Kinetic Analysis of Photochromic Systems under Continuous Irradiation. Application to Spiropyrans

Article abstract of DOI:10.1021/jp9531117

The kinetics of spiropyran photochromic systems in solution in a stirred batch reactor continuously irradiated with monochromatic light was studied by UV/visible spectrophotometry.The plots of absorbance vs time were analyzed, and the desired parameters (quantum yields, UV/visible spectrum of the unstable photomerocyanine, ...) were extracted from an iterative computation which fitted the calculated curves to the experimental ones on the basis of a representative model of the reaction mechanism.The method was applied to spiro whose corresponding photomerocyanine has a lifetime of 100 s in toluene; the two quantum yields of the direct process of photocoloration and the reverse reaction of photobleaching could be determined along with the spectrum of the corresponding photomerocyanine.To demonstrate the general nature of the photokinetic method, it was applied to the reaction of spiro, where the main photochromic process is accompanied by a photodegradation.Despite this interfering phenomenon, the photokinetic method could be used to extract the parameters of the main photochromic process.It also showed that the photodegradation products catalyzed the termal back-isomerization.The order of magnitude of the rate constant of this catalytic process and the quantum yields of photodegradation were estimated.

Full text of DOI:10.1021/jp9531117

J. Phys. Chem. 1996, 100, 4485-4490
4485
Kinetic Analysis of Photochromic Systems under Continuous Irradiation. Application to
Spiropyrans
V. Pimienta,† D. Lavabre,† G. Levy,† A. Samat,‡ R. Guglielmetti,‡ and J. C. Micheau*,†
URA 470, UniVersite´ P. Sabatier, F-31062 Toulouse, France, and URA 1320, Faculte´ des Sciences de Luminy,
UniVersite´ d’Aix-Marseille II, F-13288 Ce´dex 9 Marseille, France
ReceiVed: October 23, 1995X
The kinetics of spiropyran photochromic systems in solution in a stirred batch reactor continuously irradiated
with monochromatic light was studied by UV/visible spectrophotometry. The plots of absorbance Vs time
were analyzed, and the desired parameters (quantum yields, UV/visible spectrum of the unstable photomero-
cyanine, ...) were extracted from an iterative computation which fitted the calculated curves to the experimental
ones on the basis of a representative model of the reaction mechanism. The method was applied to spiro-
[benzothiazoline-2′,2-benzopyran] whose corresponding photomerocyanine has a lifetime of 100 s in toluene;
the two quantum yields of the direct process of photocoloration and the reverse reaction of photobleaching
could be determined along with the spectrum of the corresponding photomerocyanine. To demonstrate the
general nature of the photokinetic method, it was applied to the reaction of spiro[indoline-2′,2-benzopyran],
where the main photochromic process is accompanied by a photodegradation. Despite this interfering
phenomenon, the photokinetic method could be used to extract the parameters of the main photochromic
process. It also showed that the photodegradation products catalyzed the thermal back-isomerization. The
order of magnitude of the rate constant of this catalytic process and the quantum yields of photodegradation
were estimated.
I. Introduction
Photostationary Method. In this method, which is the best
known, only the properties of the photostationary state are taken
A system is said to be photochromic when it comprises a
reversible photochemical reaction of a single chemical species.
This reaction gives rise to the formation of one or more
photoisomers whose UV/visible absorption spectrum and hence
color differ markedly from those of the starting compound. Such
systems have numerous practical applications, and depending
on their kinetic and spectral properties, they may find applica-
tions as optical switches and memories or in variable transmis-
sion materials.1 Knowledge of all the photochromic parameters
is thus of considerable importance to design new molecules for
such applications. The determination of these parameters is not
a trivial exercise as in many cases the photoisomers are too
labile to be isolated chemically. Consequently, the relevant
parameters are not readily extracted as the spectral change
obtained on irradiation may be due to either a high quantum
yield of photoisomerization with a low molar extinction
coefficient of the photoisomer or vice versa. A kinetic analysis
is needed.
