Photophysics and kinetics of naphthopyran derivatives, part 3: A general procedure to uniquely identify the kinetic and spectroscopic parameters of ABC(2k,2φ) systems and application to naphthopyran reactions

Article abstract of DOI:10.1002/kin.20192

A new procedure has been developed with the aim of determining the kinetic and spectroscopic parameters of any photochemical systems of the generic ABC(2k,1φ) and ABC(2k,2φ) types. General expressions of the colorability and the equilibrium concentration ratio of two (of the three involved) species at a photostationary state or a state of thermal equilibrium have been derived. The procedure has been successfully applied to achieve unique identifiability for the seven sequences of the ABC(2k,1φ and 2φ) type that may represent naphthopyran dynamics. It is demonstrated that the reactivity of this particular compound is certainly described by a higher kinetic level than those (three or four reaction steps) considered in the present study.

Full text of DOI:10.1002/kin.20192

Photophysics and Kinetics
of Naphthopyran
Derivatives, Part 3:
A General Procedure
to Uniquely Identify the
Kinetic and Spectroscopic
Parameters of ABC(2k,2φ)
Systems and Application
to Naphthopyran Reactions
M. MAAFI,1 R. G. BROWN2
1Leicester School of Pharmacy, De Montfort University, The Gateway, Leicester, LE1 9BH, UK
2School of Applied Sciences, University of Glamorgan, Pontypridd, Rhondda Cynon Taff, Cardiff, CF37 1DL, UK
Received 27 June 2003; revised 5 April 2005; accepted 17 March 2006
DOI 10.1002/kin.20192
Published online in Wiley InterScience (www.interscience.wiley.com).
ABSTRACT: A new procedure has been developed with the aim of determining the kinetic
and spectroscopic parameters of any photochemical systems of the generic ABC(2k,1φ) and
ABC(2k,2φ) types. General expressions of the colorability and the equilibrium concentration
ratio of two (of the three involved) species at a photostationary state or a state of thermal
equilibrium have been derived. The procedure has been successfully applied to achieve unique
identifiability for the seven sequences of the ABC(2k,1φ and 2φ) type that may represent naph-
thopyran dynamics. It is demonstrated that the reactivity of this particular compound is cer-
tainly described by a higher kinetic level than those (three or four reaction steps) considered
in the present study. ꢀ 2006 Wiley Periodicals, Inc. Int J Chem Kinet 38: 421–430, 2006
C
Parts 1 and 2 of this series are references [1] and [2].
Correspondence to: Mounir Maafi; e-mail: mmaafi@dmu.ac.uk.
2006 Wiley Periodicals, Inc.
c
ꢀ

422
MAAFI AND BROWN
is conducted using monochromatic light whose wave-
length corresponds to that of an isosbestic point. We
also assume that a true photostationary state is attain-
able within a reasonable time period. In the subsequent
treatment the labeling of all the parameters is carried
out in conformity with that used in previous studies [1
and 2] except when otherwise specified.
We will consider two separate situations, namely
where the system is being irradiated or where the irra-
diation light is switched off and only thermal processes
operate. The quantities related to the photochemical re-
action are throughout denoted by the subscript “P” and
those characterizing the thermal reaction, when the ir-
radiation light has been removed, are denoted by the
letter “T.” The photostationary state is referred to by the
symbol “PSS,” a state of thermal equilibrium by “STE”
and initial quantities by the superscript or subscript “0”.
Changes in concentration of one or more of the species
during the course of the reaction are monitored at an
observation wavelength λobs nm.
INTRODUCTION
Many photochemical systems involving three reactive
species are characterized by three or four reaction steps
including two thermal processes and one or two pho-
tochemical reactions [3–9]. These can symbolically be
named as ABC(2k,2φ) systems. Theoretically, the sim-
plest ABC(2k,1φ) systems are dependent on eight vari-
ables, i.e. two rate constants, a photochemical yield,
two molar absorption coefficients and three concentra-
tions at the photostationary state, if we consider that the
spectral features of one compound are known as well
as the initial concentrations of all individual species.
Frequently, these system constants are not accessible
to measurement and neither are the spectra of all the
species involved. Consequently, analysis of the pho-
tochemical reaction is practically very demanding; so
that, to determine the set of parameters, the approaches
proposed generally necessitate the introduction of as-
sumptions [3–9].
In the present procedure it will be assumed that the
experimentalist knows the total initial concentration
(C0) of the starting compounds (Eq. (1)).
In this study, for the first time we propose a general
procedure aiming to analytically solve the kinetics of
such systems without introducing any simplifications.
Furthermore, only one set of parameters is generated
because the method offers the possibility to achieve
unique identifiability for each of the sequences consid-
ered. In the second part of the paper, we apply the pro-
cedure to investigate the kinetics of the photochrome
3,3-diphenyl-3H-naphtho[2,1-b]pyran.
