Thermal expansivity, isobaric

Here Q(t) denotes the heat input per unit volume accumulated up to time t, Cp is the specific heat per unit mass at constant pressure, Cv the specific heat per unit mass at constant volume, c is the sound velocity, oCp the coefficient of isobaric thermal expansion, and pg the equilibrium density. (4) The heat input Q(t) is the laser energy released by the absorbing molecule per unit volume. If the excitation is in the visible spectral range, the evolution of Q(t) follows the rhythm of the different chemically driven relaxation processes through which energy is... 

Typical values of the isobaric expansivity and the isothermal compressibility are given in Table 1.2. The difference between the heat capacities at constant volume and constant pressure is generally negligible for solids at low temperatures where the thermal expansivity becomes very small, but the difference increases with temperature see for example the data for AI2O3 in Figure 1.2. 

The effect of temperature on the equation of state is introduced through the iso-baric thermal expansivity. It is generally assumed that isobaric expansivity and iso-baric compressibility work independently of each order and the volume as a function of T and p is then expressed as... 

The heat capacity models described so far were all based on a harmonic oscillator approximation. This implies that the volume of the simple crystals considered does not vary with temperature and Cy m is derived as a function of temperature for a crystal having a fixed volume. Anharmonic lattice vibrations give rise to a finite isobaric thermal expansivity. These vibrations contribute both directly and indirectly to the total heat capacity directly since the anharmonic vibrations themselves contribute, and indirectly since the volume of a real crystal increases with increasing temperature, changing all frequencies. The constant volume heat capacity derived from experimental heat capacity data is different from that for a fixed volume. The difference in heat capacity at constant volume for a crystal that is allowed to relax at each temperature and the heat capacity at constant volume for a crystal where the volume is fixed to correspond to that at the Debye temperature represents a considerable part of Cp m - Cv m. This is shown for Mo and W [6] in Figure 8.15. 

In an alternative approach [7], applicable if the isobaric thermal expansivity is known as a function of temperature, Cp m - Cv m is given by... 

Gibbs energy minimization has also predicted negative isobaric expansion coefficients for certain crystalline zeolite framework structures, which subsequently were confirmed experimentally [6], Many solids show negative thermal expansion at very low temperatures, including even some alkali halides (Barron and White (Further reading)). Many other solids on heating expand in some directions and contract in others. 

We may first assume that isothermal compressibility fiy and isobaric thermal expansion coefficient a are independent, respectively, of T and P. Equations 1.91 and 1.99, integrated on T and P, respectively, give... 

Table 5.10 Isobaric thermal expansion of olivine componnds. Regressions for Mg2Si04 and Fe2Si04 from Fei and Saxena (1986).

Table 5.35 Molar volnme (cm /mole), isobaric thermal expansion (K" ), and isothermal bulk modulus (Mbar) of pyroxene end-members according to Saxena (1989)...

Table 5.48 summarizes thermal expansion and compressibility data for amphibole end-members according to the databases of Holland and Powell (1990) and Saxena et al. (1993). Isobaric thermal expansion (a, K ) and isothermal compressibility (jS, bar ) may be retrieved from the listed coefhcients by applying the polynomial expansions... 

Table 5.64 lists isobaric thermal expansion and isothermal compressibility coefficients for feldspars. Due to the clear discrepancies existing among the various sources, values have been arbitrarily rounded off to the first decimal place. 

Table 5.64 Isobaric thermal expansion (K ) and isothermal compressibility (bar ) of feldspars...

If the heat capacity functions of the various terms in the reaction are known and their molar enthalpy, molar entropy, and molar volume at the 2) and i). of reference (and their isobaric thermal expansion and isothermal compressibility) are also all known, it is possible to calculate AG%x at the various T and P conditions of interest, applying to each term in the reaction the procedures outlined in section 2.10, and thus defining the equilibrium constant (and hence the activity product of terms in reactions cf eq. 5.272 and 5.273) or the locus of the P-T points of univariant equilibrium (eq. 5.274). If the thermodynamic data are fragmentary or incomplete—as, for instance, when thermal expansion and compressibility data are missing (which is often the case)—we may assume, as a first approximation, that the molar volume of the reaction is independent of the P and T intensive variables. Adopting as standard state for all terms the state of pure component at the P and T of interest and applying... 

Figure 8J (A) Isobaric thermal expansion, (B) its first r-derivative, (C) isothermal compressibility, and (D) isobaric heat capacity of H2O within the critical region, based on the equation of state of Levelt Sengers et al. (1983). From Johnson and Norton (1991), American Journal of Science, 291, 541-648. Reprinted with permission of American Journal of Science.

The coefficient of isobaric thermal expansion is defined as the fractional change in volume of a liquid as temperature changes under constant pressure. 

When a value of thermal expansion is reported, it must include the pressure and temperature range for which it is valid. Thermal expansion as defined here must not be used interchangeably with the coefficient of isobaric thermal expansion defined above. 

EXAMPLE 8-6 A sample of reservoir oil was placed in a laboratory cell at 5000psig and 76°F. The volume was 54.74 cc. Temperature was increased to 220°F and pressure was held constant by increasing cell volume to 59.55 cc. Calculate the coefficient of isobaric thermal expansion and calculate the thermal expansion. 

First, calculate the coefficient of isobaric thermal expansion. 

The density of the pseudoliquid is adjusted to reservoir pressure using the coefficient of isothermal compressibility and is adjusted to reservoir temperature using the coefficient of isobaric thermal expansion. 

Fig- 11-4. Density adjustments for isobaric thermal expansion of reservoir liquids. 

Figure 11-4, page 304, Density Adjustment for Isobaric Thermal Expansion of Reservoir Liquids... 

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. 

We should remember that certain partial derivatives are quantities that can be measured conveniently the isobaric thermal expansivity (also known as the coefficient of thermal expansion) ... 

Here, Cp is the heat capacity at constant pressure, aP is the isobaric thermal expansion, and kp is the isothermal compressibility Table 1.1 shows the molar heat capacities of some gas compounds. 

Here a denotes a constant thermal expansivity at zero pressure. The exponential representation, Equation (3A-4) has been shown to yield a good fit of the zero pressure isobar only for a few polymers (Zoller, 1989). Moreover a constant thermal expansivity at zero... 

Later, the pressure-scanning technique was used to investigate the thermophysical properties, isobaric molar heat capacity Cp (J K" mol" ), and Up, over extended T and p of several fluids or their mixtures, such as quinoline, n-hexane, 1-hexa-namine, and its binary mixtures with 1-hexanol, m-cresol, and its binary mixtures with quinoline, etc. As a rule, for simple liquids without strong intermolecular interactions, such as -hexane, for example, both the C -isotherms and the pressure effects (isotherms) on the isobaric heat capacity at pressures up to 700 MPa exhibit minima. It is worth recalling that the pressure effect on the Cp is related to the iso-baiic thermal expansibility ttp by the following equation (the effect of pressure on the Up is discussed in the next section) ... 

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