Random statistics

A classical description of such a structure is of no real use. That is, if we attempt to describe the structure using the same tools we would use to describe a box or a sphere we miss the nature of this object. Since the structure is composed of a series of random steps we expect the features of the structure to be described by statistics and to follow random statistics. For example, the distribution of the end-to-end distance, R, follows a Gaussian distribution function if counted over a number of time intervals or over a number of different structures in space,... 

Consider the first three components listed for the gaseous product. Random statistics favor C as a product despite this, B is the major product. This suggests that equilibrium is achieved via the pathway, A B <= C. This is the expected pathway if isomerization occurs via the 1,3-hydrogen transfer depicted in Eqs. (17)—(19) on this basis, CH2=CH—CD2H, corresponding to B, should be the initial product. The surface product, which had more opportunity for reaction, is, as expected, somewhat closer to the statistical equilibrium. 

Dendrimer synthesis involves a repetitive building of generations through alternating chemistry steps which approximately double the mass and surface functionality with every generation as discussed earlier [1-4, 18], Random (statistical) hyperbranched polymer synthesis involves the self-condensation of multifunctional monomers, usually in a one-pot single series of covalent formation events [31], Random hyperbranched polymers and dendrimers of comparable molecular mass have the same number of branch points and terminal units, and any application requiring only these two characteristics could be satisfied by either architectural type. Since dendrimer synthesis requires many defined synthetic and process purification steps while hyperbranched synthesis may involve a one-pot synthetic step with no purification, the dendrimers will necessarily be a much more expensive material to produce. 

Wang, R, Hickner, M., Kim, Y. S., Zawodzinski, T. A. and McGrath, J. E. 2002. Direct polymerization of sulfonated poly(arylene ether sulfone) random (statistical) copolymers Candidates for new proton exchange membranes. Journal of... 

Direct Copoiymerization of Sulfonated Monomers To Afford Random (Statistical) Copolymers 4595... 

Conventional polymerization methods yield macromolecules mostly with random (statistical) or nearly statistical and only very seldom with alternating distribution of the monomer units (see Sect. 3.4.1). Special methods are required in order to synthesize block and graft copolymers (Sect. 3.4.2). 

When synthesizing random (statistical) and alternating copolymers two phenomena have to be kept in mind ... 

The complex H2Re2(CO)s exchanges about three times faster than H2Os3(CO)io, in agreement with the visual observation of greater adsorption of the former on the Florisil. Both exchanges follow random statistics within the experimental error of the analyses, as indicated in Table I. This implies that adsorption and desorption on the support is competitive with H/D exchange. 

Facility for QC sampling, how the sample taken and the pack resealed, inventory changes. Is random statistical sampling really possible ... 

Although the values of K will depend on the nature of Q, Z, T, and v, there is, for each v, a special situation, the ideal random case, where the sorting of the substituents about the central atom follows the laws of random statistics. For this case, the Z and T substituents become arranged about the Q in a completely random fashion irrespective of other substituents which are attached to Q. The K values for the ideal random case may be derived mathematically. If it is assumed that the Z/T atom ratio is p/q (with p + q = 1), the probability P of having a central atom Q with i Z substituents and (v — i) T substituents in the v sites subject to redistribution is... 

Fig. 26 Distributions of folding times for 128-unit HP copolymers with protein-like, random-block, and random statistics...

Of course, the probability is small that at any instant, the enantiomeric mixture at equilibrium is exactly equimolar the absence of observable chirality phenomena, such as optical activity, is the result of rapid cancelations of random statistical fluctuations of activity in the time domain of observation. In other words, although, at any instant, the mixture (with a high degree of probability) has an excess of one enantiomer or the other, under measurement conditions, it effectively contains an equal number of enantiomeric molecules. When 10,000,000 dissymmetric [i.e., chiral] molecules are produced under conditions which favor neither enantiomorph, there is an even chance that the product will contain an excess of more than 0.021 % of one enantiomorph or the other. It is practically impossible for the product to be absolutely optically inactive [12],... 

Martin and Synge (3) introduced the important concept of theoretical plates into chromatography. Their concept was derived from partition theory and random statistics, and was related to similar ideas developed for extraction and fractional distillation. They supposed that the column could be divided into a number of sections called theoretical plates, and that solutes (dissolved compounds) could be expected to achieve equilibrium between the two phases (mobile and stationary) that exist within each plate. The chromatographic process, like an extraction process, can be visualized to occur when mobile phase (solvent) is transferred to the next plate, where a new equilibrium is established. Theoretical plate numbers of 1000 or more are common for HPLC columns, which means that 1000 separate equilibria must be established to obtain the same degree of separation by solvent... 

Random copolymers are a special type of statistical copolymer in which the distribution of repeat units is truly random (some words of caution are necessary here because older textbooks and scientific papers often use the term random copolymer to describe both random and non-random statistical copolymers). A section of a truly random copolymer is represented below ... 

As can be seen in Figure 7.15 very different lifetimes occur and each dataset shows different lifetimes. In principle the sample material and the size distribution of pores and the open porosity component each generate an annihilation rate (f ) with some relative intensity (I ). The spectrum is then convoluted with the experimental response function (R). Random statistical noise and a certain background (B) level are added. The measured time spectmm M(t)... 

Last, but certainly not least, we need to define a parameter that measures the deviation from random statistics. This is given the symbol %, not to be confused with the Flory-Huggins interaction parameter % (Chapter II). The x we will use here is defined in Equation 6-45 ... 

Samples tested were styrene copolymers of methacrylates, acrylates, vinyl acetate, and acrylonitrile, in addition to ethyl methacrylate-butyl methacrylate copolymers. These samples were dissolved in the initial mobile phase and the injection volume was 0.05-0.2 mL. These samples were prepared by solution polymerization at low conversion and have rather narrow CCD. These samples are random (statistical) copolymers. 

---