C0 = CA0 + CB0 + CC0 = CA(t) + CB(t) + CC(t) (1)
The rate constants and photochemical yields in-
volved in the various ABC(2k,2φ) mechanisms are de-
noted respectively, ka, kb and φc, φd (where the a–d
indexes correspond each to a couple of capital letters as
depicted in Scheme 1). For each possible mechanism
it is necessary to define the basic kinetic coefficients
a1−9 as given by Eqs. (2) [1]. The aiP coefficients are
obtained from Eqs. (2) by setting all ki and φi values
to zero except for the four reaction steps involved in
the mechanism under consideration. The coefficients
representing the thermal reactions (aiT) are similar to
the aiP ones but with all quantum yields set to zero. The
light constant γ is defined experimentally.
FUNDAMENTAL BASIS
The basic model showing the total number of pho-
tochemical and thermal reactions that can link three
species together is shown in Scheme 1. The solution
of any ABC(2k,2φ) mechanism extracted from the ba-
sic model will be achieved progressively in four steps.
The theoretical description of the kinetic model and all
the expressions used here to set out the general proce-
dure are based on the formalism developed previously
for ABC(nk,mφ) systems [1]. The reader is therefore
referred to reference [1] for the derivation of the equa-
tions used in this paper. The irradiation of the solution
a1(P,T) = −kAB − φAB · γ − kAC − φAC · γ (2.1)
a2(P,T) = kBA + φBA · γ
a3(P,T) = kCA + φCA · γ
a4(P,T) = kAB + φAB · γ
(2.2)
(2.3)
(2.4)
a5(P,T) = −kBA − φBA · γ − kBC − φBC · γ (2.5)
a6(P,T) = kCB + φCB · γ
a7(P,T) = kAC + φAC · γ
a8(P,T) = kBC + φBC · γ
(2.6)
(2.7)
(2.8)
a9(P,T) = −kCB − φCB · ꢀ − kCA − φCA · γ (2.9)
γ = F ·l · I0 · εisos. = constant (2.10)
Scheme 1 General model from which ABC(2k,2φ) sys-
tems are extracted.

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
423
We proceed now to the description of the individ-
ual steps of the procedure for a single hypothetical se-
quence ABC(ka, kb, φc, φd). Each step is aimed at de-
riving one or more of the unknown parameters that are
subsequently used in the next step(s).
pair of values ϑ and ϑ
.
−T
+T
ka = −ϑ
kb = −ϑ
(3.5)
(3.6)
+T
−T
In the case where a mono-exponential trace is ob-
served, the ꢁ values can be attributed to the rate con-
stant of a thermal process if there is only one thermal
reaction in the mechanism. If there are two thermal
processes involved in the mechanism, it is possible
that the observed kinetics are wavelength-dependant
and the values of ka and kb can be derived from mono-
exponential fits as a function of λobs. (Note: the derived
ϑ value will be equal to ka, kb or ka + kb.)
Step 1: Defining the Rate Constants
The data collected from the separate photochemical
and thermal experiments are individually fitted by the
model Eq. (3.1). We stress here that the procedure is
exclusively applicable to those sequences which do not
include a reversible thermal process linking two of the
three species. This work will consider bi-exponential
kinetic traces and a general treatment for ABC(2k,2φ)
mechanisms, and hence, simpler cases can easily be
worked out from this procedure.
According to the formalism [1] the photochemical
and thermal overall reactions of any ABC(2k,2φ) se-
quence are classified in Cases I and II and well de-
scribed by Eq. (3.1).
The few mono-exponential kinetic cases which are
not solvable by the present procedure are those where
only the sum of rate constants (ka + kb) is experimen-
tally accessible, e.g., a convergent mechanism where
the spectra of species A, B and C completely overlap
each other and only compound A is isolable.
m0λ(P,T) − M0λ(P,T) · ϑ
−(P,T)
+(P,T) · t
M(λP,T)(t) = −
· eϑ
Step 2: Defining the Quantum Yields
ϑ
−(P,T) − ϑ
+(P,T)
The fitting of the trace generated by the photochemi-
mλ0(P,T) − M0λ(P,T) · ϑ
+(P,T)
cal reaction gives the overall reaction rates (ϑ and
+P
−(P,T)·t
+
· eϑ
+ θ4λ(P,T)
ϑ
−P) whose expressions incorporate combinations of
ϑ
−(P,T) − ϑ
+(P,T)
the a19P and a20P coefficients, independent of the wave-
length, as shown in Step 1 (Eqs. (3.3) and (3.4)).
