Isobaric specific volume-temperature

In Fig. 3 c the schematic volume-temperature curve of a non crystallizing polymer is shown. The bend in the V(T) curve at the glass transition indicates, that the extensive thermodynamic functions, like volume V, enthalpy H and entropy S show (in an idealized representation) a break. Consequently the first derivatives of these functions, i.e. the isobaric specific volume expansion coefficient a, the isothermal specific compressibility X, and the specific heat at constant pressure c, have a jump at this point, if the curves are drawn in an idealized form. This observation of breaks for the thermodynamic functions V, H and S in past led to the conclusion that there must be an internal phase transition, which could be a true thermodynamic transformation of the second or higher order. In contrast to this statement, most authors... 

Fig. 4. Specific volume-temperature curves at different pressures of the 1-l.c. (isobaric measurements) 37 38)...

Isobaric specific volume versus temperature curves are shown in Figure 1 for Kapton HN, PI-2540, Upiloc S and PI-2611. The temperature dependent volume at zero pressure V(0,T) was obtained fi-om an extrapolation of the Tait equation fit to the 10 to 50 MPa data. The V(0,T) data were fit with a second order pofynomial. The coefficients for the zero pressure volume, V(0,T), and the Tait parameter, B(T), for each film are listed in Table 1. 

As there exists a phase equilibrium both phases must have reached in the internal thermodynamic equilibrium with respect to the arrangement and distribution of the molecules the measuring time. Therefore, no time effects or path dependencies of the thermodynamic properties in the liquid crystalline phase should be expected. To check this point for the l.c. polymer, a cut through the measured V(P) curves at 2000 bar has been made (Fig. 6) and the volume values are inserted at different temperatures in Fig. 7, which represents the measured isobaric volume-temperature curve at 2000 bar 38). It can be seen from Fig. 7 that all specific volumes obtained by the cut through the isotherms in Fig. 6 he on the directly measured isobar. No path dependence can be detected in the l.c. phase. From these observations we can conclude that the volume as well as other properties of the polymers depend only on temperature and pressure. The liquid crystalline phase of the polymer is a homogeneous phase, which is in its internal thermodynamic equilibrium within the normal measuring time. 

This method is to be used to calculate the specific volume of a pure polymer liquid at a given temperature and pressure. This procedure uses the empirical Tait equation along with a polynomial expression for the zero pressure isobar. The method requires only the equation constants for the polymer. 

The above-mentioned method was initially developed for measuring the isobaric heat capacities of aqueous salt solutions up to 573 K and 30 MPa. For a typical run, the sample cell was loaded with the sample solution and the reference cell was loaded with a reference fluid of known heat capacity (usually water). Then, the temperature was increased from to T, at constant pressure, and the difference Q in the transferred heat was corrected taking into account both the cell s volumetric dissymmetry and the differences between the densities and specific heat capacities of the measured sample and reference fluids, respectively. Such an experiment allows the measurement of the product pCp representing the isobaric heat capacity divided by volume. In order to obtain the desired isobaric heat capacity, Cp, of the solution, it was necessary to know its density. For this purpose, the isobaric specific heat capacity and density were represented by polynomials in terms of temperature T ... 

Cp = isobaric specific heat c = isochoric specific heat e = specific internal energy h = enthalpy k = thermal conductivity p = pressure s = specific entropy t = temperature T = absolute temperature u = specific internal energy 4 = viscosity V = specific volume / = subscript denoting saturated liquid g = subscript denoting saturated vapor... 

Historically, the MNSJ equations (6.33) and (6.34) were evaluated using x-ray data on the crystalline specific volume. Thus, the isobars of linear polyethylene (LPE), poly(viny lidene fluoride) (PVDF), and poly(chlorotrifluoroethylene) (PCIF3), and an isotherm at atmospheric pressure of LPE were fitted to the theoretical dependencies [Simha and Jain, 1978 Jain and Simha, 1979a,b]. The fitting requires five characteristic parameters P T V, c/s, and 60, the last two having universal values. While in the melt, the macromolecular external degrees of freedom 3c/s 1 in the crystalline polymers, c/s l. For different crystalline species the characteristic reduced quantum temperature value adopted at 0 (K) is o = (hpvo/ksT ) 0.022, where... 

FIGURE 8.2 Temperature dependence of specific volume for EVOH-44 at the pressures indicated, as received from isobaric-mode experiments with a heating rate of 1 K/min. 

In Fig. 5 the isobaric and isothermal pVT curves of P4SC are shown. Because of the very small changes in specific volume the first phase transition can hardly be seen in the isobaric experiment. The two transitions are resolved only in the heating runs. The phase transition temperature, Td, is shifted to higher values and a decrease in AVsp is observed for both phase transitions. The two transitions are separated by only about 7°C for all pressures investigated. In the isothermal experiment (Fig. 5, right) pressure-induced crystallization can be seen for temperatures above 120°C, concurrently with a decrease in Ksp. Crystallization experiments performed with different pressure cycles show no distinct changes in the phase behavior in particular, the pure J. phase could not be observed. 

