Molecular specific volume

Pi = upstream absolute pressure p2 = downstream absolute pressure R = gas constant T = absolute temperature M,, = molecular weight Di = upstream specific volume of gas Vo = downstream specific volume of gas... 

As the temperature is decreased, free-volume is lost. If the molecular shape or cross-linking prevent crystallisation, then the liquid structure is retained, and free-volume is not all lost immediately (Fig. 22.8c). As with the melt, flow can still occur, though naturally it is more difficult, so the viscosity increases. As the polymer is cooled further, more free volume is lost. There comes a point at which the volume, though sufficient to contain the molecules, is too small to allow them to move and rearrange. All the free volume is gone, and the curve of specific volume flattens out (Fig. 22.8c). This is the glass transition temperature, T . Below this temperature the polymer is a glass. 

An artificial neural network based approach for modeling physical properties of nine different siloxanes as a function of temperature and molecular configuration will be presented. Specifically, the specific volumes and the viscosities of nine siloxanes were investigated. The predictions of the proposed model agreed well with the experimental data [41]. 

The subscripts 1,2,3 refer to the main solvent, the polymer, and the solvent added, respectively. The meanings of the other symbols are n refractive index m molarity of respective component in solvent 1 C the concentration in g cm"3 of the solution V the partial specific volume p the chemical potential M molecular weight (for the polymer per residue). The surscript ° indicates infinite dilution of the polymer. 

Equation (4) applies to a mol. of any gas, R having a constant value which can be calculated as soon as we know the specific volume of the gas at a given temperature and pressure, and its molecular weight. 

Abnormally low atomic heats were explained by Richarz on the assumption of a diminution of freedom of oscillation consequent on a closer approximation of the atoms, which may even give rise to the formation of complexes. This is in agreement with the small atomic volume of such elements, and with the increase of atomic heat with rise of temperature to a limiting value 6 4, and the effect of propinquity is seen in the fact that the molecular heat of a solid compound is usually slightly less than the sum of the atomic heats of the elements, and the increase of specific heat with the specific volume when an element exists in different allotropic forms. 

FIGURE 24.6 Specific volume as a function of temperature at the indicated pressures for 1,2-PB network (molecular weight between cross-hnks 8 kg/mol) [81]. The solid squares denote the glass transition defined from the intersection of the glassy and liquid data. 

A gas condenses to a liquid if it is cooled sufficiently. Condensation occurs when the average kinetic energy of motion of molecules falls below the value needed for the molecules to move about independently. Thus, the molecules in a liquid are confined to a specific volume by intermolecular forces of attraction. Although they cannot readily escape, liquid molecules remain free to move about within the liquid phase, hi this behavior, liquid molecules behave like the molecules of a gas. The large-scale consequences of the molecular-level properties are apparent. Like gases, liquids are fluid, so they flow easily from place to place. Unlike gases, however, liquids are compact, so they cannot expand or contract significantly. 

D = mass fraction of i xt = mole fraction of i V = partial specific volume of i c = molar concentration of mixture r = mass concentration of i M = molecular weight of i. 

The hydrodynamic properties of Ustilago cytochrome c were investigated by Thelander (86). He found the partial specific volume to be 0.721 ml/g and the molecular weight, by sedimentation equilibrium, to be 15,500. The latter value, although higher than that given by summation of the constituent amino acid residues (i.e., 11,877, see Table 3), indicates that the protein is monomeric. 

Here, Mn, is the molecular weight of the polymer chains prepared under identical conditions but in the absence of the crosslinking agent, u is the specific volume of the polymer and V is the molar volume of water. 

Volume of unit cell specific volume of a liquid volume molecular volume... 

The radius of the peptide molecule can be calculated from the molecular mass, Avogadro number, and the partial specific volume. 

If (1) the diffusion coefficient D of the polymer in the used solvent, (2) the specific volume v of the dissolved polymer, and (3) the density of the solvent solvent known, one can determine the molecular weight of the dissolved polymer according to the above equation by measuring the sedimentation coefficient (by measuring the maximum of the concentration gradient at regular time intervals). 

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