into consideration. The most cited studies are those of Fisher3
and Wyman.4 The method is applicable to compounds with
two stable isomers which interconvert photochemically (indi-
goids, azobenzenes, ...).5-7 In this case, the rate of conversion
in the photostationary state only depends on the irradiation
wavelength. The UV/visible spectra of several photostationary
mixtures are obtained by monochromatic irradiation at different
wavelengths. This effectively alters the contributions of the
two photochromic processes (direct and reverse) whose ratio
of rates of reaction is proportional to the ratio of the molar
extinction coefficients of the two photoisomers. From these
photostationary spectra, the spectra of each of the constituent
photoisomers can be recalculated algebraically without recourse
to chemical isolation from the reaction mixture.
If a thermal isomerization also accompanies the photochemi-
cal processes, the photostationary method still applies provided
the unstable isomer has a long enough lifetime.8 In this case,
the rate of photostationary conversion also depends on the
intensity of the incident light flux.9 The higher the light
intensity, the higher the rate of photoconversion at photosta-
tionary equilibrium. The quantum yield of photoisomerization
and the molar extinction coefficient of the unstable isomer can
be extracted from the relationship between the incident photon
flux and the extent of conversion.10-12
II. Methods of Photokinetic Analysis
The first report of an extraction of photochromic parameters
was that of Zimmerman et al. in 1958.2 In this study, the authors
established the differential equation describing the rate of
photoisomerization of a photochromic system in a uniformly
stirred liquid phase under continuous irradiation. Combination
of this equation with the Beer-Lambert law remains the basis
for all photokinetic analyses, although two different approaches
can be distinguished, one based on consideration of the
stationary state and the other on the reaction progress:
The extent of photostationary conversion may also be varied
by altering the temperature, which modifies the rate constant
of the spontaneous isomerization reaction. The higher the
temperature, the higher the rate constant and the lower the extent
of photostationary conversion. The above-mentioned parameters
can also be extracted by this method.13,14
* To whom correspondence should be addressed.
† Universite´ P. Sabatier.
Instead of only exploiting the displacements of the position
of the photostationary states as a function of the various
‡ Universite´ d’Aix-Marseille II.
X Abstract published in AdVance ACS Abstracts, February 15, 1996.
0022-3654/96/20100-4485$12.00/0 © 1996 American Chemical Society

4486 J. Phys. Chem., Vol. 100, No. 11, 1996
Pimienta et al.
Figure 1. Photochromic equilibrium and structures of spiropyran (closed form) A and photomerocyanine (open form) B.
experimental constraints, the reaction kinetics under continuous
irradiation can be studied. This is referred to as the photokinetic
method.
Photokinetic Method. This method can produce the same
type of information as the photostationary method, but it is more
general as it can take account of a more complex reaction
mechanism including for example a photodegradation. It can
also be employed to evaluate the validity of different hypotheses
about the reaction mechanism.
On the other hand, the photokinetic method requires continu-
ous evaluation of the photokinetic factor, a component of the
photochemical rate equation.15,16 A continuous measurement
of the absorbance at various wavelengths including the irradia-
tion wavelength λ′ must be carried out to gain complete
information about the kinetic and spectral characteristics of the
Figure 2. Photochromism of benzothiazoline 1 in solution in toluene:
(a) photocoloration under UV irradiation, (b) thermal back-isomerization
in the dark, (c) irradiation by a 50 s pulse of blue light (425 ( 60 nm).
Note acceleration of back-isomerization during irradiation.
reaction. After numerical processing of the data, all parameters
of the photochromism can be obtained.
Assuming that without photodegradation the photochromic
phenomenon can be schematized as an isomerization of a stable
isomer A into an unstable one B, a general kinetic scheme
encompasses three main processes: a photochemical isomer-
ization (ΦAB) and two back-isomerizations, whether photo-
chemical (ΦBA) or thermal (kBA):
can occur and, in a nonpolar solvent such as toluene, strong
absorption bands are observed between 550 and 650 nm. The
lifetimes of the benzothiazoline or spiro[indoline-2′,2-benzopy-
ran] type photomerocyanines range from 10-8 to 103 s,
depending on their particular structure, the nature of substituents,
and the solvent. By thermal back-isomerization or, for some
compounds, under visible irradiation, the photomerocyanines
are transformed into the starting spiropyran (closed form) by
closure of the C-O bond. Spiro[benzothiazoline-2′,2-benzopy-
ran] 1 was chosen for the present study as it combines two
opposing photochemical processes and a relatively slow thermal
back-isomerization rate constant in solution in toluene (Figure
1).
