Heat capacity classical theory

Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

The decrease in the heat capacity at low temperatures was not explained until 1907, when Einstein demonstrated that the temperature dependence of the heat capacity arose from quantum mechanical effects [1], Einstein also assumed that all atoms in a solid vibrate independently of each other and that they behave like harmonic oscillators. The motion of a single atom is again seen as the sum of three linear oscillators along three perpendicular axes, and one mole of atoms is treated by looking at 3L identical linear harmonic oscillators. Whereas the harmonic oscillator can take any energy in the classical limit, quantum theory allows the energy of the harmonic oscillator (en) to have only certain discrete values ( ) ... 

In general a crystal that contains n atoms per unit cell have a total of 3L n vibrational modes. Of these there are 3L acoustic modes in which the unit cell vibrates as an entity. The remaining 3L(n - 1) modes are optic and correspond to different deformations of the unit cell. At high temperatures where classical theory is valid each mode has an energy k T and the total heat capacity is 3R, in line with the Dulong-Petit law. 

At high F, when the spacing of vibrational energy levels is low with respect to thermal energy, crystalline solids begin to show the classical behavior predicted by kinetic theory, and the heat capacity of the substance at constant volume (Cy) approaches the theoretical limit imposed by free motion of all atoms along three directions, in a compound with n moles of atoms per formula unit limit of Dulong and Petit) ... 

The expressions in (3.72) and (3.73) are valid only for monatomic ideal gases such as He or Ar, and must be replaced by somewhat different expressions for diatomic or polyatomic molecules (Sidebar 3.8). However, the classical expressions for polyatomic heat capacity exhibit serious errors (except at high temperatures) due to the important effects of quantum mechanics. (The failure of classical mechanics to describe the heat capacities of polyatomic species motivated Einstein s pioneering application of Planck s quantum theory to molecular vibrational phenomena.) For present purposes, we may envision taking more accurate heat capacity data from experiment [e.g., in equations such as (3.84a)] if polyatomic species are to be considered. The term perfect gas is sometimes employed to distinguish the monatomic case [for which (3.72), (3.73) are satisfactory] from more general polyatomic ideal gases with Cv> nR. 

It is noteworthy that Gibbs himself was acutely aware of the qualitative failures of 19th-century molecular theory (as revealed, for example, by erroneous classical predictions of heat capacities Sidebar 3.8). In the preface to his Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics (published in the last year of his life), Gibbs wrote ... 

Both the classical and statistical equations [Eqs. (5.22) and (5.23)] yield absolute values of entropy. Equation (5.23) is known as the Boltzmann equation and, with Eq. (5.20) and quantum statistics, has been used for calculation of entropies in the ideal-gas state for many chemical species. Good agreement between these calculations and those based on calorimetric data provides some of the most impressive evidence for the validity of statistical mechanics and quantum theory. In some instances results based on Eq. (5.23) are considered more reliable because of uncertainties in heat-capacity data or about the crystallinity of the substance near absolute zero. Absolute entropies provide much of the data base for calculation of the equilibrium conversions of chemical reactions, as discussed in Chap. 15. 

If the expression for the translational partition function is inserted into equation (16.8), it is readily found, since tt, m, fc, h and V are all constant, that the translational contribution Et to the energy, in excess of the zero-point value, is equal to %RT per mole, which is precisely the classical value. The corresponding molar heat capacity at constant volume is thus f P. As stated earlier, therefore, translational energy may be treated as essentially classical in behavior, since the quantum theory leads to the s ame results as does the classical treatment. Nevertheless, the partition function derived above [[equation (16.16) [] is of the greatest importance in connection with other thermodynamic properties, as w ill be seen in Chapter IX. 

The behaviour of the heat capacities of substances both in the solid and gaseous phases was a mystery to scientists before the discovery of quantum theory. In classical mechanics where energy is regarded as continuously variable it can be shown that the energy of a system is divided equally between the various modes of motion called degrees of freedom. Furthermore, according to classical physics each degree of freedom contributed RT/ 2 to the... 

For gas molecules, the heat capacity is a constant equal to C = (n/2)pkB where n is the number of degrees of freedom for molecule motion, p is the number density, and kB is the Boltzmann constant. The rms speed of molecules is given as v = V3kBTlm, whereas the mean free path depends on collision cross section and number density as = (pa)-1. When they are put together, one finds that the thermal conductivity of a gas is independent of p and therefore independent of the gas pressure. This is a classic result of kinetic theory. Note that this is valid only under the assumption that the mean free path is limited by inter-molecular collision. 

