In this demonstration, you allow three metal cylinders to come to thermal equilibrium with a hot water bath. One cylinder is made of aluminum, the second one is iron, and the third is made of lead. Each cylinder has a mass of 50 g. After the cylinders have reached thermal equilibrium, place each one at the top of one of the tracks on the wax slide. The cylinder having the greatest heat capacity melts the most wax and travels farthest down the slide. The one with the smallest heat capacity, of course, goes the shortest distance down the slide.
“Specific heat capacity,” usually referred to as “specific heat,” is the amount of heat required per unit mass of a substance to raise its temperature one (Celsius) degree. We can write this as c = Q/(mΔT). “Heat capacity” is the amount of heat required to raise the temperature of an object made of that substance – a given mass of that substance – one degree. This is usually written as C = Q/ΔT. The total heat necessary to raise the temperature of a body of mass m, from temperature T to temperature T + ΔT, then, is Q = mcΔT. The specific heats of the different metals are 0.908 J/g·C° for aluminum, 0.473 J/g·C° for steel, and 0.130 J/g·C° for lead. Since specific heat does exhibit some temperature dependence, a temperature range is specified for each of these values. They are 17-100°C, 18-100°C and 20-100°C, respectively. (These values are from the third edition of College Physics by Francis Weston Sears and Mark W. Zemansky, p. 309.) While this demonstration is not really quantitative, the lengths of the tracks that the metal cylinders leave in the wax are at least roughly in proportion to the heat capacities of the cylinders.
It is interesting to note that while the specific heats above vary quite widely among the three metals used in this demonstration, the molar heat capacities of these metals – the amount of heat necessary to raise the temperature of one mole of atoms of the metal one C° – are really quite close to each other. The molar masses of aluminum, iron and lead are 26.98 g/mol, 55.84 g/mol and 207.2 g/mol, respectively. Their molar heat capacities, then, are the values above multiplied by the molar masses, or 24.5 J/mol·C°, 26.4 J/mol·C° and 26.9 J/mol·C°, respectively. Thus, while the amount of heat required to raise the temperature of these substances by a particular number of degrees per unit mass varies significantly, the amount per atom for the different materials is almost the same. In 1918, Dulong and Petit noticed that for almost all elemental solids, the molar heat capacity was close to 6 cal/mol·C° (≈ 25 J/mol·C°). (For a few, for example carbon (as diamond), whose molar heat capacity is 6.11 J/mol·C°, the heat capacity was much different.) This observation is known as the law of Dulong and Petit (or Dulong and Petit law, or Dulong and Petit rule). In reality, molar heat capacities vary with temperature, approaching zero as temperature goes to 0 K, and approaching the Dulong and Petit value as temperature goes to infinity.
One can arrive at the law of Dulong and Petit by treating the atoms in a material as individual oscillators. Strictly speaking, the specific heats and molar heat capacities given above are cp, the heat capacity at constant pressure, which is easier to measure than cv, heat capacity at constant volume. For solids, the difference between the two is small (the volume does not change much as the substance is heated or cooled), and if one knows the physical properties of the material, one can correct for this small difference. (The relevant physical properties are the coefficient of thermal volume expansion, β, the isothermal compressibility, κ (= -ΔV/VΔp) and the density, ρ. The correction is: cp = cv + Tβ2/κρ.) When we use a model to derive an expression for the specific heat, however, if we allow only for the internal energy of our material to change and no work to be done on it (i.e., the temperature can change, but not the volume) what we calculate is cv. If we consider each atom to be an oscillator, the equation for its energy contains a kinetic energy term and a potential energy term, each of which affords the atom a degree of freedom in each dimension. You can also think of these as representing a vibrational degree of freedom and a translational degree of freedom, respectively. For the three dimensions, then, each atom has six degrees of freedom. If we apply the theorem of equipartition of energy, the average energy for each degree of freedom is 1/2 kT, and the average total energy for each atom is 3kT. For a mole of atoms the energy is 3N0kT, or 3(6.022 × 1023/mol)(1.380× 10-23J/K)T = 3(8.310 J/mol·K)T. This is also 3RT. So cv = (∂Q/∂dt) = 3R, which equals 24.9 J/mol·K (or, to be consistent with our notation above, 24.9 J/mol·C°).
For this classical approximation to work, kT must be much greater than the energy associated with the fundamental vibrational frequency of an atom in the material lattice, that is, kT >> hν, or T >> hν/k, where hν/k is the characteristic temperature of the material, called the Debye temperature and denoted as ΘD or TD. For most substances at room temperature, this condition holds. For silicon, for which cv is 19.8 J/mol·C°, it does not, and quantum effects are important. They are even more important for diamond, whose cv, as mentioned above, is 6.11 J/mol·C°. Of course, as we decrease the temperature, quantum effects become more and more important for all materials, and cv begins to decrease, eventually approaching zero, as mentioned above. The Debye temperature is named for Peter Debye, who developed the quantum mechanical theory that enabled him to determine the relationship between cv and temperature that agreed with experiment. He found that if one plots cv versus the (dimensionless) ratio T/TD, the molar heat capacities of a wide variety of substances fall on the same curve. (Actually, he plotted cv/c∞ versus T/TD, where c∞ is the value of the heat capacity as T approaches infinity, but this yields the same sort of curve.)
References:
1) Resnick, Robert and Halliday, David. Physics, Part One, Third Edition(New York: John Wiley and Sons, 1977), pp. 477-480.
2) Sears, Francis Weston and Zemansky, Mark W. College Physics, Third Edition (Reading, Massachusetts: Addison-Wesley Publishing Company, Inc., 1960), pp. 308-310.
3) Reif, Frederick. Statistical Physics; Berkeley Physics Course, Volume 5 (New York: McGraw-Hill Book Company, 1967), pp. 250-255.