Boltzmann distribution of ions

Figure 2.13 illustrates what is currently a widely accepted model of the electrode-solution interphase. This model has evolved from simpler models, which first considered the interphase as a simple capacitor (Helmholtz), then as a Boltzmann distribution of ions (Gouy-Chapman). The electrode is covered by a sheath of oriented solvent molecules (water molecules are illustrated). Adsorbed anions or molecules, A, contact the electrode directly and are not fully solvated. The plane that passes through the center of these molecules is called the inner Helmholtz plane (IHP). Such molecules or ions are said to be specifically adsorbed or contact adsorbed. The molecules in the next layer carry their primary (hydration) shell and are separated from the electrode by the monolayer of oriented solvent (water) molecules adsorbed on the electrode. The plane passing through the center of these solvated molecules or ions is referred to as the outer Helmholtz plane (OHP). Beyond the compact layer defined by the OHP is a Boltzmann distribution of ions determined by electrostatic interaction between the ions and the potential at the OHP and the random jostling of ions and... 

Caution should be taken when calculating the doublelayer force between two parallel plates. It is clear that the force is not proportional to the excess concentration of ions at the middle distance (with respect to the concentration of ions at infinity), since this Langmuir equation involved the assumption of ions of negligible sizes. We will use instead the procedure introduced by Verwey and Overbeek,18 which is based on general thermodynamic principles, and does not imply the Boltzmann distribution of ions.19 The force, per unit area, between two parallel plates separated by a distance l is given by... 

The exponential summation term in this equation is the Boltzmann distribution of ions. The last term in the equation is the Boltzmann distribution of ions for a ssnnmetric salt (i.e., valence z = —1 1, -2 2, or -3 3). The surface integral of the dot product of the gradient of the potential, Vijj, and the unit vector, n, normal to the surface S gives the total excess charge in the solution per unit area (C m... 

Using the Poisson equation and the Boltzmann distribution of ions, the Poisson-Boltzmann equation results ... 

The Debye-Hiickel theory is a study of the equilibrium properties of electrolyte solutions, where departures from ideal behaviour are considered to be a result of coulombic interactions between ions in an equilibrium situation. It is for this reason that equilibrium statistical mechanics can be used to calculate an equilibrium Maxwell-Boltzmann distribution of ions. 

The Poisson equation can be modified by taking into account a (thermal) Boltzmann distribution of ions in the solvent. The negative ions will accumulate where the potential is positive, and vice versa, subject to a thermal fluctuation. The charge densities from a collection of ions with charges q and -q and concentration c are given by eq. (14.54). 

Equation 7 gives the so-called Boltzmann distribution of ions near a charged surface [2]. 

A damped oscillating charge distribution of the interfacial structure was also obtained by Sotnikov and Esin/ assuming a Boltzmann distribution of ion vacancies in the interface cf. the case of dilute solutions of ions in water. Additionally, they introduced a polarizability factor to account for the deformation of ion charge under the influence of the electric field within the multilayer. Their revised formula for the interfacial capacitance gave... 

If AG is small in the Marcus model (Figure 18.7) there will be a Boltzmann distribution of ion pairs and stepwise ET may become possible. The important thing... 

Equation (7) gives the so-called Boltzmann distribution of ions near a charged surface [2]. The major assumptions implicit with the derivation of this distribution are as follows ... 

Assuming that in a macroscopic sample there is a Boltzmann distribution of ions over the various 5 levels, we have that the number of ions with moment will be e p 2S H/kT). Using this and eqn 9.6 together... 

The interaction between an ion and a surface is not only electrostatic each ion experiences a further additive term that come partly firom dispersion forces, Ui. Then, we have a modified Boltzmann distribution of ions in... 

When this equation is applied to a system composed of a macromolecule immersed in an aqueous medium containing a dissolved electrolyte, the fixed partial charges of each atom of the macromolecule result in a charge density described by p, and the mobile charges of the dissolved electrolyte are described by /O , which i derived from a Boltzmann distribution of the ions and coions. 

Further simphfication of the SPM and RPM is to assume the ions are point charges with no hard-core correlations, i.e., du = 0. This is called the Debye-Huckel (DH) level of treatment, and an early Nobel prize was awarded to the theory of electrolytes in the infinite-dilution limit [31]. This model can capture the long-range electrostatic interactions and is expected to be valid only for dilute solutions. An analytical solution is available by solving the Pois-son-Boltzmann (PB) equation for the distribution of ions (charges). The PB equation is... 

From the equilibrium requirement that the chemical potential involving all ionic species be uniform throughout the phase boundary, the distribution of ions within the electrical double layer can be expressed by the Boltzmann equation ... 

The association rate data determined in this study can be used to make quite a precise binding energy estimate for the aluminum ion-benzene complex. The relation between the association rate constant and the binding energy was made with use of phase space theory (PST) to calculate as a function of E, with a convolution over the Boltzmann distribution of energies and angular momenta of the reactants (see Section VI). PST should be quite a reasonable approximation for... 

It is also possible to obtain excited neutral species by heating the molecules in a furnance. This method was employed to obtain a vibrationally excited N2 beam that was reacted with 0+ ions.127 Since the molecules undergo a large number of collisions with the walls of the furnace before escaping into the beam, a Boltzmann distribution of internal-energy states is established. With such an apparatus, the source temperature is measured by an optical pyrometer and is typically in the range 1000-3000° K. Several reactions of ions with excited neutrals are listed in Table III. 

Taking the surface potential to be xp°, the potential at a distance x as rp, and combining the Boltzmann distribution of concentrations of ions in terms of potential, the charge density at each potential in terms of the concentration of ions, and the Poisson equation describing the variation in potential with distance, yields the Pois-son-Boltzmann equation. Given the physical boundary conditions, assuming low surface potentials, and using the Debye-Huckel approximation, yields... 

Fig. 9. This Larmor spectrum was measured in the analysis trap by resonant excitation (at 104 GHz) of the transition between the two spin states (spin up and down) of the bound electron. The asymmetric line shape of the resonance curve is due to the strong magnetic inhomogeneity in the analysis trap in combination with the thermal Boltzmann distribution of the ion s axial oscillation amplitude...

The distribution of ions of species T, r,(x). is related to the bulk concentrations c0i of ions by the Boltzmann equation ... 

We assume that the equilibrium distribution of ions obeys a Boltzmann distribution so that volume charge density p r) at position r, r being the distance from the particle center (r > a), is given by... 

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