Parabolic dependence

The fonn of the classical (equation C3.2.11) or semiclassical (equation C3.2.11) rate equations are energy gap laws . That is, the equations reflect a free energy dependent rate. In contrast with many physical organic reactivity indices, these rates are predicted to increase as -AG grows, and then to drop when -AG exceeds a critical value. In the classical limit, log(/cg.j.) has a parabolic dependence on -AG. Wlren high-frequency chemical bond vibrations couple to the ET process, the dependence on -AG becomes asymmetrical, as mentioned above. 

If one assumes the potential energy curves to have a similar parabolic dependence on the displacement of the atoms, a simple relation can be deduced between activation energy, the crossing point energy of the two curves, and the reaction energy. One then finds for a ... 

FIGURE 4.4 Schematic of threshold behavior of ionization processes. Under ideal conditions, one expects a step function for photoionization, a linear variation with energy under electron impact, and a parabolic dependence for double ionization by electron impact. 

Thus, we now have a reasonable model of the interface in terms of the classical Helmholtz model that can explain the parabolic dependence of y on the applied potential. The various plots predicted by equation (2.18) are shown in Figures 2.5(a) to (c). The variation in the surface tension of the mercury electrode with the applied potential should obey equation (2.18). Obtaining the slope of this curve at each potential V (i.e. differentiating equation (2.18)), gives the charge on the electrode, [Pg.49]

Equation (3.44) (in the Arrhenius form) is usually called the Marcus equation [74,75]. A special feature of the Marcus equation is that it predicts the parabolic dependence of the activation energy AEa on the free energy change AG/, that is, AEa is related to the free energy change AG/ in a parabolic form. 

In summary, to apply the Marcus theory of electron transfer, it is necessary to see if the temperature dependence of the electron transfer rate constant can be described by a function of the Arrhenius form. When this is valid, one can then determine the activation energy AEa only under this condition can we use AEa to determine if the parabolic dependence on AG/ is valid and if the reaction coordinate is defined. 

The quadratic rate equation [Eq. (1)] of the continuum theory arises because it implicitly assumed the parabolic dependence of the free energy profile on the solvent coordinate q. One of the consequences of this quadratic equation is the generation of a maximum in the dependence of the rate of reaction on the free energy of reaction and also in current density-overpotential dependence. 

The voltammetric response depends on the equilibrium constant K and the dimensionless chemical kinetic parameter e. Figure 2.30 illustrates variation of A f, with these two parameters. The dependence AWp vs. log( ), can be divided into three distinct regions. The first one corresponds to the very low observed kinetics of the chemical reaction, i.e., log( ) < —2, which is represented by the first plateau of curves in Fig. 2.30. Under such conditions, the voltammetric response is independent of K, since the loss of the electroactive material on the time scale of the experiment is insignificant. The second region, —2 < log( ) < 4, is represented by a parabolic dependence characterized by a pronounced minimum. The descending part of the parabola arises from the conversion of the electroactive material to the final inactive product, which is predominantly controlled by the rate of the forward chemical reaction. However, after reaching a minimum value, the peak current starts to increase by an increase of . In the ascending part of the parabola, the effect of... 

Fignre 2.41 indicates that the net peak current is a parabolic function of the electrode kinetic parameter. This is illnstrated in Fig. 2.43. With respect to the electrochemical reversibility of the electrode reaction, approximately three distinct regions can be identified. The reaction is totally irreversible for log(ca) < — 2 and reversible for log(ft)) > 2. Within this interval, the reaction is qnasireversible. The parabolic dependence of the net peak cnrrent on the logarithm of the kinetic parameter asso-... 

For a given adsorption constant, the observed electrochemical reversibility depends on the kinetic parameter defined as (O = Xy, or (o =. This reveals that the inherent properties of reaction (2.208) are very close to surface electrode reactions elaborated in Sect. 2.5. The quasireversible maximum is strongly pronounced, being represented by a sharp parabolic dependence of vs. m. The important feature of the maximum is its sensitivity to the adsorption constant, defined by the following equation ... 

An approximately parabolic dependence will occur if the hetaeron forms micelles into which the eluite can partition. In that case, the eluite molecules in the mobile phase can be present as free eluite, as eluite bound to hetaeron and bound to micelles. The total eluite concentration in the eluent, [Em]r> is given by... 

