Constants for heat capacity

Table 2.1 Constants for Heat Capacity Equation 2.28 for Air and Water...

Table 1. Constants for heat capacities of gases in ideal state and liquid water.

In Fig. 1 there are three example applications from a Mettler DSC Application Description (DSC of the type illustrated in Fig. 4.4, left center). Calculate the crystallinity of the polyethylene sample (for the heat of fusion, look in the Appendbc) and the calibration constant for heat capacity measurement in J/(s V) [the aluminum oxide specific heat capacity is 1.005 J/(K g)j. Furthermore, estimate the heat of fusion of phenacetin [use Eq. (1), of Fig. 5.28 the equilibrium melting temperature of pure phenacetin is 407.6 K]. 

The enthalpy change AH for a temperature change from to T2 can be obtained by integration of the constant pressure heat capacity... 

Magee J W, Blanco J C and Deal R J 1998 High-temperature adiabatic calorimeter for constant-volume heat capacity of compressed gases and liquids J. Res. Natl Inst. Stand. Technol. 103 63... 

Hea.t Ca.pa.cities. The heat capacities of real gases are functions of temperature and pressure, and this functionaHty must be known to calculate other thermodynamic properties such as internal energy and enthalpy. The heat capacity in the ideal-gas state is different for each gas. Constant pressure heat capacities, (U, for the ideal-gas state are independent of pressure and depend only on temperature. An accurate temperature correlation is often an empirical equation of the form ... 

For the ideal-gas state there is an exact relation between the constant pressure heat capacity and the constant volume heat capacity, C, via the ideal-gas constant, R. 

Constant volume heat capacities for Hquid organic compounds were estimated with a four parameter fit (219). A 1.3% average absolute error for 31 selected species was reported. A group contribution method for heat capacities of pure soHds andHquids based on elemental composition has also been provided (159). 

An initially clean activated carbon Led at 320 K is fed a vapor of benzene in nitrogen at a total pressure of 1 MPa. The concentration of benzene in the feed is 6 mol/m. After the Led is uniformly saturated with feed, it is regenerated using benzene-free nitrogen at 400 K and 1 MPa. Solve for Loth steps. For sim-phcity, neglect fluid-phase acciimiilation terms and assume constant mean heat capacities for stationary and fluid phases and a constant velocity. The system is described by... 

For first trial on tubeside assume equal heat is transferred in each pass w ith constant fluid heat capacity. 

The constant-volume and constant-pressure heat capacities of a solid substance are similar the same is true of a liquid but not of a gas. We can use the definition of enthalpy and the ideal gas law to find a simple quantitative relation between CP and Cv for an ideal gas. 

For one mole of an ideal gas, dH = Cp dT + Md(f>, where Cp is the constant pressure heat capacity per mole. Now the adiabatic condition implies that... 

The kinetic equilibrium constant is estimated from the thermodynamic equilibrium constant using Equation (7.36). The reaction rate is calculated and compositions are marched ahead by one time step. The energy balance is then used to march enthalpy ahead by one step. The energy balance in Chapter 5 used a mass basis for heat capacities and enthalpies. A molar basis is more suitable for the current problem. The molar counterpart of Equation (5.18) is... 

The expression for heat capacity brings out the fact that it is an indefinite quantity even when mass is specified, since 8q is so. This is no longer the case when certain conditions, particularly constant volume or constant pressure conditions, are specified. The heat capacity then becomes a definite quantity as a consequence of 8q becoming a definite quantity. 

In many cases the magnitude of the last term on the right side of equation 2.2.7 is very small compared to AH°98a6. However, if one is to be able to evaluate properly the standard heat of reaction at some temperature other than 298.16 °K, one must know the constant pressure heat capacities of the reactants and the products as functions of temperature as well as the heat of reaction at 298.16 °K. Data of this type and techniques for estimating these properties are contained in the references in Section 2.3. 

This model, the Einstein model for heat capacity, predicts that the heat capacity is reduced on cooling and that the heat capacity becomes zero at 0 K. At high temperatures the constant-volume heat capacity approaches the classical value 3R. The Einstein model represented a substantial improvement compared with the classical models. The experimental heat capacity of copper at constant pressure is compared in Figure 8.3 to Cy m calculated using the Einstein model with 0g = 244 K. The insert to the figure shows the Einstein frequency of Cu. All 3L vibrational modes have the same frequency, v = 32 THz. However, whereas Cy m is observed experimentally to vary proportionally with T3 at low temperatures, the Einstein heat capacity decreases more rapidly it is proportional to exp(0E IT) at low temperatures. In order to reproduce the observed low temperature behaviour qualitatively, one more essential factor must be taken into account the lattice vibrations of each individual atom are not independent of each other - collective lattice vibrations must be considered. 

