Statistical repeat unit

Complex 69 was prepared in about 60% yield with about 15% metallation and a small drop in molecular weight (from 73,000 for PMPS to 68,000 for 69), which, allowing for the increase in statistical repeat unit, indicates some degradation. Related model compounds based on the coordination of phenyl rings in phenyltrimethylsilane and 1,2-diphenyltetramethyldisilane were compared with 69. The UV absorption maxima (224 and 337 nm) were almost unchanged from those of the precursor PMPS (223 and 338 nm), which possibly reflects the low metal loading as... 

Let us study this partition function from a different viewpoint. In order for the helices to be generated on this chain, helical sequences must be selected from the finite length n. Let be the number of helices with length f = 1,2,3,..., n (counted in the number of statistical repeat units). 

For simplicity, let us assume that the volume of a statistical repeat unit of a polymer... 

In this book, we discriminate it from the molecular theta temperature 0 defined in Chapter 1 based on the intramolecular interaction. depends on both intra- and intermolecular interaction. If the interaction between the statistical repeat units can be described by a single excluded volume parameter v in (1.71), these two are identical. In the perturbational calculation of the third virial coefficient, simple substitution of (1.71) cannot explain the observation of positive A3 > 0 at the 0 temperature. In such a case, the third cluster integral must be introduced in addition to the binary cluster integral v. 

To find the entropy of mixing [6,8-1 l],we first introduce a hypothetical lattice cells for the solution of the total volume V. We choose the unit volume to be that of the unit cell and make the customary simplifying assumption that the solvent molecules and the statistical repeat units of the polymer chains occupy the same volume a. We count the... 

Consider a binary mixture of linear polymers R A/ and R Bg carrying associative groups A and B. The number of statistical repeat units on a chain (for simplicity referred to as the degree of polymerization, DP) is assumed to be ha for R Ay chains and b for R Bg chains. Although we use the word polymer for the primary molecules before they form associated complexes, we may apply our theory to low-molecular weight molecules equally well by simply fixing ma and b at small values. 

Let n/ be the number of statistical repeat units on an /-functional primary molecule, and let Nfhe the total number of molecules in the solution. The number of repeat units per functional group R = uf/f is assumed to be independent of /. The weight fraction Pf of the associative groups carried by the molecules with specified / relative to the total number of associative groups is given by... 

The probabilities of the various dyad, triad, and other sequences that we have examined have all been described by a single probability parameter p. When we used the same kind of statistics for copolymers, we called the situation one of terminal control. We are considering similar statistics here, but the idea that the stereochemistry is controlled by the terminal unit is inappropriate. The active center of the chain end governs the chemistry of the addition, but not the stereochemistry. Neither the terminal unit nor any other repeat unit considered alone has any stereochemistry. Equations (7.62) and (7.63) merely state that an addition must be of one kind or another, but that the rates are not necessarily identical. 

When we discussed random walk statistics in Chap. 1, we used n to represent the number of steps in the process and then identified this quantity as the number of repeat units in the polymer chain. We continue to reserve n as the symbol for the degree of polymerization, so the number of diffusion steps is represented by V in this section. 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

Copolymers are built from the repetition of two (or more) "repeat units". Depending on the spatial arrangement of those units (e.g., A and B) along the chain (sequencing), various types of copolymers can be made alternating, block, random (or statistical), branched, crosslinked (see Figure 12). 

Telechelic polymers rank among the oldest designed precursors. The position of reactive groups at the ends of a sequence of repeating units makes it possible to incorporate various chemical structures into the network (polyether, polyester, polyamide, aliphatic, cycloaliphatic or aromatic hydrocarbon, etc.). The cross-linking density can be controlled by the length of precursor chain and functionality of the crosslinker, by molar ratio of functional groups, or by addition of a monofunctional component. Formation of elastically inactive loops is usually weak. Typical polyurethane systems composed of a macromolecular triol and a diisocyanate are statistically simple and when different theories listed above are... 

In this model the linear chains become the rays of a star molecule. The rays, consisting of m repeating units, can be considered stiff rods, but in most cases they will be flexible and can be described in a first approximation by Gaussian chain statistics. A star molecule has only one branching unit among/xm units which belong to Hnear chains. Their properties can be expected to show a close... 

Structural disorder resulting from the statistical co-crystallization of different configurational repeating units (see Fig. 2). 

Statistical co-crystallization of different constitutional repeating units, which may either belong to the same copolymer chains (copolymer isomorphism) or originate from different homopolymer chains (homopolymer isomorphism). 

Statistical copolymers containing repeating units each with a different functional group can be obtained using appropriate mixture of monomers. For example, a polyestermide can be synthesized from a ternary mixture of a diol, diamine, and diacid or a binary mixture of a diacd and amine-alcohol [East et al., 1989]. 

Alternating, statistical, and random copolymers are named by following the prefix poly with the names of the two repeating units. The specific type of copolymer is noted by inserting -alt-, -stat-, or -ran- in between the names of the two repeating units with -co- used when the type of copolymer is not specified, for example, poly(styrene-ofi-acrylonitrile), poly(styrene-i fat-acrylonitrile), poly(styrene-ran-acrylonitrile), and poly(styrene-co-acrylonitrile) [Wilks,... 

Figure 26 shows an analysis of the area ratios for the core and 02j levels which are also essentially core like (cf. Fig. 15). For polyacrylic acid the measured areas for the various structural features for the 0ls levels and Cts levels and the overall ratios for the Cis to 0ls levels (corrected for differing sensitivity factors) are 1.0, 2.0 and 1.6 respectively, in excellent agreement with the theoretical values of 1.0, 2.0 and 1.5 based on a statistical sampling of the polymer repeat unit. The area ratios for the individual components for the Cu levels show an excellent correlation with the number of carbon atoms in the alkyl groupsb. (Slope Experimental 0.99, Theoreti-...

For comparison purposes the dotted lines indicate the correlations expected for samples which on the ESCA depth profiling scale correspond to a statistical sampling of the appropriate repeat unit in the polymer. It is clear that there are considerable deviations from such correlations in a direction which overall suggests that the samples are oxidized. 

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