Boltzmann’s factor

The proportion of molecules having a particular energy, at temperature T is given by the Boltzmann s factor (assuming only translational energy along two degrees of freedom) as... 

The local mole fractions cannot be measured easily, but must be related to the overall composition. A simple correlation can be obtained from statistical thermod5mamics for which the quotient of the local mole fractions equals the quotient of the total mole fractions times a Boltzmann s factor according to ... 

The conclusion that the free-volume fraction at Tg is not a universal parameter for linear polymers of differing molecular structure can be qualitatively confirmed by the following arguments71. Assume that at temperatures far below Tg polymeric chains are in a state of minimum energy of intramolecular interaction, Le. the fraction of higher-energy ( flexed ) bonds is zeroS4. On the other hand, let the equilibrium fraction of flexed bonds at T> Tg obey the Boltzmann statistics and be a function of Boltzmann s factor e/kT. Thus, the fraction of flexed bonds at Tg can be estimated from the familiar expression ... 

Let us now consider a ferromagnetic system made of spins a (r) attached to the lattice sites. We shall admit that the spin-spin interaction is given by a Hamiltonian JV a) which will be assumed to be translationally invariant. At equilibrium, the weight of a configuration of the system is given by the Boltzmann s factor... 

Motivating forces of complex-formation are interactions of electrostatic charges and chemical affinity of functional groups in minerals and ions in the solution. Charges form electrostatic field, which defines Boltzmann s factor and forces ions to move according to the laws of electrostatics. 

Here kB is the Boltzmann s constant, A represents a normalization factor, W, and W2 are the activation energies for a thermally enhanced decay of the photoin-duced states, E is the temperature-independent part of the decay factor, and B represents the relative weight of the HVaclivalcd process. 

The calculated energy of interaction of an atomic moment and the Weiss field (0.26 uncoupled conduction electrons per atom) for magnetic saturation is 0.135 ev, or 3070 cal. mole-1. According to the Weiss theory the Curie temperature is equal to this energy of interaction divided by 3k, where k is Boltzmann s constant. The effect of spatial quantization of the atomic moment, with spin quantum number S, is to introduce the factor (S + 1)/S that is, the Curie temperature is equal to nt S + l)/3Sk. For iron, with 5 = 1, the predicted value for the Curie constant is 1350°K, in rough agreement with the experimental value, 1043°K. 

The Schottky-Mott theory predicts a current / = (4 7t e m kB2/h3) T2 exp (—e A/kB 7) exp (e n V/kB T)— 1], where e is the electronic charge, m is the effective mass of the carrier, kB is Boltzmann s constant, T is the absolute temperature, n is a filling factor, A is the Schottky barrier height (see Fig. 1), and V is the applied voltage [31]. In Schottky-Mott theory, A should be the difference between the Fermi level of the metal and the conduction band minimum (for an n-type semiconductor-to-metal interface) or the valence band maximum (for a p-type semiconductor-metal interface) [32, 33]. Certain experimentally observed variations of A were for decades ascribed to pinning of states, but can now be attributed to local inhomogeneities of the interface, so the Schottky-Mott theory is secure. The opposite of a Schottky barrier is an ohmic contact, where there is only an added electrical resistance at the junction, typically between two metals. 

Availability change to form embryo Initial bubble diameter Frequency factor in nucleation Enthalpy of vaporization Rate of formation of critical-sized embryos per unit volume Jacob numter [Eq. (17)] Boltzmann s constant or thermal conductivity... 

This introduces a new unknown (and free at this stage) coefficient K. In the case of a system of molecules in a thermal bath (definitely not the one we consider), there is a relation between D and K such that, at thermal equilibrium, the equilibrium density in the potential (h is given by Boltzmann s law. This requires that K = mD/hgT, where kg is Boltzmann s constant and T the absolute temperature. In Eq. (12) the factor p in front of in is to ensure that, if... 

Whenever the system finds itself at the saddle point it will pass into one of the valleys on either side in a short time. The rate at which the system moves through the saddle point can be determined in a relatively simple way because the energy of the system is nearly constant for a short distance in the direction of the path through the saddle point. The frequency factor which determines the rate of flow of systems which have reached the barrier is kT/h, where k is Boltzmann s constant,. T is the absolute temperature and h is Planck s constant. 

Here Nc is the density of states in the conduction band, g the level degeneracy factor, n the carrier concentration in the band, A the activation energy of the level, Boltzmann s constant, and T the temperature. Now, in general, except at fairly low temperatures, the occupancy for shallow levels (with/ = /s) will be small, i.e., fs 1, and consequently... 

Factor 1 The exponential of the empty water lattice Helmholtz free energy divided by kT, where k is Boltzmann s constant,... 

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