Schultz distribution

Kotlarchyk M, Stephens RB, Huang JS (1988) Study of Schultz Distribution to Model Polydispersity of Microemulsion Droplets. J Phys Chem 92 1533-1538... 

All our results will hold for general p y), subject only to the condition that the Laplace transform exists. Experimentally p(y) in general is not very well known, but is characterized by some low order moments. An often used ansatz is the so-called Schultz distribution... 

H. E. Jensen, Selectivity coefficients of mixtures of ideal cation-exchangers, Agrochimica XIX.-247 (1975). See also K. Bunzl and W. Schultz, Distribution coefficients of 137Cs and 85Sr by mixtures of clay and humic material, J. Radioanalyti. Nucl. Chem. 90(1) 23 (1985). 

Scattering techniques provide the most definite proof of micellar aggregation. Zielinski et aL (34) employed SANS to study the droplet structures in these systems. Conductivity measurements (35) and SANS (36) were also used to study droplet interactions at high volume fraction in w/c microemulsions formed with a PFPE-COO NH4 surfactant (MW = 672). Scattering data were successfully fitted by Schultz distribution of polydisperse spheres (see footnote 37). A range of PFPE-COO NH/ surfactants were also shown to form w/c emulsions consisting of equal amount of CO2 and brine (38-40). 

From a range of possible models a Schultz distribution of spherical particles is often found to give the best fits and physically reasonable parameters. This scattering law may be written... 

Pi, p2 are the core/shell SLDs and several particle shapes were used to calculate P(Q). The best fits were given by a spherical core-shell model with a Schultz distribution [3] ofparticle sizes and the aggregation number (i.e. the number of molecules per micelle, Nagg), the shell-SLD (pz) and the breadth parameter in the Schultz distribution (Z) were adjusted to fit the data. Ri and R were calculated from the fitted parameters. 

Abraham, W. H. (1970) The Flory-Schultz distribution in reversible semi-semi-batch polymerization, Chem. Eng. Sci. 25, 331-335. 

The various molecular weight distributions have been modeled. Two of the most important are the Schultz (65) distribution and the Poisson (66) distribution. Taking n to be the degree of polymerization and as the differential distribution by weight, the Schultz distribution is... 

Particle size distributions have often been approximated by various analytical functions. Note that a physically meaningful distribution is non-zero only for nonnegative radii, and for this reason the popular Gaussian distribution is not very convenient. (However, this function can still be used to approximate narrow size distributions.) An important distribution without this drawback is the generalized exponential distribution (or Schultz distribution) (6, 20), given as follows ... 

Figure 1.13 Theoretical chain length distribution curves based on the Schultz distribution O the Schultz-Flory distribution. Both distributions have X = 50.

The I(Q) data were analyzed using the multimodel FISH program [50] a Schultz distribution of spherical particles gave the best fits and most physically reasonable parameters. The scattering law can be written as... 

For the larger ring sizes, both propagation and depropagation must be taken into account (Equation 1.3b), and a certain measurable concentration of the unreacted monomer appears when the system comes to thermodynamic equilibrium ([M]e,= 1/Kp = kd/kp)- In this case, the molar mass distribution may be described by the most probable (Flory-Schultz) distribution function [92], for which ... 

The olefin distribution in the catalytic processes, on the other hand, tends to foUow the Schultz-Flory equation, where equals the number of moles of olefins having carbon number N, X equals the moles of olefins having two carbon numbers lower, andis a constant depending on the reaction conditions can range from 0.4—0.9 but usually equals 0.6—0.8. 

This distribution is known as the Schultz-Flory or most probable distribution.2S The moments of the molecular weight distribution are ... 

Held, E. E. (1963). Qualitative distribution of radionuclides at Rongelap Atoll, page 167 in Radioecology, Schultz, V. and Klement, A. W., Eds. (Reinhold Publishing Corp., New York). 

Palumbo, R.F. 1963. Factors controlling the distribution of the rare earths in the environment and in living organisms. Pages 533-537 in V. Schultz and A.W. Klement, Jr. (eds.). Radioecology. Reinhold, New York. 

Schrock alkylidyne catalysts, 26 948-949 Schrodinger s equation, 16 734-735 Schultz-Flory distribution, 20 156 Schultz-Flory equation, 17 714 Schulze—Hardy rule, 7 289, 10 121 Schweizer, M. E., 11 248 Schwenzfeier process, 3 641 Science... 

In reactions where several different outcomes are possible, the final product distribution reflects the relative free energies of each transition state when the reaction is under kinetic control (Schultz and Lemer, 1993). Baldwin s rules predict that for acid-catalysed ring closure of the hydroxyepoxide [65] the tetrahydrofuran product [66] arising from 5-exo-tet attack will be preferred... 

Schultz, D. P., Harman, P. D. Uptake, distribution, and elimination of the lampricide 21,5-dichloro-41-nitrosalicyl-anilide (Bayer 73) by largemouth bass (Micropterus salmoides). J. Agrio. Food Chem. (In press)... 

If aU propagation rate coefficients are equal and all termination rate coefficients are equal, we can obtain a simple expression for the chain length distribution, called the Schultz Flory distribution... 

While the catalysts and conditions are very different than in ZN (water is a severe poison for ZN but is a product in FT), both produce linear polymers, and the molecular weight distributions are very similar because they are controlled by relative rate coefficients of propagation kp and termination k. If these are nearly independent of molecular weight one obtains the Schultz Flory size distribution in both processes. 

Figure 2 Heaviest alcohols (wt%) vs. C2+OH (wt%) for a standard, straight-line Schultz-Flory distribution.

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