Bernoulli statistics

Figure 7.9 Fractions of iso, syndio, and hetero triads as a function of p, calculated assuming zero-order Markov (Bernoulli) statistics in Example 7.7.

In the research described in the preceding problem, Randall was able to assign the five peaks associated with tetrads in the C-NMR spectrum on the basis of their relative intensities, assuming zero-order Markov (or Bernoulli) statistics with Pm = 0.575. The five tetrad intensities and their chemical shifts from TMS are as follows ... 

On the basis of these observations, criticize or defend the following proposition Regardless of the monomer used, zero-order Markov (Bernoulli) statistics apply to all free radical, anionic, and cationic polymerizations, but not to Ziegler-Natta catalyzed systems. 

For the interbipolycondensation the condition of quasiideality is the independence of the functional groups either in the intercomponent or in both comonomers. In the first case the sequence distribution in macromolecules will be described by the Bernoulli statistics [64] whereas, in the second case, the distribution will be characterized by a Markov chain. The latter result, as well as the parameters of the above mentioned chain, were firstly obtained within the framework of the simplified kinetic model [64] and later for its complete version [59]. If all three monomers involved in interbipolycondensation have dependent groups then, under a nonequilibrium regime, non-Markovian copolymers are known to form. 

The statistical treatment of a hemiisotactic polymer can be made on the basis of a single parameter a the corresponding formulas are reported in Table 4, last column. For extreme values of a the polymer is no longer hemiisotactic but syndiotactic (for a = 0) or isotactic (for a = 1). The particular distribution existing in the hemiisotactic polymer is not reproducible with either the Bernoulli or the Markov processes expressed in m/r terms. 

Under the condition that the reaction capability is only affected by the nature of the last monomer unit of the growing polymer chain end (terminal model, Bernoulli statistics), the copolymerization equation can be transformed according to Kelen and Tudos ... 

The pioneers in mathematical statistics, such as Bernoulli, Poisson, and Laplace, had developed statistical and probability theory by the middle of the nineteenth century. Probably the first instance of applied statistics came in the application of probability theory to games of chance. Even today, probability theorists frequently choose... 

In statistical copolymers the sequence of monomeric units obeys some known statistical law, e. g. Markov statistics of zero order (Bernoulli), or of the first, second or higher orders. Such copolymers are designated as poly (M,-,yfaf-M2-sfaf- M, -- ) [3]. 

Random copolymers are a special case of statistical copolymers. The probability of finding a given monomeric unit at any place in the chain is independent of the nature of the neighbouring units (Bernoulli distribution). For such a copolymer the probability of finding the sequence M,M2M3, P[ M,M2M3 ] is given by the relation... 

In the simplest case, when the structure of the propagating chain does not affect the configuration of the generated diad, the formation probabilities of meso and racemic diads, Pm and Pr, are related as Pr = (1 — Pm). Chain structure obeys Bernoulli statistics as if the added units were selected at random from a reservoir in which the fraction Pm of the total amount is m, and the fraction (1-Pm) is r. An isotactic polymer will be formed for Pm - 1, and a syndiotactic polymer for Pm -> 0. Within these limits the chains will consists of randomly ordered m and r structures. 

Coleman and Fox published an alternative mechanism [82], According to these authors, the propagating centres exist in two forms, each of which favours the generation of either the m or r configuration. When both centres are in equilibrium, and when this equilibrium is rapidly established, the chain structure can be described by a modified Bernoulli statistics [83, 84]. The configurations of some polymers agrees better with this model than with first-or even second-order Markov models [84, 85]. 

The stereochemistry of addition to a free centre is mostly determined by interactions between the monomer and active centre during approach to the transition state. In simple cases, represented by equations (34) and (35) only the two primary components will interact, and Bernoulli statistics with a single probability parameter Pm will predominate. For Pm = 0.5, the propagation rate constants of isotactic and syndiotactic growth, kpj and k, will differ... 

Radical polymerizations usually obey this model. Methyl methacrylate and vinyl chloride have a tendency to syndiotactic propagation (Pm <0.5) [86]. Low temperatures and bulky substituents cause deviations from Bernoulli statistics. 

Wicke and Elgert [90] polymerized a-methylstyrene with BuLi in tetrahydrofuran. They observed that the distance between the ions in the pair does not affect polymer configuration. This supports the theory of Coleman and Fox [82] concerning the presence of two types of centre in mutual rapid equilibrium and yielding different tacticities. The overall order can then be described by Bernoulli statistics. 

During the nineteenth century the concepts that atoms and molecules are in continual motion and that the temperature of a body is a measure of the intensity of this motion were developed. The idea that the behavior of gases could be accounted for by considering the motion of the gas molecules had occurred to several people (Daniel Bernoulli in 1738, J. P. Joule in 1851, A. Kronig in 1856), and in the years following 1858 this idea was developed into a detailed kinetic theory of gases by Clausius, Maxw eli, Boltzmann, and many later investigators. The subject is discussed in courses in physics and physical chemistry, and it forms an important pau of the branch of theoretical science called statistical mechanics. 

When the manner of addition is affected by the growing chain end, the configurations of the added units will not obey Bernoulli statistics. In the simplest case, first-order Markov statistics will operate. Addition will be characterized by two parameters because the probability of r diad generation by monomer addition to an m end unit, will not be identical with monomer addition to an r end unit, P . The probabilities of m or r diad generation by addition to m or r chain ends will be bound by the relations P = (1 — Pmr) rm 0 rr) According to first-order Markov... 

Actually, the fixation of diad configuration in the polymer chain proceeds only when the next monomer molecule is attached. Until this moment, the end unit can still undergo isomerization becoming either planar or its mirror antipode. In the absence of isomerization, the formation of a stereoregular syndiotactic polymer should be expected. In the case when an end unit has the time (before the attack of the following monomer molecule) to adopt a planar configuration, the distribution of diads in the polymer is described by Bernoulli s statistics (binomial distribution) [42]. If the state to two end units of... 

If isomerization is absent or slow, an isotactic polymer is obtained. In the case of fast isomerization or a planar structure of the end unit, diad distribution in the polymer is described by Bernoulli s statistics. If the rates of isomerization and propagation are comparable, the precise form of the distribution function may be determined on the basis of a method of solution of multicenter problems developed by the author [6],... 

Figure 2.2 Family tree of the Bernoullis (based on Boyer, 1991). Nicholas Senior and Nicholas I were not mathematicians. James I, Nicholas II and Daniel I are particularly important in the history of statistics. Daniel II, Christopher and John-Gustave are not included by Bell (1953) but are included by Boyer (1991) and Stigler (1986).

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