A f B (ΦAB
)
)
V1
V2
V3
B f A (ΦBA
(I)
B f A (kBA
)
Under these conditions, the kinetic equation for change in [A]
is
The existence of a photochemical ring closure can be
demonstrated by performing an irradiation with a 50 s long pulse
of blue light during the thermal back-isomerization. This
irradiation is absorbed by the photomerocyanine but not by the
closed form. For the duration of this irradiation pulse, there is
a speeding up of the decay in the photomerocyanine concentra-
tion. On the other hand, in the dark, only the spontaneous ring
closure (monoexponential thermal back-isomerization) takes
place (Figure 2).
For complete analysis of this system, experiments must be
performed at two different irradiation wavelengths. Figure 3a
and b illustrates these two series of experiments (a, λ′ ) 366
nm; b, λ′ ) 400 nm) and the result of the simultaneous
numerical fitting of the equations (A-3) and (A-4) (see Ap-
pendix) over the two sets of kinetic curves. The quality of fitting
confirms that scheme I is a good model of the photochromism
of the benzothiazoline 1 in toluene solution.
d[A]/dt ) -V1 + V2 + V3
(1)
Using eq 1 (see Appendix) all the parameters of the
photochromism are extracted in a two-stage process: (a) plotting
Abs Vs t curves and (b) numerical processing of the data
obtained.
III. Application to Quantitative Study of the
Photochromism of a Spiropyran in Toluene Solution
Spiropyrans are photochromic molecules consisting of two
orthogonal heterocycles joined by a spiro junction.17-19 They
have an actinic absorption band in the near ultraviolet (300 <
λmax < 350 nm). The value of ꢀmax ranges from 1000 to 10000
L‚mol-1‚cm-1 depending on the particular structure and the
nature of the substituting groups. Irradiation in this band leads
to singlet and triplet excited states20 and induces cleavage of
the C-O bond. Several isomeric open forms (photomerocya-
nines), differing in geometric arrangement around the three C-C
bonds of the central chain, may be produced.21 In the
photomerocyanines, conjugation between the two heterocycles
We have obtained the following parameter values: ΦAB
)
B
0.11 ( 0.03, ΦBA ) 0.02 ( 0.005, ꢀ366 ) 6000 ( 400, ꢀ400
B
) 16 000 ( 1000, ꢀ646 ) 27 000 ( 1000 L‚mol-1‚cm-1, kBA
B
) (1 ( 0.03) × 10-2 s-1 at 25 °C.

Kinetic Analysis of Photochromic Systems
J. Phys. Chem., Vol. 100, No. 11, 1996 4487
Indoline 2 is a highly photodegradable molecule.22-25 Figure
5 shows the photochemical behavior of indoline 2 under
continuous irradiation: the phase of photocoloration is followed
by a phase of irreversible photodegradation. On switching off
the irradiation, a thermal monoexponential back-isomerization
was observed whose rate constant (kobs) increased slightly but
significantly as a function of the extent of photodegradation.
This indicated that the products of photolysis act as catalysts
in the thermal back-isomerization.26
In contrast to benzothiazoline 1, indoline 2 does not undergo
a photochemical back-isomerization. The photochromism of
indoline 2 can be thus described by a type I′ mechanism
combined with a photodegradation process and the thermal back-
isomerization catalyzed by the photodegradation products.
