Mixture isothermal compressibilities

In this equation c is the mixture isothermal compressibility, and the and Ki are the component mole fractions and isothermal compressibilities at the same temperature and pressure as the mixture. Values of A were derived from VE(P, T, x) in Reference 11. This quantity was quite large for all the systems studied (10,11). Whereas the maximum VE observed was on the order of 10% of the mixture V, the maximum A M... 

Three methods were tested for calculating mixture isothermal compressibilities. The first used a mole fraction average of component values ... 

In Table VII a comparison is made between root mean square deviations from experimental binary data for the three methods. Experimental mixture isothermal compressibilities were taken from Appendix B of Reference 11. Component values for use in Equation 24 came from Appendix A of the same reference. Use of values from the correlation would have yielded slightly higher deviations for the systems containing N2. 

It is generally much better to use the Equation 25 approach than the mole-fraction average of the component values to obtain mixture isothermal compressibilities. Very little is gained by trying to optimize the deviation parameters to binary data. It would appear that = 0 is a satisfactory approximation in this method. 

Table VII. Root Mean Square Deviations (GPa-1) between Calculated Mixture Isothermal Compressibilities and Experimental Values from Reference 11, pp. 166—184, and Optimum Deviation Parameters (j) for Equation 26...

The energy required to reversibly separate gas mixtures is the same as that necessary to isothermally compress each component in the mixture from the partial pressure of the gas in the mixture to the final pressure of the mixture. This reversible isothermal work is given by the familiar relation... 

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. 

Evaluations of Rd and Y necessitate a knowledge of certain physical properties of the two liquids and the mixtures. The variation of refractive index with concentration is measured readily by refractometry, if I nT, — n21 is large. The coefficient of isothermal compressibility of a mixture t2 requires specialised equipment. Alternatively, it can be determined from the heat capacity and the coefficient of isentropic compressibility87, 88, the latter being yielded from velocity of sound data88. However, provided and 02 for the pure compounds are known, j312 is evaluated most conveniently on the basis of additivity, thus ... 

The excess molar volumes of 10-40 mol % methanol/C02 mixtures at 26°C as a function of pressure has been determined. The excess molar volumes varied with composition and pressure significant interaction between CO2 and methanol was noted from the observed excess molar volumes. To better characterize the interaction and its effect on analyte solubility, the partial molar volume of naphthalene at infinite dilution in liquid 10 and 40 mol % methanol/C02 mixtures was determined. The variation of the partial molar volume at infinite dilution with pressure correlated well with isothermal compressibility of the methanol/C02 mixtures (Souvignet and Olesik, 1995). 

Recovery of solvent by isothermal compression. This method was proposed by Claude [14]. It was applied to the recovery of alcohol containing camphor which escapes during the manufacture of celluloid. With alcohol and ether this process entails compressing the vapours to 7 atm, thus causing the condensation of the alcohol and after that rapidly expanding them. Ether is condensed by intensive cooling. The necessary plant was very expensive and there was risk of explosion when the mixture of the air with alcohol and ether was compressed too rapidly. It never attained wide application. 

The mixture dimethyl sulphoxide + water has attracted a great deal of interest. The excess function HE is negative for this mixture at 298 K (Clever and Piggott, 1971 Fox and Whittingham, 1975), as also are GE (Lam and Benoit, 1974 Philippe and Jambon, 1974) and FE-quantities (Lau et al., 1970). A set of smoothed thermodynamic excess functions is shown in Fig. 54 (Kenttamaa and Lindberg, 1960). The dependence on x2 of the isothermal compressibilities of DMSO + water mixtures is quite different from that for the TA monohydric alcohols + water mixtures. The curves for the latter systems show... 

Isothermal compressibility for an ideal gas mixture k is given by HP, whereas for liquids the compressibility is negligible. The activities can be introduced to describe the deviations from the ideal behavior of solutions the activities are expressed in terms of the activity coefficients. 

FIGURE 5.3. Isothermal compressibility of TBA-water mixture as a function of TBA molar fraction at four temperatures T = 283 293 303 313 K, (a) in the whole concentration range, and (b) within the water-rich region with the anomalous behaviour of xt shown in more detail. Predictions of the RISM-KH theory in comparison with experimental data. 

Interaction parameters can also be calculated from values of the e qiansion coefficients of polymer blends using Equation-of-state theories, or from values of the isothermal compressibility of the mixture They can also be obtained from measures... 

In the above equations, the Gy are expressed in volume per mole, kr is the isothermal compressibility, Vi is the partial molar volume of component i, Xi is the mole fraction of component i, V is the molar volume of the mixture, T is the absolute temperature, R is the universal gas constant, and Yi is the activity coefficient of component i. 

The new reference state reduces to GV in the limiting case of an ideal mixture, but also satisfies the volume conservation condition. The following differences exist between the ML and SR excesses the ML excesses have non-zero values if either the partial molar volumes differ from the ideal ones or D = + Xi d a.yi/dxi)pj 7 1, where P represents the pressure and y- is the activity coefficient of component v, the SR excesses have non-zero values only if D 1. The present reference state is a hypothetical one similar to the ideal state, in which the molar volume, the partial molar volumes and the isothermal compressibility are the real ones. 

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 ... 

The present paper is devoted to the local composition of liquid mixtures calculated in the framework of the Kirkwood—Buff theory of solutions. A new method is suggested to calculate the excess (or deficit) number of various molecules around a selected (central) molecule in binary and multicomponent liquid mixtures in terms of measurable macroscopic thermodynamic quantities, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volumes. This method accounts for an inaccessible volume due to the presence of a central molecule and is applied to binary and ternary mixtures. For the ideal binary mixture it is shown that because of the difference in the volumes of the pure components there is an excess (or deficit) number of different molecules around a central molecule. The excess (or deficit) becomes zero when the components of the ideal binary mixture have the same volume. The new method is also applied to methanol + water and 2-propanol -I- water mixtures. In the case of the 2-propanol + water mixture, the new method, in contrast to the other ones, indicates that clusters dominated by 2-propanol disappear at high alcohol mole fractions, in agreement with experimental observations. Finally, it is shown that the application of the new procedure to the ternary mixture water/protein/cosolvent at infinite dilution of the protein led to almost the same results as the methods involving a reference state. 

The Kirkwood—Buff (KB) theory of solution (often called fluctuation theory) employs the grand canonical ensemble to relate macroscopic properties, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volnmes, to microscopic properties in the form of spatial integrals involving the radial distribution function. This theory allows one to obtain information regarding some microscopic characteristics of mnlti-component mixtures from measurable macroscopic thermodynamic quantities. However, despite its attractiveness, the KB theory was rarely used in the first three decades after its publication for two main reasons (1) the lack of precise data (in particular regarding the composition dependence of the chemical potentials) and (2) the difficulty to interpret the results obtained. Only after Ben-Naim indicated how to calculate numerically the Kirkwood—Buff integrals (KBIs) for binary systems was this theory used more frequently. 

The purpose of this Appendix is to provide expressions for calculating the KBIs for binary mixtures from measurable thermodynamic quantities such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volumes. 

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. 

The KBIs for the 1-propanol (l)/water (2) mixture are available in the literature,and there is agreement between the KBIs obtained in various calculations.In the present paper, the excesses (deficits) for the 1-propanol (l)/water (2) mixture have been calculated with eqn (13) by using the KBIs already calculated by us. The partial molar volumes for the 1-propanol (l)/water (2) mixture have been calculated from density data" and the isothermal compressibilities have been evaluated using the expression ... 

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