Boltzmann concentration dependence

The data in Figure 5 can now be considered in light of the conduction model developed above. As stated previously, conduction in reduced poly-I behaves like an activated process. There are two sources that potentially could be responsible for this behavior. The first is the Boltzmann type concentration dependence of the 1+ and 1- states discussed above. The number of charge carriers is expected to decrease approximately exponentially with T. The second is the activation barrier to self-exchange between 1+ and 0 sites and 0 and 1- sites. For low concentration of charge carriers both processes are expected to contribute to the measured resistance. 

Mn Mgi )2Si04 olivine mixture obtained by Morioka (1981). The concentration profiles are symmetric, indicating that the diffusivities are independent of the concentration of the diffusing ion. On the contrary, possible asymmetry in the diffusion profiles indicate that concentration depends significantly on the diffusing cations. In this case, the interdiffusion coefficient can be obtained by the Boltzmann-Matano equation ... 

This method of extracting concentration-dependent D is usually referred to as Boltzmann analysis. 

If the diffusion coefficient depends on time, the diffusion equation can be transformed to the above type of constant D by defining a new time variable a = jDdt (Equation 3-53b). If the diffusion coefficient depends on concentration or X, the diffusion equation in general cannot be transformed to the simple type of constant D and cannot be solved analytically. For the case of concentration-dependent diffusivity, the Boltzmann transformation may be applied to numerically extract diffusivity as a function of concentration. 

The analysis of these experiments follows a technique first introduced by Boltzmann.2 Under the assumption of ideal solutions the diffusion equation with concentration dependent diffiisivity is... 

Plotting the concentration profiles C y,t) as a function of this new parameter, i.e., as C (j] = yf-y/t), for different times t should therefore yield coinciding representations. Equation 12 cannot be integrated either. However, being interested only in the concentration dependence D C) of the diffusivity as the key quantity of our study, following Boltzmann [96,97] we may rewrite Eq. 12 as... 

Equation (3) can be simplified through the assumption of constant activity coefficients. Under the assumption of constant activity coefficients, Eq. (3) is in harmony with a Boltzmann distribution of electrons and holes. Such an approach is valid for p/ Nv and n/Nc less than 0.1. At higher concentrations, Fermi-Dirac statistics must be used to account for the distribution of electrons and holes as functions of energy. These effects can be treated by introduction of concentration-dependent activity coefficients for... 

In the case of electrode-electrolyte solution interfaces, the Poisson-Boltzmann equation has been modified for integrating many effects as, for example, finite ion size, concentration dependence of the solvent, ion polarizability, and so on. More often, this modification consists in the introduction of one or several supplementary terms to the energetic contribution in the distribution, which leads to modified Poisson-Boltzmann (MPB) nonlinear differential equations [52],... 

Boltzmann and Matano showed [22, 23] how the concentration dependent chemical diffusion coefficient D (c) can be determined from the data obtained in a diffusion experiment with two semi-infinite regions and the initial conditions of Fig. 5-5 for the case where the molar volume Km binary system is independent of concentration. 

On the basis of a Boltzmann-Matano analysis of measured impurity-atom concentration profiles, it was concluded that the diffusion coefficient was concentration-dependent at high As concentrations. 

Figures Universal polymerization curve of EPs. Plotted is the fraction polymerized material as a function of the dimensionless ratio hp T - Tp)/kBT with hp the net enthalpy gain of the formation of a single link, Tp the concentration-dependent polymerization temperature, and kg Boltzmann s constant. The line gives the theoretical prediction of the isodesmic model. The symbols Indicate experimental data on five chemically different ollgo(phenylene vlnyl)s In the solvent methyl cyclohexane at a concentration of 1M [38], By fitting to the data, values of hp are obtained from 24 to 70 kg T equivalent to 60 to 170 kJ mol". ...

This balance equation can also be derived from kinetic theory [ 159]. In the Maxwellian average Boltzmann equation for the species s type of molecules, the collision operator does not vanish because the momentum ntsCg is not an invariant quantity. Rigorous determination of the collision operator in this balance equation is hardly possible, thus an appropriate model closure for the diffusive force Pjr is required. Maxwell [95] proposed a model for the diffusive force based on the principles of kinetic theory of dilute gases. The dilute gas kinetic theory result of Maxwell [95] is generally assumed to be an acceptable form for dense gases and liquids as well, although for these mixtures the binary diffusion coefficient is a concentration dependent, experimentally determined empirical parameter. 

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