Isobaric thermal expansion

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... 

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... 

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. 

Measurements of the isobaric thermal expansivity of methane in the pressure range from 50 to 165 MPa and along four isotherms (303, 333,363, and 393 K) have been performed in decreasing the pressure at a low constant rate, a=0.02 MPas , in such a way as to remain at thermodynamic equilibrium. Several hundreds of data points were collected and fitted, as a function of p along each isotherm, to the following empirical equation ... 

Figure 3 Transitiometric investigation of the pressure effect on the isobaric thermal expansivities a. of gaseous, liquid, and solid substances (a) comparison between calculated ana experimental data of methane at 333 K (bl) quinoline as a simple fluid (b2) water as the associated liquid (cl) effect of temperature on medium-density polyethylene (MDPE) (c2) evolution of with the crystallinity (points represent the experimental data and the lines were obtained by the least squares fitting of data) (c3) comparison between the experimental a for three polyethylenes with different degrees of crystallinity and the predicted values for crystal and amorphous phases obtained by extrapolation from linear fitting of the experimental data...

From the MF analysis, when = 0 the model coincides with the one proposed in which gives rise to the SF scenario (Fig. 3a). When > 0 the model displays a phase diagram with a LLCP (Fig. 3b) [13]. For 0, keeping J and the other parameters constant, we find that Tc 0, and the power-law behavior of Kt and the isobaric thermal expansion coefficient Op is preserved. Further, we find for the entropy S that, for any value of J , (dS/dT)p ... 

Since the lattice parameters depend significantly on the temperature (Table 2), it is possible to estimate the coefficient of isobaric thermal expansion roughly to about 2.8x10 K ... 

Figure 11.10b shows plots of the temperature and pressure dependence of the volumes in PFE rubber [Dlubek et al., 2004e, 2005a]. The slope of the Vf versus hole size curve is constant and the same in isothermal compression and isobaric thermal expansion experiments. The same behavior was observed for PD3 [Kilburn et al., 2006b]. The temperature-dependent macroscopic volume V shows a behavior that is parallel to Vf. 

The prediction of miscibility requires knowledge of the parameters T" (the characteristic temperature), p (the characteristic pressure) and V (the characteristic specific volume) of the corresponding equation of state which can be calculated from the density, thermal expansivity and isothermal compressibility. The isobaric thermal expansivity and the isothermal compressibility can be determined experimentally from p-V-Tmeasurements where these values can be calculated from V T) and V(p)j. The characteristic temperature T is a measure of the interaction energy per mer, V is the densely packed mer volume so that p is defined as the interaction energy per... 

Discuss significance of the isobaric thermal expansivity for service performance of moving parts. From this point of view, what are the advantages of using polymer liquid crystals ... 

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