Theta factor

Heat-transfer rate kW Btu/b 0 Holland theta factor ... 

The theta factor (which turns out to be a rather complicated function) can be found5 by direct solution of the Schrodinger equation or by the use of ladder operators. The work is involved and we omit it. The spherical-top wave functions (5.60) are identical with the symmetric-top wave functions (5.68)—see Problem 5.14. 

Temperature affects all the physical properties relevant in mass transfer viscosity, density, surface tension, and diffusivity. The empirical factor most often used to account for temperature changes in all these parameters is the theta factor, 0,... 

Stenstrom M K and Gilbert R G (1981) Review paper Effects of alpha, beta and theta factor upon the design specification and operation of aeration systems, Water Research 15 643-654. 

Metal deposition occurs with sharp gradients within a catalyst pellet, usually concentrated on the outside of catalyst pellets forming a U-shaped distribution. Sato et at [3] related this metal deposition with simultaneous diffusion and reaction, and suggested a value of 8 for the Thiele modulus in a slab geometry, Tamm [4] suggested that this distribution can be characterized by a theta factor defined in a ( Undricat geometry as... 

Coyne, J. Kaufman, P. Post, T.A. Key parameters for consideration in up-pumping technology in fermentation. Proceedings INTERPHEX East, ISPE, Philadelphia, PA, March 19, 1998. Stenstrom, M.K. Gilbert, R.G. Effects of alpha, beta, theta factor upon the design, specification and operation of aeration systems. Water Res. 1981, 15, 643-654. 

Equations (5.106) and (5.104) give the explicit formula for the normalized theta factor in the angular-momentum eigenfunctions. Using (5.106), we construct Table 5.1, which gives the theta factor in the angular-momentum eigenfunctions. 

The linearity of L with N is maintained at the theta point. Relative to Eq. 5, the chains have shrunk by a factor of (a/d),/3 but the linear variation indicates that the chains are still distorted at the theta point and characteristic dimensions do not shrink through a series of decreasing power laws as do free chains [29-31]. Experimentally, Auroy [25] has produced evidence for this linearity even in poor solvents. Pincus [32] has recently applied this type of analysis to tethered polyelectrolyte chains, where the electrostatic interactions can produce even stronger stretching effects than those that have been discussed for good solvents. Tethered polyelectrolytes have also been studied by others [33-35],... 

The plateau adsorbances at constant molecular weight increased linearly with the square root of NaCl concentration. For the same NaCl concentration the adsorbance was nearly independent of the molecular weight. The thickness of the adsorbed layer was approximately proportional to the square root of the molecular weight for the Theta solvent (4.17 M NaCl). For good solvents of lower NaCl concentrations the exponent of the molecular weight dependence of the thickness was less than 0.5. At the same adsorbance and molecular weight the cube of the expansion factor at, defined by the ratio of the thicknesses for good solvent and for Theta solvent, was proportional to the inverse square root of NaCl concentration. 

Expansion of Thickness of the Adsorbed Layer. In the low salt concentration the large thickness compared with the case of the Theta solvent (4.17 M NaCl) is considered to be due to the electrostatic repulsion, i.e., the excluded volume effect of the adsorbed NaPSS chains. Usually, the expansion factor at, defined by the ratio of the thickness in good solvent and that in the Theta solvent, is used to quantitatively evaluate the excluded volume effect for the adsorbed polymers. 

On the other hand, factor analysis involves other manipulations of the eigen vectors and aims to gain insight into the structure of a multidimensional data set. The use of this technique was first proposed in biological structure-activity relationship (i. e., SAR) and illustrated with an analysis of the activities of 21 di-phenylaminopropanol derivatives in 11 biological tests [116-119, 289]. This method has been more commonly used to determine the intrinsic dimensionality of certain experimentally determined chemical properties which are the number of fundamental factors required to account for the variance. One of the best FA techniques is the Q-mode, which is based on grouping a multivariate data set based on the data structure defined by the similarity between samples [1, 313-316]. It is devoted exclusively to the interpretation of the inter-object relationships in a data set, rather than to the inter-variable (or covariance) relationships explored with R-mode factor analysis. The measure of similarity used is the cosine theta matrix, i. e., the matrix whose elements are the cosine of the angles between all sample pairs [1,313-316]. 

