Bimolecular control

We can readily extend bimolecular control to superpositions composed of more than two states. Indeed, we can introduce a straightforward method to optimize the reactive cross section as a function of am for any number of states [252], Doing so is an example of optimal control theory, a general approach to altering control parameters to optimize the probability of achieving a desired goal, introduced in Chapter 4. 

Applications of control using moderate fields are discussed in Chapters 9 to 11. These fields allow for new physical phenomena in both bound state and continuum problems, including adiabatic population transfer in both regimes, electromagneti-cally induced transparency in bound systems, as well as additional unimolecular and bimolecular control scenarios. 

Harney, M.B., Zhang, Y.H., and Sita, L.R. (2006b) Bimolecular control over polypropene stereochemical microstructure in a well-defined two-state system and a new fundamental form Stereogradient polypropene. Angewandte Chemie-International Edition, 45,6140-6144. 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

The introductory remarks about unimolecular reactions apply equivalently to bunolecular reactions in condensed phase. An essential additional phenomenon is the effect the solvent has on the rate of approach of reactants and the lifetime of the collision complex. In a dense fluid the rate of approach evidently is detennined by the mutual difhision coefficient of reactants under the given physical conditions. Once reactants have met, they are temporarily trapped in a solvent cage until they either difhisively separate again or react. It is conmron to refer to the pair of reactants trapped in the solvent cage as an encounter complex. If the unimolecular reaction of this encounter complex is much faster than diffiisive separation i.e., if the effective reaction barrier is sufficiently small or negligible, tlie rate of the overall bimolecular reaction is difhision controlled. 

There are significant differences between tliese two types of reactions as far as how they are treated experimentally and theoretically. Photodissociation typically involves excitation to an excited electronic state, whereas bimolecular reactions often occur on the ground-state potential energy surface for a reaction. In addition, the initial conditions are very different. In bimolecular collisions one has no control over the reactant orbital angular momentum (impact parameter), whereas m photodissociation one can start with cold molecules with total angular momentum 0. Nonetheless, many theoretical constructs and experimental methods can be applied to both types of reactions, and from the point of view of this chapter their similarities are more important than their differences. 

Crim F F 1996 Bond-selected chemistry vibrational state control of photodissociation and bimolecular reaction J. Phys. Chem. 100 12 725-34... 

Modem electron transfer tlieory has its conceptual origins in activated complex tlieory, and in tlieories of nonradiative decay. The analysis by Marcus in tire 1950s provided quantitative connections between the solvent characteristics and tire key parameters controlling tire rate of ET. The Marcus tlieory predicts an adiabatic bimolecular ET rate as... 

Consider a dilute solution of two reactant molecules, A and B. Inevitably an A molecule and a B molecule will undergo an encounter, the frequency of such encounters depending upon the concentrations of A and B. If, upon each encounter of A and B, they undergo bimolecular reaction, then the rate of this reaction is determined solely by the rate of encounter of A and B that is, the rate is not controlled by the chemical requirement that an energy barrier be overcome. One way to find this rate is to treat the problem as one of classical diffusion, and so this maximum possible rate of reaction is often called the diffusion-controlled rate. This problem was solved by Smoluchowski. In the following development no provision is made for attractive forces between the molecules. ... 

Schmolukowski in 1917, a diffusion-controlled bimolecular reaction in solution at 25 °C can reach a value for th second-order rate constant k as high as 7 x 109 m 1s-1. Nitrosations of secondary aliphatic amines also have rates which are relatively close to diffusion control (see Zollinger, 1995, Sec. 4.1). 

The reaction does not feature a bimolecular step, such as direct Sn2 attack of the hydroxide nucleophile on the cobalt center. Rather, hydroxide ion participates in a prior-equilibrium reaction, and the actual rate-controlling reaction is believed to be the uni-molecular expulsion of the leaving group from a species that contains a coordinated... 

