Volume and isothermal compressibility

Energies, Excess Enthalpies, Excess Volumes, and Isothermal Compressibilities of Cyclohexane + 2,3-Dimethylbutane , J. Client. Thermodyn., 6, 35-41 (1974). J. B. Ott, K. N. Marsh, and R. H. Stokes, Excess Enthalpies, Excess Gibbs Free Energies, and Excess Volumes for (Cyclohexane + n-Hexane), and Excess Gibbs Free Energies and Excess Volumes for (Cyclohexane + Methylcyclohexane) at 298.15 and 308.15 K , J. Chem. Thermodyn., 12, 1139-1148 (1980). 

Therefore, it is important to have a theoretical tool which allows one to examine (or even predict) the thickness of the LC region and the value of the LC on the basis of more easily available experimental information regarding liquid mixtures. A powerful and most promising method for this purpose is the fluctuation theory of Kirkwood and Buff (KB). " The KB theory of solutions allows one to extract information about the excess (or deficit) number of molecules, of the same or different kind, around a given molecule, from macroscopic thermodynamic properties, such as the composition dependence of the activity coefficients, molar volume, partial molar volumes and isothermal compressibilities. This theory was developed for both binary and multicomponent solutions and is applicable to any conditions including the critical and supercritical mixtures. 

Kirkwood and Buff [15] obtained expressions for those quantities in compact matrix forms. For binary mixtures, Kirkwood and Buff provided the results listed in Appendix A. Starting from the matrix form and employing the algebraic software Mathematica [16], analytical expressions for the partial molar volumes, the isothermal compressibility and the derivatives of the chemical potentials for ternary mixtures were obtained by us. They are listed in Appendix B together with the expressions at infinite dilution for the partial molar volumes and isothermal compressibility. 

Calculation of Compressed Liquid Excess Volumes and Isothermal Compressibilities for Mixtures of Simple Species... 

After expressions for the chemical potential derivatives have been obtained, one can use them to determine corresponding expressions for the partial molar volumes and isothermal compressibility. Using Equation 1.50 in Equation 1.47 with i = k = 2 followed by some rearrangement using the GD expression provides... 

The explicit expressions for the chemical potential derivatives, partial molar volumes, and isothermal compressibility become rather cumbersome for ternary systems. Experimental data are also much less common. However, there are many interesting effects that involve ternary systems (see Chapter 4). Also, we shall see that considerable simplification is obtained when one of the components is at infinite dilution (see Chapters 10 and 11). If one requires specific expressions for the various properties, it will prove convenient to define the following set of variables (Smith 2006a),... 

Ewing, M. B. Marsh, K. N. Excess Gibbs free energies, excess enthalpies, excess volumes, and isothermal compressibilities of cyclohexane + 2,3-dimethylbutane J. Chem. Thermodyn. 1974, 6, 35-41... 

The pressure dependence of the reaction rate constant is given by the following relation in terms of partial molar volumes and isothermal compressibility (90) ... 

Figure 4.14 Behavior of thermodynamic variables at Tg for a second-order phase transition (a) volume and fb) coefficient of thermal expansion a and isothermal compressibility p.

The form of equations (8.11) and (8.12) turns out to be general for properties near a critical point. In the vicinity of this point, the value of many thermodynamic properties at T becomes proportional to some power of (Tc - T). The exponents which appear in equations such as (8.11) and (8.12) are referred to as critical exponents. The exponent 6 = 0.32 0.01 describes the temperature behavior of molar volume and density as well as other properties, while other properties such as heat capacity and isothermal compressibility are described by other critical exponents. A significant scientific achievement of the 20th century was the observation of the nonanalytic behavior of thermodynamic properties near the critical point and the recognition that the various critical exponents are related to one another ... 

Figure 1.2 Molar heat capacity at constant pressure and at constant volume, isobaric expansivity and isothermal compressibility of AI2O3 as a function of temperature.

The effect of a change in volume on the entropy is given by the ratio of the iso-baric expansivity and isothermal compressibility of a compound ... 

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... 

Figure 17. Specific volume Vt and isothermal compressibility (at the glass transition temperature Tg) calculated from the LCT as a function of the inverse number l/M of united atom groups in single chains for constant pressure (P = I atm 0.101325 MPa) F-F and F-S polymer fluids. Both quantities are normahzed by the corresponding high molar mass limits (i.e., by...