Referring to the set of photodegradation products under the terms
C or D from either the closed form A or the open form B
respectively and without specification of the details of the
mechanisms involved (direct photolysis, intervention of singlet
oxygen, photooxidation of solvent, etc.) the following skeleton
kinetic scheme was drawn up (see Appendix):
A f B (ΦAB
B f A (kBA
)
V1 ) ΦABꢀ′A[A]lFI0
V3 ) kBA[B]
)
A f C (ΦAC
)
)
VAC ) ΦACꢀ′A[A]lFI0
VBD ) ΦBDꢀ′B[B]lFI0
V4 ) k4[B][C]
B f D (ΦBD
Figure 3. Absorbance of a 1.5 × 10-4 mol‚L-1 solution of benzothia-
zoline 1 in toluene (a) irradiated at 366 nm; curve 1, Abs at λ ) 366
nm; curve 2, Abs at λ ) 646 nm and (b) irradiated at 400 nm; curve
3, Abs at λ ) 400 nm; curve 4, Abs at λ ) 646 nm. The four
experimental curves were simultaneously numerically fitted.
B + C f A + C (k4)
B + D f A + D (k5)
V5 ) k5[B][D]
(II)
This mechanism referred as model II is described by the
following set of differential equations:
d[A]/dt ) -V1 + V3 - VAC + V4 + V5
d[B]/dt ) V1 - V3 - VBD - V4 - V5
d[C]/dt ) VAC
(2)
(3)
(4)
(5)
d[D]/dt ) VBD
To carry out a quantitative kinetic analysis of model II, three
experiments a, b, and c are fitted simultaneously. The experi-
mental conditions are chosen to show the relative influence of
duration of irradiation (short or long), initial concentration [A]0
(low or high), and photon flux I0 (low or high).
Figure 4. Absorption spectra of the benzothiazoline 1: closed form
A (ꢀAλ Vs λ, dotted line, experimental determination) and the open form
Figure 5 shows the results of the simultaneous fitting of the
three experiments. The curves with solid lines correspond to
the numerical fitting of mechanism II. It can be seen that this
model accounts for the three experiments. However, either of
the photodegradation processes (ΦAC or ΦBD but not both) can
be omitted without marked deterioration in the quality of the
fitting. This indicates that the localization of the photodegra-
dation processes from either the closed form A (ΦAC) or the
open form B (ΦBD) or both is not discernible from the numerical
processing of the three experiments. However, out of all
hypotheses examined, the overall quantum yield of the set of
photodegradation processes (ΦAC + ΦBD) was evaluated at
around 6 × 10-3, or ≈0.7% of the quantum yield of photocol-
oration. Furthermore, the model also demonstrated that the
photodegradation products had a catalytic influence on the
thermal back-isomerization (k4, k5 or (k4 + k5)) whose overall
λ
B (ꢀB Vs λ, full line, numerical determination).
Figure 4 shows the UV/visible spectra of the closed (mea-
sured) and open (calculated) forms of benzothiazoline 1.
The residual error obtained by fitting the data to mechanism
I (ΦBA * 0) is less than one-third those obtained by fitting to
mechanism I′ (ΦBA ) 0). However, the two mechanisms (I
and I′) can only be distinguished by taking both experiments
into consideration at the same time; taken separately, each one
can be fitted perfectly to either of the two mechanisms.
IV. Photokinetic Study of a Photodegradable
Photochromic Compound: The Case of
Spiro[indoline-2′,2-benzopyran] 2
The photokinetic method under continuous irradiation can
also take account of other chemical or photochemical processes.
second-order rate constant (k4 + k5) was around 70 L‚mol-1‚s-1
.

4488 J. Phys. Chem., Vol. 100, No. 11, 1996
Pimienta et al.
for the parameters of the main photochromic process (between
+6% and -4% with respect to the corresponding values
indicated above). To establish the mechanism of photodegra-
dation in more detail, experiments involving strong degradation
at different irradiation wavelengths need to be analyzed.
However, using our experimental setup the spectral lines of the
mercury lamp used in the present experiments do not provide
enough power (I0); the spiropyran does not have a sufficiently
large absorption coefficient (ꢀA′) to induce marked photodeg-
radation with a reasonable duration of irradiation.
