Boltzmann constant model

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,... 

K). T is the measurement temperature and Tq is the "degeneracy temperature," equal to kEo, where k is the Boltzmann constant. According to a two-dimensional electron gas model for graphitic carbons (see ref. 2a), is the energy "shift" from the Fermi level (Ep), to the top of the valence band. Small values of To ( <344 K) and consequently of Eq signify a more perfect graphite... 

The Gaussian chain model yields a spring constant even for a single bond k=3k T/ , where k is the Boltzmann constant. From Eq. 3.3 the chain extension between arbitrary points along the chain may be computed to R n) -R(m)y)= n - m ... 

Equation 5.23 is considered to be an ideal solid solution model. If we choose to extend our equations from one hydrate crystal to a large number Na (Avogadro s number) of crystals, we must replace the Boltzmann constant k with the universal gas constant R (=kN ). Ballard (2002) defined the chemical potential of water in hydrates as... 

The formulas for the statistical characteristics are given as a generalization of the expressions obtained in Section IE. These characteristics depend now on three model parameters angular well width p, mean lifetime x, and the reduced well depth u = Uo/(kBT), where fcB is the Boltzmann constant and T is the temperature. 

The basic features of ET energetics are summarized here for the case of an ET system (solute) linearly coupled to a bath (nuclear modes of the solute and medium) [11,30]. We further assume that the individual modes of the bath (whether localized or extended collective modes) are separable, harmonic, and classical (i.e., hv < kBT for each mode, where v is the harmonic frequency and kB is the Boltzmann constant). Consistent with the overall linear model, the frequencies are taken as the same for initial and final ET states. According to the FC control discussed above, the nuclear modes are frozen on the timescale of the actual ET event, while the medium electrons respond instantaneously (further aspects of this response are dealt with in Section 3.5.4, Reaction Field Hamiltonian). The energetics introduced below correspond to free energies. Solvation free energies may have entropic contributions, as discussed elsewhere [19], Before turning to the DC representation of solvent energetics, we first display the somewhat more transparent expressions for a discrete set of modes. 

The idea of a thermodynamic temperature scale was first proposed in 1854 by the Scottish physicist William Thomson, Lord Kelvin [iv]. He realized that temperature could be defined independently of the physical properties of any specific substance. Thus, for a substance at thermal equilibrium (which can always be modeled as a system of harmonic oscillators) the thermodynamic temperature could be defined as the average energy per harmonic oscillator divided by the Boltzmann constant. Today, the unit of thermodynamic temperature is called kelvin (K), and is defined as the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. 

In this method a random number generator is used to move and rotate molecules in a random fashion. If the system is held under specified conditions of temperature, volume and number of molecules, the probability of a particular arrangement of molecules is proportional to exp(-U/kT), where U is the total intermolecular energy of the assembly of molecules and k is the Boltzmann constant. Thus, within the MC scheme the movement of individual molecules is accepted or rejected in accordance with a probability determined by the Boltzmann distribution law. After the generation of a long sequence of moves, the results are averaged to give the equilibrium properties of the model system. 

---