Isothermal compressibility correlations

R. ZwanzigandR. D. Mountain, J. Chem. Phys. 43,4464 (1965) show that the modulus Goo and the isothermal compressibility are determined by similar integrals containing the pair correlation function and the interparticle potential for simple Lennard-Jones fluids. The adiabatic (zero frequency) bulk modulus Ko equals —y(0P/0P) j, which clearly is a kind... 

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). 

You need an estimate of the coefficient of isothermal compressibility of a gas at 1100 psia and 150°F. Unfortunately, you don t have the correlations in this book with you. What value do you use ... 

Estimation of Formation Volume Factor of Oil at Saturation Pressure Using Ideal-Solution Principles—Estimation of Formation Volume Factor of Oil at Saturation Pressure by Correlation—Estimation of Formation Volume Factor of Oil at Pressures Above the Bubble-Point Pressure Adjustment of Formation Volume Factor of Oil and Solution Gas- Oil Ratio for Field Derived Bubble-Point Pressure Total Formation Volume Factor The Coefficient of Isothermal Compressibility of Oil... 

Additional values of the isothermal compressibility can be estimated for the many solvents for which no values of kT or ks have been determined experimentally from a correlation with other solvent properties (Marcus and Hefter 1997) ... 

SCCF spin-correlated crystal field K isothermal compressibility... 

So, c(q = 0) is a finite quantity even when the isothermal compressibility %T tends to diverge in the critical region of any fluid. As will be seen, the former quantity is useful, because of being accessible from direct experiment [12] thanks to recent SANS measurements (see Section V). At variance from g(r), the direct correlation function c(r) in not zero inside the core region (r < a) and its knowledge is crucial in this range of distances, since it is directly related to y T so that... 

Thermodynamic properties, such as the excess energy [Eq. (4)], the pressure [Eq. (5)], and the isothermal compressibility [Eq. (7)] are calculated in a consistent manner and expressed in terms of correlation functions [g(r), or c(r)], that are themselves determined so that Eq. (17) is satisfied within 1%. It is usually believed that for the thermodynamic quantities, the values of the correlation functions B(r) and c(r), e.g.] do not matter as much inside the core. This may be true for quantities dependent on g(r), which is zero inside the core. But this is no longer true for at least one case the isothermal compressibility that depends critically on the values of c(r) inside the core, where major contribution to its value is derived. In addition, it should be stressed that the final g(r) is slightly sensitive to the consistent isothermal compressibility. 

In this framework, we present the repercussions on the physical properties of a renormalized indirect correlation function y (r) conjugated with an optimized division scheme. All the units are expressed in terms of the LJ parameters, that is, reduced temperature T = kBT/e and reduced density p = pa3. In order to examine the consequences of a renormalization scheme, the direct correlation function c(r) calculated from ZSEP conjugated with DHH splitting is compared in Fig. 7 to those obtained with the WCA separation. For high densities, the differences arise mainly in the core region for y(r) and c(r) [77]. These calculated quantities are in excellent agreement with simulation data. The reader has to note that similar results have been obtained with the ODS scheme (see Ref [80]). Since the acuracy of c(r) can be affected by the choice of a division scheme, the isothermal compressibility is affected too, as can be seen in Table III for the pkBTxT quantity. As compared to the values obtained with... 

Further attempts have been made this last decade to obtain competitive results for ppex as compared to simulation data. Recently, Bomont proposed the approximation B X)(r) = a(T, p)B(r) [98], Once the correlation functions, the excess internal energy, the pressure, and the isothermal compressibility are calculated with respect to the first thermodynamic consistency condition, the parameter cl(T, p) is iterated until p0pex/0p satisfies the second thermodynamic consistency condition within 1% [Eq. (87)]. At the end of the iteration cycle... 

These expressions are formally exact and the first equality in Eq. (123) comes from Euler s theorem stating that the AT potential u3(rn, r23) is a homogeneous function of order -9 of the variables r12, r13, and r23. Note that Eq. (123) is very convenient to realize the thermodynamic consistency of the integral equation, which is based on the equality between both expressions of the isothermal compressibility stemmed, respectively, from the virial pressure, It = 2 (dp/dE).,., and from the long-wavelength limit S 0) of the structure factor, %T = p[.S (0)/p]. The integral in Eq. (123) explicitly contains the tripledipole interaction and the triplet correlation function g (r12, r13, r23) that is unknown and, according to Kirkwood [86], has to be approximated by the superposition approximation, with the result... 

A wide variety of density- and temperature-dependent input parameters are required. These include p, the number density of the solvent, kj, the isothermal compressibility, f, the correlation length of density fluctuations, y = Cp/Cv, the ratio of specific heats, and 37, the viscosity. Very accurate equations of state for ethane (74,75,99) and CO2 (74) are available that provide the necessary input information. The necessary input parameters for fluoroform were obtained by combining information from a variety of sources (76,100,101). There is somewhat greater uncertainty in the fluoroform parameters. 

Figure 8.14 The product of the tension of the liquid-vapor surface, cr, and the isothermal compressibility, Kj-, identified by Egelstaff and Widom (1970) as proportional to the spatial correlation length. This combination was suggested as appropriate for the low density of the coexisting vapor phase.

Isothermal compressibility. As seen in [18], this quantity is a response function, and consequently it will solely depend on the connected part of the fluid-fluid correlation function. Here, as in [18] one gets,... 

In addition to the above expressions it is also possible to obtain expressions for the isothermal compressibility and other thermodynamic properties from the site-site direction correlation functions . Such expressions require assumptions about the range of the site-site direct correlation functions and also, in some cases, are restricted to certain closures of the SSOZ equation. We will consider such routes to the thermodynamic properties in Section V.C. 

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. 

The three interrelated phenomena that are observed near the critical point are (a) increase in the density fluctuation, (b) increase in the isothermal compressibility, and (c) increase in the correlation length of G(R). 

In this work component critical volumes were taken from Reid et al. (23). They are given in Table VI, along with root mean square deviations between the correlation predictions and the isothermal compressibilities tabulated in Appendix A of Reference 11 (pressures up to 50 MPa). For Ar, CH4, and C2HG the correlation is quite accurate, giving isothermal compressibilities with average deviations within 5%. Deviations for N2 are on this order for 91 and 100 K, but they become... 

Table VI. Component Critical Volumes (Vc) from Reference 23 and Root Mean Square Deviations (s) between the Brelvi—O Connell Correlation (12) and Component Isothermal Compressibilities from Reference 11, pp. 124—128...

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. 

The solubilities of small molecules in fluorous solvents are determined to a large extent by two parameters solute polarity and size. The first is an extension of the like dissolves like paradigm. The second is uniquely important to perflu-orinated solvents because of low intermolecular forces they have large cavities (free volumes) that can accommodate small molecules. The solubilities of gases in fluorocarbons are also well established and show a correlation with the isothermal compressibility of the solvent, supporting the cavity-based solubility model. 

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