forms
has been much more difficult to study. Nonetheless,
Go¨rner and Chibisov [10] have suggested on the ba-
sis of microsecond transient spectra of 3,3-diphenyl-
3H-naphtho[2,1-b]pyran that the closed form and the
trans–trans species are both photochemically reactive
to produce the other two structures, but that species B
(cis–trans isomer) reacts exclusively toward the closed
form (by both thermal and photochemical routes). It
is obvious from our experimental results on 1 that the
thermally stable isomer (C) is necessarily photoactive,
otherwise it can be accumulated over a long time and
the reaction will end by complete transformation of the
initial species into C. Accordingly, the very fact of ob-
serving a photostationary state for 1 (kBC = kCB = 0), is
a good indication for the deactivation of excited mero-
cyanine C* into at least one of its counterparts (φCB = 0
and/or φCA = 0, in Scheme 3). On the basis of this dis-
cussion, it emerges that no reaction step among the six
possible photoreactions of Scheme 3 can objectively
be ruled out with certainty.
Much less quantitative and/or qualitative results are
available in the literature on the photocoloring process
of 2H-naphthopyrans. Picosecond and submillisecond
investigation of some 2H-benzopyrans (2,4-diphenyl-
2H-benzopyran and 2,2,4-triphenyl-2H-benzopyran)
[28] concluded that the photochemical C O bond
cleavage occurs exclusively via the first excited sin-
glet state within 2 ps. It has been suggested for these
species that the photocleavage yields only the open
forms (20 ps–12 µs) that can revert to the closed form
by a single bond rotation. In a subsequent photoexci-
tation of the initially generated open forms, Kodama
et al. [29] have suggested other open forms (lifetime
>1 ms) are formed (which need two double bond ro-
tations for reversion to the closed form). Femtosecond
results dealing with the unsubstituted 3,3-diphenyl-
3H-naphtho[2,1-b]pyran derivative [6,9] in organic
media show that the ring-opening reaction proceeds
mainly from a singlet excited state and is much faster
(450 fs) than that reported for 2H-benzopyrans [29].
Aubard et al. [6] observed that the transient spectra
recorded after a delay of 10 ps do not exhibit any fur-
ther spectral changes (until 100 ps) and resemble those
obtained in continuous irradiation experiments. Given
the excitation pulse width used in these experiments
(≤100 fs) and the similarity of the visible region spec-
tra of species B and C [15], the eventuality of them
being simultaneously produced cannot be definitely
The approaches applied so far with relation to naph-
thopyrans studied under continuous irradiation have
dealt with three structures including 3,3-diphenyl-
3H-naphtho[2,1-b]pyran [11,12] and two 2H-NPYs
[26,27]. Except for the 3H-NPY derivatives for which
an ABC(1k,1φ) kinetic has been adopted [12], each
of these studies has been carried out on the basis of
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278
MAAFI AND BROWN
a preconceived ABC(1k,3φ) mechanism that ignores
some of the reaction steps discussed above. In the
case of the 2H-NPYs, it has been suggested that the
closed form species undergoes a divergent photoreac-
tion yielding both open form species, but considered
the cis–trans open form (isomer B) to be photochem-
ically inactive [26,27]. In addition, the methods used
assumptions and worked with a deficit in the number
of equations in comparison with that of the unknown
parameters. In this respect, the mathematical formal-
ism we have developed [13] is an interesting tool to
elaborate new spectrokinetic methods [15,17] without
the need to introduce any simplifying assumptions. In
a previous study [15], we demonstrated that the kinet-
ics of systems such as these naphthopyrans can only be
solved analytically for ABC(1 or 2k,2φ) mechanisms if
the only available experimental data are a photocolor-
ing fading data set. However, the formalism yields, as
described in this paper, the prerequisite number of in-
dependent equations to match the unknown parameters
and can hence fully describe an ABC(3k,6φ) system.
The solution for such cases can only be achieved by
solving this system of linearly independent (but non-
linear) equations.
We have applied the analytical solution of the
ABC(1k,2φ) kinetics to the experimental results of 1,
but no consistent set of values could be found. This is
interpreted as meaning that the mechanism governing
the reaction of 1 cannot be described by the various ki-
netic models allowed by an ABC(1k,2φ) system. This
is in agreement with the results depicted in Table II,
and are analogous to those reported for a 3H-NPY
derivative [15].
