Correlation time constants

For complex ions of similar size within a homologous series, it is expected that t//tc remains relatively constant for different solvents at a given temperature, in accordance with the Stokes-Einstein-Debye model. From the slope of the plot 5iso versus y/(Az/f/2)/A av shown in Fig. 16, the correlation time constants of a series of tra 5 -[Co(acac)2XY] complexes were estimated and the results were compared favourably with the Tc obtained from the relaxation measurements of the methene carbons (see Table 5). 

Table 5.6 Correlation time constants Too and activation energies AE for the reorientation motions in methyl-substituted naphthalene crystals. The correlation time is = Too ( = single crystals).

Table 2.2. Correlation time constants for the diffusive translational and rotational motions of a single molecule in liquid water ...

Here To = is the correlation time constant of random process e(t) e(t)... 

Figure 2.6 depicts the reorientational time correlation function (RTCF) of ranks, / = 1 and 2 for a representative composition, x, = 1.0 (upper panel) and the product of translational diffusion coefficient and rotational correlation time (D x t ) as a function of Xj (lower panel). RTCF has been calculated by Equation 2.17. The upper panel shows that the RTCF of first rank (/ = 1) decays at a rate slower than that of second rank (Z = 2). This is expected. For other compositions, this trend remains the same. Rotational correlation time constant has been obtained via time integration of RTCF as follows ... 

Unfortunately, all measurements in two last perfect works were done at room temperature, and the energetic parameters could not be determined. Also, the authors did not try to simulate ERR spectra, therefore, they could operate only with average rotational correlation time constants while it was recently proved that Tc values differ noticeably for the... 

Also we must bear in mind that the advancement of the coordinates fidfds two fiinctions (i) accurate calculation of dynamical properties, especially over times as long as typical correlation times x (ii) accurately staying on the constant-energy hypersurface, for much longer times Exact time reversibility is highly desirable (since the original equations... 

Figure 8 Effects of spin diffusion. The NOE between two protons (indicated by the solid line) may be altered by the presence of alternative pathways for the magnetization (dashed lines). The size of the NOE can be calculated for a structure from the experimental mixing time, and the complete relaxation matrix, (Ry), which is a function of all mterproton distances d j and functions describing the motion of the protons, y is the gyromagnetic ratio of the proton, ti is the Planck constant, t is the rotational correlation time, and O) is the Larmor frequency of the proton m the magnetic field. The expression for (Rjj) is an approximation assuming an internally rigid molecule.

Objective Evaluation of Color. In recent years a method has been devised and internationally adopted (International Commission on Illumination, I.C.I.) that makes possible objective specification of color in terms of equivalent stimuli. It provides a common language for description of the color of an object illuminated by a standard illuminant and viewed by a standard observer (H). Reflectance spectro-photometric curves, such as those described above, provide the necessary data. The results are expressed in one of two systems the tristimulus system in which the equivalent stimulus is a mixture of three standard primaries, or the heterogeneous-homogeneous system in which the equivalent stimulus is a mixture of light from a standard heterogeneous illuminant and a pure spectrum color (dominant wave-length-purity system). These systems provide a means of expressing the objective time-constant spectrophotometric results in numerical form, more suitable for tabulation and correlation studies. In the application to food work, the necessary experimental data have been obtained with spectrophotometers or certain photoelectric colorimeters. 

Up to now it has been tacitly assumed that each molecular motion can be described by a single correlation time. On the other hand, it is well-known, e.g., from dielectric and mechanical relaxation studies as well as from photon correlation spectroscopy and NMR relaxation times that in polymers one often deals with a distribution of correlation times60 65), in particular in glassy systems. Although the phenomenon as such is well established, little is known about the nature of this distribution. In particular, most techniques employed in this area do not allow a distinction of a heterogeneous distribution, where spatially separed groups move with different time constants and a homogeneous distribution, where each monomer unit shows essentially the same non-exponential relaxation. Even worse, relaxation... 

H NMR 38,39,42, 50-55 Hole capacity constant 205 Homogeneous distribution of correlation times 37 Humic acids 17 Humidity of plastics 119 Humins 17 Hydrogen bond 200 Hydrophilic 191, 194, 206... 

In order to find the correlation time ze = zp of rotational energy, it is necessary to eliminate the constant component Ke(00) from Eq. (1.56). 

Experimental distances from NOEs/ROEs of small molecules are recommended not to be classified into regions of small, medium, and large as it is often done in the structure determination of large molecules. As opposed to macromolecules, the overall correlation time Tc can be considered constant in small molecules. Thus, it is possible to measure distances in the range between 2 and 5 A with an accuracy of about 10%. Often distances between protons are almost exclusively used for the structure determination. This leads to the fact that molecules with small numbers of hydrogen atoms are more difficult to determine. 

The choice of capacity is sometimes a problem, and may change according to the particular circumstance. Sometimes using a definition of time constant, based on the above equations, is not very helpful and other means must be employed. For example, mixing time is a very important time constant relating to liquid mixing, and this is best obtained directly from empirical correlations of experimental data. 

Mixing time constants, Tmix are also available based on an empirical correlation and are usually closely related to the value of Xcirc (Joshi et al., 1982). A value of Tmix = 4 Tcirc is often used for stirred vessels and a value of mix = 2 to 4 Tcirc for columns. The exact value strongly depends on the degree of mixing obtained. 

Investigation of water motion in AOT reverse micelles determining the solvent correlation function, C i), was first reported by Sarkar et al. [29]. They obtained time-resolved fluorescence measurements of C480 in an AOT reverse micellar solution with time resolution of > 50 ps and observed solvent relaxation rates with time constants ranging from 1.7 to 12 ns. They also attributed these dynamical changes to relaxation processes of water molecules in various environments of the water pool. In a similar study investigating the deuterium isotope effect on solvent motion in AOT reverse micelles. Das et al. [37] reported that the solvation dynamics of D2O is 1.5 times slower than H2O motion. 

All tuning relations provide different results. Generally, the Cohen and Coon relation has the largest proportional gain and the dynamic response tends to be the most underdamped. The Ciancone-Marlin relation provides the most conservative setting, and it uses a very small derivative time constant and a relatively large integral time constant. In a way, their correlation reflects a common industrial preference for PI controllers. 

The correlation time (time constant) corresponds to the lag at which the maximum value Wxx(0) is decreased by the factor 1/e and the correlation becomes practically zero. Correlation time is an important quantity in sam-... 

Fig. 2.8. Autocorrelation function (ACF) of a stochastic process. rc is the correlation time (time constant)...

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