Heat capacity isobaric

Diagrams of isobaric heat capacity (C and thermal conductivity for carbon dioxide covering pressures from 0 to 13,800 kPa (0—2,000 psi) and 311 to 1088 K have been prepared. Viscosities at pressures of 100—10,000 kPa (1—100 atm) and temperatures from 311 to 1088 K have been plotted (9). 

The following tables of properties of carbon dioxide are available enthalpy, entropy, and heat capacity at 0 and 5 MPa (0 and 50 atm, respectively) from 273 to 1273 K pressure—volume product (PV), enthalpy, and isobaric heat capacity (C from 373 to 1273 K at pressures from 5 to 140 MPa (50-1,400 atm) (14). 

Equation (2.18) is another example of a line integral, demonstrating that 6q is not an exact differential. To calculate q, one must know the heat capacity as a function of temperature. If one graphs C against T as shown in Figure 2.8, the area under the curve is q. The dependence of C upon T is determined by the path followed. The calculation of q thus requires that we specify the path. Heat is often calculated for an isobaric or an isochoric process in which the heat capacity is represented as Cp or Cy, respectively. If molar quantities are involved, the heat capacities are C/)m or CY.m. Isobaric heat capacities are more... 

The linearity of van t Hoff plots, such as Figure 3.14, depends on the degree to which the isobaric heat capacity of the system (Cp) remains constant between the... 

Another heat capacity is Cp, the heat capacity measured at constant pressure (which is also called the isobaric heat capacity). The values of Cp and Cv will differ, by perhaps as much as 5-10 per cent. We will look at Cp in more depth in the next section. 

Table 5.24 lists selected data concerning entropy and isobaric heat capacity, covering andradite, grossular, pyrope, and almandine terms, compared with results of calculations based on the Kieffer model (Ottonello et al., 1996). 

Table 5.36 Thermodynamic properties of pure pyroxene components in their various structural forms according to Saxena (1989) (1), Berman (1988) (2), and Holland and Powell (1990) (3) database. = standard state entropy of pure component at 7) = 298.15 K and Py = bar (J/mole) Hjp p = enthalpy of formation from elements at same standard state conditions. Isobaric heat capacity function Cp is... 

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.

Besides equilibriumconstants, additional thermodynamic data were included, if available, although little emphasis was put on their completeness. The data for primary master species comprise the standard molar thermodynamic properties of formation from the elements (AfG standard molar Gibbs energy of formation AfH°m standard molar enthalpy of formation ApSm- standard molar entropy of formation), the standard molar entropy (5m), the standard molar isobaric heat capacity (Cp.m), the coefficients Afa, Afb, and Afc for the temperature-dependent molar isobaric heat capacity equation... 

Data for secondary master species and product species include the stoichiometry and logI0 K° of the formation reactions, the standard Gibbs energy of reaction (ArGm), the standard enthalpy of reaction (Athe standard entropy of reaction (Ar5m), the standard isobaric heat capacity... 

C = isobaric heat capacity per unit mass, gas D - diffusion coefficient, gas E = activation energy g = acceleration due to gravity... 

The enthalpy, internal energy and their excess quantities of the Lennard-Jones binary mixture have been determined using the PY approximation. The values obtained are in good agreement with the results of MC calculation. The enthalpy and isobaric heat capacity are calculated using the extended expression of the thermodynamic quantities in terms of pair correlation functions. 

We have calculated enthalpy, internal energy, excess molar enthalpy, and excess molar internal energy based on the integral equation theory. Validity of its use has been confirmed by the comparison of our results with those of MC calculation. Then, we have calculated the differential thermodynamic quantities of the isobaric heat capacity Cp and the excess isobaric molar heat capacity, Cp. ... 

The above-mentioned method was initially developed for measuring the isobaric heat capacities of aqueous salt solutions up to 573 K and 30 MPa. For a typical run, the sample cell was loaded with the sample solution and the reference cell was loaded with a reference fluid of known heat capacity (usually water). Then, the temperature was increased from to T, at constant pressure, and the difference Q in the transferred heat was corrected taking into account both the cell s volumetric dissymmetry and the differences between the densities and specific heat capacities of the measured sample and reference fluids, respectively. Such an experiment allows the measurement of the product pCp representing the isobaric heat capacity divided by volume. In order to obtain the desired isobaric heat capacity, Cp, of the solution, it was necessary to know its density. For this purpose, the isobaric specific heat capacity and density were represented by polynomials in terms of temperature T ... 

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) ... 

When the p-isotherms cross at a given pressure, (dup/d T)p changes its sign and this property is responsible for the above-mentioned minima. In addition, as the temperature increases, the minima are shifted to higher pressures. At higher temperatures, especially when approaching the critical temperature, the minimum becomes a shallow one. As an example for associated liquids, some selected Cp-isotherms and the pressure effects at different temperatures on the isobaric heat capacity of m-cresol are presented in Figures 2(al) and (a2), respectively. 

Figure 2 Isobaric heat capacities (al) and pressure effects on the isobaric heat capacities (a2)for m-cresol. Typical W-shaped excess molar heat capacities (b) at 298.15 K of f 1,4-dioxane-hn-alkanesjmixtures numbers refer to the n-alkane carbon numbers...

The resulting equation has uncertainties of 0.1% in density in the liquid below 450 K, 1% in density at temperatures between 450 and 500 K, 3% in density at temperatures above 500 K, 0.5% in density in the vapor phase and at supercritical conditions below 10 MPa and 450 K, 0.5% in vapor pressure, and 2% in isobaric heat capacity. 

The uncertainty in density is 0.2%, that for vapor pressure is 0.3%, and that for the isobaric heat capacity is 2%. The uncertainties are higher in and above the critical region. 

The uncertainty in density is 0.2% up to 400 K and 1% at higher temperatures. The vapor pressure uncertainty is 1.5%. In the liquid phase, the uncertainty in isobaric heat capacity is 3%. 

Drastic increase of C with approaching the liquid-liquid phase transition indicates close proximity of the liquid-liquid critical point, located at negative pressures. Similar increase of C (without the step increase, however) we may expect also for the ST2RF water model, where the liquid-liquid critical point is located at slightly positive pressure. Strong temperature dependence of C leads to the strong increase of the isobaric heat capacity upon cooling, which is clearly seen both in experiment and in simulations. ... 

Fig. 2. Isobaric relationship between enthalpy and temperature in the liquid, glassy, and crystalline states. is the melting temperature, and Fg the glass transition temperature. The lower diagram shows the behavior of the isobaric heat capacity. The arrow indicates the -function singularity due to latent heat at a first-order phase transition. (From Debenedetti, 1996.)...

The quantities a, c, f, F, r, and p are the thermal diffusivity, sound speed, heat capacity ratio, bulk viscosity coefficient, shear viscosity coefficient, and density of the sample, respectively and Eo, a, P and Cp are the energy fluence of the laser beam, the optical absorption coefficient, the volume expansion coefficient, and the isobaric heat capacity, respectively, of the fluid. Tlie first and second terms in Eq. 2 describe the time dependences of the thermal and acoustic modes of wave motion, respectively. Since the decays of the acoustic and thermal mode densities back to their ambient values take place on such different time scales (microsecond time scale for acoustic mode and millisecond time scale for thermal mode), they were recorded on the oscilloscope using different time bases. 

For slightly metastable states of superheated water no problems arise in describing its thermophysical properties. They differ little Ifom properties on the saturation line. But a problem will arise at the approach of the spinodal, when isothermal compressibility, thermal expansion and isobaric heat capacity tend to infinity. 

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