Mass exponent models

Experimental values of the molar mass exponent close to 2 have been obtained. For example, for poly(methyl methacrylate), a value of 2.45 has found (see P. Prentice, Polymer, 1983, 24, 344—350). As with values of selfdiffusion coefficient, this has been regarded as close enough to 2 for reptation to be considered a good model of the molecular motion occurring at the crack tip. 

For small molecules (metabolites, monomers, and small oligomers) the mobility equation may be empirically approached with the Offord model (linear relation to charge-to-size ratio, the charge being obtained directly from the ionization constants and the size being approached with the molecular mass exponent a factor a) [see Offord (1966)]. 

Inherent to all mass bias models stemming from Eq. (5.7) is the built-in variable (discrimination exponent) that distinguishes the various mass discrimination phenomena. Hence we have linear, exponential, equilibrium, power, and other discrimination models. This variable, in turn, is often used to identify the presence of a particular discrimination model [32, 41], Consequently, considerable effort has been spent in extracting the numerical value of the mass bias discrimination... 

The Freundlich isotherm (or Freundlich model) is an empirical description of species sorption similar to the K, approach, but differing in how the ratio of sorbed to dissolved mass is computed. In the model, dissolved mass, the denominator in the ratio, is raised to an exponent less than one. The ratio, represented by the Freundlich coefficient Kf, is taken to be constant, as is the exponent, denoted f, where 0< f <1. As before, the masses of dissolved and sorbed species are entered, respectively, in units such as moles per gram of dry sediment and moles per cm3 fluid. Since the denominator is raised to an arbitrary exponent rif, the units for Kf are not commonly reported, and care must be taken to note the units in which the ratio was determined. 

Alternate mass-core hard potential channel In the two billiard gas models just discussed there is no local thermal equilibrium. Even though the internal temperature can be clearly defined at any position(Alonso et al, 2005), the above property may be considered unsatisfactory(Dhars, 1999). In order to overcome this problem, we have recently introduced a similar model which however exhibits local thermal equilibrium, normal diffusion, and zero Lyapunov exponent(Li et al, 2004). 

Model simulations of particle volume concentrations in the summer as functions of the particle production flux in the epilimnion of Lake Zurich, adapted from Weilenmann, O Melia and Stumm (1989). Predictions are made for the epilimnion (A) and the hypolimnion (B). Simulations are made for input particle size distributions ranging from 0.3 to 30 pm described by a power law with an exponent of p. For p = 3, the particle size distribution of inputs peaks at the largest size, i.e., 30 pm. For p = 4, an equal mass or volume input of particles is in every logaritmic size interval. Two particle or aggregate densities (pp) are considered, and a colloidal stability factor (a) of 0.1 us used. The broken line in (A) denotes predicted particle concentrations in the epilimnion when particles are removed from the lake only in the river outflow. Shaded areas show input fluxes based on the collections of total suspendet solids in sediment traps and the composition of the collected solids. 

While the Rouse model predicts a linear time evolution of the mean-square centre of mass coordinate (Eq. 3.14), within the time window of the simulation t<9 ns) a sublinear diffusion in form of a stretched exponential with a stretching exponent of (3=0,83 is found. A detailed inspection of the time-dependent mean-squared amplitudes reveals that the sublinear diffusion mainly originates from motions at short times t[Pg.39]

It can be seen that a theoretical prediction of values is not possible by any of the three above-described models, because none of the three parameters - the laminar film thickness in the film model, the contact time in the penetration model, and the fractional surface renewal rate in the surface renewal model - is predictable in general. It is for this reason that the empirical correlations must normally be used for the predictions of individual coefficients of mass transfer. Experimentally obtained values of the exponent on diffusivity are usually between 0.5 and 1.0. 

For the initiation by azo initiators only the dependence kp / kt0 5=f ([M]) has to be considered in a kinetic model [10]. Accordingly, an initiator exponent of 0.5 and a monomer exponent of 2 are valid. By adding amine the decomposition velocity of APS is increased by an orders of magnitude. The chain side reactions with the monomer and termination by chlorine atoms are then significantly suppressed which results in a monomer exponent of 2 and higher molar masses of the homopolymer [11]. The kinetics of 2.3 order in monomer and 0.47 order in initiation [59], explained by partial cyclization and termination of cyclized radicals, could not be confirmed. 

A mathematical model has been proposed to account for the mutual synergistic action of either particle component on the other in increasing the value of the dimensionless time 0 as shown in Fig. 9b. Thus, the dimensionless time 0X for the coarse particles could be assumed to exert a mass-fraction-based influence on the fine particles, proportional to 0 1 — x2), which is affected by certain interaction by the fines, inclusive of their ability to adhere to the surface of the coarse and form clusters among themselves, lumped in certain appropriate form, for instance, [1 + /(x2)], where the function /(x2) may again be assumed to possess certain appropriate form, for instance, exponential, x2, where n1 may be called the interactive exponent. This results in an overall contribution by the coarse particles, suitably corrected for the interaction of the fine particles, 0 1 — x2Xl + x2l). This function has the property of accommodating the following boundary conditions ... 

When this is done, some parametric form for the mass spectrum has to be assumed. The initial approximation, that of a power law, is referred to as Salpeter mass function, following Salpeter. This approximation, of course, cannot apply over the entire range of possible masses, since the lower masses produce divergence in the total population. It is usual to specify three parameters the upper and lower mass cutoffs and the exponent. While not useful in a fundamental way for explicating the origin of the mass spectrum, it is a convenient parametrization for models of star formation and the populations of external galaxies. 

---