The numerical values of a19P and a20P can be ob-
tained by using Eqs. (3.2) and the fitting results:
(3.1)
where M0 and m0 represent the initial absorbances and
rates of the reactions, and the θ4 coefficients are con-
stantsdefinedataspecificwavelength. Theoverallther-
mal rate constants are given by the following expres-
sions [1]:
a19P = ϑ+P + ϑ
(4.1)
(4.2)
−P
a20P = −ϑ+P · ϑ
−P
ꢀ
ꢂ
ꢁ
a129(P,T) + 4a20(P,T) (3.2)
As we already know the values of ka and kb from
Step 1, we are then left with a pair of simultaneous
equations involving the two unknowns φc and φd. In
this situation the solution is simple, however, it can
lead to several solutions if the expressions in Eqs. (3.3)
and/or (3.4) are quadratic. In this case all positive roots
should be retained, and each pair of roots (φc·γ, φd·γ)
tested further on. This indicates that cases might exist
where unique identifiability is not possible because of
these different solutions.
ϑ
(P,T) = 0.5 · a19(P,T)
where the coefficients a19 and a20 are defined by
a19P = a1P + a5P + a9P
a20P = a7P(a3P − a2P) + a8P(a1P − a3P) + a9P(a2P − a1P)
(3.3)
(3.4)
The fitting results of the purely thermal reaction im-
mediately supply the values of the thermal rate con-
stants as in Eqs. (3.5) and (3.6). However, when the
numbers“a”and“b”arespecifiedforaparticularmech-
anism, the values of ka and kb take either of the rate
At this stage it is important to note that the solutions
obtained in the former two steps will allow all the a1–a9
coefficients to be numerically defined. It is also worth
noting that the procedure allows the determination of
all kinetic parameters (k’s and specially φ’s) before any
quantitative knowledge of the spectroscopic features of
compounds A, B, and C or their concentrations at the
equilibrium state has been established.
constants ϑ and ϑ−T. In other words, if the relative
+T
magnitude of the rate constants (ka and kb) is not known
with certainty, each of them should be tested against the

424
MAAFI AND BROWN
Also, by using the mass balance equation (Eq. (1)), we
define the last concentration of compound A:
Step 3: Defining the PSS Concentrations
We will use in the following another set of equations
given in reference [1]. The latter formalism has been
developed for a specific case where the absorbance
(MPλ,T(t) as given by Eq. (3.1)) reflects the variation
of the absorbance of both B and C isomers. We use
the derived equations unchanged, however, it is worth
noting that the conclusion (as can be seen below) will
be applicable to any ABC(2k,2φ) case (i.e. the nature
of the signal analyzed does not matter).
CAPSS,STE = C0 − CPSS,STE − CCPSS,STE
(5.7)
B
The numerical values of the three expressions can be
worked out since the values of each aiT (Steps 1 and
2), aiP (Step 2) and C0 are known.
As we can see from Eqs. (5.5)–(5.7), the concen-
tration expressions at the (photostationary or thermal)
equilibrium state are exclusively dependant on the ki-
netic parameters, do not incorporate specific molar ab-
sorptioncoefficientsandarenotwavelengthdependent.
Accordingly, these represent general formulae that are
valid for all sequences independently of the nature of
the signal that has been analyzed (containing one, two
or three cumulative absorbances).
Furthermore, because these results are true as well
for the ABC(6k,6φ) general model, a more important
andgeneraldefinitionemergeswithrelationtotheequi-
librium concentration ratio (Keq). Depending on the
specific dynamics involved, the relationship (Eq. (5.8))
is applicable to purely thermal, purely photochemical
or combined thermal/photochemical reactions in order
to define the concentration ratio at an equilibrium state.
The absorbance at the photostationary state (PSS) or
at a state of thermal equilibrium (STE), which we call
here MPλSS,STE, is given by the relationships Eqs. (5):
a2λ1P
a20P
θ4λP = −
θ4λT = −
= MPλSS
= MSλTE
(5.1a)
(5.1b)
a2λ1T
a20T
where the coefficient a2λ1 has the following explicit
form:
ꢃ
a2λ1(P,T) = (a3(P,T) · a4(P,T) − a1(P,T) · a6(P,T)) · εBλ
ꢄ
+ (a2(P,T) · a7(P,T) − a1(P,T) · a8(P,T)) · ελC · C0 (5.2)
CBPSS,STE a3(P,T) · a4(P,T) − a1(P,T) · a6(P,T)
PSS,STE
eq
K
=
=
CCPSS,STE
a2(P,T) · a7(P,T) − a1(P,T) · a8(P,T)
By introducing Eq. (5.2) into Eq. (5.1) we find:
(5.8)
a3(P,T) · a4(P,T) − a1(P,T) · a6(P,T)
MPλSS,STE
=
· C0 · ελB
−a20(P,T)
An immediate consequence of the definition of the
concentrations in analytical forms (Eqs. (5.5)–(5.7)) is
found in solving the identifiability of those ABC sys-
tems possessing an equilibrium state and whose kinetic
parameters (ai coefficients) are known (see Step 4).
