Having briefly discussed atomic structure, bonding and the classification of solids, we now consider their thermal properties.

Transcription

1 Thermal Energy of Solids. Introduction. Having briefly discussed atomic structure, bonding and the classification of solids, we now consider their thermal properties. The thermally induced vibrations of the atoms that comprise a crystal are responsible for such physical properties as heat capacity, temperature coefficient of thermal expansion and thermal conductivity. The most important ways in which a solid may absorb thermal energy are: Stimulation of atomic vibration Stimulation of electronic motion or excitation Stimulation of molecular rotation Of these, atomic vibration is common to all solids and is the most important of the three with respect to the behavior of solids. Internal Energy and Heat Capacity. The total energy of a solid consists of two parts, the thermal energy and any other energy which might exist at 0 K; the sum of these components is termed the internal energy, U. The internal energy is a well-defined bulk property, which is dependent upon temperature. Of equal or perhaps greater importance, is the temperature derivative of U, the heat capacity at constant volume (C v ): in which C v has units of cal / mole / K 1 of 11

2 A precise calculation of C v for a solid is extremely difficult because it requires knowledge of every oscillating atom in the solid. However, a successful result can be obtained by the use of extensive approximations about the behavior of the oscillating atoms. Classical Theory: Approach of Dulong and Petit Dulong and Petit showed that the heat capacities of many substances were related to their atomic weights, i.e. the product of their specific heats and atomic weights being approximately constant at -- 6 cal / mole / K. In fact, this is only an approximation so that for many elements the heat capacity lies between 5-7 cal /mol / K at 0 o C with an average value of 6.2. The heat capacities are not constant but increase by approximately 0.04%/ o C for temperatures above 0 o C. Curves have a common sigmoidal shape where in the low temperature range C v varies as T 3. (a) The heat capacities of some elements. (From F. K. Richtmyer, E. H. Kennard, and T. Lauritsen, Introduction to Modern Physics, 5th ed., p. 410, McGraw-Hill, New York, 1955.) 2 of 11

3 The "universal" curve (from F. Seitz, Modern Theory of Solids, p. 109, McGraw-Hill, New York, 1940). Classical approach. The classical theory assumed that all the internal energy of a solid could be considered to reside in the ion cores of a solid. A solid was thought of as being composed of an assembly of non-interacting ion cores that behaved as simple harmonic oscillators, vibrating about an equilibrium position, in thermal equilibrium at a given temperature. This approach neglects any contribution of the valence electron to the internal energy of the solid. The thermal equilibrium of the ion cores is treated, as in the case of ideal gases, as though the energy distribution is continuous and makes use of Maxwell-Boltzmann statistics. The ions were considered to have three degrees of freedom that correspond to their energies of translation parallel to the three Cartesian coordinates. 3 of 11

4 Therefore _ U i = KE + PE = kt + kt = kt (cal / ion) _ 6 U I = N A U i = N A kt (cal / mole) 2 _ 6 U I = RT 2 Letting R 2 (cal / mole) / K and taking the definition of heat capacity as du / dt: C v 6 (cal / mole) / K This result is in agreement with the findings of Dulong and Petit but fails to show a temperature dependence for the heat capacity. This arises because the solid is treated as an assembly of independent oscillators The very close proximities of the ions in the three dimensional arrays that constitute crystalline solids cause this assumption to give a great oversimplification. In fact, the oscillations of a given ion affect those of its neighbors. These in turn influence their neighbors and so on. In addition, if the internal energy of a solid resides primarily in the ions, their amplitudes of oscillation must be expected to vary with temperature. Thus, a more realistic description of heat capacity must take these factors in account. 4 of 11

5 Einstein model. Einstein studied the classical prediction and observed that significant deviations were found at low temperatures. Einstein utilized a quantum mechanics approach and incorporated Plank's hypothesis of discrete vibrational frequencies. He assumed that the internal energy of a solid was associated only with the ion core, i.e., the energy of the electrons was not taken into account. Hence, the solid was treated as an assembly of independent simple harmonic oscillators in thermal equilibrium whereby each oscillated with the same frequency, v. The energy of a single quantized oscillator, E = hv, is called a phonon. The average energy of such an oscillator is: where h is Plank s constant. This expression is quite different from that given by classical mechanics: E = kt. Now, if each ion of the solid: independently oscillated at the same frequency, v o, and has three translational degrees of freedom, 5 of 11

6 then the internal energy of a mole of the solid is given by: where N A is Avogadro's number. Therefore, Now multiplying the numerator and denominator by k. At very high temperatures kt becomes much larger than hv o, and so, where the term [exp (-hv o / kt)] can be expanded using: 6 of 11

7 e -x = 1 x + and keeping only the lowest order terms in hv o / kt. This gives C v = 3R in accord with classical calculations. At low temperatures C v decreases and approaches zero in an exponential fashion so that the model accounts qualitatively for the lowering of the heat capacity at low temperatures. Einstein's model is a simplification because it requires all atoms in the solid to oscillate with the same, single frequency whereas they actually vibrate with a range of frequencies. At very high temperatures kt becomes much larger than hv o, and so, where the term [exp (-hv o / kt)] can be expanded using: e -x = 1 x + and keeping only the lowest order terms in hv o / kt. This gives C v = 3R in accord with classical calculations. At low temperatures C v decreases and approaches zero in an exponential fashion so that the model accounts qualitatively for the lowering of the heat capacity at low temperatures. Einstein's model is a simplification because it requires all atoms in the solid to oscillate with the same, single frequency whereas they actually vibrate with a range of frequencies. 7 of 11

8 Debye Theory. In the Debye approach, the solid is treated as an isotropic, homogeneous medium instead of an ensemble of oscillating particles. In effect, the oscillations are averaged over the frequencies present. As a consequence of this approach, a temperature, the Debye temperature, θ D, is defined by equating the classical and quantum expressions for energy: kθ D = hv c or θ D = hv c /k where v c is the maximum frequency. This effectively defines θ D as the maximum temperature at which quantum mechanics must be used. At higher temperatures, classical mechanics can be utilized. 8 of 11

9 Comparison of theories. 9 of 11

10 A more common way of presenting heat capacity results is: 10 of 11

11 At 0 K, the heat capacity is zero. At temperatures above 0 K it climbs rapidly and is proportional to T 3 in this region. At high temperatures it reaches a nearly constant value of approximately 6 cal/mole/k. The constant θ D has a different value for each solid. When T = θ D, C v reaches approximately 96% of its final value. Representative values of θ D are given in the following table of Debye Temperatures θ D in Kelvin for a range of different materials. 11 of 11