Quantitative determination of photokinetic parameters for reversible photoisomerizations in light scattering adsorbent-adsorbates

Article abstract of DOI:10.1524/zpch.2000.214.2.149

Based on an approximate solution of the differential equations describing the rate of a reversible photoisomerization and the attenuation of the actinic light in a powdered adsorbent-adsorbate a method is presented permitting us to predict the irradiation

Full text of DOI:10.1524/zpch.2000.214.2.149

Zeitschrift für Physikalische Chemie, 214, 2, S. 149Ϫ167 (2000)
by Oldenbourg Wissenschaftsverlag, München

Quantitative Determination
of Photokinetic Parameters
for Reversible Photoisomerizations
in Light Scattering Adsorbent-Adsorbates
By R. Gade and Th. Porada
Institut für Phys. Chemie, Universität Jena, Steiger 3, D-07743 Jena, Germany
(Accepted February 22, 1999)
Photoisomerization / Light Scattering / Absorption Coefficients /
Quantum Yields / Azobenzene
Based on an approximate solution of the differential equations describing the rate of a
reversible photoisomerization and the attenuation of the actinic light in a powdered ad-
sorbent-adsorbate a method is presented permitting us to predict the irradiation time de-
pendence of the sample reflectance and to determine both the quantum yields of the
partial reactions and the absorption coefficients of the reactant and the product from the
reflectance-time-curves measured at two different wavelengths. The present method, thus,
is an improved version of the peviously reported procedure [1] which enables the quantum
yield of a simple photoreaction to be calculated from the initial slope of the R(t)-curve.
To test the method proposed here it has been used for calculating the photokinetic param-
eters of azobenzene adsorbed on disperse silica.
1. Introduction
Based on the recent progress in the photocatalytic purification of waste
waters [2, 3] and in the photoelectrochemical conversion of solar energy [3,
4
] the photochemistry of molecules adsorbed on heterogeneous systems has
received increasing attention during the last few years [5, 6].
However, the spectroscopic investigations of heterogeneous systems are
much more complicated than those of homogeneous ones. This is firstly
due to the circumstance that the light propagation in heterogeneous systems
is strongly influenced by scattering effects. Since, in addition, the adsorbed
molecules are relatively fixed within the light scattering medium the attenu-
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50
R. Gade and Th. Porada
ation of the actinic light owing to absorption and scattering gives rise to the
formation of concentration gradients varying with time. In contrast to stirred
solutions, hence, the concentrations of all species involved in the photoreac-
tion depend on both the irradiation time and the penetration depth of the
light so that the rate equations in general become nonlinear, partial differen-
tial equations. On these conditions the common way to determine the de-
sired kinetic parameters, e.g. quantum yields or rate constants then consists
in fitting a numerical solution of these differential equations to available
spectroscopic data. Using this method Oelkrug and coworkers [7] deter-
mined the isomerization quantum yields of various organic compounds ad-
sorbed on alumina. With this procedure being rather complicated Simmons
and coworkers [8, 9] presented several models which enable the quantum
yields of some solid state reactions in powders to be directly calculated
from diffuse reflectance data. All of these models, however, are founded on
drastic and inconsistent assumptions so that they can only be employed in
a limited degree. That is why we have recently proposed another method
[1] which in contrast to Simmons’ approach strictly uses the Kubelka-Munk
theory without any additional presumptions to describe the light propagation
in the scattering medium. Based on the power series expansions of both the
concentrations and the intensities of the light-fluxes we succeeded in relat-
ing the quantum yield of the photoreaction to the slope of the measured
reflectance-time curve. Hence, the afore-mentioned procedure may be re-
garded as a generalization of the “initial-slope method’’ [10] developed for
stirred solutions to the case of light scattering materials.
Employing our method to the photobleaching of an azo-dye adsorbed
on nanodisperse titania [11] we could quantitatively determine the quantum
yield of this TiO -catalysed photodegradation from diffuse reflectance data.
2
It must be noted, however, that the knowledge of the absorption coefficient
of the photoproduct either at the irradiation wavelength λ or at some detec-
i
tion wavelength λ is a prerequisite for applying this method. If, in general,
d
the absorption coefficient of the product is unknown at any wavelength and
should be determined together with the quantum yields the procedure out-
lined above will fail. On the other hand, a generally applicable method for
determining both the quantum yields and the absorption coefficients of the
photoproducts did also not exist for stirred solutions. Since in this case the
relation between the absorbance A (t,λ ) of the irradiated solution and the
λ
i
d
kinetic parameters desired was much simpler than in the case of light scat-
tering adsorbent-adsorbates we dealt with this problem first and developed
a procedure [12] which combines the method of Fischer [13] with tech-
niques for calculating the absorbance A (ϱ,λ ) in the photostationary state,
λ
i
d
particularly with the “method of transformed time’’ [14]. The needed ab-
sorbance values A (ϱ,λ ) are then determined from a least-squares fit of the
λ
i
d
irradiation-time function [15]
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Quantitative Determination of Photokinetic Parameters for Reversible ...
151
λ λ
Ϫp I Θ(t)
i i
A i
λ
(t,λ
d
) ϭ A i
λ
(ϱ,λ
d
) Ϫ (A i
λ
(ϱ,λ
d
) Ϫ A i
λ
(0,λ
d
)) e
(1)
to the points {Θ(t
n
),A i(t ,λ ) of the experimental absorbance-time curve
λ
n
d
where
t
i(t′,λ
λ i
ϪA )
1
Ϫ10
A i(t′,λ )
Θ(t) ϭ
͐
dt′
(2)
0
λ
i
denotes the so-called “transformed time” [14] which may easily be calcu-
lated from measured reaction spectra of the illuminated solution.
