Entropy of phase transition

If the heat capacity is not constant, it must be used as a function of temperature for Equation (1.70), for which the integration must then be carried out. When the temperature nears the absolute zero temperature, Cp=aT3 where a = 2.27x 10 4cal mol1 K-4. If there are some phase changes before reaching the temperature T, the entropy of phase transitions must be incorporated into the calculation for the absolute entropy ... 

In a displacive phase transition, the positions of atoms are ordered in both phases, but the position changes from a less symmetric site to a more symmetric one as the crystal undergoes the phase transition. Order-disorder transitions are distinguished from displacive transitions by, among other properties, a large entropy of phase transition, dielectric dispersion at low frequencies, and directly, by crystal structure revealing two or more sites fractionally occupied by the same atom. 

This extensive review gives heat capacities and entropies for approximately 1400 organic compounds in the liquid and solid phases at or near 25 °C. Values for enthalpies and entropies of phase transitions are also included. The literature coverage is from 1881 through most of 1982 and there are detailed references given to the source literature. 

Table 2, containing results derived from measurements of heat capacities of some pure organic compounds published since 1961, gives values of temperatures, enthalpies, and entropies of phase transitions, and also of standard entropies. In the Table, (t) refers to crystal-crystal transitions (m) to crystal-liquid transitions (c), (1), and (g) to crystal, liquid, and gas G to glass transition and m, s to metastable, stable crystal. 

Of course, Eq. (16) is approximate and cannot be used to calculate the entropies of phase transitions. It is useful from the point of view of the selection of more reliable experimental data because it shows the tendency of parameter variations in the series of compoimds imder consideration. We used this equation to select several phase transition parameters not reported by Gaune-Escard et al. (1996) and Rycerz and Gaune-Escard (2002a.b,c) from the set of data available in the Gmelin Handbook (1982). These data included experimental values reported for TbCla (Dworkin and Bredig, 1971), H0CI3 (T = 993 K, Drobot et al., 1968 ... 

Note that to use this formula we use the enthalpy of vaporization at the boiling temperature. Table 2.1 lists the entropy of vaporization of several substances at 1 atm. For the standard value, AyapS, we use data corresponding to 1 bar. Because vaporization is endothermic for all substances (with one exception of little relevance to biology helium), all entropies of vaporization are positive. The increase in entropy accompanying vaporization is in line with what we should expect when a compact liquid turns into a gas. To calculate the entropy of phase transition at a temperature other than the transition temperature, we have to do additional calculations, as shown in the following brief illustration. 

Thermodynamic properties of ionic liquids, such as heat capacity Cp,m, glass transition temperature melting temperature Tm, thermal decomposition temperature Ta, enthalpy and entropy of phase transitions are important data for the basic understanding of these materials and their application in academia and industry. These thermodynamic properties can be determined using adiabatic calorimetry and thermal analysis techniques (DSC, TG-DTG). 

Table 1.2 Temperatures, the molar enthalpies, and entropies of phase transitions of adamantane.

The entropy change AS/ - and the volume change AV/ - are the changes which occur when a unit amount of a pure chemical species is transferred from phase I to phase v at constant temperature and pressure. Integration of Eq. (4-18) for this change yields the latent heat of phase transition ... 

When we compared the viscosities of solutions of natural rubber and of guttapercha and of other elastomers and later of polyethylene vs.(poly)cis-butadiene, with such bulk properties as moduli, densities, X-ray structures, and adhesiveness, we were greatly helped in understanding these behavioral differences by the studies of Wood (6) on the temperature and stress dependent, melting and freezing,hysteresis of natural rubber, and by the work of Treloar (7) and of Flory (8) on the elasticity and crystallinity of elastomers on stretching. Molecular symmetry and stiffness among closely similar chemical structures, as they affect the enthalpy, the entropy, and phase transitions (perhaps best expressed by AHm and by Clapeyron s... 

