Boltzmann exponential expression

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. 

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. 

Different forms of (60) derived for a variety of systems have the exponential temperature dependence in common. In practice 4>a is invariably much larger than kT. The Boltzmann factor therefore increases rapidly with temperature and the remainder of the expression may safely be treated as constant over a small temperature range. The activation energy may be determined experimentally from an Arrhenius plot of In R vs 1/T, which should be a straight line of slope —A/k. 

This argument shows simply where the Arrhenius temperature dependence of reaction rates originates. Whenever there is an energy barrier that must be crossed for reaction, the probability (or rate) of doing so is proportional to a Boltzmann factor. We will consider the value of the pre-exponential factor and the complete rate expression later. 

The next problem is to express the charge density as a function of the potential so the differential equation (26) can be solved for f/. The procedure is to describe the ion concentrations in terms of the potential by means of a Boltzmann factor in which the work required to bring an ion from infinity to a position at which the potential p is given by z,e p. The probability of finding an ion at this position is given by the Boltzmann factor, with this work appearing as the exponential of energy ... 

In case that the energy can be expressed as the sum of two terms, one depending only on the coordinates (the potential energy) and the other only on the momenta (the kinetic energy), the Boltzmann distribution law for coordinates can be discussed separately from that for momenta, because the Boltzmann factor can be split into the product of two exponential terms, one involving only the coordinates and the other only the momenta. 

The form of distribution (17) recalls a Boltzmann expression with modulus of distribution 7. Attempts at a direct physical explanation of this result are thwarted by the obvious dependence of 7, not only on the state of the surface, but also on the nature of the gas whose adsorption proceeds according to equation (1). Nevertheless, formula (17) makes very plausible the experimentally observed constancy of the functional dependence A(Q) itself which leads to equation (1). It seems natural that with training or sintering of the surface, the liberation or destruction of points with different heats of adsorption may proceed in such a way as to preserve the exponential relation between A and Q, changing only the constants D, Q0, and especially 7. 

Here A is the pre-exponential factor, e0 the excess energy of the complex activated compared with the energy of the initial particles, K the Boltzmann constant, Zj the fraction of the surface occupied by the 7-type adsorbed particles, z0 the free surface fraction, p, the partial pressures of gaseous substances, and ml the number of elementary sites occupied by the activated complex. An expression to calculate the pre-exponential factor A has been given elsewhere [36]. ... 

Essentially different situation is encountered in the case of a high potential barrier or, equivalently, at low temperatures both these conditions are expressed by the relation ct 1. Applying the Boltzmann law (4.24) to the two-well potential (4.33), we arrive at the conclusion that the orientational probability is almost totally localized in exponentially small vicinities of the directions If = 0 and vf = re. It is also obvious that in a system with the energy function (4.33) at full equilibrium, the populations of both wells are equal. 

In the last expression, Vo is a pre-exponential factor, kB—Boltzmann constant, T— annealing temperature, Emi is the migration energy of H atoms over the z -th scenario . It corresponds to their activation energy, Eai, in a case of spatial redistribution of H atoms between the (tetrahedral) interstices (Ea Emi). Therefore, the temperature dependence of x follows the so-called Arrhenius law ... 

The reactant and the transition state represent two states that have different energy. Therefore, the population of the state of higher energy is determined by the Boltzmann distribution law, and the rate constant is expressed by the exponential equation known as the Arrhenius equation,... 

Allison el al. (1991) state that the activity difference between ions near the surface and those far away is the result of electrical work in moving the ions across the potential gradient between the charged surface and the bulk solution, The activity change of an ion moved from the surface to the bulk solution is described by EDL theory with an exponential Boltzmann expression... 

The exponentially decreasing term in the Boltzmann expression would seem to favour the very lowest energy states. However, this would lead to the paradoxical situation in which everything in the Universe should be at zero enthalpy. This can be resolved as follows. 

It has become customary to represent G i) as a series or continuous spectrum of exponential terms as expressed in Eq. (14). Historically the reason for representations using exponentials is that a single exponential term represents the form of a model that had been proposed by Maxwell [M14] in the 1860s prior to the publication of Boltzmann [B26].This model has the form of a differential equation which is equivalent to... 

Equation (7.2) expresses the net minority-carrier density/unit area as the product of the bulk minority-carrier density/unit volume nj/Ns, with the depth of the minority-carrier distribution diNv multiplied in turn by the customary Boltzmann factor exp(g(0s — Vs)/kT) expressing the enhancement of the interface density over the bulk due to lower energy at the interface. The depth diNv is related to the carrier distribution near the interface using the approximation (valid in weak inversion) that the minority-carrier density decays exponentially with distance from the oxide-silicon surface. In this approximation, diNv is the centroid of the minority-carrier density. For example, for a uniform bulk doping of 10 dopant ions/cm at 290 K, using Eq. (7.2) and the surface potential at threshold from Eq. (7.7) (0th = 0.69 V), there are Qp/q = 3 x 10 charges/cm in the depletion layer at threshold. This Qp corresponds to a diNv = 5.4 nm and a carrier density at threshold of JVinv = 5.4 x 10 charges/cm. ... 

---