So as to be able to analyse photoreactions in light scattering systems in
a similar way as those in stirred solutions we searched for an approximate
solution of the differential equations describing the reaction rate and the
attenuation of the actinic light. Based on the approximation found an ana-
lytic expression for the time-dependence of the diffuse reflectance R (t,λ )
λ
i
d
could be derived whose form is analogous with Eq. (1). Accordingly, in the
present case the extent of conversion β (ϱ) in the photostationary equilib-
λ
i
rium can be obtained from a least-squares fit of the points {t
n
,R i
λ
(t
n
,λ
d
); n ϭ
0
, ....N} to this analytic expression. Once the quantity β (ϱ) is determined
λ
i
the quantum yields and the absorption coefficient of the photoproduct are
easy to calculate by means of the method developed below.
2. Theory
2
.1 Rate equations for photoisomerizations
in powdered adsorbent-adsorbates
As it is illustrated in Fig. 1 the surface (z ϭ 0) of a powdered adsorbent-
adsorbate sample is homogeneously illuminated with monochromatic light
having the wavelength λ
i
and the irradiance I
0
i
(λ ). Due to the irradiation
with light a reversible photoisomerization
hν
A Y=y B
is supposed to be induced in the disperse adsorbent-adsorbate. According
to a preceding paper [1] the rate equations of this photoreaction can be
written as
∂
cAλi
A
B
ϭ (Ϫ Φ ε (λ )c (z,t) ϩ Φ ε (λ )c (z,t)) δ[I ϩ I ]
(3)
(4)
B
A
i
Aλ
i
A
B
i
Bλ
i
ϩλ
i
Ϫλ
i
∂
c
∂
t
∂
Bλi
∂cAλi
ϭ Ϫ
t
∂t
where ε
A
(λ
i
) and ε
B
(λ
i
) designate the molar absorption coefficients of the
A
species A and B, respectively, at the wavelength λ . Furthermore, Φ de-
i
B
notes the partial quantum yield for the production of B when A is excited
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R. Gade and Th. Porada
Fig. 1. Illustration of the light fluxes penetrating the volume element dV inside a light
scattering sample.
B
A
and Φ the corresponding one for the reverse case. In the case considered
here the concentrations c (z,t) and c (z,t) of the species A and B, respec-
Aλ
i
Bλ
i
tively, depend not only on the exposure time t but also on the irradiation
wavelength λ and the penetration depth z. The irradiances I and IϪλi of
i
ϩλ
i
the penetrating (ϩ) and backscattered (Ϫ) irradiation light fluxes obey the
Kubelka-Munk equations
∂
I
ϩλi
ϭ Ϫ (K ϩ S) Iϩλi(z,t) ϩ S IϪλi(z,t)
ϭ Ϫ S I (z,t) ϩ (K ϩ S) I (z,t)
(5)
(6)
∂
z
∂
IϪλi
ϩλ
i
Ϫλ
i
∂
z
where K and S designate the absorption and the scattering coefficient, re-
spectively, of the powdered adsorbent-adsorbate. The absorption coefficient
K is assumed to be composed additively of contributions of the particulate
substrate and the adsorbed species A and B, viz:
K ϭ K
S
ϩ K
A
ϩ K
B
(7)
where K and K
A
B
in each case are proportional to the absorption coefficient
of the corresponding species and its concentration, i.e.
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153
K
A
(z,t,λ
i
) ϭ δε
A
(λ
i
) cAλi(z,t)
(8)
(9)
K (z,t,λ ) ϭ δε (λ ) c (z,t)
B
i
B
i
Bλ
i
The proportionality factor δ can easily be determined by taking into account
that Eqs. (7)Ϫ(9) hold for any irradiation time t, especially also for t ϭ 0
where
c
Bλi(z,0) ϭ 0
and hence
K(z,0,λ
(10)
i
) ϭ K
S
(λ
i
) ϩ δε
A
(λ
i
) c .
0
(11)
Due to the spatial constancy of K at t ϭ 0 the Kubelka-Munk relationship
2
(
1 Ϫ R0)
K(t ϭ 0)
F(R
0
) ϭ
ϭ
(12)
2
R
0
S
may be applied, finally yielding
S[F(R (λ )) Ϫ F(R (λ ))]
0
i
S
i
δ ϭ
.
(13)
c
0
ε
A
(λ )
i
Here, R (λ ) denotes the diffuse reflectance of the unirradiated sample while
0
i
R (λ ) designates that of the pure adsorbent and both are taken at the wave-
S
i
length λ
balance equation
Aλi(z,t) ϩ cBλi (z,t) ϭ c
i
. Because of Eq. (4) the concentrations of A and B satisfy the local
c
0
(14)
allowing one concentration to be eliminated in Eqs. (3) and (4) so that the
remaining rate equation can be rewritten in terms of the extent of conversion
c (z,t)
Bλ
i
β i
λ
(z,t) ϭ
(15)
c
0
which leads to
β i(z,t)
∂
λ
ϭ (m i
λ
Ϫ p i
λ
λ
β i) δ[Iϩλi ϩ IϪλi]
(16)
∂
t
where c is the total concentration of the adsorbate and the quantities mλi
0
and p i
λ
are given by
A
m i
λ
ϭ Φ
B
ε
A
(λ
i
)
(17)
(18)
A
B
p ϭ Φ ε (λ ) ϩ Φ ε (λ ) .
