Boltzmann relation

Boltzmann equation Boltzmann relation Boltzman statistics Bolvidon Bolzano process Bombesin [31362-50-2]... 

The potential of mean force is a useful analytical tool that results in an effective potential that reflects the average effect of all the other degrees of freedom on the dynamic variable of interest. Equation (2) indicates that given a potential function it is possible to calculate the probabihty for all states of the system (the Boltzmann relationship). The potential of mean force procedure works in the reverse direction. Given an observed distribution of values (from the trajectory), the corresponding effective potential function can be derived. The first step in this procedure is to organize the observed values of the dynamic variable, A, into a distribution function p(A). From this distribution the effective potential or potential of mean force, W(A), is calculated from the Boltzmann relation ... 

According to the Boltzmann relation, the entropy change AS for the process of compression is given by... 

For molar quantities, the general Boltzmann relation (Equation 5.3) shows that ... 

This is immedicately recognizable as the Boltzmann relation. Moreover, since r 0 + r c = 1, then... 

The wave equation is built from V E cx pext/ - Because electrostatic double-layer equations are easier to think about in terms of potentials rather than electric fields E = -V0, we set up the problem of ionic-charge-fluctuation forces in terms of potentials. Charges pext come from the potential 0 through the Boltzmann relation... 

Let us derive an expression for the entropy of mixing of the large particles with a fluid containing small particles. The procedure employed is similar to that of Ref. [17]. The entropy of mixing is given by the Boltzmann relation... 

Substituting Eq. (A.4) into the Boltzmann relation, and using the Stirling approximation for the factorials, one obtains... 

It is worth remembering that we are still working with the one-electron picture, and that we have applied the Boltzmann relation in order to approximate Fermi and quasi-Fermi distribution functions, assuming the quasi-free electron and hole densities of states in the bands. 

The work gained in bringing a hydrogen ion from the interior of a solution to the surface is —c . Hence, by the Boltzmann relation,... 

Saturation leads to equalization in the populations of the energy levels, contrary to the Boltzmann distribution. On the other hand, a number of NMR techniques can be employed to increase the population difference well beyond that given by the Boltzmann distribution. In some instances it is convenient to retain the formalism of the Boltzmann relation by defining a spin temperature Ts that satisfies Eq. 2.19 for a given ratio n /na. For times much less than T, it is meaningful to have Ts = T, the macroscopic temperature of the sample, because the spin system and lattice do not interact in this time frame.Viewed in this way, saturation... 

Many more-sophisticated models have been put forth to describe electrokinetic phenomena at surfaces. Considerations have included distance of closest approach of counterions, conduction behind the shear plane, specific adsorption of electrolyte ions, variability of permittivity and viscosity in the electrical double layer, discreteness of charge on the surface, surface roughness, surface porosity, and surface-bound water [7], Perhaps the most commonly used model has been the Gouy-Chapman-Stem-Grahame model 8]. This model separates the counterion region into a compact, surface-bound Stern" layer, wherein potential decays linearly, and a diffuse region that obeys the Poisson-Boltzmann relation. 

Electrostatic theories improved via a treatment in which the distribution of the neutral solute is evaluated around the ions of the electrolyte and compared with that around solvent molecules. The electrostatic polarization energy suffered by the neutral solute, because of the ion electrostatic field, generates a Boltzmann relation that sanctions the salting effects [41]. 

The fundamental relation between the entropy S of an assembly of molecules and the total number of distinguishable degenerate (i.e., of equal energy) arrangements of the molecules is given by the well-known Boltzmann relation ... 

Because of assumption 1, however, we need not find an explicit expression for U but can concentrate on the entropy expression. For this we shall again invoke the Boltzmann relation [Appendix 2, equation (g)], as we did for the isolated chain ... 

If the infrared radiation from an emitting body can be captured and transduced into a usable signal, it can be made the basis for measuring the temperature of the emitting surface. Such a method is particularly attractive for the measurement of the temperature at a rubbing interface, since it does not require the insertion of a probe which might introduce a major perturbation into the system. However the method does require that the behavior of the system obeys the Stefan-Boltzmann relation... 

Phonon velocity is constant and is the speed of sound for acoustic phonons. The only temperature dependence comes from the heat capacity. Since at low temperature, photons and phonons behave very similarly, the energy density of phonons follows the Stefan-Boltzmann relation oT lvs, where o is the Stefan-Boltzmann constant for phonons. Hence, the heat capacity follows as C T3 since it is the temperature derivative of the energy density. However, this T3 behavior prevails only below the Debye temperature which is defined as 0B = h( DlkB. The Debye temperature is a fictitious temperature which is characteristic of the material since it involves the upper cutoff frequency ooD which is related to the chemical bond strength and the mass of the atoms. The temperature range below the Debye temperature can be thought as the quantum requirement for phonons, whereas above the Debye temperature the heat capacity follows the classical Dulong-Petit law, C = 3t)/cb [2,4] where T is the number density of atoms. The thermal conductivity well below the Debye temperature shows the T3 behavior and is often called the Casimir limit. 

A common method to determine voltage sensitivity is shown in Figure 5A, in which the data for the larger channel from one experiment are fitted to a form of the Boltzmann relation that can be plotted as a linear function of... 

Figure 5. Voltage dependence of P0 for the larger (130-pS) connexin-32 channels. A, Fit to ln( P0/Pj) as a function of voltage. B, Fit to P0 as a function of voltage. The solid line is calculated with the Boltzmann relation (SSE — 0.007) and the dashed line is an independent fit (SSE = 0.001).

Suppose that the spin populations are now disturbed so that some additional nuclei in state a are promoted to state b. While the system may no longer be in thermal equilibrium with the lattice, we can still describe the populations by the Boltzmann relation provided that we change the definition of T to keep the relationship correct. Since the ratio is now... 

In NMR, we say that the spins are saturated and this corresponds to the situation immediately after a n/2 pulse. From the Boltzmann relation, Tg= and what s more Tg=- is a valid... 

There are certain consequences of the small size of the energy gap involved in a nuclear-spin transition. The Boltzmann relation gives the populations of nuclear spins in the upper energy state N and in the lower energy state Nj) in terms of the energy gap A between them ... 

The electric field in solution is given by the Poisson-Boltzmann relation. 

The value of O at the distance r = 1 / c, where the ionic atmosphere is most densely populated, gives an estimate of the validity of the Debye-Huckel approximation. Neglecting factors of the order of unity, it turns out that q = [ zeYlsrkT) K < 1 hence at 25°C, / c) < 5.10 v/z. Thus the linearised Poisson-Boltzmann relation underestimates the electrostatic interactions in polyvalent electrolytes and even for (1-1) salts in solvents of low dielectric constant. 

---