First-order reactions polymerization

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

Fig. 10.—Benzoyl peroxide-initiated polymerization of vinyl-i-j3-phenyl-butyrate in dioxane at 60°C plotted as a first-order reaction. [M]o and [M ] represent concentrations of monomer initially and at time t, respectively. In experiments 1, 2, and 3, respectively, [M]o = 2.4, 7.28, and 5.97 g. of monomer per 100 cc. of dioxane. (Results of Marvel, Dec, and Cooke obtained po-larimetrically.)...

It is well known that the base hydrolysis of polyacrylamide is catalyzed by OH ions (first order reaction) and obeys autoretarded kinetics due to the electrostatic repulsion between the anionic reagent and the polymeric substrate(3-5). In the range of slightly acid pH (3 < pH < 5), Smets and Hesbain(6) have demonstrated a... 

Some examples, such as thermal polymerization of styrene and decomposition of di-f-butyl peroxide, are given in [194], both treated as first-order reactions. The activation energy found for the decomposition of di-f-butyl peroxide agrees well with the literature value. From the pressure data, it appears that the initial pressure rise is caused by the evaporation of toluene, present as a solvent. At higher temperatures, the gases generated by decomposition are the main contributors to the pressure rise. 

A plug flow or tubular flow reactor is tubular in shape with a high length/diameter (1/d) ratio. In an ideal case (as in the case of an ideal gas, this only approached reality) flow is orderly with no axial diffusion and no difference in velocity of any members in the tube. Thus, the time a particular material remains within the tube is the same as that for any other material. We can derive relationships for such an ideal situation for a first-order reaction. One that relates extent of conversion with mean residence time, t, for free radical polymerizations is ... 

Malkin s autocatalytic model is an extension of the first-order reaction to account for the rapid rise in reaction rate with conversion. Equation 1.3 does not obey any mechanistic model because it was derived by an empirical approach of fitting the calorimetric data to the rate equation such that the deviations between the experimental data and the predicted data are minimized. The model, however, both gives a good fit to the experimental data and yields a single pre-exponential factor (also called the front factor [64]), k, activation energy, U, and autocatalytic term, b. The value of the front factor k allows a comparison of the efficiency of various initiators in the initial polymerization of caprolactam [62]. On the other hand, the value of the autocatalytic term, b, describes the intensity of the self-acceleration effect during chain growth [62]. 

For each initiator there is a useful temperature range for which the initiator decomposition rate constant, kd, will produce radicals at suitable rates for polymerization. The initiation rate is usually controlled by the decomposition rate of the initiator, which depends directly on its concentration (first-order reaction). The temperature window can be enlarged by the use of catalysts such as a tertiary amine (Eq. (2.80)), or an organometallic compound in a redox reaction (Eqs (2.81) and (2.82)). 

Generally speaking, the values vhp and vhh depend on the conformation of the macroradical as a whole. In the first approximation, however, they can be considered as constant, vhp and vhh- hi this case, the polymerization process is described on the basis of the first-order reaction kinetic equations. In particular, one can define the following parameter characterizing the composition change along the chain [86]... 

Example 5.7 A CSTR is commonly used for the bulk polymerization of styrene. Assume a mean residence time of 2h, cold monomer feed (300 K), adiabatic operation (UAext = 0), and a pseudo-first-order reaction with rate constant... 

The polysaccharides of wood (holocellulose) may be hydrolyzed by two general methods (1) by strong acids, such as 70-72 percent sulfuric acid or 40-45 percent hydrochloric acid or (2) by dilute acids, such as 0.5-2.0 percent sulfuric acid. The hydrolysis by strong acids is constant, proceeds as a first-order reaction, and is independent of the degree of polymerization. The reaction may be represented as follows ... 

These values are not absolute velocity constants since transfer to alkyl-aluminum occurs during the reaction. However, the transfer reactions will be proportional to the concentration of active centers. Hence, at similar conversions the fractions of active alkylmetal bonds are likely to be approximately the same, and the constants will be proportional to the true velocity constants. Determining relative propagation velocity constants is complicated by the participation of a termination reaction. At low temperatures the polymer is insoluble and the catalyst is embedded in a semi-solid mass, resulting in very slow rates of polymerization. At temperatures of 41°-60° C. reasonably good first-order reactions with respect to monomer are found, but at higher temperatures there is a rapid fall-off in reaction rate with time (Figure 4). The velocity constants in Table III were calculated from the linear portion of the rate-time curve, and no account was taken of termination reactions. 

