Chemical KineticsTaylor Series

Mentor teacher: Grosu CorinaStudent:Negoescu OanaGroup:1114Series:BFirst year

Chemical KineticsChemical kinetics describes the change of species with time under the conditions of nonequilibrium. For qualitative statements, we require a reaction mechanism and the associated kinetic constants of the individual reactions. For special areas, like pyrolysis and flames, there are already sophisticated computer programs, which develop automatically kinetic models [1] and perform a data reduction [2]. Also in the field of geochemistry, such programs are available [3]. A large number of kinetic data are well known and compiled in individual tables, like Landolt Bornstein [4], and further in the review literature [5]. Also machinereadable databases were arranged, including the NIST database for chemical kinetics [6]. Version 7.0 contains over 38,000 records on over 11,700 reactant pairs. It also includes reactions of excited states of atoms and molecules [7]. Kinetic calculations are substantially more complex in comparison to the field of equilibrium thermodynamics. Formal KineticsOften, the laws of chemical kinetics, expressed in terms of differential equations, can be considerably simplified by the introduction of dimensionless variables. We show examples of dimensionless variables with regard to chemical kinetics in Sect. 11.3.8. Nowadays systems of differential equations can be solved by the use of computer programs, even by a simple spreadsheet calculator. Here, we exemplify a simple consecutive reaction of first order. We modify the scheme of Eq. (11.5) in a way that we force an addition of the component A to the reactor in periodic time intervals t0.(1)Here (t) is the Dirac delta function. The first equation of Eq. (1) has on the right-hand side two terms. If kA = 0, then we have for the first term in the sum of the Dirac delta functions

The Dirac delta function integrates to the unit step function at t = t0. Thus, the concentration of A jumps by [A] at t = t0. The same holds in general for the further terms in the sum for t = lt0; l = 2, 3, . . . . The system of Eq. (1) can still be solved by analytic methods, but initially we demonstrate the use of a numerical method. To implement a numerical method, recall the Taylor expansion

When we at first omit the sum of the Dirac delta functions, we can convert the differential equations in Eq. (1) in a system of difference equations, as(2)From this system, [A](t + t) and [B](t + t) can be readily calculated from terms that contain only terms at time t. We set discrete time steps in sufficient small fraction of t0, e.g., t0/24. The time step depends on the rate constants kA and kB. The relative change of the concentration should be in the range of a few percent in the course of a single time step. To simulate the periodic addition of A, we increase [A] by [A ], whenever the time reaches an integer multiple of t0. The numeric calculation can be performed with a spreadsheet calculator. An example for time dependence of the concentration is shown in Fig.1. The intermediate B starts from zero concentration and stabilizes after some time, hereby still fluctuating. If we change the dosage from 1 concentration units every 24 h to 0.5 concentration units every 12 h, the same amount is added to the system, however, in comparison the half amount twice in the original period. The simulation is shown in Fig.2. As expected, the stationary value of the intermediate B remains the same; however, the fluctuation becomes less in comparison to the simulation in Fig.1. We proceed now with some attempt for an analytical solution in the stationary state. The stationary state is characterized not by the common conditions d[A](t)/dt = 0 and d[B](t)/dt = 0 but by a periodic fluctuation of the concentrations.Assuming that the concentration [A](t) is periodic in time with a period of t0 in the stationary state, we can calculate the fluctuations in a simple way. Immediately

Numeric simulation of the time dependence of Eq. (1), dosage 1 unit every 24 h

Numeric simulation of the time dependence of Eq. (1), dosage 0.5 units every 12h

after feed the concentration should be [A](t). After the period t0, there is a decrease to the value [A](t + t0) = [A](t) exp(kAt0), just before the next portion [A] is added. After the addition, the concentration returns to that of [A](t). We inspect now a single period in the stationary state. For convenience, we set the time at the beginning of the period as zero, i.e.,t 0. In this notation, we get the ratio of lower to upper limiting value in the stationary state as

Further, we solve the differential equation for the intermediate [B](t) for one period with nonzero starting concentration of B, i.e., [B](0) = 0. The solution is

The concentration of B should be the same at the beginning and at the end of the period [B](0) = [B](t). From this condition, the relation arises.

Finally, we show a parametric plot of [A](t) and [B](t) from the data plotted in Fig. 18.2 in Fig. 18.3. The trajectory starts at [A] = 0.5; [B] = 0 and moves up, until it approaches to a limit cycle, or periodic orbit.