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. 

Some of the heat transferred to the surroundings during an exothermic reaction are absorbed by the calorimeter and its parts. In order to account for this heat, a calorimeter constant or heat capacity of the calorimeter is required and usually expressed in J °C 1. 

In addition to the intermolecular potential, there is an intramolecular portion of the Helmholtz free energy. Cheetah uses a polyatomic model to account for this portion including electronic, vibrational, and rotational states. Such a model can be expressed conveniently in terms of the heat of formation, standard entropy, and constant-pressure heat capacity of each species. 

Results in Table I illustrate some of the strengths and weaknesses of the ST2, MCY and CF models. All models, except the MCY model, accurately predict the internal energy, -U. Constant volume heat capacity, Cv, is accurately predicted by each model for which data is available. The ST2 and MCY models overpredict the dipole moment, u, while the CF model prediction is identical with the value for bulk water. The ratio PV/NkT at a liquid density of unity is tremendously in error for the MCY model, while both the ST2 and CF models predictions are reasonable. This large error using the MCY model suggests that it will not, in general, simulate thermodynamic properties of water accurately (29). Values of the self-diffusion coefficient, D, for each of the water models except the CF model agree fairly well with the value for bulk water. 

The SI unit for heat capacity is J-K k Molar heat capacities (Cm) are expressed as the ratio of heat supplied per unit amount of substance resulting in a change in temperature and have SI units of J-K -moC (at either constant volume or pressure). Specific heat capacities (Cy or Cp) are expressed as the ratio of heat supplied per unit mass resulting in a change in temperature (at constant volume or pressure, respectively) and have SI units of J-K -kg . Debye s theory of specific heat capacities applies quantum theory in the evaluation of certain heat capacities. 

Equation (3.7) is true for heat capacities at either constant pressure or constant volume. However, the heat capacity will be different if the system is at constant pressure (Cp) or constant volume (Cv). For the case of constant pressure, combining Eqs (3.5) and (3.7) gives... 

There is a well-known precedent for the difference in the results predicted by stiff and rigid models. The constant-volume heat capacity of a diatomic ideal gas is predicted to be jkT per molecule for a stiff classical model and jkT for a classical rigid rotor. A quanmm mechanical analysis of a diatomic gas yields the... 

Equation (4.36) provides a simple method for estimating an important heat transfer dimensionless group called the Prandtl number. Recall from general chemistry and thermodynamics that there are two types of molar heat capacities, C , and the constant pressure heat capacity, Cp. For an ideal gas, C = 3Cpl5. The Prandtl number is... 

The terms in Equation 1.2 are described in Nomenclature. The condition of constant heat capacity can be relaxed if accurate data is available for heat capacity as a function of both conversion and temperature. 

In the remainder of this section it is desired to obtain the relative, constant-pressure heat capacity of the liquid at x=j and the concentration fluctuation factor for all compositions. Since the latter equation is complicated, it is not written out in full here. This has been done in Eqs. (37)-(45) of the paper by Liao et al. (1982) for the special case that 14 = 34 = 0 and / l3 is the only nonzero cubic interaction term, i.e., the version of the model applied here to the Ga-Sb and In-Sb binaries. Bhatia and Hargrove (1974) have given equations for the composition fluctuation factor at zero wave number for the special cases of complete association or dissociation and only quadratic interaction coefficients. 

Evaluate the translational, rotational, and vibrational contributions to the constant volume heat capacity Cv for 0.1 moles of the A127C135 molecule at 900°C and a pressure of 1 mBar. The molecular constants needed are given in the previous problem. 

It is frequently required to examine the combined performance of two or more processes in series, e.g. two systems or capacities, each described by a transfer function in the form of equations 7.19 or 7.26. Such multicapacity processes do not necessarily have to consist of more than one physical unit. Examples of the latter are a protected thermocouple junction where the time constant for heat transfer across the sheath material surrounding the junction is significant, or a distillation column in which each tray can be assumed to act as a separate capacity with respect to liquid flow and thermal energy. 

Most equations for heat capacities of substances are empirical. Heat capacity at constant pressure is generally expressed in terms of temperature with a power,series type formula ... 

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