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Method for the determination of absorption coefficients, reaction rate constants and thermodynamic data in the system
1. Photo&em.
Photobiol. A: Chem., 71 (1993) 205-212
Method
for the determination
205
of absorption
coefficients, reaction hV
rate constants
and thermodynamic
data in the system A T--)B? A
G. Gauglitz
and
E. Scheerer
Imtitut fiir Pkysikalische und Theoretische Ckemie, Auf der Motgemtelle 8, W-7400 Tiibingen (Germany) (Received
July 20, 1992; accepted
December
3, 1992)
Abstract A method
is presented
for the kinetic
analysis
of the reaction
system
A TB
which
uses the dependence
of
the photostationary state on the temperature. One kinetic measurement yields the reaction rate constants of the thermal reverse reaction at various temperatures, thermodynamic data and the spectrum of photoproduct B. The knowledge of the absorption coefficient of the photoproduct allows the photochemical quantum yield to be calculated by numerical integration of the rate law. To obtain an indication of the significance of the data, the method is tested using sets of simulated data and is finally applied to the photochromic systems of some dihydroindolizines.
1. Introduction
Many
photochromic systems follow the mechh” anism A e-B. In spite of this simple mechanism A
the photokinetic factor prevents the differential equation from being solved in a closed form. For this reason, a least-squares approach is used to solve the overdetermined system of linear equations by numerical (‘formal’) integration [l]. If a pure photochemical mechanism is considered, the product of the photochemical quantum yield and the absorption coefficient of the starting material A at the wavelength of irradiation gives the element of the Jacobi matrix. Since the thermal reverse reaction is superimposed, its reaction rate constant must also be included in the Jacobi matrix. The concentrations are determined from the absorbance signals. The elements of the Jacobi matrix are calculated by multilinear regression. In order to avoid problems arising from the correlation of two of these elements [2], a slight change in the model equation, i.e. an c1priori knowledge of the absorption coefficient of the photoproduct, yields satisfactory results. Unfortunately, for one of the compounds examined there is still correlation of fDedicated
to Professor
75th birthday.
lOlO-6030/93/$6.CO
Dr. H. Mauser
on the occasion
of his
the elements. Another rearrangement of the model equation allows significant evaluation. The absorption coefficients of the starting material A can be determined without problems taking the spectrum of the non-irradiated substance_ However, this is not the case for the absorption coefficients of the photoproduct, since B ‘cannot be isolated in pure form because of the thermal reversibility of the system. Approaches for the determination of the absorption coefficient of nonisolable products have been proposed [3]. However, these are only applicable under special conditions. Often the measurements are realized at low temperatures or at high intensities to reduce the effects of the thermal reverse reaction on the evaluation. The experimental data can then be evaluated according to a pure photochemical equation at a first approximation. However, low-temperature measurements are disadvantageous because of the necessary experimental expenditure and the fact that the absorption coefficients determined at low temperatures differ from those at room temperature. Measurements at high intensities cause other problems. Tbe reaction solution exhibits inhomogeneities, since the gradient of the components in the cell cannot be homogenized by stirring during the time interval between measurements. Furthermore, high irradiation intensities may cause local heating of the solution.
0 1993 - Elsevier
Sequoia.
All rights reserved
206
G. Gauglitz, E. Scheerer I Kinetic analysis
&
of the system A
B
4
For these reasons, a new approach is proposed which takes advantage of the thermal reverse reaction, since a change in temperature can shift the photostationary state. This can be used to obtain an equation which allows the determination of the absorption coefficient of the photoproduct, the energy of activation and the frequency factor of the thermal reverse reaction. The absorption coefficients of B are inserted in the numerically integrated differential equation. Thus the differential photochemical quantum yield, defined by PO= +a absA (where LA is the amount of light absorbed by the reactant that undergoes the photoreaction [l]), can be calculated using the absorbance-time curve of the photoreaction. Spectra are taken for evaluation at various wavelengths. During the recording of the spectrum the temperature must be stable, otherwise local inhomogeneities will cause numerical problems with curve fitting. For this reason, a diode array was used which allows fast spectral measurements. Because of the numerical problems in the evaluation, simulated data sets were used to obtain an indication of the significance of the proposed equation and the main influences on the correctness of the constants obtained. These results were used to interpret the experimental data obtained for the photochromic dihydroindolizine derivatives l’H-2’,3’-dimethoxycarbonyl-spiro[fluoren-9,1’pyrrolo(l,Zb)pyridazine] and its methyl forms. Their structures are given in Table 1. The slightly yellow starting materials, stable in the dark, react at long irradiation wavelengths (404 nm) by ring opening to the coloured betaines. TABLE
1. Structures
of the dihydroindolizines
examined
These thermally react backwards to the closed form. In Scheme 1 a mechanism is given which was proposed from mechanistic examination [4]. In addition to this intended reaction, further irreversible parallel and/or consecutive reactions take place, especially at shorter irradiation wavelengths.
&
( E -
!,
’
Isomer)
COOCH,
i e
=I
H
H,COOC
(
Z
Scheme 1. Photoreaction tochromic system.
-
.
u- \ N-N
isomer)
and thermal
qclization
of the pho-
However, under normal conditions the graphical matrix rank analyses [l] of the ring opening and cyclization show uniform reactions. This means that a single reaction step is found. Thus the overall reaction can be considered to follow the proposed mechanism. This assumption can be proven by the reaction spectrum given in Fig. 1 which shows the absorbance VS. wavelength for many reaction times. The absorbance data at the wavelengths 450, 470, 550 and 388 nm are plotted vs. the absorbance at 620 nm at all reaction times in Fig. 2 to demonstrate the straight lines obtained in this type of diagram (absorbance (E) diagram).
absorbance
COOCH, Compound
Substituent
in position
6
7
8
8a
I
H
II III isomer L
CH, H H
H H
H H H
H H H n
mixture)
CH,
CH, H H
‘=f, H
CI%
358
400
450
588
Fig. 1. Typical reaction spectrum compound IV in tetrahydrofuran direction of cyclization.