Parametric plot of [B] and [A] from the data plotted in Fig. 2

The example with periodic feed is relevant in chemical reactors. Further, it is the starting point for the kinetics of drug administration. However, in real live the situation is much more complicated. For example, in the drug administration for insulin the intake of carbohydrates interferes the kinetics [8].Stochastic Nature of Chemical KineticsIf we deal with an individual particle, then we cannot predict whether this particle will react in the next instant of time or not. In contrast, we can make statements for a large number of particles, e.g., how the concentration of a certain species in a chemical reaction changes in time.Simple Reaction with First OrderA simple reaction with first order is the decomposition of a particle, e.g., azoisobutyronitril. We inspect now a certain time interval, which we divide into small time stepst.We observe now the molecule at the times t, t+t, t+2t, . . . , t+nt. It could happen that in between the observations the molecule decomposes according to the reaction in Fig.3. The probability for this decomposition is p.(3)Decomposition of azoisobutyronitrilMarkov ChainsWe can formalize the above process in a certain way.StatesWe define the possible states: 1. The molecule is intact.2. The molecule has decomposed already.Transition MatrixDuring a small time interval t in between the observations the state of the system can change. The molecule changes from state (1) into state (2). We define now the transition probabilities in Table 1. The transition probabilities can be summarized in a scheme, as shown in Eq. (3).

(3)

According to the lower row a backward reaction is impossible, because of p21 = 0. If the molecule has decomposed, then it remains decomposed. Further, the probabilities are not dependent on the time. No matter, whether the molecule is in the condition (1) or (2), to a further transition the same probabilities apply. This is a so-called homogeneous Markov1 chain.State After Two Time IntervalsWe presume the molecule is intact at the beginning of the observations. If we find the molecule after two time intervals decomposed, then this can have happened in two kinds:1. The molecule is already decomposed within the first time interval.2. The molecule decomposed only within the second time interval.We can inspect also the possibilities to find the molecule after two intact time intervals:1. the molecule remained intact the whole time; 2. or the molecule decomposed in the meantime into state (2), therefore it has decomposed and then again regenerated.However, for the backward reaction the probability (p21) in the special example is equal to zero. In general, it could also be that the molecule changes more than twice the state within the two time intervals. Later we will make the time interval so small that no transition can escape the eyes of the observer. Therefore, we can exclude this possibility. The mathematical formulation of statements above is given in Eq. (4).(4)Here the terms are not explicit functions of the time steps n, at which we are checking the state of the system. However, the probabilities are a function of the length of the time interval t. In a similar way, we obtain the expressions for the other transition probabilities. In fact, the procedure turns out as a matrix multiplication. After two time units, we find in the general case

and for the special example

The State After Multiple Time IntervalsThe general result after multiple time intervals arises simply as a result of repetition of the process

For the special example discussed in this section we obtain

Dependence of the Transition Probability on the Time IntervalThe transition probability p depends on the length of the selected time interval. The expansion into a Taylor series results in

If the time interval limits to zero t 0, then the transition probability should also tend to zero p 0. Thus, we presume that the chemical process is not infinitely fast. Therefore, we postulate p(0) = 0.Transition Probability After Multiple Time StepsThe probability that the molecule did not decompose after n time steps yet is p11(n)t. We set(5)and get(6)If the limit x 0 in Eq. (6), then we obtain

p11(t) is the probability that a particle does not react and remains intact at least until t. This probability decreases exponentially with time. If many particles react in this way independently, then we can identify this probability with the frequency, or the fraction of molecules that are still there as such. The well-known law of a first-order reaction emerges as

The derivation given above shows clearly that the kinetic laws are actually no deterministic laws, but have rather a probabilistic character. In the common sense, it is strongly engraved that by the development of quantum mechanics the interpretation of states as probabilities is forced. In fact, the interpretation of states emerging from nondeterministic laws is sound even in the classical statistical mechanics, in addition, like here, in chemical kinetics. By the way, if we identify in the above derivation the time as the path of a particle, light quantum, etc., through an absorbing medium, and the rate constant as the absorption coefficient, then we obtain the law of Lambert2 and Beer.3Opposing Reactions of First OrderThe transition probabilities of opposing reactions of first order can be summarized in a scheme given as

