Occupied volume

Figure 4.15 Geometrical representation of the temperature variation of the actual volume (solid line) and the occupied volume (broken line). The shaded difference indicates the free volume which decreases to a critical value at T .

The authors of [40] used L. L. Blyler s and T. K. Kwei s formula to process experimental data [41, 8] and obtained good correlation between theory and experiment. In all the processed experiments viscosity was established in accordance with pressure at channel input. To describe data presented by C. J. Ma and C. D. Han [2-5], who1 studied freon-containing polymer melts, the same paper supposed that the entire volume of gas is expended in part on the increase of the free volume of the composition, and that the occupied volume also changes in its presence. This consideration made it possible for the authors of [40] to attain fair correlation between theory and practice. This makes, in our opinion, the ideas expressed in [39, 40] worthy of the most serious attention, however critical the following evaluation of these works may appear to the reader. 

To account for the variation of the dynamics with pressure, the free volume is allowed to compress with P, but differently than the total compressibility of the material [22]. One consequent problem is that fitting data can lead to the unphysical result that the free volume is less compressible than the occupied volume [42]. The CG model has been modified with an additional parameter to describe t(P) [34,35] however, the resulting expression does not accurately fit data obtained at high pressure [41,43,44]. Beyond describing experimental results, the CG fit parameters yield free volumes that are inconsistent with the unoccupied volume deduced from cell models [41]. More generally, a free-volume approach to dynamics is at odds with the experimental result that relaxation in polymers is to a significant degree a thermally activated process [14,15,45]. 

A gas will obey the ideal gas equation whenever it meets the conditions that define the ideal gas. Molecular sizes must be negligible compared to the volume of the container, and the energies generated by forces between molecules must be negligible compared to molecular kinetic energies. The behavior of any real gas departs somewhat from ideality because real molecules occupy volume and exert forces on one another. Nevertheless, departures from ideality are small enough to neglect under many circumstances. We consider departures from ideal gas behavior in Chapter if. 

Cohen and Turnbull [20,21] laid down the foundation for the free volume concept in modeling self-diffusion in simple van der Waals liquids. They considered that the volume in a liquid is composed of two parts, the actual volume occupied by the liquid molecules and the free volume surrounding these molecules opened up by thermal fluctuation. Increasing temperature increases only the free volume and not the occupied volume. The average free volume per molecule, vf, can be defined as... 

The conclusion from the monomer solvent studies is that, in nearly equal molar solutions, DMT and 4,4 -BPDC compete for absorption of the 298 nm radiation. However, the results also show that, even in equal concentrations, the DMT emission, when excited by 298 nm light, is several times as intense as the 4,4 -BPDC emission at 472 nm. It must be emphasized that these studies do not preclude the existence of energy transfer from excited DMT to 4,4 -BPDC. From the volume calculation used above, it can be shown that a concentration of v 0.1 M 4,4 -BPDC is needed to assume an occupied volume with radius of 15 8, the required distance for the exchange mechanism. 

Following Flory (1969), a 0 solvent is a thermodynamically poor solvent where the effect of the physically occupied volume of the chain is exactly compensated by mutual attractions of the chain segments. Consequently, the excluded volume effect becomes vanishingly small, and the chains should behave as predicted by mathematical models based on chains of zero volume. Chain dimensions under 0 conditions are referred to as unperturbed. The analogy between the temperature 0 and the Boyle temperature of a gas should be appreciated. 

It is well known that the transfer of nonpolar molecules from nonpolar to polar surroundings results in a decrease in the partial molar volume of the solute. The dimerization studies also show that there is a similar volume decrease when two monomers form a dimer. This volume decrease is of the order of 20 cm3 mol-1. It is difficult to understand how there can be first a volume decrease when the nonpolar molecules are transferred from the nonpolar to the polar environment and then a further volume decrease when two molecules come together and partly reverse the first transfer. It is a little dangerous to speak of the partial reversal of a process we know so little about. It is believed that the hydrophobic hydration is a cooperative phenomenon, in which the exact microstructure of water is very important for the occupied volume. How this microstructure changes when two molecules associate in a hydrophobic interaction is not par-... 

