Volume variation with pressure

For most studies of structural phase transitions at high pressures the simplest available EoS, introduced by Mumaghan (1937), provides a sufficiently accurate representation of the volume variation with pressure. It can be derived by assuming that the bulk modulus varies linearly with pressure, K=K + K P with K, being a constant. Integration yields the P-F relationship ... 

As for volume variations with pressure, there is no fundamental thermodynamic basis for specifying the form of cell parameter variations with pressure. It is therefore not unusual to find in the literature cell parameter variations with pressure fitted with a polynomial expression such as a= aQ+a P+ a P, even when the P-V data have been fitted with a proper EoS function. Use of polynomials in P is not only inconsistent, it is also unphysical in that a linear expression implies that the material does not become stiffer under pressure, while a quadratic form will have a positive coefficient for implying that at sufficiently high pressures the material will expand with increasing... 

FIGURE 1.1 Variation of pressure-volume product with pressure. 

Volume variations with conversion are large for constant-pressure gas-phase reactions with change in mole number. Here, as a rule, operation at constant volume poses no difficulties. Liquid-phase reactions may also entail volume contraction or expansion. However, these are not related to changes in mole number and can be predicted only if information on partial molar volumes is at hand. Because liquids are essentially incompressible, even at elevated temperature, it is unsafe to conduct liquid-phase reactions without a gas cap in a closed reactor. Some variation of liquid-phase volume with conversion therefore is apt to occur. Fortunately, the variation at constant temperature is usually so small that it can be neglected in the evaluation or accounted for by a minor correction. 

V, T curves respectively. The variation of the specific heat with volume, or with pressure is therefore greater the more the substance differs in its behaviour from the perfect gas. It is interesting to note that Cp varies with the pressure for gases which obey van der Waals equation. 

The equations derived in 30c, 30d thus also give the variation with pressure and temperature of the fugacity of a constituent of a liquid (or solid) solution. In equation (30.17), Vi is now the partial molar volume of the particular constituent in the solution, and in (30.21), i is the corresponding partial molar heat content. The numerator — fti thus represents the change in heat content, per mole, when the constituent is vaporized from the solution into a vacuum (cf. 29g), and so it is the ideal" heat of vaporization of the constituent i from the given solution, at the specified temperature and total pressure. 

Reservoir engineers describe the relationship between the volume of fluids produced, the compressibility of the fluids and the reservoir pressure using material balance techniques. This approach treats the reservoir system like a tank, filled with oil, water, gas, and reservoir rock in the appropriate volumes, but without regard to the distribution of the fluids (i.e. the detailed movement of fluids inside the system). Material balance uses the PVT properties of the fluids described in Section 5.2.6, and accounts for the variations of fluid properties with pressure. The technique is firstly useful in predicting how reservoir pressure will respond to production. Secondly, material balance can be used to reduce uncertainty in volumetries by measuring reservoir pressure and cumulative production during the producing phase of the field life. An example of the simplest material balance equation for an oil reservoir above the bubble point will be shown In the next section. 

Meters are accurate within close limits as legislation demands. However, gas is metered on a volume basis rather than a mass basis and is thus subject to variation with temperature and pressure. The Imperial Standard Conditions are 60°F, 30inHg, saturated (15.56°C, 1913.7405 mbar, saturated). Gas Tariff sales are not normally corrected, but sales on a contract basis are. Correction may be for pressure only on a fixed factor basis based on Boyle s Law or, for larger loads, over 190,000 therms per annum for both temperature and pressure using electronic (formerly mechanical) correctors. For high pressures, the compressibility factor Z may also be relevant. The current generation of correctors corrects for pressure on an absolute basis taking into account barometric pressure. 

For a dilute solution at high pressure, the variation of activity coefficient with pressure cannot be neglected. But when x2 is small, it is often a good approximation to assume, as above, that the activity coefficient is not significantly affected by composition. If we also assume that v2 the partial molar volume of the solute, is independent of both pressure and composition... 