in that p is the probability for the forward reaction and q is the probability for the backward reaction in a certain time interval t. The probabilities should be independent on the entire number of the steps n, but dependent on the time interval t. We substitute now z = p + q and obtain (7)By repeated application of the matrix multiplication we obtain for p11(n)t

and for p12(n)t)(8)is obtained. Further, we substitute

Inserting these shorthands into Eq. (7) results in

Next, we go to the limit x 0 and obtain(9)Equation (18.10) can be transformed [9, p. 789] into(10)In Eq. (10), [A] refers to the concentration of the reacting species and [A0] to the initial concentration of the species at the beginning of the reaction.Generalization of the SchemeAfter the preceding comments, the generalization of the scheme should not cause problems. Consider a reaction scheme as given in Eq. (11).(11)We have designated the individual compounds in place of A, B, C, . . . with (1), (2), (3),. . . . Further, the individual components are numbered in such a way that only compounds with successive numbers can be converted into one another. The transition matrix reads thereby

Into this transition matrix now the individual transition probabilities are inserted. There are a lot of zeros in the transition matrix. For example, p13 = 0. This arises from the kinetic scheme. It is not allowed that a species is directly transformed from (1) into (3). Rather the reaction runs via the intermediate compound (2). Therefore, the matrix is a tridiagonal one. Usually the required transition probabilities are not very well known. For this reason, as a first approximation, we set them all equal to a certain number, which will be later multiplied by an appropriate factor.With progressive reaction, the transition matrix rises to a certain power. We find the coefficients then directly in the first row of the matrix. These coefficients refer to the probability that the reaction has been taken place, e.g., from compound (1) to compound (n) and these coefficients are to be weighted. If we raise this matrix to a sufficiently large power, then we approximate the equilibrium state. All elements in the first line become then equivalently large. The weighting factors for the probabilities are chosen now in this way that in equilibrium the correct concentrations will turn out. If we know that a certain reaction step with a clearly smaller reaction rate takes place, then we can correct the scheme in this way for this particular pi j , of course. However, in this case it is compulsory that the individual terms in the transition matrix are again properly standardized. Each element of the transition matrix for a certain epoch out of the equilibrium is then divided by the corresponding elements for the equilibrium, so that the weighting factors for the equilibrium tend to approach again to 1.

Infinite Number of CompoundsIn the limit of an infinite number of compounds the transition probabilities p1,k (t), the reaction scheme approaches a mathematical expression similar to the diffusion equation. Starting with Eq. (11), we get the conventional scheme as follows:(12)If all the rate constants are equal, i.e., ki, j = k, Eq. (12) reduces to(13)We presume that a certain function exists for that f (i ) = ci . We treat now i as a continuous variable. Taylor expansion yields

and consequently(14)Inserting (18.15) with k = 1 in the last equation of Eq. (13) yields

as an approximation for large i. Here we crossed over to the notation. On the other hand, set (13) can be consecutively multiplied by x, x2, x3 to result in(15)Summing Eq. (15) and defining a generating function

