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 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): 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 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
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 tempera￾ture. )D 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. 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 depen￾dence of the molar heat capacity Cv for some materials.