Parabolic energy barrier

There are some reservations about the strict applicability of the treatment outlined above to actual reactions. In the first place a parabolic energy barrier becomes unrealistic for configurations far removed from the transition state, and a more appropriate type of barrier is shown by the full curve in Figure 22(a), in which the broken curve represents a parabola. There is one potential energy function of this kind for which an explicit expression for the permeability can be obtained, commonly known as the Eckart barrier. For a symmetrical barrier (AH = 0) the equations are... 

In Equation 21, T is the absolute temperature, h is Planck s constant, is Boltzmann constant, and AG is the free energy barrier height relative to infinitely-separated reactants. The temperature-dependent factor r(7) represents quantum mechanical tunneling and the Wigner approximation to tunneling through an inverted parabolic barrier ... 

In the absence of supersaturation (pt = j.K) the W(r) dependence is parabolic, W(r)=4nr2a (Fig. IV-2) rcr °°, and also Wcr - 00. Upon penetration into the metastable region (p > pK), the maximum appears in the W(r) curve, i.e. Wcr and rcr both have finite values that decrease as the supersaturation, Ap, increases. The work of the critical nucleus formation, Wcr, can be thus viewed as the height of the energy barrier that one needs to overcome in order to make further spontaneous growth of nuclei possible. 

The density of reactive states p°( ) was fit by a sum of terms KTpT( ), as given in Eqs. (14) and (15), appropriate to scattering by parabolic potential energy barriers. (Note that the use of the parabolic barrier is the simplest barrier shape for understanding p°( )... 

A question often asked is whether the parabolic energy wells as predicted by Pieranski have an activation barrier that prevents the particle from falling in spontaneously. One can argue that, especially for a large spherical particle, upon its approach to the soft interface, the interface needs to deform and liquid has to drain. This event adds an activation barrier that needs to be overcome for the particle not to bounce off the interface, and clearly the interfacial tension between the two soft bulk phases (liquid-liquid and liquid-air) and the viscosity of both phases play key roles. Note that a potential hydrophobic effect [28] can counterbalance such a barrier because the dewetting of the liquid between a hydrophobic particle and the hydrophobic liquid phase, or air, stimulates long-range attraction and eases the adhesion process. 

Figure 5.8 shows the parabolic relationships between the energy ( ) of the donor and acceptor species and the average displacement (x) of the solvent molecules from their most stable configuration. The intersection of these two parabolas is the energy barrier to the electron transfer step (i.e. free energy of activation, AG ). The free energy of activation is a function of the... 

The transition is fully classical and it proceeds over the barrier which is lower than the static one, Vo = ntoColQl- Below but above the second cross-over temperature T 2 = hcoi/2k, the tunneling transition along Q is modulated by the classical low-frequency q vibration. The apparent activation energy is smaller than V. The rate constant levels off to its low-temperature limit k only at 7 < Tc2, when tunneling starts out from the ground state of the initial parabolic term. The effective barrier in this case is neither V nor Vo,... 

Fig. 1.22. Angular-dependent potential U for one-dimensional libration over barriers of height U0. The arrows show the way of libration below barriers and random translations from one well to another due to high-energy fluctuations. The broken line presents the approximation of the parabolic well valid at the bottom.

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