Transition state theory thermodynamic functions

The activation parameters from transition state theory are thermodynamic functions of state. To emphasize that, they are sometimes designated A H (or AH%) and A. 3 4 These values are the standard changes in enthalpy or entropy accompanying the transformation of one mole of the reactants, each at a concentration of 1 M, to one mole of the transition state, also at 1 M. A reference state of 1 mole per liter pertains because the rate constants are expressed with concentrations on the molar scale. Were some other unit of concentration used, say the millimolar scale, values of AS would be different for other than a first-order rate constant. 

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. 

The thermodynamic formulation of the transition state theory is useful in considerations of reactions in solution when one is examining a particular class of reactions and wants to extrapolate kinetic data obtained for one reactant system to a second system in which the same function groups are thought to participate (see Section 7.4). For further discussion of the predictive applications of this approach and its limitations, consult the books by Benson (59) and Laidler (60). Laidler s kinetics text (61) and the classic by Glasstone, Laidler, and Eyring (54) contain additional useful background material. 

Comparison of collision theory, the partition function form and the thermodynamic form of transition state theory... 

Theoretical calculations are less fundamental and rigorous for solution reactions. This is a consequence of the difficulty of calculating partition functions in solution. The main focus for solution reactions has been on the thermodynamic formulation of transition state theory. 

The rate constant can be expressed by using molecular, statistical mechanical, and thermodynamical quantities, functions, and formulations. For instance, in the transition state theory of chemical reactions for an elementary reaction... 

Rel. (9) indicates that the kinetics of a catalytic reaction described by equation (7) does not depend on the concentration of the reacting gas (is of zero order as a function of gas concentration). For rel. (9) the evaluation of k is based on statistical thermodynamics applied to transition state theory for chemical reactions [12]. This theory shows that k has the following expression ... 

Reuter, Frenkel, and Scheffler have recently used DFT-based calculations to predict the CO turnover frequency on RuO2(110) as a function of 02 pressure, CO pressure, and temperature.31 This was an ambitious undertaking, and as we will see below, remarkably successful. Much of this work was motivated by the earlier success of ab initio thermodynamics, a topic that is reviewed more fully below in section 3.1. The goal of Reuter et al. s work was to derive a lattice model for adsorption, dissociation, surface diffusion, surface reaction, and desorption on defect-free Ru02(l 10) in which the rates of each elementary step were calculated from DFT via transition state theory (TST). As mentioned above, experimental evidence strongly indicates that surface defects do not play a dominant role in this system, so neglecting them entirely is a reasonable approach. The DFT calculations were performed using a GGA full-potential... 

At the same time transition state theory requires information about the activated complexes, assumes equilibrium only for reactants, but not products and requires introduction of a special partition function (minus one degree of freedom). Another question which remains is the applicability of statistical thermodynamics, if the life time of activated complexes is ca. 10 13 s. For instance the application of transmission coefficient contradicts the basic principles of TST, namely statistical equilibrium between reactants and activated complexes. 

Rate constants can be estimated by means of transition-state theory. In principle all thermodynamic data can be deduced from the partion function. The molecular data necessary for the calculation of the partion function can be either obtained from quantum mechanical calculations or spectroscopic data. Many of those data can be found in tables (e.g. JANAF). A very powerful tool to study the kinetics of reactions in heterogeneous catalysis is the dynamic Monte-Carlo approach (DMC), sometimes called kinetic Monte-Carlo (KMC). Starting from a paper by Ziff et al. [16], several investigations were executed by this method. Lombardo and Bell [17] review many of these simulations. The solution of the problem of the relation between a Monte-Carlo step and real time has been advanced considerably by Jansen [18,19] and Lukkien et al. [20] (see also Jansen and Lukkien [21]). First principle quantum chemical methods have advanced to the stage where they can now offer quantitative predictions of structure and energetics for adsorbates on surfaces. Cluster and periodic density functional quantum chemical methods are used to analyze chemisorption and catalytic surface reactivity [see e.g. 24,25]. 

The insert in Figure 7.6 shows a distribution of states for the activated complexes as a function of their distance from the lowest energy transition state along the reaction coordinate. This is important because now the transition state can be considered to represent molecules with a distribution of various energies. Thus, we can rationalize the idea of an equilibrium of states for the activated complex, allowing for a thermodynamic analysis that connects the energy of the reactant with the products, just as is done in transition state theory. 

With reactions in solution which are not controlled by diffusion, similar chemical considerations are involved to those given earlier, and again we resort to TST. However, because of problems, in particular, of standard states and activities, there are difficulties in formulation of partition functions for species solution. The best way of applying TST in this case involves the thermodynamic formulation of CTST (Conventional Transition State Theory). Within this, the reaction rate is directly proportional to the concentration of the activated complexes. 

Table 10.4 lists the rate parameters for the elementary steps of the CO + NO reaction in the limit of zero coverage. Parameters such as those listed in Tab. 10.4 form the highly desirable input for modeling overall reaction mechanisms. In addition, elementary rate parameters can be compared to calculations on the basis of the theories outlined in Chapters 3 and 6. In this way the kinetic parameters of elementary reaction steps provide, through spectroscopy and computational chemistry, a link between the intramolecular properties of adsorbed reactants and their reactivity Statistical thermodynamics furnishes the theoretical framework to describe how equilibrium constants and reaction rate constants depend on the partition functions of vibration and rotation. Thus, spectroscopy studies of adsorbed reactants and intermediates provide the input for computing equilibrium constants, while calculations on the transition states of reaction pathways, starting from structurally, electronically and vibrationally well-characterized ground states, enable the prediction of kinetic parameters. 

The measurement of exchange rates is important, since it gives us vital information on the transition state between reagents and products. Absolute rate theory states that the rate is given by eq. (1), in which k, h and R are Boltzmann s, Planck s and the gas constants, and T is the absolute temperature. The transmission coefficient, k, is usually taken as 1. The thermodynamic functions AG, AH and AS represent the change between the initial and transition states. 

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