a2(P,T) · a7(P,T) − a1(P,T) · a8(P,T)
+
· C0 · ελC
(5.3)
−a20(P,T)
On the other hand, the absorbance at the photosta-
tionary state (or at the thermal equilibrium) is equal to
the absorbance at the beginning of the thermal (as the
light is switched off) or the photochemical (as the light
is switched on) reaction, as defined [1] by
Step 4: Defining the Molar
Absorption Coefficients
In order to quantify the spectroscopic features of the
species, the final step of the procedure brings in the use
of the initial velocity equations relating to the photo-
chemical as well as the thermal reactions. These quan-
tities have the following forms [1]:
MPλSS,STE = CPSS,STE · εBλ + CPSS,STE · ελC
(5.4)
B
C
Finally, let us compare these latter equations. It can
be seen that by comparison of the terms of Eqs. (5.3)
and (5.4), the formulae for the concentrations at the
equilibrium states for the species B and C can be ex-
tracted as
ꢅ
ꢆ
m0λ(P,T) = a1(P,T) · εAλ + a4(P,T) · εBλ + a7(P,T) · ελC
ꢅ
· CA0,PSS,STE + a2(P,T) · εAλ + a5(P,T) · εBλ
ꢆ
ꢅ
+ a8(P,T) · ελC · CB0,PSS,STE + a3(P,T) · ελA
+ a6(P,T) · ελB + a9(P,T) · ελC · CC0,PSS,STE
a3(P,T) · a4(P,T) − a1(P,T) · a6(P,T)
CBPSS,STE
CCPSS,STE
=
=
· C0 (5.5)
· C0 (5.6)
−a20(P,T)
a2(P,T) · a7(P,T) − a1(P,T) · a8(P,T)
−a20(P,T)
ꢆ
(6.1)

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
425
where CX0 (or CXSTE) and CXPSS, expressing the concen-
trations (X = A, B, or C) at different specific times, are
correlated to the labels “P” and “T,” respectively.
The numerical values of m0P and m0T are obtained
graphically from the treatment of the experimental ki-
netic traces (or available from the fitting in the cases
where Eq. (3.1) is applied). We then have two equations
to identify the two unknowns εBλ and εCλ. However, for
the cases in which no information on the spectroscopic
properties of the species (A, B, and C) involved are
available (including those situations where the three
species are characterized by very similar absorption
spectral shapes), we would require one more equation
to obtain the three unknown extinction coefficients. At
this stage, we make use of the equations for the initial
absorbance before any reaction takes place (M0), at
the photostationary state (MPSS) and at the equilibrium
(completion) of the thermal reaction (MSTE). These can
be written in a general form as shown in Eq. (6.2), but
each case will have a specific expression according to
the occurrence of the concentrations, molar absorption
coefficients and/or the number of species responsible
for the signal measured.
temperature, the concentration of the starting material,
the spectral composition of the signal (e.g. Eq. (6.3) is
applicable whether one, two or all three species con-
tribute to the spectrum at λobs), the absorbance at the
irradiation wavelength (that of the isosbestic point) and
the incident light intensity. It should also provide the
same results for a system under flash or continuous
irradiation conditions.
mλ0P
CA0 · γ
COL =
= − (φ4 + φ12) · ελA + φ4 · εBλ + φ12 · ελC
(6.3)
Since the quantities mλ0P, γ and CA0 = C0 involved in
Eq. (6.3) are all available experimentally, the colorabil-
ity may be calculated even though the mechanism ex-
ceeds the ABC(2k,2φ) model, or is unknown altogether
(provided that the conditions set above are valid). This
represents an efficient and an easily evaluated criterion
to compare with the colorability of analogous systems;
e.g., for different (naphtho or spiro) pyran derivatives
or between a set of identical experiments carried out in
different media. Finally, it is worth noting that Eq. (6.3)
reduces to the exact definition for the colorability of
an AB system as the one already established in the
literature [7].
Mαλ = ελA · CAα + ελB · CBα + εCλ · CCα
(6.2)
where α = 0, PSS or STE.
Algorithm of the Procedure
Therefore, in all cases, Eq. (6.2) will generate extra
equations (two or more) that will allow us to complete
the parameter-defining procedure. Any spare equations
(not used in Step 4) can be employed to validate the
whole parameter set.
In the following we provide a flowchart of the different
steps of the procedure.
1. Write down the sequence that represents the pho-
tochemical/thermal ABC(2k,2φ) or ABC(2k,1φ)
dynamics under study.
