Mayo-Walling Equation

Using Eq. (9.39) for the rate of polymerization, derive the following Mayo-Walling equation for Ziegler-Natta polymerization ... 

Under steady-state conditions [M ] can be substituted using Equation (1.5) to give the following expression, known as the Mayo or Mayo-Walling equation, for the reciprocal of x ... 

The first component of these equations is the known Mayo-Walling equation that describes the specific rate co of copolymerization in liquid MP by the classic kinetic model of polymerization at quadratic chain decay ... 

Thus simultaneous measurements of r and Rp at known [M] enable the ratio kpikt to be evaluated. Normally these measurements are made under non-steady-state conditions using a technique known as the rotating sector method. Knowledge of kplkf and kpikt allows the value of the individual rate constants to be determined (see Table 2.6). Once kp is known, values of ktr can be evaluated by application of the Mayo-Walling Equation (2.41) to experimental data. This also enables the determination of (x )o from which q can be calculated. 

The variation of x with temperature also depends upon the temperature dependence of the transfer constants, C, in the Mayo-Walling Equation... 

This is the cationic polymerization equivalent of the Mayo-Walling Equation (2.41). Additional terms can be added to the right-hand side to take account of other chain transfer reactions, e.g. for chain transfer to solvent the additional term is A . 5-[S]/A p[M]. In each case3c is independent of initiator concentration and in the absence of chain transfer is given by... 

Mayo and Walling, who have given a penetrating critique of the Q,e scheme, point out that it represents in essence merely a transcription to equation form of the reactivity series of Table XX and the po-larity series of Table XXII. Regardless of the manner of interpretation adopted, it is apparent that monomer reactivity in copolymerization depends on two factors. One of these relates to the intrinsic characteristics of the monomer (and of the activated complex produced from it as well) as they tend to favor its addition to a radical. As we have seen, the capacity for resonance stabilization in the transition state is of foremost importance in determining the general level of monomer reactivity. The second factor has to do with the specificity... 

It should be noted that the monomer consumption in the initiation step is neglected in the above equations owing to the small amount of initiator used in most systems. Furthermore, Mayo and Walling suggested that radical concentrations (i.e., and... 

The copolymer equation, which expresses the composition of growing chains at any reaction time t, was developed in the 1930s by a group of investigators including Wall, Mayo, Simha, Alfrey, Borstal, and Lewis (for instance, and j4-0. 

Equation (2.38), which relates the instantaneous composition of the copolymer d[M /d[M2 ) to the prevailing monomer concenttations, can be used to determine the values of ry and r2- Many such values have been recorded (Ham and Alfrey, 1964 Mayo and Walling, 1950 Greenley, 1999 Eastmond, 1976a). Typical values of these parameters for styrene copolymerizations are shown in Table 2.11, which illustrates the wide variations that prevail. The relative reactivity actually expresses the relative reactivity of each of the monomers shown toward the styrene radical compared to the reaction with styrene monomer. Tlius, the ry and r2 values permit some conclusions about the expected composition of the copolymer obtained at any given monomer ratio. 

Therefore the three "criteria" for simulation are identical values of kL, kg and a/e in the industrial and the laboratory absorber. "Simulation" means that, if the bulk ccm x)sitions of gas and liquid in the laboratory absorber are the same as in a volume element of the industrial absorber, the absorption rate per unit interfacial area in this elanent will be the same as in the laboratory model determined experimentally as a function of Cgo> or p, and the above balance equations can be integrated numerically step by step between the limit compositions at the entrance and the exit of the industrial absorber to find the length h of the absorber. The third criterion (a/e) is required whenever the reaction between dissolved gas and a reactant in solution is slow. Indeed reactions will proceed in the bulk liquid, and the rate of absorption in industrial or laboratory absorber will depend iqx>n the volume of bulk liquid available per unit area of interface. Comparison of Tables (I) and (II) leads to the choice of the laboratory equipment (62, 110) for a specified gas-liquid contactor. For example it mayoe seen that any wetted wall or stirred vessel can simulate a packed column (with respect to kL and kg) for reactions occuring in the liquid film. 

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