Gu, Z. and Brennecke, J.F., Volume expansivities and isothermal compressibilities of imidazolium and pyridynium based ionic liquids, ]. Chem. Eng. Data, 47, 339, 2002. 

Hofman, T. et al.. Densities, excess volumes, isobaric expansivity, and isothermal compressibility of the (l-ethyl-3-methylimidazolium ethylsulfate + methanol) system at temperatures (298.15 to 333.15) K and pressures from (0.1 to 35) MPa, /. Chem. Thermodyn., 40, 580, 2008. 

Some properties are directly connected with mass and packing density (or its reciprocal specific volume), thermal expansibility and isothermal compressibility. Especially the mechanical properties, such as moduli, Poisson ratio, etc., depend on mass and packing. In this chapter we shall discuss the densimetric and volumetric properties of polymers, especially density and its variations as a function of temperature and pressure. Density is defined as a ratio ... 

J,. Generally, volume expansivity and isothermal compressibility k depend on T and P. Prove that ... 

The energy Z7 of a body is the sum of two parts, viz. the heat energy and the so-called volume energy, the increment of which is the work done in an isothermal compression. As the volume and the compressibility are both finite at the absolute zero, while the heat energy approaches zero, it follows that the total energy U must be finite at the absolute zero, and hence that... 

For binary mixtures, Kirkwood and Buff [15] obtained the following expressions for the partial molar volumes, the isothermal compressibility and the derivatives of the chemical potentials with respect to concentrations. 

It was shown previously [5,14] that the KB theory of solution can be used to relate the thermodynamic properties of ternary mixtures, such as the partial molar volumes, the isothermal compressibility and the derivatives of the chemical potentials to the KB integrals. In particular for the derivatives of the activity coefficients one can write the following rigorous relations [5] ... 

In 2002, Morrow and Maginn presented an all-atom force field for [C4mim][PF6] using a combination of DFT calculations (B3LYP/6-311+G ) and CHARMM 22 parameter values [13]. MD simulations were carried out in the isothermal-isobaric ensemble at three different temperatures. The calculated properties contained infrared frequencies, molar volumes, volume expansivities, isothermal compressibilities, self-diffusivities, cation-anion exchange rates, rotational dynamics, and radial distribution functions. These thermodynamic properties were found to be in good agreement with available experimental values [13]. 

Selected mechanical properties of liquid COFj have been estimated from molecular data [1683]. The calculated values for the molar volume, thermal expansion coefficient (a), and isothermal compressibility (3), are recorded as a function of temperature in Table 13.16. 

Owing to the great forces set up when liquids are heated in closed vessels, the direct determination of the specific heat at constant volume, cv, is very difficult, and it has usually been determined indirectly. The ratio of specific heats cp/cv has been determined from the adiabatic and isothermal compressibilities, from the velocity of sound, U, and by Kundt s method ( 3.VIII D). 

Standard molal volumes of minerals together with the coefficients of isobaric expansion and isothermal compressibility are available in several compilations (8,9). Standard partial molal volumes of uncomplexed ions are available at 25°C and 1 atm. (1) however, data are sparse for the coefficients of isobaric expansion and isothermal compressibility. 

The molar volume change in ionization reactions at higher temperatures and pressures cannot be calculated for most of the aqueous complexes because of a lack of data on isobaric expansion and isothermal compressibility coefficients. Entropy and heat capacity correlations have recently been used to generate equation of state parameters for estimating molal volumes of aqueous complexes at elevated temperatures and pressures (Sverjensky, 12). These coefficients are available for aqueous complexes only of univalent anions and, therefore, the pressure dependence of ionization constants at elevated temperatures cannot be estimated using Equation 4. 

Table I. Component Liquid Molar Volumes (Vs) and Isothermal Compressibilities (ks) at 100 K and Vapor Pressure (Ps) Which Were Used along with Shape Factors (c or a) to Determine Equation-of-State Parameters a and b...

To examine ability of the relation to describe PVT dependencies, the author used experimental data of eight polymers (T-span = 30°C, P-span = 0-20 MPa). For comparison, FOV, S-L, S-S, and D-W relations were also used. The evaluation was performed computing errors in describing the volume, thermal expansivity and isothermal compressibility. As in the previous evaluations, S-S dependence performed the best. For the... 

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