Heat capacity classical calculation

The Geothermal Response Test as developed by us and others has proven important to obtain accurate information on ground thermal properties for Borehole Heat Exchanger design. In addition to the classical line source approach used for the analysis of the response data, parameter estimation techniques employing a numerical model to calculate the temperature response of the borehole have been developed. The main use of these models has been to obtain estimates in the case of non-constant heat flux. Also, the parameter estimation approach allows the inclusion of additional parameters such as heat capacity or shank spacing, to be estimated as well. 

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. 

Giauque, whose name has already been mentioned in connection with the discovery of the oxygen isotopes, calculated Third Law entropies with the use of the low temperature heat capacities that he measured he also applied statistical mechanics to calculate entropies for comparison with Third Law entropies. Very soon after the discovery of deuterium Urey made statistical mechanical calculations of isotope effects on equilibrium constants, in principle quite similar to the calculations described in Chapter IV. J. Kirkwood s development showing that quantum mechanical statistical mechanics goes over into classical statistical mechanics in the limit of high temperature dates to the 1930s. Kirkwood also developed the quantum corrections to the classical mechanical approximation. 

From classic thermodynamics alone, it is impossible to predict numeric values for heat capacities these quantities are determined experimentally from calorimetric measurements. With the aid of statistical thermodynamics, however, it is possible to calculate heat capacities from spectroscopic data instead of from direct calorimetric measurements. Even with spectroscopic information, however, it is convenient to replace the complex statistical thermodynamic equations that describe the dependence of heat capacity on temperature with empirical equations of simple form [15]. Many expressions have been used for the molar heat capacity, and they have been discussed in detail by Frenkel et al. [4]. We will use the expression... 

For a given uptake and temperature T, dSs/dT = Cp where Cp is the differential molar heat capacity of sorbed fluid. This expression can be approximated by Tm ASs/AT = Cp where Tm is the mean temperature corresponding with the interval AT over which ASs is the entropy change, and where Cp refers to the temperature Tm. For classical oscillators Cp should be 24.9 J/mole/deg, and thus it is interesting to compare Cp calculated as above with this value. A5S/AT did not vary much with amount sorbed, so that Cp found for one uptake is typical. Several values of Cp are given below. All are near but a little below the classical oscillator value. 

The classical heat capacity values for 2T + IV, IT + 2V, and 3V are, respectively, 16.5, 20.7, and 24.9 J/mole/°K, and may be compared with the values given earlier for the three H-zeolites, for which the calculations of AS6 suggest localization of Kr (i.e., 3V). The values of the heat capacity previously discussed lie between those expected for IT + 2V and 3V, respectively, in reasonable agreement with 3V from the entropy. Similar magnitudes for heat capacity values have been observed for Kr and Xe in other zeolites 24) ... 

Example 2. Calculate the classical constant-pressure heat capacity of CH4. Compare this with the literature value of 35.46 J/mol K at 15°C. 

Both the classical and statistical equations [Eqs. (5.22) and (5.23)] yield absolute values of entropy. Equation (5.23) is known as the Boltzmann equation and, with Eq. (5.20) and quantum statistics, has been used for calculation of entropies in the ideal-gas state for many chemical species. Good agreement between these calculations and those based on calorimetric data provides some of the most impressive evidence for the validity of statistical mechanics and quantum theory. In some instances results based on Eq. (5.23) are considered more reliable because of uncertainties in heat-capacity data or about the crystallinity of the substance near absolute zero. Absolute entropies provide much of the data base for calculation of the equilibrium conversions of chemical reactions, as discussed in Chap. 15. 

ISe. Classical Calculation of Heat Capacities.— For a diatomic molecule two types of rotation are possible, as seen above, contributing RT per mole to the energy. Since there are two atoms in the molecule, i.e., n is 2, there is only one mode of vibration, and the vibrational energy should be RT per mole. If the diatomic molecules rotate, but the atoms do not vibrate, the total energy content E will be the sum of the translational and rotational energies, i.e., RT + RT = RT, per mole hence,... 

At 300 K, the translational and rotational contributions to the heat capacity will be classical, i.e., f and R, respectively, making a total of or 4.97 cal. deg. mole . If the vibrational contribution 1.12 is added, the total is 6.09 cal. deg. mole . (The experimental value which is not very accurate, is close to this result some difference is to be expected, in any case, because the calculations given above are based on ideal behavior of the gas. The necessary corrections can be made by means of a suitable equation of state, 21d.)... 

Thermal energy is mainly taken up as the vibrations of the atoms in the solid. Classically, the calculation of the heat capacity of a solid was made by assuming that each atom vibrated quite independently of the others. The heat capacity was then the sum of all of the identical atomic contributions, and independent of temperature. The result was... 

In equilibrium thermodynamics model A and in model B not far from equilibrium (and with no memory to temperature) the entropy may be calculated up to a constant. Namely, in both cases S = S(V, T) (2.6)2, (2.25) and we can use the equilibrium processes (2.28) in B or arbitrary processes in A for classical calculation of entropy change by integration of dS/dT or dS/dV expressible by Gibbs equations (2.18), (2.19), (2.38) through measurable heat capacity dU/dT or state Eqs.(2.6>, (2.33) (with equilibrium pressure P° in model B). This seems to accord with such a property as in (1.11), (1.40) in Sects. 1.3, 1.4. As we noted above, here the Gibbs equations used were proved to be valid not only in classical equilibrium thermodynamics (2.18), (2.19) but also in the nonequilibrium model B (2.38) and this expresses the local equilibrium hypothesis in model B (it will be proved also in nonuniform models in Chaps.3 (Sect. 3.6), 4, while in classical theories of irreversible processes [12, 16] it must be taken as a postulate). 

Senyshyn et al. (2005b) calculated the above-mentioned properties and determined the thermal expansion coefficient of rare earth gallates using a semi-classical approach. Ideal (X-ray) density, Griineisen parameter, isohoric heat capacity Cy, bulk and shear moduli, and thermal expansion coefficient were calculated for RGaOa (R = La-Gd) at 300 K are listed in Table 47. 

At the times when the classical calorimeters were built, no computers existed and all evaluation was done by hand. Therefore, there was a need for simple formulas to calculate the quantities of interest from the measured curves. The construction of the calorimeters was such to give a signal strictly proportional to the heat flow rate into the sample itself with a calibration factor almost not influenced by the heat transfer to the sample and its heat capacity. The price to be paid for this comfort was a rather low sensitivity of the calorimeter with a need for large samples and large time constants in the range from some seconds up to many minutes in the case of very sensitive microcalorimeters (see Section 7.9.2). 

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