Instantaneous distributions

Example 13.5 Determine the instantaneous distributions of chain lengths by number and weight before and after termination by combination. Apply the quasi-steady and equal reactivity assumptions to a batch polymerization with free-radical kinetics and chemical initiation. 

James and Guth showed rigorously that the mean chain vectors in a Gaussian phantom network are affine in the strain. They showed also that the fluctuations about the mean vectors in such a network would be independent of the strain. Hence, the instantaneous distribution of chain vectors, being the convolution of the distribution of mean vectors and their fluctuations, is not affine in the strain. Nearly twenty years elapsed before his fact and its significance came to be recognized (Flory, 1976,... 

FIGURE 6.4 An instantaneous distribution of electrons that leads to polarity in two atoms. There will be a force of attraction between the atoms (or molecules) even though they do not have permanent polarity. The number of electrons and their ability to be moved will determine the magnitude of the attractive force. 

To put the previous statement into perspective it is necessary to stipulate that any macrosystem with well-defined values of its extensive parameters is made up of myriads of individual particles, each of which may be endowed with an unspecified internal energy, within a wide range consistent with all external constraints. The instantaneous distribution of energy among the constituent particles, adding up to the observed macroscopic energy, defines a microstate. It is clear that any given macrostate could arise as the result of untold different microstates. 

An assumed instantaneous distribution between the adsorbed and free solute is modelled by a Langmuir-type adsorption isotherm, where... 

Hydrogen bonds are continually breaking and reforming by thermal agitation and, according to the association model, there exist instantaneous distributions of free monomers Bj, hydrogen-bonded dimers B2 and multimers Bj, which obey the reaction scheme ... 

Experiment indicates the existence of completely idiosyncratic three-dimensional regimes of propagation of so-called spin of a detonation wave in which the instantaneous distribution also depends on the angle in the cylindrical system of coordinates, coaxial with the tube, in spite of the complete symmetry of the initial conditions. 

If the voltage is high enough, the noise of isolated contacts can be considered as white at frequencies at which the distribution function / fluctuates. This allows us to consider the contacts as independent generators of white noise, whose intensity is determined by the instantaneous distribution function of electrons in the cavity. Based on this time-scale separation, we perform a recursive expansion of higher cumulants of current in terms of its lower cumulants. In the low-frequency limit, the expressions for the third and fourth cumulants coincide with those obtained by quantum-mechanical methods for arbitrary ratio of conductances Gl/Gr and transparencies Pl,r [9]. Very recently, the same recursive relations were obtained as a saddle-point expansion of a stochastic path integral [10]. 

From these expressions it follows that the instantaneous distribution of the M units along the backbone (and consequently that of the grafts) formed instanteneously follows the Bernouillian statistics, provided the molar amount of the macromonomer in the reaction mixture is small enough. Fluctuations in composition generally remain within rather narrow limits, provided the conversion is not too high consequently, the content of grafts in a graft copolymer sample remains within narrow limits, too. 

The third component to the electrostatic interaction is caused by the motion of the electron cloud, which creates an oscillating field. It couples to the oscillating field of the neighboring molecules, which gives an attractive contribution to the total energy. This should be obvious, from the following example. Consider the frozen electron distribution of a nonpolar molecule (e.g., a noble gas). The instantaneous distribution possesses a dipole moment, which for the same reason as described above, induces a dipole in neighboring molecules, which in turn act on the first molecule, etc. This contribution is denoted the dispersion term or the London term [9]. Note that this contribution is only approximately pairwise additive. 

The system of our interest is a dilute solution containing a single solute molecule. Even when the solute concentration is finite, our development is valid by viewing one of the solute molecules as the solute and the others as part of mixed solvent. To describe completely the configuration of a solvent molecule relative to the solute, the position and orientation need to be specified simultaneously. The complete set of the position and orientation is called the full coordinate and is denoted collectively by x. If the solute and/or solvent are flexible, the intramolecular degrees of freedom are also incorporated into x. In the full coordinate representation, the instantaneous distribution pf is introduced as... 

It is then useful to reduce the information content by introducing a projected coordinate. With projection, some information of x is retained, while the others are disregarded. When the projection is implemented with respect to a function P(x), the corresponding distribution functions are generated from the instantaneous distribution given by... 

Another quantity to be determined is the frozen electron density distribution n r) that appears in Eq. (17-60). Of course, the optimal distribution ii(r) is the one that minimizes the contribution of the density fluctuation to the free energy change expressed by Eq. (17-61). Here, we propose to take the ensemble average of the instantaneous distribution n r) that fluctuates according to the solvent molecular motion as a most natural and practical choice of n(r), thus,... 

According to Eq. (17-69), the pure solvent refers in the computation of 8p, to the pairwise additive potential system with the solute-solvent interaction Eqm/mm (h, X). On the other hand, the solution system involves the solute-solvent interaction EqM/mm( , X) with the QM energy term of Edist(n) - E. From Eq. (17-69), the energy coordinate to formulate 8 fx, in the energy representation is associated with the instantaneous distribution p (f) through... 

In many emulsion polymerizations, the monomer/polymer ratio is kept constant during Interval II, and the accumulated MWD is approximately equal to the instantaneous distribution. Equation 84 shows that the Cm value can be determined from the slope of the InN(P) versus P plot. 

When termination occures by combination as well as disproportination, the initial instantaneous distribution of molecular weights is give by [18, 19]... 

Digital tools provide instantaneous distributed access to original experience. In a paperless world, one that is mobile, wireless, and extremely portable, customers seek data versus documents. Documents have become data embedded in inseparable multiple layered architectures. Technological and economic forces have essentially redefined document formats and purposes. The goal is more and more to preserve or re-create the immediacy of a communicative act, just-in-time delivery, real-time updates. 

According to Bel F dhila Simonin [11], the average of the relative velocity between each particle and the surrounding fluid can be expressed as a function of the mean relative velocity and a drift velocity due to the correlation between the instantaneous distribution of the particles and the large scale turbulent fluid motion with respect to the particle diameter ... 

Thus, (33) has a close resemblance to the conventional expression in (3). Equation (33) takes the distribution of atomic positions into account as well as the finite duration of the X-ray probe pulse. Clearly, if the temporal duration of the X-ray pulse is short compared to the molecular dynamics, the diffraction signal corresponds to diffraction from the instantaneous distribution of atomic positions, p(R, tp). at time tp. Structure determination of short-lived intermediates can be accomplished with long probe pulses, i.e., long on the timescale of atomic motion but short compared to the lifetime of the population. This limit corresponds to current experiments where fast kinetics is studied by X-ray pulses from synchrotrons. 

The distribution transport rate (r is ) is a measure of how quickly drug molecules are exchanged between the plasma and the tissues. A rapid distribution transport rate causes the plasma and tissues to come quickly into equilibrium with each other, whereas a slower rate will cause a prolonged approach to equilibrium. As with the rate of absorption, different types of PK modeling approaches can be employed to approximate distribution rates. In the case of distribution there are essentially two types of models, instantaneous distribution and first-order distribution. The difference between the two types of models is in the number of compartments used to represent the drug disposition in the body. 

Instantaneous distribution is represented by considering the body as a single compartment. This type of PK model can have parameters associated with how fast drug reaches the systemic circulation by absorption... 

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