Pressure derivative

Ruoff (1967) first showed how the coefficients of the shock-wave equation of state are related to the zero pressure isentropic bulk modulus, and its first and second pressure derivatives, K q and Kq, via... 

The detailed first principles study of the three stable polymorphs has been performed recently using the LCAO technique The main drawback of that work is that no cell optimization was performed for anatase or brookite. The energy-volume curves that were used to calculate the bulk modulus, B, and its pressure derivative, B, have been produced by varying the volume with the c/a ratio and fractional atomic coor nates being fixed at experimental values which makes results unreliable. 

It has been stated above that the difference of partial molar volumes of the LS and HS isomers AK° can be obtained from the relaxation amplitude A of ultrasonic absorption. An independent method for the determination of AE° is based on the pressure dependence of the equilibrium constant. The pressure derivative of being determined by ... 

The rheological behaviour of a typical surfactant at atmospheric pressure, derived using this technique, has been reported and illustrated in various figures in an earlier communication 7). 

Ehlig-Economides, C. "Use of the Pressure Derivative for Diagnosing Pressure-Transient Behavior," J. Petr. Technol.. October 1988, 1280-1282. 

The pressure derivative of the Gibbs energy (eq. 1.68) at constant temperature is V ... 

Many attempts (130,131,132) have been made to derive an equation of state from molecular theory, but none of them have resulted in an adequate equation capable of expressing the results of direct measurements to within the experimental error. To meet this need, it has been necessary to employ empirical equations the major justification for each is that it fits the experimental data. It is also desirable to choose an equation that gives reliable concentration, temperature, and pressure derivatives. 

At atmospheric pressure (101.325 kPa), a quartz is in equilibrium with (3 quartz at 847 + 1.5K [8]. The enthalpy change of the transition from a quartz to (3 quartz is 728 + 167 J mol Berger et al. [9] measured the volume change by an X-ray diffraction method, and they reported a value of 0.154 + 0.015 cm mol Use these values to calculate the slope at atmospheric pressure of the equilibrium curve between a quartz and 3 quartz in Figure 13.1. Compare your results with the value of the pressure derivative of the equilibrium temperature dT/dP, which is equal to 0.21 K (Mpa) [10]. 

Systematic density measurements at a wide range of femperafure and pressure [28,31,63,68,69] were helpful to obfain isofhermal compressibility, which is calculated using the isothermal pressure derivative of density according to Equation 1.2... 

From this equation, it can be proved that co is negative and thus the outlet pressure derived from the Ergun equation is always higher than the actual one, or in other words, the pressure drop is lower than the actual one. This is shown in Figure 5.12, where... 

The direction of differencing for the two first-order equations, Eqs. 7.60 and 7.61, is opposite that is, the pressure derivative involves information at points j + 1 and j, while the radial-coordinate equation involves information at points j and j — 1. This differencing is essential to the implicit-boundary condition specification that sets the radial coordinate at both boundaries. The opposite sense of the differencing permits information to be propagated from both boundaries into the interior of the domain. 

Taken together, the system of equations represents a set of stiff ordinary differential equations, which can be solved numerically. Because more than one dependent-variable derivative can appear in a single equation (e.g., the momentum equation has velocity and pressure derivatives), it is usually more convenient to use differential-algebraic equation (DAE) software (e.g., Dassl) for the solution rather than standard-form ODE software. 

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... 

Subatmospheric pressure usually is expressed in reference to perfect vacuum or absolute zero pressure, lake absolute zero temperature (the concept is analogous), absolute zero pressure cannot be achieved, but it does provide a convenient reference datum. Standard atmospheric pressure is 14.695 psi absolute, 30 inches of mercury absolute, or 760 mmHg of density 13.595 g/cm3 where acceleration due to gravity is g = 980.665 emir. ] mmHg. which equals 1 torr. is the most commonly used unit of absolute pressure. Derived units, the million or micrometer, representing 1/1000 of 1 mmHg or 1 torr, are also used for subtorr pressures. 

Other thermodynamic variables can be related to the voltage E and its temperature and pressure derivatives. They are as follows ... 

Figure 17.5 summarizes temperatures and pressure derivatives of H and for (cyclohexane + hexane). Figure 17.5a gives C m, the temperature derivative of HThat is,... 

Table A7.10 Volumetric properties and the pressure derivatives of Pitzer coefficients ...

The first term in Eq (7) describes transport related to the Henry s law environment, while the second term is related to the Langmuir environment The tendency for the permeability to asymptotically approach the limiting value of kDDD at high pressures derives from the fact that after saturation of the upstream Langmuir capacity at high pressures, additional pressure increases result in additional flux contributions only from the term related to Henry s law which continues to increase as upstream pressure increases ... 

The values of the chemical potentials in each phase, together with their pressure derivatives (available through the measured number densities), can be exploited to home in on the coexistence pressure. The method has been successfully applied to calculate the phase diagrams of a number of simple fluids and fluid mixtures [79, 80]. 

The amount of hydrogen chemisorbed on to a nickel surface at 600 K is approximately proportional to the square root of its pressure. Derive a two-dimensional equation of state for the adsorbed gas and comment on its significance. 

The excess quantities in each equation represent the corresponding temperature and pressure derivatives of the excess chemical potential of mixing. When the values of the temperature and pressure derivatives of Apfx for the reference state or Ap x for the standard state are defined, then all quantities in Equations (8.77)—(8.80) are also defined. 

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