The osmotic second virial coefficient A2 is another interesting solution property, whose value should be zero at the theta point. It can be directly related with the molecular second virial coefficient, expressed as B2=A2M /N2 (in volume units). For an EV chain in a good solvent, the second virial coefficient should be proportional to the chain volume and therefore scales proportionally to the cube of the mean size [ 16]. It can, therefore, be expressed in terms of a dimensionless interpenetration factor that is defined as... 

Fig. 11. A log-log plot of the expansion factor, a, vs reduced excluded volume, z% for MC data of 12-arms stars with N=25-109 units triangles N=25 crosses N=49 asterisks N=85 squares N=109. The top and bottom solid lines and figures represent slopes corresponding to the predicted asymptotic behaviors for the EV and sub-theta regimes, respectively. Reprinted with permission from [143]. Copyright (1992) American Chemical Society...

Fig. 16. Generalized Kratky plot of the form factor of a star from numerical data of a MG simulation [161]. x=q . e/kgT=0.1 corresponds to EV chains ande/kgT=0.3 is close to the theta state...

The first principal component values (Theta 1) for each sample were determined and these values were correlated with the total PCB concentration (Figure 14) recorded for each sample in a separate computer data base that contained other environmental data such as hydrology and sediment texture. The results indicated that certain samples deviated by factors of about two. Upon examining the sample records, the recorded dilution values... 

Ratio of a dimensional characteristic of a macromolecule in a given solvent at a given temperature to the same dimensional characteristic in the theta state at the same temperature. The most frequently used expansion factors are expansion factor of the mean-square end-to-end distance, Ur = (/o) expansion factor of the radius oj gyration, as = (/0) relative viscosity, = ([ /]/[ /]o), where [ ] and [ /]o are the intrinsic viscosity in a given solvent and in the theta state at the same temperature, respectively. 

These results indicate that the presence of the wave theta is revealed by the value of the expected visibility. If the visibility is one, the 0 waves do not exist, meaning that the quantum waves are mere mathematical probability waves devoid of any physical meaning. If the fringe pattern is blurred and the visibility decreased by a factor of, then it would imply that quantum waves, just like any ordinary wave, are real. 

In principle, intrinsic viscosities used for estimating branching should be measured under conditions where the expansion factor a is unity, but as indicated in Section 6, it is not easy to identify such conditions. Some authors, e.g. Moore and Millns (40) have measured [tf at the theta-temperature of the corresponding linear polymer, but it is doubtful whether a is unity at that temperature for either linear or branched polymer, if the theories of Casassa or of Candau et al. are valid. If a were the same for both linear and branched polymers under the same conditions g would be unaffected and g could be measured at any convenient temperature some authors have presented data suggesting that g is nearly the same in good and poor solvents, e.g. Hama (42) and Graessley (477), but other authors, e.g. Berry (43) have found g to vary. The best that can be done at present would appear to be to measure g at the theta-temperature on the assumption that this ratio will be less temperature-sensitive than either intrinsic viscosity, and that even if this temperature is not the correct one it will be near it. Errors in estimates of branching due to this effect are likely to be much less serious than those due to the use of an incorrect relation between g and g0. 

The radius of gyration is expected to be different under theta and nontheta conditions since the extent of coil swelling due to imbibed solvent changes with solvent goodness. We define a coil expansion factor a as follows ... 

The standard method for normalisation of diffracted intensity data into electron units, is to compute both the mean square atomic scattering factor and the mean incoherent scatter for the particular molecular repeat over a range of high two theta values (say 40°-60°) where their total value can be considered to be equivalent to the actual diffraction from the molecular system concerned. An appropriate normalisation factor is then applied to the experimental intensity data after geometrical correction and, finally, incoherent scatter is subtracted ( 1 ). 

Figure 1. Corrected equatorial trace in electron units for a PET specimen normalized over the full two theta range (]2 mean-square atomic scattering factor C incoherent scatter t total scatter)...

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