Bodenstein hypothesis

The Bodenstein hypothesis [13] can also be called a hypothesis of the stationary or quasi-stationary state. It makes the following assumptions  

In contrast to thermal reactions, any photochemical reaction is accompanied by a number of photophysical processes which all have to be taken into account in the reaction scheme. Most of these processes are thermal reactions. The mechanism of the photoreaction and - as we will see in the examples - the photochemical quantum yield depends on these photophysical steps. One of the most simple photoreactions is a photo-isomerisation 

As in the case of thermal reactions, the reaction scheme introduced in Section 2.1.1.1 can be used to set up the differential equations. However, the degrees of advancement are primed, since the number of steps can be reduced as will be demonstrated by use of the Bodenstein hypothesis. In the last column of this scheme, the number of moles of light quanta are written for a photochemical step, which are absorbed by the reactant starting this photochemical step. According to this assumption and the different photophysical relaxation processes discussed in Section 1.3 the primary exited molecule A completely deactivates into the lowest level of vibrational energy of the first exited singlet state. Three further steps are possible  

Reactant A reacts by step 1 and will be re-formed by reactions 3 and 4. In consequence, the change of A in dependence on time is (summation of all the products stoichiometric coefficient in the first row times the element of the last row reactions 1,3,4) 

During this process all the molecules of reactant A will react, since they absorb light. The amount of light absorbed is given according to the definition 7a in mole 1 S . The rate laws for the photophysically exited molecules can be given accordingly 

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These derivations use matrix equations and approximations as, for example, the Bodenstein hypothesis, as well as other relationships typical of formal kinetics. All these approaches are found in Chapters 2 and 3. The intention is to introduce this formalism to the reader by help of examples and by rather simple mechanisms. 

Bodenstein hypothesis can be applied to instable intermediates are complex photoreactions. 

Under these conditions n is the number of reactants which start any photoreaction and n is the total number of all reactants, is the number of photoreactions, sr - s the number of thermal reactions to which the Bodenstein hypothesis (see Section 2.1.3.2) cannot be applied. 

The approaches to use the reaction scheme introduced in thermal kinetics can be very helpful and allow the transformation of the mechanism into a differential equation. All the elementary processes have been included in the scheme. For this reason, constants of these elementary processes can be explicitly determined from the quantum yields by these means. However, many of these steps cancel out by using the Bodenstein hypothesis (see Section 2.1.3.2) and only effective steps of the reaction are found. Nevertheless, the total scheme explains dependencies of quantum yields on concentration as discussed in Chapter 3 and in the following example. 

Conclusion Most of the photophysical steps are thermal reactions, for which the Bodenstein hypothesis is valid. The photochemical quantum yield depends on these elementary steps. 

A thermal reaction with just a single elementary step is call simple uniform in the case of a photochemical reaction elementary reactions for which the intermediates can be omitted because of the Bodenstein hypothesis are taken also as a simple one. Examples are both the photoreactions given in Section 2.1.3.3. 

The reduction of the reaction scheme because of linear dependencies between partial steps works for photoreactions in the same way as for thermal reactions. In addition it has to be considered that intermediates are negligible and allow the application of the Bodenstein hypothesis. Therefore the number of steps reduces to one per photoreactive step. Between these steps, linear relationships can happen. Photoreactions and true thermal reactions (not the photophysical steps) can exhibit linear dependencies. The simplest case is a photoisomerisation superimposed on a thermal back-reaction. 

Conclusion (a) Photochemical steps of reaction include many thermal degradation processes, which do not need to be considered in the rate law, if the Bodenstein hypothesis is valid, (b) The mechanism has to be reduced as far as possible to avoid linear dependencies, (c) Thermal and photochemical reactions can be treated in principle by the same formalism. Rate constants times concentrations have to be substituted by the product of the partial photochemical quantum yield times the amount of light absorbed, which contains the concentration of the reactant which starts the photoreaction. 

A more general solution is offered by the Bodenstein hypothesis, which was introduced in Section 2.1.3.2, and this is treated in more deUiil next. 

This solution obtained by the application of the Bodenstein hypothesis can be used in a more generalised form given in Section 2.5.1. It is true that the concentration of B is very small. But it has to be multiplied by the rate constants 2 which can reach very high values. 

