Onsager formula

To compute the above expression, short molecular dynamics runs (with a small time step) are calculated and serve as exact trajectories. Using the exact trajectory as an initial guess for path optimization (with a large time step) we optimize a discrete Onsager-Machlup path. The variation of the action with respect to the optimal trajectory is computed and used in the above formula. 

Equation (2.39) leads to the prediction that AA should be proportional to p2. For a bulk solvent, this can be considered as a molecular equivalent of the well-known Onsager formula derived for the continuum dielectric model [12],... 

With this distribution, the escape probability is calculated using the Onsager (1938) formula (see Chapter 9),... 

Over the temperature interval 165 K to 300 K, the calculations of Silinsh and Jurgis (1985) indicate that the thermalization rate in pentacene decreases from 3 X 1012 to 0.8 X 1012 s 1. The trend is opposite to what would be expected in liquid hydrocarbons and may be attributed to the rapid increase of mcff with temperature. The calculated mean thermalization distance increases with incident photon energy fairly rapidly, from 3 nm at 2.3 eV to 10 nm at 2.9 eV, both at 204 K. With increasing temperature, (r) decreases somewhat. These thermalization distances have been found to be consistent with the experimental photogeneration quantum efficiency when Onsager s formula for the escape probability is used. 

Taking Ay = 15 nm, . = 5 eV, and other values as before, the b value for LAr is evaluated as 1400 nm, which is much larger than 133 nm, obtained by fitting the free-ion yield to the Onsager formula (vide supra). Similar calculations for LKr and LXe give b values of the gaussian thermalization distribution... 

Equation (9.2) shows the effect of the reaction radius rl on the escape probability, which, remarkably, is free of the diffusion coefficient. Normally rc r which reduces Eq. (9.2) to the celebrated Onsager formula 0 = exp(- r /rg) as given by Eq. (9.1). 

FIGURE 9.1 Simplified derivation of Onsager s escape probability formula. In the stationary state a unit electron current at ro partitions as I toward the reaction radius and as Is toward the sink at infinity the latter is the escape probability. Reproduced from Mozumder (1969a), with the permission of John Wiley Sons, Inc. ... 

Onsager s (1938) formula for the probability of escaping geminate-ion recombination in the presence of an external field E may be written as... 

Continuum models have a long and honorable tradition in solvation modeling they ultimately have their roots in the classical formulas of Mossotti (1850), Clausius (1879), Lorentz (1880), and Lorenz (1881), based on the polarization fields in condensed media [32, 57], Chemical thermodynamics is based on free energies [58], and the modem theory of free energies in solution is traceable to Bom s derivation (1920) of the electrostatic free energy of insertion of a monatomic ion in a continuum dielectric [59], and Kirkwood and Onsager s... 

The calculation of J 18b) is made in the same fashion. When the two results are added, we recover the well-known Onsager formula ... 

While Onsager s formula has been widely used, there have also been numerous efforts to improve and generalize it. An obvious matter for concern is the cavity. The results are very sensitive to its size, since Eqs. (33) and (35) contain the radius raised to the third power. Within the spherical approximation, the radius can be obtained from the molar volume, as determined by some empirical means, for example from the density, the molar refraction, polarizability, gas viscosity, etc.90 However the volumes obtained by such methods can differ considerably. The shape of the cavity is also an important issue. Ideally, it should be that of the molecule, and the latter should completely fill the cavity. Even if the second condition is not satisfied, as by a point dipole, at least the shape of the cavity should be more realistic most molecules are not well represented by spheres. There was accordingly, already some time ago, considerable interest in progressing to more suitable cavities, such as spheroids91 92 and ellipsoids,93 using appropriate coordinate systems. Such shapes... 

Onsager reciprocity relations as well as the Green-Kubo and Einstein formulas for these coefficients ... 

In the case where the reaction distance may be assumed much shorter than the range of the Coulomb interaction R r ), Eq. (14) reduces to the well-known Onsager formula [2]... 

An important difference between the Bom and Kirkwood-Onsager formulae is that the... 

An immediate application of this theorem is to derive the Onsager-Yang formula for the spontaneous magnetization M0(T) of an Ising ferro-magnet.4,6 In terms of the correlation functions in Eq. (10), the long-range order T(oo), and hence M0(T) can be defined by... 

These jumps follow from the Onsager relations. Unlike the 2-slit geometry, the closed ABI requires many reflections of the electron waves from the forks connecting the ring with the leads. Each such reflection adds a term to the interference sum of amplitudes, and modifies the simple 2-slit formula. In fact, unitarity (conservation of current) and time reversal symmetry imply that = Q 4>) [11], and therefore (3 (as well as 7 etc.) must be equal to 0 or 7T. The additional reflections also explain the need for higher harmonics near resonances. Below we include these many reflections, and replace the 2-slit formula by a new one - which can be used to extract olqd from the closed interferometer data [12]. 

In water, at ordinary concentrations, the hydrogen chloride is practically all present as the hydrated ions. The infrared absorption bands characteristic of HCl, and shown by the liquid hydride and its solutions in nonionizing solvents do not appear in the aqueous solutions.451 In dilute solutions, the conductivities agree with the Debye-Huckel-Onsager formula. 

Born s idea was taken up by Kirkwood and Onsager [24,25], who extended the dielectric continuum solvation approach by taking into account electrostatic multipole moments, Mf, i.e., dipole, quadrupole, octupole, and higher moments. Kirkwood derived the general formula ... 

Still starting from the Onsager s theory, Mirone and co-workers [29,36,38] proposed a relation for the ratio between Raman intensities in solution and in vacuo given by the following formula ... 

From the point of view of theory, the formulae of Table 2.6 are equally applicable to both gas and condensed phase samples, as they include the local field factors, which account for local modifications to the Maxwell fields due to bulk interactions within the Onsager-Lorentz model. 

In the case of strong electrolytes with a = 1 the last, mentioned equation changes to the form of Onsager s equation, while for very weak electrolytes or substantially diluted solutions (o, si 0, /A 1) it changes to the classical formula of Arrhenius. If we consider the problem of electrolytes from the point of view of the equation (III-31) we can see that there is no sharp dividing line between strong and weak electrolytes. It can equally be seen that the Debye-Hiickel-Onsager theory does not replace the theory of Arrhenius but merely corrects and suitably supplements it. 

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