550
788
for the thermal cyclization of (THF). The arrows mark the
G. Gauglitz, E. Scheerer / Kinetic analysis of the system A +
absorbance b=
B
207
E-a(okA EB-EA
and a=
E-a(O)+
= a(O)+,-E
lA-e,
lB-eA
Both equations
inserted
in eqn. (3) give
4’)~~ -E(s)
ko
E:(s)- ‘WJ) = Fig. 2. Absorbance lines are observed wavelengths.
2. Fundamental
diagrams of compound IV in THF. Straight for combinations of the absorbances at two
Using E(O)=a(O)c, be rearranged to
a(o)(eE’ - Ed _ E(s)
kinetics
2.1. Determination of the reaction constants and absoFtion coefficients es The rate law for the reaction
A %B
is given
+k,b(t)
b(s)
to
e --Et/R7
+~10001,,&F(E ‘(s))
1 E(s) -E(O) -E&T+
p10001,~~F(E’(s))
1
e
1 40)(%--5%)
(6) The difference between EL and lA should be noted. Whereas EA is the molar decadic absorption coefficient of A at the irradiation wavelength, eA is the absorption coefficient at the observation wavelength. In the above equation the photokinetic factor is dependent on the temperature. However, it can be calculated for each temperature. If l/[E(s)-E(O)] is plotted OS. l/T we obtain an exponential function which can be used to determine the kinetic constants by parameter estimation. A monoexponential fitting procedure according to Gauss--Newton results in the parameters PO, PI and Pz, defined by eqn. (7) 1
1
ePllT
+
V’(s))
cp10001,~~F(E’(s))
Any change in temperature by the use of the Arrhenius in the above equation
4s) -=
ko
=
E(s)--E(O)
k-r
-EJRT (4)
or
(1)
where cpis the differential photochemical quantum yield, 1, is the intensity of monochromatic radiation (mol photons cm e-2 s-l), EL is the molar decadic absorption coefficient of compound A at the irradiation wavelength, F(E’) = (1 - 10mE’)/E’ is the photokinetic factor with absorbance E’(t) at the irradiation wavelength at time t, a(t) and b(r) are the concentrations of A and B and a(0) is the concentration of compound A at the beginning of the reaction. The rate of the thermal reverse reaction is characterized by Jc+. Furthermore, the factor of 1000 accounts for the fact that the concentrations are given in moles per litre and the path length in the cell is given in centimetres. In the photostationary state we obtain
e
allows the above equation
ko
du/dt= - q~1000I,,e~F(E’)a(t)
b(s)
1
-NO
bY
4s) -=
(~10001~ @(E’(s))
ko plooo&cp(E’(s))
must be considered relationship for kT p,=_
e -EdRT
2
(3)
where k, represents the frequency factor, E, is the energy of activation and R is the gas constant. The Lambert-Beer law and the conservation of mass result in
Pz=
1
(d=l
a(O)(% - %)d This equation can be used to calculate the energy of activation, the absorption coefficients of B and
208
G. Gauglitz,E. ScheererI Kineticanalyst of the ~stem A &B
the frequency factor quantum yield,
for a known
photochemical
2.2. Determination of the quantum yield of the photochemical reaction Thus it is necessary to determine the photochemical quantum yield, It can be obtained by transformation of eqn. (1) into the absorbance domain using the Lambert-Beer law dE = - ~10001,E~F(E’)E dt + 4W
+ (p10001,&(0)E,F(E’)
-E)
(8)
Numerical integration (“formal integration”) [l] allows us to solve the above equation. An overdetermined system of differential equations using all the absorbances measured at many reaction times
+ cploooI~r~a(o)e,
+kT ‘* (E(0) -E) s II can be written
F(E’) dt s II dt
in matrix
(9)
notation
AE=INT.Z The vector AE contains the differences in absorbance during the reaction at a chosen observation wavelength; the vector Z contains the parameters z,,= (plOOO&-,e~, z1 = ~100OL,~~a(O)~n and z2=kT. In the matrix INT the three integrals given above are found. The parameters z0 and z1 correlate, and therefore eqn. (9) was rearranged to eqn. (lo), in which the absorption coefficient ln must be known. El2
(11) In the latter case /cT is calculated using the data obtained for a thermal reverse reaction at the same temperature as the photoreaction. The parameter z,, allows us to calculate the photochemical quantum yield.
2.3. Simulation of data sets To demonstrate the significance and limitations of the method, simulated data sets were tested. Important information on the relevance of eqn. (7) was obtained. (1) The larger the change in absorbance during the measurement, the better the results. PObecomes very unreliable if the absorbance change is very small in a noisy measurement. The determination of k0 is then impossible. However, the activation energy and absorption coefficient of B can be obtained with a small standard deviation (less than a few per cent). The best change in absorbance is obtained if the irradiation intensity is chosen such that the photochemical and thermal rates are of the same order. This optima1 temperature T, can be determined using Fig. 3. The concentrations of A and B in the photostationary state depend on the temperature. T, represents the
‘2
dE=(plOoO~& J El1 -]F(B’)E II
E&(O) I
F(P)
elf
J
*,+,j,,.,,
dt
(10)
,I
By multilinear regression the interesting parameters z0 and z2 are determined. In the case of compound IV z0 and z2 still correlate. For this reason, the above equation is rearranged to
Fig. 3. Concentrations of A and B in the photostationary at various temperatures.
state
G. GQ&IZ, E. Scheem I Kinetic adysri
of the system A
&B
209
d
temperature at which the two curves of concentration cross. The window marked represents the optimal temperature range. For high thermal rates, a high intensity of irradiation must be supplied to find an optimal temperature range. Under these conditions photodegradation to side products and inhomogeneities of the solution must be excluded. (2) Unstable lamp intensities make any evaluation impossible. (3) At high temperatures the data are noisier than at lower temperatures. For this reason, low absorbance values should not be included in the evaluation. Errors from too rapid a change in temperature in between the measurements can be reduced if high temperatures are measured first, since at low temperatures the reciprocals of high absorbances cause less deviations between the ideal curve and the measured data. (4) Parameters P, and Pz are relatively insensitive to poor reaction conditions. However, for a reliable determination of PO the measured curve should not be affected by the factors mentioned above.
