Specific Heat of a Solid

11 Key excerpts on "Specific Heat of a Solid"

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  Cryogenic Engineering, Revised and Expanded

  • Thomas Flynn(Author)
  • 2004(Publication Date)
  • CRC Press

    (Publisher)

  5 Transport Properties of Solids 1. THERMAL PROPERTIES The thermal properties of materials at low temperatures of most interest to the process engineer are specific heat, thermal conductivity, and thermal expansivity. (We shall use the term ‘‘specific heat’’ to refer to a material property and the term ‘‘heat capacity’’ to refer to the atomic scale contributions to specific heat.) It will be shown that each of these properties depends on the intermolecular potential of the lattice and accordingly that they are interrelated. 1.1. Specific Heat 1.1.1. Lattice Heat Capacity Nearly all the physical properties of a solid (e.g., specific heat, thermal expansion) depend on the vibration or motion of the atoms in the solid. Specific heat is often measured at low temperatures for design purposes. However, specific heat measure-ments are important in their own right, because the variation of specific heat with temperature shows how energy is distributed among the various energy-absorbing modes of the solid. Thus specific heat measurements give important clues to the structure of the solid. Finally, because other properties also depend on the lattice structure and its vibration, specific heat measurements are used to predict or corre-late other properties, such as thermal expansion. Therefore, an understanding of the temperature dependence of specific heat not only gives useful design information but also is helpful in predicting other thermal properties. The specific heat of any material is defined from thermodynamics as C V ¼ @ U @ T V ð 5 : 1 Þ where U is the internal energy, T is the absolute temperature, and V is the volume. C V is the property more useful to theory than C P , because it directly relates internal energy, and hence the microscopic structure of the solid, to temperature. However, it must be remembered that most solids expand when they are heated at constant pressure. As a result, the solid does work against both internal and external forces.

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  Physics for O.N.C. Courses

  • R.A. Edwards(Author)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER 7

  The Measurement of Heat Quantity

  Publisher Summary

  This chapter describes different methods of measurement of heat quantity. The specific heat of a substance at any temperature is defined as the quantity of heat required to raise the temperature of unit mass of that substance by one degree. The thermal capacity of a body or mass of material is the quantity of heat required to raise the temperature of that body or mass of material by one degree. The quantity of heat required to convert unit mass of the solid into liquid without change of temperature is called its latent heat of fusion. The latent heat of vaporization, or evaporation, is the quantity of heat required to convert unit mass of liquid into vapor without change of temperature, that is, at the boiling point. The latent heat of fusion of ice can be measured by a simple method of mixtures. An electrical method could be used using a calorimeter similar to that used for the electrical method for the determination of the specific heats of solids. The calorific value of a fuel is the quantity of heat liberated as a result of the complete combustion of unit mass of the fuel. A gas calorimeter is used to measure the calorific value of gas fuels or of liquid fuels, which easily vaporize.

  7.1 Specific Heat

  The specific heat of a substance at any temperature is defined as the quantity of heat required to raise the temperature of unit mass of that substance by one degree . For example, if an infinitesimal quantity of heat dH is required to raise a mass m of a substance from a temperature θ to a temperature θ + dθ , then the specific heat s of the substance is defined as

  (7.1)

  Modern practice is to express specific heat in joules per gram per degC (J g−1 degC−1 ) or alternatively in joules per kilogram per degC (J kg−1 degC−1 ). Over quite considerable ranges of temperature the value of s for a material often varies only slightly, but in accurate work its variation may be regarded as significant. If s may be considered constant over a finite change Δθ in temperature, then the heat ΔH gained or lost by a mass m of the substance in rising or falling through the temperature range Δθ

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  Electronic Materials

  Principles and Applied Science

  • Yuriy Poplavko(Author)
  • 2018(Publication Date)
  • Elsevier

    (Publisher)

  kl tensor.

  3.3 Crystal Heat Capacity

  The heat capacity of body is a physical quantity defined as a ratio of the amount of heat dQ obtained by the body corresponding to an increase in its temperature dT :

  C = dQ / dT .

