Solution concentration Boltzmann distribution

The effect of surface potential on interfacial ionic concentration is given by a Boltzmann distribution relating the total solution concentration to that at the interface. For charged, amphiphilic species, binding constants replace these ionic concentrations, and the expression... 

In Eq. 30, Uioo and Fi are the activity in solution and the surface excess of the zth component, respectively. The activity is related to the concentration in solution Cioo and the activity coefficient / by Uioo =fCioo. The activity coefficient is a function of the solution ionic strength I [39]. The surface excess Fi includes the adsorption Fi in the Stern layer and the contribution, f lCiix) - Cioo] dx, from the diffuse part of the electrical double layer. The Boltzmann distribution gives Ci(x) = Cioo exp - Zj0(x), where z, is the ion valence and 0(x) is the dimensionless potential (measured from the Stern layer) obtained by dividing the actual potential, fix), by the thermal potential, k Tje = 25.7 mV at 25 °C). Similarly, the ionic activity in solution and at the Stern layer is inter-related as Uioo = af exp(z0s)> where tps is the scaled surface potential. Given that the sum of /jz, is equal to zero due to the electrical... 

It is interesting to consider a simple model for noble gas solution in liquids. From the Boltzmann distribution theorem, the ratios of equilibrium concentrations of solute molecules in two phases (here we consider liquid and gas phases) can be written as... 

The solute concentration profile within the accessible part of the pore will be determined by the Boltzmann distribution law. 

Here s(r) is the dielectric constant. We use s(r) to emphasize that the constant is a function of position for example, it is different inside a solvent and inside a solute. The electrostatic potential to be determined and p(r) is the charge distribution of the solute. Equation (25) is an exact equation of the electrostatic potential in a dielectric. If the solvent contains some dissolved salt (i.e., the concentration of positive and negative charges is the same) and we assume that the distribution of salt ions follows a Boltzmann distribution, then the electrostatic potential of the system is described by the Poisson-Boltzmann equation... 

Another limit for which an analytic solution is readily obtained is that of large Debye length. In this case the ion concentrations are uniform across the pore and are given by the Boltzmann distribution of Eq. (6.5.15) that is. 

This equation yields the mass-average molar mass. We can derive it from thermodynamics exclusively, using the Boltzmann distribution for the centrifugal potential field. It can be shown that the ratio D/s depends on concentration, because of nonideality in the solutions, through a relation similar to Eq. (35.20). 

The fact that the reaction in basic solutions is observed to be first-order indicates that the slow step involves only a molecule of i—(CH3)3CBr. The second step, the addition of OH" to the i—(CH3)3C carbocation, is fast under these conditions. At low concentrations of OH", the second step in the process shown in Case I may not be fast compared to the first. The reason for this is found in the Boltzmann Distribution Law. The transition state represents a high-energy state populated according to a Boltzmann distribution. If a transition state were to be 50 kj/mol higher in energy than the reactant state, the relative populations at 300 K would be... 

A space-charge region is also formed, and the bands bent, when a potential is apphed to the electrode. As above, the band edges remain pinned at the electrode/solution interface, which arises because the potential drop between the bulk semiconductor and the solution is essentially entirely across the space-charge region rather than at the semiconductor interface. As a consequence, the intrinsic electron transfer rate constant is independent of applied potential. Nevertheless the current (and hence the effective rate constant) does depend on the apphed potential because the concentration of electrons (the majority carriers) at the electrode surface relative to its bulk concentration has a Boltzmann dependence on the energy difference between the band edge and the interior of the electrode. (The Fermi Dirac distribution reduces to a Boltzmann distribution when E > Fp-)... 

To obtain Eqs. 5-10, it was assumed that the concentration of solute within the adsorption boundary layer is related to the solute-surface interaction energy by a Boltzmann distribution. The essence of the thin-layer polarization approach is that a thin diffuse layer can still transport a significant amoimt of solute molecules so as to affect the solute transport outside the diffuse layer. For a strongly adsorbing solute (e.g., a surfactant), the dimensionless relaxation parameter fila (or Kid) can be much greater than imity. If all the adsorbed solute were stuck to the surface of the particle (the diffuseness of the adsorption layer disappears), then L = 0 and there would be no diffusiophoretic migration of the particle. In the limit of [l/a 0 (very weak adsorption), the polarization of the diffuse solute in the interfacial layer vanishes and Eq. 5 reduces to Eq. 1. 

This analysis leads us to the conclusion that ionic surfactants in salt-free solutions undergo kinetically limited adsorption. Indeed, dynamic surface tension curves of such solutions do not exhibit the diffusive asymptotic time dependence of non-ionic surfactants, depicted in Fig. 1. The scheme of Section 2, focusing on the diffusive transport inside the solution, is no longer valid. Instead, the diffusive relaxation in the bulk solution is practically immediate and we should concentrate on the interfacial kinetics, Eq. (21). In this case the subsurface volume fraction, t, obeys the Boltzmann distribution, not the Davies adsorption isotherm (15), and the electric potential is given by the Poisson-Boltzmann theory. By these observations Eq. (21) can be expressed as a function of the surface... 

The free counterions form an electrical double layer in which the counterion concentrations around each micelle decrease in a Poisson-Boltzmann distribution into the aqueous phase. Figure 6 illustrates the double layer and the radial distributions of counterions at different salt concentrations obtained by solving the Poisson-Boltzmann equation. Note that the thickness of the double layer depends on the ion salt concentration. The graph also illustrates a two-site model for ion distribution used in the pseudophase models to describe measured ion distributions in solutions of ionic association colloids, that is, counterions are either bound or free (see below). However, explanations based only on coulombic interactions between headgroups and counterions fail to account for commonly observed trends in ion-specific effects, for example, the Hofmeister... 

In the GG model (Eigure 15.5a), the ions are not surface adsorbed in a condensed layer as considered by Helmholtz, but remain in solution because of their thermal motion. At equilibrium, the ion concentration profiles can be described in first approximation by the Boltzmann distribution. 

The expression for Teiec embodies two competing trends of the solution phase potential at the reaction plane. An increase in solution phase potential results in a larger driving force for electron transfer in cathodic direction. This effect is proportional to the cathodic transfer coefficient c. At the same time, a more positive value of (y)- (l) - (po) corresponds to lower proton concentration at the reaction plane, following a Boltzmann distribution (Equation 3.77). The magnitude of this effect is determined by the reaction order yh+ K is> therefore, of primary interest to know the difference of kinetic parameters, - yh+ ... 

The diffuse double layer can be described by the Gouy-Chapman equation, which is a solution of the Poisson-Boltzmann equation for a planar diffuse double layer. The Poisson-Boltzmann equation relates the electrical potential to the distribution and concentration of charged species. The distribution of charges in the electrolyte solution is described by Boltzmann distributions. 

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