Isothermal compressibility method

This second step is accomplished using the coefficient of isothermal compressibility. Methods of estimating values of this compressibility coefficient will be discussed later in this chapter. However, we now will show the use of compressibility in computing density changes corresponding to pressure changes. 

Here, the concept of particle scaling introduced in SPT is combined with equation (30) to evaluate Gc. This kind of relationship has been validated by a molecular dynamic study made by Postma et al. who performed a computer simulation to create five cavities of different sizes in water using simple point charge water molecules (see Free Energy Changes in Solution and Monte Carlo Simulations for Liquids for a review of computer simulation techniques). The numerical results from molecular dynamics simulation correlate very well with the cavity volume (computed from cavity thermal radius. Figure 3). A comparison between the isothermal compressibility method and SPT is given in Section 3. 

Certainly, the isothermal compressibility method is limited by the availability of molecular dynamics simulation data for other solvents. Postma et al. showed that the simulation data are well reproduced by SPT. Hence, an alternative is to use SPT for hard spheres to obtain parameters for different solvents ... 

First, the performance of the isothermal compressibility method (Section 2.2.3) is investigated by comparing TCF for... 

Table 3 Comparison Between TCP Calculated with SPT and the Isothermal Compressibility Method (ITC), and Between Experimental (E) and Calculated Solvation Thermodynamics Water (hrst row), Benzene (second row), Carbon Tetrachloride (third row), and Cyclohexane (fourth row)... 

Other Refrigeration Methods. Cryocoolers provide low temperature refrigeration on a smaller scale by a variety of thermodynamic cycles. The Stirling cycle foUows a path of isothermal compression, heat transfer to a regenerator matrix at constant volume, isothermal expansion with heat transfer from the external load at the refrigerator temperature, and finally heat transfer to the fluid from the regenerator at constant volume. 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

Several methods are also available for determination of the isothermal compressibility of materials. High pressures and temperatures can for example be obtained through the use of diamond anvil cells in combination with X-ray diffraction techniques [10]. kt is obtained by fitting the unit cell volumes measured as a function of pressure to an equation of state. Very high pressures in excess of 100 GPa can be obtained, but the disadvantage is that the compressed sample volume is small and that both temperature and pressure gradients may be present across the sample. 

To liquefy hydrogen, it has to be cooled down to about -253 °C. The thermodynamically ideal liquefaction method consists of isothermal compression and subsequent... 

To generate isothermal compressibilities from sound speeds, it is necessary to have reliable expansibility and heat capacity data (equation 18). We have developed an iterative method to convert high pressure sound speed to isothermal compressibilities (84). The effect of pressure on the volume of a solution (3V/3P)T at a constant pressure is given by... 

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. 

Next, procedures for estimating values of the coefficient of isothermal compressibility and oil viscosity are discussed. The chapter ends with methods of estimating hydrocarbon liquid-gas interfacial tension. 

The normal procedure for estimating formation volume factor at pressures above the bubble point is first to estimate the factor at bubble-point pressure and reservoir temperature using one of the methods just described. Then, adjust the factor to higher pressure through the use of the coefficient of isothermal compressibility. The equation used for this adjustment follows directly from the definition of the compressibility coefficient at pressures above the bubble point. 

Partial molar volumes and the isothermal compressibility can be calculated from an equation of state. Unfortunately, these equations require properties of the components, such as critical temperature, critical pressure and the acentric factor. These properties are not known for the benzophenone triplet and the transition state. However, they can be estimated very roughly using standard techniques such as Joback s modification of Lyderson s method for Tc and Pc and the standard method for the acentric factor (Reid et al., 1987). We calculated the values for the benzophenone triplet assuming a structure similar to ground state benzophenone. The transition state was considered to be a benzophenone/isopropanol complex. The values used are shown in Table 1. 

The study of glass transition is an important subject in current research, and simulations may well be suited to help our understanding of the phenomenon. An example is the application of Monte Carlo techniques by Wittman, Kremer, and Binder.The authors employed a lattice method in two dimensions to model the system. The glass transition was determined by monitoring the free volume changes as well as isothermal compressibility. The glasslike behavior was determined by evaluating the bond autocorrelation function. The authors found that both the dynamic polymer structure factor and the orienta-... 

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

Compression may take place at a filling station, receiving hydrogen from a pipeline. The energy requirement depends on the compression method. The work required for isothermal compression at temperature T from pressure P, to Pj is of the form... 

The isothermal compressibilities have been calculated with eq 5, using for the isothermal compressibilities of the pure substances the data from refs 53—56 (only the value for 2-butanol was taken as that for isobutanol). The data have been fitted using the Redlich—Kister equation.The values of D have been obtained from the activity coefficients or total pressure data by the sliding polynomials method. To check the accuracy of our calculations, the D values have been... 

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. 

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. 

Another method suggested by the authors for predicting the solubility of gases and large molecules such as the proteins, drugs and other biomolecules in a mixed solvent is based on the Kirkwood-Buff theory of solutions [18]. This theory connects the macroscopic properties of solutions, such as the isothermal compressibility, the derivatives of the chemical potentials with respect to the concentration and the partial molar volumes to their microscopic characteristics in the form of spatial integrals involving the radial distribution function. This theory allowed one to extract some microscopic characteristics of mixtures from measurable thermodynamic quantities. The present authors employed the Kirkwood-Buff theory of solution to obtain expressions for the derivatives of the activity coefficients in ternary [19] and multicomponent [20] mixtures with respect to the mole fractions. These expressions for the derivatives of the activity coefficients were used to predict the solubilities of various solutes in aqueous mixed solvents, namely ... 

The thermal pressure coefficients of the pure polymers can be estimated from the ratio of thermal expansion coefficient, a, to isothermal compressibility, Px, as y = a/px. In the case where y is not available from the literature it can be calculated from solubility parameters which themselves are related to the cohesive energy density (C.E.D.) and hence to the strength of the internal pressure of the structural molecules u5). The binary parameter, S2/S, is obtainable from the method of group contribution given by Bondi116. It can alternatively be calculated by casting shadows of models of the molecules for various orientations, where the area for the monomer unit is estimated from the area of the projections. 

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

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