3. Experimental
details
A diode array spectrometer HP 8452 A (Hewlett Packard, Waldbronn) was modified according to Fig. 4. Thereby, the photoreaction could be followed spectroscopically during irradiation. An HPK 125 (Philips) or an HBO 100 W/2 (Osram) was used for irradiation at 404 nm (interference filter) via a fibre placed perpendicular to the optical pathway of measurement. Therefore the irradiation did not disturb the absorbance measurement. A beam splitter allowed the irradiation intensity to be controlled via a second fibre and the photodiode.
The compound l’H-1’,3’-dimethoxycarbonylspiro[fluoren-9,1’-pyrrolo(l,2-b)pyridazine] and its methyl derivatives were prepared according to refs. 4 and 5. The irradiation intensity was determined by Parker actinometry [6]. Methanol was purified according to ref. 7, and tetrahydrofuran (THF) was supplied by Baker. The measurement procedure was as follows. Initially, the photostationary state was produced at constant temperature and constant intensity of monochromatic irradiation. The absorbance was measured and then the temperature was varied during irradiation. The temperature intervals depended on the change in absorbance. The larger the absorbance change, the smaller the temperature interval (for the compounds studied intervals of 0.5-2°C were suitable). During this temperature variation, the stationary state was shifted producing a new absorbance value. Therefore after each temperature change a new measurement was necessary. To ensure that the system had reached the photostationary state, after each variation in temperature the absorbance was measured until no change was observed. The temperature-dependent spectra were evaiuated according to eqn. (7). The absorption coefficient of compound B and the activation energy could be determined immediately. Then component A was irradiated without temperature variation and the spectra were evaluated using eqn. (10) or eqn. (11). The absorption coefficient of B ilsbsorbanre
lltenperature
Fig. 4. Cell holder: place via a fibre (3) of measurement (4) beam; 4, observation
monochromatic irradiation of 404 nm takes placed perpendicular to the optical pathway (1, Temperature probe; 2, cell; 3, irradiation beam; 5, to thermostat; 6, magnetic stirrer).
(K)
xlo-x
Fig. 5. Measurement with temperature variation: compound lV in THF was irradiated during temperature variation; l/E(s) w. l/Tisplotted. For evaluation, the wavelength at the peak maximum of the betaine band was taken, since at this wavelength l,,=O.
2lO
G. GaugIitz; E. Scheerer I Kinetic ondysti
of the
system A &B A
TABLE
2. Excitation
of I (1 mol-’
l
A (nm)
coefficients cm-‘)
l
In THF
In methanol A
hv
A
404” 388” 51oC 5145 5w
8000 lOOMI
7200 6000 20300
6600 8200
A (nm)
6 0f
404’ 382b 384b 396b 444= 4w .528= 53oc 608’ 62O=
hv 6200 5300
m (1 mOrl CIII-*)
4
In THF hv
Insoluble methanol
in
A 5300 6700
hv 3700 2.500
E of Iv (I mol-’ cm-‘) In THF
hv
5600
4800
7700
2800
A
In methanol hv
5200 7900
5200 4800
In THF hv
A
A
hv
10400
8800
10100
6300
11100
7200 14uKl
10800
4900 14800
14400 14500 8400 12300
“Wavelength of irradiation. bPeak maximum of the non-irradiated Teak maximum of the photoproduct.
3. Quantum
component.
yields
Compound
cp
1 II
0.66f0.06 0.56 f 0.05 0.62f0.03 0.56 + 0.06
IV
In methanol
22500
A
m
cm-‘)
20300
In methanol
TABLE
of II (1 mol-’
must be inserted in these equations. The photochemical quantum yield obtained allowed the frequency factor k. to be determined using PO4. Results and discussion Compound N in THF was irradiated during temperature variation; l/E(s) VS. l/T is plotted in Fig. 5. For evaluation, the wavelength at the peak maximum of the bet&e band is taken since, at this wavelength, E._,= 0 and the absorbance change AE is maximal. The points represent the measured data; the curve is the result of the parameter eqn. (7) using the fitting according to Gauss-Newton aigorithm (a program written by Dr. S. Weiss was used).
The measured absorbances must be corrected due to the thermal expansion of the solvent (producing a change in concentration) and the temperature dependence of the refractive index. A temperature variation of 10 “C caused a change in absorbance of approximately 2%. The necessary correction of the absorbance values can be given by
E
~ E =o=r= e4"-T)
(12)
where (Y= In(1 +XX lo-*)/lo and n represents the percentage absorption change of the solution at a temperature change of 10"C x was determined experimentally. The absorption coefficients of the dihydroindolizines are independent of temperature in the observed temperature range of 10-50 “C. This was checked by measuring the unirradiated components and the betaine bands at various temperatures in a region in which A does not absorb. No shift in the absorbance maxima was obtained. The corrected absorbance data were inserted into eqn. (7). The evaluation yields PO, P, and Pz
G. Cm&
TAl3LE
4. Frequency
E. Scheerer i Kinetic
factor and thermodynamic
analysis of the system A
+
B
211
data
AGk
& (kT mol-‘)
LLY” (kJ m-1)
As+
(s-l) Solvent: methanol P rn’ Iv
5.0 f 32% 5Sf48% 16.6f20%
84.8*0.9% 84.7 * 1.4% 95.8ztO.5%
82.3 f 0.9% 82.2 + 1.4% 93.2 f 0.5%
- 10.4 f 36% -9.5+38% -0.4+221%
85.4ztO.11% 85.lf0.09% 93.3 *O.M%
Solvent: THF I” lla IlIb Iv
4.9 * 24% 45.9*39% 6.6 f 22% 12.9*23%
83.3 f 0.7% 89.0*1.1% 84.2*0.3% 89.4* 1.3%
80.8 f 0.7% 86.5 f 1.1% 81.7f0.3% 86.9 f 1.3%
-10.5*31% 8.lfXI% -8.1*31% -2.4*115%
S3.9kO.o6%
Compound
“In temperature bin temperature
b
(x10-‘*)
(J mol-’
K-l)
(kJ mol-‘)
84.0+0.05% 84.1 f0.07% 87.6f0.05%
range 11-57 “C!. range 3-57 “C.