Because only a few higher energy levels will be populated by electrons as the temperature rises, the electrons do not contribute appreciably to the specific heat. Thus, the electronic heat capacity is almost negligible, and a major drawback of the classical theory has been corrected. 

In equilibrium thermodynamics model A and in model B not far from equilibrium (and with no memory to temperature) the entropy may be calculated up to a constant. Namely, in both cases S = S(V, T) (2.6)2, (2.25) and we can use the equilibrium processes (2.28) in B or arbitrary processes in A for classical calculation of entropy change by integration of dS/dT or dS/dV expressible by Gibbs equations (2.18), (2.19), (2.38) through measurable heat capacity dU/dT or state Eqs.(2.6>, (2.33) (with equilibrium pressure P° in model B). This seems to accord with such a property as in (1.11), (1.40) in Sects. 1.3, 1.4. As we noted above, here the Gibbs equations used were proved to be valid not only in classical equilibrium thermodynamics (2.18), (2.19) but also in the nonequilibrium model B (2.38) and this expresses the local equilibrium hypothesis in model B (it will be proved also in nonuniform models in Chaps.3 (Sect. 3.6), 4, while in classical theories of irreversible processes [12, 16] it must be taken as a postulate). 

If a solid were classical, the heat capacity would be 3 Alfc. This is indeed the case at high temperatures and is called the law of Dulong and Petit. However, the experimental heat capacity goes to zero at low temperatures. This can be explained by regarding the solid as a collection of quantized oscillators. The only difficulty is to determine the spectrum of frequencies of the oscillators. For many purposes, the solid can be regarded as an elastic continuum. The result is the Debye theory. If something more sophisticated is needed one must solve for the normal modes of the crystal, i.e., the method of lattice dynamics. 

As shown above, the heat capacity Cy (or at any rate that part of it which is due to the vibrations) may be expected to have a value of 3Jt whenever hvjkT 1. This would be so, even at the lowest temperatures, if Planck s constant h were zero, and this is the case in the classical or pre-quantum mechanics. In fact, classical theory leads to the expectation that, for any crystalline substance, Cy has the constant value of 3R per mole. This is contrary to experiment, and it is known that Cy usually diminishes below 3A, with fall of temperature, and seems to approach zero at the absolute zero. One of the early successes of the quantum theory consisted in finding the reason for this decrease in Cy which is quite inexplicable in classical theory. The explanation is implicit in the previous equations and is due to the fact that the oscillators can only take up finite increments of energy. When a system of oscillators is held at low temperature, most of them are in their lowest energy level, and a small rise oftemperature is insufficient to excite them to the next higher level. Therefore Cy, which measures the intake of energy per unit increase of temperature, is smaller than at higher temperature. 

Heat capacities are of fundamental importance in glass transition theories, because the measure of the heat absorbed provides a direct measure of the increase in molecular motion. The use of classical thermodynamics leads to an easy introduction of the pure-component heat capacities at constant pressure, Cpi and Cp2 ... 

Although the fundamental function principle of nanocalorimeters is not changed, the theory of such calorimeters and the mathematics for the deconvolution of the sample properties from the measurements are much more complicated than for common calorimeters. The pathway of the heat flow cannot be approximated by a one-dimensional model, and the heat capacity of the sample is often in the same order of magnitude as the heat capacity of the calorimeter system, which in many chips consists only of a very thin silicon membrane. In classical calorimeters, the calorimeter system is much larger in mass and heat capacity than the sample. This is desirable to avoid an influence of the sample on the sensitivity of the calorimeter and to keep the calibration factor a device property and free it from sample properties. 

Fio. 9.3. Constant-volume heat capacity at a fixed density equal to the critical density, as a function of the temperature near the critkai temperature T°. (a) Non-dassical critical point (A-point), as in a real fluid (b) classic critical point, as in mean-field theory. 

Metals provided additional problems for the classical theory of heat capacity. Metals are generally much better conductors of heat than nonmetals because most of the heat is carried by the free electrons. According to the classical theory, this electron gas should contribute an additional (3/2)R to the heat capacity. But the measured heat capacity of metals approached nearly the same 3R Dulong-Petit limit as the nonmetals. How can the electrons be a major contributor to the thermal conductivity and not provide significant additional heat capacity ... 

Careful measurements at low temperatures indicated that metals do have a small electronic contribution to heat capacity that approaches 0 with a first power dependence on T, as will be shown later. In fact, at very low temperatures, the electronic contribution can be greater than the lattice contribution. In order to explain these departures from classical theory, we must reformulate the problem quantum mechanically. 

---