Photokinetic equations

Formal integration of these equations is possible using 

The different coefflcients allow us to calculate the partial photochemical quantum yields knowing the absorption coefficients. Using the absorption coefficients for the three reactants at the wavelength of irradiation (313 nm) =15900, = 1320 l mol cm the three quantum yields are obtained as 

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A different approximation is based on the use of very dilute solutions in which the absorbance is smaller than 0.02 units (A <0.02). In this case, the photokinetic factor is considered constant as an expansion into series. Accordingly, the factor exhibits a value of 2.303, and the rate depends on the concentration in a similar way as observed in first-order (thermal) reactions. Assuming a partial absorption during the photoreaction and taking into account that the photoproducts will also absorb at the actinic wavelength, the photokinetic equations become more complicated. There are a large number of such different equations, each tailored to a specific problem, as demonstrated in the various examples in the following sections. 

This quantity appears in most of the photokinetic equations. Part of it was named the photokinetic factor. This factor can be defined using different absorption coefficients. 

As seen in Chapter 5 giving examples of application of the photokinetic equations, absorption spectroscopy in the UV/vis is the general method to monitor the reactions. Since in this spectral range the molar absorption coefficient is used in decadic units, in practice this photokinetic factor is used as... 

Nevertheless it becomes obvious that these equations become more complex. Therefore the details are handled in total in the next chapter. In Section 2.5 approximations are introduced which simplify some of the equations given and allow reduction in the expenditure of setting up photokinetic equations. 

In contrast to thermal kinetics the photokinetic equations given above cannot be solved in a closed form, since the photokinetic factor depends on the absorbance of the solution itself and is consequently dependent on time. In the literature approximations either with respect to total absorbance or to very dilute solutions are used. The consequences are discussed in detail in Section 3.3.3. Using the conditions... 

Photokinetic equations for different reactions 5.5.2.7 Uniform reactions... 

Since the layer is extremely thin, Bernoulli s differential equations (see Section 3.4.1) reduce to the normal photokinetic equations (see eq. (5.107)). The boundary conditions of constant /abs.A as well as constant a z,t) within the z-direction of the measurement beam can be used as a first approximation. 

If the fulgide given in the reaction scheme in Section 5.6.3 is connected to an oligomere backbone and mesomeric groups, liquid crystals are formed which exhibit photoreactivity. Application of photokinetic equations and methods given in Sections 5.6.3, 5,7.2.1 as well as 5.7.3.1 allows an evaluation even above the glass temperature. According to eq. (5.172) the system of differential equations... 

Since photokinetic equations cannot be solved in a closed form, equations are preferably rearranged due to allow formal (numerical) integration of the integrals. It is best to choose a matrix representation. The elements of the Jacobi matrix ate obtained by... 

This equation contains three terms (i) AB oeA which is constant under given conditions of irradiation, (ii) the concentration of the photosensitive reagent, [A], and (iii) the photokinetic factor16-19... 

Approximate integration of Equation 3.22 is possible by linear interpolation of the reciprocal photokinetic factor,222 F(Xia,t) (Equation 3.23). 

The next paragraphs focus on the most recent advances in electron photodetachment processes in aqueous ionic solutions. Interesting results on ultrafast UV-IR spectroscopy of photoexcited aqueous chloride ions are presented in Figure 5-8. A complex photokinetic model of time-resolved data has been considered and explained in detail in recent pubhcations (85, 86). The primary photophysical and photochemical events triggered by one- or two-photon processes can be summarized with the following equations ... 

Rate constants are determined by variation of experimental conditions and chemical composition of the reaction mixture. Data are measured by application of a variety of modem analytical methods. Modem numerical approaches of curve fitting and/or solution of differential equations are applied. Results and consequences influence chemical reaction engineering as well as production costs. Many books cover these formal thermal kinetics in detail, but most are restricted to simple mechanisms. In contrast, analogous treatments of photochemical reactions are restricted to publications of special reactions and examinations. Therefore this book aims to supply an overall treatment of formal photokinetics beyond the scope of normal books on kinetics. 

The change in concentration of reactants is at the centre of interest in photokinetics as well as the determination of these partial photochemical quantum yields. The time laws cannot be integrated in a closed form. Therefore to avoid the problems with solving these differential equations, the integrals are numerically calculated - a procedure named formal integration. This method also turns out to be advantageous in thermal and photochemical examinations. 

The photokinetic differential equations for the changes of the concentration of the reactant with time cannot be integrated in the given form. The reason is that the amount of the light absorbed I, of a reactant Aj per time and volume unit depends on the volume element where the reaction takes place and on the time of irradiation r in a very complex manner. This is one of the essential difficulties in photokinetics. This /, only can be calculated if the... 

Derivation of photochemical equations can become easier using the Napierian molar absorption coefficient. This is discussed in detail in Section 5.1.2.2. The result is the photokinetic factor given by... 

EXPLICIT DIFFERENTIAL EQUATIONS IN PHOTOKINETICS 3.2.1 Rates of simple uniform photoreactions... 

Based on the derivation in Section 1.4.3 of the amount of light absorbed and the equations derived therein for a simple photokinetic formalism, rates and related photochemical quantum yields are discussed in further detail in the following sections. 

The equation above gives only the simplest case where the irradiation is incident perpendicularly and the medium is not absorbing. However, in photokinetics normally conditions where the sample shows absorption are of interest. For this reason another equation has to be used, given by... 

The true differential quantum yields can no longer be localised in the integrated equations of photokinetics. The different rate constants are distributed to different terms in the equation. On the other hand one can correlate quantum yields in the differential equations. If the quantum yield does not depend on the intensity of the irradiation source on finds according to eq. (3.35)... 

Many photochromic systems can be handled by the equations given in the previous section. They represent essential examples of photochemical reactions. Their principles and their applications are discussed in detail in the literature. A good review is given in Ref. [162], Typical examinations deal with the photokinetics of fulgides [163-165] and spiropyrans [166-168]. 

The non-selectivity of UVA is-spectra with respect to the reactants participating in a reaction causes problems in the photokinetic evaluation. That means the number of partial reaction steps together with the number and sequence of the reactions products has to be determined indirectly. A mechanism has to be assumed and differential equations have to be set up. These have to be verified by a correlation of simulated data according to the pro-... 

After derivation of the principles of kinetic examinations and especially the fundamentals of photokinetics in this final chapter a large number of examples have been discussed based on the equations derived in Chapters 2 and 3. These examples cover a wide field of types of photochemical reactions that take place in various applications. By use of different types of equipment, it was demonstrated how relevant data can be obtained during the reaction. This knowledge was applied to calculate reaction constants as partial photochemical quantum yields or at least the data for a turn over, if the spectroscopic or other characteristics of the compounds involved in the reaction are not known in detail. 

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