14 Thermal Properties of Materials 14.1·Fundamentals The thermal properties of materials are important whenever heat- ing and cooling devices are designed.Thermally induced expan- sion of materials has to be taken into account in the construc- tion industry as well as in the design of precision instruments. Heat conduction plays a large role in thermal insulation,for ex- ample,in homes,industry,and spacecraft.Some materials such as copper and silver conduct heat very well;other materials,like wood or rubber,are poor heat conductors.Good electrical con- ductors are generally also good heat conductors.This was dis- covered in 1853 by Wiedemann and Franz,who found that the ratio between heat conductivity and electrical conductivity (di- vided by temperature)is essentially constant for all metals. The thermal conductivity of materials varies only within five orders of magnitude (Figure 14.1).This is in sharp contrast to the variation in electrical conductivity,which spans about twenty-five orders of magnitude (Figure 11.1).The thermal con- ductivity of metals and alloys can be readily interpreted by mak- ing use of the electron theory,elements of which were explained in previous chapters of this book.The electron theory postulates that free electrons perform random motions with high velocity over a large number of atomic distances.In the hot part of a metal bar they pick up energy by interactions with the vibrating lattice atoms.This thermal energy is eventually transmitted to the cold end of the bar. In electric insulators,in which no free electrons exist,the con- duction of thermal energy must occur by a different mechanism. This new mechanism was found by Einstein at the beginning of
14 The thermal properties of materials are important whenever heating and cooling devices are designed. Thermally induced expansion of materials has to be taken into account in the construction industry as well as in the design of precision instruments. Heat conduction plays a large role in thermal insulation, for example, in homes, industry, and spacecraft. Some materials such as copper and silver conduct heat very well; other materials, like wood or rubber, are poor heat conductors. Good electrical conductors are generally also good heat conductors. This was discovered in 1853 by Wiedemann and Franz, who found that the ratio between heat conductivity and electrical conductivity (divided by temperature) is essentially constant for all metals. The thermal conductivity of materials varies only within five orders of magnitude (Figure 14.1). This is in sharp contrast to the variation in electrical conductivity, which spans about twenty-five orders of magnitude (Figure 11.1). The thermal conductivity of metals and alloys can be readily interpreted by making use of the electron theory, elements of which were explained in previous chapters of this book. The electron theory postulates that free electrons perform random motions with high velocity over a large number of atomic distances. In the hot part of a metal bar they pick up energy by interactions with the vibrating lattice atoms. This thermal energy is eventually transmitted to the cold end of the bar. In electric insulators, in which no free electrons exist, the conduction of thermal energy must occur by a different mechanism. This new mechanism was found by Einstein at the beginning of Thermal Properties of Materials 14.1 • Fundamentals
272 14.Thermal Properties of Materials Rubber, Cork H20 Sulfur Wood, Glass, Fe Cu Asbestos Nylon Concrete NaCl Ge Si Al Ag Diamond 10-1 101 102 103 Lm·K Phonon conductors Electron conductors FIGURE 14.1.Room- the 20th century.He postulated the existence of phonons or lat- temperature thermal tice vibration quanta,which are thought to be created in large conductivities for numbers in the hot part of a solid and partially eliminated in the some materials.See cold part.The transfer of heat in dielectric solids is thus linked also Table 14.3. to a flow of phonons from hot to cold.Figure 14.1 indicates that in a transition region both electrons as well as phonons may con- tribute,in various degrees,to thermal conduction.Actually, phonon-induced thermal conduction occurs even in metals,but its contribution is negligible compared to that of electrons. Other thermal properties are the specific heat capacity,and a related property,the molar heat capacity.Their importance can best be appreciated by the following experimental observations: Two substances with the same mass but different values for the specific heat capacity require different amounts of thermal en- ergy to reach the same temperature.Water,for example,which has a relatively high specific heat capacity,needs more thermal energy to reach a given temperature than,say,copper or lead of the same mass.Specifically,it takes 4.18 J!to raise 1 g of water by 1 K.But the same heat raises the temperature of 1 g of cop- per by about 11 K.In short,water has a larger heat capacity than copper.