Isothermal compressibility pressure dependence

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. 

An important factor for the well-to-wheel energy balance is the energy required to compress hydrogen to 700 bar, which is often considered to be an impediment for this pressure level. This assumption neglects that, that (for an ideal gas) the energy needed for (adiabatic or isothermal) compression solely depends on the relation of the starting and the final pressure. The compression from 10 to 20 bar for example requires the same energy as the compression from 350 to 700 bar. 

The dynamic surface tension of a monolayer may be defined as the response of a film in an initial state of static quasi-equilibrium to a sudden change in surface area. If the area of the film-covered interface is altered at a rapid rate, the monolayer may not readjust to its original conformation quickly enough to maintain the quasi-equilibrium surface pressure. It is for this reason that properly reported II/A isotherms for most monolayers are repeated at several compression/expansion rates. The reasons for this lag in equilibration time are complex combinations of shear and dilational viscosities, elasticity, and isothermal compressibility (Manheimer and Schechter, 1970 Margoni, 1871 Lucassen-Reynders et al., 1974). Furthermore, consideration of dynamic surface tension in insoluble monolayers assumes that the monolayer is indeed insoluble and stable throughout the perturbation if not, a myriad of contributions from monolayer collapse to monomer dissolution may complicate the situation further. Although theoretical models of dynamic surface tension effects have been presented, there have been very few attempts at experimental investigation of these time-dependent phenomena in spread monolayer films. 

Compression of hydrogen consumes energy depending on the thermodynamic process. The ideal isothermal compression requires the least amount of energy (just compression work) and the adiabatic process requires the maximum amount of energy. The compression energy W depends on the initial pressure p and the final pressure pf, the initial volume V and the adiabatic coefficient y ... 

Free Volume Versus Configurational Entropy Descriptions of Glass Formation Isothermal Compressibility, Specific Volume, Shear Modulus, and Jamming Influence of Side Group Size on Glass Formation Temperature Dependence of Structural Relaxation Times Influence of Pressure on Glass Formation... 

Knowledge of temperature and pressure dependence of physical-chemical properties is very useful to estimate the values of derived parameters, such as the thermal expansion coefficient, a, and the isothermal compressibility, Kj. 

Let us first examine the pressure dependence of the compressibility factor under isothermal conditions. Table 2.2 and Fig. 2.2 exhibit some representative P-dependent values of Z for gaseous C02 (at fixed temperature 40°C = 313K) in tabular and graphical form. 

The value of the molar volume of a solvent at other temperatures and pressures, not too far from the ambient, can be obtained by employing the isobaric thermal expansibility, ap, and the isothermal compressibility, kt. The former of these expresses the relative increase in volume on raising the temperature at a constant pressure and the latter expresses the relative decrease of the volume on raising the pressure at a constant temperature. These quantities are also temperature and pressure dependent, but over a limited range of these variables near ambient conditions they can be taken as being constant. 

The pressure dependence of equilibrium constants in this work are estimated with Eq. 2.29, which requires knowledge of the partial molar volumes and compressibilities for ions, water, and solid phases. For ions and water, molar volumes and compressibilities are known as a function of temperature (Table B.8 Eqs. 3.14 to 3.19). Molar volumes for solid phases are also known (Table B.9) unfortunately, the isothermal compressibilities for many solid phases are lacking (Millero 1983 Krumgalz et al. 1999). 

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 ... 

The dependence of the average excess pressure in foam (capillary pressure of bubbles) on its specific area is established by Derjaguin [105]. The mechanical work W done under isothermal compression (or decompression) of foam equals... 

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... 

Straightforward measurements of elastic properties of materials can be made via high-pressure static compression experiments, in which X-ray diffraction (XRD) is used to measure the molar volume (V), or equivalently the density (p), of a material as a function of pressure (P). The pressure dependence of volume is expressed by the incompressibility or isothermal bulk modulus (Kt), where Kp = —V(bP/bV)p. 

The first three terms of equation 4 can be easily calculated from the available data (fi). The fifth term is zero. The temperature and pressure dependency of the coefficient of thermal expansion, Opif and the isothermal compressibility, up to... 

Such a derivation was originally developed and used by Kirchoff [1858] and Rankine [1849] ( ) to express the temperature dependence of vapor pressure. It was also successfully used by Brostow (O to express the temperature dependence of the isothermal compressibility of a wide variety of organic liquids, some metallic liquids and water. By a similar analogy, we have used it to express the viscosity of liquid mold fluxes. 

Figure 9.9 Exceptional physical properties of liquid water (solid lines) temperature dependences (upper diagrams) of the density d (45) and isothermal compressibility Xt (adapted from Refs. (45 7)) pressure dependences (lower drawings) of the shear viscosity 7] at various temperatures (adapted from Ref. (48)) and of the isothermal diffusion coefficient Z) at 0 (adapted from Ref. (49)). Dashed lines sketch typical dependences displayed by almost all other liquids. Note that at —15 °C no value is given for 17 at/ > 300MPa, because of a phase transition towards ice V (Figure 8.5).

The pressure dependence of the melt viscosity (r ) can be estimated by using Equation 13.19 (derived from classical thermodynamics to relate the pressure and temperature coefficients of r [V]), where p is the hydrostatic pressure, K is the isothermal compressibility, a is the coefficient of volumetric thermal expansion, and d is a partial derivative. The sign of the pressure coefficient of the viscosity is opposite to the sign of the temperature coefficient. Consequently, since r decreases with increasing T, it increases with increasing p. Equation 13.20 is obtained by integrating Equation 13.19. 

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