In this paper, the problem of product non-uniformity is placed in perspective as we consider the simple case of flow between parallel plates with a flat velocity profile accompanied by a first order reaction. The system equations involved are non-dimensionalized and four parameters identified. A systematic attempt is made to explore a host of constrained and unconstrained wall temperature profiles that alleviate the problem of product non-uniformity at the reactor outlet. Work is already underway to study similar problems in complex polymerization systems, and some preliminary results are reported here. 

Up to relatively high conversions, the rate of polymerization can be satisfactorily represented as a first-order reaction [5] according to the following equation ... 

We assumed that the polymerization proceeds by a normal coordination polymerization. The effect of the catalyst concentration on the polymerization was examined by polymerization at different ratios of catalyst to styrene (Figure 17.13). The reaction rate increased in proportion to the catalyst ratio. However, the decay of the polymerization reaction was too fast to explain it as a first-order reaction. 

The macromolecular silyl chloride reacts with sodium in a two-electron-transfer reaction to form macromolecular silyl anion. The two-electron-trans-fer process consists of two (or three) discrete steps formation of radical anion, precipitation of sodium chloride and generation of the macromolecular silyl radical (whose presence was proved by trapping experiments), and the very rapid second electron transfer, that is, reduction to the macromolecular silyl anion. Some preliminary kinetic results indicate that the monomer is consumed with an internal first-order-reaction rate. This result supports the theory that a monomer participates in the rate-limiting step. Thus, the slowest step should be a nucleophilic displacement at a monomer by macromolecular silyl anion. This anion will react faster with the more electrophilic dichlorosilane than with a macromolecular silyl chloride. Therefore, polymerization would resemble a chain growth process with a slow initiation step and a rapid multistep propagation (the first and rate-limiting step is the reaction of an anion with degree of polymerization n[DP ] to form macromolecular silyl chloride [DP +J, and the chloride is reduced subsequently to the anion). 

Microtubule assembly in cells differs in some ways from assembly in vitro. In cells, nucleation of microtubules requires a third type of tubulin, which is called y-tubulin, that functions in concert with other proteins in the form of a y-tubulin ring complex. In most animal cells, the y-tubuIin ring complex is located at the pericentriolar region of the microtubule organizing center (or centrosome) where it nucleates microtubule assembly at the minus ends (7). The y-tubulin does not become incorporated into the microtubule, but rather it only localizes to the minus ends. Assembly of tubulin to form microtubules during the early stages of polymerization in vitro can be considered a pseudo first-order reaction. A steady state is eventually attained in which both the soluble tubulin concentration and the microtubule polymer mass attain stable plateaus (8). The critical concentration at apparent equilibrium (actually a steady state, see below) is the concentration of soluble tubulin in apparent equilibrium with the microtubule polymers. 

Representative for systems exhibiting sigmoidal conversion curves Fig. 1 shows experimental results for the rate constant of the reaction of TS, evaluated from thermal and y-polymerization data according to K = (1 — X) dX(t)/dt, and normalized to the rate constant in the low conversion limit. It is obvious, that at low conversion K depends on X, contrary to what is to be expected for a simple first order reaction. The functional form of KPC) is different for the two modes of polymerization. The overall increase of K with increasing X reveals an autocatalytic reaction enhancement. A measure for its efficiency is the ratio K(X = 0.5)/K(X = 0) which tirnis out to be about 200 for TS under thermal polymerization conditions. This effect is often observed with disubstituted diacetylenes albeit with different kinetic... 

The thermal cis-trans isomerization of crotonitrile has been studied in the gas phase at pressures from 0.2 to 20 torr and temperatures from 300° to 560° C . The isomerization is a homogeneous unimolecular reversible first order reaction, the rate coefficient for the direction cis trans being given by exp((-51.3 3.7)//i7 ) sec. Calculated thermodynamic parameters are = 0.17 + 0.12 kcal.mole and AS = —0.39+0.19 eu. The only side reaction with an appreciable rate was a surface polymerization. 

The intercept in Fig. 14 gives, therefore, the ion-pair rate coefficient and the slopes of the lines yield kpKj. Conductance measurements can be used to determine and hence a value for kp is obtained. The validity of these concepts can be checked by carrying out the polymerization in the presence of sodium tetraphenylboride. This salt dissociates to a much greater extent than polystyrylsodium and its presence suppresses the ionization of the latter by a common ion effect. Under appropriate conditions a simple first order reaction in active centres can be observed in its presence with a rate equal to that measured by extrapolation to infinite concentration of active centres. 