Summaries of the properties of gases, particularly the variation of pressure with volume and temperature, are known as the gas laws. The first reliable measurements of the properties of gases were made by the Anglo-Irish scientist Robert Boyle in 1662 when he examined the effect of pressure on volume. A century and a half later, a new pastime, hot-air ballooning, motivated two French scientists, Jacques Charles and Joseph-Louis Gay-Lussac, to formulate additional gas laws. Charles and... 

To account for the variation of the dynamics with pressure, the free volume is allowed to compress with P, but differently than the total compressibility of the material [22]. One consequent problem is that fitting data can lead to the unphysical result that the free volume is less compressible than the occupied volume [42]. The CG model has been modified with an additional parameter to describe t(P) [34,35] however, the resulting expression does not accurately fit data obtained at high pressure [41,43,44]. Beyond describing experimental results, the CG fit parameters yield free volumes that are inconsistent with the unoccupied volume deduced from cell models [41]. More generally, a free-volume approach to dynamics is at odds with the experimental result that relaxation in polymers is to a significant degree a thermally activated process [14,15,45]. 

At constant temperature and composition, the variation of free energy with pressure is related to the volume of the system as... 

The Flenry s law constant data calculated as the ratio of vapor pressure to solubility in Figure 1.7.13 are quite scattered. There is little systematic variation with molar volume. Most values of log H lie between -0.1 to -0, i.e., H lies between 0.8 and 0.08, and the resulting air-water partition coefficient KAW or H/RT thus lies between 3 x 10-4 and 3 x 10-5. 

From equation 2.43 in Volume 1, the variation of pressure with height is given by ... 

The variation of the phase transition temperature with pressure can be calculated from the knowledge of the volume and enthalpy change of the transition. Most often both the entropy and volume changes are positive and the transition temperature increases with pressure. In other cases, notably melting of ice, the density of the liquid phase is larger than of the solid, and the transition temperature decreases... 

For condensed phases (liquids and solids) the molar volume is much smaller than for gases and also varies much less with pressure. Consequently the effect of pressure on the chemical potential of a condensed phase is much smaller than for a gas and often negligible. This implies that while for gases more attention is given to the volumetric properties than to the variation of the standard chemical potential with temperature, the opposite is the case for condensed phases. 

When the Krichevsky-Kasarnowsky equation fails it may be because of either changing activity coefficient of the solute gas with composition, changing partial molal volume of the gas with pressure, or both. The Krichevsky-Ilinskaya equation takes into account the variation in the activity coefficient of the solute gas with mole fraction by means of a two-suffix Margules equation. 

All these volume changes, AV4, are associated with the observed variation of the kinetic constants K( with pressure. They cannot always be interpreted in terms of single reactions, but must be analyzed according to the explicit expressions of the KjS for the mechanism in question. In Eq. (74), v, W, and the KjS must be evaluated at the same pressure as the derivatives, AVjJ. 

In high-pressure biochemistry the buffer system should be chosen with care. As should be clear from Section IV,A, ionizations are followed by negative volume changes and therefore increase with pressure. The variation of pH with pressure depends on the volume of ionization AVZ, which can differ as much as —30 cm3 mol-1 from one buffer to another. 

The bomb method is quite similar to the bubble method except that the constant volume condition causes a variation in pressure. One must, therefore, follow the pressure simultaneously with the flame front. 

By differentiating Eq.(6) with respect to time, considering that the variations of the volume and total pressure are negligible, and using the enthalpy definition. Hi = CpiT, where Cp is the heat capacity of i-reactant (kJ/mol °C), Eq.(5) can be written as follows ... 

Eontanella and co-workers studied the effect of high pressure variation on the conductivity as well as the H, H, and O NMR spectra of acid form Nafionl 17 membranes that were exposed to various humidities. Variation of pressure allows for a determination of activation volume, A V, presumably associated with ionic and molecular motions. Conductivities (a) were obtained from complex electrical impedance diagrams and sample geometry, and A V was determined from the slope of linear isothermal In a versus p graphs based on the equation A E = —kJ d In a/d/j] t, where p is the applied pressure. At room temperature, A Ewas found to be 2.9 cm mol for a sample conditioned in atmosphere and was 6.9 cm mol for a sample that was conditioned in 25% relative humidity, where the latter contained the lesser amount of water. 

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