Boltzmann distributions condensed phases

Most of the chemical reactions occur in the condensed phase or in the gas phase under conditions such that the number of intermolecular collisions during the reaction time is enormous. Internal energy is quickly distributed by these collisions over all the molecules according to the Maxwell-Boltzmann distribution curve. 

Implicit solvation models developed for condensed phases represent the solvent by a continuous electric field, and are based on the Poisson equation, which is valid when a surrounding dielectric medium responds linearly to the charge distribution of the solute. The Poisson equation is actually a special case of the Poisson-Boltzmann (PB) equation PB electrostatics applies when electrolytes are present in solution while the Poisson equation applies when no ions are present. Solving the Poisson equation for an arbitrary equation requires numerical methods, and many researchers have developed an alternative way to approximate the Poisson equation that can be solved analytically, known as the... 

The simplest model for the electrical double layer is the Helmholtz condenser. A distribution of counterions in the bulk phase described by a Boltzmann distribution agree with the Gouy-Chapman theory. On the basis of a Langmuir isotherm Stem (1924) derived a generalisation of the double layer models given by Helmholtz and Gouy. Grahame (1955) extended this model with the possibility of adsorption of hydrated and dehydrated ions. This leads to a built-up of an inner and an outer Helmholtz double layer. Fig. 2.14. shows schematically the model of specific adsorption of ions and dipoles. 

Implicit solvation models developed for condensed phases represent the solvent by a continuous electric field, and are based on the Poisson equation, which is valid when a surrounding dielectric medium responds linearly to the charge distribution of the solute. The Poisson equation is actually a special case of the Poisson-Boltzmann (PB) equation PB electrostatics applies when electrolytes are present in solution, while the Poisson equation applies when no ions are present. Solving the Poisson equation for an arbitrary equation requires numerical methods, and many researchers have developed an alternative way to approximate the Poisson equation that can be solved analytically, known as the Generalized Born (GB) approach. The most common implicit models used for small molecules are the Conductor-like Screening Model (COSMO) [96,97], the Dielectric Polarized Continuum Model (DPCM) [98], the Conductor-like modification to the Polarized Continuum Model (CPCM) [99], the Integral Equation Formalism implementation of PCM (lEF-PCM) [100] PB models and the GB SMx models of Cramer and Truhlar [52,57,101,102]. The newest Miimesota solvation models are the SMD (universal Solvation Model based on solute electron Density [57]) and the SMLVE method, which combines the surface and volume polarization for electrostatic interactions model (SVPE) [103-105] with semiempirical terms that account for local electrostatics [106]. Further details on these methods can be found in Chapter 11 of reference 52. 

A common situation found in condensed phases under illumination is for all levels, except electronic levels, to be thermally equilibrated. Thus, under constant illumination, the sample is a mixture of thermally/vibrationally-equilibrated ground-state(s) with a very small, non-Boltzmann population of the excited electronic state, but which is itself thermally and vibrationally Boltzmann distributed. So the situation is similar to two non-equilibrated chemical species each of which is thermally equilibrated a thermally equilibrated ground-state, and a thermally equilibrated high energy excited-state. 

It is perhaps worth stating at the outset the conditions under which single-exponential decay should be anticipated. Considering a single emitting component in the condensed phase, electronic excitation will be followed by rapid equilibrium of vibrational energy to produce the Boltzmann distribution of levels from which emission occurs. 

Later, in a model where each cylindrical polymer rod was confined to a concentric, cylindrical, electroneutral shell whose volume represents the mean volume available to the macromolecule, the concept was extended to the macroscopic system itself which was considered as an assembly of electroneutral shells at whose periphery the gradient of potential goes to zero and the potential itself has a constant value. Closed analytical expressions which represent exact solutions of the Poisson-Boltzmann equation can be given for the infinite cylinder model. These solutions, moreover, were seen to describe the essence of the problem. The potential field close in to the chain was found to be the determining factor and under most practical circumstances a sizable fraction of the counter-ions was trapped and held closely paired to the chain, in the Bjerrum sense, by the potential. The counter-ions thus behave as though distributed between two phases, a condensed phase near in and a free phase further out. The fraction which is free behaves as though subject to the Debye-Hiickel potential in the ordinary way, the fraction condensed as though bound . 

One other relevant theoretical set of work concerns the counterion distribution particularly in the dilute limit. Manning solved the Debye-HUckel equation for a single infinitely thin polyelectrolyte. He found that when a < Xb the counterions condense onto the hne polymer reducing the charge density until the charge separation becomes equal to the Bjerrum length. The details are altered when the Poisson-Boltzmann approximation is used for a cylindrical polyelectrolyte, " but the basic point of condensation occuring for A > 1 remains. In a similar vein, Oosawa proposed a two-phase model of bound and free counterions. These results are especially relevant, since many prototypical polyelectrolytes, such as DNA and NaPSS, have A 3,... 

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