Compressibility data, isothermal

B. One Atmosphere Compressibilities. Most isothermal compressibility (6) measurements are made over extended pressure ranges and very few direct measurements have been made near 1 atm. It is difficult to obtain reliable values of 8 at 1 atm from high pressure PVT data due to the problems of extrapolating the compression [k = (v - v )/v P, where v is the specific volume] to zero applied pressure. If the compressions, volumes, or densities are fit to functions of P, the compressibility... 

Table 5.48 summarizes thermal expansion and compressibility data for amphibole end-members according to the databases of Holland and Powell (1990) and Saxena et al. (1993). Isobaric thermal expansion (a, K ) and isothermal compressibility (jS, bar ) may be retrieved from the listed coefhcients by applying the polynomial expansions... 

If the heat capacity functions of the various terms in the reaction are known and their molar enthalpy, molar entropy, and molar volume at the 2) and i). of reference (and their isobaric thermal expansion and isothermal compressibility) are also all known, it is possible to calculate AG%x at the various T and P conditions of interest, applying to each term in the reaction the procedures outlined in section 2.10, and thus defining the equilibrium constant (and hence the activity product of terms in reactions cf eq. 5.272 and 5.273) or the locus of the P-T points of univariant equilibrium (eq. 5.274). If the thermodynamic data are fragmentary or incomplete—as, for instance, when thermal expansion and compressibility data are missing (which is often the case)—we may assume, as a first approximation, that the molar volume of the reaction is independent of the P and T intensive variables. Adopting as standard state for all terms the state of pure component at the P and T of interest and applying... 

Fig. 7.16. Experimental and calculated equation of state for (a) MgO and (b) CaO taken from Bukowinski (1985). Both calculated isothermal compression data and Hugoniot data (with shading representing uncertainty due to possible errors in the Gruneisen parameter) are shown. See original text for details of sources of experimental data (after Bukowinski, 1985 reproduced with the publisher s permission).

All calculations were carried out at T = 313.15 K. The vapor-liquid equilibrium (VLB) data for the ternary mixture and the corresponding binaries were taken from [32]. The excess volume data for the ternary mixture A,A-dimethylformamide-methanol-water and binary mixtures A, A-dimethylformamide-methanol and methanol-water were taken from [33], and the excess volume data for the binary mixture A,A-dimethylformamide-water from [34]. There are no isothermal compressibility data for the ternary mixture, but the contribution of compressibility to the binary KBls is almost negligible far from the critical point [6]. For this reason, the compressibilities in binary and ternary mixtures were taken to be equal to the ideal compressibilities, and were calculated from the isothermal compressibilities of the pure components as follows ... 

Fig. 16. Isothermal compressibility data of DPPC-cholesterol mixtures as a function of cholesterol concentration and pressure at T= 50 °C [98].

In the present approach, we apply an accurate and numerically efficient equation of state for the exp-6 fluid based on Zerah and Hansen s hypemetted-mean spherical approximation (HMSA) [111] equations and Monte Carlo calculations to detonation, shocks, and static compression. Thermal effects in the EOS are included through the dependence of the coefficient of thermal expansion on temperature, which can be directly compared to experiment. We find that we can replicate shock Hugoniot and isothermal compression data for a wide variety of solids with this simple form. 

We have also calibrated the CO model to static compression data. In Figure 8 we compare isotherms of CO to experimental data. Good agreement is found. 

The form of Equations 10b, 14, and 15a, b were obtained by fitting the isothermal compressibility data for the pure noble gases over all available ranges of reduced temperature and density to the relation... 

Then, the isothermal compressibility data on other pure substances, generally for V/Vc < Vi, were used to obtain their T and V values. Table II reports the results for the systems here. More extensive tables are reported elsewhere (5,6). In general, the fitting is most sensitive to the value of V, so when few data are available, T can be estimated as the critical temperature and a one-parameter fit is used. A consequence of this is that the binary parameter, K12, will compensate for any erroneous... 

Figures 2a and 2b are a selection of experimental isotherm residuals, Pv-RT)v x 10 . On the abscissa, critical density is about 15.6 x 10 g-mole/cm. Open circles are for parahydrogen and filled circles represent data for normal hydrogen from various sources. Compressed liquid isotherms are shown in Fig. 2b only near and below the critical temperature of about 32.98 K. While small differences between normal and parahydrogen can be seen, no research has been done on the reliability of such comparisons over the range of these figures. P-V-T data for normal hydrogen above 98°K were published by Michels et al. in 1959 p].

