High-temperature series

The truncation of the high temperature series in Eq. (4) at order is a vahd concern when applying the theory at low temperatures. This concern also extends to GD theory, which implicitly involves a truncation at order p. At some point, these perturbative treatments must simply fail, but we expect the lattice theories to identify faithfully the location of the entropy crisis at low temperatures, based on numerous previous comparisons between measurements and GD theory. Experience [96] with the LCT in describing equation of state [97] and miscibility [98] data indicates that this approach gives sensible and often accurate estimates of thermodynamic properties over wide ranges of temperatures and pressures. In light of these limitations, we focus on the temperature range above Tg, where the theory is more reliable. 

Here, the Weiss temperature 9 is a function of the exchange energies /ex that can be obtained from a high-temperature series expansion ... 

Exact solutions do not exist for the exchange Hamiltonian of two-dimensional layers and three-dimensional networks. These situations can, however, be described using approximation methods such as molecular field approximation and high-temperature series expansion. 

For the case of ferromagnetic coupling, a high-temperature series expansion has been proposed ... 

Although there are no exact results to compare with the three-dimensional heat capacity, tentative extrapolations of high-temperature series have indicated that ... 

Generally the critical temperature is lower in the more refined approximations than it is in the mean field approximation. It is not very different from the one obtained using high temperatures series expansions (Elliott et al., 1971). A numerical comparison of different approximations is shown in fig. 17.28. Somewhat subtle is the question of the order of the phase transition. The decoupling approximations going beyond the RPA-MF approximation lead to first order phase transitions for some or even all values of the relevant parameters. 

Fig. 238. Cu(HCOO)2 4H2O. The experimental paramagnetic susceptibility is compared with the calculated one resulting from the use of high temperature series expansions (HTS),...

Fig. 240. Cu(HCOO)2 -4H20. Temperature dependence of l/fC, and 1/ 2- Measuring points experimental solid lines high temperature series expansion solution to the Heisenberg two-dimensional antiferromagnetic lattice with J/hc=-54cm, g =2.34, gj =2.04. The calculated X T and values have been corrected for TIP=120 10" ...

The plagioclase structures are triclinic and similar to albite, i.e., essentially a sanidine framework distorted to conform fairly closely around the smaller Ca and Na ions substituting in place of K in sanidine. They show both low- and high-temperature forms, as well as intermediate structural states. There is evidence of complete solid solution in the high-temperature series from albite to anorthite, but there are a number of complex structural changes or discontinuities in the corresponding low-temperature series (Figure 6). If all of these structures... 

It is detemrined experimentally an early study was the work of Andrews on carbon dioxide [1], The exact fonn of the equation of state is unknown for most substances except in rather simple cases, e.g. a ID gas of hard rods. However, the ideal gas law P = pkT, where /r is Boltzmaim s constant, is obeyed even by real fluids at high temperature and low densities, and systematic deviations from this are expressed in tenns of the virial series ... 

Figure A2.5.6 shows a series of typical p, Fisothemis calculated using equation (A2.5.1). (The temperature, pressure and volume are in reduced units to be explained below.) At sufficiently high temperatures the pressure decreases monotonically with increasing volume, but below a critical temperature the isothemi shows a maximum and a minimum.

A schematic of a continuous bulk SAN polymerization process is shown in Figure 4 (90). The monomers are continuously fed into a screw reactor where copolymerization is carried out at 150°C to 73% conversion in 55 min. Heat of polymerization is removed through cooling of both the screw and the barrel walls. The polymeric melt is removed and fed to the devolatilizer to remove unreacted monomers under reduced pressure (4 kPa or 30 mm Hg) and high temperature (220°C). The final product is claimed to contain less than 0.7% volatiles. Two devolatilizers in series are found to yield a better quaUty product as well as better operational control (91,92). 

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