The Heat Capacity of Solids

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... 

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. 

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) ... 

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. 

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 ... 

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... 

The Debye theory of the heat capacities of solid elements W) yields an expression for their entropies. 

The heat capacity of most of the solid crystalline elements is approximately 6 (cal)/(g mole)(K) at or near room temperature. The heat capacities of solid compounds are higher. Tables 8.6 and 8.7 give heat capacities for various solid compounds. 

Kopp s rule is a simple empirical method for estimating the heat capacity of a solid or liquid at or near 20°C. According to this rule, Cp for a molecular compound is the sum of contributions (given in Table B.IO) for each element in the compound. For example, the heat capacity of solid calcium hydroxide. Ca(OH)2, would be estimated from Kopp s rule as... 

TEST 1. Estimate the heat capacity of solid calcium carbonate (CaCOr) using Kopp s rule and... 

The heat capacity is assumed constant and equal to the heat capacity of solid BOgC at the estimated melting point. 

The main value of Debye s theory is that it provides a reasonably satisfactory treatment of the heat capacity of solids. 

Since as we have already seen, the heat capacities of solids can be accounted for theoretically, it is of interest to examine whether the heat capacities of liquids can also be predicted in a similar way. We observe in the first place, that in the neighbourhood of the melting point the specific heats of simple solids and liquids are generally nearly equal. J Thus for solid mercury at 234 c = 6 77 cal./deg. mole, while for... 

Most of the equations for the heat capacities of solids, liquids, and gases are empirical. We usually express the heat capacity at constant pressure Cp as a function of temperature in a power series, with constants a, b, c, and so on for example. 

To evaluate the suitability of Kopp s rule for the heat capacities of solids, compute the heat capacities of sodium sulfate (Na2S04), dextrose (CeHi206), and copper ammonium sulfete [CuS04(NH4)2 SO4 6H2O] and compare your values with the experimental ones at 25°C. 

The thermodynamic functions of solids I and II are recorded in Table 6.5 [752]. Those of solid III have not been determined, but it is considered that the heat capacity of solid III is similar to the values for solids I and II. The earlier values of Cp [751] for solid II are valid below 118.3 K, and are valid for solid I above 118.3 K. 

Further work on similar types of cells has been carried out, in which not only is use made of the Nernst Theorem but likewise of the Einstein theory of atomic heat of solids (as modified by Nernst and Lmdemann) This will be taken up after we have discussed Planck s Quantum Theory of radiation and Einstein s application of it to the heat capacity of solids (Vol. Ill)... 

D. Correlation for the heat capacity of "solid" polymers at room temperature... 

Cp(T) is the property normally measured by calorimetry. However, CV(T) is also important, since it is directly calculated by theoretical models which express the heat capacity of a material in terms of the vibrational motions of its atoms. Vibrational motions approximated by the harmonic oscillator model are commonly accepted to be the source of the heat capacities of solids. CV(T) is estimated by integrating over the frequency spectrum of these vibrations. 

The heat capacities of polymers can be classified into two types, namely, the heat capacities of "solid" polymers, which will be denoted by Cps, and the heat capacities of "liquid" polymers, which will be denoted by Cp. ... 

D. Correlation for the Heat Capacity of "Solid" Polymers at Room Temperature... 

Values of CP may be measured down to a few degrees Kelvin, but a means of extrapolation must be found for the last few degrees. The Debye theory of the heat capacity of solids gives, at low temperatures ... 

The development of the quantum theory was at first slow. It was not until 1905 that Einstein2 suggested that the quantity of radiant energy hv was sent out in the process of emission of light not in all directions but instead unidirectionally, like a particle. The name light quantum or photon is applied to such a portion of radiant energy. Einstein also discussed the photoelectric effect, the fundamental processes of photochemistry, and the heat capacities of solid bodies in terms of the quantum theory. When light falls on a metal plate, electrons arc emitted from it. The maximum speed of these photoelectrons, however,... 

So we see that the DuLong-Petit law gives reasonable though not exact values for the heat capacities of solids. 

The last term, e-, is usually present in expressions for the heat capacity of solids and is included here for generality. 

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