  The unit of heat capacity in SI is [J/K]. The concept of heat capacity is applicable to substances that are in various states of aggregation (solid, liquid, or gas) as well as to ensembles of particles and even quasiparticles (e.g., the heat capacity of electronic gas in metals or heat capacity of phonons in a crystal lattice). The value of heat capacity depends on the nature of a substance.

  Specific heat is the heat capacity per given unit of substance, which can be measured in kilograms, cubic meters, and moles. Depending on the quantification of unit heat applied, the mass, volume, and molar value of specific heat are distinguished. The mass specific heat is the amount of heat necessary to increase the temperature of a unit mass of material by one temperature unit; in SI, it is [J kg− 1  K− 1 ]. The volumetric specific heat , C V , is the amount of heat that is necessary to be applied to a unit volume of material to heat it by one temperature unit; in SI, it is measured in [J m− 3  K− 1 ], that is, joules per cubic meter and Kelvin. The molar specific heat , C μ , is the amount of heat that is necessary for 1 mol of the substance to be heated by 1°; in SI, it is [J/(mol K)], while in the Gaussian system, this specific heat is determined in [cal/(g-mol K)]. The vast majority of solids have specific heat close to 1 kJ/(kg K); for example, water has a relatively high heat capacity: 4.2 kJ/(kg K).

  In solids, both the crystal lattice and electrons contribute to the specific heat. For reasons that are explained further on, the electronic specific heat in metals at normal conditions is rather small; therefore the mechanisms of the lattice for specific heat are mainly considered (Chapter 4

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  Important Concepts and Components of Introductory Physics

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter- 1 Heat Capacity Heat capacity (usually denoted by a capital C , often with subscripts) is the measurable physical quantity that characterizes the amount of heat required to change a body's temperature by a given amount. In the International System of Units, heat capacity is expressed in units of joules per kelvin. Derived quantities include the molar heat capacity , which is the heat capacity per mole of a pure substance. Similarly, specific heat capacity (also called more properly mass-specific heat capacity or more loosely specific heat), is the heat capacity per unit mass of a body. These quantities are intensive quantities. That is, they are not dependent on amount of material, but directly reflect the type of material, as well as the physical conditions of heating. Temperature reflects the average total kinetic energy of particles in matter. Heat is transfer of thermal energy; it flows from regions of high temperature to regions of low temperature. Thermal energy is stored as kinetic energy and, in molecules and solids, also as potential energy in the modes of vibration or phonons. These represent degrees of freedom of movement for atoms. These degrees of freedom, and sometimes others, contribute to the heat capacity of a thermodynamic system. As the temperature approaches absolute zero, the specific heat capacity of a system also approaches zero. Quantum theory can be used to quantitatively predict specific heat capacities in simple systems. Background Before the development of modern thermodynamics, it was thought that heat was a fluid, the so-called caloric . Bodies were capable of holding a certain amount of this fluid, hence the term heat capacity , named and first investigated by Joseph Black in the 1750s. Today one instead discusses the internal energy of a system. This is made up of its microscopic kinetic and potential energy.

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  Fundamentals of Physics, Extended

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Heat capacity C has the unit of energy per degree or energy per kelvin. The heat capacity C of, say, a marble slab used in a bun warmer might be 179 cal/C°, which we can also write as 179 cal/K or as 749 J/K. The word “capacity” in this context is really misleading in that it suggests analogy with the capacity of a bucket to hold water. That analogy is false, and you should not think of the object as “containing” heat or being limited in its ability to absorb heat. Heat transfer can proceed without limit as long as the necessary temperature differ- ence is maintained. The object may, of course, melt or vaporize during the process. Specific Heat Two objects made of the same material — say, marble — will have heat capacities proportional to their masses. It is therefore convenient to define a “heat capacity per unit mass” or specific heat c that refers not to an object but to a unit mass of the material of which the object is made. Equation 18-13 then becomes Q = cm ∆T = cm(T f ‒ T i ). (18-14) Through experiment we would find that although the heat capacity of a particular marble slab might be 179 cal/C° (or 749 J/K), the specific heat of marble itself (in that slab or in any other marble object) is 0.21 cal/g · C° (or 880 J/kg · K). 524 CHAPTER 18 TEMPERATURE, HEAT, AND THE FIRST LAW OF THERMODYNAMICS We are led then to this definition of heat: Heat is the energy transferred between a system and its environment because of a temperature difference that exists between them. From the way the calorie and the British thermal unit were initially defined, the specific heat of water is c = 1 cal/g · C° = 1 Btu/lb · F° = 4186.8 J/kg · K. (18-15) Table 18-3 shows the specific heats of some substances at room temperature. Note that the value for water is relatively high. The specific heat of any substance actually depends somewhat on temperature, but the values in Table 18-3 apply reasonably well in a range of temperatures near room temperature.

