Temperature dependence of heat capacity

The temperature dependency of heat capacity can usually be described by a polynomial expression, e.g.,... 

To use equation 2.10 correctly, we need to know how the heat capacities vary in the experimental temperature range. However, these data are not always available. A perusal of the chemical literature (see appendix B) will show that information on the temperature dependence of heat capacities is much more abundant for gases than for liquids and solids and can be easily obtained from statistical mechanics calculations or from empirical methods [11]. For substances in condensed states, the lack of experimental values, even at a single temperature, is common. In such cases, either laboratory measurements, using techniques such as differential scanning calorimetry (chapter 12) or empirical estimates may be required. 

Fig. 4.10 Temperature dependence of heat capacity and different contributions to heat capacity...

Fig. 4.11 Temperature dependence of heat capacity for fullerene C60 and hydrofullerenes CH2n...

As discussed in Section n, for monomeric organic and inorganic and glassforming liquids, the temperature dependence of heat capacity change is often described by hyperbolic temperature dependence [17,43] ... 

Assuming the validity of Adam-Gibbs equation for relaxation dynamics and the hyperbolic temperature dependence of heat capacity, the strength parameter is found to be inversely proportional to the change in heat capacity [see Eq. (2.10)] at the glass transition temperature [48,105]. 

Solubility in mole fractions. AG in kJ mol-1 AS in J K l mol-1 rs in °C. The properties of the last three substances in their hypothetical liquid state under standard conditions were estimated by appropriate conversion from gas to dissolved state thermodynamic properties. Temperature dependence of heat capacity change described by exponential scaling (see p. 217). 

Remark 7.6. The analysis framework we presented is also applicable if an inert component is used to increase the heat capacity of the reaction mixture. In this case, the model (7.2f) would be augmented by the equations corresponding to the model of the separation unit. However, the stoichiometric matrix S and reaction rates r would remain unchanged, since the inert component does not partake in any reaction. Furthermore, the analysis can be applied if more complex correlations are used for the physical parameters of the system (e.g., temperature dependence of heat capacities and densities), as long as the basic assumptions (7.27), (7.29), and (7.30) apply. 

Abstract. The statistical calculation of the temperature dependence of heat capacity of ordering two-component fullerite has been fulfilled in the approximation of pair interaction between fullerenes by the method of average energies in the model of spherically symmetrical stiff balls. 

Experimental investigation of the temperature dependence of heat capacity of solid-phase fullerite showed the availability of abrupt peak of capacity in the region of temperature To=249-260 K [1-5] (Fig. 1). 

Figure 1. Experimental plots of the temperature dependence of heat capacity for fiillerite according to [21 (a) and [81 (b). Tn is the temperature of scl <-> feel structure phase transition.

As an example, we will consider the molecular dynamical behavior of egg white lysozyme. The temperature dependence of mobility of fluorescence, spin and Mossbauer labels attached to lysozyme was found to be similar to other investigated proteins the monotonic increase typical for rigid polymers in dry states and in samples with water content (wt) was less than the critical value (wtcr) and drastically burst when wt > wtcr at T > 200 K took place (Frolov et al., 1978 Likhtenshtein, 1979). At similar conditions, experiments on the temperature dependence of heat capacity indicated only a monotonic steady increase for rigid organic material. Recently, in the fully dried lysozyme crystal, similar monotonic behavior of heat capacity was observed in temperatures between 8 and 30°C. At D20 content more than 24 wt %, a slight deviation from the monotony was observed at temperatures above approximately 185 K, which most probably is due to the eutectic melting of NaCl/2H20 present in the samples to prevent water crystallization (Miyazaki et al., 2000). 

Entropy decreases with decreasing temperature due to an increase of the local order following the hydrogen bond formation. Similar arguments to those developed above explain the minimum of the temperature dependence of heat capacity. 

Twin models. Figure 2 illustrates the temperature dependence of heat capacity for the two twin models and Table I gives the corresponding numerical data. Figure 2 typifies the Cp(T) curve of conventional glasses with a well defined enthalpy relaxation peak and smooth solid and liquid lines. From the extrapolated solid and liquid lines we can measure the heat capacity jump at Tg, by equation 1. Within our experimental range, the data fit a straight line with slopes (B) as listed in Table n. 

Fig. 6.8 Temperature dependencies of heat capacity (a), and inverse value of nematic-like structural susceptibility (b) in the vicinity of N-I phase transition...

To interpret the results of experimental investigations within the theory of corresponding states, one needs to determine experimentally the temperature dependence of heat capacity and density, the thermal coefficients of expansion and pressure of the individual component.s, and such properties of mixtures as the chemical potentials and enthalpy of mixing (Cowie et al., 1971), i.e. measuring a number of quantities on a specialized, sophisticated, and expyensivc equipment is required (Blanks, 1977). 

For heats (and therefore temperature changes) that are not too large, the temperature change is directly proportional to the exchanged heat. The proportionality constant is the heat capacity of the calorimeter substance (previously referred to as the water value"). However, if the temperature change exceeds a few Kelvin, the temperature dependence of heat capacity stands in the way of a linear relationship, and a knowledge of the temperature function of the particular heat capacity in question is required in order to determine heat on the basis of a measured temperature difference. 

Fig.2. Schematic representation of temperature dependence of heat capacity divided by P (,ClP) for amorphous and crystalline materials at low temperature regions...

Data in [1.39] indicate that the temperature dependence of heat capacity, enthalpy, and entropy are somewhat steeper. The values of C , — // ) and S%... 

The techniques used in data processing and trends revealed thereby are of key importance for determining approaches to solving the problem of analytically describing the temperature dependence of heat capacity over a fairly broad temperature interval. We therefore describe the essence of these studies. 

The thermal expansion of crystals, that is, the transition to the quasiharmonic approximation in calculations of the temperature dependence of heat capacity (Demlow et al., 1998), was included using Lord-Ahlberg-Andrews equation (Lord et al., 1937)... 

The temperature dependence of heat capacity over the temperature range T = 0 to or Ttr was written as the sum of three contributions (see Eq. (11)). Two of these (Ciat(T)and Cd, cor(T)) were described analytically... 

The heat capacities of LUCI3 measured experimentally and smoothed by a cubic spline function were reported by Tolmach et al. (1987c). We calculated 6r>, 6 1, 0e2, 0e3, and a on the basis of all these experimental data, except three values in the temperature range 254.24-262.41 K. The characteristic parameters listed in Table 22 were obtained at a comparatively low = 0.05497 value. Comparison with the corresponding data on hexagonal lanthanide trichlorides shows that an increase in the molar volume causes an insignificant decrease in a, noticeable increases in 0 2 and 0E3, and substantial decreases in 0 and 0ei- Such modifications of these parameters change the temperature dependence of heat capacity in a quite definite way. Namely, because of the low 0 and 6ei values, heat... 

FIGURE 16 Temperature dependence of heat capacity of ( ) LaCls (Sommers and Westrum, 1976), (A) GdCls (Sommers and Westrum, 1977), and ( ) LuCls (Gorbunov et aL, 1986). 

Our analytic description of the temperature dependence of heat capacities over the temperature range 298.15-Tm (in certain instances, over the temperature ranges 298.15-Ttr and Tf -T separately) and the heat capacities in the hquid state at the temperatures of melting as well as of the enthalpies of phase transitions (including the temperatures of these transitions) can be used to calculate the thermodynamic functions of the compounds imder consideration. The calculations were performed according to the procedure described in the previous section, and the resulting functions are listed in Table A6 in the form of the/i coefficients (see Appendix). 

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