Monomer was polymerized by a first order reaction but it interferes with the formation of the active catalyst, and the active centre concentration falls inversely with increase in initial monomer concentration. Polymer molecular weights increase with initial monomer concentration and the monomer transfer coefficient (fetr.M/ p) is small (8x10 ). Catalyst efficiency is low (ca. 0.05), in agreement with the observations of Dawes and Winkler [146] for the polymerization of butadiene using the same catalyst. 

When all the monomer has been imbibed by the particles, the monomer concentration decreases with conversion and the reaction becomes first-order, provided that the number of radicals per particle remains constant—i.e., at low initiation rate in small particles. Such a polymerization system is shovm in Figure 12, where the logarithm of the reaction rate is plotted against time. From about 60 to 90% conversion, this plot gives a straight line, as is required by a first-order reaction. 

The f-transition metal catalysts were first described by von Dohlen [98] in 1963, Tse-chuan [99] in 1964 and later by Throckmorton [100]. In the 1980s Bayer [14] and Enichem [101] developed manufacturing processes based on neodymium catalysts. The catalyst system consists of three components [102] a carboxylate of a rare earth metal, an alkylaluminum and a Lewis acid containing a halide. A typical catalyst system is of the form neodymium(III) neodecanoate/diisobutylaluminum hydride/butyl chloride [103]. Neodymium(III) neodecanoate has the advantage of very high solubility in the nonpolar solvents used for polymerization. The molar ratio Al/Nd/Cl = 20 1 3. Per 100 g of butadiene, 0.13 mmol neodymium(III) neodecanoate is used. With respect to the monomer concentration, the kinetics are those of a first-order reaction. 

In the nonsterilized systems in which abiotic and biotic processes were not differentiated, Larson and Hufnal (1980) also reported that the Mn(IV) oxides are the most efficient, among the metal oxides, in oxidative polymerization of dissolved phenols. Ono et al. (1977) investigated the rate of radical formation on MnOi at high pH (pH 9) and proposed a mechanism that involves the removal of protons from hydroquinone before electron transfer. Stone and Morgan (1984a) reported that the reduction of Mn oxide by hydroquinone is a first-order reaction (with respect to oxide loading) and must occur on the oxide surface, i.e., phenols must form a surface complex prior to electron transfer. 

Example 11-11 An ethylene stream is fed to a polymerization reactor in which the catalyst is poisoned by acetylene. The ethylene is prepared by catal3d ically dehydrogenating ethane. Hence it is important that the dehydrogenation catalyst be selective for dehydrogenating rather than C2H4. The first-order reactions are... 

Another claim of a thermal and photochemical homogeneous (or topotactic) polymerization of the diacetylene 30 a (Scheme 2.1.9) is questionable. The claim had to be modified several times, it was admitted later that the polymorph required crystal solvent (e.g. 0.5 dioxane) [57], and 30 a was no longer discussed in the review of the same research group [58] that dealt with 30 b and listed 32 further diacetylenes. First order reaction was claimed in order to try to substantiate topochem-ical behavior [58]. However, inspection of the published kinetic curves indicates zero order up to 90 % conversion after an induction period instead of first order for... 

The application of this method to the study of THF polymerization on BF3 + propylene oxide (bulk polymerization, T- 20 °C) has made it possible to establish the following facts in this system the active centers are deactivated in accordance with the first order reaction k A = 1.9 x 10 4 s 1). Propylene oxide is consumed... 

This is a first-order reaction. The half-life of benzoyl peroxide at 100°C is 19.8 min. (a) Calculate the rate constant (in min ) of the reaction, (b) If the half-life of benzoyl peroxide is 7.30 h, or 438 min, at 70°C, what is the activation energy (in kJ/mol) for the decomposition of benzoyl peroxide (c) Write the rate laws for the elementary steps in the above polymerization process, and identify the reactant, product, and intermediates, (d) What condition would favor the growth of long, high-molar-mass polyethylenes ... 

Pure Monomers. The first pulse radiolysis study of a polymerizing system was with isobutylene, which gave an absorption with a peak at 297 mfi, which disappeared in a fast first-order reaction (6). The absorption may be attributed to the trimethylcarbonium ion, which, from independent work, appears to have an absorption maximum at 292 m/ and an extinction coefficient close to 6.3 X 103M 1 cm. 1 (28). 

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