The heat capacity of crystals

Heat capacity was introduced in Section 3.4.3 as the heat energy capable of heating a body by 1 grad. For an ideal gas in molecular kinetic theory the mole heat capacity appeared to be equal to 

Apply these representations to a solid. For this purpose we should first analyze the character of possible movements of the molecules in it Clearly, translational motion is excluded in this case. The rotation of molecules in a aystal is basically possible. For example, in crystal NH4CI a group NH4+ at different tanperatures can make rotations around axes of different symmetry and sometimes exhibit a free rotation. However, the contribution of rotation to the thermal capacity of solids is appreciably less than oscillatory degrees of freedom. As a result only oscillations of atoms basically define the crystal thermal capacity. Furthermore, the three-dimensional character of a crystal should be taken into account. 

The molar internal energy of a crystal U will then be given as a product 

The derivative from internal energy on temperature gives a mole thermal crystal heat capacity C. As the volume of a crystal does not vary appreciably with temperature, we exclude index V. We shall then obtain 

Denoting the ratio (TiojIk) by 6, the last expression can be rewritten as 

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Since co2 =K/m, the mean potential and kinetic energy terms are equal and the total energy of the linear oscillator is twice its mean kinetic energy. Since there are three oscillators per atom, for a monoatomic crystal U m =3RT and Cy m =3R = 2494 J K-1 mol-1. This first useful model for the heat capacity of crystals (solids), proposed by Dulong and Petit in 1819, states that the molar heat capacity has a universal value for all chemical elements independent of the atomic mass and crystal structure and furthermore independent of temperature. Dulong-Petit s law works well at high temperatures, but fails at lower temperatures where the heat capacity decreases and approaches zero at 0 K. More thorough models are thus needed for the lattice heat capacity of crystals. 

The last term is evaluated from experimental measurements of heat capacities and heats of transition (Sec. 6-1). The extrapolated term is usually evaluated with the aid of the Debye equation for the heat capacity of crystals. The Debye equation for the heat capacity of crystals at low temperature is... 

Debye was responsible for theoretical treatments of a variety of subjects, including molecular dipole moments (for which the de-bye is a non-SI unit). X-ray diffraction and scattering, and light scattering. His theories relevant to thermodynamics include the temperature dependence of the heat capacity of crystals at a low temperature (Debye crystal theory), adiabatic demagnetization, and the Debye-Huckel theory of electrolyte solutions. In an interview in 1962, Debye said that he... 

Scientists attempted to use statistical thermodynamics to understand the heat capacities of crystals at low temperatures. Given the success of statistical... 

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... 

How do we determine the Einstein temperature Be without knowing the characteristic vibrational frequency of the atoms in the crystal Typically, experimental data is fitted to the mathematical expression in equation 18.64 and a value of the Einstein temperature is used to allow for the best possible fit to experimental results. For example, a plot of experimental measurements ofthe heat capacity versus Tdividedby E (which isproportional to T, whereas 0g/r is inversely proportional to T and less easy to graph as T —> 0 K) is shown in Figure 18.4. Notice that there is reasonable agreement between experiment and theory, suggesting that Einstein s statistical thermodynamic basis of the heat capacity of crystals has merit. Table 18.6 lists a few experimentally determined Einstein temperatures for crystals. 

This expression shows that the low-temperature heat capacity varies with the cube of the absolute temperature. This is what is seen experimentally (remember that a major failing of the Einstein treatment was that it didn t predict the proper low-temperature behavior of Cy), so the Debye treatment of the heat capacity of crystals is considered more successful. Once again, because absolute temperature and dy, always appear together as a ratio, Debye s model of crystals implies a law of corresponding states. A plot of the heat capacity versus TIdo should (and does) look virtually identical for all materials. 

The law of Dulong and Petit states that the CyOf materials approaches 3A//c(which equals 3R) at high temperatures. Can you show that both Einstein s and Debye s expressions for the heat capacity of crystals agree with this generalization at high temperatures ... 

Let us consider as an example the weU-known Debye theory of the heat capacity of crystals. The internal energy of a bulk crystal minus zero energy is given in this theory by an equation... 

The variation of Cp for crystalline thiazole between 145 and 175°K reveals a marked inflection that has been attributed to a gain in molecular freedom within the crystal lattice. The heat capacity of the liquid phase varies nearly linearly with temperature to 310°K, at which temperature it rises more rapidly. This thermal behavior, which is not uncommon for nitrogen compounds, has been attributed to weak intermolecular association. The remarkable agreement of the third-law ideal-gas entropy at... 

A number of other thermodynamic properties of adamantane and diamantane in different phases are reported by Kabo et al. [5]. They include (1) standard molar thermodynamic functions for adamantane in the ideal gas state as calculated by statistical thermodynamics methods and (2) temperature dependence of the heat capacities of adamantane in the condensed state between 340 and 600 K as measured by a scanning calorimeter and reported here in Fig. 8. According to this figure, liquid adamantane converts to a solid plastic with simple cubic crystal structure upon freezing. After further cooling it moves into another solid state, an fee crystalline phase. 

The heat capacity of the crystal was measured in the 0.06-0.28 K temperature range using a relaxation method (see Section 12.2.3). A small power supplied by the heater rose the temperature Tc of the crystal above TB by a few millikelvin. When the thermal equilibrium was reached, the heating power was switched off, and the exponential decay of the crystal temperature was recorded by means of an LR700 bridge (see Section 10.8), at a rate of 5 sample/s. 