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  Halliday and Resnick's Principles of Physics

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Heat capacity C has the unit of energy per degree or energy per kelvin. The heat capacity C of, say, a marble slab used in a bun warmer might be 179 cal/C°, which we can also write as 179 cal/K or as 749 J/K. The word “capacity” in this context is really misleading in that it suggests analogy with the capacity of a bucket to hold water. That analogy is false, and you should not think of the object as “containing” heat or being limited in its ability to absorb heat. Heat transfer can proceed without limit as long as the necessary temperature differ- ence is maintained. The object may, of course, melt or vaporize during the process. Specific Heat Two objects made of the same material — say, marble — will have heat capacities proportional to their masses. It is therefore convenient to define a “heat capacity per unit mass” or specific heat c that refers not to an object but to a unit mass of the material of which the object is made. Equation 18-13 then becomes Q = cm ∆T = cm(T f ‒ T i ). (18-14) Through experiment we would find that although the heat capacity of a particular marble slab might be 179 cal/C° (or 749 J/K), the specific heat of marble itself (in that slab or in any other marble object) is 0.21 cal/g · C° (or 880 J/kg · K). 452 CHAPTER 18 TEMPERATURE, HEAT, AND THE FIRST LAW OF THERMODYNAMICS We are led then to this definition of heat: Heat is the energy transferred between a system and its environment because of a temperature difference that exists between them. From the way the calorie and the British thermal unit were initially defined, the specific heat of water is c = 1 cal/g · C° = 1 Btu/lb · F° = 4186.8 J/kg · K. (18-15) Table 18-3 shows the specific heats of some substances at room temperature. Note that the value for water is relatively high. The specific heat of any substance actually depends somewhat on temperature, but the values in Table 18-3 apply reasonably well in a range of temperatures near room temperature.

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  Physics of Matter

  • George C. King(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  10 Thermal and transport properties of solids In this chapter, various thermal and transport properties of solids are discussed. Einstein’s theory of the specific heat of solids is described as how its predictions compare to the experimental data. The modified version of the theory due to Debye is described as well. Also discussed is diffusion in solids, which is very different to diffusion in gases. This is because, unlike the molecules in a gas, the atoms in a solid are confined by the forces due to their neighbours. The other transport properties discussed are electrical and thermal conduction in metals, corresponding to the transport of electric charge and thermal energy, respectively. Some of the properties of solids can be described in terms of classical physics. However, as we will see, there are certain properties that require a quantum mechanical description. 10.1 Molar specific heats of solids We will usually consider one mole of a solid and so deal with molar specific heats. The atoms in a solid are bound to their lattice sites. Consequently, they do not have translational energy but only vibrational energy. So more precisely, we are dealing with the specific heat associated with the vibrations of the atoms in the crystal lattice. Moreover, solids expand by only a small amount when they are heated, and so the difference between the specific heats at constant volume and that at constant pressure can usually be ignored. An oscillator, such as a mass on a spring, has potential energy and kinetic energy and the total energy E is the sum of the two. For a one-dimensional oscillator moving along the x-axis, we have E 1 2 αx 2 + 1 2 m dx dt 2 , (10.1) where α is the spring constant and m is the oscillator mass. According to the classical equipartition theorem, each term on the right-hand side of this equation has a mean energy of ½kT. Hence, the mean total energy of a one-dimensional oscillator is kT.