In the case of the present kinetics of compound 1, the
number of fundamental equations exceeds by one the
number of parameters (see the appendix). However,
despite this fact, it is not possible to reach a unique
solution (the true solution). This situation arises from
the nonlinearity of the equations, with the consequence
that the kinetic solution is degenerate, i.e., multiple
solutions are found that fit the experimental results.
In this study, solving the set of equations for the
experimental traces of compound 1 (Fig. 2) yields
three different generic mechanisms, i.e., ABC(1k,4φ),
ABC(1k,5φ), and ABC(1k,6φ) (as given in Table II).
These mechanisms are the only ones that represent a
realistic physical meaning (other options, which in-
volved negative values for any of the system parame-
ters or values over unity for the quantum yields, were
discarded).
the true sequence, the kinetics are said to be nondistin-
guishable [14,16,20–23]. Moreover, for each individ-
ual mechanism our results show that several different
parameter sets can be found. This outcome underlines
the nonidentifiability of the solution [14,16,20–23].
These two problems of distinguishability and identi-
fiability prove to be central for such systems and out-
line the increased difficulty of determination of the true
solution (if achievable).
It is important to note that tens of plausible se-
quences can often be postulated for such kinetics. Ex-
amples in the literature show that the few kinetic stud-
ies available use preconceived mechanisms that opt
for simplicity. They involve three or less phototrans-
formations (ABC(1k,1φ) [12], ABC(1k,2φ) [26], and
ABC(1k,3φ) [27]) to circumvent a common problem
in the methods employed, namely a lack of indepen-
dent equations that describe the reaction to match the
number of unknown parameters.
The present treatment shows that it is possible to
avoid postulating a mechanism for the reaction prior
to the analysis of the experimental data and that the
sequences proposed earlier in the literature seem to
be unsuitable for 1 since the simplest option in our
study involves at least four photochemical reactions
ABC(1k,4φ). Such discrepancies underline two gen-
eral but important issues. First, it is possible to reach
a unique solution for a given complex kinetic system
by means of introducing a number of “reasonable”
assumptions. However, as demonstrated here for 1
(and compared to the literature findings), the achieved
unique solution might not be the true solution. This
then casts doubts on the usefulness of the results ob-
tained and raises the question of their validity. Second,
reproducing the experimental traces by using the pa-
rameter values of a unique solution (through a model
or numerical integration) is a necessary but not a suf-
ficient condition to conclude that the unique solution
at hand is the true solution. This stems from the de-
generacy of the solution where all the options of the
degenerate solution would allow a faithful reconstruc-
tion of the experimental traces. This is the case for a
good correlation between the options in Table II and
the experimental traces of Fig. 2. Hence, no solution
can be deemed acceptable for such systems (as the true
solution) before the degeneracy issue has been properly
addressed.
In the present study, the analysis of the kinetic data
shows that the true mechanism describing the reaction
of compound 1 is to be found among six options in-
volving between four and six photoreactions (Table II).
This represents the best result that can be achieved us-
ing kinetic data alone. A definitive decision about the
mechanism will need extra independent data. However,
Without imposing any prerequisite conditions or as-
sumptions, our treatment generates only a few options
that represent possible mechanisms for the dynamics
of 1. Since there is not a means to select among them
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PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
279
we can obtain more details about the reaction of 1 by
looking at the common features of the options laid out
in Table II.
low values of φAB dispute the findings of earlier studies,
where this quantum yield has been found to approach
unity (φAB > 0.8) [10–12]. The reason for this latter
finding can be traced to the treatment used therein,
which was originally developed for AB systems. Those
Inspection of the results presented in Table II re-
veals that the A ↔ B (i.e., φBA and φAB) and C → A
(φCA) photochemical reactions are always predicted to
occur. The occurrence and efficiency of the other three
photochemical reactions varies from case to case but
it is very noticeable that φAC is always very low or
zero. The A → C photochemical reaction, therefore,
only constitutes a small percentage (0%–12%) of the
photoreaction profile of the closed form, which is dom-
inated by the ring-opening reaction to form species B
(φAB). This is unsurprising given the geometric rear-
rangement that needs to take place in each case. In
those parameter sets where φAC = 0, it is reassuring to
observe that φBC is nonzero and vice versa. In other
words, there is always a (one or two photon) photo-
chemical route predicted for the formation of species
C from the initial closed form. If it is possible to draw
the conclusion that the A → C direct photochemical
reaction is negligible, which corroborates the experi-
mental data corresponding to several unsubstituted and
substituted derivatives of benzopyran [30,31], then pa-
rameter sets Sb, Sd, and Sf may be eliminated from con-
sideration (as φBC ≈ 0) leaving three (Sa, Sc, and Se)
mechanisms to account for the kinetics of compound
1. It is worth noting that two among these sequences
(Sa and Sc) represent a similar generic ABC(1k,5φ)
mechanism (as φAC ≈ 0) and can be seen as an exam-
ple of the unidentifiability of the solution.