2. Identify the initial concentration values (Eq. (1):
total C0 and individual CA0 , CB0 and CC0 ).
3. Write down the a1P–a9P coefficients for the spec-
ified sequence (Eqs. (2)).
4. Fit the experimental kinetic traces to the appro-
priate bi-exponential equation (Eq. (3.1)).
5. Calculate the rate constants for the thermal reac-
tion(s) (ka, kb) using Eqs. (3.5) and (3.6).
6. Write down the equations for a19P and a20P
(Eqs. (3.3) and (3.4)) and calculate their values
(Eqs. (4)).
A by-product of this procedure is that we can es-
tablish a definition of the colorability of photochromic
ABC systems such as chromenes, where the photo-
chemical reaction starts with a single, thermally stable
species. In the case of AB(1φ) systems, the colorability
has been defined as the product of quantum yield and
the extinction coefficient of the B (colored) species.
This is also applicable to AB(1k,1φ) systems if the
measurements are carried out so as to cancel out the
thermal process; for illustration, many spiropyrans un-
der flash photolysis conditions have been used as ap-
plication examples [7].
For the systems studied here, Eq. (6.1) reduces to
an expression made up exclusively by extinction coeffi-
cients and quantum yields of the colored species times
the initial concentration. Accordingly, the expression
forthecolorability(COL)forthesecasescanbederived
from the initial velocity of the photochemical reaction,
on the basis of Eq. (6.1). The colorability is given by
Eq. (6.3), which is defined specifically at the obser-
vation wavelength but is independent of the medium
7. Calculate the quantum yields (φc · γ, φd · γ) by
solving Eqs. (3.3) and (3.4).
8. Calculate the numerical values of the a1P–a9P and
a1T–a9T coefficients.
9. Define the species concentrations at the photo-
stationary state (CAPSS, CBPSS and CCPSS) and at
the state of thermal equilibrium (CASTE, CBSTE and
CCSTE) by applying Eqs. (5.5)–(5.7).

426
MAAFI AND BROWN
10. Write down the expressions of the initial quanti-
ties M0λP, MPλSS, MSλTE, mλ0P and m0λT according to
Eqs. (6.1) and (6.2), and calculate their numerical
values from the experimental traces.
mechanisms can be found to describe the NPY kinet-
ics (Scheme 2). Among this series there is a single
sequence which incorporates only three reaction steps
(S3 is an ABC(2k,1φ) system).
11. Define the molar absorption coefficients (ελA, εBλ
and ελC) by solving the appropriate system of
equations from point 10.
Experimental kinetic traces of the photo-
chemical and thermal reactions of 3,3-diphenyl-
3H-naphtho[2,1-b]pyran have been recorded in
5.87 × 10−5 M ethyl acetate solutions at 14◦C (Fig. 1).
The same experimental set-up and reaction conditions
as reference [2] have been used here except when
otherwise specified. The irradiation wavelength corre-
sponds to the isosbestic point situated at λirr = 358 nm.
The observation wavelength was that of the maximum
absorption of the photoisomers (λobs = 423 nm). The
fitting has been performed using Eq. (3.2) because
the analytical signal recorded in the visible region is
due to the appearance of the isomers B and C when
the initial species A is irradiated. The curves are fitted
with high accuracy with the corresponding parameter
values given in Table I.
12. Retrieve the complete individual spectra of the
(three) compounds by calculating the molar ab-
sorption coefficients at different wavelengths us-
ing each time the operations described in points
10 and 11.
APPLICATION EXAMPLE: NAPHTHOPYRAN
KINETICS
The dynamics of 3,3-diaryl-3H-naphtho[2,1-b]pyrans
(NPY) are known to obey a kinetic sequence derived
from the general ABC(4k,6φ) frame model (two rate
constants have been deleted as the compounds are
not thermochromic at 14◦C). However, their dynamics
have not been elucidated with certainty. In our previous
paper [2], we have shown that even for preconceived
mechanisms the difficult issue of identifiability is en-
countered in elucidating the parameter set. This means
that it is possible to find a number of different solu-
tions for the kinetic and spectroscopic parameter set,
with in each case, the model exactly reproducing the
experimental traces.
In the light of our interest in NPY dynamics, we
apply the procedure developed in the present work in
order to solve the identifiability problem for all possible
ABC(2k,2φ) NPY sequences. Through this investiga-
tion, we will be able to demonstrate whether or not
the kinetics of the NPY system can be represented by
an ABC(2k,2φ) mechanism; a result that has not been
demonstrated to date.