The KB inversion process involves the extraction of KBIs from the available experimental data. The experimental data required for this process—derivatives of the chemical potentials, partial molar volumes, and the isothermal compressibility—are all generally obtained as derivatives of various properties of the solution. Obtaining reliable derivatives can be challenging and will depend on the quality of the source data and the fitting function. Unfortunately, the experimental data often appear without a reliable statistical analysis of the errors involved, and hence the quality of the data is difficult to determine. Matteoli and Lepori have performed a fairly rigorous analysis of a series of binary mixtures and concluded that, for systems under ambient conditions, the quality of the resulting KBIs is primarily determined by the chemical potential data, followed by the partial molar volume data, whereas errors in the compressibility data have essentially no effect on the KBI values (Matteoli and Lepori 1984). Excess chemical potentials are typically obtained from partial pressure data, either isothermal or bubble point determinations, and from osmotic pressure or even electrochemical measurements. The particle number... 

Fig. 7. Density fluctuations, measured by saxs, for BPA-PC (o) and TMPC ( ) as a function oiT—Tg. The dotted line is the calculated density fluctuation ofTMPC from the isothermal compressibility data of this study obtained in a PVT experiment. The dashed line represents the calculated density fluctuation of BPA-PC from the literature compressibility data.

The Tait equation was first developed over 120 years ago [10] in order to fit compressibility data of freshwater and seawater. Modifications of the Tait equation are used for fitting liquid density data worldwide. The constant C in Equation (2.27) is taken as 0.0894 and is a universal constant. The zero pressure isotherms are given by... 

EOS theories for polymer liquids were grouped into three categories (1) lattice-fluid theory, (2) the hole model, and (3) the cell model. The Tait equation was first developed 121 years ago in order to fit compressibility data of freshwater and seawater. The Tait equation is a four-parameter representation of PVT behavior of polymers. Expressions for zero pressure isotherms and Tait parameters were provided, and values of Tait parameters for 16 commonly used polymers were tabulated. 

Figure 3.28 shows the calculated compressibilities and sonic velocity for a mixture of C1/C3 (30 mole% 70 mole% C3) at 130°F. In Fig. 3.28a, Cj and Cg are plotted us. pressure. This figure indicates that there is a discontinuity in both isothermal and isentropic compressibilities, when the phase boundaries are crossed. From a pressure of 1200 psia to a bubblepoint pressure of about 977 psia, there is a small increase in Cg of the undersaturated liquid the increase is, however, more noticeable. At. the bubblepoint, there is a sudden increase in both Cj and Cg. Similar behavior is also observed at the dewpoint of about 453 psia. It is interesting to note that the compressibilities in the two-phase region approaching the dewpoint are higher than the corresponding gas-phase compressibilities. Figure 3.28a also reveals that the variation of Cg in the two-phase region is less than the variation of C. This figure also provides the experimental isothermal compressibility data of Sage et al. (1933). The results in Fig. 3.28a are for a flat interface...

Equation of state data for cesium and rubidium are accurate enough to yield the behavior of the isothermal compressibility xt the iso-baric thermal expansivity up near the critical point. The compressibility data for cesium are displayed in Fig. 3.19 as a function of density. This quantity increases rapidly in the density range between about 0.22 g cm (p/Pc 0-6) and 0.48 g cm (p/Pc 1-30) for temperatures not far from T. Steep increases in xt nd are typical for fluids in the critical region. Their appearance in the data for cesium clearly demonstrates the importance of critical density fluctuations in the range 0.22 < p < 0.48 g cm . ... 

FIGURE 7.3 The hydration numbers h(c) of aqueous salts at 25°C obtained from isothermal compressibility data for LiCl (-0-) and NaOH (- -). At the right-hand side are shown the BET parameters r (number of water-binding sites per formula unit of the salt) of these salts (large filled symbols) (From Ref. 21 with permission from the pubhsher, ACS). 

Isothermal bulk moduli, (GPa), and pressure derivatives Kg determined from high-pressure X-ray diffraction, for the lanthanide (Ln) pnictides and chalcogenides of the B1 structure type. For each compound the table lists, from top to bottom the bulk modulus, in GPa, the pressure derivative, and references. Errors as indicated in the original publications. (x means compound unknown, - studied under pressure, but no compressibility data known, -l- compound exists, but no HPXRD study known and constr. ... 

Fig. 12.13 Pressure-volume equation of state for amorphous GeSe2. Experimental compression in a hydrostatic medium, a 4 1 methanoEethanol mixture [79] solid black circles)-, simulation data from [83] red open diamond)-, third-order isothermal Birch-Murnaghan equation of state fit to the experimental compression data solid black line)...

Hardly any compressibility data for hydrated molten salts have been found. StUl, for calcium nitrate hexahydrate (more precisely, Ca(N03)2 5.98H20) the isothermal compressibility Kt at 0.1 MPa is 0.195 0.016 GPa at 298 K diminishing to 0.188 0.015 GPa at 423 K spanning the melting point, 315 K and the corresponding temperature, l.lTm [78] (but note the wrong tmits in the publication). 