methods consider the overall kinetic rate (such as ϑ
+P
in Table I) as if it is due to the exclusive reaction of
species A, resulting in an overevaluation of φAB
.
The final observation that we can make about the
photoreactions relates to the photochemical paths open
to species C. The parameter sets in Table II make it
clear that the C → A photoreaction is always predicted
to take place (and with good efficiency). One could,
therefore, intuitively expect the C → B photoreaction
to also take place. If this argument holds, then the
ABC (1k,4φ) mechanism (Se) can be eliminated since
it predicts φCB = 0.
With regard to the nonphotochemical parameters
provided in Table II, it is interesting to note that εB530
is always predicted to be higher than εC530 by a signif-
icant factor of between 6 and 10. This is somewhat
surprising given the similarity of their structures. The
photostationary state, under our experimental condi-
tions, is determined by the closed form—the maximum
conversion to the open forms being 15.6% for the pa-
rameter set Sa. The relative amount of B and C, which
are present at the PSS, is quite similar for all the cases
presented in Table II. However, the combination of the
extinction coefficients with the stationary state concen-
trations indicates that the experimental absorbance at
the 530-nm observation wavelength is dominated by
the contribution from species B (92%).
It is noticeable that the photochemical recycliza-
tion reactions of both open form isomers are predicted
to take place irrespective of the number of other pro-
cesses that are also occurring in the sequence. These
steps are always significantly more efficient than the
corresponding ring-opening reactions and also occur
with a higher yield than their cis–trans isomerizations
(φBA > φBC and φCA > φCB). For the A ↔ B reactions,
the relative efficiencies of the ring-closure and the ring-
opening reactions lie in the range 2–10, whereas for
A ↔ C the lowest ratio is 9. In the case of the A ↔ B
reactions, this observation may indicate that the inter-
nal conversion of A∗ to the ground state represents
significant competition to the ring-opening reaction. A
parallel example can be found in the case of a nitro-
substituted spiropyran [32] for which a femtosecond
vibrational investigation showed that up to 90% of the
excited singlet state population decays to the ground
state. If this proves to be true for NPYs as well, it gives
a first clue as to how to improve the performance of the
system (i.e., by targeting the design of structures capa-
ble of higher reactionrates of theinitial species or byre-
ducing the efficiency of the internal conversion). These
It is worth noting that similar results for the photore-
actions efficiencies have been found for the 3H-NPY
that has been investigated using monoexponential pho-
tochemical kinetics and a specific mathematical devel-
opment [16]. This indicates that the photochemistry of
both 2H- and 3H-NPY derivatives is achieved through
similar routes despite geometry alteration.
CONCLUSION
The descriptive spectrokinetic method developed in
this paper is fully applicable to the investigation of
complex ABC(3k,6φ) reactive systems, and easily
adapted to the study of photochromic naphthopyran
kinetics. It also proves the versatility of the basic for-
malism and its ability to address a number of reaction
systems.
For the first time, this study has shown that it is pos-
sible to derive as many linearly independent equations
as there are unknown parameters that define the system
under consideration.
International Journal of Chemical Kinetics DOI 10.1002/kin

280
MAAFI AND BROWN
ꢅ
ꢆ
εCλ
εBλ
εBλ
εCλ
1
ꢀελ
The results of the present method have also un-
derlined for the first time the importance of the dis-
tinguishability/identifiability problems with relation to
the kinetics of 2H-NPY compounds. This conclusion
confirms previous experience with the homologous
3H-NPY systems. It also raises a crucial point of
interest; namely that whether the mechanism is pre-
conceived or not, unavoidable and fundamental issues
of distinguishability, and/or identifiability have to be
solved to achieve any consistent progress in under-
standing (and improving) naphthopyran reactions and
similar systems. Effort is being expended in our labo-
ratory to increase our understanding of these issues.