We have applied the step-by-step procedure de-
scribed here to each of the sequences S1–S7 and ob-
tained their kinetic and spectroscopic parameters. The
results showed that except for three mechanisms (S1, S5
and S7), in all other cases at least one of the determined
parameters has a negative value. According to our pro-
cedure and the outcome of its application, only positive
parameter values are expected to be found. This means
that the possible reaction mechanisms (S2–S4 and S6)
showing such features cannot represent the NPY dy-
namics and must be ruled out. Hence, our method can
provide a useful means to investigate the distinguisha-
bility of ABC(2k,2φ) mechanisms. The three remain-
ing candidates (divergent S1 and cyclic S5, S7) have
been analyzed according to the procedure using the
data of Table I. We have employed the resulting values
of a19P and a20P (= −0.01444 and −1.77331 × 10−4
,
Inside the general ABC(2k,2φ) model, seven possi-
ble options including divergent, consecutive and cyclic
respectively) to define the quantum yields, and in order
to extract the molar absorption coefficients the initial
Scheme 2 NPY sequences (S1–S7) obeying the ABC(2k,2φ) and ABC(2k,1φ) kinetic models (here the symbol “ꢀ” indicates
the thermal process).

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
427
Figure 1 Experimental (circles) and fitting (line) kinetic traces of the coloring and thermal bleaching of NPY (structures aside)
in ethyl acetate (λirr = 357 nm, λobs = 423 nm, C0 = 5.87 × 10−5 mol L−1, T = 14◦C).
m0P and M0T (Eqs. (7)) have been used. The resulting
parameter sets are reported in Table II.
alistic to consider a mechanism that contains the cis–
trans → NPY reaction without simultaneously allow-
ing the trans–trans → cis–trans reaction step. These
criteria are not contained within the divergent sequence
(S1), and hence we feel it should be excluded from con-
sideration. Mechanism S5 is excluded on the basis that
species B cannot be represented by the trans–trans iso-
mer because of the unlikely occurrence of such ther-
mal reaction (trans–trans → NPY, as discussed for
S1 above) and hence S5 would indicate that the pho-
toreactions of both the NPY and the cis–trans species
exclusively yield the trans–trans species. This situa-
tion is physically improbable for that it supposes the
system reacting through strained photochemical path-
ways (NPY → trans–trans ← cis–trans) while omit-
ting much simpler ones (notably NPY → cis–trans and
its reverse transformation).
ꢅ
ꢆ
423
0P
m
= φAB · εB423 + φAC · εC423 · γ · C0 (7.1)
423
0T
M
= ε4B23 · CBPSS + εC423 · CCPSS
(7.2)
The similarity of the results obtained for S1 and
S7 (Table II) comes from a close correspondence be-
tween the series of expressions defining the a1–a9 coef-
ficients. This feature underlines the distinguishability
problem in relation to such kinetics. However, in the
specific case of NPY kinetics, we consider that it is
possible to rule out two of the three remaining pos-
sible mechanisms. First, even if we had not attributed
molecular structures to isomers B and C, it is rather un-
likely that both the trans–trans and the cis–trans open
form isomers react identically as is suggested by S1.
Specifically, that their thermal routes exclusively lead
to the closed form. If this is acceptable for the trans–
cis isomer, it seems very improbable for its trans–trans
homologue. Since the latter needs to overcome two
energy barriers in order to revert to the closed form
and since one of these barriers most probably corre-
sponds to the isomerization of the trans–trans species
into the cis–trans isomer, we believe that it is not re-
On this basis, S7 seems the only mechanism that
plausibly fits NPY kinetics if it is assumed to be an
ABC(2k,2φ) system. The attribution of B and C struc-
tures for this sequence is carried out in agreement
with the latter discussion, i.e. the cis–trans isomer is
the intermediate species (B) in the thermal process
(C → B → NPY).
The kinetic behavior of NPY as predicted by S7
suggests that only the closed form is photochemically
Table I Fitting Parameters Respectively for the Photocoloring and Thermal Bleaching Kinetic Traces of NPY in Ethyl
Acetate
Parameter
m0(T,P)(s−1
)
M0(T,P)
θ4(T,P)
ν
+(T,P)(s−1
)
ν
−(T,P)(s−1
)
r2 b
a
Reaction
Photochemical
Thermal
0.01518
−0.01109
0
0.3756
0
−0.04606
−0.00385
0.999
0.998
0.3690
−0.03528
−0.00019
a These constants, indicating the values of the absorbances at t = ∞, are obtained by the fitting without imposing any constraints on their
values.
b Squared correlation coefficient values obtained on the experimental vs. calculated absorbance graphs.