which are used to calculate the frequency factor, the energy of activation and the absorption coefficients of the compounds. Spectral measurement allows the determination of the absorption coefficients of the photoproduct for many evaluation wavelengths. At wavelengths with small changes in absorbance, the best PO and PI values taken at the maximum of the betaine band were inserted into eqn. (7). Then P2 can be determined. In Table 2, the absorption coefficients of the photoproducts in THF and methanol are given. The betaine bands of compounds I and IV show negative solvatochromy, where the maximum of compound I exhibits a small (4 nm) and compound IV a stronger (12 run) bathochromic shift. Accordingly, the excited state is less polar than the ground state. Compound III does not exhibit an obvious solvatochromy (only 2 nm). Increasing methyl substitution shifts the betaine band to longer wavelengths. This can be explained by hyperconjugation of the methyl group and the resulting mesomeric effect. Thus these bands possess a small charge transfer character. However, from the + M effect (positive mesomeric effect: electron donating) of the methyl groups, the electron affinity of the heterocycle acceptor is reduced and additional methyl substitution should cause a hypsochromic shift. Error estimation according to Gauss demonstrated that an error in parameter Pz supplies the largest amount and an error in parameter Pr, a small amount to the total error sum. For this reason, P2 can be determined exactly, whereas the calculation PO is problematic. The calculated lrr values are used to determine the photochemical quantum yields according to eqns. (10) and (11). The results are given in Table 3. The values of k0 and E, are calculated using PO and PI. The results are listed in Table 4.
The thermodynamic cording to ln(k,lT)
= - AH”I(RlJ
data
are
calculated
ac-
+ ln(k$h) + AS”IR
where k, is the Boltzmann constant, h is the Planck constant, AG# is the free enthalpy of activation, AH” is the enthalpy of activation and AS* the entropy of activation. The result for k0 strongly depends on the accuracy of determination of PO, which is limited by the accuracy of the values IO, & and 9. Increasing the polarity of the solvent causes a stronger solvation of the betaine. For this reason, the betaines cyclize in methanol slower than in THF. The monomethyl-substituted compounds exhibit a small effect on the thermocyclization compared with the non-substituted compound. However, dimethyl substitution significantly reduces the rate of the thermal reverse reaction. This can be explained by steric as well as electronic effects. 5. Conclusions The method presented here can be used to solve the photochemical problem of unknown absorption coefficients of photoproducts. An advantage of this method is the integrated determination of the thermodynamic data of the thermal reverse reaction. Thus photochromic systems can be characterized and tested with respect to potential applications. In contrast with a conventional Arrhenius evaluation, k, and E, can be determined using only one dynamic measurement. For this reason, the method is less time consuming and requires a smaller amount of compound. Even fast systems, showing a thermal reverse reaction so rapid that an Arrhenius evaluation requires lowtemperature approaches or photoflash experiments, can be examined by this new method. The measurements can be carried out in the optimal temperature range by a suitable choice of light
212
G. GaugIitz, E. Scheerer f Kinetic analysti of the system A &B d
intensity. This method can be used to examine photochromic systems reacting according to the mechanism A 5
B. A
Acknowledgments
We thank Dr. S. Weiss for writing the evaluation program, and Dr. A. Beck and M. Schweitzer for the preparation of the compounds.
References 4 H. Mauser, Formale Kinerik, Bertelsmann Universitiitsverlag, 1974. P. Valki, and S. Vajda, Advanced Scientific Computing in Basic with Applications in Chemictry, Biology and Pharmaco[ogy Vol. 4, Elsevier, Amsterdam, 1989. E. Fischer, J. Phys. Chem., 71 (1967) 3704. J. Blanc and D. L. Ross, J. Phys. Chem., 72 (1968) 2817. H. Rau, J. Photochem., 26 (1984) 221.
5 6 7
G. M. Wyman and W. R. Brode, _I. Am. Chem. Sot., 73 (1951) 1487. H.-D. Ilge and R. Paetzold, Z. Phys. Chem., Leezig 264 (1983) 5. T. B. Krasieva, Ya. N. Malkin and V. A. Kuz’min, IN. Akad. Nauk SSSR, Ser. Khim., 6 (1988) 1405. M. B. Gordin and M. A. Gal’bershtam, Khitn. GeterotFikL Soedin, 7 (1971) 425. R. C. Bertelson, Photochromic processes involving heterolytic cleavage, in G. H. Brown (ed.), Photochmmism, Wiley-Interscience, New York, 1971. H. Rau, G. Greiner, G. GaugIitz and H. Meier, J. Phys. Chem., 94 (1990) 6523. F. Gregoire, D. Lavabre, J. C. Micheau, M. Gimenez and I. P. Laplante, I. Photochem., 28 (1985) 261. Y. Bershtein and Y. Kaminskii. Opt. Spektmsk, I5 (5) (1963) 381. A. K. Dioumaev, V. V. Savransky, N. V. Tkachenko and V. I. Chukharev, J. Photo&em. Phofobiol. B: Biol., 3 (1989) 385. H. Dilrr, 4n+Zsystems based on 1,5-electrocyclization, in H. Dirrr and H. Bows-Laurent (eds.), Photochromism, Molecules and Systems, Elsevier, Amsterdam, 1990, and Ph.D. Theses cited therein (C. Dorweiler, T. Milnzmay, G. Klauck, J. Mosbach and P. Spang). G. Hauck, Ph.D. Thestr, Saarbriicken, 1980. C. A. Parker, Photoluminescence of Solutions, Elsevier. Ansterdam, 1968. H. Mauser and G. Gauglitz, Z. Phys. Chem. N. F., 69 (1970) 258.
