Dependence of volume

All gases closely resemble each other in the dependence of volume on amount, temperature, and pressure. 

Volume is directly proportional to absolute temperature. The dependence of volume (V) on the Kelvin temperature (T) is shown in Figure 5.3b (p. 106). The graph is a straight line through the origin. The equation of the line is... 

This reasoning also means that we were not really describing a thermodynamic measurement of the glass transition in a polymer melt but instead a macroscopic determination of the temperature dependence of volume-related internal relaxation processes, i.e., a dynamic measurement in the disguise of a thermodynamic measurement. 

Figure 18.1. Linear dependence of volume on concentration for dilute solutions of glycolamide in water. Based on data of Gucker et al. [1].

Figure 18.2. Nonlinear dependence of volume on molality for dilute solutions of sulfuric acid (H2SO4) in water (H2O). Based on data of I. M. Klotz and C. F. Eckert, J. Am. Chem. Soc. 64, 1878 (1942).

Fig. 2. The time dependency of volume, Young s modulus, long period and weight of the swollen PVA film during soaking in the iodine solution...

Fig. 5.4 The figure shows dependence of volume gain AV versus number of hydrogen atoms in fullerane molecule X. Volume gain per hydrogen atom in is calculated per molecule of Cf/i as AV= (VC60Hx - Vca))/X, VC60 is taken as volume per molecule in FCC structure of pristine Cm (711 A3) (From Talyzin et al. 2004a)...

Fig. 10.4. Supposed model for the time dependence of volume concentration Nv of M/SC nanoparticles during polymerization of co-deposited M/SC organics layers with different M/SC contents (pi (curve 1) and (p2 (curve 2) at (p > (p2- Curve 3 characterizes the increase in layer viscosity during polymerization.

The dependence of volumes of activation, AV, on solvent composition for the hydrolysis of benzyl chloride in aqueous mixtures (Golinkin et al., 1966 Hyne et al., 1966) has been analysed using the solvent dependence of the partial molar volume of the initial state (Golinkin et al., 1967). In aqueous alcohols, as x2 increases, AV decreases to a minimum value at a mole fraction characteristic of the alcohol, e.g. x2 =0-1 for t-butyl alcohol, the intensity of the extremum increasing on going from methyl alcohol to t-butyl alcohol. The dependence of AV on x2 is a consequence of quite marked changes in the quantities 5m Vf and 8m V (Dickson and Hyne, 1971). For example, in ethyl alcohol + water mixtures, 5m V for benzyl chloride in the initial state has a maximum near x2 = 0-3, whereas 5m V has a shallow minimum near x2 = 0-1 (cf. V2 for Me4N+Cl- in these mixtures Lee and Hyne,... 

Figure 51. Dependence of volume parameters for the solvolysis of benzyl chloride on mole fraction of alcohol in ethyl alcohol + water mixtures at 323 K (Mackinnon et al., 1970).

Typical strain dependences of volume resistivity are shown in Figure 12.7 the results are related to an ethylene-propylene-diene rubber (EPDM) (supplied by ExxonMobil Chemical under the trade name Vistalon 5601) filled with 6 phr of MWNTs that is at a filler content well above the percolation threshold (determined around... 

Figure 12.4 Dependence of Volume on Temperature at Constant Pressure...

A variety of solid materials, including metals, semiconductors, molecular solids, and ionic compounds, show a remarkable deviation from the simple Hooke s law in the pressure dependence of volume at 1-lOGPa. The deviation is quite often a prelude to a pressure-induced transition to a denser structure. - " For example, Fe metal changes its strac-ture from cubic (a-phase) to hexagonal (e-phase) at 11 GPa... 

FIG. 8-1. A simple apparatus for demonstrating Boyle s law of the dependence of volume of a gas on the applied pressure. 

You will note that the dependence of volume on the absolute temperature is a direct proportionality, whereas the volume is inversely proportional to the pressure. The nature of these two relations is illustrated in Figure 8-3. 

Straightforward measurements of elastic properties of materials can be made via high-pressure static compression experiments, in which X-ray diffraction (XRD) is used to measure the molar volume (V), or equivalently the density (p), of a material as a function of pressure (P). The pressure dependence of volume is expressed by the incompressibility or isothermal bulk modulus (Kt), where Kp = —V(bP/bV)p. 