λ
i
B
A
i
A
B
i
Since because of Eq. (10) the extent of conversion β (z,t) obeys the initial
λ
i
condition
β i
and its time derivation has a positive value at this time the function β (z,t)
(
λ
(z,t))tϭ0 ϭ 0
(19)
λ
i
increases with increasing t while (∂β /∂t) decreases. Hence, in the photo-
λ
i
stationary equilibrium, i.e. as t → ϱ the time derivative of β i(z,t) vanishes
λ
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R. Gade and Th. Porada
∂
β i(z,t)
λ
0
ϭ
ϭ (m i
λ
Ϫ p i
λ
β i
λ
(ϱ)) δ[Iϩλi ϩ IϪλi
]
(20)
ͩ
ͪ
∂
t
t→ϱ
and for arbitrary values of z the function βλi(z,t) tends to the limit
mλi
β (ϱ) ϭ
.
(21)
λ
i
p i
λ
Eq. (16) manifests that the time derivative of β i
λ
depends on the irradiances
I
ϩλi and IϪλi obeying the Kubelka-Munk Eqs. (5) and (6). On the other hand,
according to Eqs. (7)Ϫ(9) the absorption coefficient K of the powdered
medium is a linear function of β . For this reason the rate Eq. (16) and the
λ
i
Kubelka-Munk Eqs. (5) and (6) are coupled to each other so that the kinetic
description of a photoisomerization in a powdered absorbent-adsorbate re-
quires solving the following set of differential equations
∂
∂
∂
βλi
ϭ p i
λ
[β i
λ
(ϱ) Ϫ β i
λ
] δ[Iϩλi ϩ IϪλi
]
(22)
(23)
(24)
∂
t
Iϩλi
ϭ Ϫ [b
1
(λ
i
) ϩ b
2
(λ
i
) β i] Iϩλi ϩ S IϪλi
λ
∂
z
IϪλi
ϭ Ϫ S Iϩλi ϩ [b
1
(λ
i
) ϩ b
2
(λ
i
) β i] IϪλi
λ
∂
z
where for the sake of clearness the following abbreviations have been used
b
1
(λ
(λ
In order that well-determined and unique solutions β i
i
) ϭ S ϩ K
S
ϩ δc
0
ε
A
(λ
i
)
(25)
(26)
b
2
i
) ϭ δc (ε (λ
0
B
i
) Ϫ ε
A
(λ )) .
i
λ
, Iϩλi, and IϪλi are
obtained, of course, the boundary-value problem of Eqs. (22)Ϫ(24) needs
to be solved, i.e. in addition to the differential Eqs. (22)Ϫ(24) these func-
tions are required to satisfy the boundary conditions
(
(
(
β i
λ
)
tϭ0 ϭ 0
(27)
(28)
(29)
I ) ϭ I0λi
ϩλ zϭ 0
i
I ) ϭ 0
Ϫλ
i
z→ϱ
the physical content of which can easily be understood. With the right-hand
sides of Eqs. (22)Ϫ(24) obviously exhibiting product termes of the func-
tions to be calculated these equations must be classified as a set of nonlinear
partial differential equations. That is why standard methods developed for
solving linear differential equations fail in this case so that any analytical
solution cannot be obtained in general.
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2
.2 Approximate solution of the differential equations
So as to obtain at least an approximate solution of Eqs. (22)Ϫ(24) being
A
B
suitable for determining the quantum yields Φ and Φ and the absorption
B
A
coefficient ε
B
i
(λ ) the differential equations will be transformed into a set of
equivalent integral equations which may subsequently be solved by the
method of successive approximation.
For this purpose the variables in Eq. (22) are separated and both sides
of the resulting equation are integrated from 0 to t with respect to time
n
which gives
t
n
β i
λ
(ϱ) Ϫ β i
λ
(z,t )
n
Ϫln
ϭ
͐
dt p δ[I (z,t) ϩ I (z,t)]
ͩ
ͪ
λ
i
ϩλ
i
Ϫλ
i
β i
λ
(ϱ) Ϫ β i
λ
(z,0)
0
Solving this equation for β and using Eq. (19) finally leads to
λ
i
t
n
Ϫp
λ
δ
͐
dt(Iϩλ ϩ IϪλi)
i
0
β (z,t ) ϭ β (ϱ)
ͭ1 Ϫ e
i
ͮ
(30)
λ
i
n
λ
i
On the other hand the integration of the differential equations (23), (24)
yields the integral equations
z
I
ϩλi(z,t) ϭ I0λi ϩ S
͐
dz′IϪλi(z′,t)
(31)
0
z
Ϫ
͐
dz′[b
1
(λ
i
2
) ϩ b (λ
i
) β i(z′,t)] Iϩλi(z′,z)
λ
0
and
z
I
Ϫλi(z,t) ϭ I0λi R i
λ
(t,λ
i
) Ϫ S
͐
dz′Iϩλi(z′,t)
(32)
0
z
ϩ
͐
dz′[b
1
(λ
i
) ϩ b
2
(λ
i
) β i(z′,t)] IϪλi(z′,z)
λ
0
describing the propagation of the penetrating and the backscattered irra-
diation light within the powdered sample the optical properties of which
have been altered by the photoreaction. As the alterations caused by the
light-induced transformation in the irradiated sample vary with the irra-
diation wavelength λ both the extent of conversion and the light intensities
i
depend not only on z and t but also on λ
i
. The last-mentioned dependence
on λ which results from the spectral behaviour of the absorption coefficient
i
is expressed by an index λ at the corresponding observables throughout this
i
paper.
