Exponentially relations between

An argument similar to this establishes the exponential relation between the cation flux through the oxide and the field F. 

Figure 9 shows the minimum lubricant him thickness as a function of velocity, with or without micro-polarity. The him thickness curves with micro-polarity are larger than those with the nonpolar molecules. This means that the microstructure and microrotation will have an influence on the him thickness. The simple exponential relation between him thickness and velocity, which holds in EHL, is no longer valid for thin him lubrication if the microstructure and the microrotation are taken into account. However, if the minimum him thickness is sufficiently large, as the velocity increases, the discrepancy between results with and without the consideration of the polar effect is very small. With an increase in both the characteristic length Z and coupling number N, the minimum him thickness becomes much larger than that of the nonpolar case. This reveals a size-dependent effect which accords well with experimental re-... 

Figure 4. Illustration of mass-dependent fractionation of Mg isotopes, cast in terms of 5 values. 5 Mg and 5 Mg values based on Mg/ Mg and Mg/ Mg ratios, respectively. A common equilibrium fractionation model, as defined by exponential relations between a values (fractionation factors) for different isotope ratios, is shown in the gray line. A simple linear relation, where the slope is proportional to the mass difference of the isotope pair, is shown in the black line. Additional mass-dependent fractionation laws may be defined, and all are closely convergent over small ranges (a few per mil) in isotope compositions at 5 values that are close to zero.

The effect on complex formation of structure defects, in our case the acrylate groups, is clearly shown in Figure 3. There is an exponential relation between degree of complexation 0, and acrylate ratio p = exp(-A.p). The acrylate ratio p is the total concentration of GGG over the total concentration of PAA, so p is equal to a plus the quantity of acrylate groups due to polyacid dissociation. The constant A is characteristic to the system and always higher than zero. The higher the complexation power of polybase the lower the A value (Figure 3). 

Relations between g and go are semi empirical and approximate (19,20). It is assumed that g is independent of solvent conditions and that a theta solvent for a linear polymer is also a theta solvent for its branched analogues. Neither of these assumptions is well founded (19). In practical applications, exponential relations between g and go of the form... 

Fig. 1.27. Quantitative results from STM images of Al(lll). An exponential relation between the corrugation amplitude and the tip-sample distance is observed. The best corrugation observed is more than 20 times greater than the maximum corrugation amplitude expected from the Fermi-level LDOS contour. (Reproduced from Wintterlin et al., 1989, with permission.)...

The form of distribution (17) recalls a Boltzmann expression with modulus of distribution 7. Attempts at a direct physical explanation of this result are thwarted by the obvious dependence of 7, not only on the state of the surface, but also on the nature of the gas whose adsorption proceeds according to equation (1). Nevertheless, formula (17) makes very plausible the experimentally observed constancy of the functional dependence A(Q) itself which leads to equation (1). It seems natural that with training or sintering of the surface, the liberation or destruction of points with different heats of adsorption may proceed in such a way as to preserve the exponential relation between A and Q, changing only the constants D, Q0, and especially 7. 

Butler, John Alfred Valentine — (Feb. 14,1899, Winch-combe, Gloucestershire, England - July 16,1977). Butler greatly contributed to theoretical electrochemistry, particularly, to connection of electrochemical kinetics and thermodynamics [i,ii]. The famous Butler-Volmer equation (1924) showing the exponential relation between current and potential was named after him (and... 

Using the Freundlich isotherm, an exponential relation between sorbed and dissolved molecules is used. 

Fig. 4.71. Under high-field condi-tons, there is an exponential relation between ionic current density and the field across an oxide.

Estimates of Covalent Character. Coulson (446) reviews attempts to determine the amount of covalent contribution to the H bond by the variation approximation scheme. Five valence bond structures which might be considered are shown in Fig. 8-3. Coulson and Danielsson (448) utilized variation trial functions appropriate to structures and j/c, together with the assumed exponential relation between bond length, r, and bond order,/ ... 

Gottesbiiren, 1991). Similarly, Parker and Doxtader (1983) found that the rate of transformation of 2,4-D in a sandy loam increased with decreasing soil moisture tension, but observed an exponential relation between the two variables that was different from that implied by the Walker equation. 

The Effect of Heat on the Active Constituents of a Solution. The thermal stability of components of a solution may determine the type of evaporator to be used and the conditions of its operation. If a simple solution contains a hydrolyzable material and the rate of its degradation during evaporation depends on its concentration at any time, an exponential relation between the remaining fraction, F, and the time, t, characteristic of a first-order reaction, is obtained, as shown in Eq. (2). 

D. An Exponential Relation Between the Microcanonical Temperature and Average Lifetimes... 

The n quantities can be calculated and the exponential relation between k and / has been verified.26... 

Then there should be an exponential relation between the rate constant and the sum of the free valences of the atoms where reaction takes place.16 The kinetic data which are available now are not sufficient for a quantitative consideration of this problem, but the qualitative agreement is very satisfactory. Figure 11 shows that, for example, in anthracene the sum of the free valences of the meso carbon atoms is greater than the sum of the frc e valences of other pairs of para positions. In fact, addition of mrueic anhydride takes place at the meso position.11 13... 

The distance dependence of J plays a crucial role for intersystem crossing of radical pairs. Usually, an exponential relation between J and the distance r of the radicals is assumed [17,18]... 

The electronic and atomic polarisation can not be frozen-in and cause the small depolarisation current after removing of the electrical field at time t2. Thus, the measured discharge current is the sum of dipole and space charge relaxation effects. The exponential relation between the relaxation times of these effects and the temperature makes it possible to shorten the discharge time drastically by thermal stimulation. 

There is an exponential relation between the reduction of carbonyl groups [chromophores formed] and discoloration [53,... 

In some cases such as electro-osmosis of 0.1 NaCl or KCl solution exponential relation between (yv)A/>=o A i.e.. 

The concept of free volume (empty space that is not occupied by polymer molecules) can be used for theoretical analysis of penetrant transport in polymers. The free volume model by Lee (1980) proposes an exponential relation between permeability and specific free volume ... 

The thermodynamic treatment of electrochemical processes presented in Sec. 2.2 describes the equilibrium condition of a system but does not present information on nonequilibrium conditions such as current flow resulting from electrode polarization (overvoltage) imposed to effect electrochemical reactions. Experimental determination of the current-voltage characteristics of many electrochemical systems has shown that there is an exponential relation between current and applied voltage. The generalized expression describing this relationship is called the Tafel equation. 

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