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  Halliday's Fundamentals of Physics, 1st Australian & New Zealand Edition

  • David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  (18.18) Table 18.3 shows the specific heats of some sub- stances at room temperature. Note that the value for water is relatively high. The specific heat of any sub- stance actually depends somewhat on temperature, but the values in the table apply reasonably well in a range of temperatures near room temperature (30°C). Molar specifc heat TABLE 18.4 Some molar specifc heats at room temperature Substance J/mol ⋅ K Lead 26.5 Tungsten 24.8 Silver 25.5 Copper 24.5 Aluminium 24.4 In many instances the most convenient unit for specifying the amount of a substance is the mole (mol), where 1 mol = 6.02 × 10 23 elementary units (18.19) of any substance. Thus 1 mol of aluminium means 6.02 × 10 23 atoms (the atom is the elementary unit), and 1 mol of aluminium oxide means 6.02 × 10 23 molecules (the molecule is the elementary unit of the compound). When quantities are expressed in moles, specific heats must also involve moles (rather than a mass unit); they are then called molar specific heats. Table 18.4 shows the values for some elemental solids (each consisting of a single element) at room temperature. An important point In determining and then using the specific heat of any substance, we need to know the conditions under which energy is transferred as heat. For solids and liquids, we usually assume that the sample is under constant pressure (usually atmospheric) during the transfer. It is also conceivable that the sample is held at constant volume while the heat is absorbed. This means that thermal expansion of the sample is prevented by applying external pressure. For solids and liquids, this is very hard to arrange experimentally, but the effect can be calculated, and it turns out that the specific heats under constant pressure and constant volume for any solid or liquid differ usually by no more than a few per cent.

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  Practical Chemical Thermodynamics for Geoscientists

  • Bruce Fegley Jr., Bruce Fegley, Jr.(Authors)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  σ for liquid metals, molten oxides, and silicates is negligible until very high temperatures.

  Solids

  The heat capacity (

  CP

  ) of solids is the sum of several different contributions that vary with temperature and with the type of material (metals such as iron, semiconductors such as pyrite, and insulators such as forsterite). In general, the heat capacity of a solid can be written as

  (4-2)

  although not every component contributes to the heat capacity of every solid. The lattice heat capacity (

  Clat

  ) is generally the largest term. It arises from vibrations of the atoms (or molecules) that make up a solid. To first approximation,

  Clat ∼

  3R per gram atom (i.e., per mole of atoms) for solids at room temperature, with notable exceptions being B, Be, diamond, and graphite, which have much smaller

  Clat

  values. The thermal expansion term (

  CP  − CV

  ) is the next most important contribution to the heat capacity for most nonmetals and is ∼5% of

  CP

  at room temperature. The electronic heat capacity (

  CE

  ) or electronic component of solids generally does not become significant until high temperatures, but metals have significant electronic heat capacities at low temperatures. Other contributions to the heat capacity of solids can be made by magnetic transitions (

  CM

  ), ordering and disordering effects in crystal lattices (Cλ ), and transitions between electronic energy levels with ΔE ∼ kT in compounds with d and f electrons such as those formed by transition metals, lanthanides, and actinides. These transitions are known as Schottky transitions, and the Schottky heat capacity contribution is

  CSch

  . Schottky transitions generally occur at temperatures close to absolute zero and are difficult to identify. For example, Krupka et al. (1985) found that bronzite (Mg0.85 Fe0.15 SiO3 ) has a heat capacity anomaly at 10–15 K. Other measurements ruled out a magnetic transition at these temperatures, and the anomaly can be fit with transitions between electronic energy levels of the d electrons in iron. Thus, it is believed to be a Schottky anomaly. We will discuss some of these heat capacity contributions in more detail in Section III

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  Physical Properties of Materials for Engineers

