Thermal Expansion and Compressibility

SO that to calculate the effect of P on Gj we must know how Vi varies as a function of P. Now the variation in volume with temperature and pressure is most commonly measured as the thermal expansion at constant P 

Similarly, the (so-called) free energy of formation from the elements is 

Should it be necessary to calculate apparent enthalpies or entropies of minerals at elevated P and T, the assumption that (dV°/dT)p = 0 plus the general equations 

All these equations are for solid phases for which V° has been assumed to be constant. For gases and supercritical fluids, this assumption is obviously not reasonable. The variation of the free energies of these gases and fluids is most conveniently handled in a completely different way, that is, by the introduction of a new function, the gacity (see Chapter 11). 

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Estimate, by means of Eq. III-41, the surface tensions of CCI4. CHCI3 and of water at 20°C. Look up the necessaiy data on thermal expansion and compressibility. 

In lambda transitions, no discontinuity in enthalpy or entropy as a function of T and/or P at the transition zone is observed. However, heat capacity, thermal expansion, and compressibility show typical perturbations in the lambda zone, and T (or P) dependencies before and after transition are very different. 

Gibbs Free Energy of a Phase at Higb P and T, Based on tbe Functional Forms of Heat Capacity, Thermal Expansion, and Compressibility... 

Combining thermal expansion and compressibility data for compounds in the (Mg,Fe)2Si04 mixture, Hazen (1977) proposed the following equation of state for olivines of the series Mg2Si04-Fe2Si04 ... 

Table 5.48 Thermal expansion and compressibility of some amphibole end-members according to Saxena et al. (1993) (1) and Holland and Powell (1990) (2).

We may envisage the almost linear variation of thermal expansion and compressibility with the amount of anorthite in the plagioclase mixture. We also see that the three polymorphs of the KAlSi308 component have substantially similar compressibilities, within uncertainties. 

If the heat capacity functions of the various terms in the reaction are known and their molar enthalpy, molar entropy, and molar volume at the 2) and i). of reference (and their isobaric thermal expansion and isothermal compressibility) are also all known, it is possible to calculate AG%x at the various T and P conditions of interest, applying to each term in the reaction the procedures outlined in section 2.10, and thus defining the equilibrium constant (and hence the activity product of terms in reactions cf eq. 5.272 and 5.273) or the locus of the P-T points of univariant equilibrium (eq. 5.274). If the thermodynamic data are fragmentary or incomplete—as, for instance, when thermal expansion and compressibility data are missing (which is often the case)—we may assume, as a first approximation, that the molar volume of the reaction is independent of the P and T intensive variables. Adopting as standard state for all terms the state of pure component at the P and T of interest and applying... 

Molar Volume, Thermal Expansion, and Compressibility of Silicate Melts... 

The density of the hydroxyfunctional hyperbranched polyesters is 1.295 g cm" and pressure-volume-temperature (PVT) measurements show that the thermal expansion and compressibility are slightly lower compared to polar linear polymers, such as PVC, poly(e-caprolactone), and poly(epichlorohydrine). ... 

Here Cp, a and are the heat capacity, volume thermal expansivity and compressibility respectively. First-order transitions involving discontinuous changes in entropy and volume are depicted in Fig. 4.1. In this figure curves G Gu represent variations in free energies of phases I and II respectively, while // Hu and F, represent variations in... 

In this connection, it is very interesting that the volume and intrachain changes obtained by various experimental methods 24,29,85) [Eq. (101)] agree well with Eq. (56) following from the Tobolsky-Shen semiempirical equation of state or the related phenomenological Eq. (76). The values of y determined from the data are rather small (0.1-0.3). As has been mentioned above, according to the semiempirical approach by Tobolsky and Shen one can formally suggest that the front-factor in Eq. (28) is pressure dependent. If it is really so, then the parameter y for rubbers can be considered as an experimental coefficient similar to the coefficient of thermal expansion and compressibility 29). 

Atomic and molecular displacement under constraint. Thermal expansion and compressibility are large and anisotropic. Sometimes structural data have been extrapolated from the room temperature (RT) down to low temperature (LT) simply by considering changes in lattice dimensions. This has led to disappointing results since, even in the absence of a phase transition, molecular shapes and orientations may change substantially. Similarly, if we find an isostatic pressure at room temperature whose effect is equivalent to a given temperature decrease at ambient pressure for, say, the chain contraction, the equivalence will not usually match for, say, the... 