finally yields

Mechanical Chemical AnalogyThe mechanical chemical analogy can predict mechanisms of degradation and pathways of degradation. The thermal degradation of polymers in the course of heavy thermal stress will result usually in the formation of small molecules that can be analyzed by means of conventional methods of organic chemistry, namely infrared spectroscopy or gas chromatography coupled with mass spectrometry. The observation of these products will serve to the chemist to establish a reaction mechanism concerning the chemical process of degradation. The reaction mechanism is proposed on the basis of certain common rules that are undoubtedly sound and founded by the general accepted up-to-date opinion common in organic chemistry. The experienced expert is able to sketch from scratch several different mechanisms of degradation, including a series of postulated unstable molecules or radicals that are not accessible by chemical analysis. It is often just a challenge to postulate certain intermediates in order to describe the experimental findings most elegantly. One of the favorite intermediates is, for example, the six-membered ring. Because of the complexity of thermal degradation it happens in fact that it is difficult to decide according to which proposed mechanism the degradation process will follow in reality, or which pathway would be the preferred degradation path. Setting up the complete set of elementary reaction steps may become a task beyond the difficulty of playing chess. Computer programs [1014] that can do the job for special fields have been developed, e.g., CHEMKIN and EXGAS. The authors suggest combining the results with thermodynamic and kinetic data. For example, in the oxidation of cyclohexane, 513 intermediates and 2446 reactions have been identified [15].Extremal Principles in MechanicsHere we will present a most simple tool that will offer more certainty to decide in between possible reaction mechanisms of degradation, apart from the pure intuition. The starting point is the mechanics of a system. It is well established that the mechanical equilibrium is associated with a minimum of energy. There is a complete analogue to chemical equilibrium.Crossing over from mechanical and chemical equilibrium to kinematics in mechanical sciences and to kinetics in chemistry the analogy is not established as well. Descriptive chemical kinetics is a very different concept than the concept of kinematics. Kinematics is governed by the Hamiltonian equations or other extremal principles. A simple procedure to calculate the energies of the intermediates in a complex chemical degradation along the reaction coordinate, using the method of increments by van Krevelen, will be developed. From the energies along the reaction coordinate we will discriminate among various proposed reaction mechanisms. The equilibrium of mechanical systems and chemical systems is described just in the same way. The concept of virtual work can be applied in both cases successfully to obtain the equilibrium state. The path of mechanical systems has been described by extremal principles. We emphasize the principles of Fermat,4 Hamilton.5 The principle of least action is named after Maupertuis6 but this concept is also associated with Leibnitz,7 Euler, and Jacobi.8 For details, cf. any textbook of theoretical physics, e.g., the book of Lindsay [16, p. 129]. Further, it is interesting to note that the importance of minimal principles has been pointed out in the field of molecular evolution by Davis [17]. The Principle of Least Action in Chemical KineticsThe principle of least action has also been applied to systems where the motion is not readily seen, e.g., for acoustic and electric phenomena. Therefore, it would be intriguing to establish a mechanical theory of chemical kinetics. In a heuristic way we could relate the kinetic energy of an ensemble of molecules to the rate at which the move along a given path of reaction. If the ensemble has a constant total energy, the path with a maximum kinetic energy would be a path with lowest average potential energy. Pictorially speaking, the preferred reaction path would be characterized with a minimum of elementary reaction steps and with minimum potential energy of the respective intermediates. Even when such a principle seems to be suggestive, we will not been work out it rigorously. Indeed, there may appear conceptual difficulties. Actually, chemical kinetics approaches the problem in a different way [18, pp. 372412]. The use of potential energy surfaces is established in general. However, potential energy surfaces have been calculated for the reaction paths for rather simple systems. In the case of composite reactions, the theory of consecutive reactions is used to identify the rate-controlling steps. The rate-controlling step is that elementary reaction step with the highest energy of activation in the whole reaction path. In this way, it can be concluded from a variety of proposed reaction mechanisms whether a certain reaction path has sufficient throughput to be regarded as a reliable reaction mechanism. The theory of consecutive reactions treats the elementary steps in an obviously isolated manner. In a sequence of elementary reactions

namely the kinetic equations for a first-order sequence are

We focus on the third equation of the set above. The form of the equation suggests that we could assess the kinetic constant k2 by measuring the decay of the species Y , no matter whether this species is poured from a reagent bottle into the reagent glass and brought to temperature; or whether the species Y is formed as an intermediate in the course of the reaction sequence given above. If the species Y is obtained in the latter mode, then it may enter additional energy from the previous elementary step into the reaction, in contrary to the first experimental setup. Therefore, the kinetic constant k2 may depend on the history of the formation of the intermediate. This aspect is not pointed out readily in the literature. In polymer origin and history of reactive intermediates [19].Calculation of the Energies Along the Reaction CoordinatesWe will use the energy profile along the reaction coordinates to decide which mechanism would be the most favorable one. For complicated systems, the calculation of the energy of a molecule or radical in terms of the mutual distances of the atoms is a tedious procedure and therefore it seems that this approach is often dismissed, when a reaction mechanism is set up. An example that illustrates the expenditure is the detailed discussion of the thermal degradation of polyvinylchloride by Bacaloglu and Fisch [20, 21]. There is a simplified procedure available, in particular for degradation reactions. We will focus instead on unstable intermediates of the reaction instead on the activated complexes themselves. The energy of the intermediates can be obtained by the method of the increments given by van Krevelen [22]. van Krevelen summarizes in tables increments of the Gibbs energy of formation for certain groups that would constitute an organic molecule. From these increments, the Gibbs energy of a certain molecule, fragment, or radical can be calculated. We illustrate the procedure to calculate the Gibbs energy of formation of a polystyrene unit in Table 2.The calculated Gibbs energy of formation is valid strictly for gas phase species, but van Krevelen gives for some cases correction for other states.Degradation of a Polystyrene ChainA simple example should illustrate the concept. Consider the degradation of a polystyrene chain. We have chosen a segment with head-to-tail links including one head-to-head link.We consider now three mechanisms of degradation that can result in the formation of styrene. These mechanisms are shown in Figs.4, 5, and 6. We are starting from the same polystyrene chain in each mechanism.We are also ending essentially with the same products. In between, however, there are different intermediates. Mechanism 1 starts with a chain scission at the head-to-head link (Intermediate 12). After this an unzipping reaction starts, resulting in the formation of one styrene unit (13). Finally, a recombination is taking place, leaving a polystyrene chain, now one unit shorter behind (14). Mechanism 2 is very similar in comparison to mechanism 1. The only difference is that the chain scission is now taking place in a regular head-to-tail sequence. Mechanism 3 suggests the formation of a tertiary polymer radical and a hydrogen radical (32). An intermolecular reaction like disproportionation results in the formation of intermediate (33). From this structure, a depolymerization step occurs to result in (34). The polymer chain is then restored (35) and the hydrogen radical finds its way back to the main chain (36). Mechanism 3 has two more steps than the other mechanisms discussed.

