The specific heat (molar heat capacity) is the amount of heat needed to raise the temperature of a g mol of material by 1 degree at constant pressure (Cp) or constant volume (CV). Thermodynamically, they can be written as

where U and H are the internal energy and enthalpy of the system, respectively. It is experimentally more suitable to evaluate the specific heat at constant pressure Cp (atmospheric). However, it is theoretically easier to predict Cv. Cp is always greater than Cv; at room temperature or below; the difference is very small for solids

(whereas for ideal gas, CP — CV = R). The SI unit of specific heat is generally given in J mol—1 K—1.

Petit and Dulong [12] conducted experiments and suggested an empirical rule that states that all solid elements have CV of 3R (where R is the universal gas con­stant, that is, ~24.9 J mol—1 K—1). Kopp [13] introduced an empirical rule that the molar heat capacity of a solid chemical compound is approximately equal to the sum of molar heat capacities of its constituent elements. Even though the molar heat capacity values for elements tend to be close to 3R near room temperature, they increase with temperature and become smaller than 3R at lower temperatures. Figure 5.52 shows CV for four pure elements (lead, copper, silicon, and diamond) as a function of temperature. Only metals, lead, and copper show specific heat close to 3R. However, for other two elements, they are quite different.

The classical physics could not explain why the specific heat changes with increasing temperature. During the early part of the twentieth century, the concept of specific heat was explained with the help of quantum theory. The problem was first treated by Einstein in 1907 by assuming that atoms in a crystal (called Einstein crystal) behave like independent harmonic oscillators. But Einstein’s derivation could not explain the experimental data near the absolute zero. Later in 1912, Debye assumed that the range of frequencies available to the oscillators is the same as that available in a homogeneous elastic continuum. Basically, Debye assumed that the vibration occurs in a coordinated way (because of the presence of atomic bonding) as opposed to Einstein’s assumption of independent oscillation. They were considered as elastic lattice waves (much like sound waves) that have high frequency with small amplitude. The lattice vibrational energy can be quantized (phonon) in a similar way as done for light waves (photon). An expression for con­stant volume specific heat developed by Debye provided good agreement with

where x = hv/kT is dimensionless, 0D = hvDfkT is Debye temperature, T is the tem­perature in K, R is the universal gas constant, and vD is the maximum frequency of atomic vibration (called Debye frequency). We should be cognizant of two facts regarding the function given in Eq. (5.80): (i) At higher temperatures up to room temperature (i. e., a low value of 0D/T), Cv tends to reach the value as predicted by Dulong and Petit. (ii) The expression also gives good accord to the experimental data at very low temperatures when Debye’s original expression is reduced to

(5.81)

This equation correctly captures the rapid approach toward zero as the temperature reaches the absolute zero. Figure 5.53 shows the Cv values as a function of temper­ature for four different elements. If the data are normalized with respect to 0D, all data come together and fall on a single master curve.

It should be noted that Cp rather than Cv is measured more often experimentally. In order to relate the theories to experiments, (Cp — Cv) needs to be known from an expression as shown in Eq. (5.82):

a2VT

Cp — Cv = ь ; (5.82)

where a is the coefficient of thermal expansion, ((dln V)/@T)p, and b is the coefficient of compressibility, — ((d ln V)=@P) T.

However, at a temperature much above the room temperature, Cv gets larger than that predicted by Dulong and Petit. The nonapplicability of Debye’s theory in

this temperature range stems from the fact that electrons of the atoms not only absorb energy but also increase their energy and this electronic contribution was not considered in Debye’s analysis. However, this is only possible for free electrons; that is, the electrons that have been excited from the filled states above the Fermi energy level. This type of situation contribution is certainly valid for metals/ alloys that have free electrons. However, they are still a small fraction of total elec­trons. In insulating and semiconducting materials, the electronic contribution is quite insignificant. The electronic contribution to specific heat bears a linear pro­portionality with temperature (i. e., CV/ T). There are also few other contributions to specific heat depending on the system, such as randomization of electron spins in a ferromagnetic material (like alpha-iron) as it is heated through the Curie tem­perature. A large spike in the specific heat can be observed at that temperature in a differential scanning calorimetry (DSC) curve.

Often the experimentally measured variation in the constant pressure specific heat (CP) is described by empirical fitting of the form shown below:

CP = a + bT + cT-2, (5.83)

where a, b, and c are curve fitting constants. Figure 5.54 shows a number of CP versus temperature curves for various elements and compounds that exhibit poly­morphism. Note the sharp changes at the temperature where the polymorphic transformations take place.