The Reaction Coordinate

As is seen from Eq. 11.20, the high temperature limit of the isotope effect is given by /v2A t the ratio of the frequencies along 

Slater has carried out a classical investigation of unimolecular reaction rates at high pressures. The model used is that the molecule decomposes whenever a given coordinate q reaches a critical extension q0. He obtains for the velocity constant 

Slater s result may now be compared with the high-temperature limit resulting from the calculations of the previous section, 

It seems worthwhile to examine critically this transcription of the Slater method into the standard absolute reaction rate theory. In the simple unimolecular bond break, it does appear reasonable that the coordinate q between the tvfo atoms A and B must reach and go beyond a critical extension q0 in order that decomposition takes place. In Slater s calculations account is taken of the different energies involved in stretching q to q0. In regarding q as the mode of decomposition in the transition state method, one must, however, first look at the potential energy surface. The decomposition path involves passage over the lowest possible barrier between reactants and products. It does not seem reasonable to assume that this path necessarily only involves motion of the atoms A and B at the activated complex. Possibly, a more reasonable a priori formulation in a simple decomposition process would be to choose q as the coordinate which tears the two decomposition fragments apart. Such a coordinate would lead roughly to the relation 

The use of the Slater (/w2//ti)W as the temperature independent factor does not obviously follow from a consideration of the potential energy surface, and thus its use must be regarded as an intermingling of the Eyring and the Slater approaches. 

For tile reaction represented by Eq. (13.1), tlie changes in the numbers of moles of the species present are in direct proportion to tlie stoicliiometric nnmbers. Thus for tlie preceding reaction, if 0.5 mol of CH4 disappears by reaction, 0.5 mol of H2O must also disappear simultaneously 0.5 mol of CO and 1.5 molof H2 are formed. Applied to a differential amount of reaction, tliis principle provides tlie equations  

The list continues to include all species. Comparisonof tliese equations yields  

All terms being equal, tliey can be identified collectively by a single quantity representing an amount of reaction. Thus a definition of de is given by tlie equation  

The general relation between a differential change dnt in tlie number of moles of a reacting species and ds is tlierefore  

Tliis new variable e, called the reaction coordinate, characterizes the extent or degree to winch a reaction has taken place. Equations (13,2) and (13.3) define changes in s with respect to clianges in the numbers of moles of tire reacting species. The definition of e itself is completed for each application by setting it equal to zero for the initial state of the system prior to reaction. Thus, integrationof Eq. (13.3) from an initial unreacted state where e = 0 and n, = ,(, to a state reached after an aibitrary amount of reaction gives  

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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...

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... 

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. 

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 A3.12.10. Schematic diagram of the one-dimensional reaction coordinate and the energy levels perpendicular to it in the region of the transition state. As the molecule s energy is increased, the number of states perpendicular to the reaction coordinate increases, thereby increasing the rate of reaction. (Adapted from [4].)...

Fukui K 1970 A formulation of the reaction coordinate J. Phys. Chem. 74 4161... 

If the reaction is elementary, there is only a single transition state between A and B. At this point the derivative of the total electronic wave function with respect to the reaction coordinate Qa b vanishes ... 

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