Root mean square separation

Root mean square separation one measure of the average separation of the ends of a random coil. It is the square root of the mean value of R, where R is the. separation of the two ends of the coil. This mean value is calculated by weighting each possible value of R- with the probability, / (eqn 19.27), of that value of R occurring. It is proportional to ftf length of each unit (eqn 19.31). 

Radius of gyration the radius of a thin hollow spherical shell of the same mass and moment of inertia as the macromolecule. In general, it is not easy to visualize this distance geometrically. However, for the simple case of a molecule consisting of a chain of identical atoms this quantity can be visualized as the root mean square distance of the atoms from the center of mass. It also depends on but is smaller than the root mean square separation by a factor of (1/6) (eqn 19.33). 

In the Gaussian statistical theory of rubbery behavior the important parameters can be summarized as follows. In the initial unstretched state the root-mean-square separation ro of chain ends is... 

There are several measures of the geometrical size of a three-dimensional random coil. The root mean square separation, is a measure of the average separation of the ends of the coil ... 

We see that as the number of residues N (each of length /) increases, the root mean square separation of its ends increases as and consequently the volume... 

Distinguish between contour length, root-mean-square separation, and radius of gyration of a random coil. 

Calculate the contour length (the length of the extended chain) and the root mean square separation (the end-to-end distance) for... 

The dumbbell relaxation time (t) in the preceding model is coil deformation dependent. Neglecting Brownian forces, the dumbbell relaxation time is given by t ssf H/fs. Equation (45) is then tantamount to saying that t increases approximately in proportion to the root mean square end-to-end separation distance R [52] ... 

The root-mean-square distance Vr separating the ends of the polymer chain is a convenient measure of its linear dimensions. The dissymmetry coefficient will be unity for (VrV 0< l and will increase as this ratio increases. 

On the other hand, the correction factor by which W r) is altered through this refined treatment, namely, exp[ —(9n/20)(r/r, ) ] from Eq. (16), depends both on n and on r/Vm If the distance of separation of the ends of the chain lies in the vicinity of its root-mean-square value, i.e., if r / then... 

As noted earlier, the diffraction of X-rays, unlike the diffraction of neutrons, is primarily sensitive to the distribution of 00 separations. Although many of the early studies 9> of amorphous solid water included electron or X-ray diffraction measurements, the nature of the samples prepared and the restricted angular range of the measurements reported combine to prevent extraction of detailed structural information. The most complete of the early X-ray studies is by Bon-dot 26>. Only scanty description is given of the conditions of deposition but it appears likely his sample of amorphous solid water had little or no contamination with crystalline ice. He found a liquid-like distribution of 00 separations at 83 K, with the first neighbor peak centered at 2.77 A. If the pair correlation function is decomposed into a superposition of Gaussian peaks, the area of the near neighbor peak is found to correspond to 4.23 molecules, and to have a root mean square width of 0.50 A. 

Root Mean Square Error of Prediction (RMSEP) Plot (Model Diagnostic) Prediction error is a useful metric for selecting the optimum number of factors to include in the model. This is because the models are most often used to predict the concentrations in future unknown samples. There are two approaches for generating a validation set for estimating the prediction error internal validation (i.e., cross-validation with the calibration data), or external validation (i.e., perform prediction on a separate validation set). Samples are usually at a premium, and so we most often use a cross- validation approach. 

Authors found a root mean square error of 28.3°C with one derivation set of 979 compounds with no hydrogen bonding (Yalkowsky et al., 1994). A standard error of 28.1 K was reported for the 1,425-compound derivation set including hydrogen bonding compounds. A separate test set of 39 compounds yielded an average absolute error of 23.9 K (6.6%) (Walters et al., 1995). 

In multilayered lattices, even in such ones, for which the macroscopic characteristics are not distinguished by an appreciable anisotropy (as, for example, HTSC type 1-2-3), the interaction between separate atoms or atomic groups can be strongly anisotropic. The "damping" interaction propagation between layers inherent in substances of the specified class may result in appreciable manifestation of such local anisotropy both in the phonon spectrum [15] and in the behaviour of some vibrationary characteristics, in particular the root-mean-square displacement of atoms from separate layers along various crystal directions. 

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