The Stationary State Method

The rather long lifetimes of many triplet states often allow an appreciable buildup of the triplet-state population under constant illumination. Already in 1954 Craig and Ross 33) measured reliable T—T spectra on a single-beam recording spectrophotometer. The observation of new absorption bands in the visible part of the spectrum is not difficult and there is satisfactory agreement between the peak positions reported from different laboratories. The determination of triplet extinction coefficients requires additionally a knowledge of the stationary triplet concentration, which is much harder to obtain. The corresponding literature values are widely scattered. 

Simultaneous measurements of the T—T spectrum and of the (Am =2) ESR signal have been performed by Brinen under steady-state excitation. The triplet concentration can be determined independently from the spectral measurements by comparing the integrated (Am =2) signal with a calibration radical signal. 

A spectrophotometer which allows spectroscopic and kinetic measurements to be made on a light irradiated sample has been developed by Ranalder et al. 5). The instrument is completely controlled by a small PDP-8/I computer. Great flexibility is introduced through software control. Several data collection routines have been written, and methods for determining molar absorption coefficients of metastable states have been discussed. 

The method of photoselection to study the polarization of triplet-triplet transitions has been applied by El-Sayed and Pavlopoulos 6) to several poly-acenes. Let us discuss some of the results obtained on napthalene, where the -axishas been chosen along the long molecular axis and they-axis along the short one. The first very weak Si- - 5o absorption band x-polarized) is 

The investigation of the 4300—6000 A region revealed four weak bands at 4560,4916, 5312, and 5750 A which have positive and negative polarization values. It is now believed that they all belong to the expected i 2u transition, 

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If we apply the stationary-state method to the free radical species H, CH3, CHO, CH3O, and CHaOCH and assume that the chains arc sufficiently long (that is, > 10) that initiation and termination rates may be neglected in comparison to chain steps, we can calculate the following stationary-state relations, where Me = CHs,... 

In general, whenever a complicated pH dependence of k is observed which corresponds to eqns. (5) and (6), or similar equations, it is caused either by acid-base equilibria of the substrate or by a change of the rate-determining step, or by combination of both. Rate equations for systems involving a change of the rate-determining step can be formulated with the aid of the stationary state method. 

Two conditions must be met if the proton transfer step is to be slow and rate-determining (a) k must be smaller than the rate coefficient of a diffusion-controlled reaction, and (b) fe i must be smaller than ku. The first condition is fulfilled in most proton transfer processes to or from carbon. It may be met also in a proton transfer among oxygen, sulfur, or nitrogen atoms if either the acidity of the proton donor or the basicity of the acceptor is extremely low. The second condition, e.g. <1kn, can be fulfilled if the intermediate is sufficiently unstable with respect to decomposition toward the products. In several examples, ku and ft, are of the same order of magnitude, e.g. k [ ku, and the proton transfer step is partially rate-determining. For these cases, the theoretical rate equation must be derived with the aid of the stationary state method Vol. 2, pp. 352-354). 

Thus, the truly spontaneous reaction involves complex I while the product catalyzed reaction involves complex II. Applying the stationary state method for the formation and decomposition of these two complexes, the following velocity constants are obtained ... 

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

A General Method of Calculation for the Stationary States of any Molecular System S. F. Boys... 

Boys, S. F., Proc. Roy. Soc. London) A200, 542, Electronic wave function. I. A general method of calculation for the stationary state of any molecular systems." a. 

In line with the general asymptotic method, Mitropolsky introduced a further generalization18 by means of an additional independent variable r (the slow time ), whereby it becomes possible to investigate not only the stationary state, but also the transient one. 

This enlarges the scope of problems that can be treated by these asymptotic methods. For example, the important problem of nonlinear resonance could otherwise be solved only in the stationary state. With this extension it is possible to determine what happens when the zone of resonance is passed at a certain rate. Likewise, with the additional extension for the slow time it is possible to attack the problem of modulated oscillations, which has previously remained outside the scope of the general theory. 

We proceed by generalizing the stationary state approach of the previous subsection to the time domain. To that end we first consider the effects of a finite laser pulsewidth and next explore the effect of a finite time delay between the pulses. The latter signal reduces in the case of identical excitation pathways to the method of two-photon interferometry that has been studied extensively in the literature of solids and surfaces [79]. 

As was already noted in [9], the primary effect of the YM field is to induce transitions (Cm —> Q) between the nuclear states (and, perhaps, to cause finite lifetimes). As already remarked, it is not easy to calculate the probabilities of transitions due to the derivative coupling between the zero-order nuclear states (if for no other reason, then because these are not all mutually orthogonal). Efforts made in this direction are successful only under special circumstances, for example, the perturbed stationary state method [64,65] for slow atomic collisions. This difficulty is avoided when one follows Yang and Mills to derive a mediating tensorial force that provide an alternative form of the interaction between the zero-order states and, also, if one introduces the ADT matrix to eliminate the derivative couplings. 

Larsson and co-workers have used relation (18) to calculate Tjb for organic molecules in which two centers are bridged by saturated groups [65,66], and for mixed valence systems [67]. The stationary states /i and /2 are determined by a CNDO/S method, with extensive configuration interaction and use of semi-empirical parameters. The nuclear configuration Q where relation (18) is valid is adjusted so as to satisfy the delocalization property expressed by (17). These... 