The validity of the Bodenstein hypothesis has been discussed in great detail (see e.g. [2]). However, it has proven its applicability in some cases. Besides, its correctness has been proven in systems which can be mathematically treated in a closed form. This point can be demonstrated in the example given above. The exact solution is derived by eq. (2.92). Using the eigenvalues r, and r, in the form... 

The differential equation makes it obvious that the Bodenstein hypothesis becomes valid just after a certain period of induction. In the model reaction examined above one obtains under the conditions f = 0, a = aQ, fe = c = 0 the following time-dependencies ... 

These equations violate the second assumption of the Bodenstein hypothesis (changes in the concentration of intermediates are negligible compared with those of the initial or final products). A further disadvantage of the Bodenstein hypothesis is that one cannot predict how large the difference between the two rate constants has to be to restrict the error to a limited value by using the approximation. For further details see Section 2.5.4. 

If B and E are unstable short-lived intermediates and the Bodenstein hypothesis can be applied, the following relationships are obtained, since b = Q ... 

The second assumption (z = 0) of the Bodenstein hypothesis as a consequence of the first (e = 0) is reasonable. Just arguing from a mathematical point of view assumption 1 is neither a necessary nor sufficient condition for the validity of assumption 2, For this reason in the case of the method of quasi-stationary variables one avoids assumption 1 and takes for the changing concentration... 

If the given criterion is applied to the chain reaction of Section 2.5.2.1, some very complex expressions are obtained. They make it obvious that the Bodenstein hypothesis fails at the end of the reaction for Cq>Oq. For this reason the chosen approximate solution of the system can no longer describe the progress of the reaction for large turnovers. For this reason the reaction does not come to an end in a finite time. 

In contrast to a treatment according to the Bodenstein hypothesis the changes b of concentration with time are finite. That means the factor... 

If either the Bodenstein hypothesis can be applied to a reaction system or one of the steps is an isomeric equilibrium, the number of linear independent steps of reaction is reduced. The concentration of an unstable intermediate is according to the Bodenstein hypothesis with z = 0 ... 

B and E are intermediates in the radical chain reaction treated in Section 2.S.2.1. The Bodenstein hypothesis can be applied. By use of... 

The photoaddition via the singlet state results by application of the Bodenstein hypothesis to the photophysical intermediates in the rate law... 

The first line of the scheme notes x, = rl/. This term takes into account that the radicals D which are produced in the photochemical primary process stay in a solvent cage and can recombine to form A. For this reason the fraction of quanta causing photoreaction becomes r< 1. Applying the Bodenstein hypothesis to all intermediates the following relationships can be derived ... 

In this type of reaction a single reactant which is excited by absorption of light is able to start more than one gross reaction. The requirement is that the Bodenstein hypothesis can be applied to all intermediates. Whereas in the previous section in the case of the treated parallel photoreactions the gross... 

Furthermore according to the Bodenstein hypothesis it is assumed that the short living intermediates do not absorb light and the gross reactions are formed as described in the previous sections. 

Derivations assume that reactant A absorbs the radiation and starts the photoreaction. Napierian units are used. Intense stirring of the solution is required. The Bodenstein hypothesis is valid for all the photophysical intermediates. 

Photoreactions are called complicated if more than one component starts the photo-reaction by absorption of light and/or if slow dark reactions have to be taken into consideration, to which the Bodenstein hypothesis cannot be applied. These complicated reactions can be treated straightforwardly, if... 

Only two of the four reactions are linear independent according to Section 2.1.4. The reaction can even be described by a single degree of advancement, if for example B is a short lived intermediate, which can be treated by use of the Bodenstein hypothesis. The experimentally determined concentration-difference matrix contains the following elements (each column correlates to a different time of reaction) ... 

A, fra 5-azobenzene B, cw-azobenzene C, dihydrobenzo[c]cinnoline D, benzo[c]cinnoline E, hydrazobenzene F, benzidine all the compounds are protonised). The Bodenstein hypothesis can be applied to the intermediates C and E, since the thermal reactions 4 and 5 are very fast. Accordingly... 

Therefore as in the case of the singlet pathway, one partial reaction can be omitted in the scheme. Application of the Bodenstein hypothesis to the intermediates A and A" gives... 

It is assumed that intermediates have very short lifetimes and the Bodenstein hypothesis is valid. The scheme reduces because of the linear dependency of the first two steps according to Appendix 6.3.1 to... 

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