Photobiol. A: Chem., 71 (1993) 205-212
Method
for the determination
205
of absorption
coefficients, reaction hV
rate constants
and thermodynamic
data in the system A T--)B? A
G. Gauglitz
and
E. Scheerer
Imtitut fiir Pkysikalische und Theoretische Ckemie, Auf der Motgemtelle 8, W-7400 Tiibingen (Germany) (Received
July 20, 1992; accepted
December
3, 1992)
Abstract A method
is presented
for the kinetic
analysis
of the reaction
system
A TB
which
uses the dependence
of
the photostationary state on the temperature. One kinetic measurement yields the reaction rate constants of the thermal reverse reaction at various temperatures, thermodynamic data and the spectrum of photoproduct B. The knowledge of the absorption coefficient of the photoproduct allows the photochemical quantum yield to be calculated by numerical integration of the rate law. To obtain an indication of the significance of the data, the method is tested using sets of simulated data and is finally applied to the photochromic systems of some dihydroindolizines.
1. Introduction
Many
photochromic systems follow the mechh” anism A e-B. In spite of this simple mechanism A
the photokinetic factor prevents the differential equation from being solved in a closed form. For this reason, a least-squares approach is used to solve the overdetermined system of linear equations by numerical (‘formal’) integration [l]. If a pure photochemical mechanism is considered, the product of the photochemical quantum yield and the absorption coefficient of the starting material A at the wavelength of irradiation gives the element of the Jacobi matrix. Since the thermal reverse reaction is superimposed, its reaction rate constant must also be included in the Jacobi matrix. The concentrations are determined from the absorbance signals. The elements of the Jacobi matrix are calculated by multilinear regression. In order to avoid problems arising from the correlation of two of these elements [2], a slight change in the model equation, i.e. an c1priori knowledge of the absorption coefficient of the photoproduct, yields satisfactory results. Unfortunately, for one of the compounds examined there is still correlation of fDedicated
to Professor
75th birthday.
lOlO-6030/93/$6.CO
Dr. H. Mauser
on the occasion
of his
the elements. Another rearrangement of the model equation allows significant evaluation. The absorption coefficients of the starting material A can be determined without problems taking the spectrum of the non-irradiated substance_ However, this is not the case for the absorption coefficients of the photoproduct, since B ‘cannot be isolated in pure form because of the thermal reversibility of the system. Approaches for the determination of the absorption coefficient of nonisolable products have been proposed [3]. However, these are only applicable under special conditions. Often the measurements are realized at low temperatures or at high intensities to reduce the effects of the thermal reverse reaction on the evaluation. The experimental data can then be evaluated according to a pure photochemical equation at a first approximation. However, low-temperature measurements are disadvantageous because of the necessary experimental expenditure and the fact that the absorption coefficients determined at low temperatures differ from those at room temperature. Measurements at high intensities cause other problems. Tbe reaction solution exhibits inhomogeneities, since the gradient of the components in the cell cannot be homogenized by stirring during the time interval between measurements. Furthermore, high irradiation intensities may cause local heating of the solution.
0 1993 - Elsevier
Sequoia.
All rights reserved
206
G. Gauglitz, E. Scheerer I Kinetic analysis
&
of the system A
B
4
For these reasons, a new approach is proposed which takes advantage of the thermal reverse reaction, since a change in temperature can shift the photostationary state. This can be used to obtain an equation which allows the determination of the absorption coefficient of the photoproduct, the energy of activation and the frequency factor of the thermal reverse reaction. The absorption coefficients of B are inserted in the numerically integrated differential equation. Thus the differential photochemical quantum yield, defined by PO= +a absA (where LA is the amount of light absorbed by the reactant that undergoes the photoreaction [l]), can be calculated using the absorbance-time curve of the photoreaction. Spectra are taken for evaluation at various wavelengths. During the recording of the spectrum the temperature must be stable, otherwise local inhomogeneities will cause numerical problems with curve fitting. For this reason, a diode array was used which allows fast spectral measurements. Because of the numerical problems in the evaluation, simulated data sets were used to obtain an indication of the significance of the proposed equation and the main influences on the correctness of the constants obtained. These results were used to interpret the experimental data obtained for the photochromic dihydroindolizine derivatives l’H-2’,3’-dimethoxycarbonyl-spiro[fluoren-9,1’pyrrolo(l,Zb)pyridazine] and its methyl forms. Their structures are given in Table 1. The slightly yellow starting materials, stable in the dark, react at long irradiation wavelengths (404 nm) by ring opening to the coloured betaines. TABLE
1. Structures
of the dihydroindolizines
examined
These thermally react backwards to the closed form. In Scheme 1 a mechanism is given which was proposed from mechanistic examination [4]. In addition to this intended reaction, further irreversible parallel and/or consecutive reactions take place, especially at shorter irradiation wavelengths.
&
( E -
!,
’
Isomer)
COOCH,
i e
=I
H
H,COOC
(
Z
Scheme 1. Photoreaction tochromic system.
-
.
u- \ N-N
isomer)
and thermal
qclization
of the pho-
However, under normal conditions the graphical matrix rank analyses [l] of the ring opening and cyclization show uniform reactions. This means that a single reaction step is found. Thus the overall reaction can be considered to follow the proposed mechanism. This assumption can be proven by the reaction spectrum given in Fig. 1 which shows the absorbance VS. wavelength for many reaction times. The absorbance data at the wavelengths 450, 470, 550 and 388 nm are plotted vs. the absorbance at 620 nm at all reaction times in Fig. 2 to demonstrate the straight lines obtained in this type of diagram (absorbance (E) diagram).
absorbance
COOCH, Compound
Substituent
in position
6
7
8
8a
I
H
II III isomer L
CH, H H
H H
H H H
H H H n
mixture)
CH,
CH, H H
‘=f, H
CI%
358
400
450
588
Fig. 1. Typical reaction spectrum compound IV in tetrahydrofuran direction of cyclization.
