Isothermal expansion below

Fig. 2-3. Typical phasediagram of a pure substance wjth two lines of isothermal expansion 123 below critical temperature, 45 above critical temperature.

By combining the thermodynamic equation of stat e with the van der Waals equation, we can show that dU/dV)x = ajV. Below the critical temperature, the van der Waals equatiop predicts, approximately, a liquid volume equal to b, and a gas volume equal to RT/p. Assuming that the substance follows the van der Waals equation, what increase in energy attends the isothermal expansion of one mole of a substance from the liquid volume to the gaseous volume ... 

The fluorine content, density, critical surface energy, glass transitions, thermal expansion coefficient above and below the glass transition, and 300°C isothermal thermogravimetric stabilities of the fluoromethylene cyanate ester resin system with n = 3, 4, 6, 8, 10 are summarized Table 2.2. Also included for the purpose of comparison are the corresponding data for the aromatic cyanate ester resin based on the dicyanate of 6F bisphenol A (AroCy F, Ciba Geigy). 

The approach to the critical point, from above or below, is accompanied by spectacular changes in optical, thermal, and mechanical properties. These include critical opalescence (a bright milky shimmering flash, as incident light refracts through intense density fluctuations) and infinite values of heat capacity, thermal expansion coefficient aP, isothermal compressibility /3r, and other properties. Truly, such a confused state of matter finds itself at a critical juncture as it transforms spontaneously from a uniform and isotropic form to a symmetry-broken (nonuniform and anisotropically separated) pair of distinct phases as (Tc, Pc) is approached from above. Similarly, as (Tc, Pc) is approached from below along the L + G coexistence line, the densities and other phase properties are forced to become identical, erasing what appears to be a fundamental physical distinction between liquid and gas at all lower temperatures and pressures. 

The expressions in brackets are the expansivities above and below Tg. The constant K3 is a function of bond type in chains and is really constant for every class of polymers. The physical interpretation of this equation may be consistent with the iso-free-volume concept. However, we believe that the introduction of this equality is in practise a denial of the concept. There are also other arguments against this concept. Kastner56 found, for example, that dielectric losses diminish during the isothermal volume contraction, which indicates a dependence of relaxation times on free-volume. However, if we assume that relaxation time depends exclusively on free-volume, the calculated reduction factor differs from the experimental one. 

Shah et al. carried out a Monte-Carlo simulation in the isothermal-isobaric (NPT) ensemble of [C4mim][PF6] [12]. The authors calculated the molar volume, cohesive energy density, isothermal compressibility, cubic expansion coefficient, and liquid structure as a function of temperature and pressure. A united atom force field was developed using a combination of ab initio calculations and literature parameter values were also developed. Calculated molar volumes were within 5% of experimental values, and a reasonable agreement was obtained between calculated and experimental values of the isothermal compressibility and cubic expansion coefficient. [PF6] anions were found to cluster preferentially in two favorable regions near the cation, namely around the C2 carbon atom, both below and above the plane of the imidazole ring [12],... 

The possible existence of an endpoint for the supercooled liquid locus is particularly Interesting in view of the experiments of Angell and coworkers (7,8,9,10). They find that pure water at ordinary pressures (even very finely dispersed) cannot apparently be supercooled below about —40 "C, and that virtually all physical properties manifest an impending "lambda anomaly at T, = —45 . The most striking features of this anomaly are the apparent divergences to infinity of isothermal compressibility, constant-pressure heat capacity, thermal expansion, and viscosity. We now seem to have in hand a qualitative basis for explaining these observations. 

This method has the disadvantage at temperatures below the critical temperature because of the significant decrease in the pressure range within which measurements along an isotherm can be made. As a result, it has become general practice to use a relatively high temperature Burnett expansion as a base,... 

The rapid variations (rise or fall) in the value of the thermodynamic response fimctions, namely the specific heat, the isothermal compressibility (both increase), and the coefficient of thermal expansion (which decreases with temperature when the latter is lowered below the freezing point), are some of the known spectacular anomalies of liquid water. These variations have till now eluded a fully satisfactory understanding [6]. Many computer simulation studies have been done and several theoretical approaches have been developed but they are still not universally accepted. 

These coefficients are themselves functions of pressure and temperature, but for liquids, the effect of pressure is quite weak we may take them to depend on temperature only. The coefficient of isothermal compressibility expresses the degree to which the volume of a liquid responds to pressure. Since liquids at temperatures well below the critical are nearly incompressible, the coefficient of isothermal compressibility is approximately zero. The coefficient of thermal expansion is usually small, as liquids expand much less than gases, but it is not zero. Perry s Oiemiccd Engineers Handbook provides data on the thermal expansion of selected liquids. 

Additional data The volume expansivity and the isothermal compressibility of liquid n-butane are given below and may be assumed to be constant ... 

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