Sanchez et al. re-examined polymer bulk data in a rigorous classical thermodynamic analysis (7,8). A new principle of temperature-pressure (T-P) superposition of compression response was foimd. Stated briefly, a dimensionless pressure variable is used to superpose compression data as a function of temperature into a universal curve. The governing parameter of compression is the first pressure coefficient B (= CiB/QP)p, of the bulk modulus B. It is related to the asymmetry of the free energy aroimd its minimum, between dilation and compression. For polymers, Bi is aroimd 10, and universal. A new isothermal equation of state was formulated through a Fade analysis of the pressure dependence of the bulk modulus (8). Both the Tait and the Fade equations describe almost perfectly the isothermal pressure dependence of volume. Both can be used to smoothen experimental PVT data. 

The monolayer resulting when amphiphilic molecules are introduced to the water—air interface was traditionally called a two-dimensional gas owing to what were the expected large distances between the molecules. However, it has become quite clear that amphiphiles self-organize at the air—water interface even at relatively low surface pressures (7—10). For example, x-ray diffraction data from a monolayer of heneicosanoic acid spread on a 0.5-mM CaCl2 solution at zero pressure (11) showed that once the barrier starts moving and compresses the molecules, the surface pressure, 7T, increases and the area per molecule, M, decreases. The surface pressure, ie, the force per unit length of the barrier (in N/m) is the difference between CJq, the surface tension of pure water, and O, that of the water covered with a monolayer. Where the total number of molecules and the total area that the monolayer occupies is known, the area per molecules can be calculated and a 7T-M isotherm constmcted. This isotherm (Fig. 2), which describes surface pressure as a function of the area per molecule (3,4), is rich in information on stabiUty of the monolayer at the water—air interface, the reorientation of molecules in the two-dimensional system, phase transitions, and conformational transformations. 

A wide variety of physical properties are important in the evaluation of ionic liquids (ILs) for potential use in industrial processes. These include pure component properties such as density, isothermal compressibility, volume expansivity, viscosity, heat capacity, and thermal conductivity. However, a wide variety of mixture properties are also important, the most vital of these being the phase behavior of ionic liquids with other compounds. Knowledge of the phase behavior of ionic liquids with gases, liquids, and solids is necessary to assess the feasibility of their use for reactions, separations, and materials processing. Even from the limited data currently available, it is clear that the cation, the substituents on the cation, and the anion can be chosen to enhance or suppress the solubility of ionic liquids in other compounds and the solubility of other compounds in the ionic liquids. For instance, an increase in allcyl chain length decreases the mutual solubility with water, but some anions ([BFJ , for example) can increase mutual solubility with water (compared to [PFg] , for instance) [1-3]. While many mixture properties and many types of phase behavior are important, we focus here on the solubility of gases in room temperature IFs. 

Wunderlich30 and Zubov33 suppose that ECC under high pressures occur as a result of an isothermal thickening of folded-chain lamellae. However, this contradicts the later data of Wunderlich and of Japanese authors31 who have shown that folded-chain crystals (FCC) are formed after ECC, when the melt is cooled. According to Kawai22, crystallization under hydrostatic compression can he considered as a variant of the bicomponent crystallization. 

Berthelot showed that the mean compressibility between 1 and 2 atm. does not differ appreciably from that between 0 and 1 atm. in the case of permanent gases, and either may be used within the limits of experimental error. But in the case of easily liquefiable gases the two coefficients are different. According to Berthelot and Guye the value of aJ can be determined from that of aj by means of a small additive correction derived from the critical data, and the linear extrapolation then applied Gray and Burt consider, however, that this method may lead to inaccuracies, and consider that the true form of the isothermal can only be satisfactorily ascertained by the experimental determination of a large number of points, followed by graphical extrapolation. 

Traditional amphiphiles contain a hydrophilic head group and the hydrophobic hydrocarbon chain(s). The molecules are spread at molecular areas greater (-2-10 times) than that to which they will be compressed. The record of surface pressure (II) versus molecular area (A) at constant temperature as the barrier is moved forward to compress the monolayer is known as an isotherm, which is analogous to P-V isotherms for bulk substances. H-A isotherm data provide information on the molecular packing, the monolayer stability as de-... 

Typical values of the isobaric expansivity and the isothermal compressibility are given in Table 1.2. The difference between the heat capacities at constant volume and constant pressure is generally negligible for solids at low temperatures where the thermal expansivity becomes very small, but the difference increases with temperature see for example the data for AI2O3 in Figure 1.2. 

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