V. Conclusion
The kinetics of photochemical isomerization according to
scheme I can be analyzed by numerical fitting of the parameters
of the model. Extraction of ΦAB, ΦBA, and ꢀB requires
measurements using two different irradiation wavelengths. In
each case, recordings must be made at two wavelengths, one
being the irradiation wavelength and the other a wavelength
giving a large enough difference in absorbance. Simultaneous
fitting of these two experiments demonstrates the existence of
the photochemical back-isomerization and provides an estimate
of its quantum yield. This method can be generalized to all
type I photochemical systems whatever the molecular structures
under consideration (azobenzenes, spiroxazines, etc.). The
speeding up of the reaction during the irradiation pulse of visible
light cannot be exploited rigorously to obtain ΦBA as it requires
prior knowledge of the spectrum of B, which is precisely one
of the unknowns of the problem. Nonetheless, it can detect
the presence of ΦBA in a qualitative way. For spiro[benzothia-
zoline-2′,2-benzopyran] 1, the method produces the molar
Figure 5. Experimental results and simultaneous numerical fitting using
model II of the absorbance of the three solutions of indoline 2
(photocoloration and photodegradation + thermal back-isomerization
reaction) (solvent: toluene; irradiation at 366 nm). (a) [A]0 ) 8.94 ×
10-5 mol‚L-1 (high), I0 ) 1.22 × 10-6 mol‚L-1‚s-1 (high), and then I0
) 0 at t ) 320 s (short); (b) [A]0 ) 5.42 × 10-5 mol‚L-1 (low), I0 )
1.50 × 10-6 mol‚L-1‚s-1 (high), then I0 ) 0 at t ) 4220 s (long); (c)
[A]0 ) 8.03 × 10-5 mol‚L-1 (high), I0 ) 2.23 × 10-7 mol‚L-1‚s-1
(low), and then I0 ) 0 at t ) 4000 s (long); A, monitoring at 604 nm;
B, monitoring at 366 nm.
extinction coefficient of the open form ꢀ646 and the quantum
B
yields of the two photochemical processes.
In the presence of a photodegradation reaction as for spiro-
[indoline-2′,2-benzopyran] 2, the photochromic system may be
described by a type II scheme. Provided that several experi-
ments under selected initial conditions are fitted simultaneously,
the photokinetic method is still applicable. It enables the
determination of the parameters of the main photochromic
process and demonstrate catalysis of the thermal back-isomer-
ization reaction by one or more products of the photolysis. The
increase in the rate constant kobs as a function of the extent of
photodegradation can thus be interpreted in a quantitative way.
TABLE 1: Experimental and Calculated Values of
Apparent Rate Constant kobs of the Thermal
Back-Isomerization as a Function of the Extent of
Photodegradation
kobsb (s-1
)
% photodegrada
calcd
exptl
calcd
VI. Experimental Section
a
b
c
1.4
21
3.8
5.97 × 10-3
6.30 × 10-3
6.01 × 10-3
5.87 × 10-3
6.27 × 10-3
6.03 × 10-3
Synthesis of the 3-(isopropyloxy)-8-methoxy-3′-methyl-6-
nitrospiro[benzothiazoline-2′,2-2H-benzopyran] 1. Com-
pound 1 is obtained by condensation of the quaternary salt
(tosylate or iodide) of 2-(isopropyloxymethyl)-3-methylben-
zothiazolium with 2-hydroxy-3-methoxy-5-nitrobenzaldehyde in
alkaline ethanol.27
a % Photodegrad (calcd) ) ([C]∞ + [D]∞)/[A]0; [C]∞ and [D]∞ are
concentrations of the photodegradation products at the end of irradiation;
they are evaluated from the results of the simulation. b The values of
obs(calc) were obtained from the following relationship: kobs(calc) )
kBA + k4[C]∞ + k5[D]∞.
k
Compound 2 has been synthesized according to previous work
(see ref 17).
Inclusion of the catalytic process by the photodegradation
products can account for the increase in kobs as a function of
the extent of photodegradation (Table 1).
For the set of photodegradation products (C + D) the mean
extinction coefficient at 366 nm was evaluated by the model at
around 3500 L‚mol-1‚cm-1. Their absorption in fact interferes
with that of the closed form.
