Macroscopic density

Fig. 1 Annealing time dependences of the crystallization isotherm 0 (below) and the macroscopic density (above) of PET annealed at 115 °C [6]... 

It has been shown above that during the induction period no change occurs in DSC, macroscopic density, and WAXD. Does nothing change in this period In order to answer this question, we made real-time SAXS measurements. 

Fig. 12. Transport coefficient ( ) for [3H]PVP360 as a function of the initial concentration difference of dextran T10 that exists across the boundary, C -C", where C is the dextran concentration outside the capillary and C" the dextran concentration inside the capillary, the mean concentration of the dextran being 135.0kgm-3. The corresponding initial macroscopic density difference across the boundary Ae, u ( ) is shown on the left-hand ordinate 511...

Another approach employed to establish the occurrence of a density nversion between the two solutions subsequent to boundary formation involves dialysis between the two solutions s0>. The dialysis membrane is impermeable to the polymer solutes but permeable to the micromolecular solvent, H20. Transfer of water across the membrane occurs until osmotic equilibrium involving equalization of water activity across the membrane is attained. Solutions equilibrated by dialysis would only undergo macroscopic density inversion at dextran concentrations above the critical concentration required for the rapid transport of PVP 36 0 50). The major difference between this type of experiment and that performed in free diffusion is that in the former only the effect of the specific solvent transport is seen which is equivalent to a density inversion occurring with respect to a membrane-fixed or solute-fixed frame of reference. Such restrictions are not imposed on free diffusion where equilibration involves transport of all components in a volume-fixed frame of reference. The solvent flow is governed specifically by the flow of the polymer solutes as described by Eq. (3) which, on rearrangement, gives... 

Evaluate the attraction between the molecule and the /th layer by integrating this expression over all values of y between 0 and o°. The assumption that p 8 = p(d + 8) ensures that the blocks have the correct macroscopic density. If the separation between layers is the same as the separation of the surfaces of the blocks (i.e., z = d), then the equivalent of Equation (59) results from summing the 4>, values for all Ps between 0 and oo and taking the limit of 8 - 0. Derive the analog of Equation (59) for matter with this hypothetical structure. Note that E,(l + 0 4 = 1-082. Equation (68) is obtained by assuming a similar structure for the second block and continuing along these lines. 

So far, only microscopic densities have been considered. The observables are, however, the macroscopic density, N, of the diffusing species and the... 

As a simple approximation to reality, the macroscopic density, N, and the reaction strength, Q, and hence qx 1 also, may be presumed to be constant throughout the volume, V. If there are no intermolecular forces, except for a hard core repulsion between sink (so that they cannot overlap), the densities become constant and... 

The average (macroscopic) density of excited fluorophors in a volume V is... 

A specific expression for k(p) was developed by Reck and Prager [507] which satisfies the constraint on the macroscopic density. They considered the probability that the excited fluorophor was at least a certain distance from the nearest quencher and from this found the upper bound... 

To develop a lower bound on the steady state, Reck and Prager [507] again considered the variational integral of eqn. (265). In this case, however, let the approximate solution j/ satisfy the diffusion equation (263) rather than the equation defining the macroscopic density M as previously done. Multiply eqn. (263) by j5(r), a Lagrangian undetermined multiplier and add it to the variational integral to give... 

The fit of the model packing fraction to the macroscopic density is the essential point of our model, as already mentioned. That is why we chose a Monte-Carlo method to obtain two-dimensional liquid-like distance statistics of hard discs. The procedure we used is exactly the same as used by Metropolis et al. [14] with the addition of averaging a large number of system configurations. 

It is an essential test of our model that the number density of CH2 groups corresponds to the macroscopic density. To verify this, we calculate the experimental CH2 density from the macroscopic density using... 

Fig. 6. Unit-cell density vs. macroscopic density for bulk-crystallized linear polyethylene62 ...

The next reasonable step in studying our chemical games is to consider ensembles of A s and B s (e.g., topers and policemen), when they are randomly and homogeneously distributed in the reaction volume and are characterized by macroscopic densities of a number of particles. The peculiarity of the A + B -y B reaction is that the solution of a problem with a single A could be extrapolated for an ensemble of A s (in other words, a problem is linear in particles A). As it was said above, it is analytically solvable for Da = 0 but turns out to be essentially many-particle for Db = 0. It is useful to analyze a form of the solution obtained for the particle concentration tia (t) in terms of the basic postulates of standard chemical kinetics (i.e., the mean-field theory). 

It is useful to find a quantity that could serve us as a measure of these density fluctuations. Its simplest characteristic is the dispersion of a number of particles N in some volume V i.e., (N2) — (N)2. The distinctive feature of the classical ideal gas is a simple relation between the dispersion and macroscopic density (TV2) - (TV)2 = (IV) = nV. Moreover, all other fluctuation characteristics of the ideal gas, related to the quantity (Nm, could also be expressed through (TV) or density n. Therefore, in the model of ideal gas the density n is the only parameter characterizing the fluctuation spectrum. Such the particle distribution is called the Poisson distribution. It could be easily generalized for the many-component system, e.g., a mixture of two ideal gases. Each component is characterized here by its density, nA and nB density fluctuations of different components are statistically independent, (IVAIVB) = (Na)(Nb). 

Due to the spatial homogeneity it is independent of fj allowing us to introduce the simplest spatial characteristics - macroscopic densities of particles (concentrations)... 

The Gb function hereafter is calculated quite similarly.) Equation (7.1.4) for macroscopic density contains the integral... 

First equations (8.2.7) and (8.2.8) of the set are not affected by the superposition approximation and thus yield the exact equations for the time development of the dimensionless macroscopic densities (concentrations) ... 

In appraising the average accuracy one must bear in mind that experimentally the mean polarizabilities are usually obtained from the refractive index n (at 5893 A, the sodium D-line) and the Lorenz-Lorentz equation (with M molecular weight, p macroscopic density, Vav Avogadro s number) ... 

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