Boltzmann exponential

For small deviations from electroneutrality, the charge density at x is proportional to -(x)/kT9 where < is the difference of the electrostatic potential from its (constant) value when there is no charge density (the density of a species of charge z is proportional to 1 - zkT on linearizing the Boltzmann exponential). Then the Poisson equation [Eq. (44)] becomes the linearized Poisson-Boltzmann equation ... 

To obtain a useful approximate solution of the PB equation (S8.6-4), we consider the dilute limit in which the electric potential O is weak compared with the ambient thermal energy kT. In this limit, the Boltzmann exponential can be linearized by retaining only the leading term in the power series expansion... 

Where several species are in equilibrium, the ratios are proportional to their Boltzmann exponential factors. For example, if the relative free energies G of A, B and C are 0, 5.0 and 20.0 kJ mol1 (here G for species A has been set to zero and B and C he 5.0 and 20.0 kJ mol1 higher) then... 

Because of the v dependence, blue light is scattered more efficiently than red (a fact that accounts for the blue color of the sky). However, the Boltzmann exponential factor is dominant in Eq. (6), and the anti-Stokes features are always much weaker than the corresponding Stokes lines. 

There is considered to be one state for every independent wave function J (a). The exponential factor, called the Boltzmann exponential factor, is the same as in the classical Boltzmann distribution law, which differs from Equation 49-1 only in the way the state of the system part is described. The constant k is the Boltzmann constant, with the value 1.3709 X 10 1 erg deg-1. The absolute temperature T occurring in Equation 49-1... 

Pitts used an approach in which the Poisson-Boltzmann exponential term was expressed as a series, but where only a few terms in the exponential were retained though more than in Debye-Hiickel. This also gave higher order terms in concentration. 

Perikinetic and Orthokinetic Flocculation. Particles in a dispersion are subjected under any circumstances to Brownian motion (the average velocity is distributed according to the Boltzmann exponential law) and hence, it is the absolute barrier height compared to the thermal energy kpT) that determines stability. This means that larger particles are more stable against thermal motion (perikinetic flocculation) than smaller particles. However, of practical importance during heterophase polymerization is the behavior of particles in dependence on the stirrer speed (orthokinetic flocculation), where their velocity depends on the... 

The subscript (o) emphasizes the difference between this type of average (where all states are equally weighted) and a thermal average (where they are weighted by the Boltzmann exponential exp (— /t) thermal averages are written ( ) (without a subscript). The relationship between the two types of averages is, for any fimction G(Si... S<... ) of the spins ... 

If ionic size correlations between ionic species i and j are neglected, c fnnctions vanish. Consequently, Equation 3.6 becomes the widely known Boltzmann exponential expression. 

The characteristic Boltzmann exponential dependence of conductivity on temperature was observed only on samples with crystal 3-D-polymerized structure as, for example, in sample No. 2 (Figure 16.18a). As seen in Figure 16.18a, the curve... 

The experimentally observed dependence (a in a wide temperature range (4-300 K) may indicate that the temperature dependence of mobility is of exponential character and compensates the Boltzmann exponential component. It is known, for example, that in diamonds the mobility of holes depends as T on temperature, where the parameter s smoothly changes from 1.5 below 400 K up to 3 above 400 K. In the assumption of two types of charge carriers the total conductivity is their superposition ... 

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