550
788
for the thermal cyclization of (THF). The arrows mark the
G. Gauglitz, E. Scheerer / Kinetic analysis of the system A +
absorbance b=
B
207
E-a(okA EB-EA
and a=
E-a(O)+
= a(O)+,-E
lA-e,
lB-eA
Both equations
inserted
in eqn. (3) give
4’)~~ -E(s)
ko
E:(s)- ‘WJ) = Fig. 2. Absorbance lines are observed wavelengths.
2. Fundamental
diagrams of compound IV in THF. Straight for combinations of the absorbances at two
Using E(O)=a(O)c, be rearranged to
a(o)(eE’ - Ed _ E(s)
kinetics
2.1. Determination of the reaction constants and absoFtion coefficients es The rate law for the reaction
A %B
is given
+k,b(t)
b(s)
to
e --Et/R7
+~10001,,&F(E ‘(s))
1 E(s) -E(O) -E&T+
p10001,~~F(E’(s))
1
e
1 40)(%--5%)
(6) The difference between EL and lA should be noted. Whereas EA is the molar decadic absorption coefficient of A at the irradiation wavelength, eA is the absorption coefficient at the observation wavelength. In the above equation the photokinetic factor is dependent on the temperature. However, it can be calculated for each temperature. If l/[E(s)-E(O)] is plotted OS. l/T we obtain an exponential function which can be used to determine the kinetic constants by parameter estimation. A monoexponential fitting procedure according to Gauss--Newton results in the parameters PO, PI and Pz, defined by eqn. (7) 1
1
ePllT
+
V’(s))
cp10001,~~F(E’(s))
Any change in temperature by the use of the Arrhenius in the above equation
4s) -=
ko
=
E(s)--E(O)
k-r
-EJRT (4)
or
(1)
where cpis the differential photochemical quantum yield, 1, is the intensity of monochromatic radiation (mol photons cm e-2 s-l), EL is the molar decadic absorption coefficient of compound A at the irradiation wavelength, F(E’) = (1 - 10mE’)/E’ is the photokinetic factor with absorbance E’(t) at the irradiation wavelength at time t, a(t) and b(r) are the concentrations of A and B and a(0) is the concentration of compound A at the beginning of the reaction. The rate of the thermal reverse reaction is characterized by Jc+. Furthermore, the factor of 1000 accounts for the fact that the concentrations are given in moles per litre and the path length in the cell is given in centimetres. In the photostationary state we obtain
e
allows the above equation
ko
du/dt= - q~1000I,,e~F(E’)a(t)
b(s)
1
-NO
bY
4s) -=
(~10001~ @(E’(s))
ko plooo&cp(E’(s))
must be considered relationship for kT p,=_
e -EdRT
2
(3)
where k, represents the frequency factor, E, is the energy of activation and R is the gas constant. The Lambert-Beer law and the conservation of mass result in
Pz=
1
(d=l
a(O)(% - %)d This equation can be used to calculate the energy of activation, the absorption coefficients of B and
208
G. Gauglitz,E. ScheererI Kineticanalyst of the ~stem A &B
the frequency factor quantum yield,
for a known
photochemical
2.2. Determination of the quantum yield of the photochemical reaction Thus it is necessary to determine the photochemical quantum yield, It can be obtained by transformation of eqn. (1) into the absorbance domain using the Lambert-Beer law dE = - ~10001,E~F(E’)E dt + 4W
+ (p10001,&(0)E,F(E’)
-E)
(8)
Numerical integration (“formal integration”) [l] allows us to solve the above equation. An overdetermined system of differential equations using all the absorbances measured at many reaction times
+ cploooI~r~a(o)e,
+kT ‘* (E(0) -E) s II can be written
F(E’) dt s II dt
in matrix
(9)
notation
AE=INT.Z The vector AE contains the differences in absorbance during the reaction at a chosen observation wavelength; the vector Z contains the parameters z,,= (plOOO&-,e~, z1 = ~100OL,~~a(O)~n and z2=kT. In the matrix INT the three integrals given above are found. The parameters z0 and z1 correlate, and therefore eqn. (9) was rearranged to eqn. (lo), in which the absorption coefficient ln must be known. El2
(11) In the latter case /cT is calculated using the data obtained for a thermal reverse reaction at the same temperature as the photoreaction. The parameter z,, allows us to calculate the photochemical quantum yield.
2.3. Simulation of data sets To demonstrate the significance and limitations of the method, simulated data sets were tested. Important information on the relevance of eqn. (7) was obtained. (1) The larger the change in absorbance during the measurement, the better the results. PObecomes very unreliable if the absorbance change is very small in a noisy measurement. The determination of k0 is then impossible. However, the activation energy and absorption coefficient of B can be obtained with a small standard deviation (less than a few per cent). The best change in absorbance is obtained if the irradiation intensity is chosen such that the photochemical and thermal rates are of the same order. This optima1 temperature T, can be determined using Fig. 3. The concentrations of A and B in the photostationary state depend on the temperature. T, represents the
‘2
dE=(plOoO~& J El1 -]F(B’)E II
E&(O) I
F(P)
elf
J
*,+,j,,.,,
dt
(10)
,I
By multilinear regression the interesting parameters z0 and z2 are determined. In the case of compound IV z0 and z2 still correlate. For this reason, the above equation is rearranged to
Fig. 3. Concentrations of A and B in the photostationary at various temperatures.
state
G. GQ&IZ, E. Scheem I Kinetic adysri
of the system A
&B
209
d
temperature at which the two curves of concentration cross. The window marked represents the optimal temperature range. For high thermal rates, a high intensity of irradiation must be supplied to find an optimal temperature range. Under these conditions photodegradation to side products and inhomogeneities of the solution must be excluded. (2) Unstable lamp intensities make any evaluation impossible. (3) At high temperatures the data are noisier than at lower temperatures. For this reason, low absorbance values should not be included in the evaluation. Errors from too rapid a change in temperature in between the measurements can be reduced if high temperatures are measured first, since at low temperatures the reciprocals of high absorbances cause less deviations between the ideal curve and the measured data. (4) Parameters P, and Pz are relatively insensitive to poor reaction conditions. However, for a reliable determination of PO the measured curve should not be affected by the factors mentioned above.