Note that two H+ and one CU ions are formed. The rate-controlling step may be a bimolecular reaction of the intermediates so formed note that the composition of its transition state does, indeed, conform to the data ... 

The second term suggests that one of the metal ions may associate with chloride ions. The complex so formed reacts with the other in the bimolecular, rate-controlling step. One scheme is... 

A minor component, if truly minute, can be discounted as the reactive form. To continue with this example, were KCrQ very, very small, then the bimolecular rate constant would need to be impossibly large to compensate. The maximum rate constant of a bimolecular reaction is limited by the encounter frequency of the solutes. In water at 298 K, the limit is 1010 L mol-1 s"1, the diffusion-controlled limit. This value is derived in Section 9.2. For our immediate purposes, we note that one can discount any proposed bimolecular step with a rate constant that would exceed the diffusion-controlled limit. 

Two of these forms suggest possible bimolecular rate-controlling steps. The kinetic data cannot be used to decide between them, although other tests might. For example, one could test whether dimethyl oxalate reacts with OCl , and the monomethyl ester with HOC1. 

The rate of the bimolecular reaction in this limit is -d[A]/dt = kdclA][B]0 (where dc means diffusion-controlled), which is also equal to the flux of B toward A, multiplied by [A], That is,... 

A characteristic of free radicals is the bimolecular radical-radical reaction which in many cases proceeds at the diffusion-controlled limit. These radical-radical reactions can occur either between two identical radicals or between unlike radicals, the two processes being known as self-termination and cross-termination reactions, respectively. 

Relation (18) for the potential-dependent PMC signal is a reasonably good approximation only for the depletion region, where the space charge layer is controlled by the presence of fixed electron donors (Afo). It would become even more complicated if bimolecular or even more complicated kinetic reaction steps were considered. 

As the polymerization reaction proceeds, scosity of the system increases, retarding the translational and/ or segmental diffusion of propagating polymer radicals. Bimolecular termination reactions subsequently become diffusion controlled. A reduction in termination results in an increase in free radical population, thus providing more sites for monomer incorporation. The gel effect is assumed not to affect the propagation rate constant since a macroradical can continue to react with the smaller, more mobile monomer molecule. Thus, an increase in the overall rate of polymerization and average degree of polymerization results. 

Bimolecular Reactions. Models of surface-catalyzed reactions involving two gas-phase reactants can be derived using either the equal rates method or the method of rate-controlling steps. The latter technique is algebraically simpler and serves to illustrate general principles. 

This chapter has provided a brief overview of the application of optimal control theory to the control of molecular processes. It has addressed only the theoretical aspects and approaches to the topic and has not covered the many successful experimental applications [33, 37, 164-183], arising especially from the closed-loop approach of Rabitz [32]. The basic formulae have been presented and carefully derived in Section II and Appendix A, respectively. The theory required for application to photodissociation and unimolecular dissociation processes is also discussed in Section II, while the new equations needed in this connection are derived in Appendix B. An exciting related area of coherent control which has not been treated in this review is that of the control of bimolecular chemical reactions, in which both initial and final states are continuum scattering states [7, 14, 27-29, 184-188]. 

In principle, these approaches are very attractive because they probe multiple pathways in the critical regions where the pathways are separated, but in practice these are extremely challenging experiments to conduct, and the interpretation of results is often quite difficult. Furthermore, these experiments are difficult to apply to bimolecular collisions because of the difficulty of initiating the reaction with sufficient time resolution and control over initial conditions. 

A characteristic reaction of free radicals is the bimolecular self-reaction which, in many cases, proceeds at the diffusion-controlled limit or close to it, although the reversible coupling of free radicals in solution to yield diamagnetic dimers has been found to be a common feature of several classes of relatively stable organic radicals. Unfortunatly, only the rate constants for self-termination of (CH3)jCSO (6 x 10 M s at 173 K) and (CH3CH2)2NS0 (1.1 X 10 M s at 163K) have been measured up to date by kinetic ESR spectroscopy and consequently not many mechanistic conclusions can be reached. 

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