Rate expressions propagation

The propagation rate expression can be used to describe simple dissociative schemes of the type... 

However, the mechanisms by which the initiation and propagation reactions occur are far more complex. Dimeric association of polystyryllithium is reported by Morton, al. ( ) and it is generally accepted that the reactions are first order with respect to monomer concentration. Unfortunately, the existence of associated complexes of initiator and polystyryllithium as well as possible cross association between the two species have negated the determination of the exact polymerization mechanisms (, 10, 11, 12, 13). It is this high degree of complexity which necessitates the use of empirical rate equations. One such empirical rate expression for the auto-catalytic initiation reaction for the anionic polymerization of styrene in benzene solvent as reported by Tanlak (14) is given by ... 

In this work, the characteristic "living" polymer phenomenon was utilized by preparing a seed polymer in a batch reactor. The seed polymer and styrene were then fed to a constant flow stirred tank reactor. This procedure allowed use of the lumped parameter rate expression given by Equations (5) through (8) to describe the polymerization reaction, and eliminated complications involved in describing simultaneous initiation and propagation reactions. 

LDPE polymerization reaction consists of various elementary reactions such as initiation, propagation, termination, chain transfer to polymer and monomer, p-scission and so forth [1-3], By using the rate expression of each elementary reaction in our previous work [4], we can construct the equations for the rate of formation of each component. 

In 1919 Christiansen (25), Herzfeld (26), and Polanyi (27) all suggested the same mechanism for this reaction. The key factor leading to their success was recognition that hydrogen atoms and bromine atoms could alternately serve as chain carriers and thus propagate the reaction. By using a steady-state approximation for the concentrations of these species, these individuals were able to derive rate expressions that were consistent with that observed experimentally. 

A reaction rate expression that is proportional to the square root of the reactant concentration results when the dominant termination step is reaction (4c), that is, the termination reaction occurs between two of the radicals that are involved in the unimolecular propagation step. The generalized Rice-Herzfeld mechanism contained in equations 4.2.41 to 4.2.46 may be employed to derive an overall rate expression for this case. 

In order for the overall rate expression to be 3/2 order in reactant for a first-order initiation process, the chain terminating step must involve a second-order reaction between two of the radicals responsible for the second-order propagation reactions. In terms of our generalized Rice-Herzfeld mechanistic equations, this means that reaction (4a) is the dominant chain breaking process. One may proceed as above to show that the mechanism leads to a 3/2 order rate expression. 

FRET is a nonradiative process that is, the transfer takes place without the emission or absorption of a photon. And yet, the transition dipoles, which are central to the mechanism by which the ground and excited states are coupled, are conspicuously present in the expression for the rate of transfer. For instance, the fluorescence quantum yield and fluorescence spectrum of the donor and the absorption spectrum of the acceptor are part of the overlap integral in the Forster rate expression, Eq. (1.2). These spectroscopic transitions are usually associated with the emission and absorption of a photon. These dipole matrix elements in the quantum mechanical expression for the rate of FRET are the same matrix elements as found for the interaction of a propagating EM field with the chromophores. However, the origin of the EM perturbation driving the energy transfer and the spectroscopic transitions are quite different. The source of this interaction term... 

The obvious challenge in the interpretation of the data is to find a suitable explanation for the independence of the third term of the rate law, Eq. (102), on the concentrations of HSO3 and 02. The rate expression determined experimentally could be modeled quantitatively by combining the following propagation steps with the uncatalyzed reaction mechanism ... 

When developing this equation in terms of the rate expressions for the individual reactions it has been the usual practice to assume that there is only one propagating species in the system - an assumption which will be discussed in Section 6, and that the propagation reaction is of second order ... 

The shape and length of the wave that propagates in the bed is related to the mathematical form of the uptake rate expression that is substituted into the last term in Eq. (9.10). Equation (9.15) is the pde describing diffusion of an adsorbate from the gas phase into the adsorbent This is but one of many diffusion based uptake models that might be substituted into the uptake rate term of Eq. (9.9) or (9.10). 

This expression of [M+] is substituted into the propagation rate expression Equation 5.5 giving the following equation ... 

The Mayo equation (Equation 6.42) that gives positive slopes when the data is plotted (such as Figure 6.3) is the reciprocal relationship derived from the expression cited earlier. The ratio of the rate of cessation or termination by transfer to the rate of propagation is called the chain transfer constant (Cs). 

The rate of propagation, and therefore the rate of polymerization, is the sum of many individual propagation steps. Since the rate constants for all the propagation steps are the same, one can express the polymerization rate by... 

The rate of living cationic polymerization is expressed as the rate of propagation of ion pairs... 

The rate of polymerization in nonterminating systems can be expressed as the rate of propagation... 

The need for solvation in anionic polymerization manifests itself in some instances by other deviations from the normal reaction rate expressions. Thus the butyllithium polymerization of methyl methacrylate in toluene at — 60°C shows a second-order dependence of Rp on monomer concentration [L Abbe and Smets, 1967]. In the nonpolar toulene, monomer is involved in solvating the propagating species [Busson and Van Beylen, 1978]. When polymerization is carried out in the mixed solvent dioxane-toluene (a more polar solvent than toluene), the normal first-order dependence of Rp on [M] is observed. The lithium diethylamide, LiN(C2H5)2, polymerization of styrene at 25°C in THF-benzene similarly shows an increased order of dependence of Rp on [M] as the amount of tetrahydrofuran is decreased [Hurley and Tait, 1976]. 

The rate expression for propagation is given by the rate of monomer disappearance ... 

The number average degree of polymerization, x , may be expressed as a function of the rate of propagation, Rp, and the sum of the rates of transfer and termination, jRtr and Rt, respectively ... 

The alkylaluminum component combined with V(acac)3 has an influence on the kinetic behavior of the propylene polymerization at —78 °C47 82 83). In the polymerization with V(acac)3 and dialkylaluminum monohalide like Al(i-C4H9)2C1, Al(n-C3H7)2C1, A1(C2HS)2C1 or Al(C2H5)2Br, M of polypropylene increased proportionally to the polymerization time, and the polydispersity (M Mj was as narrow as 1.15 0.05 (see Fig. 9). This is the case of living polymerization. As can be seen from Fig. 9, the rate of increase of Mn, i.e. the rate of propagation of living chains as expressed by lvln/(42 t), is influenced by the kind of aluminum component and decreases in the series... 

Note. The overall rate of reaction must not be expressed in terms of production of C2H6 as this is only a minor side reaction and not a rate of propagation. 

The expressions for the rates of propagation vp, termination vt, and transfer are then better written in the form... 

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