The types of values reported in the database standard enthalpies of formation at 298.15 K and 0 K, bond dissociation energies or enthalpies (D) at any temperature, standard enthalpy of phase transition—fusion, vaporization, or sublimation—at 298.15 K, standard entropy at 298.15 K, standard heat capacity at 298.15 K, standard enthalpy differences between T and 298.15 K, proton affinity, ionization energy, appearance energy, and electron affinity. The absence of a check mark indicates that the data are not provided. However, that does not necessarily mean that they cannot be calculated from other quantities tabulated in the database. 

The conducting ion sublattice in FICs is generally considered molten . The molten sublattice model for fast ion conduction was first proposed by Strock (1936) on the basis of structural and thermodynamic data for Agl. In most FICs, the entropy of the phase transition to the FIC state is larger than the entropy of melting. For example, in Agl the entropy of the transition at 420 K from the -form to the a-form (FIC state) is 14.7 J deg mol , whereas the entropy of melting at 861 K is only 11 J deg mol . ... 

The common characteristics of phase transitions are that the Gibbs energy is continuous. Although the conditions of equilibrium and the continuity of the Gibbs energy demand that the chemical potential must be the same in the two phases at a transition point, the molar entropies and the molar volumes are not. If, then, we have two such phases in equilibrium, we have a set of two Gibbs-Duhem equations, the solution of which gives the Clapeyron equation (Eq. (5.73))... 

One of the most challenging tasks in the theory of liquids is the evaluation of the excess entropy Sex, which is representative of the number of accessible configurations to a system. It is well known that related entropic quantities play a crucial role, not only in the description of phase transitions, but also in the relation between the thermodynamic properties and dynamics. In this context, the prediction of Sex and related quantities, such as the residual multiparticle entropy in terms of correlation functions, free of any thermodynamic integration (means direct predictive evaluation), is of primary importance. In evaluating entropic properties, the key quantity to be determined is the excess chemical potential (3pex. Calculation of ppex is not straightforward and requires a special analysis. 

Not too many theories have been formulated from this point of view and some of the more interesting cases are at the speculative stage of development. Even so, it is remarkable how some of the most enigmatic of natural phenomena have no convincing explanation apart from broken-symmetry theories. Included are the initiation or nucleation of phase transitions, superconductivity (T4.5.1), the arrow of time (entropy) and the cosmic imbalance between matter and antimatter. The beauty of the world, indeed seems to lie in approximate symmetries. 

The second thermodynamic quantity controlling propagation is the entropy. Changes of phase transitions, of concentration, and addition itself... 

One of the important problems in thermodynamics involves the determination of the entropy of any material. Before considering this matter we must examine the characteristics of phase transitions. 

Using these relationships, the enthalpy and/or entropy changes associated with the occurrence of phase transitions, melting, sublimation or decomposition of the sample can be determined. Such processes generally result in considerable changes in the heat capacity of the sample, which appear on the DSC trace as marked deviations from the extrapolated baseline. 

As the temperature of a substance increases, the particles vibrate more vigorously, so the entropy increases (Figure 15-14). Further heat input causes either increased temperature (still higher entropy) or phase transitions (melting, sublimation, or boiling) that also result in higher entropy. The entropy of a substance at any condition is its absolute entropy, also called standard molar entropy. Consider the absolute entropies at 298 K listed in Table 15-5. At 298 K, any substance is more disordered than if it were in a perfect crystalline state at absolute zero, so tabulated values for compounds and elements are always positive. Notice especially that g of an element, unlike its A// , is not equal to zero. The reference state for absolute entropy is specified by the Third Law of Ther-... 

In a number of publications [12], classification of phase transitions in small systems has been presented. This scheme is based on the distribution of zeroes of the canonical partition function in the complex temperature plane. Among others. Gross has suggested a microcanonical treatment [13], where phase transitions of different order are distinguished by the curvature of the entropy 5 = In According to this scheme, a back-bending in the micro-... 

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