Melting temperatures, heats and entropies of fusion

A key quantity necessary to carry out the thermodynamic analysis is the enthalpy of fusion per repeating unit. This quantity is an inherent property of a polymer chain. 

The validity of this equation for polymer-diluent mixmres has been amply demonstrated. The only parameters that are needed to analyze the experimental data are the respective molar volumes of the diluent and polymer repeating unit. 

The other direct thermodynamic method that leads to reliable values for AHu is the application of the Clapeyron equation to the change in the equilibrium melting temperature with applied hydrostatic pressirre p. Accordingly 

In order to apply Eq. (6.1) the volume of the repeating unit for the liquid and crystal (unit cell) needs to be known as a function of pressure at the melting temperature. From the experimentally determined and A V (the latent volume change per unit 

There are several indirect methods that also yield values of AH . They all require the determination of the enthalpy of fusion as well as the degree of crystallinity of the system. The degree of crystallinity can be obtained by different experimental techniques such as infra-red, wide-angle x-ray diffraction and density measurement among others. Quite often the enthalpy of fusion is measured as a fiinction of density and the data extrapolated to the value of the unit cell to yield AH . The directly measured enthalpy of fusion, as well as the methods used to determine the crystallinity level, are dependent on morphological and structural detail. Moreover, all of the methods usually have different sensitivities to the phase structures. In our 

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Further information on the effect of polymer structure on melting points has been obtained by considering the heats and entropies of fusion. The relationship between free energy change AF with change in heat content A// and entropy change A5 at constant temperature is given by the equation... 

It is possible to determine C quantitatively using Hildebrand s theory of microsolutes. An example of the accuracy that can be achieved is provided by the calculation of the solubilities of a series of p-aminobenzoate esters in hexane (17,18). Michaels, et al. (19) used this approach to estimate the solubility of steroids in various polymers. The solubilities of seven steroids in six polymers were calculated from the steroid melting points, heats of fusion, and solubility parameters. Equation 8 was derived, where Jjj is the maximum steady state flux, h is the membrane thickness, x is the product of V, the molar volume of the liquid drug, and the square of the difference in the solubility parameters of the drug and polymer, p is the steroid density, T is melting point (°K), T is the temperature of the environment, R is the gas constant, and AH and ASf are the enthalpy and entropy of fusion, respectively. 

Polyethylene data are shown in Fig. 2.23. At the equilibrium melting temperature of 416.4 K, the heat of fusion and entropy of fusion are indicated as a step increase. The free enthalpy shows only a change in slopes, characteristic of a first-order transition. Actual measurements are available to 600 K. The further data are extrapolated. This summary allows a close connection between quantitative DSC measurement and the derivation of thermodynamic data for the limiting phases, as well as a connection to the molecular motion. In Chaps. 5 to 7 it will be shown that this information is basic to undertake the final quantitative step, the analysis of nonequilibrium states as are common in polymeric systems. 

In practice, heat capacity measurements are usually made from about 10 K. Values of C, below the lowest temperature of the measurements are obtained by extrapolation. For organic compounds with melting temperatures below the highest temperature of the Cj, measurements, values for the enthalpy and entropy of fusion are obtained in the course of the measurements and it is usual to calculate the equilibrium temperature (Tm) in the... 

We typicallyknowthe enthalpy (heat) and entropy offiision at aspecifted temperature— the normal melting point, T. Therefore, we need to construct a thermodynamic pathway to find the enthalpy and entropy of fusion at any T. Figure 8.19 illustrates a path for the calculation of Ahfus- Adding together the three steps, we get ... 

The entropy of fusion in cal deg-1 g-1 is easily computed from the data of Table 3 by dividing the heats of fusion by the absolute temperature of the melting point. For theoretical reasons in many cases it is interesting to know the entropy of fusion at constant volume. Thus, the total measured entropy of fusion at constant pressure and composition, ASf, must be corrected for the gain in entropy due to the volume increase on fusion. The fusion process may be imagined as occurring at... 

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,... 

If the pressure is above the triple point, the solid will first melt, then vaporize. In this case, wo can proceed in a similar way. On melting, the entropy increases by the entropy of fusion, determined from the latent heat of fusion and temperature of fusion by the relation... 

Calculate the molar entropies of fusion and vaporization for benzene. Having a molecular weight of 78.1, benzene melts at 5.5°C with a heat of fusion of 2350 cal/(g mol). Its normal boding point is 80.1°C, and its heat of vaporization at that temperature is 94.1 cal/g. 

The melting temperature for PAN is extremely high as compared, e. g., with that of isotactic polystyrene (230 °C) or high density polyethylene (137—140 °C). Generally, a high melting point, T , can be caused by a high heat of fusion, AHf, and/or by a low entropy of fusion, ASf ... 

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