3. Experimental
details
A diode array spectrometer HP 8452 A (Hewlett Packard, Waldbronn) was modified according to Fig. 4. Thereby, the photoreaction could be followed spectroscopically during irradiation. An HPK 125 (Philips) or an HBO 100 W/2 (Osram) was used for irradiation at 404 nm (interference filter) via a fibre placed perpendicular to the optical pathway of measurement. Therefore the irradiation did not disturb the absorbance measurement. A beam splitter allowed the irradiation intensity to be controlled via a second fibre and the photodiode.
The compound l’H-1’,3’-dimethoxycarbonylspiro[fluoren-9,1’-pyrrolo(l,2-b)pyridazine] and its methyl derivatives were prepared according to refs. 4 and 5. The irradiation intensity was determined by Parker actinometry [6]. Methanol was purified according to ref. 7, and tetrahydrofuran (THF) was supplied by Baker. The measurement procedure was as follows. Initially, the photostationary state was produced at constant temperature and constant intensity of monochromatic irradiation. The absorbance was measured and then the temperature was varied during irradiation. The temperature intervals depended on the change in absorbance. The larger the absorbance change, the smaller the temperature interval (for the compounds studied intervals of 0.5-2°C were suitable). During this temperature variation, the stationary state was shifted producing a new absorbance value. Therefore after each temperature change a new measurement was necessary. To ensure that the system had reached the photostationary state, after each variation in temperature the absorbance was measured until no change was observed. The temperature-dependent spectra were evaiuated according to eqn. (7). The absorption coefficient of compound B and the activation energy could be determined immediately. Then component A was irradiated without temperature variation and the spectra were evaluated using eqn. (10) or eqn. (11). The absorption coefficient of B ilsbsorbanre
lltenperature
Fig. 4. Cell holder: place via a fibre (3) of measurement (4) beam; 4, observation
monochromatic irradiation of 404 nm takes placed perpendicular to the optical pathway (1, Temperature probe; 2, cell; 3, irradiation beam; 5, to thermostat; 6, magnetic stirrer).
(K)
xlo-x
Fig. 5. Measurement with temperature variation: compound lV in THF was irradiated during temperature variation; l/E(s) w. l/Tisplotted. For evaluation, the wavelength at the peak maximum of the betaine band was taken, since at this wavelength l,,=O.
2lO
G. GaugIitz; E. Scheerer I Kinetic ondysti
of the
system A &B A
TABLE
2. Excitation
of I (1 mol-’
l
A (nm)
coefficients cm-‘)
l
In THF
In methanol A
hv
A
404” 388” 51oC 5145 5w
8000 lOOMI
7200 6000 20300
6600 8200
A (nm)
6 0f
404’ 382b 384b 396b 444= 4w .528= 53oc 608’ 62O=
hv 6200 5300
m (1 mOrl CIII-*)
4
In THF hv
Insoluble methanol
in
A 5300 6700
hv 3700 2.500
E of Iv (I mol-’ cm-‘) In THF
hv
5600
4800
7700
2800
A
In methanol hv
5200 7900
5200 4800
In THF hv
A
A
hv
10400
8800
10100
6300
11100
7200 14uKl
10800
4900 14800
14400 14500 8400 12300
“Wavelength of irradiation. bPeak maximum of the non-irradiated Teak maximum of the photoproduct.
3. Quantum
component.
yields
Compound
cp
1 II
0.66f0.06 0.56 f 0.05 0.62f0.03 0.56 + 0.06
IV
In methanol
22500
A
m
cm-‘)
20300
In methanol
TABLE
of II (1 mol-’
must be inserted in these equations. The photochemical quantum yield obtained allowed the frequency factor k. to be determined using PO4. Results and discussion Compound N in THF was irradiated during temperature variation; l/E(s) VS. l/T is plotted in Fig. 5. For evaluation, the wavelength at the peak maximum of the bet&e band is taken since, at this wavelength, E._,= 0 and the absorbance change AE is maximal. The points represent the measured data; the curve is the result of the parameter eqn. (7) using the fitting according to Gauss-Newton aigorithm (a program written by Dr. S. Weiss was used).
The measured absorbances must be corrected due to the thermal expansion of the solvent (producing a change in concentration) and the temperature dependence of the refractive index. A temperature variation of 10 “C caused a change in absorbance of approximately 2%. The necessary correction of the absorbance values can be given by
E
~ E =o=r= e4"-T)
(12)
where (Y= In(1 +XX lo-*)/lo and n represents the percentage absorption change of the solution at a temperature change of 10"C x was determined experimentally. The absorption coefficients of the dihydroindolizines are independent of temperature in the observed temperature range of 10-50 “C. This was checked by measuring the unirradiated components and the betaine bands at various temperatures in a region in which A does not absorb. No shift in the absorbance maxima was obtained. The corrected absorbance data were inserted into eqn. (7). The evaluation yields PO, P, and Pz
G. Cm&
TAl3LE
4. Frequency
E. Scheerer i Kinetic
factor and thermodynamic
analysis of the system A
+
B
211
data
AGk
& (kT mol-‘)
LLY” (kJ m-1)
As+
(s-l) Solvent: methanol P rn’ Iv
5.0 f 32% 5Sf48% 16.6f20%
84.8*0.9% 84.7 * 1.4% 95.8ztO.5%
82.3 f 0.9% 82.2 + 1.4% 93.2 f 0.5%
- 10.4 f 36% -9.5+38% -0.4+221%
85.4ztO.11% 85.lf0.09% 93.3 *O.M%
Solvent: THF I” lla IlIb Iv
4.9 * 24% 45.9*39% 6.6 f 22% 12.9*23%
83.3 f 0.7% 89.0*1.1% 84.2*0.3% 89.4* 1.3%
80.8 f 0.7% 86.5 f 1.1% 81.7f0.3% 86.9 f 1.3%
-10.5*31% 8.lfXI% -8.1*31% -2.4*115%
S3.9kO.o6%
Compound
“In temperature bin temperature
b
(x10-‘*)
(J mol-’
K-l)
(kJ mol-‘)
84.0+0.05% 84.1 f0.07% 87.6f0.05%
range 11-57 “C!. range 3-57 “C.
