Heat capacity atomic, solids

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 ... 

Other methods of determining atomic weights X-ray method heat capacity of gases heat capacity of solids isomorphism chendcal analogy. 

Notice in Table 1 that the same about 25 joule rule also applies to the molar heat capacities of solid ionic compounds. One mole barium chloride has three times as many ions as atoms in 1 mol of metal. So, you expect the molar heat capacity for BaCl2 to be C = 3 x 25 J/K mol. The value in Table 1,75.1 J/K mol, is similar to this prediction. 

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)... 

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. 

Heat capacities of solids are always functions of temperature, as illustrated for several ceramics in Figure 34.1. Note the units are J g-atom internal energy of a solid is capacity is zero. As the temperature rises the heat capacity increases, which is indicative of the various mechanisms by which energy is absorbed. The heat capacity approaches... 

The dependence of the heat capacity of solids under standard conditions (298 K) on the atomic numbers of lanthanides is a smooth curve for... 

The vibrations of the atoms in the crystalline lattice are important in understanding the thermal properties of both metallic and nonmetallic solids. The energy involved in these vibrations represents thermal energy hence lattice vibrations are primarily resporrsible for the heat capacity of solids. Also, these vibrations are able to transport heat and are the dominant source of thermal conductivity in nonmetals. Therefore, in order to understand thermal properties of solids, it is necessary to start with a general understanding of the nature of lattice d5mamics. 

By the late 1800s, study of heat capacities had led to the perplexing observation that the heat capacities of solids at quite low temperatures were very much below those expected from the Dulong-Petit rule. Some measurements gave heat capacities only 1% of the predicted value. About 1907, Einstein tied quantum behavior to heat capacities. He showed that if the vibrational energies of atoms in a solid were quantized, heat capacity would diminish sharply at low temperature. The high-temperature limit of Einstein s theory was the result achieved earlier by Boltzmann. Einstein s theory proved not as quantitatively accurate in its application to the heat capacity curves of solids as it was to those of diatomic gases. 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

The heat capacity can be computed by examining the vibrational motion of the atoms and rotational degrees of freedom. There is a discontinuous change in heat capacity upon melting. Thus, different algorithms are used for solid-and liquid-phase heat capacities. These algorithms assume different amounts of freedom of motion. 

The explanation of the hydrogen atom spectmm and the photoelectric effect, together with other anomalous observations such as the behaviour of the molar heat capacity Q of a solid at temperatures close to 0 K and the frequency distribution of black body radiation, originated with Planck. In 1900 he proposed that the microscopic oscillators, of which a black body is made up, have an oscillation frequency v related to the energy E of the emitted radiation by... 

There are no reliable prediction methods for solid heat capacity as a function of temperature. However, the atomic element contribution method of Hurst and Harrison,which is a modification of Kopp s Rule, provides estimations at 298.15 K and is easy to use ... 

Cps = solid heat capacity at 298.15 K, J/mol K n = number of different atomic elements in the compound N, = number of atomic elements i in the compound Agi = numeric value of the contribution of atomic element i found in Table 2-393... 

Example 15 Estimate solid heat capacity of dibenzothiophene, Ci2HsS. The required atomic element contributions from Table 2-393 are C = 10.89, H = 7.56, and S = 12.36. Substituting in Eq. (2-63) ... 

TABLE 2-393 Atomic Element Contributions to Estimate Solid Heat Capacity at 298.15 K... 

One of the first attempts to calculate the thermodynamic properties of an atomic solid assumed that the solid consists of an array of spheres occupying the lattice points in the crystal. Each atom is rattling around in a hole at the lattice site. Adding energy (usually as heat) increases the motion of the atom, giving it more kinetic energy. The heat capacity, which we know is a measure of the ability of the solid to absorb this heat, varies with temperature and with the substance.8 Figure 10.11, for example, shows how the heat capacity Cy.m for the atomic solids Ag and C(diamond) vary with temperature.dd ee The heat capacity starts at a value of zero at zero Kelvin, then increases rapidly with temperature, and levels out at a value of 3R (24.94 J-K -mol-1). The... 

The high-temperature contribution of vibrational modes to the molar heat capacity of a solid at constant volume is R for each mode of vibrational motion. Hence, for an atomic solid, the molar heat capacity at constant volume is approximately 3/. (a) The specific heat capacity of a certain atomic solid is 0.392 J-K 1 -g. The chloride of this element (XC12) is 52.7% chlorine by mass. Identify the element, (b) This element crystallizes in a face-centered cubic unit cell and its atomic radius is 128 pm. What is the density of this atomic solid ... 

ALL CHANGES IN PHASE involve a release or absorption of calories. One reason for this is that each solid has its own heat capacity. That is, there is a characteristic heat content for each material which depends upon the atoms composing the solid, the nature of the lattice vibrations within it, and its structure. The total heat content, or enthalpy, of each solid is defined by ... 

When the temperature is such that hv kT, neither of the limiting cases described earlier can be used. For many solids, the frequency of lattice vibration is on the order of 1013 Hz, so that the temperature at which the value of the heat capacity deviates substantially from 3R is above 300 to 400 K. For a series of vibrational energy levels that are multiples of some fundamental frequency, the energies are 0, hv, 2hv, 3hi/, etc. For these levels, the populations of the states (n0, nu n2, etc.) will be in the ratio 1 e hl T e Jh,/Ikr e etc. The total number of vibrational states possible for N atoms is 3N... 

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 decrease in the heat capacity at low temperatures was not explained until 1907, when Einstein demonstrated that the temperature dependence of the heat capacity arose from quantum mechanical effects [1], Einstein also assumed that all atoms in a solid vibrate independently of each other and that they behave like harmonic oscillators. The motion of a single atom is again seen as the sum of three linear oscillators along three perpendicular axes, and one mole of atoms is treated by looking at 3L identical linear harmonic oscillators. Whereas the harmonic oscillator can take any energy in the classical limit, quantum theory allows the energy of the harmonic oscillator (en) to have only certain discrete values ( ) ... 

We now distinguish solid state transformations as first-order transitions or lambda transitions. The latter class groups all high-order solid state transformations (second-, third-, and fourth-order transformations see Denbigh, 1971 for exhaustive treatment). We define first-order transitions as all solid state transformations that involve discontinuities in enthalpy, entropy, volume, heat capacity, compressibility, and thermal expansion at the transition point. These transitions require substantial modifications in atomic bonding. An example of first-order transition is the solid state transformation (see also figure 2.6)... 

At high F, when the spacing of vibrational energy levels is low with respect to thermal energy, crystalline solids begin to show the classical behavior predicted by kinetic theory, and the heat capacity of the substance at constant volume (Cy) approaches the theoretical limit imposed by free motion of all atoms along three directions, in a compound with n moles of atoms per formula unit limit of Dulong and Petit) ... 

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... 

A general reference often consulted today for the physical and chemical properties of common chemicals is Lange s Handbook of Chemistry (Dean 1999), which lists many chemical compounds and their most important properties. It is organized into separate chapters of Physical constants of organic molecules with 4300 compounds and Physical constants of inorganic molecules, and lists each compound alphabetically by name. Some of these properties are very sensitive to temperature, but less sensitive to pressure, and they are listed as tables, or more compactly as equations of the form /(T) for example, liquid heats of evaporation, heat capacities of multi-atom gases, vapor pressures over liquids, liquid and solid solubilities in liquids, and liquid viscosities. Some of these properties are sensitive both to temperature and pressure. 

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