Heat Capacities of Solids

The average energy in the canonical ensemble of the whole system is 

Besides, the canonical partition function [11] of the system of oscillators is [13] 

We will now attempt an analysis of Equation 1.21 for nmol of a metallic, ionic, or covalent crystal, with 1 ion per lattice site, that is, for an Avogadro number, NA, of ions at a high temperature. At these conditions, C/ Tu.op, and, consequently, 

Since the heat capacity at constant volume is defined as 

However, we need to know the behavior of solids at all temperatures. Einstein, in 1907, to deal with the problem, assumed that all the normal vibration modes have the same angular frequency co . As a result, Equation 1.21 will take the following form [12]  

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However, the possibility that might not go to zero could not be excluded before the development of the quantum theory of the heat capacity of solids. When Debye (1912) showed that, at sufficiently low... 

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 absolute value of the entropy of a compound is obtained directly by integration of the heat capacity from 0 K. The main contributions to the heat capacity and thus to the entropy are discussed in this chapter. Microscopic descriptions of the heat capacity of solids, liquids and gases range from simple classical approaches to complex lattice dynamical treatments. The relatively simple models that have been around for some time will be described in some detail. These models are, because of their simplicity, very useful for estimating heat capacities and for relating the heat capacity to the physical and chemical... 

Sections 3.1 and 3.2 describe heat capacity and explain how it may be determined at constant pressure Cp or at constant volume Cy. Most chemists need to make calculations with Cp, which represents the amount of energy (in the form of heat) that can be stored within a substance - the measurement having been performed at constant pressure p. For example, the heat capacity of solid water (ice) is 39 JK-1 mol-1. The value of Cp for liquid water is higher, at 75 JK-1 mol-1, so we store more energy in liquid water than when it is solid stated another way, we need to add more energy to H20(i) if its temperature is to increase. Cp for steam (H20(g)) is 34 JK-1 mol-1. Cp for solid sucrose (II) - a major component of any jam - is significantly higher at 425 JK-1 mol-1. 

A marginal but very important application of the drop calorimetric method is that it also allows enthalpies of vaporization or sublimation [162,169] to be determined with very small samples. The procedure is similar to that described for the calibration with iodine—which indeed is a sublimation experiment. Other methods to determine vaporization or sublimation enthalpies using heat flow calorimeters have been described [170-172], Although they may provide more accurate data, the drop method is often preferred due to the simplicity of the experimental procedure and to the inexpensive additional hardware required. The drop method can also be used to measure heat capacities of solids or liquids above ambient temperature [1,173],... 

The versatility of the DSC method and the high speed of the experiments have costs in terms of accuracy. For example, the best accuracy in the determination of heat capacities of solids by DSC is typically 1% [3,248-250], at least one order of magnitude worse than the accuracy of the corresponding measurements by adiabatic calorimetry [251]. This accuracy loss may, however, be acceptable for many purposes, because DSC experiments are much faster and require much smaller samples than adiabatic calorimetry experiments. In addition, they can be performed at temperatures significantly above ambient, which are outside the normal operating range of most adiabatic calorimeters. 

Figure 11.7 Experimental and calculated heat capacities of solid and liquid PTT [49], From Heat capacity of poly(trimethylene terephthalate), Pyda, M., Boiler, J., Grebowicz, J., Chuah, H., Lebedev, B. V. and Wunderlich, B., J. Polym. Sci., Polym. Phys. Ed., 36, 2499-2511 (1998), Copyright (1998 John Wiley Sons, Inc.). Reprinted by permission of John Wiley Sons, Inc...

Figure 4.3. Molar heat capacities of solid sodium (o) and palladium ( ). [Data from G. L. Pickard and F. E. Simon, Proc. Phys. Soc. 61, 1 (1948).]...

Molar heat capacities of solid n-heptane are listed in Table A.4. 

Jones, W.M. and Giauque, W.F. The entropy of nitromethane. Heat capacity of solid and liquid. Vapor pressure, heats of fusion and vaporization, / Am. Chem. Soc., 69(5) 983-987, 1947. 

Holland (1989) reconsidered the significance of constant K in fight of Einstein s model for the heat capacity of solids (see eq. 3.35 and 3.45) ... 

Cps specific heat capacity of solid phase at constant pressure... 

Calorimeters of Historical and Special Interest Around 1760 Black realized that heat applied to melting ice facilitates the transition from the solid to the liquid stale at a constant temperature. For the first time, the distinction between the concepts of temperature and heat was made. The mass of ice that melted, multiplied by the heal of fusion, gives the quantity of heal. Others, including Bunsen, Lavoisier, and Laplace, devised calorimeters based upon this principle involving a phase transition. The heat capacity of solids and liquids, as well as combustion heats and the production of heat by animals were measured with these caloritnelers. 

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. 

One problem that arises in calculating absolute entropies is that heat capacities of solids are not known to OK thus, the first integral in Eq. (5) cannot be performed. The T in the denominator of this integrand suggests that this contribution to the entropy might be very large. However, heat capacities also... 

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). 

As with gases, data for the heat capacities of solids and liquids come from experiment. The temperature dependence of CP for solids and liquids can also be expressed by equations of the form of Eq. (4.4). Data for a few solids are given in Table 4.2, and for a few liquids, in Table 4.3. Data for specific heats (CP on a unit-mass basis) of many solids and liquids are given by Perry and Green.I... 

Einstein Theory of Low-Temperature Heat Capacity of Solids [2], When we consider the heat capacity of solids, we realize that they consist of vibrating atoms or molecules. Their vibrations are quantized, of course, and have the nice name of phonons. Einstein considered a single vibration of an oscillator, along with its partition function ... 

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