Electronic heat capacity theory

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. 

However, it is possible to observe the electron heat capacity of metals at very low temperatures. Since theory predicts that only electrons within an energy range kT of the Fermi level of electron energy states will accept thermal energy from the lattice, the electronic heat capacity will be linearly proportional to the absolute temperature. 

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. 

From a computational point of view, the heat of formation, which is derived from the electronic energy of the molecule molecule> is the most difficult thermochemical quantity to predict accurately. Entropies and heat capacities are derived from vibration and rotational constants, all of which can be predicted with considerable accuracy using relatively low levels of theory. Thus, the development of ab initio methods appropriate for a new class of compounds focuses primarily on identifying a level of theory and the basis set(s) needed to achieve sufficient accuracy in the electronic energy [67,68]. 

In practice, then, it is fairly straightforward to convert the potential energy determined from an electronic structure calculation into a wealth of thennodynamic data - all that is required is an optimized structure with its associated vibrational frequencies. Given the many levels of electronic structure theory for which analytic second derivatives are available, it is usually worth the effort required to compute the frequencies and then the thermodynamic variables, especially since experimental data are typically measured in this form. For one such quantity, the absolute entropy 5°, which is computed as the sum of Eqs. (10.13), (10.18), (10.24) (for non-linear molecules), and (10.30), theory and experiment are directly comparable. Hout, Levi, and Hehre (1982) computed absolute entropies at 300 K for a large number of small molecules at the MP2/6-31G(d) level and obtained agreement with experiment within 0.1 e.u. for many cases. Absolute heat capacities at constant volume can also be computed using the thermodynamic definition... 

Lorentz1 advanced a theory of metals that accounts in a qualitative way for some of their characteristic properties and that has been extensively developed in recent years by the application of quantum mechanics. He thought of a metal as a crystalline arrangement of hard spheres (the metal cations), with free electrons moving in the interstices.. This free-electron theory provides a simple explanation of metallic luster and other optical properties, of high thermal and electric conductivity, of high values of heat capacity and entropy, and of certain other properties. 

The heat capacity at constant volume Cv from the translational and rotational degrees of freedom are determined via Eqs. 8.124 and 8.128, the vibrational contributions to Cv are calculated by Eq. 8.129, and the electronic contribution to Cv is from Eq. 8.123. For an ideal gas, Cp = Cu + R, so Cp=41.418 J/mole/K. The experimental value is Cp=38.693 J/mole/K. Agreement with experiment gets better at higher temperature. At 1000 K, Cp from our calculation is 59.775 J/mole/K, compared to a value of 58.954 from the NIST-JANAF Tables. The difference between theory and experiment is due entirely to our use of the vibrational frequencies obtained from the ab initio results, rather than using the experimental frequencies. 

This is a heat capacity proportional to the temperature, and in Sec. 5, Chap. V, we computed it for a particular case, showing that it amounted to only about 1 per cent of the corresponding specific heat of free electrons on the Boltzmann statistics, at room temperature. In Table XXIX-2 we show the value of the electronic specific heat at 300° abs., computed from the values of Wi which we have already found, in calories per mole. We verify the fact that this specific heat is small, and for ordinary purposes it can be neglected, so that the specific heat of a metal can be found from the Debye theory, considering only the atomic vibrations. At low temperatures, however, Eq. (2.4) gives a specific heat varying as the first power of the temperature, while Debye s theory, as given in Eq. (3.8),... 

The development of the quantum theory was at first slow. It was not until 1905 that Einstein2 suggested that the quantity of radiant energy hv was sent out in the process of emission of light not in all directions but instead unidirectionally, like a particle. The name light quantum or photon is applied to such a portion of radiant energy. Einstein also discussed the photoelectric effect, the fundamental processes of photochemistry, and the heat capacities of solid bodies in terms of the quantum theory. When light falls on a metal plate, electrons arc emitted from it. The maximum speed of these photoelectrons, however,... 

MO bond order One-half the difference between the numbers of electrons in bonding and antibonding MOs. (336) model (also theory) A simplified conceptual picture based on experiment that explains how an aspect of nature occurs. (9) molality (m) A concentration term expressed as number of moles of solute dissolved in 1000 g (I kg) of solvent. (403) molar heat capacity (C) The quantity of heat required to change the temperature of I mol of a substance by 1 K. (187) molar mass [M) (also gram-molecular weight) The mass of 1 mol of entities (atoms, molecules, or formula units) of a substance, in units ofg/mol. (72)... 

A model system for DSC was developed by Flynn (16) in which the electronic response of the instrument is coupled with the heat flow across an interface. Equations are derived that relate the time constants for this two-step process with the thermal properties of the sample and the amplitudes, areas, slopes, and dwell time of the DSC curves. Flynn (17) has also developed a simple theory to utilize DSC for the determination of heat capacities, glass transition, and enthalpies of transition. 

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