Thermal expansion and heat capacity

The state of polarization, and hence the electrical properties, responds to changes in temperature in several ways. Within the Bom-Oppenheimer approximation, the motion of electrons and atoms can be decoupled, and the atomic motions in the crystalline solid treated as thermally activated vibrations. These atomic vibrations give rise to the thermal expansion of the lattice itself, which can be measured independendy. The electronic motions are assumed to be rapidly equilibrated in the state defined by the temperature and electric field. At lower temperatures, the quantization of vibrational states can be significant, as manifested in such properties as thermal expansion and heat capacity. In polymer crystals quantum mechanical effects can be important even at room temperature. For example, the magnitude of the negative axial thermal expansion coefficient in polyethylene is a direct result of the quantum mechanical nature of the heat capacity at room temperature." At still higher temperatures, near a phase transition, e.g., the assumption of stricdy vibrational dynamics of atoms is no... 

Liu X, Wang Y, Liebermann RC, Maniar PD, Navrotsky A. (1991) Phase transition in CaGeOs perovskite evidence from X-ray powder diffraction, thermal expansion and heat capacity. Phys Chem Minerals 18 224-230... 

This expression reduces to the classical Clausius-Clapeyron equation when the difference in compressibility, thermal expansion and heat capacity vanish as is observed for most phase transitions in lipids [80]. 

Furthermore, using the above expressions in Equation 1.5 through Equation 1.7, and Equation 1.29, and then evaluating the derivatives leads to expressions for the compressibility, thermal expansion, and heat capacity. The results expressed in terms of fluctuations in the isothermal-isobaric ensemble are... 

It is important to note that the above formulas represent fluctuations (8X=X - (X)) in the properties of the whole system, that is, bulk fluctuations. They are useful expressions but provide no information concerning fluctuations in the local vicinity of atoms or molecules. These latter quantities will prove to be most useful and informative. One can also derive expressions for partial molar quantities by taking appropriate first (to give the chemical potential) and second (to give partial molar volume and enthalpy) derivatives of the expressions presented in Equation 1.28. However, these do not typically lead to useful simple formulas that can be applied directly to theory or simulation. For instance, while it is straightforward to calculate the compressibility, thermal expansion, and heat capacity from simulation, the determination of chemical potentials is much more involved (especially for large molecules and high densities). 

Order of the Paramagnetic-Antiferromagnetic Transition. A second-order transition is indicated by various studies with neutrons, Ott, Kjems [14], and the behavior of thermal expansion and heat capacity around the N6el temperature [5].The temperature dependence of critical scattering definitely excludes the existence of a smeared-out first-order phase transition [14]. How-... 

Temperature increases the equilibrium interionic separation. This is because the amplitude of vibration increases with temperature. But it brings about a more symmetrical arrangement. In other words, the entropy decreases as temperature is increased. The increase in thermal expansion and heat capacity is parallel with an increase in temperature, as shown in Figure 16.4. Both result from an increase in the internal energy of the material. 

Materials with molecular networks, such as cross-linked elastomers and crystalline polymers, do not flow and so are classified as viscoelastic solids. Shear stresses do not decay to zero with time, ie, equilibrium stresses can be supported. Above Tg, such amorphous materials are still classified as solids, but most of their physical properties such as thermal expansivity, thermal conductivity, and heat capacity are liquid-like. 

Specific volume v J, thermal expansion coefficient, heat capacity at constant pressure (cj, heat capacity at constant volume (cj, enthalpy, entropy, and thermal conductivity (X) are thermal characteristics relevant for processing and applications. They depend on PE grade, temperature, pressure, average molecular weight, branching, crystallinity or density, stretching ratio, heating rate, spherulite size, and so on. 

In the present section some other properties of epoxy polymers will be considered briefly, namely thermal expansion, anharmonicity, heat capacity and microhardness. 

A wide variety of physical properties are important in the evaluation of ionic liquids (ILs) for potential use in industrial processes. These include pure component properties such as density, isothermal compressibility, volume expansivity, viscosity, heat capacity, and thermal conductivity. However, a wide variety of mixture properties are also important, the most vital of these being the phase behavior of ionic liquids with other compounds. Knowledge of the phase behavior of ionic liquids with gases, liquids, and solids is necessary to assess the feasibility of their use for reactions, separations, and materials processing. Even from the limited data currently available, it is clear that the cation, the substituents on the cation, and the anion can be chosen to enhance or suppress the solubility of ionic liquids in other compounds and the solubility of other compounds in the ionic liquids. For instance, an increase in allcyl chain length decreases the mutual solubility with water, but some anions ([BFJ , for example) can increase mutual solubility with water (compared to [PFg] , for instance) [1-3]. While many mixture properties and many types of phase behavior are important, we focus here on the solubility of gases in room temperature IFs. 

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. 

Equation 2.47 describes the interdependence of thermal expansion, compressibility, and heat capacity of a first-order transition and furnishes a precise tool for the evaluation of the internal consistency of experimental data in solid state transition studies (see Helgeson et al., 1978 for a careful application of eq. 2.47). 

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

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... 

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... 

Some physical properties of water are shown in Table 7.2. Water has higher melting and boiling temperatures, surface tension, dielectric constant, heat capacity, thermal conductivity and heats of phase transition than similar molecules (Table 7.3). Water has a lower density than would be expected from comparison with the above molecules and has the unusual property of expansion on solidification. The thermal conductivity of ice is approximately four times greater than that of water at the same temperature and is high compared with other non-metallic solids. Likewise, the thermal dif-fusivity of ice is about nine times greater than that of water. 

The glass-transition temperature Tt is thought by some to be a second-order transition, so some data relevant to it are shown in the middle two sketches. The volume and entropy merely show a change in slope at Tr The second derivatives are shown in the bottom pair of sketches, with the expected discontinuities in the thermal expansion coefficient and heat capacity. 

Lattice vibrations are fundamental for the understanding of several phenomena in solids, such as heat capacity, heat conduction, thermal expansion, and the Debye-Waller factor. To mathematically deal with lattice vibrations, the following procedure will be undertaken [7] the solid will be considered as a crystal lattice of atoms, behaving as a system of coupled harmonic oscillators. Thereafter, the normal oscillations of this system can be found, where the normal modes behave as uncoupled harmonic oscillators, and the number of normal vibration modes will be equal to the degrees of freedom of the crystal, that is, 3nM, where n is the number of atoms in the unit cell and M is the number of units cell in the crystal [8],... 

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