Irradiation and Data Recording. The irradiation was
derived from a 200 W high-pressure mercury lamp equipped
with interference filters enabling transmission of a selected
single emission line. Monochromatic light intensity was
determined directly in the reactor using an aqueous solution of
potassium ferrioxalate (I0366 ) (3.4 ( 0.1) × 10-6 mol‚L-1‚s-1
;
I0 ) (7.2 ( 0.1) × 10-6 mol‚L-1‚s-1). The reactor was a
quartz cylinder closed at either end with Teflon bungs. The
contents were stirred continuously with a magnetic bar driven
by a stepper motor whose speed could be controlled. The
internal volume of the reactor was 2.1 mL. The whole setup
was enclosed in a thermostated copper block (T ) 10 or 25
°C) placed inside the sample chamber of a HP8451 diode array
400
The parameters of the main photochromic process are ΦAB
) 0.84 ( 0.02, ꢀ366 ) 8000 ( 300, ꢀ604 ) 32 500 ( 1000
B
L‚mol-1‚cm-1, and kBA ) (5.8 ( 0.04) ×B10-3) s-1 at 10 °C.
Experiments b and c cannot be interpreted if photodegradation
is neglected. However, by considering solely experiment a
where photodegradation is weak, acceptable values are obtained

Kinetic Analysis of Photochromic Systems
J. Phys. Chem., Vol. 100, No. 11, 1996 4489
spectrophotometer. The reactor had an optical path length of
1.1 cm. Pure toluene was employed as reference. The sampling
time was chosen to deliver at least 25 points over the curved
portions of the plots of absorbance. For each experiment,
absorbances at two wavelengths (one being the irradiation
wavelength) were recorded together.
Using eqs A-3 and A-4, the parameters ꢀB, ΦAB, ΦBA, ꢀ′B,
and kBA can be extracted from the kinetic analysis of the plots
of absorbance Vs time under continuous irradiation, kBA being
obtained from a previous analysis of the thermal back-
isomerization. The experiments must be carried out under the
following conditions: the reaction mixture in solution is well
stirred and irradiated with a monochromatic photon flux I0 at
the irradiation wavelength λ′. The photon flux is directed onto
the cuvette via an optical fiber perpendicular to the measurement
beam of a diode array spectrophotometer. The alterations in
absorbance at all wavelengths, including the irradiation wave-
length, can be readily determined in this setup.
To prime the iterative calculation of the parameters, realistic
starting values are estimated and the differential equation (A-
3) is then integrated numerically, followed by application of
the Beer-Lambert law (A-4) to obtain simulated curves of Abs
Vs t. Comparison of the simulated curves with the experimental
ones produces a residual error: the sum of squares of the
differences between the calculated and experimental values. The
numerical values of these parameters are then automatically
optimized in an iterative procedure designed to minimize this
residual error.
Analysis and Processing of Data. The data were transferred
to and stored on a HP9000/380 workstation. To determine the
desired parameters, the residual error (RE) is minimized. RE
) ∑p∑j(Yjcalc - Yjobs)2/pj, where p is the number of plots fitted
simultaneously and j is the number of points on each plot. The
values of Yjcalc are obtained by numerical integration (Runge-
Kutta semi-implicit procedure) of eq 1 followed by application
of eq A-4 for the benzothiazoline 1 or integration of eqs 2-5
and application of the Beer-Lambert law for the indoline 2. It
was supposed that the products of photodegradation do not
absorb significantly at 604 nm. Yjobs are the corresponding
experimental absorbances. The minimization algorithm is of
the Powell type. It consists of fitting the parameters until a
minimum RE is obtained. kBA constants are extracted before-
hand from the kinetics of the back-isomerization reaction in
the dark and at the same temperature. We showed that the
values of the extracted parameters do not depend on the initial
values used to prime the iterative computation.
Our objective was to propose a reliable method for evaluating
a photochromic reaction scheme and to define the nature and
minimum number of experiments required to obtain the desired
parameters. Two typical cases will be examined:
Appendix: Theoretical Basis of Photokinetic Method
(a) System with No Photochemical Back-Isomerization
(Scheme I′).
