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The deviations in copolymer composition are greater when homopropagation is faster than cross-propagation (r1 , r2 > 1) than when cross-propagation is faster than homopropagation (r1 , r2 < 1) 6-2d Types of Copolymerization Behavior.

Different types of copolymerization behavior are observed depending on the values of the monomer reactivity ratios. Copolymerizations can be classi ed into three types based on whether the product of the two monomer reactivity ratios r1 r2 is unity, less than unity, or greater than unity. 6-2d-1 Ideal Copolymerization: r1 r2 1



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Jul 13, 2015 · Convert PDF to Word Using C# The first step will be to get the PdfBox package using the Nuget Package Manager. Now, import the following DLLs into your .cs file: The third step will be to install the DocX NuGet Package from the NuGet Package Manager: Let's read a PDF file and try to get the text from it.

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we have to be careful when interpreting individual terms as radiation reaction and vacuum contributions. For although the total vector potential operator commutes with atomic operators at all times, the same cannot be said of each of the two terms on the right-hand side of equation (6.8) taken separately. Apparently, any ordering of atomic and split field operators is possible in (6.14) as long as it is used in both the term involving only As and in that involving Avac only [3,453]. Nevertheless, Dalibard et al. [131] pointed out that to be able to assign a physical meaning to such terms in (6. 14),each of them has to be Hermitian; otherwise, they cannot be associated with observables. The Hermiticity requirement allows only one ordering [131]: the totally symmetric ordering. So we can write





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A copolymerization is termed ideal when the r1 r2 product is unity. Ideal copolymerization * * occurs when the two types of propagating species M1 and M2 show the same preference for

Hatom) vac'

k22 k12 k21 k11 or r2 1 r1 6-28

(6.17)

and the relative rates of incorporation of the two monomers into the copolymer are independent of the identity of the unit at the end of the propagating species. For an ideal copolymerization Eq. 6-28 is combined with Eq. 6-12 or 6-15 to yield the copolymerization equation as

where the first term on the right-hand side of (6.17) is the radiation reaction contribution,

change from speci c acid (i.e., H ) catalysis to general-acid catalysis. The proton concentration is relatively high and [H ] is a more effective catalyst than the unionized carboxylic acid in the polar, low conversion region. HA in Eqs. 2-12, 2-16, and 2-17 is replaced by H . The reaction rate becomes rst-order each in carboxyl and hydroxyl groups and H . The proton concentration is given by

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4~e {:s .As(O) + As(O) :S},

H KHA HA 1=2 2-24

(6.18)

where KHA is the ionization constant for the carboxylic acid. Combination of Eqs. 2-17 and 2-24 yields

and the second term is the vacuum fluctuation contribution, (6.19)

d COOH 1=2 k3 KKHA COOH 3=2 OH dt 2-25

for the polymerization rate The reaction medium becomes less polar after the lowconversion (>80%) region and the major catalyst in the reaction system is the unionized carboxylic acid the reaction is second-order in carboxylic acid and third-order overall (Eqs 2-18 and 2-19) Another problem is the very high concentrations of reactants present in the lowconversion region The correct derivation of any rate expression such as Eqs 2-20 and 222 requires the use of activities instead of concentrations The use of concentrations instead of activities assumes a direct proportionality between concentration and activity This assumption is usually valid at the dilute and moderate concentrations where kinetic studies on small molecules are typically performed However, the assumption often fails at high concentrations and those are the reaction conditions for the typical step polymerization that proceeds with neat reactants.

Now, we are going to calculate the expectation values of (6.18) and (6.19) up to first order in the fine structure constant. We start with the radiation reaction contribution (6.18). Because As is already proportional to q, we can approximate all atomic operators by the corresponding unperturbed atomic operators. The unperturbed atomic operators are those of the uncoupled atom. We will denote them by the subscript a. They satisfy the following Heisenberg equations of motion:

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