(The large heat capacity of water is,incidentally,the rea- son for the balanced climate in coastal regions and the heating of North European countries by the warm water of the Gulf Stream.)We need to define the various versions of heat capaci- ties for clarification. The heat capacity,C',is the amount of heat,do,that needs to be transferred to a substance in order to raise its temperature by a certain temperature interval.The unit for the heat capacity is J/K. IThe unit of energy is the joule;see Appendix II.Obsolete units are the calorie (1 cal =4.18 J),or the British thermal unit(BTU)which is the heat required to raise the temperature of one pound of water by one de- gree fahrenheit (1 BTU 1055 J)
the 20th century. He postulated the existence of phonons or lattice vibration quanta, which are thought to be created in large numbers in the hot part of a solid and partially eliminated in the cold part. The transfer of heat in dielectric solids is thus linked to a flow of phonons from hot to cold. Figure 14.1 indicates that in a transition region both electrons as well as phonons may contribute, in various degrees, to thermal conduction. Actually, phonon-induced thermal conduction occurs even in metals, but its contribution is negligible compared to that of electrons. Other thermal properties are the specific heat capacity, and a related property, the molar heat capacity. Their importance can best be appreciated by the following experimental observations: Two substances with the same mass but different values for the specific heat capacity require different amounts of thermal energy to reach the same temperature. Water, for example, which has a relatively high specific heat capacity, needs more thermal energy to reach a given temperature than, say, copper or lead of the same mass. Specifically, it takes 4.18 J1 to raise 1 g of water by 1 K. But the same heat raises the temperature of 1 g of copper by about 11 K. In short, water has a larger heat capacity than copper. (The large heat capacity of water is, incidentally, the reason for the balanced climate in coastal regions and the heating of North European countries by the warm water of the Gulf Stream.) We need to define the various versions of heat capacities for clarification. The heat capacity, C, is the amount of heat, dQ, that needs to be transferred to a substance in order to raise its temperature by a certain temperature interval. The unit for the heat capacity is J/K. FIGURE 14.1. Roomtemperature thermal conductivities for some materials. See also Table 14.3. 272 14 • Thermal Properties of Materials Sulfur Wood, Asbestos Rubber, Cork Nylon H2O Glass, Concrete NaCl SiO2 Fe Ge Si Al Cu Ag Diamond 103 102 101 10–1 1 K W m · K Phonon conductors Electron conductors 1The unit of energy is the joule; see Appendix II. Obsolete units are the calorie (1 cal 4.18 J), or the British thermal unit (BTU) which is the heat required to raise the temperature of one pound of water by one degree fahrenheit (1 BTU 1055 J)
14.1·Fundamentals 273 The heat capacity is not defined uniquely,that is,one needs to specify the conditions under which the heat is added to the sys- tem.Even though several choices for the heat capacities are pos- sible,one is generally interested in only two:the heat capacity at constant volume C'and the heat capacity at constant pressure Cp.The former is the most useful quantity because Cv is ob- tained immediately from the energy,E,of the system.The heat capacity at constant volume is defined as: (14.1) On the other hand,it is much easier to measure the heat capac- ity of a solid at constant pressure than at constant volume.For- tunately,the difference between Cp and Cv for solids vanishes at low temperatures and is only about 5%at room temperature. The specific heat capacity is the heat capacity per unit mass: c=C (14.2) 17n where m is the mass of the system.Again,one can define it for constant volume or constant pressure.It is a material constant and is temperature-dependent.Characteristic values for cp are given in Table 14.1.The unit of the specific heat capacity is J/g.K.We note from Table 14.1 that the cp values for solids are considerably smaller than the specific heat capacity of water. Combining Egs.(14.1)and (14.2)yields: △E=△Tncv, (14.3) which expresses that the thermal energy (or heat)which is trans- TABLE 14.1.Experimental thermal parameters of various substances at room temperature and ambient pressure Specific heat Molar Molar heat Molar heat capacity,(cp)(atomic)mass capacity (Cp)capacity (Cv) Substance J g J 8·K mol mol·K ol·K Al 0.897 27.0 24.25 23.01 Fe 0.449 55.8 25.15 24.68 Ni 0.456 58.7 26.8 24.68 Cu 0.385 63.5 24.48 23.43 Pb 0.129 207.2 26.85 24.68 Ag 0.235 107.9 25.36 24.27 C(graphite) 0.904 12.0 10.9 9.20 Water 4.184 18.0 75.3
