Heat capacity below

None of the papers describing the studies below 20 K gives the measured values of they report only the values of y and 0d derived from the relation 

Since the heat capacity values from 4 to at least 20 K are very poorly defined, we shall discuss the relevant publications in some detail. The values of y, relating to the electronic contribution and 0d, relating to the contribution from lattice vibrations, from these studies are summarised in Table V-2. 

There are also two papers which report values of 0d from other sources. Reese et al. [1973REE/S1N] describe a neutron diffraction study (at room temperature) to give the phonon distribution (distribution of vibration frequencies), from which the variation of 0D as a function of temperature is derived, and Rosengren et al. [1975ROS/EBB] 

although the best values of 0d above 4 K are not clear, it does seem certain that the Debye temperature has a minimum just below 20 K, and that the use of a constant Debye temperature, particularly that derived from measurements in the range 2 -10 K is not appropriate. 

Bergman at the Glushko Thermocenter, Moscow has kindly provided us with the detailed heat capacity data from 1.374 to 298.15 K selected in [1982GLU/GUR], together with following snmmaiy of these selections  

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Product Specific heat capacity above freezing Highest freezing point (°C) Latent heat of freezing Specific heat capacity below freezing... 

Equation (4.2) requires that the total area above 0 Kelvin be obtained, but heat capacity measurements cannot be made to the absolute zero of temperature. The lowest practical limit is usually in the range from 5 K to 10 K, and heat capacity below this temperature must be obtained by extrapolation. In the limit of low temperatures, Cp for most substances follows the Debye low-temperature heat capacity relationship11 given by equation (4.4)... 

Molar heat capacities Cp "(crystalline reactant) can be determined down to about 10 K, and the Debye equation that applies at very low temperatures can be used to estimate heat capacities below 10 K. The Debye equation is Cpm ° = kT Heat of combustion measurements can be used to obtain AfH ° (298.15 K) of the crystalline substance at 298.15 K, and the heat of solution makes it possible to calculate AfH ° (aq soln,298.15 K). When third law measurements have been made, the standard Gibbs energy of formation of the substance in dilute aqueous solution can be calculated using... 

A glass transition is assumed at 800 K. The heat capacity below 800 K is obtained from the heat capacity of the crystal. Above 800 K the heat capacity is adopted as 50.0 cal k" mol", based on the enthalpy measurements In the range 1169-1250 K reported by Janz et al. (1 ). The entropy is calculated in a manner analogous to that used for the enthalpy of formation. 

The high temperature enthalpy of calcium fluoride has been reported by Naylor (9) to 1789 K, Krestnikov and Karetnikov (10) to 1273 K, and Lyashenko (11) to 1490 K. All the data are in approximate agreement and the more extensive results of Naylor on a very pure sample are adopted. The heat capacities, below 1424 K, were derived from a polynomial fit of the enthalpy data of the form H = aT + bT + c + d/T + e/T. ... 

K). The heat capacities below 51 K were obtained from Stout ( ). In that paper, the C values, 15-58 K, were plotted... 

The heat capacities of NaP(t) are derived from the enthalpy data, 1287.6 - 1746.5 K, determined by O Brien and Kelley (1 ). A glass transition temperature is assumed at 900 K. The heat capacities below 900 K are taken from those for NaF(cr). The C values above 1746.5 K are obtained by extrapolation. The entropy is calculated in a manner analogous to that used for the enthalpy formation. 

The adopted heat capacities for NaOH(t) in the temperature range 596 to 1000 K are from the enthalpy measurements of Douglas and Dever ( ). The heat capacities below the melting point and above 1000 K are extrapolated from the experimental heat capacity... 

The heat capacity is estimated by comparison with those for Na SiOg(t), Na O(t) and Li OCt). A glass transition is assumed at 1000 K i.e., the heat capacities below 1000 K are taken to be the same as those for LigSiOgCcr). 

The uncertainties in the equation are 0.05% in the saturated-liquid density between 280 and 335 K and 6.2% in density in the liquid phase below 430 K and 10 MPa. The uncertainty increases to 0.3% up to 100 MPa and 0.5% up to 800 MPa. In the vapor phase and at supercritical state points, the uncertainty in density is 1%, whereas in the liquid phase between 430 K and the critical point it is 0.5% in density. Other uncertainties are 0.2% in vapor pressure between 300 and 430 K, 0.5% in vapor pressure at higher temperatures, 2% in heat capacities below 550 K, 5% at higher temperatures, and 1% in the liquid-phase speed of sound below 430 K. The estimated uncertainty in viscosity is 1.0% along the saturated-liquid line, 5% elsewhere. Uncertainty in thermal conductivity is 3%, except in the supercritical region and dilute gas which have an uncertainty of 5%. 

Subsequently, Huntelaar et al. [95HUN/COR] measured the heat capacity and related thermochemical properties of SrZrSi207(s). The heat capacity was measured between 10 and 320 K using adiabatic calorimetry whereas enthalpy increments relative to 298.15 K were determined between 400 and 850 K using drop calorimetry. After correction for a zirconia impurity and estimation of the heat capacity below 13 K using the function A.T (where A is 0.000251 J-moP ), the heat capacity and entropy at... 

From another common point of view heat capacity is only a name for the expression on the left in line (24.13), regardless of whether or not it describes in reality a temperature-related quantity. We will return to the subject in a somewhat different context in the subsection Heat capacities below. 

After the usual assumption of Ce+Cl= Cp(a-La), Lounasmaa and Sundstrom (1967) found Cm T over the region 2.5 to 10 K, a temperature dependence which does not correspond to any known theory. If, on the other hand, analysis is made of the heat capacity below 2.3 K, where Cm is expected to be relatively small and to follow the long wave length limit behaviour for an antiferromagnet, one finds y = 12.4 mJ/mole-K and the leading nuclear term... 

The higher y values obtained during the 1960 s in investigations of heat capacities below 4.2 K are clearly due to impurity effects, which are fortunately negligible at the very low temperatures where Cn is investigated, but which are enough to disturb any low temperature extrapolation of the type discussed in section 1.5 above (Morrison and Newsham 1968). 

The kinetic theory defines Tg as the temperature at which the relaxation time for the segmental motions in the main polymer chain is of the same order of magnitude as the time scale of the experiment. The kinetic theory is concerned with the rate of approach to equilibrium of the system, taking the respective motions of the holes and molecules into account. The kinetic theory provides quantitative information about the heat capacities below and above the glass transition temperature and explains the 6 to 7°C shift in the glass transition per decade of time scale of the experiment. 

Solution Since the molar heat capacity below -258°C K) is followed by... 

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