Moreover it is noted that the unknown reflectance value R (t,λ ) of the
λ
i
i
infinitely thick irradiated sample which appears on the right-hand side of
Eq. (32) will only be determined by the boundary condition (29). So as to
spectroscopicly detect the photoinduced concentration changes within the
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R. Gade and Th. Porada
irradiated adsorbent-adsorbate the powdered sample is to be analysed by
means of monochromatic light of appropriate wavelength. This is usually
done with a common spectrophotometer equipped with a diffuse reflectance
accessory. In this case the irradiance I0λd of the analysing light is choosen
so low that in this way the sample reflectance
I
Ϫλi(0,t,λ )
d
R (t,λ ) ϭ
(33)
λ
i
d
I
ϩλi(0,t,λ )
d
may be measured at the observation wavelength λ but no photoinduced
d
transformation of the sample will be caused.
Naturally, the intensities I (z,t,λ ) and I (z,t,λ ) of the penetrating
ϩλ
i
d
Ϫλ
i
d
and the backscattered analysing light respectively also satisfy Kubelka-
Munk equations of the form (23, 24) but in contrast to the latter the coef-
ficients b and b now have to be taken at the observation wavelength λ .
1
2
d
So as to get the approximate solution desired of the integral Eqs. (30)Ϫ
(
32) we have to proceed as follows. As the starting point of the iteration
procedure the state of the unirradiated sample is choosen where the extent
of conversion is given by
(
0)
i
β
λ
(z,t) ϭ 0 .
(34)
Inserting this approximation in the Kubelka-Munk Eqs. (23) and (24) speci-
fied for the analysing light and taking into account the boundary conditions
(28, 29) we obtain the following expressions for the forward and backward
directed fluxes of the analysing light
(
0)
d
Ϫzκ(λ )
I
I
(z,0,λ ) ϭ I
e
(35)
(36)
ϩλ
i
d
0λ
d
(0)
Ϫzκ(λd)
Ϫλi(z,0,λ
d
) ϭ I0λd
R
0
(λ ) e
d
i.e. the Kubelka-Munk solutions for an homogeneous semiinfinite medium.
Here R (λ ) is the diffuse reflectance of the unirradiated sample at the detec-
tion wavelength λ whereas
0
d
d
κ(λ
denotes the attenuation coefficient and
(λ ) ϭ K ϩ δc (λ ) ϭ b (λ ) Ϫ S
d
) ϭ
Ί
K
0
(λ
d
)(K
0
(λ
d
) ϩ 2 S)
(37)
K
0
d
S
0
ε
A
d
1
d
(38)
the absorption coefficient of the unirradiated powder material at this wave-
length. On the other hand, according to the Kubelka-Munk relationship
2
K
0
(λ
d
)
[1 Ϫ R
0
(λ
(λ
) may be expressed by the reflectance value
R (λ ) and the scattering coefficient S of the pure adsorbent so that we
d
)]
ϭ F(R
0
(λ
d
)) ϭ
(39)
S
2R
(λ
0
d
)
the absorption coefficient K
0
d
0
d
finally have
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2
S[1ϪR (λ )]
0
d
κ(λ
d
) ϭ
.
(40)
2
R
0
(λ )
d
In an analogous way, i.e. by inserting Eq. (34) into the Kubelka-Munk
Eqs. (23) and (24) we get the zero-order approximation for the intensities
of the scattered irradiation light fluxes
(
0)
i
Ϫzκ(λ )
I
I
(z,0,λ ) ϭ I
e
(41)
ϩλ
i
i
0λ
i
(0)
i
Ϫzκ(λ )
Ϫλi(z,0,λ
i
) ϭ I0λi
R
0
(λ
i
)e
(42)
where R
irradiation wavelength λ
0
(λ
i
) is the diffuse reflectance of the unirradiated sample at the
i
and the attenuation coefficient κ(λ
i
) at this wave-
length is given by
2
0
S[1ϪR (λ )]
i
κ(λ
i
) ϭ
Ί
K
0
i
(λ)(K
0
(λ
i
) ϩ 2 S) ϭ
.