  • Daniel D. Pollock(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  The heat capacity of a substance at constant volume is defined as C V = (∂ U ∂ T) V (4-1) where U is the internal energy and T is the absolute temperature. This is the quantity of heat required to raise the temperature of 1 mol of a substance 1 K. The heat capacity at constant pressure is C P = (∂ H ∂ T) P = (∂ (U + P V) ∂ T) P (4-2) in which H is the enthalpy, P is the pressure, and V is the volume. Performing the indicated differentiation C P = (∂ U ∂ T) P + P (∂ V ∂ T) P + V (∂ P ∂ T) P and noting that ∂P/∂T = 0 at constant pressure C P = (∂ U ∂ T) P + P (∂ V ∂ T) P (4-3) The quantity ∂U can be expressed as ∂ U = (∂ U ∂ V) T ∂ V + (∂ U ∂ T) V ∂ T and when divided by ∂T this. becomes ∂ U ∂ T = (∂ U ∂ V) T (∂ V ∂ T) P + (∂ U ∂ T) V (4-4) Thus, C P (Equation 4-3) becomes C P = (∂ U ∂ V) T (∂ V ∂ T) P + (∂ U ∂ T) V + P (∂ V ∂ T) P (4-5) When this is used with Equation. 4-1 C P − C V = (∂ U ∂ V) T (∂ V ∂ T) P + (∂ U ∂ T) V + P (∂ V ∂ T) P − (∂ U ∂ T) V C P − C V = (∂ U ∂ V) T (∂ V ∂ T) P + P (∂ V ∂ T) P = [ (∂ U ∂ V)[--=PLGO-SE. PARATOR=--]T + P ] (∂ V ∂ T) P (4-6) For most solids, especially metals at normal pressures, (∂ V /∂T) P is relatively small, Δ V /ΔT being on the order of 10 −4 to 10 −5 cm 3 /°C. Thus, for many purposes it can be assumed that this quantity is negligible and that C P ≅ C V (4-7) This approximation will be employed for solids where applicable and convenient. It gives a maximum error of <0.5 cal/mol · K. Dulong and Petit (1818) showed that the heat capacities of many elements were related to their atomic weights; the product of their specific heats (the quantity of heat required to raise the temperature of 1 g 1 K) and atomic weights being approximately constant, about 6 cal per mol-degree. This is only a good approximation. For many metals, the heat capacity lies between 5 and 7 cal/mol-deg at 0°C, and has an average value of 6.2. The heat capacities are not constant but increase about 0.04%/°C for temperatures above 0°C

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  Solid State Physics

  • Gerald Burns(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  [G. Shaefer, J. Phys. Chem. Solids 12, 233 (I960).] 480 500 Frequency ( c m -1 ) 520 Notes For the first law of thermodynamics and the definitions and relations between C P and C v see any textbook on thermodynamics. J. DeLaunay [Solid State Physics 2, 219 (1956)] has a nice histor-ical review of the early heat capacity work including references to Dulong and Petit's work; Einstein [Ann. Physik 22, 180 (1907); 31, 679 (1911)]. Also see: Debye [Ann Physik 39, 789 (1912); an Eng-lish translation of this paper is in The Collected Papers of Peter J. W. Debye Interscience, 1954]; Born and von Karman [Physik Ζ. 13, 297 (1912)]; Blackman [Reports of Progress in Physics 8, 11 (1941)]. 400 CHAPTER 11 SOME T H E R M A L EFFECTS IN SOLIDS The book by E. S. R. Gopal, Specific Heats at Low Tempera-tures (Plenum Press, 1966) is a nice reference. Many other solid state texts have a chapter on specific heat. T. Muto and Y. Takagi's article in Solid State Physics 1, 193 (1955) is a fine article on the theory of order-disorder transition of the type discussed in this chapter. Dekker and Kittel (especially the earlier editions) also have discussions of this topic. Debye's little book Polar Molecules (Dover Publications, 1945) is a delight to thumb through. See his Figs. 4 and 5 for a picture of a quadrupole and octupole moment. The material in Section 11-7 is adapted from Debye's book and dates back to his original work in this field in 1912. See also the earlier editions of Kittel. Two related books also worth looking at are C. J. F. Böttcher, Theory of Electric Polarization (Elsevier, 1952), H. Fröhlich, Theory of Dielectrics (Oxford, 1949), and C. P. Smyth, Dielectric Constant and Molecular Structure (Chem. Catalog, 1931). Brown has a large chapter devoted to color centers in alkali halide crystals and is a good reference for point defects. Two earlier review articles by F. Seitz [Rev. Mod. Phys. 18, 384 (1946); 26, 7 (1954)] should also be looked at as well as B.

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