The general kinetic scheme of the photochromic phenomenon
involving an isomerization of a stable isomer A into an unstable
one B can be schematized as
A f B (ΦAB
B f A (kBA
)
A f B (ΦAB
)
)
V1 ) ΦABꢀ′A[A]lFI0
V2 ) ΦBAꢀ′B[B]lFI0
V3 ) kBA[B]
)
(I′)
B f A (ΦBA
(I)
In this case, ΦBA ) 0 and the sum (ΦABꢀ′AI0lF + kBA) acts
as an “apparent rate coefficient” under continuous irradiation.
Since kBA is measured from the thermal back-isomerization in
the dark and ꢀ′A from the spectrum of the starting compound
A, both I0 and F are required to determine the quantum yield
B f A (kBA
)
where
Φ
AB. I0 is obtained by actinometry, while F depends on the
F ) (1 - 10-Abs′)/Abs′
(A-1)
absorbance at the irradiation wavelength and therefore on ꢀ′B.
One must measure the change in Abs′ which then gives a value
for the photokinetic factor F at any instant.
F is a dimensionless term referred to as the photokinetic factor.
It is solely a function of Abs′ (namely of [A] and [B] and thus
of the progress of the reaction); it ranges from 2.3 to 0 for values
of Abs′ ranging from 0 to ∞.28,29 Abs′, ꢀ′A, and ꢀ′B are
respectively the total absorbance of the solution and the molar
extinction coefficients of A and B at the irradiation wavelength
λ′; l is the optical path; [A] and [B] are the concentrations of A
and B, and they are related by the equation of conservation of
matter:
To obtain enough information to extract all the parameters
of the system without photochemical back-isomerization, the
following must be recorded: (i) the change in absorbance under
continuous irradiation at two different wavelengths:30 the
irradiation wavelength (enabling calculation of the photokinetic
factor F) and, for example, the λmax of B (to provide an adequate
difference in absorbance), (ii) the initial spectrum of A and the
complete spectrum of the reaction mixture at one time t near
the final irradiation time to calculate the spectrum of the unstable
photoisomer (ꢀ Vs λ) from eq A-4, once the previous parameters
have been extracted), and (iii) the kinetics of the thermal back-
[A] + [B] ) [A]0
(A-2)
isomerization in the dark to determine kBA
.
where [A]0 is the initial concentration of the stable isomer A.
Under these conditions, the kinetic equation governing the
concentration of the stable isomer A is (see section II):
(b) System with Photochemical Back-Isomerization (Scheme
I). Similar reasoning shows that in this case the “apparent rate
coefficient” is [(ΦABꢀ′A + ΦBAꢀ′B)I0lF + kBA] which contains
an additional parameter ΦBA. This parameter cannot be
d[A]/dt ) -[(ΦABꢀ′A + ΦBAꢀ′B)I0lF + kBA][A] +
(ΦBAꢀ′BI0lF + kBA)[A0] (A-3)
extracted by the above procedure, and a further set of measure-
ments is required at a second irradiation wavelength which must
be chosen such that the ratio of the two molar extinction
coefficients ꢀ′A/ꢀ′B differs in the two experiments. This
effectively alters the relative efficiency of the two photochemical
processes (direct and reverse).31
Then, the absorbance at any wavelength, Abs, which is a
linear function of [A] is obtained by combining the Beer-
Lambert law and the equation of conservation of matter (A-2):
To extract all the parameters from the system with photo-
chemical back-isomerization, the above-mentioned experiments
Abs ) [(ꢀA - ꢀB)[A] + ꢀB[A0]]l
(A-4)

4490 J. Phys. Chem., Vol. 100, No. 11, 1996
Pimienta et al.
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must be carried out, but at two different irradiation wavelengths.
This provides sufficient information assuming that the quantum
yields are not modified by the change in irradiation wavelength.
This last hypothesis is generally justified if the two irradiation
wavelengths lie in the same absorption bands for both A and
B. The molar extinction coefficient of the unstable photoisomer
B is obtained as described above for the system without
photochemical back-isomerization.