(43)
2
R
0
i
(λ )
Substituting for the intensity profiles from Eqs. (41) and (42) into Eq. (30)
within the next iteration cycle we find the following expression for the first-
order approximation of βλi
(
1)
i
Ϫtu exp(Ϫzκ(λ ))
λ
i
β
λ
(z,t) ϭ β i
λ
(ϱ) {1Ϫe
where the abbreviation
u i ϭ p i δI0λi[1 ϩ R (λ )]
i
}
(44)
λ
λ
0
i
(45)
has been used. So as to calculate the first-order approximations of the ir-
radiances of the scattered analysing light then the zero-order approximations
(35, 36) of these intensity profiles as well as the first-order approximation
(
44) of the extent of conversion are to be inserted in the right-hand sides of
the integral Eqs. (31) and (32) where the coefficients b
must be taken at the detection wavelength λ . Thus we get
1
and b
2
, of course,
(46)
d
z
(
1)
I
i(z,t,λd)
ϩλ
d
Ϫz′κ(λ )
ϭ 1 ϩ (SR
0
(λ
d
) Ϫ b
1
(λ
d
))
͐
dz′ e
I
0λd
0
z
Ϫz′κ(λ
d
)
iexp(Ϫz′κ(λ
λ i
Ϫtu ))
Ϫ b (λ )β (ϱ)
͐
dz′e
[1 Ϫ e
]
2
d
λ
i
0
and
z
(
1)
I (z,t,λ )
λ
i
d
d
Ϫz′κ(λ )
ϭ R i
λ
(t,λ
d
) ϩ (b
1
(λ
d
)R
0
(λ
d
) Ϫ S) 0
͐
dz′e
(47)
I
0λd
z
Ϫz′κ(λ
d
)
iexp(Ϫz′κ(λ
λ i
Ϫtu ))
ϩ b
2
(λ
d
)R (λ
0
d
)β i
λ
(ϱ) 0
͐
dz′e
[1 Ϫ e
]
In the same way also the first-order approximations for the intensity pro-
(
1)
(1)
files I (z,t,λ ) and I (z,t,λ ) of the irradiation light may be calculated
ϩλ
i
i
Ϫλ
i
i
which enable the second-order approximation of the extent of conversion
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58
R. Gade and Th. Porada
to be subsequently obtained from Eq. (30). The hitherto unknown reflec-
tance value R i(t,λ ) in Eq. (47) can be determined if the limiting approach
z → ϱ is applied to this equation and the boundary condition (29)
λ
d
lz i→m ϱ IϪλi(z,t,λ ) ϭ 0
d
is taken into account. At the end of this iteration cycle, thus, the time-
dependence of the diffuse reflectance R (t,λ ) can be expressed as
λ
i
d
R i
λ
(t,λ
d
) ϭ Q1λi(λ
d
) Ϫ Q2λi(λ
d
)P(v,u i
λ
,t)
(48)
(49)
where the function P(v,uλi,t) has the integral representation
ϱ
1
Ϫκ(λ
d
)z′ Ϫtu
e
λ
iexp(Ϫκ(λ
i
)z′)
vϪ1 Ϫtu
λ
iw
P(v,u i
λ
,t) ϭ κ(λ
i
)
͐
dz′e
ϭ
͐
dw w
e
0
0
v denotes the ratio
κ(λ
d
)
v ϭ
κ(λ
i
)
and the parameters Q1λi and Q2λi are given by
SϪR (λ )[b (λ ) ϩ b (λ )β (ϱ)]
0
d
1
d
2
d
λ
i
Q (λ ) ϭ
(50)
1
λ
i
d
κ(λ )
d
R (λ )b (λ )β (ϱ)
0
d
2
d
λ
i
Q
2λi(λ
d
) ϭ Ϫ
.
(51)
κ(λ)
i
The illustrative meaning of the parameters Q1λi and Q2λi becomes clear if
we let t approach 0 and ϱ, respectively, in Eq. (48). Because of
lim P(v,u ,t) ϭ 0
λ
i
t→ϱ
we get in the former case
Q (λ ) ϭ R (ϱ,λ )
(52)
1
λ
i
d
λ
i
d
whereas taking into consideration that
1
lt i→m 0 P(v,uλi,t) ϭ
v
in the latter the relationship
Q
2λi(λ
d
) ϭ v[R i
λ
(ϱ,λ
d
) Ϫ R i
λ
(0,λ )]
d
(53)
follows.
Hence, formula (48) discussed above has a similar form as Eq. (1)
describing the irradiation-time dependence of the absorbance of a stirred
solution in which a reversible photoisomerization proceeds.
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159
2
.3 Determining quantum yields and absorption coefficients
of the products from reflectance-time curves
According to a proposal previously made by Fischer [13] the quantum
yields of a reversible photoisomerization in a stirred solution as well as the
absorption coefficients of the photoproduct can be calculated if the sample
absorbances in two different photostationary states are known. Unfortu-
nately, the photostationary states of many photoisomerizations, however,
cannot be attained because of the occurrence of side reactions. In such a
case we can gain the spectroscopic data needed for a kinetic analysis by
illuminating the powdered absorbent-adsorbate successively with light of
two different wavelengths λ ϭ λ ,λ and measuring in each case a set of
i
1
2
diffuse reflectance spectra R (t ,λ) of the irradiated powder sample for a
λ
i
n
sequence of increasing exposure times t (n ϭ 1, ...,N). From each of the
n
two series of spectra assigned to different irradiation wavelengths λ ϭ λ ,λ
i
1
2
afterwards two reflectance-time curves R i
λ
(t
n
,λ
d
) were extracted at the detec-
tion wavelengths λ ϭ λ , λ so that altogether four such curves are avail-
d
1
2
able. A least-squares fit of the points {t , R (t ,λ )} of these experimental
n
λ
i
n
d
reflectance-time curves to the analytical expression (48) describing the ir-
radiation-time dependence of the diffuse reflectance then enables the un-
known parameters Q (λ ), Q (λ ) and u of this expression to be deter-
1
λ
i
d
2λ
i
d
λ
i
mined. Taking into account the definition (51) of the parameter Q2λi and, in
addition, Eq. (21) for the parameters β (ϱ) appearing there we are able to
λ
i
establish the following six equations
Q (λ )κ(λ )
2
λ
1
1
1
ϭ [ε
ϭ [ε
ϭ [ε
ϭ [ε
A
A
A
A
(λ
1
) Ϫ ε
) Ϫ ε
) Ϫ ε
B
B
B
B
(λ
1
)]β 1
λ
(ϱ)
(ϱ)
(ϱ)
(ϱ)
(54)
(55)
(56)
(57)
(58)
(59)
R
0
(λ
1
)δc
0
Q
2λ2(λ
2
)κ(λ
)δc
Q (λ )κ(λ )
2
)
(λ
2
(λ
2
)]β 2
λ
R
0
(λ
2
0
2
λ
1
2
1
(λ
2
(λ
2
)]β 1
λ
R
0
(λ
2
)δc
0
Q (λ )κ(λ )
2
λ
2
1
2
(λ
1
) Ϫ ε
(λ
1
)]β 2
λ
R
0
(λ
1
)δc
0
B
A
Ϫ1
ε (λ ) Φ
B
1
β 1(ϱ) ϭ 1 ϩ
λ
ͫ ͩ ͪͬ
ͫ ͩ ͪͬ
A
B
A
ε (λ ) Φ
1
B
A
Ϫ1
ε (λ ) Φ
B
2
λ
β 2(ϱ) ϭ 1 ϩ
A
B
A
ε (λ ) Φ
2
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60
R. Gade and Th. Porada
obviously not being linearly independent. Dividing Eq. (54) by Eq. (57),
Eq. (56) by Eq. (55), and adding the resulting equations we arrive at
Q
2λ1(λ
2λ2(λ
1
)
Q
2λ1(λ
2
) κ(λ1)
β 1
λ
(ϱ)
(ϱ)
ϩ
ϭ 2
.
