Boltzmann distributions transition state theory

Transition state theory assumes an equilibrium energy distribution among all possible quantum states at all points along the reaction coordinate. The probability of finding a molecule in a given quantum state is proportional to which is a Boltzmann... 

To understand how collision theory has been derived, we need to know the velocity distribution of molecules at a given temperature, as it is given by the Maxwell-Boltzmann distribution. To use transition state theory we need the partition functions that follow from the Boltzmann distribution. Hence, we must devote a section of this chapter to statistical thermodynamics. 

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... 

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.)... 

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. 

Effect of Solvent on Arrhenius Plots. If water is a substrate, then the presence of an organic solvent, which may disrupt the structure and/or orientation of water, may alter the Arrhenius plot. For example, a linear plot is seen with fumarate hydratase in the presence of 10% methanol. However, the plot is biphasic in the presence of 10% ethanol . See Boltzmann Distribution Collision Theory Temperature Dependency, Transition-State Theory Energy of Activation On... 

ARRHENIUS EQUATION PLOT ARRHENIUS EQUATION PLOT BOLTZMANN DISTRIBUTION COLLISION THEORY TEMPERATURE DEPENDENCY, TRANSITION-STATE THEORY... 

So the rate constant ks may be found as the ratio between j and na. This expression for the rate constant is equivalent to a one-dimensional version of Eq. (5.49) when the Boltzmann distribution is replaced by P(r, v). It is, however, important to notice that Eq. (11.2) allows for recrossings across the barrier, since the lower integration limit is —oo and not 0 as in the previous presentations of transition-state theory where the possibility of recrossings was neglected. 

The derivation of the transition state theory expression for the rate constant requires some ideas from statistical mechanics, so we will develop these in a digression. Consider an assembly of molecules of a given substance at constant temperature T and volume V. The total number N of molecules is distributed among the allowed quantum states of the system, which are determined by T, V, and the molecular structure. Let n, be the number of molecules in state i having energy e,- per molecule. Then n, is related to e,- by Eq. (5-17), which is known as the Boltzmann distribution. 

This set of approximations is essentially similar to that for the more familiar canonical transition state theory, apart from the final assumption, that the state is described by a microcanonical distribution (i.e. fixed energy) rather than the Boltzmann distribution (thermal equilibrium) of canonical transition state theory. 

More importantly, a molecular species A can exist in many quantum states in fact the very nature of the required activation energy implies that several excited nuclear states participate. It is intuitively expected that individual vibrational states of the reactant will correspond to different reaction rates, so the appearance of a single macroscopic rate coefficient is not obvious. If such a constant rate is observed experimentally, it may mean that the process is dominated by just one nuclear state, or, more likely, that the observed macroscopic rate coefficient is an average over many microscopic rates. In the latter case k = Piki, where ki are rates associated with individual states and Pi are the corresponding probabilities to be in these states. The rate coefficient k is therefore time-independent provided that the probabilities Pi remain constant during the process. The situation in which the relative populations of individual molecular states remains constant even if the overall population declines is sometimes referred to as a quasi steady state. This can happen when the relaxation process that maintains thermal equilibrium between molecular states is fast relative to the chemical process studied. In this case Pi remain thermal (Boltzmann) probabilities at all times. We have made such assumptions in earlier chapters see Sections 10.3.2 and 12.4.2. We will see below that this is one of the conditions for the validity of the so-called transition state theory of chemical rates. We also show below that this can sometime happen also under conditions where the time-independent probabilities Pi do not correspond to a Boltzmann distribution. 

Traditional literature treats enzyme catalyzed reactions, including hydrogen transfer, in terms of transition state theory (TST) [4, 34, 70]. TST assumes that the reaction coordinate may be described by a free energy minimum (the reactant well) and a free energy maximum that is the saddle point leading to product. The distribution of states between the ground state (GS, at the minimum) and the transition state (TS, at the top of the barrier) is assumed to be an equilibrium process that follows the Boltzmann distribution. Consequently, the reaction s rate is exponentially dependent on the reciprocal absolute temperature (1/T) as reflected by the Arrhenius equation ... 

The essential feature of transition state theory is that there is a concentration of the species at the saddle point, the transition state or activated complex, that is in equihbrium with reactants and products. The Boltzmann Distribution Law governs the concentration of that transition state, and the rate of reaction is proportional to its concentration. Since the concentration... 

For chemical kinetics, transition state theory is most useM in the form that starts from reactants in thermal equilibrium. For our purpose we want a more detailed version, that of reactants with a total energy in the range E oE + AE. If we know how to do that, we can and will average over a Boltzmann distribution in E to obtain the thermal results. The first task at hand is to define what is meant by reactants at equilibrium at a total energy within the range (and at given values of any other conserved quantum numbers). It is the foundation of statistical mechanics that equilibrium under such conditions means that all possible quantum states of the reactants are equally probable. ... 

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 ... 

Finally, (2J + 1) is the degeneracy of the initial state and the expression involving the refractive index n is known as Lorentz s local-field correction. Calculations of transition probabilities within file frame of JO theory are usually made assuming that all Stark sublevels within the ground level are equally populated and that the material under investigation is optically isotropic. The former hypothesis is only reasonable in some cases, e.g., when transitions initiate from non-degenerate states such as Eu( Fo). Otherwise, there is a Boltzmann distribution of the population among the crystal-field sublevels. The second assumption is not valid for uniaxial or biaxial crystals, but, of course, holds for solutions. 

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