The prediction of miscibility requires knowledge of the parameters T" (the characteristic temperature), p (the characteristic pressure) and V (the characteristic specific volume) of the corresponding equation of state which can be calculated from the density, thermal expansivity and isothermal compressibility. The isobaric thermal expansivity and the isothermal compressibility can be determined experimentally from p-V-Tmeasurements where these values can be calculated from V T) and V(p)j. The characteristic temperature T is a measure of the interaction energy per mer, V is the densely packed mer volume so that p is defined as the interaction energy per... 

It is used to describe the glass transition. We illustrate the glass transition by an isobaric volume-temperature diagram as given in Figure 11 in chapter 5 V = V P,T) symbolizes the specific volume, V= vim. The diagram reveals ... 

Plastic melts are also compressible. Thus, a temperature and pressure change of200°C and 50 MPa, respectively, causes a 10 to 20 percent difference in density, depending on whether the polymer is amorphous or semicrystalline. Specific volume (v), the inverse of density, is often used to relate density to temperature and pressure. As shown in the typical pressure-volume-temperature (pvT) curves presented in Fig. 5.6, specific volume increases with temperature but decreases with increasing pressure. The isobars exhibit significant changes as the polymer passes through its transition temperature this is Tg for amorphous... 

Following injection of the plastic into the cavity (at a constant temperature, TJ its specific volume has diminished, and the compressed plastic is subject to further isobaric cooling at the holding pressure. The measure of compression of the plastic is the increase in its density. 

For plastics with a subsequent semi-crystalline structure, e.g., HOPE in Figure 3.4b, the plastic should set under constant pressure (isobaric procedure) down to the crystallisation temperature T. Release of the heat of crystallisation causes a retardation of the temperature drop, while at the same time there is a rapid fall in specific volume. There is a partial arrangement of molecules into ordered structures called crystallites. 

Figure 8 Specific volume v of poly(vinyl acetate) vs. temperature T at different pressures, as indicated, for isobaric formed at 1 atm note definition of rj(P, 0) for this type of glass (after ref. 37, with permission)...

From thermodynamic reasons follows that the change AmH of the molar (or specific or segment molar) enthalpy in an isothermal-isobaric mixing process is also the molar (or specific or segment molar) excess enthalpy, ff, of the mixture. The dependence of bf upon temperature, T, and pressure, P, permits the correlation of such data with excess heat capacities, Cf, and excess volumes,... 

The first coefficient describes the most common case, namely how much entropy AS flows in if the temperature outside and (also inside as a result of entropy flowing in) is raised by AT and the pressure p and extent of the reaction are kept constant. In the case of the secmid coefficient, volume is maintained instead of pressure (this only works well if there is a gas in the system). In the case of J = 0, the third coefficient characterizes the increase of entropy during equilibrium, for example when heating nitrogen dioxide (NO2) (see also Experiment 9.3) or acetic acid vapor (CH3COOH) (both are gases where a portion of the molecules are dimers). Multiplied by T, the coefficients represent heat capacities (the isobaric Cp at constant pressure, the isochoric Cy at constant volume, etc.). It is customary to relate the coefficients to the size of the system, possibly the mass or the amount of substance. The corresponding values are then presented in tables. In the case above, they would be tabulated as specific (mass related) or molar (related to amount of substance) heat capacities. The qualifier isobaric and the index p will... 

The specific heat capacity of an ideal gas is defined as the heat per amount of substance in the ideal gas state necessary to obtain a certain temperature change. It must be distinguished between the specific isobaric heat capacity c p (at constant pressure) and the specific isochoric heat capacity c[f (at constant volume). For ideal gases, both quantities are related [52] via... 

Since there is a limit on the number of degrees of freedom of a system with different phases existing in equilibrium, it is possible to use a two-dimensional plot (e.g., an x-y plot) to show the possible behavior. In the beginning of this text, P-V isotherms were drawn as two-dimensional plots of the allowed behavior of a gas. We could instead have drawn V-T isobars to show the relation between volume and temperature at specific pressures. To understand and follow phase behavior, it is a plot of pressure versus temperature that is normally the most useful. This is because temperature and pressure, not volume and entropy, are most easily varied in the laboratory. Volume is not independent when the variables of interest are pressure and temperature, and so we could consider drawing "constant-volume" curves on a P-T plot. More interesting to follow is the pressure and temperature at which phase equilibrium is maintained, regardless of the volume. 

In reviewing the data in Table 6.3 it should be kept in mind that the heats of vaporization for a mixture evaluated under constant pressure conditions are considerably different from those for the same mixture determined under constant temperature conditions. For constant-pressure processes, the heat of vaporization of a specific mixture is the difference between the saturated-vapor enthalpy, /Zy, and the saturated-liquid enthalpy, /z, at the specified composition. For constant-temperature processes, the total pressure of the system must be adjusted continuously, resulting in a somewhat different value for the heat of vaporization. This is analogous to the difference in specific heats at constant volume and at constant pressure. Since evaporation is normally performed under constant pressure, the isobaric heat of vaporization is generally implied when the unqualified term heat of vaporization is used. 

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