Figure 15.8 shows the thermal scheme of one detector there are six lumped elements with three thermal nodes at Tu T2, r3, i.e. the temperatures of the electrons of Ge sensor, Te02 absorber and PTFE crystal supports respectively. C), C2 and C3 are the heat capacity of absorber, PTFE and NTD Ge sensor respectively. The resistors Rx and R2 take into account the contact resistances at the surfaces of PTFE supports and R3 represents the series contribution of contact and the electron-phonon decoupling resistances in the Ge thermistor (see Section 15.2.1.3). 

Because the heat capacities of crystalline solids at various T are related to the vibrational modes of the constituent atoms (cf section 3.1), they may be expected to show a functional relationship with the coordination states of the various atoms in the crystal lattice. It was this kind of reasoning that led Robinson and... 

OB that the heat capacity of the vitreous state Cpf is very similar to the heat capacity of the crystal Cpf and that both, T being equal, are notably lower than the heat capacity of the molten state (C y). 

A recent review of the experimental situation has been given by Honig(1985). It is pointed out that the electrical properties, particularly near to the transition, are very sensitive to purity and specimen preparation, and that much of the extensive experimental work is therefore open to doubt. None the less, the broad features of the behaviour of this material are clear. The history of the so-called Verwey transition in this material goes back to 1926, when Parks and Kelly (1926) detected an anomalous peak near 120 K in the heat capacity of a natural crystal of magnetite. The first detailed investigations were those of Verwey and co-workers (Verwey 1939, Verwey and Haayman 1941, Verwey et al. 1947), who showed that there was a near discontinuity in the conductivity at about 160K. The conductivity as measured by Miles et al. (1957) is shown in Fig. 8.1. 

Stockmayer and Hecht (1953) have developed an additional mathematical theory of the heat capacity of chain polymeric crystals. Their theory is based on the concept of strong valence forces between atoms in the polymeric chain and of weak (non-zero) coupling between chains. This model corresponds to that also proposed by Tarassov (1952). There are not many low temperature specific heat data on polymers, but the Stockmayer-Hecht theory can be tested by calculating the Tm constant... 

The intramolecular contributions to the heat capacity of the C60H2n crystals are found from the scaled B3LYP/6-31G vibrational frequencies of the molecules. 

We must also consider the conditions that are implied in the extrapolation from the lowest experimental temperature to 0 K. The Debye theory of the heat capacity of solids is concerned only with the linear vibrations of molecules about the crystal lattice sites. The integration from the lowest experimental temperature to 0 K then determines the decrease in the value of the entropy function resulting from the decrease in the distribution of the molecules among the quantum states associated solely with these vibrations. Therefore, if all of the molecules are not in the same quantum state at the lowest experimental temperature, excluding the lattice vibrations, the state of the system, figuratively obtained on extrapolating to 0 K, will not be one for which the value of the entropy function is zero. 

The heat capacity of the trivalent lanthanide trihalides consists of a lattice component, arising mainly from the vibrations of the ions in the crystal, and an excess component (Westrum Jr. and Grpnvold, 1962 Westrum Jr., 1970 Flotow and Tetenbaum, 1981 Westrum Jr., 1983) ... 

The data shown in fig. 10 are not the values reported by Gorbunov et al. (1986) and Tolmach et al. (1987, 1990a, 1990b, 1990c), because they did not extrapolate their measurements to 0 K in all cases. To derive S° (298.15 K) we have assumed that the heat capacity of L11CI3 represents the lattice component, and Am at the lower temperature limit is derived from the results for this compound. The excess contribution at this temperature is calculated from the crystal field energies (see table 5) derived from spectroscopic studies of the ions in transparent host crystals (Dieke et al., 1968 Morrison and Leavitt, 1982 ... 

Clearly one obtains the best performance for a given time constant with a detector that has the lowest possible heat capacity. The heat capacity of a crystal varies like C oc (T/0 )3, where On is the Debye temperature. Diamond has the highest Debye temperature of any crystal, so FIRAS used an 8 mm diameter, 25 fim thick disk of diamond as a bolometer (Mather et al., 1993). Diamond is transparent, so a very thin layer of gold was applied to give a surface resistance close to the 377 ohms/square impedance of free space. On the back side of the diamond layer an impedance of 267 ohms/square gives a broadband absorbtion. Chromium was alloyed with the gold to stabilize the layer. The temperature of the bolometer was measured with a small silicon resistance thermometer. Running at T = 1.6 K, the FIRAS bolometers achieved an optical NEP of about 10 14 W/y/IIz. 

Figure 3.21 Effect of total heat capacity differences between sample and reference on baseline position in a DTA/DSC trace. See text for discussion. In addition, this sketch of glass crystallization shows the baseline shifted in the exothermic direction after crystallization In the same region of temperature, the heat capacity of a crystal is less than the corresponding glass above Tg (see sections 3.7.2 and 7.6).

Debye Theory of the Heat Capacity of Solids. Debye assumed that a cubic crystal of side L and volume V = Lr can be taken as a vacuum (German Hohlraum) that supports a set of standing waves, each with form... 

Differential scanning calorimetry (DSC) can be used to determine experimentally the glass transition temperature. The glass transition process is illustrated in Fig. 1.5b for a glassy polymer which does not crystallize and is being slowly heated from a temperature below Tg. Here, the drop which is marked Tg at its midpoint, represents the increase in energy which is supplied to the sample to maintain it at the same temperature as the reference material. This is necessary due to the relatively rapid increase in the heat capacity of the sample as its temperature is increases pass Tg. The addition of heat energy corresponds to the endothermal direction. 

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