Equilibrium heat of fusion

The Kirchhoff equation as derived above riiould be applicable to both chemical and physical processes, but one highly important limitation must be borne in mind. For a chemical reaction there is no difficulty concerning (dAH/dT)p, i.e., the variation of AH with temperature, at constant pressure, since the reaction can be carried out at two or more temperatures and AH determined at the same pressure, e.g., 1 atm., in each case. For a phase change, such as fusion or vaporization, however, the ordinary latent heat of furion or vaporization (AH) is the value under equilibrium conditions, when a change of temperature is accompanied by a change of pressure. If equation (12.7) is to be applied to a phase change the AH s must refer to the same pressure at different temperatures these are consequently not the ordinary latent heats. If the variation of the equilibrium heat of fusion, vaporization or transition with temperature is required, equation (12.7) must be modified, as will be seen in 271. 

In connection with the variation of the equilibrium heat of fusion with temperature, this simplification is not permissible. However, d AV)/dT ]p is usually small for the solid-liquid phase change, and so equation (27.31) reduces to... 

Physical properties of PHAs are determined by monomer units, which are predominantly responsible for the molecular interactions, the molecular weight, and the molecular weight distribution. In addition, different crystalline modifications and processing conditions have a considerable effect on the achievable property level of the samples. For this reason, only the basic material data are listed and compared the glass transition temperature (7 ), the equilibrium melting temperature of an infinite crystal (T ), the equilibrium heat of fusion (AH ), and the densities of the amorphous (yj and crystalline (yc) parts (Table 1). 

Equilibrium melting temperature of an infinite crystal (7 ) Equilibrium heat of fusion (A/7 )... 

Further study shows that this LC-PI exhibits double melting behavior, which can be induced by annealing or crystallization at lower temperatures [81]. Double melting phenomenon is also observed in some other thermotropic liquid crystalline polymers [82,83]. Literature indicated that the crystallization in solid polymers was the same as that from melt. To obtain the metastable equilibrium heat of fusion, an extrapolation of versus (logfa) for the newly... 

Thermodynamic approaches provide powerfiil tools to characterize the properties in identifying these metastable states to imderstand the effects of phase size, dimensionality, and composition on the materials properties. One well-known example is the density gradient column method to determine densities of semicrystalline polymers. Based on known equilibrium crystalhne and amorphous densities, the crystallinity of a semicrystalline sample can be calculated by using equations 4 and 5. However, it should be noted that the determination of these equilibrium density data is not trivial. Proper extrapolations are necessary to ensure the equilibrium nature of the results. Detailed issues discussed can be foimd in Reference 146. Another commonly used method is to measure the heat of crystallization or fusion by using dsc. By knowing the equilibrium heat of fusion, the crystallinity of a sample can be easily calculated. 

The heat of fusion of 100% crystalline polymer, or as it is sometimes called, the equilibrium heat of fusion (AHl), is the heat of fusion of the equilibrium polymeric crystals at the equilibrium melting point (the heat of fusion of 100% crystalline polymer depends somewhat on the melting temperature that is why AHf is given at T ). 

The Equilibrium Heat of Fusion Another equilibrium quantity is the heat of fusion for 100% crystalline polymer. The value of this parameter must be known for determining the crystallinity of polymers by DSC. 

Here, Tm is the melting point of the polymer in the solvent, is the equilibrium melting point of the polymer, AHl is the equilibrium heat of fusion (J/mol) of the repeating unit, V2 is the molar volume of the polymer, Vi is the molar volume of the solvent Vi is the volume fraction of the solvent, %i is the polymer-solvent interaction parameter, and R is the universal gas constant. This equation can be simphfied to the following form ... 

Method (b) for the estimation of the equilibrium heat of fusion is based on melting point depression by a diluent and the Flory-Huggins equation [3,30] ... 

The method (c) for the estimation of the equilibrium heat of fusion A//f (100%) is based on the determination of low molar mass analog heat of fusion. Figure 9.9 shows the linear increase in heat of fusion with molecular chain length. The typical behavior of analogs of PE [1, 20] is observed, such that bigger heat of fusion A//t corresponds to larger molar mass... 

The estimation of the equilibrium heat of fusion A//t (100%) from method (f) is based on using the Clausius-Clapeyron equation [31] ... 

Fig. 1. Schematic thermal analysis results on fusion and devitrification (top and bottom curves, respectively). The area under the peak in the top curve represents the heat of fusion Hf and can be used to calculate the entropy of fusion ASf = AHf/Tm in case of equilibrium melting. The increase in heat capacity ACP at Tg is related to the number of motifs that gain mobility...

A Hn is the heat of fusion per mole of 4-units, is the smallest length of crystallites existing at equilibrium at the temperature T, and is given by the equation,... 

The line-decomposition analysis of the equilibrium spectrum of the a-meth-ylene carbon was carried out using the elementary line shapes thus obtained for the three phases. The result is shown in Fig. 24. The composite curve of the decomposed components reproduces well the experimentally observed spectrum. The mass fraction of the crystalline component was estimated as 0.60 that is described in the figure. Based on the heat of fusion of 8.13 KJ/mol of this sample and the value of 14.2 KJ/mol for the crystalline material of this polymer the crystalline fraction was estimated to be 0.57. Here the heat of fusion for the crystalline material was obtained from the effect of diluent on the melting temperature with use of the relationship developed by Flory [91 ]. The crystalline fraction estimated from the NMR analysis is in good accord with the value estimated from the heat of fusion, supporting the rationality of the NMR analysis. 

By calculating the enthalpy of the reaction and assuming no significant entropy change, the AH (based on heats of formation and heats of fusion) of Equation 3 is approximately +5.2 kcal/mole and of Equation 4 is approximately +10.5 kcal/mole. While both reactions are not favored thermodynamically, at high temperatures (i.e., 375°C), these reactions will establish an equilibrium where significant amounts of KOH and/or K+ and OH" may exist. 

The concept of melting point may be put on a more quantitative basis. The process of melting is a thermal one and is characterized by the collapse of molecular units in a crystal to a disordered array. Energy is required to rupture the crystal lattice so that the heat of fusion, A/7fus, is positive. In addition, the solid-liquid transition involves an increase of randomness and the entropy of fusion, ASfts, is also positive. Melting point is the temperature of fusion (7 ) at which solid and liquid phases are in equilibrium. Therefore,... 

This idea has been further developed by Sutherland,2 who concludes that ordinary water vapour is monohydrol, H20 liquid water is an equilibrium mixture of tnhydrol, (H20)3, and dihydrol, (H20)2 whilst ice consists entirely of trihydrol. The high value found for the latent heat of fusion of ice thus receives explanation, for it is not due merely to the heat absorption consequent upon physical change of solid to liquid it is enhanced by the heat required to effect the simultaneous dissociation or depolymerisation of a large proportion of trihydrol molecules to the dihydrol form, as indicated by the thermal equation which Sutherland writes as ... 

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