Ideal randomness

Table II also demonstrates the discrepancy existing between E0/RTe calculated by the Yang-Li quasi-chemical theory and the experimental ratio. E0 is the energy difference between a fully ordered superlattice and the corresponding solid solution with an ideally random atom species distribution. It is a quantity that can only be estimated from existing experimental information, but the disparity between theory and experiment is beyond question.

The exponent a is a function of solvent power usually a > 1/2, but for an idealized random-flight polymer, a = 1/2. 

Fig. 10 Equilibrium radius of gyration of a molecule plotted as a function of temperature the molecule is composed of 1000 beads. The radius of gyration shows a steep increase and a large fluctuation above 700 K. The insets show typical chain conformations at indicated temperatures. Note that the ideal random coil state of this fully flexible chain should have the mean-square radius of gyration R2 = 1000 x (1.54/3.92)2/6 = 25.7, the value is around 800 K...

What is the value of riV2 for an ideal random copolymer ... 

The degree of orientation [91] was defined by an orientation index, S, which has an upper limit indicative of perfect orientation along a given axis and a lower limit corresponding to an ideally random orientation. A particularly useful definition for S is ... 

In ideal random crosslinking polymerization or crosslinking of existing chains, the reactivity of a group is not influenced by the state of other groups all free functionalities, whether attached or unattached to the tree, are assumed to be of the same reactivity. For example, the molecular weight distribution in a branched polymer does not depend on the ratio of rate constants for formation and scission of bonds, but only on the extent of reaction. Combinatorial statistics can be applied in this case, but use of the p.g.f. simplifies the mathematics considerably. 

The calculated root-mean-square displacement for a general sequence of jumps has two terms in Eq. 7.31. The first term, NT(r2), corresponds to an ideal random walk (see Eq. 7.47) and the second term arises from possible correlation effects when successive jumps do not occur completely at random. 

For a random walk, f = 1 because the double sum in Eq. 7.49 is zero and Eq. 7.50 reduces to the form of Eq. 7.47. In principle, f can have a wide range of values corresponding to physical processes relating to specific diffusion mechanisms. This is readily apparent in extreme cases of perfectly correlated one-dimensional diffusion on a lattice via nearest-neighbor jumps. When each jump is identical to its predecessor, Eq. 7.49 shows that the correlation factor f equals NT.6 Another extreme is the case of f = 0, which occurs if each individual jump is exactly opposite the previous jump. However, there are many real diffusion processes that are nearly ideal random walks and have values of f 1, which are described in more detail in Chapter 8. 

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... 

Relative Distribution of Molecules for the Ideally Random Case for Two Exchanging Substituents on One Type of Central Moiety at the Overall Composition R—vl2... 

Equilibrium Constants for the Ideally Random Redistribution of Two Kinds of Substituents on a Central Moiety... 

Fig. 1. Equilibrium distribution of molecules for the ideal random case in systems QZV vs QT as a function of the composition parameter R. A, v=2 B, v=3 C, v=4.

For the ideal random case, the AH term is zero, thus giving the following equation for the free energy ... 

One is generally interested in the enthalpies of scrambling reactions since these may be measured thermochemically and thus brought into relationship with equilibrium constants. Real scrambling equilibria, however, deviate more or less from the ideal random case. And, in order to be able to calculate enthalpies, it is assumed that for large values of the enthalpy, (JS)real (JS)rand. Accordingly, we may now estimate the enthalpy of a real scrambling reaction. 

The good agreement of the equilibrium constants with the values for ideal randomness (Table II) confirms Calingaert s conclusions. Analyses of tetramethyl- and tetraethyllead redistribution mixtures today may be performed conveniently by gas chromatography (20, 67, 232). 

In boron trihalide adducts and tetrahaloborate ions, halogen redistribution equilibria are reasonably close to this ideal random case when chlorine, bromine, and iodine are involved (27, 28, 80, 100, 112), as are equilibria in the uncomplexed heavier boron trihalides (111). 

Statistical errors that cannot be prevented. Even for an ideal random mixture the quantitative distribution in samples of a given magnitude is not constant but is subject to random fluctuations. It is the only sampling error, which cannot be suppressed and occurs in ideal sampling. It can be estimated beforehand and reduced by increasing the sample size. 

Unfortunately, for some solvent-polymer combinations, even for nearly ideal random coils such as polystyrene, the coefficients are not constant but vary with molar mass. 

The needle-like crystallites, when packed into a flat sample, will also tend to align parallel to the surface. However, the preferred orientation axis, which in this case coincides with the elongated axes of the needles, will be parallel to the sample surface. In addition to the nearly unrestricted distribution of needles axes in the plane parallel to the sample surface (which becomes nearly ideally random when the sample spins around an axis perpendicular to its surface), each needle may be freely rotated around its longest direction. Hence, if the axis of the needle coincides, for example, with the vector d. then reflections from reciprocal lattice points with vectors parallel to will be suppressed to a greater extent and reflections from reciprocal lattice points with vectors perpendicular to d / will be strongly increased. This example describes the so-called in-plane preferred orientation. 

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