Entropy phase transformation

Driving forces for solid-state phase transformations are about one-third of those for solidification. This is just what we would expect the difference in order between two crystalline phases will be less than the difference in order between a liquid and a crystal the entropy change in the solid-state transformation will be less than in solidification and AH/T will be less than AH/T . 

The solid product, BaO, was apparently amorphous and porous. Decomposition rate measurements were made between the phase transformation at 1422 K and 1550 K (the salt melts at 1620 K). The enthalpy and entropy of activation at 1500 K (575 13 kJ mole-1 and 200 8 J K"1 mole-1) are very similar to the standard enthalpy and entropy of decomposition at the same temperature (588 7 kJ and 257 5 J K-1, respectively, referred to 1 mole of BaS04). The simplest mechanistic explanation of the observations is that all steps in the reaction are in equilibrium except for desorption of the gaseous products, S02 and 02. Desorption occurs over an area equivalent to about 1.4% of the total exposed crystal surface. Other possible models are discussed. 

One of the consequences of accepting the presence of multiple magnetic states is an additional contribution to the entropy and, therefore, several authors have considered the inclusion of multiple states in their description of low-temperature phase transformations in Fe and its alloys (Kaufman et al. 1963, Miodownik 1970, Bendick and Pepperhoff 1978). However, most authors have, in the end, preferred to describe the magnetic effects in Fe using more conventional temperature-independent values for the magnetic moments of the relevant phases. This is partly linked to the absence of any provision for the necessary formalism in current... 

From the dynamic mechanical investigations we have derived a discontinuous jump of G and G" at the phase transformation isotropic to l.c. Additional information about the mechanical properties of the elastomers can be obtained by measurements of the retractive force of a strained sample. In Fig. 40 the retractive force divided by the cross-sectional area of the unstrained sample at the corresponding temperature, a° is measured at constant length of the sample as function of temperature. In the upper temperature range, T > T0 (Tc is indicated by the dashed line), the typical behavior of rubbers is observed, where the (nominal) stress depends linearly on temperature. Because of the small elongation of the sample, however, a decrease of ct° with increasing temperature is observed for X < 1.1. This indicates that the thermal expansion of the material predominates the retractive force due to entropy elasticity. Fork = 1.1 the nominal stress o° is independent on T, which is the so-called thermoelastic inversion point. In contrast to this normal behavior of the l.c. elastomer... 

An isothermal change in phase (phase transformation) of a substance produced by input of energy as heat always leads to an increase in the entropy of the substance. 

If q is the heat exchanged reversibly per mole of the substance during the phase transformation at temperature T, then the change of entropy (AS) in this process is given by AS = q/T dP q... 

Compute the entropy change due to phase transformation. In this case,... 

Chemical behaviour depends on chemical potential and electromagnetic interaction. Both of these factors depend on the local curvature of space-time, commonly identified with the vacuum. Any chemical or phase transformation is caused by an interaction that changes the symmetry of the gauge field. It is convenient to describe such events in terms of a Lagrangian density which is invariant under gauge transformation and reveals the details of the interaction as a function of the symmetry. The chemically important examples of crystal nucleation and the generation of entropy by time flow will be discussed next. The important conclusion is that in all cases, the gauge field arises from a symmetry of space-time and the nature of chemical matter and interaction reduces to a function of space-time structure. 

By way of illustrations we display in Fig. 1.17.2a plot of the molar heat capacity of oxygen under standard conditions. The plot of Cp vs. In T is then used to determine the entropy of oxygen from the area under the curves. Note that the element in the solid state exists in three distinct allotropic modifications, with transition temperatures close to 23.6 and 43.8 K the melting point occurs at 54.4 K, and the boiling point is at 90.1 K. All the enthalpies of transition at the various phase transformations are accurately known. An extrapolation procedure was employed below 14 K, which in 1929 was about the lower limit that could conveniently be reached in calorimetric measurements. 

In other cases, an unfavorable enthalpy term was found to be compensated by a favorable entropy term, thus rendering negative the free energy change associated with a particular phase transformation. Lamivudine can be obtained in two forms, one of which is a 0.2-hydrate obtained from water or from methanol that contains water and the other of which is non-solvated... 