which are used to calculate the frequency factor, the energy of activation and the absorption coefficients of the compounds. Spectral measurement allows the determination of the absorption coefficients of the photoproduct for many evaluation wavelengths. At wavelengths with small changes in absorbance, the best PO and PI values taken at the maximum of the betaine band were inserted into eqn. (7). Then P2 can be determined. In Table 2, the absorption coefficients of the photoproducts in THF and methanol are given. The betaine bands of compounds I and IV show negative solvatochromy, where the maximum of compound I exhibits a small (4 nm) and compound IV a stronger (12 run) bathochromic shift. Accordingly, the excited state is less polar than the ground state. Compound III does not exhibit an obvious solvatochromy (only 2 nm). Increasing methyl substitution shifts the betaine band to longer wavelengths. This can be explained by hyperconjugation of the methyl group and the resulting mesomeric effect. Thus these bands possess a small charge transfer character. However, from the + M effect (positive mesomeric effect: electron donating) of the methyl groups, the electron affinity of the heterocycle acceptor is reduced and additional methyl substitution should cause a hypsochromic shift. Error estimation according to Gauss demonstrated that an error in parameter Pz supplies the largest amount and an error in parameter Pr, a small amount to the total error sum. For this reason, P2 can be determined exactly, whereas the calculation PO is problematic. The calculated lrr values are used to determine the photochemical quantum yields according to eqns. (10) and (11). The results are given in Table 3. The values of k0 and E, are calculated using PO and PI. The results are listed in Table 4.
The thermodynamic cording to ln(k,lT)
= - AH”I(RlJ
data
are
calculated
ac-
+ ln(k$h) + AS”IR
where k, is the Boltzmann constant, h is the Planck constant, AG# is the free enthalpy of activation, AH” is the enthalpy of activation and AS* the entropy of activation. The result for k0 strongly depends on the accuracy of determination of PO, which is limited by the accuracy of the values IO, & and 9. Increasing the polarity of the solvent causes a stronger solvation of the betaine. For this reason, the betaines cyclize in methanol slower than in THF. The monomethyl-substituted compounds exhibit a small effect on the thermocyclization compared with the non-substituted compound. However, dimethyl substitution significantly reduces the rate of the thermal reverse reaction. This can be explained by steric as well as electronic effects. 5. Conclusions The method presented here can be used to solve the photochemical problem of unknown absorption coefficients of photoproducts. An advantage of this method is the integrated determination of the thermodynamic data of the thermal reverse reaction. Thus photochromic systems can be characterized and tested with respect to potential applications. In contrast with a conventional Arrhenius evaluation, k, and E, can be determined using only one dynamic measurement. For this reason, the method is less time consuming and requires a smaller amount of compound. Even fast systems, showing a thermal reverse reaction so rapid that an Arrhenius evaluation requires lowtemperature approaches or photoflash experiments, can be examined by this new method. The measurements can be carried out in the optimal temperature range by a suitable choice of light
212
G. GaugIitz, E. Scheerer f Kinetic analysti of the system A &B d
intensity. This method can be used to examine photochromic systems reacting according to the mechanism A 5
B. A
Acknowledgments
We thank Dr. S. Weiss for writing the evaluation program, and Dr. A. Beck and M. Schweitzer for the preparation of the compounds.
References 4 H. Mauser, Formale Kinerik, Bertelsmann Universitiitsverlag, 1974. P. Valki, and S. Vajda, Advanced Scientific Computing in Basic with Applications in Chemictry, Biology and Pharmaco[ogy Vol. 4, Elsevier, Amsterdam, 1989. E. Fischer, J. Phys. Chem., 71 (1967) 3704. J. Blanc and D. L. Ross, J. Phys. Chem., 72 (1968) 2817. H. Rau, J. Photochem., 26 (1984) 221.
5 6 7
G. M. Wyman and W. R. Brode, _I. Am. Chem. Sot., 73 (1951) 1487. H.-D. Ilge and R. Paetzold, Z. Phys. Chem., Leezig 264 (1983) 5. T. B. Krasieva, Ya. N. Malkin and V. A. Kuz’min, IN. Akad. Nauk SSSR, Ser. Khim., 6 (1988) 1405. M. B. Gordin and M. A. Gal’bershtam, Khitn. GeterotFikL Soedin, 7 (1971) 425. R. C. Bertelson, Photochromic processes involving heterolytic cleavage, in G. H. Brown (ed.), Photochmmism, Wiley-Interscience, New York, 1971. H. Rau, G. Greiner, G. GaugIitz and H. Meier, J. Phys. Chem., 94 (1990) 6523. F. Gregoire, D. Lavabre, J. C. Micheau, M. Gimenez and I. P. Laplante, I. Photochem., 28 (1985) 261. Y. Bershtein and Y. Kaminskii. Opt. Spektmsk, I5 (5) (1963) 381. A. K. Dioumaev, V. V. Savransky, N. V. Tkachenko and V. I. Chukharev, J. Photo&em. Phofobiol. B: Biol., 3 (1989) 385. H. Dilrr, 4n+Zsystems based on 1,5-electrocyclization, in H. Dirrr and H. Bows-Laurent (eds.), Photochromism, Molecules and Systems, Elsevier, Amsterdam, 1990, and Ph.D. Theses cited therein (C. Dorweiler, T. Milnzmay, G. Klauck, J. Mosbach and P. Spang). G. Hauck, Ph.D. Thestr, Saarbriicken, 1980. C. A. Parker, Photoluminescence of Solutions, Elsevier. Ansterdam, 1968. H. Mauser and G. Gauglitz, Z. Phys. Chem. N. F., 69 (1970) 258.