Reactions Boltzmann distribution

Tine force field was then used to predict the results for fhe addition of the E and Z isomers c Ihe enol boronate of butanone (R = Me) to ethanol (R = Me). The relevant transitio. Iructures are shown in Figure 11.34. A Boltzmann distribution, calculated at the ten perature of the reaction (—78°C), predicted that the Z isomer would show almost complel syn selectivity syn anti = 99 1) and that the E isomer would be selective for the an product anti syn = 86 14). These results were in good agreement with the experunenti... 

The MEP is defined as the path of steepest descent in mass-weighted Cartesian coordinates. This is also called intrinsic reaction coordinate (IRC). In reality, we know that many other paths close to the IRC path would also lead to a reaction and the percentage of the time each path is taken could be described by the Boltzmann distribution. 

Direct dynamics trajectory calculations at the MP2/6-31-FG level of theory were then used to explore the reaction dynamics of this system [63]. Sixty-four trajectories were started from the central barrier shown at A in Fig. 11, with initial conditions sampled from a 300 K Boltzmann distribution. Of the 31 trajectories that moved in the direction of products, four trajectories followed the MEP and became trapped in the hydrogen-bonded [CH3OH ... 

The relative velocity between the molecules not only determines whether A and B collide, but also if the kinetic energy involved in the collision is sufficient to surmount the reaction barrier. Velocities in a mixture of particles in equilibrium are distributed according to the Maxwell-Boltzmann distribution in spherical coordinates ... 

As we are particularly interested in surface reactions and catalysis, we will calculate the rate of collisions between a gas and a surface. For a surface of area A (see Fig. 3.8) the molecules that will be able to hit this surface must have a velocity component orthogonal to the surface v. All molecules with velocity Vx, given by the Max-well-Boltzmann distribution f(v ) in Cartesian coordinates, at a distance v At orthogonal to the surface will collide with the surface. The product VxAtA = V defines a volume and the number of molecules therein with velocity Vx is J vx) V Vx)p where p is the density of molecules. By integrating over all Vx from 0 to infinity we obtain the total number of collisions in time interval At on the area A. Since we are interested in the collision number per time and per area, we calculate... 

The initial state-specific reaction rate constant for both diatom-diatom and atom-triatom reactions is calculated by averaging the corresponding cross-section over a Boltzmann distribution of translational energy ... 

The frequency with which the transition state is transformed into products, iT, can be thought of as a typical unimolecular rate constant no barrier is associated with this step. Various points of view have been used to calculate this frequency, and all rely on the assumption that the internal motions of the transition state are governed by thermally equilibrated motions. Thus, the motion along the reaction coordinate is treated as thermal translational motion between the product fragments (or as a vibrational motion along an unstable potential). Statistical theories (such as those used to derive the Maxwell-Boltzmann distribution of velocities) lead to the expression ... 

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

The rate constant is measured in units of moles dnr3 sec /(moles dnr3)", where n = a + b. Time may also be in minutes or hours. It should be noted that in case where the reaction is slow enough, the thermal equilibrium will be maintained due to constant collisions between the molecules and k remains constant at a given temperature. However, if the reaction is very fast the tail part of the Maxwell-Boltzmann distribution will be depleted so rapidly that thermal equilibrium will not be re-established. In such cases rate constant will not truly be constant and it should be called a rate coefficient. 

We consider a simple reaction composed of only a single elementary step of reacting particles that obey the Boltzmann distribution function. Then, the reaction rate, v, is given in Eqn. 7-13 [Rysselberghe, 1963] ... 

In Chapter 7 general kinetics of electrode reactions is presented with kinetic parameters such as stoichiometric number, reaction order, and activation energy. In most cases the affinity of reactions is distributed in multiple steps rather than in a single particular rate step. Chapter 8 discusses the kinetics of electron transfer reactions across the electrode interfaces. Electron transfer proceeds through a quantum mechanical tunneling from an occupied electron level to a vacant electron level. Complexation and adsorption of redox particles influence the rate of electron transfer by shifting the electron level of redox particles. Chapter 9 discusses the kinetics of ion transfer reactions which are based upon activation processes of Boltzmann particles. 

In this article we use transition state theory (TST) to analyze rate data. But TST is by no means universally accepted as valid for the purpose of answering the questions we ask about catalytic systems. For example, Simonyi and Mayer (5) criticize TST mainly because the usual derivation depends upon applying the Boltzmann distribution law where they think it should not be applied, and because thermodynamic concepts are used improperly. Sometimes general doubts that TST can be used reliably are expressed (6). But TST has also been used with considerable success. Horiuti, Miyahara, and Toyoshima (7) successfully used theory almost the same as TST in 66 sets of reported kinetic data for metal-catalyzed reactions. The site densities they calculated were usually what was expected. (Their method is discussed further in Section II,B,7.)... 

MSN.l. I. Prigogine, Sur la perturbation de la distribution de Maxwell-Boltzmann par des reactions chimiques (On the perturbation of the Maxwell-Boltzmann distribution by chemical reactions), Suppl Nuovo Cimento 6, 289-295 (1949). 

The formidable problems that are associated with the interpretation of LP kinetic data for nonstatistical IM reactions can be entirely avoided if the reactions can be studied in the HPL of kinetic behavior. In the HPL, the energy content of the initially formed species, X and Y, in reaction (2) would be very rapidly changed by collisions with the buffer gas so that the altered species, X and Y, would have normal Boltzmann distributions of energy. Furthermore, those Boltzmann energy distributions would be continuously refreshed as the most energetic X and Y within the distributions move forwards or backwards along the reaction coordinate. The interpretation of rate constants measured in the HPL is expected to be relatively straightforward because conventional transition-state theory can then be applied. 

The Boltzmann distribution helps us understand how intermediates can become trapped in energy wells between successive transition states in a multistep reaction mechanism. This behavior forms the basis for a special field of enzymology known as cryoenzymology. By appropriate choice of water-miscible solvents, enzymes can be studied at ultra-low temperatures where the rates of interconversion of enzyme species can be greatly retarded. See Cryoenzymology... 

We will not prove the Arrhenius relationship here, but it falls out nicely from statistical thermodynamics by considering that all molecules in a reaction must overcome an activation energy before they react and form products. The Boltzmann distribution tells us that the fraction of molecules with the required energy is given by tx (—Ea/RT), which leads to the functional dependence shown in Eq. (3.12). 

---