Ubbelohde [19] has emphasized the sparsity of data on the properties of solids and liquids in the vicinity of the melting point. Fusion is accompanied by a loss of stmctural order, which always contains a positional contribution and sometimes also contains an orientational contribution, so that the entropy increase may include more than a single term. In some solids the increasing disorder, resulting from the temperature rise, may involve one or more phase transformations below the melting point. 

As supersaturation is increased, the entropy of phase transformation is lowered, thereby lowering AG for formation of the condensed crystalline phase (nucleation), as shown in Fig. 4-4. In the (3-carotene/THF/water example of Fig. 4-3, S > 8, the critical-sized embryo is about one or two molecules. 

The selection of entropy and heat capacity data for trigonal selenium in the temperature range 298.15 to 494.2 K and for the liquid in the temperature range 494.2 to 1500 K. The thermodynamic properties of the metastable monoclinic phase and the supercooled liquid are assessed for use as auxiliary data. The major source of information is the review by Gaur, Shu, Mehta, and Wunderlich [81GAU/SHU] which has been combined with other information for phase transformations. 

The standard entropy of a-SnSe was evaluated to be 86.93 J-K" -mol , corresponding to Af5° (SnSe, a, 298.15 K) = - 6.3 J-K -mor, in the thermodynamic optimisation and assessment of the Sn-Se system in [96FEU/MAJ]. The value originates mainly from the modelling and assumptions made about the liquid phase in the system and the recalculation to 298.15 K by the use of enthalpies of phase transformations and heat capacities. The only experimental determination of the entropy at low temperatures was made by Melekh, Stepanova, Fomina, and Semenkovich [71MEL/STE] who performed emf measurements on the galvanic cells... 

Statistical Mechanics of the Harmonic Oscillator. As has already been argued in this chapter, the harmonic oscillator often serves as the basis for the construction of various phenomena in materials. For example, it will serve as the basis for our analysis of vibrations in solids, and, in turn, of our analysis of the vibrational entropy which will be seen to dictate the onset of certain structural phase transformations in solids. We will also see that the harmonic oscillator provides the foundation for consideration of the jumps between adjacent sites that are the microscopic basis of the process of diffusion. 

This chapter has shown how the zero-temperature analyses presented earlier in the book may be extended to incorporate finite-temperature effects. By advancing the harmonic approximation we have been able to construct classical and quantum mechanical models of thermal vibrations that are tractable. These models have been used in the present chapter to examine simple models of both the specific heat and thermal expansion. In later chapters we will see how these same concepts emerge in the setting of diffusion in solids and the description of the vibrational entropies that lead to an important class of structural phase transformations. 

Another implication of Eq. (5.16) is that if the vibrational frequency of the atoms changes from, say, a frequency a to a, as a result of a phase transformation or the formation of defects, e.g., the associated entropy change is... 

Given that (see Fig. 9.8) at the glass transition temperature, the specific volume Vs and entropy S are continuous, whereas the thermal expansivity a and heat capacity Cp are discontinuous, at first glance it is not unreasonable to characterize the transformation occurring at Tg as a second-order phase transformation. After all, recall that, by definition, second-order phase transitions require that the properties that depend on the first derivative of the free energy G such as... 

Many of the changes which occur in solids are not profound enough to impose a new space lattice and thereby to reveal themselves as polymorphic phase transformations. They may depend upon alterations of configuration of the statistical kind exemplified by the copper-zinc alloy and the abnormality m the specific heat with which they are associated is connected with the increased potential energy imposed by the more random arrangement. The heat absorption is due primarily, not to the excitation of new degrees of freedom, but to the increase in configurational entropy. 

In some cases crystallisation sequences are observed where a first formed phase transforms after extended reaction times to other phases. This kind of behaviour is in accord with Ostwald s law, which states that under kinetically controlled conditions, the first phases to form will be those with higher entropy, and these may transform towards the thermodynamically most stable phase via phases with progressively lower entropy and lower free energy. The sequential synthesis of zeolite Na-Y and then the denser zeolite Na-P from the same... 

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