Theory of heat capacity

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 

For temperatures well above the Debye temperature, the value of Cy is 3R, where R is the gas constant, 8.3145 JK moP Thus, all solids would be expected to have a high-temperature heat capacity of about 25JK mol , equal to the classical value. As the value of the Debye temperature is below room temperature for many solids, the room-temperature heat capacity can be approximated to 3R (Table 15.1). 

At temperatures well below the Debye temperature, the value of Cy is given by  

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The first law of thermodynamics leads to a broad array of physical and chemical consequences. In the following Sections 3.6.1-3.6.8, we describe the formal theory of heat capacity and the enthalpy function, the measurements of heating effects that clarified the energy and enthalpy changes in real and ideal gases under isothermal or adiabatic conditions, and the general first-law principles that underlie the theory and practice of thermochemistry, the measurement of heat effects in chemical reactions. 

One example of an experimental problem that can usefully be solved by adjusting theory to yield linear equations is the example of the determination of heat capacities, Cp at low temperatures, especially temperatures where experimental values are simply inaccessible. Because Cp cannot be measured experimentally down to absolute zero then an appropriate extrapolation needs to be made (see Frame 16). This latter possibility arises because the Debye theory of heat capacities at low temperatures predicts that as T —> 0 ... 

This difficulty does not arise in solid, where we have an energy spectrum corresponding to its 3N - 6 modes of vibration. If this spectrum is discrete, then the addition of the weight functions for each of these 3N - 6 modes of vibration gives the entropy of a solid (Einstein s theory of heat capacity). 

Such a complicated character of F(6/T) is clarified in the theory of heat capacity for linear polymers, established by Dole [94] and described in detail by Wunderlich and, Baur [95]. It is based on a concept advanced by Tarasov [96] and developed by Lifshitz [97] for chain solids. 

Heat capacity is the basic quantity derived from calorimetric measurements, as presented in Sects. 4.2-4A, and is used in the description of thermodynamics, as shown in Sects. 2.1 and 2.2. For a full description of a system, heat capacity information is combined with heats of transition, reaction, etc. hi the present section the measurement and the theory of heat capacity are discussed, leading to a description of the Advanced THermal Analysis System, ATHAS. This system was developed over the last 30 years to increase the precision of thermal analysis of linear macromolecules. 

For a description of the theory of heat capacities of liquid macromolecules, see Loufakis K, Wunderlich B (1988) J Phys Chem 92 4205 209 and Pyda M, Wunderlich B (1999) Macromolecules 32 2044-2050. 

The description of the theory of heat capacity and the application of heat capacity measurements have been given by Wunderlich and other researchers [2,3-17]. The most comprehensive and updated heat capacity data are collected in the ATHAS data bank (Advanced THermal Analysis) which has been developed over the last 25 years by Wunderlich (Chemistry Department, The University of Tetmessee), and coworkers. 

It follows from the introduction that the amplitude of oscillations of crystal atoms situated close to the surface is larger by than in the bulk. As a consequence, the Debye temperature of surface layers is lower by than that of the bulk material. Since the heat capacity is related to the values specified in accordance with the Debye quantum theory of heat capacity, the heat capacity of surface layers should be higher than the heat cap>acity of bulk materials. The amplitudes of atomic oscillations are higher in the liquid than in the solid phase, and the temperature dependence of the heat capacity of the liquid phase is in the majority of cases steeper compared with the solid phase. Extending this conclusion to the surface layer, we are led to suggest that the difference between the temp>erature dep>endences of the heat capacities of the surface layer in the liquid and solid state (that is, ACp(T)) increases compared with the bulk material. 

It is clear that one needs to know the heat capacities of a substance as a function of temperature and pressure in order to calculate the entropy and other thermodynamic quantities. A detailed understanding of the theory of heat capacities (which requires statistical mechanics) is beyond the scope of this book. Here we shall only give a brief outline of Peter Debye s theory for the heat capacities of solids, an approach that leads to an approximate general theory. The situation is more complex in liquids because there is neither complete molecular disorder, as in a gas, nor is there long-range order, as in a solid. 

FIGURE 18.3 Measurement of the heat capacity of crystals at very low temperatures shows a curve that looks like a y = kT curve. Any theory of heat capacities of crystals should predict this kind of... 

FIGURE18.4 The Einstein theory of heat capacity of crystals agrees reasonably well with experimental measurements. 

Metals provided additional problems for the classical theory of heat capacity. Metals are generally much better conductors of heat than nonmetals because most of the heat is carried by the free electrons. According to the classical theory, this electron gas should contribute an additional (3/2)R to the heat capacity. But the measured heat capacity of metals approached nearly the same 3R Dulong-Petit limit as the nonmetals. How can the electrons be a major contributor to the thermal conductivity and not provide significant additional heat capacity ... 

Debye s theory of heat capacities of monatomic crystals is built on transverse and longitudinal frequencies of oscillation in the whole crystal... 

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