The Barrier between Reactants and Products

We have now seen that energy is both the carrot and the cart of chemical reactions, and so can finally unwrap the meaning of my delphic remark at the end of Chapter 2. Energy, its dispersal in disorder, is the carrot the driving power of chemical reactions. Energy, the need to overcome the barriers between reactants and products, is also the cart, in the sense of holding back free unrestrained flight towards the carrot. 

In qualitative terms, the reaction proceeds via an activated complex, the transition state, located at the top of the energy barrier between reactants and products. Reacting molecules are activated to the transition state by collisions with surrounding molecules. Crossing the barrier is only possible in the forward direction. The reaction event is described by a single parameter, called the reaction coordinate, which is usually a vibration. The reaction can thus be visualized as a journey over a potential energy surface (a mountain landscape) where the transition state lies at the saddle point (the col of a mountain pass). 

As a first step toward a TST treatment of the stochastically driven dynamics, it is crucial to assume, just as in the autonomous case, that the deterministic dynamics has a fixed point that marks the location of an energetic barrier between reactants and products. In the case of Eq. (13), the fixed point is given by a saddle point q0 of the potential U(q). The reaction rate is determined by the... 

Eb are the energy barriers for forward and reverse reactions, A Hr is the heat of the reaction to be discussed later, and the horizontal scale is called the reaction coordinate, an iU-defmed distance that molecules must travel in converting between reactants and products. Polanyi and Wigner lirst showed from statistical mechanics that the rates should be described by expressions of the form as given in the boxed equation by a Boltzmann factor, exp( —E / R T), which is the probabihty of crossing a potential energy barrier between reactant and product molecules. In fact, it is very rare ever to fmd reaction-rate coefficients that are not described with fair accuracy by expressions of this form. 

Here N(x) is the potential energy barrier between reactant and product state of the hydrogen and E is the particle energy. For a static square barrier the theory predicts a huge non-realistic isotope effect and its non-sensitivity to temperature. The thermal fluctuations produce a thermal distribution of the transfer distance, /. For a rectangular barrier and low frequency vibration of substrate and medium and harmonic behavior of l ... 

Activation energy is the minimum amount of energy required for reactants to be transformed into products (i.e., to overcome the energy barrier between reactants and products). When the activation energy is high, the reaction is slower. 

The electronic coupling matrix element (//ab) reflects the strength of the interaction between reactants and products at the nuclear configuration of the transition state. Square-barrier ET tunneling models predict that the coupling will... 

In order to visualize the energy barrier between reactants and products, it is assumed that each system can be represented as a classical harmonic oscillator along the reaction coordinate. This is illustrated in fig. 7.15. The left-hand parabola gives the Gibbs energy of the reactants and the right-hand parabola, that of... 

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... 

Potential of mean force The thermodynamic quantity needed to estimate equilibrium constants is the AG° between reactants and products. By sampling a reaction coordinate r a potential of mean force (pmf) can be obtained. From the frequency of occurrence of different r values a distribution function g(r) is calculated that is related to w(r), the relative free energy, or the pmf, by w(r) = —kT In g r). By using an additional constraining or biasing potential (umbrella) a system can be forced to sample a reaction coordinate region which would be infrequently sampled in the absence of the umbrella potential because of high barriers in w(r). 

What happens if we add a catalyst to a chemical system that is at equilibrium As shown in FIGURE 15.14, - (Figure 14.23) a catalyst lowers the activation barrier between reactants and products. The activation energies for both the forward and reverse reactions are lowered. The catalyst thereby increases the rates of both forward and reverse reactions. Since K is the ratio of the forward and reverse rate constants for a reaction, you can predict, correctly, that the presence of a catalyst, even though it changes the reaction rate, does not affect the numeric value of K (Figure 15.14). As a result, a catalyst increases the rate at which equilibrium is achieved but does not change the composition of the equilibrium mixture. 

The BA essentially treats the interaction potential as a first order perturbation on the motion of the non-interacting reactant and product collision systems. This approximation is not expected to be reliable for chemical reactions at low collision energies. Consider, for example, a reaction with a large barrier between reactants and products, such as H+H2. The distortion of the incident plane wave by the barrier is clearly an important physical process. This suggests a better description of the reactive scattering might be obtained if the distortion of the relative motion in the initial and final channels is taken into account. This is the aim of the DWBA. 

A transition structure is a saddle point on a potential energy (enthalpy) surface for a reaction. A transition state corresponds to a free energy maximum on the path between reactant and product. The geometry of the transition structure corresponds closely to the geometry of the transition state if the barrier is relatively high and the entropy of the system does not vary rapidly near the geometry of the transition structure. If either of these conditions is not met, then the geometry of the transition structure may be very different from that of the transition state. For discussions, see references 17 and 18. Also see Bauer, S. H. Wilcox, C. F., Jr. /. Chem. Educ. 1995, 72,13. 

What happens if we add a catalyst to a chemical system that is at equilibrium As shown in T Figure 15.14, a catalyst lowers the activation barrier between reactants and products. The activation energies for both the forward and reverse reactions are lowered. The... 

A catalyst cannot alter the position of equilibrium. It merely lowers the activation energy barrier between reactants and products so that equilibrium is attained more rapidly than in its absence. Even with the simplest starting materials, many reaction paths leading to different products are possible. A catalyst should be able to select one of these paths to the exclusion of all others. High selectivity of this type is desirable in any industrial process. Different... 

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