For a CSTR the stationary-state relationship is given by the solution of an algebraic equation for the reaction-diffusion system we still have a (non-linear) differential equation, albeit ordinary rather than partial as in eqn (9.14). The stationary-state profile can be determined by standard numerical methods once the two parameters D and / have been specified. Figure 9.3 shows two typical profiles for two different values of )(0.1157 and 0.0633) with / = 0.04. In the upper profile, the stationary-state reactant concentration is close to unity across the whole reaction zone, reflecting only low extents of reaction. The profile has a minimum exactly at the centre of the reaction zone p = 0 and is symmetric about this central line. This symmetry with the central minimum is a feature of all the profiles computed for the class A geometries with these symmetric boundary conditions. With the lower diffusion coefficient, D = 0.0633, much greater extents of conversion—in excess of 50 per cent—are possible in the stationary state. 

The stationary state concentrations of intermediates in a static system are very small to be detected by common spectroscopic techniques. But if a strong flash oflight is used, a large concentration of the intermediates may be generated which can easily be subjected to spectroscopic analysis. Essentially it is a relaxation method, the flash duration must suitably match the decay constant of the intermediates. The technique was developed by Norrish and Porter in early fifties. 

Arnold et al.24 have, on the other hand, actually used the stationary state established to compare the quenching efficiency of added gases to that of the wall. The method will be discussed further in Section V. 

The data on kp and kt as reported in the literature differ considerably. Therefore, we conducted new studies on methyl methacrylate (MMA), benzyl methacrylate (BMA), and styrene (St) as monomers. The constants were obtained by applying the method of intermittent illumination (rotating sector) combined with stationary state methods. The viscosity of the solvents varied between 0.5 and 100 cP. No mixed solvents composed of low- and high-molecular components were used but pure solvents only, the molecules of which did not deviate very much from a spherical form (methyl formate, diethyl phthalate, diethyl malonate, dimethyl glycol phthalate, etc.). 

To give an example, Fig. 20 shows a diagram for E = 10 and various selected values of Kp fi. The dashed lines indicate the range over which multiple steady states of t/(ip) occur. Here, by means of numerical methods it is not possible to determine a unique solution of the effectiveness factor of the pellet for given conditions at the external pellet surface [91]. Which operating point will be observed in a real situation again depends upon the direction from which the stationary state is approached [91]. 

Bimolecular deactivation (pathway vii, Fig. 1) of electronically excited species can compete with the other pathways available for decay of the energy, including emission of luminescent radiation. Quenching of this kind thus reduces the intensity of fluorescence or phosphorescence. Considerable information about the efficiencies of radiative and radiationless processes can be obtained from a study of the kinetic dependence of emission intensity on concentrations of emitting and quenching species. The intensity of emission corresponds closely to the quantum yield, a concept explored in Sect. 7. In the present section we shall concentrate on the kinetic aspects, and first consider the application of stationary-state methods to fluorescence (or phosphorescence) quenching, and then discuss the lifetimes of luminescent emission under nonstationary conditions. 

A review of the semidassical method is given by Cottrell and McCoubrey [9] and by Rapp and Kassal [13]. In this method, the translational motion is treated classically, while the molecule BC is assumed to have quantized vibrational levels. By converting the force V (x) on the oscillator due to the incident atom to V (t) by utilization of the classical trajectory x(t), one may apply time dependent perturbation theory. The wave function for the perturbed system is written as a sum of the stationary-state wave functions Y (y)exp( —icojf), with coefficients ck given by... 

The most direct method of measuring the initiation rate is the determination of the incorporation rate of initiator fragments into the polymer. Experimentally this is a complicated method. The evaluation is usually based on the assumption that, in the stationary state at slow initiator consumption, i>p, uinil and v (mean kinetic chain length) remain almost constant. 

This specific feature of the stationary state of chemical systems that undergo their evolution via an arbitrary combination of only monomolec ular (or reduced to monomolecular) transformations, as well as transforma tions that are linear in respect to the intermediates, is of practical importance to simplify the analysis of complex stepwise chemical processes with the use of methods of nonequilibrium thermodynamics. 

Practically, the stationary state of the systems where the mentioned con ditions are met can be found using numerical methods by minimizing functionals of type (3.6). In some cases, this way of derivation of the sta tionary concentrations of intermediates may be preferable compared to the direct solution of differential equations (3.7). 

Let us apply the thermodynamic method for finding the conditions of arising instability of the stationary state. The kinetics of changes in the concentration of intermediate X are described by the equation... 

In a general case of catalytic stepwise reactions with a large number of catalytic intermediates, the kinetic methods are indeed preferable to ana lyze the stability of the stationary state of the system. This can be done by considering the evolution of minor fluctuations in the concentrations... 

Classifying variables into fast or slow is a typical approach in chemical kinetics to apply the method of (quasi)stationary concentrations, which allows the initial set of differential equations to be largely reduced. In the chemically reactive systems near thermodynamic equihbrium, this means that the subsystem of the intermediates reaches (owing to quickly changing variables) the stationary state with the minimal rate of entropy production (the Rayleigh Onsager functional). In other words, the subsys tern of the intermediates becomes here a subsystem of internal variables. 

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