The heat capacity is not defined uniquely, that is, one needs to specify the conditions under which the heat is added to the system. Even though several choices for the heat capacities are possible, one is generally interested in only two: the heat capacity at constant volume Cv and the heat capacity at constant pressure Cp. The former is the most useful quantity because Cv is obtained immediately from the energy, E, of the system. The heat capacity at constant volume is defined as: Cv E T v . (14.1) On the other hand, it is much easier to measure the heat capacity of a solid at constant pressure than at constant volume. Fortunately, the difference between Cp and Cv for solids vanishes at low temperatures and is only about 5% at room temperature. The specific heat capacity is the heat capacity per unit mass: c C m (14.2) where m is the mass of the system. Again, one can define it for constant volume or constant pressure. It is a material constant and is temperature-dependent. Characteristic values for cp are given in Table 14.1. The unit of the specific heat capacity is J/g K. We note from Table 14.1 that the cp values for solids are considerably smaller than the specific heat capacity of water. Combining Eqs. (14.1) and (14.2) yields: E Tm cv, (14.3) which expresses that the thermal energy (or heat) which is trans- 14.1 • Fundamentals 273 TABLE 14.1. Experimental thermal parameters of various substances at room temperature and ambient pressure Specific heat Molar Molar heat Molar heat capacity, (cp) (atomic) mass capacity (Cp) capacity (Cv) Substance g J K m g ol mo J l K mo J l K Al 0.897 27.0 24.25 23.01 Fe 0.449 55.8 25.15 24.68 Ni 0.456 58.7 26.80 24.68 Cu 0.385 63.5 24.48 23.43 Pb 0.129 207.2 26.85 24.68 Ag 0.235 107.9 25.36 24.27 C (graphite) 0.904 12.0 10.90 9.20 Water 4.184 18.0 75.30
274 14.Thermal Properties of Materials ferred to a system equals the product of mass,increase in tem- perature,and specific heat capacity. A further useful material constant is the heat capacity per mole. It compares materials that contain the same number of mole- cules or atoms.The molar heat capacity is obtained by multi- plying the specific heat capacity cv(or cp)by the molar mass,M (see Table 14.1): C,=G=cw·M, (14.4) where n is the amount of substance in mol. We see from Table 14.1 that the room-temperature molar heat capacity at constant volume is approximately 25 J/mol.K for most solids.This was experimentally discovered in 1819 by Du- long and Petit.The experimental molar heat capacities for some materials are depicted in Figure 14.2 as a function of tempera- ture.We notice that some materials,such as carbon,reach the Dulong-Petit value only at high temperatures.Some other ma- terials such as lead reach 25 J/mol.K at relatively low tempera- tures. All heat capacities are zero at T=0K.The Cy values near T= O K climb in proportion to T3 and reach 96%of their final value at a temperature Op,which is defined to be the Debye tempera- ture.Op is an approximate dividing point between a high- temperature region,where classical models can be used for the interpretation of Cv,and a low-temperature region,where quan- tum theory needs to be applied.Selected Debye temperatures are listed in Table 14.2. 25 Pb Cu Cv mol.K Carbon FIGURE 14.2.Temperature depen- dence of the molar heat capacity 100 200 300 400 500 Cy for some materials. T [K]
ferred to a system equals the product of mass, increase in temperature, and specific heat capacity. A further useful material constant is the heat capacity per mole. It compares materials that contain the same number of molecules or atoms. The molar heat capacity is obtained by multiplying the specific heat capacity cv (or cp) by the molar mass, M (see Table 14.1): Cv C n v cv M, (14.4) where n is the amount of substance in mol. We see from Table 14.1 that the room-temperature molar heat capacity at constant volume is approximately 25 J/mol K for most solids. This was experimentally discovered in 1819 by Dulong and Petit. The experimental molar heat capacities for some materials are depicted in Figure 14.2 as a function of temperature. We notice that some materials, such as carbon, reach the Dulong–Petit value only at high temperatures. Some other materials such as lead reach 25 J/mol K at relatively low temperatures. All heat capacities are zero at T 0 K. The Cv values near T 0 K climb in proportion to T3 and reach 96% of their final value at a temperature )D, which is defined to be the Debye temperature. )D is an approximate dividing point between a hightemperature region, where classical models can be used for the interpretation of Cv, and a low-temperature region, where quantum theory needs to be applied. Selected Debye temperatures are listed in Table 14.2. 274 14 • Thermal Properties of Materials 25 Cv J mol. K Pb Cu Al Carbon 100 200 300 400 500 T [K] FIGURE 14.2. Temperature dependence of the molar heat capacity Cv for some materials.