Degradation of polystyrene to yield styrene. Mechanism 1: chain scission at the headto- head link

Of course, the expert will know for this simple example what is themore probable mechanism of degradation without doing any calculation. However, our goal rather is to illustrate the general concept, with a simple example. There may be cases that are more complicated, where the result will not be as obvious as here. The Gibbs energy of formation of the intermediates has been calculated and is presented in Table 18.3. For the calculation of the Gibbs energy of formation at Degradation of polystyrene to yield styrene. Mechanism 2: chain scission at the headto- tail link600 K the first approximation of Ulich has been used, i.e., the standard enthalpy of formation and the standard entropy of formation have been used. Actually, it is not necessary to calculate the Gibbs energy of the whole molecule, only the reacting parts are needed. This is a consequence of the increment method. On the other hand, it is not allowed to omit or to add some parts of the structure Degradation of polystyrene to yield styrene. Mechanism 3: hydrogen abstraction followed by chain scissionin the course of the mechanism. This is often done in the abbreviated notation of reaction mechanisms, e.g., from somewhere hydrogen radicals are coming and introduced in the scheme. Further, we can subtract the Gibbs energy of the initial structure from all others. In this case, the table would start with zero. It is also possible to form the difference of the Gibbs energies between two subsequent species occurring in the mechanism. This will give a picture of the energetic situation of a single elementary reaction. In Fig. 7, a plot of the energy profiles of the three reaction mechanisms is shown.Inspection of Fig.7 shows that, as expected, the mechanism involving free hydrogen radicals has higher Gibbs energy values than the other two proposed mechanisms. These are running nearly at the same energy profile. Mechanism (1) is preferred over mechanism (2), because it is elementary knowledge in organic chemistry that tertiary radicals will be more stable than secondary radicals. Therefore, the head-to-head link is regarded as a weak link. Inspection of Table 18.3 shows that the structures 12 and 13 of mechanism (1) have slightly lower Gibbs energy than the corresponding structures 22 and 23 of mechanism (2) at 298K but the situation is reversed at 600 K. This reflects the temperature dependence of the Gibbs energy. The effect is small, however, and the data are not as precise that this effect should be discussed seriously. In other situations, such effects may be more pronounced. Summarizing, it may be stated that the calculation of the energy profile for different reaction mechanisms is helpful to discriminate between the proposed mechanisms. Of course, it is obvious that the method is suitable in general for the discussion of the reliability of proposed reaction mechanism. One big advantage

Free Gibbs enthalpy of the species involved in the reaction mechanisms. The plot is reduced by the free Gibbs enthalpy of the reactant. Therefore, the plot starts at zerois that the method is easy to apply. On the other hand, the tables of increments, which can be regarded as a partial structure library, are containing structural elements of stable groups and some free radicals. For an extensive application of the method, however, other entries, like partial structures with delocalized electrons, ionic groups, would be helpful.