(16) Polster, J.; Mauser, H. J. Photochem. Photobiol., A: Chem. 1988,
43, 109-18.
(17) Bertelson, R. C. In Photochromism; Brown, G. H., Ed.; J. Wiley
and Sons Inc.: New York, 1971; Chapter III.
(18) Guglielmetti, R. In Photochromism, Molecules and Systems; Du¨rr,
H., Bouas-Laurent, H., Eds.; Elsevier: Amsterdam, 1990; Chapters 8 and
23.
(19) Samat, A.; de Keukeleire, D.; Guglielmetti, R. Bull. Soc. Chim.
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(1) Crano, J. C.; Kwak, W. S.; Welch, C. N. Applied Photochromic
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(3) Fisher, E. J. Phys. Chem. 1967, 71 (11), 3704-6.
(4) Wyman, G. M. Mol. Photochem. 1974, 6, 81-90.
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(6) Blanc, J.; Ross, D. L. J. Phys. Chem. 1968, 72, 2817-24.
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(8) The lifetime τ must be sufficiently long for the reaction mixture to
be made macroscopically homogeneous by the stirring system. In general,
τ ≈ 10 s is the limiting value below which the reaction mixture is not
adequately stirred. For the short-lived photomerocyanines (10-10 < τ <
10-1 s), laser or flash photolysis are the methods of choice for kinetic
analysis, but only of the thermal back-isomerization reactions. See: (a)
Lenoble, C.; Becker, R. S. J. Phys. Chem. 1986, 90, 62-5. (b) Tamai, N.;
Masuhara, H. Chem. Phys. Lett. 1992, 191 (1-2), 189-94. (c) Zhang, J.
Z.; Schwartz, B. J.; King, J. C.; Harris, C. B. J. Am. Chem. Soc. 1992, 114,
10921-7. For the longer-lived forms (τ > 10 s), quantum yields, rate
constants, and the spectrum of the most stable photomerocyanine can be
obtained under continuous irradiation.
(21) Ernsting, N. K.; Arthen-Engeland, T. J. Phys. Chem. 1991, 97,
5502-9.
(22) Baillet, G.; Campredon, M.; Guglielmetti, R.; Giusti, G.; Aubert,
C. J. Photochem. Photobiol., A: Chem. 1994, 83, 147-53.
(23) Baillet, G.; Giusti, G.; Guglielmetti, R. J. Photochem. Photobiol.,
A: Chem. 1993, 70, 157-61.
(24) Baillet, G.; Lokshine, V.; Guglielmetti, R.; Giusti, G. C. R. Acad.
Sci., Ser. II 1994, 319, 41-6.
(25) Salemi, C.; Giusti, G.; Guglielmetti, R. J. Photochem. Photobiol.,
A: Chem. 1995, 86, 247-52.
(26) Baillet, G.; Guglielmetti, R.; Giusti, G.; Mol. Cryst. Liq. Cryst. 1994,
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(27) (a) Samat, A. Thesis, Brest (France), 1976. (b) Samat, A.; Kyster,
J.; Garnier, F.; Metzger, J.; Guglielmetti, R. Bull. Soc. Chim. Fr. 1975,
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(28) Borderie, B.; Lavabre, D.; Micheau, J. C.; Laplante, J. P. J. Phys.
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(30) If the photokinetic curve is close to first order, Abs′ changes little
during irradiation and F is practically constant. On the other hand, if the
photokinetic curve departs from first order, F may sometimes (if the change
is sufficient) be estimated by fitting this curve alone without having to
monitor the irradiation wavelength. See: Misra, G. P.; Lavabre, D.;
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(31) This approach is akin to the temperature variation method used in
the study of thermochromic equilibria. See: Tan-Sien-Hee, L.; Lavabre,
D.; Levy, G.; Micheau, J. C. New J. Chem. 1989, 13, 227-33.
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(12) Rau, H. EPA Newslett. 1991, 41, March, 40-55.
(13) Gauglitz, G.; Scheerer, E. J. Photochem. Photobiol., A: Chem. 1993,
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