Low-temperature heat capacity

Brunauer and co-workers [129, 130] found values of of 1310, 1180, and 386 ergs/cm for CaO, Ca(OH)2 and tobermorite (a calcium silicate hydrate). Jura and Garland [131] reported a value of 1040 ergs/cm for magnesium oxide. Patterson and coworkers [132] used fractionated sodium chloride particles prepared by a volatilization method to find that the surface contribution to the low-temperature heat capacity varied approximately in proportion to the area determined by gas adsorption. Questions of equilibrium arise in these and adsorption studies on finely divided surfaces as discussed in Section X-3. 

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)... 

For Cy/T to approach zero as T approaches zero, CV must go to zero at a rate at least proportional to T. Earlier, we summarized the temperature dependence of Cy on T for different substances and showed that this is true. For example, most solids follow the Debye low-temperature heat capacity equation of low T for which... 

Thus Cp m and CVm differ little from one another at low temperatures. The Debye low-temperature heat capacity equation (and other low-temperature relationships) we have summarized calculates Cp.m, as well as CV. m, without significant error. 

Thus, values for C°p m T, S°m T, (H°m T - H°m 0) and (G°mT H°m0) can be obtained as a function of temperature and tabulated. Figure 4.16 summarizes values for these four quantities as a function of temperature for glucose, obtained from the low-temperature heat capacity data described earlier. Note that the enthalpy and Gibbs free energy functions are graphed as (// , T - H°m 0)/T and (G T — H q)/T. This allows all four functions to be plotted on the same scale. Figure 4.16 demonstrates the almost linear nature of the (G°m T H°m 0)/T function. This linearity allows one to easily interpolate between tabulated values of this function to obtain the value at the temperature of choice. 

E4.1 Show that at very low temperatures where the Debye low temperature heat capacity equation applies that the entropy is one third of the heat capacity. 

Graph the above data in the form Cp,m/T against T2 to test the validity of the Debye low-temperature heat capacity relationship [equation (4.4)] and find a value for the constant in the equation, (b) The heat capacity study also revealed that quinoline undergoes equilibrium phase transitions, with enthalpies as follows ... 

G. Brodale and W. F. Giauque, "The Heat of Hydration of Sodium Sulfate. Low Temperature Heat Capacity and Entropy of Sodium Sulfate Decahydrate", J. Am. Chem. Soc., 80, 2042-2044 (1958). 

Figure 10.14 Graph showing the limiting behavior at low temperatures of the heat capacity of (a), krypton, a nonconductor, and (b). copper, a conductor. The straight line in (a) follows the prediction of the Debye low-temperature heat capacity equation. In (b), the heat capacity of the conduction electrons displaces the Debye straight line so that it does not go to zero at 0 K.

Data obtained from L. Finegold and N. E. Phillips. Low-Temperature Heat Capacities of Solid Argon and Krypton". Plus. Rev.. 111. 1383-1391 (1969). 

The obtained value AHf(PUF3,c) = - 1585.7 + 2.9 kJ.mol-1 cannot be considered as entirely satisfactory as the reliability of the adopted value for the enthalpy of dehydration is not demonstrated. The only experimentally known enthalpies of formation for the actinide trifluorides are AHf(PuF3,c) and AHf(UF3,c) accuracy is therefore essential if these two data are used to estimate the enthalpies of formation of the other trifluorides. The low temperature heat capacity measurements of Osborne et al. (22) using 242PuF3(c) yield S°(PuF3,c 126.11+0.38 J.K"l.mol"i. 

C] 7.38 At low temperatures, heat capacities are proportional to T5. Show that, near 1=0, the entropy of a substance is equal to one-third of its heat capacity at the same temperature. 

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. 

Low temperature heat capacity experiments with huckminsterfullerene, Ceo, indicate that this substance does not follow the Dehye equation, but requires a more complex model. See W. P. Beyermann, et al. Phys. Rev. Lett. 68, 2046 (1992). 

Getting, F.L. Low-temperature heat capacity and related thermodynamic functions of propylene oxide, / Chem. Phys., 41(1) 149-153, 1964. 

Haselton H. T. and Westrum E. F. Jr. (1980). Low-temperature heat capacities of synthetic pyrope, grossular, and pyropegogrossular4Q. Geochim. Cosmochim. Acta, 44 701-709. 

Unfortunately, there axe at this time no low temperature heat capacity data on polymers of known crystallinity so that absolute entropies at 25° C can be calculated. This gap in our knowledge of polymers represents a scientific vacuum that will rapidly be filled1. 

NdRu4Sbi2 is metallic and undergoes some type of magnetic transition near 1.3 K. The magnetic susceptibility follows a Curie-Weiss law above 50 K with an effective moment of 3.45/u.b and a Weiss temperature of -28 K. Crystal fields likely effect the susceptibility and magnetic interactions for temperatures below 50 K. Low temperature heat capacity data confirm the bulk nature of the magnetic transition (Takeda and Ishikawa, 2000b). 

EuRu4Sbn is metallic and becomes ferromagnetic for temperatures below 3.3 K (Takeda and Ishikawa, 2000b). The low temperature saturation moment is about 6.2/xb, 89% of the Eu+2 value. Low temperature heat capacity measurements indicate that the magnetic entropy removed due to magnetic order is also only about 90% of its expected value (Rln8). It is likely that the lanthanide site is not completely filled in this compound although mixed valence behavior can not be ruled out with the available data. 

Low-temperature heat capacity and standard entropy of the solid trihalides... 

The standard molar entropies at 298.15 K derived from the low-temperature heat capacity measurements are summarized in table 3. Similar to the heat capacity, the entropy can be described as the sum of the lattice and excess components (Konings, 2001, 2002) ... 

Fig. 14. The reduced enthalpy increment of LaF3 (in J-K 1-mol 1) o, Henderson (1970) Lyon et al. (1978) , value at 298.15 K derived from the low-temperature heat capacity measurements of Lyon et al. (1978).

Valentine, R.H., Brodale, G.E., Giauque, W.F. (1962) Trifluoromethane Entropy, low temperature heat capacity, heats of fusion and vaporization, and vapor pressure. J. Phys. Chem. 66, 392-395. 

L.J. Sundstrom, Low temperature heat capacity of the rare earth metals 379... 

Scott, D.W., Finke, H.L., Gross, M.E., Guthrie, G.B., Huffman, H.M. (1950) 2,3-Dithiabutane low temperature heat capacity, heat of fusion, heat of vaporization, vapor pressure, entropy and thermodynamic functions. J. Am. Chem. Soc. 72, 2424—2430. 

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