The lower cycle represents the chemical changes occurring during polymerization and relates them to the free volume of the system. In general, free volume of a polymer system is the total volume minus the volume occupied by the atoms and molecules. The occupied volume might be a calculated van der Waals excluded volume [139] or the fluctuation volume swept by the center of gravity of the molecules as a result of thermal motion [140,141]. Despite the obscurity in an exact definition for the occupied volume, many of the molecular motions in polymer systems, such as diffusion and volume relaxation, can be related to the free volume in the polymer, and therefore many free volume based models are used in predicting polymerization behavior [117,126,138]. 

Level swell is the mechanism by which runaway chemical reactions vent a two-phase mixture. When a runaway chemical reaction generates gas or vapour, bubbles are formed throughout the bulk of the liquid. Because the bubbles are buoyant, they wijl tend to rise through the liquid in order to disengage at the surface. However, whilst they remain in the liquid, they occupy volume and so cause the... 

The fat globules of milk reduce the conductivity by occupying volume and by impeding the mobility of ions. Thus the conductivity of whole milk is less than that of skim milk by about 10%, and that of cream varies with the fat content (Gerber 1927 Muller 1931 Prentice 1962). Homogenization of milk does not measurably influence conductivity (Prentice 1962). The conductivity of whey and ultrafiltrate is slightly greater than that of skim milk (Schulz 1956 Schulz and Sydow 1957). A possible relationship between the electrical conductivity and physical stability of evaporated milk and concentrated infant milk products has been reported (Hansson 1957). Samples of poor physical stability tended to have relatively low conductivity values compared to those of the more stable products. 

The volume v2 in Fig. 1 is the occupied volume in WLF theory. As may be seen from Fig. 1, the value v2 is obtained by extrapolation of the liquid volume line at temperature T2 parallel to AB to Tg (portion BJ). Temperature T2 is most conveniently determined from the viscosity temperature plot for a supercooled liquid near its Tg23,26 ... 

Difficulties arise, however, concerning the interpretation of T0 in terms of the free-volume concept. In order to overcome these difficulties, Miller31 suggested that the occupied volume could vary with temperature. 

The dependence of f% on filler content was carefully investigated for filled polystyrene87 and the values for/g were calculated in different ways, using various values for the occupied volume v0. The results of these calculations have shown that the values for /g do not coincide when calculated in different ways. Nor are these values constant for the different amounts of filler incorporated. This shows once more that the glass temperature is not a temperature of constant fractional free-volume. 

Now consider two phases in coexistence as one joint system. Assume that they occupy volumes V(0 and respectively, and contain and M2) particles. Following literally the prescription in Eq. (12), the free energy of the joint system of = N particles is found from... 

It is, therefore, not surprising that there exists a definite relationship between Aand the enthalpy of vaporization, Av H, the former constituting a fraction between 0.2 and 0.3 of the latter, as is readily obtained from the data in Tables 3.1 and 3.9. The pressure dependence of the viscosity is also closely related to the free volume of the solvent. The fluidity (O = l/r ) is proportional to the ratio between the free and the occupied volume, the former, as mentioned above, being the difference between the actual molar volume and the intrinsic molar volume (Tables 3.1 and 3. 4) (Hildebrand 1978). In fact, the logarithm of the viscosity of liquids was found (Marcus 1998) to be described well for some 300 liquids by the empirical relationship ... 

The latter can be taken as either Vx, or Evdw, or VL, related by Eqs. (3.19), (3.20), and (3.21), respectively, to the constitution of the solvents. It is obvious that the free volumes defined according to these choices of the intrinsic volume are not the same, and caution must be exercised when this notion is applied to concrete problems. The fluidity O = l/r of solvents depends on the free volume O = B[(V - V0)/V0J, according to (Hildebrand 1978), where B is a temperature-independent constant and V0 is the occupied volume, that may be equated with the intrinsic volume, see also Eq. (3.33). As mentioned in Chapter 3, the compressibilities of solvents appear to depend mainly on their free volumes, according to Eq. (3.8), so that there exists a relationship between the compressibilities of solvents and their fluidities (Marcus 1998). Two non-linear curves result from plots of log O v s kt, one for non-associated liquids and the... 

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