428
MAAFI AND BROWN
Table II Determined Parameter Valuesa Characterizing the Uniquely Identifiable Solutions for Mechanisms S1, S5
and S7
ελB (mol−1
ελC (mol−1
φi · γ (s−1
φAB · γ
)
φi · γ (s−1
φAC · γ
)
CAPSS(mol L−1
2.219 × 10−6
)
CBPSS(mol L−1
6.072 × 10−7
)
CCPSS (mol L−1
5.587 × 10−5
)
L cm−1
)
L cm−1
)
S1
S5
S7
23637
6347
9.65 × 10−3
φBC · γ
4.78 × 10−3
φAC · γ
2.219 × 10−6
2.219 × 10−6
2.322 × 10−7
9.082 × 10−7
5.625 × 10−5
5.557 × 10−5
70050
23638
6271
6254
1.07 × 10−2
φAB · γ
3.69 × 10−3
φAC · γ
9.68 × 10−3
4.76 × 10−3
a The thermal rate constants for these mechanisms are kBA (S1, S5 and S7) = 0.03528 s−1 and kCB (S5 and S7) = kCA (S1) = 0.00019 s−1
.
active, and forms species B twice as efficiently as
species C (Table II). The thermal back-recyclization
reaction ofB(characterizedby kBA)ismuchfasterthan
the isomerization of C into B (rate constant kCB). As a
consequence, the concentration of compound B at the
photostationary state is around 60 times less than that
of compound C at the photostationary state (Scheme 3).
On the basis of the latter figures, the contribution of C
to the observed spectrum of the mixture (at the PSS)
is expected to be much more important than that of B
because its molar absorption coefficient is estimated to
be only four times less than that of B (which does not
compensate for the difference in the concentrations)
(Fig. 2).
Although the results given in Table II seem satis-
factory, it is however compulsory that before S7 is def-
initely accepted as the true mechanism, all equations
characterizing both the photochemical and the thermal
reactions are checked for consistency. For example, we
canconsiderEqs. (8), whichrepresentstheinitialveloc-
ity and the pre-exponential terms of the fading process.
Their values can be retrieved from the fitting results
(Table I).
Figure 2 Calculated spectra of individual species B ( ) and
◦
C (ꢀ) for mechanism S7 and the experimental spectrum of
their mixture ( ) in the visible region. Inset: correlation line
ꢁ
(r2 = 0.99) between calculated and experimental data of the
mixture.
423
m
− M423 · ϑ
−T
423
1T
0T
0T
θ
θ
= −
(8.2)
(8.3)
ϑ
−T − ϑ
+T
ꢅ
ꢆ
423
0T
= −k2 · CBPSS · ε4B23 + k5 · εB423 − ε4C23 · CCPSS
423
0T
m
− M423 · ϑ
m
+T
423
2T
0T
=
ϑ
−T − ϑ
+T
(8.1)
It becomes apparent from the calculated values of
these expressions, by exclusively using the data from
Table II, that these do not correspond to those obtained
from the fitting of the traces. This means also that de-
pending on the pair of equations we use to fulfill Step 4
of the procedure—here we can pick any two from the
three Eqs. (7.1), (7.2) and (8.1) available—we find a
different set of ε values. It is worth noting the power
of our procedure specifically because it affords these
auto-checking options. This should be possible for any
situation (this might generally be true owing to the fact
that Eqs. (6) in Step 4 generates more equations than the
Scheme 3
ABC(2k,2φ) sequence (S7).
Predicted kinetic parameters of the NPY

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
429
two needed for the determination of the molar absorp-
tion coefficients). In the case of our 3H-naphtho[2,1-
b]pyran derivative, these results prove incontestably
that its kinetics does not obey any of the sequences
presented in Scheme 2 and, therefore, it should be de-
scribed by a higher level mechanism than ABC(2k,2φ).
This conclusion is consistent with the fact that there is
no objective reason to rule out the photochemical re-
activity of the NPY photoisomers as has been proven
for its homologue 2H-naphthopyran structure [10] and
some spiro-naphthopyrans [7].
Because the ratios of any two of these three quanti-
ties(mλ0T, m0λP and M0λT)areconstantforallwavelengths
(see correlation equations in Fig. 3), we can demon-
strate (see Appendix) that within the wavelength range
considered the spectra of the NPY photoisomers B and
C are characterized by an identical shape.
These results are in agreement with the transient
and permanent spectra obtained from nanosecond ex-
periments on the trans–trans isomer photoreaction of
a 2H-NPY analogue [10].
However, some information about the mechanism
may be obtained from a close analysis of the experi-
mental data. Indeed, when the initial velocities (m0λT
and mλ0P) are plotted against the absorbance of the
mixture (M0λT) at the PSS for each wavelength (380–
580 nm) as obtained from the fitting of the traces,
straight lines are obtained (Fig. 3) with intercepts equal
to zero. Although, the actual mechanism taking place
in the dynamics of NPY is unknown (and we do not
assume one), the experimental data in Fig. 3 collected
at the PSS are strictly dependant on the variation of
the extinction coefficients of B and C (which are also
unknown) as the real kinetic parameters and concentra-
tions are constant (at the PSS). On the other hand, we
observe that for any possible thermal sequence (which
can involve between two and four reaction steps) [2],
theinitialvelocityshouldhavetheformofEq. (9). Also,
the absorbance at the PSS of the mixture (Eq. (7.2)) and
the initial velocity of the photochemical reaction (Eq.