The apparently parabolic dependence of z on 57a = (7a T)/Ta in Fig. 7 is actually quantitative. For both classes of polymers, z can be described over a broad temperature range below 7a by the simple expression... 

Generally, the transport parameter used is the logarithm of the partition coefficient of the bas (bioactive substance) or some quantity derived from it. The partition coefficient is almost always determined between water and 1-octanol. Parameters obtained by chromatographic methods are being used with increasing frequency however. The term in t2 is introduced to account for the frequently observed parabolic dependence of a data set of bas on the transport parameter. Models other than... 

Let us now consider the case where A = 111. From the above relations, we can calculate ZA = 47.90. In Figure 2.6, we show the actual masses of the nuclei with A = 111. The expected parabolic dependence of mass upon Z is observed. The most stable nucleus has Z = 48 (Cd). All the A = 111 nuclei that have more neutrons than mCd release energy when they decay by (3 decay while the nuclei with fewer neutrons than nlCd release energy when they decay by (3+ or EC decay. 

This parabolic dependence of the nuclear mass upon Z for fixed A can be used to define a nuclear mass surface (Fig. 2.8). The position of the minimum mass for each A value defines what is called the valley of 3 stability. (3 decay is then depicted as falling down the walls of the valley toward the valley floor. 

It is desirable to avoid errors in pH measurement, which limit the accuracy of the above NMR pH titration. If an NMR titration is applied to a mixture of two acids, HA and HA, each of whose chemical shifts, 6 and S, follows Equation (13), it is possible to eliminate pH from the two equations and replace it with n, the number of equivalents of titrant added. Thus Ellison and Robinson obtained Equation (14), where A = 6-6, A = 6 -6, A° = <5° -S °, and R = KJK1 20 Moreover, it is not necessary to prepare solutions of exact molarity, because n can be evaluated more readily as (6-6 )/(6° -6 ), from the variation of the chemical shift of HA during the titration. When R is near 1, this is approximately a parabolic dependence of A on n. The titration of a mixture of formic acid and 1802-formic acid permitted the evaluation of the lsO IE on acidity from the 13C NMR chemical shifts 6 and S of the carboxyl carbons. In practice, this involved fitting to the three parameters A-, A°, and R. This same equation was used with 31P chemical shifts to evaluate the lsO IE on the acidity of phosphoric acid and alkyl phosphates.21... 

Let 0 be angular deflection of a dipole from the symmetry axis of the potential 1/(0), let p be a small angular half-width of the well (p Ci/2), and let (/0 be the well depth its reduced value u Uo/(kgT) is assumed to be 1. Since in any microscopically small volume a dipole moment of a fluid is assumed to be zero, we consider that two such wells with oppositely directed symmetry axes arise in the interval [0 < 0 < 2ji]. For brevity we consider now a quarter-arc of the circle. The bottom of the potential well is flat at 0 < 0 form factor/is defined as the ratio of this flat-part width to the whole width of the well. Thus, the assumed potential profile is given by... 

It exhibits maxima at 0 = 0 and 0 = it, with pm being the dimensionless resonance frequency of angular harmonic declinations 0 performed near the bottom of the well. In the EB model, small declinations 0 are assumed. Then Eq. (475) yields the parabolic dependence on 0 ... 

The major difference between the two equations is in the intercept Of greater interest, though, is the difference in nQ, which describes the optimal lipophilicity. Whereas there was a non-linear (parabolic) dependence of antibacterial activity on lipophilicity in the case of the sensitive strains [ti0 = 0.94 (0.29-1.28)], the activity against the resistant strain linearly increased with lipophilicity at least up to n of 3.2. 

Fig. 1.5. Experimental determination of reaction-diffusion constants from a linear-parabolic dependence between the layer thickness, x, and time, t, of interaction of initial substances tan a = kom (a), tan p = kW ...

Clearly, when calculating the physical (diffusional) constants, first of all it is necessary to check out whether the layer indeed grows in the diffusion controlled regime where the conditions 0sbi isbiA and k0A12 kXA 2/x must be satisfied. For this, the experimental data should be treated using the parabolic dependence (see equation (1.33))... 

The layer thickness-time kinetic relationships are in general rather complicated, not merely parabolic. Depending on the values of the chemical and physical (diffusional) constants, their different portions can be described by linear, paralinear, asymptotic, parabolic and other laws. 

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