Minimum, kinetic

For large molecules, there is a possibility that, due to the synthesis conditions, the molecule is trapped in a local minimum (kinetic minimum), which is different from the global minimum of the free energy (thermodynamic minimum)-, see Fig. 7.5. 

For the same reason that the diamonds (kinetic minimum) in your safe do not change spontaneously into graphite (thermodynamic minimum), a molecule imprisoned in the kinetic minimum may rest there for a very long time (compared to the length of the experiment). 

Fig. 7.5. The basins of the thennodynamic minimum (T), of the kinetic minimum (K) and of the global minimum (G). The deepest basin (G) should not conespond to the thermodynamically most stable conformation (T). Additionally, the system may be caught in a kinetic minimum (Kl. from which it may be difficult to tunnel to the thermodynamic minimum basin. Explosives and fullerenes may serve as examples of K.

Jacobi coordinate system (p. 341) kinetic minimum (p. 353) Langevin dynamics (p. 371)... 

The numbering-up method also introduces another potential advantage. The scaled model operates as a continuous bioreactor rather than batch while providing the operator with the same control advantages of the batch operation at the same time. Therefore, the process time is expected to be shorter with miniaturized bioreactors since most standardized bioreactors use a process time that is longer than the kinetic minimum. Safety is also increased tremendously since the process can be stopped at any point in the process flow (Ehrfeld et al., 2000). These controls can be instituted automatically without the need for human supervision. 

Thus the kinetic and statistical mechanical derivations may be brought into identity by means of a specific series of assumptions, including the assumption that the internal partition functions are the same for the two states (see Ref. 12). As discussed in Section XVI-4A, this last is almost certainly not the case because as a minimum effect some loss of rotational degrees of freedom should occur on adsorption. 

In the previous section, non-equilibrium behaviour was discussed, which is observed for particles with a deep minimum in the particle interactions at contact. In this final section, some examples of equilibrium phase behaviour in concentrated colloidal suspensions will be presented. Here we are concerned with purely repulsive particles (hard or soft spheres), or with particles with attractions of moderate strength and range (colloid-polymer and colloid-colloid mixtures). Although we shall focus mainly on equilibrium aspects, a few comments will be made about the associated kinetics as well [69, 70]. 

Fast transient studies are largely focused on elementary kinetic processes in atoms and molecules, i.e., on unimolecular and bimolecular reactions with first and second order kinetics, respectively (although confonnational heterogeneity in macromolecules may lead to the observation of more complicated unimolecular kinetics). Examples of fast thennally activated unimolecular processes include dissociation reactions in molecules as simple as diatomics, and isomerization and tautomerization reactions in polyatomic molecules. A very rough estimate of the minimum time scale required for an elementary unimolecular reaction may be obtained from the Arrhenius expression for the reaction rate constant, k = A. The quantity /cg T//i from transition state theory provides... 

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques... 

To obtain the Hamiltonian at zeroth-order of approximation, it is necessary not only to exclude the kinetic energy of the nuclei, but also to assume that the nuclear internal coordinates are frozen at R = Ro, where Ro is a certain reference nucleai configuration, for example, the absolute minimum or the conical intersection. Thus, as an initial basis, the states t / (r,s) = t / (r,s Ro) are the eigenfunctions of the Hamiltonian s, R ). Accordingly, instead of Eq. (3), one has... 

Chemists are interested not only in the thermodynamics of a process (the relative stability o the various species) but also in its kinetics (the rate of conversion from one structure tc another). Knowledge of the minimum points on an energy surface enables thermodynamic data to be interpreted, but for the kinetics it is necessary to investigate the nature of the... 

Analytes A and B react with a common reagent R with first-order kinetics. If 99.9% of A must react before 0.1% of B has reacted, what is the minimum acceptable ratio for their respective rate constants ... 

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