References1. Battin Leclerc, F., Glaude, P.A., Warth, V., Fournet, R., Scacchi, G., Come, G.M.: Computer tools for modelling the chemical phenomena related to combustion. Chem. Eng. Sci. 55(15), 28832893 (2000) 2. Handle, D.J., Broadbelt, L.: Mechanism reduction during computer generation of compact reaction models. AlChE J. 43(7), 18281837 (1997) 3. Baverman, C., Stromberg, B., Moreno, L., Neretnieks, I.: Chemfronts: A coupled geochemical and transport simulation tool. J. Contam. Hydrol. 36(34), 333351 (1999) 4. Eucken, A., Landolt, H., Bornstein, R. (eds.): Zahlenwerte und Funktionen aus Physik, Chemie, Astronomie, Geophysik und Technik, 13, (1985)5. Nichipor, H., Dashouk, E., Yermakov: A computer simulation of the kinetics of high temperature radiation induced reduction of NO. Radiat. Phys. Chem. 54(3), 307315 (1999)6. Chase, M.W., Sauerwein, J.C.: NIST Standard Reference DATA Products Catalog 1994, NIST Chemical Kinetics Database: Version 2Q98 Standard Reference DATA Program. No. 782 in NIST Special Publication. National Institute of Standards and Technology, Gaithersburg, MD (1994)7. N.N.I. Technology of Standards: Chemical kinetics database on the web, standard reference database 17, version 7.0 (web version), release 1.3 a compilation of kinetics data on gasphase reactions. [electronic] http://kinetics.nist.gov/kinetics/index.jsp (2000)8. Makroglou, A., Li, J., Kuang, Y.: Mathematical models and software tools for the glucoseinsulin regulatory system and diabetes: An overview. Appl. Numer. Math. 56(3), 559573 (2006)9. Atkins, P.W.: Physical Chemistry, 4th edn. Oxford University Press, Oxford (1990)10. Dahm, K.D., Virk, P.S., Bounaceur, R., Battin-Leclerc, F., Marquaire, P.M., Fournet, R., Daniau, E., Bouchez, M.: Experimental and modelling investigation of the thermal decomposition of n-dodecane. J. Anal. Appl. Pyrolysis 71(2), 865881 (2004)11. Edwards, K., Edgar, T.F., Manousiouthakis, V.I.: Kinetic model reduction using genetic algorithms. Comput. Chem. Eng. 22(12), 239246 (1998)12. Kee, R.J., Rupley, F.M., Miller, J.A.: Chemkin-II: A Fortran Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics. Report SAND-89-8009, Sandia Natl. Lab., Livermore, CA, USA (1989)13. Klinke, D.J., Broadbelt, L.J.: Mechanism reduction during computer generation of compact reaction models. AlChE J. 43(7), 18281837 (1997)14. Warth, V., Battin-Leclerc, F., Fournet, R., Glaude, P.A., Come, G.M., Scacchi, G.: Computer based generation of reaction mechanisms for gas-phase oxidation. Comput. Chem. (Oxford) 24(5), 541560 (2000)15. Buda, F., Heyberger, B., Fournet, R., Glaude, P.A., Warth, V., Battin-Leclerc, F.: Modeling of the gas-phase oxidation of cyclohexane. Energy Fuels 20, 14501459 (2006)16. Lindsay, R.B.: Concepts and Methods of Theoretical Physics. Dover Publications, Inc., New York (1969)17. Brian, K.D.: The forces driving molecular evolution. Prog. Biophys. Mol. Biol. 69, 83150 (1998)18. Laidler, K.J., Meiser, J.H.: Physical Chemistry. The Benjamin Cummings Publishing Company, Menlo Park, CA (1982)19. Tudos, F.: Kinetics of radical polymerization. X. The assumption of hot radicals in inhibition processes. Makromol. Chem. 79(1), 824 (1964)20. Bacaloglu, R., Fisch, M.: Degradation and stabilization of poly(vinyl chloride). IV. Molecular orbital calculations of activation enthalpies for dehydrochlorination of chloroalkanes and chloroalkenes. Polym. Degrad. Stabil. 47(1), 932 (1995)21. Bacaloglu, R., Fisch, M.: Degradation and stabilization of poly(vinyl chloride). V. Reaction mechanism of poly(vinyl chloride) degradation. Polym. Degrad. Stabil. 47(1), 3357 (1995)22. van Krevelen, D.W.: Properties of Polymers. Their Correlation with Chemical Structure. Their Numerical Estimation and Prediction from Additive Group Contributions, 3rd edn., pp. 627639. Elsevier, Amsterdam (1990)