Eigen-mobilities

Also, since rik is independent of the molar fractions, all components of the system save i and k may be removed without changing Ujc If then the components i and k are renamed -f and —, one finds for the eigen mobility. of the cations with respect to the anions... 

If we put the masses of LiCl (m+ = 6.5, m. = 35.5) into the formula for the pure ion gas, we get /x li = —0.42, a value that is only by 20% larger than the experimental value for molten LiCl, /x Li = —0.34. Also the temperature independence of /x + in the gas corresponds to the low temperature dependence found in the condensates. Thus, many of the big differences between gases and condensates in structure and transport mechanisms seem to cancel out when relative differences of the eigen mobilities of isotopes are considered. 

It is also interesting to compare the eigen mobilities of Li and Li with the mobilities Uq and W7 in mixtures of the lithium isotopes. For liquid LiCl (23, 42) and solid Li2S04 (5i) the difference is... 

This indicates a cooperative motion of the lithium ions. If the concept is adopted that a lithium ion always moves as a member of a group of n lithium ions, and that the momentary mobility of an ion is the mean of the eigen mobilities of the momentary members of the group it belongs to, the mean isotopic lithium mobilities in a mixture will be... 

Eigen pointed out that the mobility of a proton in ice at 0°C is about 50 times larger than in water. In ice, H20 molecules already occupy fixed positions suitable for accepting a proton, so that the proton mobility is directly proportional to the rate of tunnelling. 

Ionic mobility refers to the velocity of an ion moving toward an oppositely charged electrode when a 1-volt potential is applied across a 1-centimeter electrochemical cell, strongly hydrated molecular cluster, such as [H-(OH2)4], is probably a more realistic representation (M. Eigen (1964) Angew. Chem. (Int. Eng. Edn.) 3, 1). 

Structural diffusion is provided by various complexes bare hydronium ion, Eigen complexes, and - Zundel complexes. Structural diffusion of bare hydronium ion and Eigen complexes occurs by proton hops between two water molecules. Two or more protons and several water molecules are involved in the structural diffusion of Zundel complexes. The contribution of mechanisms to the overall mobility depends on the temperature. Eigen and Zundel complexes prevail at room temperature, whereas bare hydronium ions dominate at high temperatures. Excess proton mobility of water has Arrhenius-like (-> Arrhenius equation) temperature dependence with the - activation energy about 0.11 eV. 

System peaks arise when one or several components of the mobile phase are adsorbed on the stationary phase, and because their concentrations are such that their adsorption isotherms are not linear. Injection of the sample in the multicomponent mobile phase perturbs the equilibrium of the strong solvent(s) or other additives of the mobile phase, which is nonlinear. As a result of this perturbation, more peaks may be recorded than there are components in the injected sample. For example, the injection of a sample of the pure weak solvent may generate as many peaks as there are additives in the mobile phase, if these additives are all adsorbed by the stationary phase. These extra peaks have received a variety of names in the early literature. They have been called system peaks, pseudopeaks, ghost peaks, eigen peaks, vacancy peaks, induced peaks, etc. We call them system peaks. Note that not all system peaks are recorded this depends on the type of detector used. Also, system peaks may be positive or negative, depending on the experimental conditions. 

The measurements of conductivities and dielectric constants furnish data for the computation of concentrations of the diflFerent types of defects as a function of solute concentration and of temperature, as well as interpretations in terms of lattice position, thermodynamics, and kinetics of these defects (77, 79). The quantitative evaluation of these measurements depends critically on the determination of the proton mobility, ion concentration, and dissociation constant in pure ice (Table IV) made by Eigen and coworkers (46, 47). 

Hall-Effect Measurements. Bullemer and Riehl (16) made these measurements. The experimental difficulties are great because of high electrode resistance, polarization effects, surface conductivity, and a low signal-to-noise ratio. Special palladium-black electrodes were used. The majority carriers were found to have a positive sign, as they should if they are protons. At —2°C. and —8°C., respectively, their concentration was 1.0 and 0.4 X 10 cm."", and the Hall mobility was 0.8 and 1.4 cm. /volt-sec. These values are 10 to 20 times lower and higher, respectively, than the results found by Eigen and De Maeyer from the saturation current and dissociation field effect (Table IV). 

When the ion-state concentrations determined by Eigen et al. (1964) are combined with measured static conductivities, the mobility of the positive ion state is found to have the anomalously large value of 0-075 cm s. Such a high mobility can only be the result of some sort of quantum-mechanical tunnelling process, as we shall see later. For such a process i/ o so that, from... 

Fast proton mobility in water attracted theoretical attention early, beginning with the works of von Grotthuss [7], at a time when the existence of the proton was not known, the chemical formula of water not settled, the notion of molecules was new, and little was known about the electricity laws. Modern landmarks were set by Bernal and Fowler [85], Eigen and de Mayer [86], Conway et al. [87], and Zundel and Metzger [88]. This was followed by more detailed molecular mechanisms, and analytical and computational models, see [68,77,79,84,89-93] and a conceptual essay by Agmon [ 1 ] which stimulated a new round of activities in this area. 

Fig. 2 Eigen-Zundel-Eigen (EZE) proton-mobility mechanism. The positive charge is either located at the hydronium ion (left and right structure), which is the center of the Eigen complex HsO/ (equal to HsO (1120)3), or delocalised over the Zundel ion HsOi " (central structure). Hydrogen bonds are depicted as dashed lines.

See also acid-base theories, -> Eigen complex, - pH, Zundel complex, prototropic charge transport, proton transfer, proton mobility. 

It is known from Eigen s technique of temperature jumps that a water molecule remains coordinated to rare earth aqua ions for roughly 10 s for R = La to Sm and for roughly 10 s for yttrium and R = Dy to Lu. There is a marked break from europium to terbium. Unfortunately, these results do not tell us everything we want to know. The problem is that the measured rate may refer to one (or a few) particularly mobile ligands. Thus, copper(II) aqua ions exchange water more... 

Characteristic rate constants (s ) for substitution of inner-sphere water molecuies on various metal ions. Adapted from the work of Eigen " with added data from other sources. Dashed line estimated threshold for cation mobility. "... 

---