ͩ
ͪ
Q
1
)
Q2λ2(λ
2
) κ(λ
2
)
β 2
λ
Thus, by combining the last equation with Eqs. (54), (55), (58), (59) we
may extract the following five linearly independent equations
a 1
λ
(λ
1
) ϭ [ε
A
(λ
1
) Ϫ ε
B
(λ
1
)]β 1
λ
(ϱ)
(60)
a (λ ) ϭ [ε (λ ) Ϫ ε (λ )]β (ϱ)
(61)
λ
2
2
A
2
B
2
λ
2
ε (λ )
B
1
1
1
ϭ 1 ϩ
φ β (ϱ)
(62)
ͫ
ͬ
λ
1
ε
A
1
(λ )
ε (λ )
B
2
ϭ 1 ϩ
φ β (ϱ)
(63)
(64)
ͫ
ͬ
λ
2
ε
A
2
(λ )
β 1
λ
(ϱ) ϭ rβλ2(ϱ)
where for the sake of clearness the abbreviations
Q
2λ1(λ
1
)κ(λ
1
)
a 1
λ
(λ
1
) ϭ
) ϭ
(65)
(66)
R
0
(λ
1
)δc
0
Q2λ2(λ
2
)κ(λ
2
)
a 1
λ
(λ
1
R
0
(λ
2
)δc
0
1
Q
2λ
1
(λ
1
)
Q
2λ
1
(λ
2
) κ(λ1)
r ϭ
ϩ
(67)
(68)
ͩ
ͪ
2
Q2λ2(λ
1
)
Q2λ2(λ
2
) κ(λ
2
)
B
A
B
φ ϭ (Φ /Φ )
A
have been introduced. Solving the nonlinear, algebraic Eqs. (60)Ϫ(64) for
the unknown variables ε
B
(λ
1
), ε
B
(λ
2
), β 1
λ
(ϱ), β 2
λ
(ϱ), and φ by means of a
substitution method finally yields
(
r Ϫ 1)ε (λ )ε (λ )
A
1
A
2
φ ϭ
(69)
a 1
λ
(λ
1
)ε
A
(λ
2
) Ϫ ra 2
a 1(λ
(λ ) Ϫ ε
λ
(λ
2
)ε
A
(λ
1
)
λ
1
)ε
A
(λ
2
) Ϫ a 2
λ
(λ
2
)ε
A
(λ
1
)
β 2
λ
(ϱ) ϭ
(70)
(71)
(72)
ε
A
(λ
2
)(a 1
λ
1
A
(λ
1
)) Ϫ rε (λ
A
1
)(a 2
λ
(λ
2
) Ϫ ε
A
2
(λ ))
β (ϱ) ϭ rβ (ϱ)
λ
1
λ
2
a (λ )
λ
1
1
ε
B
B
(λ
1
) ϭ ε
A
A
(λ
1
) Ϫ
) Ϫ
β 1
λ
(ϱ)
a (λ )
λ
2
2
ε
(λ
2
) ϭ ε
(λ
2
.
(73)
β 2
λ
(ϱ)
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Quantitative Determination of Photokinetic Parameters for Reversible ...
161
Here the absorption coefficients ε
calculated from the reflectance spectrum R
A
(λ) (λ ϭ λ
1
, λ
2
) of the reactant are directly
0
(λ) of the unirradiated adsor-
bent-adsorbate sample by using a previously reported statistical model of
coated particles [16]. Provided that the absorption coefficients ε (λ ) and
B
1
ε
B
2
(λ ) of the photoproduct have already been obtained from Eqs. (72), (73)
a favourable way to determine the quantum yields consists in using the
quantity
A
B
A
p i
λ
ϭ ε
A
(λ
i
)Φ ϩ ε
B
(λ
i
)Φ ϭ Φ [ε
A
(λ
i
) ϩ ε
B
(λ
i
)φ]
(74)
B
A
B
which may be calculated according to Eq. (45) from the parameter u i
λ
of
the experimental reflectance-time curves. Taking into account Eq. (68) and
the right-hand side of Eq. (74) we find for the quantum yields the relation-
ships
p i
λ
A
Φ ϭ
(75)
B
ε
A
(λ
i
) ϩ ε
B
i
(λ )φ
B
A
A
B
Φ
ϭ φΦ
(76)
B
A
A
B
where the ratio φ ϭ Φ
/Φ
is assumed to be already known from Eq. (69).
