Compressibility determination, isothermal

Self-Test 7.16B Determine AS, ASsllrr, and AStot for the reversible, isothermal compression of 2.00 mol of ideal gas molecules from 1.00 atm and 4.00 L to... 

The monolayer stability limit is defined as the maximum pressure attainable in a film spread from solution before the monolayer collapses (Gaines, 1966). This limit may in some cases correspond directly to the ESP, suggesting that the mechanism of film collapse is a return to the bulk crystalline state, or may be at surface pressures higher than the ESP if the film is metastable with respect to the bulk phase. In either case, the monolayer stability limit must be known before such properties as work of compression, isothermal compressibility, or monolayer viscosity can be determined. 

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. 

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 ... 

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). 

EXAMPLE 6-4 The following table gives volumetric data at 150°F for a natural gas. Determine the coefficient of isothermal compressibility for this gas at 150°F and 1000 psia. 

We recognize that the equation of state for an ideal gas does not describe adequately tire behavior of gases at temperatures and pressures normally encountered in petroleum reservoirs. However, Equation 6-6 does illustrate that we can expect the coefficient of isothermal compressibility of a gas to be inversely proportional to pressure. Equation 6-6 can be used to determine the expected order of magnitude of gas compressibility. 

Determine the value of the coefficient of isothermal compressibility of the black oil of Exercise 8-10 for use between 3500 and 4000 psig. 

The following physical properties can be determined from the results of a black oil reservoir fluid study bubble-point pressure, formation volume factor of oil, solution gas-oil ratio, total formation volume factor, coefficient of isothermal compressibility of oil, and oil viscosity,... 

Since the compressibility is proportional to the pressure derivative of the volume, any experiment that establishes the P-V-T relation of a gas with sufficient accuracy also yields data for the isothermal compressibility. For obtaining the adiabatic compressibility from the P-V-T relation, some additional information is necessary see section (c). Tor instance specific heat data in the perfect gas slate of the substance considered. A more direct way of determining ihe adiabatic compressibility is by measuring the speed of sound i1. the two quantities being related by... 

A gaseous pure component can be defined as supercritical when its state is determined by values of temperature T and pressure P that are above its critical parameters (Tc and Pc). In the proximity of its critical point, a pure supercritical fluid (or a dense gas as it is alternatively known) has a very high isothermal compressibility, and this makes possible to change significantly the density of the fluid with relatively limited modifications of T and P. On the other hand, it has been shown that the thermodynamic and transport properties of supercritical fluids can be tuned simply by changing the density of the medium. This is particularly interesting for... 

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) ... 

The last term, the fluid s isothermal compressibility, can be reasonably predicted from a two-parameter, cubic equation of state (EOS) such as the Redlich-Kwong EOS or the Peng-Robinson EOS.(14,15) The fluid s isothermal compressibility was determined using the Redlich-Kwong EOS to evaluate the derivative ( Vmmob/3P) in eq. 12 due to the ease of finding an analytical solution,... 

The data on the volume properties of PMS liquids (i.e. coefficient of volumetric expansion, relative volume variation, coefficient of isothermal compressibility) are essential for the performance characteristics of oli-godimethylsiloxanes in hydraulic systems, hydraulic shocks and dampers they allow one to determine the working characteristics of these systems with some brands of PMS liquids at different temperatures and pressures. 

There are several routes to the determination of (3P/ p from the structure. One is given by Eq. (5). A second possibility is offered by the isothermal compressibility... 

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. 

The condition of phase stability for such a system is closely related to the behavior of the Helmholtz free energy, by stating that the isothermal compressibility yT > 0. The positiveness of yT expresses the condition of the mechanical stability of the system. The binodal line at each temperature and densities of coexisting liquid and gas determined by equating the chemical potential of the two phases. The conditions expressed by Eq. (115) simply say that the gas-liquid phase transition occurs when the P — pex surface from the gas... 

Here, b is a constant and is a function of T only. For this gas, determine expressions for the isothermal compressibility k and the thermal pressure coefficient (dP/dT)v. These expressions should contain only T, P, 0, d9/dT, and constants. 

Five kilograms of steam in a piston/cylinder device at 150 kPa and 150°C undergoes a mec cally reversible, isothermal compression to a final pressure such that the steam is just saturate Determine Q and W for the process. 

An ideal gas at 50°C and 1 atm is heated to 500°C at constant pressure, and then isothermally compressed to 10 atm. It is then isobarically cooled to 50°C, and finally is isothermally expanded to back to its initial state. For the overall process, determine AH and AU. 

Fig. 9.4. Specific volume of Snase as a function of pressure at 40°C [15], The protein is folded up to 50 MPa, and the slope up to that pressure is indicative of the isothermal compressibility of the folded state. The arrow at 100 MPa indicates the volume change of unfolding assuming constant compressibility of the folded state and nearly complete unfolding by 100 MPa. Unfortunately, the high-pressure densitometer was limited to 100 MPa, so the compressibility of the unfolded state could not be determined...

We can determine some of the partial derivatives above by using the following Maxwell relations, thermal expansion [3, and isothermal compressibility k ... 

Two characteristics of liquid water that are relevant to the present answer are the equation of state characteristics shown in Figs. 8.11 and 8.12. The isothermal compressibilities shown in Fig. 8.11 indicate that water is stiffer than organic solvents, and that the stiffness is only weakly temperature-dependent. We don t propose here a detailed explanation of that stiffness - it is due to intermolecular interactions among solvent molecules, hydrogen bonding in the case of liquid water (Debenedetti, 2003) - but the present empirical theory of hydrophobicity merely exploits those results. This stiffness is the principal determinant of the low solubility of inert gases in liquid water. In the simplest information models this stiffness, and its temperature dependence, is expressed by the experimental n (n — l))o, which is distinctive of liquid water. 

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). 

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