Kg/p = Vp — (4/3)V. This new parameter (sometimes thought of as the P-wave velocity of an equivalent fluid, for which G = 0) can be determined directly from static compression data V = Kg/p = (1 + Tay)(bP/bp)p. The bulk sound velocity possesses another desirable feature, in that it can also be constrained indirectly through chemical equilibrium experiments. Chemical equilibria describe free energy minima the pressure dependence of free energy is described by the molar volume, and the pressure dependence of volume (or density) is described by Kp and hence V. Thus, experimental determinations of equilibrium phase boundaries can... 

Figure 22. Temperature dependence of volume for hexagonal ice (solid line), empty hydrate I (dotted line) and occupied hydrate I by xenon (dash-dot line). The volumes are scaled so that they are unity at zero temperature.

Figure 12. Time dependence of volume of copper for run C (Tables 1 and 2).

Figure 7.9 Dependency of volume specific productivity on number of stages and loading factor.

FIG. 7 Dependence of volume phase transition temperature (Tv) of NIPA gel (cy-lindrically shaped wet sample with d0 = 0.7 mm) on initial surfactant concentration. (From Ref. 26.)... 

One can see in the figure that for both gel systems the temperature dependence of volume change is not continuous an abrupt change occurs when the temperature exceeds 30 °C. It is also seen that within the experimental accuracy no difference was observed between PNIPA and mPNIPA gel beads. The presence of magnetic nanoparticles influences neither the measure of volume change nor the collapse transition temperature (abbreviated as 7c). A careful analysis based on derivatives of the qr-T curves has shown that for both kinds of PNIPA beads, Tq was found to be 32 °C. We have also studied the effect of cross-linking density on the volume phase transition. Figure 39 shows these results. 

Volume and Kelvin temperature vary directly. This means that if the Kelvin temperature of a gas doubles, its volume will double. If the Kelvin temperature is cut in half, its volume will be cut in half. Of course, this exact relationship requires that both the amount of gas and the pressure of the gas remain constant. The dependence of volume on temperature can be shown mathematically in this useful form of Charles law. 

A very important aspect of kinetic theories now surfaces. Equation (5-23) is concerned with the time dependence of volume recovery after a temperature jump. For the volume to change, the molecules must move. However, as discussed in Chapters 3 and 4, the rate at which the molecules can move is a sensitive function of the free volume, and the free volume is varying with time in Figure 5-9. Thus, it is clear that the retardation time in equation (5-23) must itself depend upon 8 and vary with time. This situation can be treated in quite a straightforward way by modifying equation (4-16) as... 

The foremost thermodynamic property associated with any phase boundary is the location of its surface in the p-V-T phase diagram. Most laboratory experiments of glass formation are carried out in a particular V-T plane, usually for atmospheric pressure, and the temperature dependence of volume through the transition is determined. If the glass transition is indeed dictated by the repulsive part of the potential, we expect, at least for simple steeply repulsive systems, that it will occur at the same molecular-reduced volume for many real and model systems and that this will be largely insensitive to the strength of the attractive component of the pair potential relative to kT. 

In supercooled isotropic melts, crystallite growth occurs after nucleation. Initially, the lateral size of the lamellae increases at a constant rate, but as soon as the spherulites start to touch each other, the growth rate decreases. Overall, the sigmoidal time dependence of volume fraction of the crystalline regions is well described by the... 

Dependence of volume swelling in benzene on the fraction of DVB per V3 = 8.5%, M= 10,000Da. (Reprinted from [126] with kind permission of... 

Figure 1.8 Dependence of volume swelling in benzene of model networks vs. molecular weight M of chain precursors for three sets of networks each set prepared at a constant concentration and constant portion of DVB per living end. (After [126]).

Figure 3.13 Dependence of volume fraction of pores, Vp, in dry styrene-DVB copoiymers prepared in the presence of 20 % linear polystyrene (/M=87000) on the concentration of divinylbenzene, wt%. Reprinted from [266] with kind permission of Elsevier.

Figure 7.5 Dependence of volume swelling in (1) n-hexane (2) methanol and (3) water on the crosslinking degree of networks prepared by crosslinking linear polystyrene with (a) p-xylylene dichloride and (b) methylal.

FIGURE 2.9 Dose dependence of volume fraction of a -phase, C, in the Fe-9at%Cr alloy calculated by CD in the case of ion irradiation and determined by SANS in the case of neutron irradiation. (First peak of size distribution function in Scripta Materialia, 61, Bergner, R, Ulbricht, A., Heintze, C., 1060, Copyright 2009, with permission from Elsevier.)... 

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