3. Experimental details
To test the method developed above we have used it to determine the quan-
tum yields of the photoinduced cis-trans-isomerization of azobenzene ad-
sorbed on disperse silica gel as well as the absorption coefficients of the
photoproduct cis-azobenzene. Here, azobenzene has been choosen as an
appropriate adsorbate because its photoisomerization can be sufficiently
well observed in the adsorbed state too and, on the other hand, no fluores-
cence could be measured “brightening’’ the powdered sample and, hence,
enhancing the diffuse reflectance.
The adsorbate azobenzene was obtained from Merck (Darmstadt) and
used without further purification. To prepare the adsorbent-adsorbate
samples needed the azobenzene was dissolved in n-hexane (Merck, UVSOL
grade) and adsorbed on the disperse silica (Merck, particle size 15Ϫ20 µm)
from this solution. For this purpose, hexanic solutions of azobenzene with
different concentrations were mixed with 4.8 g of the adsorbent and the
resulting suspensions have been stirred for 2 hours. After the adsorption
equilibrium had been established the solution was sucked from the suspen-
sion and the powder was dried at 40 °C.
The diffuse reflectance measurements were performed by means of a
SPECORD M40 UV/VIS-spectrophotometer (Carl Zeiss Jena) equipped
with a (45°/0°) diffuse reflectance accessory using a pressed powder sample
of pure MgO (Merck) as reference.
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R. Gade and Th. Porada
Fig. 2. Adsorption isotherm of trans-azobenzene adsorbed on silica gel from n-hexane.
To study the photoisomerization of the adsorbed azobenzene the powder
samples were monochromaticly illuminated with a home-made photochem-
ical irradiator consisting of a mercury lamp HBO200, a heat protection
accessory and a interference filter for selecting the monochromatic light of
the corresponding wavelength. The light intensities required for the calcu-
lation of the quantum yields were determined by using both a calibrated
pyroelectric detector RS5900 (Polytec) and a chemical actinometer.
4. Results and discussion
A prerequisite for the application of the statistical model [16] used in sect.
.3 to determine the absorption coefficients of the adsorbed azobenzene is
2
the existence of a monomolecular layer of the adsorbate on the surface of
the adsorbent particles. That is why we have first measured the adsorption
isotherm shown in Fig. 2 to elucidate the way in which the azobenzene is
adsorbed on the disperse silica gel. Since all our adsorbent-adsorbate
samples prepared as described in sect. 3 are within the linear range of the
adsorption isotherm the adsorbent particles are assumed to be coated with
a submonomolecular azobenzene layer so that the afore-mentioned pre-
requisite for using the statistical model [16] is fulfilled.
After that the powdered adsorbent-adsorbate samples were step by step
irradiated with monochromatic light of the wavelengths λ ϭ 297 nm and
1
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Quantitative Determination of Photokinetic Parameters for Reversible ...
163
Fig. 3. Diffuse reflectance spectra of an adsorbent-adsorbate sample irradiated with light
of λ ϭ 297 nm wavelength (dotted line: spectrum of the pure silica gel; ∆t ϭ 20 s).
i
λ
2
ϭ 313 nm respectively and N diffuse reflectance spectra were measured
of each of the irradiated samples characterizing its state after each irra-
diation step t ; n ϭ 1,...,N. In Fig. 3 and Fig. 4 two such sets of spectra
n
are illustrated referring to the samples with the highest adsorbate molality.
Finally, two R (t,λ )-curves were extracted at the detection wavelengths
λ
i
d
λ ϭ λ ,λ from both these series of spectra assigned to the irradiation
d
1
2
wavelengths λ
i
ϭ λ
1
2
,λ so that a total of four such reflection-time curves
are obtained which are shown in Fig. 5. Using these data and the procedure
described in sect. 2.3 we could then calculate the quantum yields and ab-
sorption coefficients which are summarized in Table 1.
From Table 1 it can be seen that the absorption coefficients of the ad-
sorbed azobenzene isomers calculated at the wavelengths λ ϭ 297 nm and
1
λ ϭ 313 nm hardly differ from those ones measured in solution. In contrast
2
to this the photoisomerization quantum yields calculated in the adsorbed
state differ remarkably from those ones determined in n-hexane as it is
shown in Table 2.
Studying the photoisomerization of aromatic azocompounds adsorbed
on silica Kerzhner and coworkers [17] arrived at similar results. They found
that in the adsorbed state both the reaction rate of the trans-cis-photoisomer-
ization and the extent of conversion in the photostationary equilibrium are
considerably higher than in a nonpolar solvent like pentane. Since in con-
trast to pentane the surface of the silica gel particles represents a strongly
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64
R. Gade and Th. Porada
Fig. 4. Diffuse reflectance spectra of an adsorbent-adsorbate sample irradiated with light
of λ ϭ 313 nm wavelength (dotted line: spectrum of the pure silica gel; ∆t ϭ 5 s).
i
Fig. 5. Reflectance-time-curves of the adsorbent-adsorbate samples irradiated with light
Ϫ7
of different wavelengths λ (azobenzene molality ϭ 1.12 · 10 mol/g).
i
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Quantitative Determination of Photokinetic Parameters for Reversible ...
165
Table 1. The calculated quantum yields of the partial reactions and the absorption coef-
ficients of the reactant and the product respectively at the wavelengths λ ϭ 297 nm and
1
Ϫ10
Ϫ2 Ϫ1
Ϫ9
Ϫ2 Ϫ1
λ ϭ 313 nm (I0λ ϭ 4.4 · 10 Einst cm s , I0λ ϭ 1.15 · 10 Einst cm s ).
