Boltzmann distribution, solution

Block relaxation, 61 Bogoliubov, N., 322,361 Boltzmann distribution, 471 Boltzmann equation Burnett method of solution, 25 Chapman-Enskog method of solution, 24... 

We consider a biological macromolecule in solution. Let X and Y represent the degrees of freedom of the solute (biomolecule) and solvent, respectively, and let U(X, Y) be the potential energy function. The thermal properties of the system are averages over a Boltzmann distribution P(X, Y) that depends on both X and Y. To obtain a reduced description in terms of the solute only, the solvent degrees of freedom must be integrated out. The reduced probability distribution P is... 

In metalloproteins, the paramagnet is an inseparable part of the native biomacromolecule, and so anisotropy in the metal EPR is not averaged away in aqueous solution at ambient temperatures. This opens the way to study metalloprotein EPR under conditions that would seem to approach those of the physiology of the cell more closely than when using frozen aqueous solutions. Still the number of papers describing metalloprotein bioEPR studies in the frozen state by far outnumbers studies in the liquid state. Several additional theoretical and practical problems are related to the latter (1) increased spin-lattice relaxation rate, (2) (bio)chemical reactivity, (3) unfavorable Boltzmann distributions, (4) limited tumbling rates, and (5) undefined g-strain. 

Maximum working-solution capacity, 14 46 Maximum work obtainable, for a change of state involving mixtures, 24 690-692 Maxixe beryl, color, 7 337 Maxon sutures, 24 222 Maxwell-Boltzmann distribution, silicon-based semiconductors and, 22 235, 236, 238... 

These equations are valid in the absence of degeneracy. Like Eqs. (10) they are characterized by the fact that elements with the same difference of indices m — m are coupled among themselves. Their equilibrium solution is a 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 the same way as described in Sec. 5.2 for a diifiise layer in aqueous solution, the differential electric capacity, Csc, of a space charge layer of semiconductors can be derived from the Poisson s equation and the Fermi distribution function (or approximated by the Boltzmann distribution) to obtain Eqn. 5-69 for intrinsic semiconductor electrodes [(Serischer, 1961 Myamlin-Pleskov, 1967 Memming, 1983] ... 

Fig. 6-46. Differential capacity observed and computed for an n-type semiconductor electrode of zinc oxide (conductivity 0. 59 S cm in an aqueous solution of 1 M KCl at pH 8.5 as a function of electrode potential solid curve s calculated capacity on Fermi distribution fimction dashed curve = calculated capacity on Boltzmann distribution function. [From Dewald, I960.]...

The third point implies that it is possible to develop a physical theory for ionic interactions that is relatively simple and still useful. The most frequently used is the Poisson-Boltzmann (P-B) equation, which combines the Poisson equation from classical electrostatics with the Boltzmann distribution from statistical mechanics. This is a second-order nonlinear differential equation and its solution depends on the geometry and the boundary conditions. 

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... 

Energy states in solution, 1462 1463 Boltzmann distribution law, 1466 distribution, 1462 1464 ground state, 1464, 1468 number of, 1462 Enthalpy... 

Let us note one vital point, which is of methodological importance. It has been traditionally accepted in electrochemistry to choose the positive direction of the electrode potential

positive electrode charge. Here the zero potential is assumed to be that of the reference electrode, which coincides, within a constant, with the potential in the solution bulk (— oo). On the other hand, in physics of semiconductor surface the potential is usually reckoned from the value in the semiconductor bulk ( ) the enrichment of the surface with electrons, i.e., the formation of a negative space charge, corresponding to the positive potential of the surface. In particular, this statement directly follows from the Boltzmann distribution for electrons and holes in the space-charge region in a semiconductor ... 

However, if the probability that there is a B reactant a distance r away from the A reactant is correlated by the Boltzmann factor (as may occur in the case of photo lytic and radiolytic ionization of solutes), the Boltzmann distribution is... 

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