Pressure and Temperature Derivation of

An estimate of the density deficit in the core is —5-10% (Boehler, 2000 Anderson and Isaak, 2002) the uncertainty in this estimate is dominantly a function of uncertainties in the pressure and temperature derivatives of EOS data for candidate core materials and knowledge of the temperatures conditions in the core. A tighter constraint on this number will greatly help to refine chemical and petrological models of the core. A density deficit estimate for the inner core is 4-5% (Hemley and Mao, 2001). 

So far we have considered only the volume as a partial molar quantity. But calculations involving solutes will require knowledge of all the thermodynamic properties of dissolved substances, such as H, S, Cp, and of course G, as well as the pressure and temperature derivatives of these. These quantities are for the most part derived from calorimetric measurements, that is, of the amount of heat released or absorbed during the dissolution process, whereas V is the result of volume or density measurements. 

Pressure and temperature derivative of yi. Let us divide Eq. (1.138) by T and take the derivative of the resulting expression with respect to temperature while holding P and x constant. 

The sudden increase in Cp with temperature at the highest pressure and high temperatures is interesting, although care should be taken in relying too heavily on this result, since it represents a prediction of the third derivative of the chemical potential near both the pressure and temperature limits of the reliability of the equations. 

Kumazawa, M., and O. L. Anderson (1969). Elastic moduli, pressure derivatives, and temperature derivatives of single crystal olivine and single crystal for-sterile. Journ. Geophys. Rs. 74, 5961-72. 

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. 

By analysing the parameters obtained from the FR in-phase and out-of-phase curve fits of the non-isothermal diffusion model, it has been found that the pressure and temperature dependence of the heat transfer coefficients, the heat exchange rates, the non-isothermahty of the system, the heat of adsorption, and the K values derived from the model are all physically rational [38,55]. One can, therefore, conclude that the low frequency FR spectral data can be attributed to the dissipation of the heats of adsorption between sorbent and the surroundings in the system. 

It will be assumed here that the equations for 2ero-pressure specific heat, vapor pressure, and the derivative of the vapor pressure with respect to temperature are all available. For example, the following forms have been found to be adequate for some fluids ... 

Table 6.2 The fugacity, fugadty coefficient, activity, and activity coefficient are equivalent representations of the chemical potential those equivalences extend to their pressure and temperature derivatives.

We need to know which way reactions will go when solutions are involved, such as minerals dissolving/precipitating, or electrolytes dissociating. We have a thermodynamic potential (the Gibbs energy) which fills this role for pure phases, so we just need to know how to determine it for dissolved substances. In addition, we need to know how it changes with T, P, and composition of the solution, which involves knowing how to determine the temperature, pressure, and compositional derivatives of this quantity. 

Other thermodynamic properties are simply related to volume and temperature derivatives of F (Callen 1960). For example, the pressure (negative of the volume derivative) can be calculated by computing the Helmholtz free energy at two different values of the volume. 

Table 27.5 Parameters derived from theoretical reproduction of the size, pressure, and temperature dependence of the bulk modulus and the Raman shift for TiOa [151]...

The extent of decarboxylation primarily depends on temperature, pressure, and the stabihty of the incipient R- radical. The more stable the R- radical, the faster and more extensive the decarboxylation. With many diacyl peroxides, decarboxylation and oxygen—oxygen bond scission occur simultaneously in the transition state. Acyloxy radicals are known to form initially only from diacetyl peroxide and from dibenzoyl peroxides (because of the relative instabihties of the corresponding methyl and phenyl radicals formed upon decarboxylation). Diacyl peroxides derived from non-a-branched carboxyhc acids, eg, dilauroyl peroxide, may also initially form acyloxy radical pairs however, these acyloxy radicals decarboxylate very rapidly and the initiating radicals are expected to be alkyl radicals. Diacyl peroxides are also susceptible to induced decompositions ... 

The vessel nozzle diameter (inside) or net free area for relief of vapors through a rupture disk for the usual process applications is calculated in the same manner as for a safety relief valve, except that the nozzle coefficient is 0.62 for vapors and liquids. Most applications in this category are derived from predictable situations where the flow rates, pressures and temperatures can be established with a reasonable degree of certainty. 

The ideal gas law is readily applied to problems of this type. A relationship between the variables involved is derived from this law. In this case, pressure and temperature change, while n and V remain constant. 

We now have the foundation for applying thermodynamics to chemical processes. We have defined the potential that moves mass in a chemical process and have developed the criteria for spontaneity and for equilibrium in terms of this chemical potential. We have defined fugacity and activity in terms of the chemical potential and have derived the equations for determining the effect of pressure and temperature on the fugacity and activity. Finally, we have introduced the concept of a standard state, have described the usual choices of standard states for pure substances (solids, liquids, or gases) and for components in solution, and have seen how these choices of standard states reduce the activity to pressure in gaseous systems in the limits of low pressure, to concentration (mole fraction or molality) in solutions in the limit of low concentration of solute, and to a value near unity for pure solids or pure liquids at pressures near ambient. 

The concept of chemical potentials, the equilibrium criterion involving chemical potentials, and the various relationships derived from it (including the Gibbs phase rule derived in Chapter 5) can be used to explain the effect of pressure and temperature on phase equilibria in both a qualitative and quantitive way. 

In Section 4.4, we used a molecular model of a gas to explain qualitatively why the pressure of a gas rises as the temperature is increased as a gas is heated, its molecules move faster and strike the walls of their container more often. The kinetic model of a gas allows us to derive the quantitative relation between pressure and the speeds of the molecules. 

This important formula, which can be derived more formally from the laws of thermodynamics, applies when any change takes place at constant pressure and temperature. Notice that, for a given enthalpy change of the system (that is, a given output of heat), the entropy of the surroundings increases more if their temperature is low than if it is high (Fig. 7.16). The explanation is the sneeze in the street analogy mentioned in Section 7.2. Because AH is independent of path, Eq. 10 is applicable whether the process occurs reversibly or irreversibly. 

The ideal gas law says that the molar density is determined by pressure and temperature and is thus known and constant in the reactor. Setting the time derivative of molar density to zero gives an expression for Qom at steady state. The result is... 

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