Coordinate reaction

It was mentioned already in the preceding sections, that the initial state of every closed system is unambigously determined by the number of moles of the initial constituents. This determines the overall number of gramatoms of the individual basic constituents, which must remain constant in the course of chemical conversions (provided no nuclear processes take place). 

The linkage between individual constituents can then be characterized by a newly selected quantity for every linearly independent reaction this is called reaction coordinate (also extent of reaction, in German Reaktionslaufzahl). This selected quantity is a quantitative measure of the depth to which chemical conversion, corresponding to a given stoichiometric description, has proceeded. Together with balance relationships the reaction coordinate elucidates the actual concentration of individual constituents with absolute accuracy. 

For some of the simples cases, the mutual relationship between reacting constituents may be expressed by graphical means  

On introduction of the reaction coordinate the actual concentration of the two components will be defined by the equations 

Chemical conversion of the pure component to the pure product A2 can be illustrated by the straight line MN in Fig. 4. Clearly, any equilibrium composition must lie on this straight line, irrespective of whether the chemical equilibrium is more favourable, due to the numerical value of the equilibrium constant, to the formation of products or vice versa. 

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The gas phase reaction shows a double minimum and a small barrier along the reaction coordinate which is the difference between the two C-CL distances. The minima disappear in aqueous solution and this is accompanied by an increase in the height of the barrier. The behaviour in dimethyl fonnamide is intennediate between these two. 

Figure A3.4.6. Potential energy along the reaction coordinate r for an unimolecular isomerization (left) and a...

The free energy as a fiinction of reaction coordinates has been explicitly represented by Quack and Troe [36, 112] for the reaction... 

A reactive species in liquid solution is subject to pemianent random collisions with solvent molecules that lead to statistical fluctuations of position, momentum and internal energy of the solute. The situation can be described by a reaction coordinate X coupled to a huge number of solvent bath modes. If there is a reaction... 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The dependence of k on viscosity becomes even more puzzling when the time scale of motion along the reaction coordinate becomes comparable to that of solvent dipole reorientation around the changing charge distribution... 

Larson R S and Kostin M D 1982 Kramers theory of chemical kinetics curvilinear reaction coordinates J. Chem. Phys. 77 5017-25... 

The barrier on the surface in figure A3,7,1 is actually a saddle point the potential is a maximum along the reaction coordinate but a minimum along the direction perpendicular to the reaction coordinate. The classical transition state is defined by a slice tlirough the top of tire barrier perpendicular to the reaction coordinate. 

The photoelectron spectrum of FH,is shown in figure A3.7.6 [54]. The spectrum is highly structured, showing a group of closely spaced peaks centred around 1 eV, and a smaller peak at 0.5 eV. We expect to see vibrational structure corresponding to the bound modes of the transition state perpendicular to the reaction coordinate. For this reaction with its entrance chaimel barrier, the reaction coordinate at the transition state is... 

Figure A3.8.1 A schematic diagram of the PMF along the reaction coordinate for an isomerizing solute in the gas phase (frill curve) and in solution (broken curve). Note the modification of the barrier height, the well positions, and the reaction free energy due to the interaction with the solvent.

In this equation, m. is the effective mass of the reaction coordinate, q(t -1 q ) is the friction kernel calculated with the reaction coordinate clamped at the barrier top, and 5 F(t) is the fluctuating force from all other degrees of freedom with the reaction coordinate so configured. The friction kernel and force fluctuations are related by the fluctuation-dissipation relation... 

The GLE can be derived by invoking the linear response approximation for the response of the solvent modes coupled to the motion of the reaction coordinate. 

The key feature of A3.8.18 is that the centroids of the reaction coordinate Feymnan paths are constrained to be at the position q. The centroid g particular reaction coordinate path q(x) is given by the zero-frequency Fourier mode, i.e. 

Haynes G R and Voth G A 1995 Reaction coordinate dependent friction in classical activated barrier crossing dynamics when it matters and when it doesn t J. Chem. Phys. 103 10 176... 

This equation represents the reaction rate at total energy E with a fixed energy in the reaction coordinate and may be written as... 

Variational RRKM theory is particularly important for imimolecular dissociation reactions, in which vibrational modes of the reactant molecule become translations and rotations in the products [22]. For CH —> CHg+H dissociation there are tlnee vibrational modes of this type, i.e. the C—H stretch which is the reaction coordinate and the two degenerate H—CH bends, which first transfomi from high-frequency to low-frequency vibrations and then hindered rotors as the H—C bond ruptures. These latter two degrees of freedom are called transitional modes [24,25]. C2Hg 2CH3 dissociation has five transitional modes, i.e. two pairs of degenerate CH rocking/rotational motions and the CH torsion. 

Figure A3.12.5. A model reaction coordinate potential energy curve for a fluxional molecule. (Adapted from [30].)...

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