P. The latter have to be interpreted in terms of thermal expansion and compressibility, respectively. Both quantities are described by a second-rank deformation tensor U which can be directly computed from the six sets of parameters versus constraint [37,107] ... 

Generally, cation-anion bonding is soft for large cations and anions. In other words, the coefficients of thermal expansion and compressibility are both larger for the A X bond than for the B-X bond. Thus, not only temperature but also pressure should be considered to affect the bond length matching to be estimated in terms of tolerance factor. 

As for the volumes of the atoms, the thermal expansion and compressibility is composed of two main terms, the cavity and the hydration. An estimate of the contribution of each factor relies on assumptions that are not easy to check. An estimate of the expansion or compression of the cavities should be possible with positron annihilation lifetime spectroscopy. This approach has proven to be a useful tool for determining the size of cavities and pores in polymers and materials. The lifetime is sensitive to the size of the cavity in which it is localized. A number of empirical relations correlate the distribution of the lifetime and the free volume [33]. Data on the pressure effect on the lifetime are only available for polymers. The results suggest that there may be a considerable contribution of the reduction in cavity size to the compressibility of a protein. 

The equations of state discussed so far, the ideal gas law, the van der Waals equation, and others, were relations between p, V, and T obtained from empirical data on the behavior of gases or from speculation about the effects of molecular size and attractive forces on the behavior of the gas. The equation of state for a liquid or solid was simply expressed in terms of the experimentally determined coefficients of thermal expansion and compressibility. These relations applied to systems at equilibrium, but there is a more general condition of equilibrium. The second law of thermodynamics requires the relation, Eq. (10.19),... 

The thermal expansion and compressibility of rhombohedral NaN3 and tetragonal KN3 are highly anisotropic. The largest values in both cases are perpendicular to the azide axis, suggesting that ionic forces within the azide layers are stronger than those between the layers. The results of Parsons and Yoffe [85] also indicate that the lattice parameters increase linearly with temperature between 300 and 600°K. 

Mitra [86] determined the pressure dependence of the Raman- and infrared-active lattice modes in KN3 and found that the librational mode is highly pressure dependent (2 cm"Vkbar pressure). The Raman-active translational mode involving motions of the ions only appears to be insensitive to pressure. The infrared-active modes, however, show a pressure coefficient of 0.5-0.7 cm Vkbar. Raman measurements at high temperature on KN3 by Iqbal [87] indicate that the librational modes are also very temperature dependent (Figure 17). Within the quasiharmonic approximation, correlation of the data with thermal expansion and compressibility measurements indicates a sizable anharmonic or selfenergy contribution to the librational mode. The mode Griineisen parameter 7y(q) of mode cjy(q), defined as... 

In order to have the first term of Eq. (3.28) negative, the pressure should decrease with increasing volume. The second term in brackets does not play any role in the statement of the second law. Because of the negative sign and the square, it is always negative. This means that the statement of the second law in this form explicitly allows any sign of the coefficient of tension and related coefficients such as the ratio of thermal expansion and compressibility coeflBcient. Recall the relation... 

The variables in Eq. (6.46) are either defined there or are the same as in Eq. (6.39). The two equations of state offer similar predictions regarding EVT behavior, thermal expansion, and compressibility. The modification suggests higher free-volume content, primarily due to the greater number of external degrees of freedom. However, the modified relation provides better prediction of the polymer-solvent miscibility and P dependence of the critical mixing temperature. 

Several attempts to estimate the hole density from a comparison of the mean hole volume with the macroscopic volume are described in the literature. The drawback of such approaches is that assumptions must be made as to the value of or on the thermal expansion and compression of the volume that is not detected by o-Ps. Frequently, it is assumed that that this volume, denoted as occupied or bulk volume, expands Uke an amorphous polymer in the glassy state [Hristov et al., 1996 Dlubek et al., 1998c Band ch et al., 2000 Shantarovich et al., 2007]. Another assumption is that no variation with temperature or pressure is shown [Bohlen and Kirchheim, 2001]. Both assumptions are intuitive but physically not proved. The most successful attempt to estimate hole densities comes from a calculation of the hole free volume with... 

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