(7.1)) can be rearranged to the same algebraic form.
CONCLUSION
A new procedure to solve analytically the kinetics of
ABC(2k,2φ) systems has been presented here in four
progressive steps. It is valid for any photochemical sys-
tem independent of the signal composition or the trends
of the intensity variation, provided that its kinetic traces
exhibit a bi-exponential trend. For the determination
of the unknowns (at least eight), it is only necessary
to know the initial total concentration. The output will
encompass all the kinetic parameters of the reactions
as well as the spectroscopic details of the three species
(i.e. nine parameters). Henceforward, the method pro-
vides an easy way to retrieve the individual spectra
of the transient structures that may be involved in the
dynamics. It affords options to auto-check the results
found without a need for further external data. It solves
finally the identifiability problem for the whole class
of ABC(2k,2φ) systems. In the course of establishing
this procedure on the basis of our previous formalism,
we have produced two general expressions for the col-
orability of ABC systems and for the concentration
equilibrium ratio of the species involved in the reaction
(Keq) as a function of the kinetic parameters, which is
useful for any ABC(6k,6φ) system.
Qλ = αi · ελB + βi · ελC
(9)
where Q is the quantity considered (m0T, m0P or M0T),
αi and βi are constants depending on rate constants,
photochemical yields and concentrations at the PSS.
We have shown an example of the application of
the procedure by considering NPY kinetics. For this
particular compound it was possible to rule out many
possible mechanisms because some of the parameter
values generated by applying our method were found
to be negative. Furthermore, it was then argued that
when the procedure was fully conducted, we reached
the conclusion that in the NPY kinetics there are cer-
tainly more photochemical processes than the two con-
sidered in the ABC(2k,2φ) sequence. We address these
issues in the next paper in this series.
APPENDIX
Figure 3 Linear correlation between initial velocities of the
thermal (mλ0T) and photochemical (m0λP) reactions with the
absorbance (M0λT) of the medium at the PSS (or the beginning
of the thermal reaction). Wavelength range 380–580 nm.
Foranysinglemechanism, wecanwritetheexpressions
of m0T, m0P and M0T in the form given by Eq. (9), which

430
MAAFI AND BROWN
is valid for each single wavelength of the considered
range. The ratios of any two of these quantities at any
wavelength, denoted here as Q1 and Q2, are given by
Eq. (A1). The ratios spanning over all wavelengths are
equal to a single experimentally defined constant (ꢁ)
as depicted by the straight lines in Fig. 3. The αi and
βi are constants as discussed above.
reactivity of the starting material (NPY); it can involve
either one process (φ4 or φ12 = 0) or two processes (φ4
and φ12 = 0) as given by Eq. (7.1).
BIBLIOGRAPHY
1. Maafi, M.; Brown, R. G. Int J Chem Kinet 2005, 37(3),
162–174.
2. Maafi, M.; Brown, R. G. Int J Chem Kinet 2005, 37(12),
717–727.
Qλ1 α1 · ελB + β1 · ελC
∀ελB and ελC ⇒
=
= ꢁ (A1)
Qλ2 α2 · ελB + β2 · ελC
3. Turro, N. J. Modern Molecular Photochemistry;
Benjamin-Cummings: Menlo Park, CA, 1978.
4. Cowen, D. O.; Drisko, R. L. Elements of Organic
This expression can be rearranged as follows:
α1 + β1 · (ελC/ελB)
ꢁ =
Photochemistry;
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Plenum
Press:
New
York,
(A2)
α2 + β2 · (ελC/ελB)
5. Durr, H.; Bouas-Laurent, H. Photochromism: Molecular
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from which the formula of the extinction coefficients
ratio is defined as:
ελC ꢁ · α2 − α1
∀λ; 380 nm < λ < 580 nm ⇒
=
= n
(A3)
εBλ
β1 − ꢁ · β2
Since all the coefficients in the right-hand side term
are constants, the ratio is a constant “n” for all wave-
lengths. This means that the spectra of isomers B and
C have the same shape.
The above consideration is true for all the equations
of m0P irrespective of the mechanism. Indeed, the latter
quantity is defined depending on the photochemical
8. Borderie, B.; Lavabre, D.; Micheau, J. C.; Laplante,
J. P.; J Phys Chem 1992, 96, 2953.
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