Poisson-type distribution

The catalytic single-step Alfen process has a good space-time yield, and the process engineering is simple. The molecular weight distribution of the olefins of the single-step process is broader (Schulz-Flory type of distribution) than in the two-step Alfen process (Poisson-type distribution) (Fig. 2). As a byproduct 2-alkyl-branched a-olefins also are formed, as shown in Table 6. About... 

This is a Poisson-type distribution, narrower than that from eqn 10.47. Any chain transfer that does not terminate the kinetic chain reduces the molecular weight, but does not alter the shape of the distribution, which remains given by eqn 10.48. 

R=C2H5i i-C H,) system used for low temperature (from —65 °C to —78 °C) propylene polymerization giving a syndiotactic polymer. Such a result, which is quite close to that of a Poisson type distribution, can be experimentally justified by the polymerization, in absence of transfer and termination reactions, taking on the characteristics of a living one. 

The density of the 7=3 states determined from the experimental data (Krasznahorkay et al. 1999) is close to a Wigner quasi-probability distribution (Brody et al. 1981) but the mixing-in of some Poisson type distribution is also visible. The density of / = 3 states has also been calculated using the back-shifted Fermi-gas description with parameters determined by Rauscher et al. (1997). The calculated curve had to be shifted by 2.7 MeV to reproduce the experimental values (Krasznahorkay et al. 1999). The energy of the ground state in the third well is obtained to be Em = (3.1 0.4) MeV. 

At low temperatures (90-120T), using ethene, insertion is very much faster than transalkylation. Each of the alkyl chains grows by successive insertions of ethene. The rates of insertion are essentially independent of chain length. A Poisson type distribution of chain lengths is produced in which the average length depends on the number of ethene molecules added per metal atom. 

As was discussed in Sect. 6.2, the complex shear modulus G co) of the simplest ladder model follows a behavior [100], so Eq. 156 gives us tmax — tiadMu c( = 2). Note that the same dependence of tmax on the size of a ladder network was also proved numerically [100]. Therefore, using a Poisson-type distribution, P(Nu) exp[- (Na/Ni)] and setting 5 = 1, one has from Eq. 159 [151] ... 

The chain length distribution in the original Ziegler Aujbaureaktion is adjusted by the ratio of ethylene to aluminum in the growth reaction in which the provided ethylene competes for the available Al-centers. After elimination in the second step this results in a Poisson-type distribution (Figure 6.16.1) that has the highest probability at the chain length that corresponds to an even distribution of all ethylene to the coordination sites at the aluminum. 

The samples obtained under such conditions exhibit a Poisson-type distribution of their molar masses. This type of distributions is obtained when one distributes in a random way m objects in n boxes, with m>>>n. 

Conventional radical polymerization usually produces polymers with a broad distribution in DP. The polymers are mixtures of the instantaneous polymers with DPw/DPn of at least 1.5 for the termination by recombination or 2.0 either for the termination by disproportionation or for the chain transfer to small molecules. In this respect, any living polymerization with rapid initiation will afford polymers with a narrow DP distribution of the Poisson type. Ring-opening met-hathesis polymerization of norbornenyl-terminated macromonomers, 8, 15, and 16, appears promising in this regard [22,23]. 

Some information can be obtained about the rate coefficients of ion-pair dissociation from measurements of molecular weight distribution. If the lifetime in any of the ionic states is short compared with its rate of reaction with monomer, the molecular weight distribution will be the simple narrow Poisson type expected for rapid initiation, propagation via a single species and no termination. A slower ion-pair dissociation process... 

Since the process represented by reactions (4a) and (4b) is a simple addition polymerization it leads to a distribution of molecular weights of the Poisson type [2]. At any st e of the reaction the concentration of polymer molecules containing n a-amino acid units [P ] is given by... 

This type of distribution is called a Poisson s distribution... 

If a radioactive sample is measured repeatedly with a constant measuring time, the results will scatter, showing a Poisson-type frequency distribution. The precision of a measurement method can be characterized by the relative standard deviation ... 

However, for the conducting Cu sample, simulation results indicate that the temperature is usually lower than that of AI2O3 case, whereas the temperature difference inside the sample at steady state is higher for Cu than for AI2O3. Figure 6.28 shows the radial mechanical displacement distribution at steady state, as the system is applied with a current of 1000 A and a pressure of 140 MPa for the case of alumina and copper sample. The corresponding radial and axial components of the stress distribution field, due to thermal and Poisson-type expansion, are illustrated in Fig. 6.29, for the AI2O3 (a and c) and Cu (b and d) samples [41]. 

In the plunger-die-sample region, the radial displacement increases relatively uniformly with increasing radius, and thus, the highest displacement is observed near the surface of the die. These radial displacements are due to the synergistic effect of the thermal expansion and a Poisson-type expansion due to the pressure applied in the vertical direction. In contrast, the radial distribution, in the case of the Cu sample (Fig. 6.29b), does not increase uniformly with the radial position. Instead, the portion where the copper is in contact with the die wall has larger... 

The electric potential is related to the bulk charge density through the Poisson equation (Landau and Lifshitz 1960). If the Boltzmann type distribution (14) is substituted into the Poisson equation and the result is integrated with respect to z, starting from the middle plane, the following first integral in the lubrication approximation takes place ... 

One of the common types of information available to safety advisers is the number of accidents sustained by employees in a factory or department. Statistically, this information is referred to as the distribution of accidents over the group, and it is usual to find that most of the group have no accidents, some have only one, while certain individuals have four, five or more. Where a high number is found, the obvious action is to investigate the reasons for them, but to save effort it is useful to have a method for determining whether or not an observed distribution of accidents comes within the compass of chance variation. The method makes use of Poisson s distribution ° which enables the expected distribution of the number of... 

In turbulent flow regime (J Cp>I00), the fluid entering each void is considered to be fully mixed and overall dispersion phenomenon can be well described by a tanks-in-series model, where the residence time in a tank is equated with the residence time of flowing fluid in a void of the length Pidf, Pidfl u. Then the tanks-in-series model gives the residence time distribution of the Poisson type (Aris and Amundson, 1957), which can be approximated by the impulse response of the dispersion model by equating... 

Fig. 1.3. Molecular weight distributions of the Schulz-Zimm type for / = 2 (left) and of the Poisson-type (right). Both correspond to the same number average degree of polymerization, Nn = 10 ...

There are a number of ways to model the interfacial electric field in SCMs, involving various combinations of a Poisson-Boltzmann distribution and/or one or more parallel-plate capacitors [21]. Originally, modelers treated essentially all the building blocks of an SCM - electrostatic model, site types, site densities, and equilibrium constants - as fitting parameters. It was soon discovered, however, that one could usually fit potentiometric titration data with multiple SCMs that posit widely diverging molecular-scale pictures of the interface [22],... 

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