2
1
2
A
B
A
B
Molality
Φ
Φ
ε
B
(λ
cm
1
)
ε
B
(λ
cm
2
)
ε
A
(λ
cm
1
)
ε
A
(λ )
cm
2
Ϫ7
2
2
2
2
[10 mol/g]
ͫ
ͬ ͫ ͬ ͫ ͬ ͫ ͬ
mmol
mmol
mmol
mmol
0.9354
1.1243
1.3054
1.6725
1.8694
0.20
0.20
0.20
0.21
0.19
0.17
0.13
0.15
0.10
0.15
6100
7900
7300
7800
7200
1500
18900
20000
19400
20000
19200
31000
32800
31700
33000
31600
3300
2600
3000
1900
Table 2. Comparison of the photoisomerization quantum yields of azobenzene solved in
n-hexane with those of azobenzene adsorbed on silica gel.
a
b
Azobenzene adsorbed on silica gel
Azobenzene solved in n-hexane
A
B
B
A
A
B
B
A
Φt→c ϭ Φ
Φ
c→t ϭ Φ
Φt→c ϭ Φ
Φ
c→t ϭ Φ
0.2
0.14
0.1
0.4
a
see Table 1.
see Ref. [12].
b
polar milieu it seems to be reasonable to explain the observed differences
of the quantum yields by polarity effects. Usually the polarity of solvents
may be characterized not only by the dielectric constant but also by a num-
ber of other parameters among them e.g. by the Kossower number Z [18].
Based on UV-spectroscopic investigations of various solvatochromic com-
pounds adsorbed on silica gel Nicholls and Leermakers [19] determined a
Kossower number Z ϭ 88 for weakly hydrated silica gel. This Z-value is
higher than that of methanol (Z ϭ 83.6) and lower than that of water (Z ϭ
9
6.4). On the other hand studying the solvent effects on the photoisomeri-
zation of azobenzene Bortolus and Monti [20] determined the following
values of the photoisomerization quantum yields Φt→c and Φc→t which are
compiled in Table 3.
This table clearly points out that with increasing polarity of the solvent
the trans-cis-photoisomerization quantum yield Φt→c increases whereas the
cis-trans-photoisomerization quantum yield Φc→t decreases. Since according
to the above considerations the silica gel is expected to have a similar po-
larity as the ethanol-water mixture does the photoisomerization quantum
yields Φt→c and Φc→t calculated by means of our method (cf. Table 1) fit in
with the polarity dependence of Table 3.
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66
R. Gade and Th. Porada
Table 3. Dependence of the photoisomerization quantum yields of azobenzene on the
solvent polarity according to Bortolus and Monti [20].
A
B
B
A
Solvent
Dielectric
constant
Φ
t→c ϭ Φ
Φc→t ϭ Φ
Ethylbromide
Ethanol
Acetonitrile
9.2
24.3
37.5
74
0.115
0.150
0.158
0.210
0.255
0.240
0.210
0.150
2
H O/Ethanol
The dependence of the quantum yields Φt→c and Φc→t on the polarity of
solvents found by Bortolus and Monti [20] can be explained by the fact that
cis-azobenzene has a nonzero electric dipole moment whereas that of trans-
azobenzene vanishes by reason of symmetry. In a nonpolar solvent the
ground state of the cis-isomer is known to be above that of the trans-isomer
because of steric reasons. With increasing polarity of the solvent, however,
the ground state of the cis-isomer will be more and more stabilized relative
to that of the trans-isomer due to the polarization effect of the molecular
dipole. Hence, in polar solvents the rate of the trans-cis-photoisomerization
becomes greater and the rate of the reverse process becomes smaller so that
the quantum yields of the photoisomerizations Φt→c and Φc→t will alter in
the way exhibited in Table 3.
On the other hand, according to Kerzhner et al. [17] the different ad-
sorptive interactions between the azobenzene isomers and the adsorbent ac-
count for the observed difference in the quantum yields shown in Table 2.
Apparently, in solution of nonpolar solvents both the isomers interact only
weakly with the solvent molecules so that the relative positions of their
energy levels will hardly be altered and particularly the trans-isomer re-
mains the more stabilized one. In the adsorbed state, however, the cis- and
the trans-isomer show a quite different behaviour. While the adsorption of
the planar-shaped trans-isomer is realized by the weak interaction of the π-
electrons of the azobenzene with the electron acceptor centers of the silica
gel surface the adsorption of the nonplanar cis-isomer is due to the forma-
tion of strong hydrogen bonds between the n-orbitals of the nitrogen atoms
and the proton-donor centers of the adsorbent. These hydrogen bonds can
also be considered as an acid-base interaction between the azo-group of the
adsorbate having a rather high basicity in this case and the Broensted acid
centers on the surface of the silica gel.
Since the adsorptive interaction of the cis-isomer is much stronger than
that of the trans-isomer the molecular state of the former will be lowered
with respect to the latter. Consequently, the quantum yield of the trans-cis-
photoisomerization will be greater and that of the cis-trans-photoisomeriz-
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Quantitative Determination of Photokinetic Parameters for Reversible ...
167
ation will be smaller than the corresponding quantum yields measured in
solution of nonpolar solvents. This prediction is in good agreement with the
quantum yields collected in Table 2.
Acknowledgement
The research work described herein was supported by a grant of the Deut-
sche Forschungsgemeinschaft which is gratefully acknowledged.
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