14.2.Interpretation of the Heat Capacity by Various Models 275 TABLE 14.2.Debye temperatures of some materials Substance ep(K) Pb 95 Au 170 Ag 230 270 Cu 340 Fe 360 Al 375 Si 650 1850 GaAs 204 InP 162 14.2.Interpretation of the Heat Capacity by Various Models The classical (atomistic)theory for the interpretation of the heat capacity postulates that each atom in a crystal is bound to its site by a harmonic force similar to a spring.A given atom is thought to be capable of absorbing thermal energy,and in doing so it starts to vibrate about its point of rest.The amplitude of the oscillation is restricted by electrostatic repulsion forces of the nearest neighbors.The extent of this thermal vibration is there- fore not more than 5 or 10%of the interatomic spacing,de- pending on the temperature.In short,we compare an atom with a sphere which is held at its site by two springs [Figure 14.3(a)]. The thermal energy that a harmonic oscillator of this kind can absorb is proportional to the absolute temperature of the envi- ronment.The proportionality factor has been found to be the Boltzmann constant kB(see Appendix II).The average energy of the oscillator is then: E=kBT. (14.5) Now,solids are three-dimensional.Thus,a given atom in a cu- bic crystal also responds to the harmonic forces of lattice atoms in the other two directions.In other words,it is postulated that each atom in a cubic crystal represents three oscillators [Figure 14.3(b)],each of which absorbs the thermal energy kBT.There- fore,the average energy per atom is: E=3kBT. (14.6)
The classical (atomistic) theory for the interpretation of the heat capacity postulates that each atom in a crystal is bound to its site by a harmonic force similar to a spring. A given atom is thought to be capable of absorbing thermal energy, and in doing so it starts to vibrate about its point of rest. The amplitude of the oscillation is restricted by electrostatic repulsion forces of the nearest neighbors. The extent of this thermal vibration is therefore not more than 5 or 10% of the interatomic spacing, depending on the temperature. In short, we compare an atom with a sphere which is held at its site by two springs [Figure 14.3(a)]. The thermal energy that a harmonic oscillator of this kind can absorb is proportional to the absolute temperature of the environment. The proportionality factor has been found to be the Boltzmann constant kB (see Appendix II). The average energy of the oscillator is then: E kBT. (14.5) Now, solids are three-dimensional. Thus, a given atom in a cubic crystal also responds to the harmonic forces of lattice atoms in the other two directions. In other words, it is postulated that each atom in a cubic crystal represents three oscillators [Figure 14.3(b)], each of which absorbs the thermal energy kBT. Therefore, the average energy per atom is: E 3kBT. (14.6) 14.2 • Interpretation of the Heat Capacity by Various Models 275 TABLE 14.2. Debye temperatures of some materials Substance )D (K) Pb 95 Au 170 Ag 230 W 270 Cu 340 Fe 360 Al 375 Si 650 C 1850 GaAs 204 InP 162 14.2 • Interpretation of the Heat Capacity by Various Models