For compound 1, the predictions of our method
point clearly to the reaction steps that are an essen-
tial part of the true mechanism. The phototransforma-
tion of the closed form is dominated by the forward
A → B reaction (A → C is only a negligible event),
and the divergent conversion of species B is likely to
be more efficient in the direction of the recyclization
to the closed form. Similarly, the C → A reaction is
always predicted and hence is considered to be active
within the true mechanism. The occurrence of the fi-
nal possible photochemical step (C → B) is not clearly
confirmed. The spectrokinetic method developed in
this paper identifies an ABC(1k,4φ) mechanism as the
minimum possibility for the kinetics of compound 1.
a1λ2
=
a5 + a8 − a6 − a9
·
· C0
ꢅ
ꢆ
a2P
εBλ
a3P
εCλ
1
ꢀελ
a1λ3 = a1P
−
−
ꢅ
ꢆ
a3P
εCλ
ꢅ
a2P
εBλ
a3P
1
a1λ4
a1λ5
=
=
−
−
ꢆ
εCλ ꢀελ · εCλ
a2P
εBλ
a2P C0
εCλ
−
ꢀελ
Fundamental Equations
Photochemical Dynamics
ꢀ
ꢁ
mλ0,P = εBλ · a4P + εCλ · a7P · C0 = 0.00483
a19P = a1P + a5P + a9P = ϑ+P + ϑ = −0.03206
−P
a20P = a7P · (a3P − a2P) + a8P · (a1P − a3P) + a9P
· (a2P − a1P) = −ϑ+P · ϑ = −1.2 × 10−4
−P
ꢀ
a21P = (a3P · a4P − a1P · a6P) · εBλ
ꢁ
+ (a2P · a7P − a1P · a8P) · εCλ · C0
= −a20P · MPλSS = 2.22 × 10−5
a4P · CAPSS + a5P · CBPSS + a6P · CCPSS = 0
a7P · CAPSS + a8P · CBPSS + a9P · CCPSS = 0
a3P · a8P − a2P · a9P
APPENDIX
Definitions
CPASS
CPBSS
CPCSS
=
=
=
· C0
· C0
· C0
ϑ
+P · ϑ
a3P · a4P − a−1PP · a6P
ϑ
+P · ϑ
a2P · a7P − a−1PP · a8P
a1 = −φAB · γ − φAC · γ a2 = +kBA + φBA · γ
a3 = +φCA · γ a4 = φAB · γ
a5 = −kBA − φBA · γ − φBC · γ a6 = +φCB · γ
a7 = +φAC · γ a8 = +φBC · γ
ϑ
+P · ϑ
−P
C0
CAPSS
MPλSS
εCλ · ꢀελ
APBSS,λ
=
−
−
ꢀελ
ꢀελ
a1λ0 · CAPSS + a1λ1 · MPλSS + a1λ2 = 0
a9 = −φCB · γ − φCA.γ a19 = ϑ + ϑ
+
−
a20 = −ϑ · ϑ
a21P = −a20P · MPλSS
+
−
εCλ − εBλ
Thermal Dynamics
ꢀελ =
εCλ · εBλ
mλ0,T = −kBA · εBλ · CBPSS = −0.00317
a19T = −kBA = −0.01858
a1λ0 = εBλ · a4 + εCλ · a7
ꢅ
ꢆ
εCλ
εBλ
εBλ
εCλ
1
ꢀελ
−
a5 + a8 − a6 − a9
Supplementary Equations
ꢅ
ꢆ
εBλ
εCλ
εBλ
εBλ
a1λ1 = a6 + a9 − a5 + a8 − a6 − a9
εCλ
εCλ
a1P + a4P + a7P = 0
a5P + a2P + a8P = 0
1
·
ꢀελ · εCλ
International Journal of Chemical Kinetics DOI 10.1002/kin

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
281
a9P + a3P + a6P = 0
Molecular Science; Martin, M. M.; Hynes, J. T. (Eds.);
Elsevier: Amsterdam, 2004; p. 283.
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a1λ3 · CAPSS + a1λ4 · MPλSS + a1λ5 = 0
a1P · CAPSS + a2P · CBPSS + a3P · CCPSS = 0
MPλSS = εBλ · CBPSS + εCλ · CCPSS = 0.1838
APCSS = CCPSS · εC530 = 0.0143
C0 = CAPSS + CBPSS + CCPSS = 7.52 × 10−5
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