Intermediate partition function

For ammonia synthesis, we still need to determine the coverages of the intermediates and the fraction of unoccupied sites. This requires a detailed knowledge of the individual equilibrium constants. Again, some of these may be accessible via experiments, while the others will have to be determined from their respective partition functions. In doing so, several partition functions will again cancel in the expressions for the coverage of intermediates. 

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. 

Equilibrium constants in the model were evaluated from the partition functions of the intermediates, assuming a uniformity of sites. The molecular partition function... 

Occasionally, the rates of bimolecular reactions are observed to exhibit negative temperature dependencies, i.e., their rates decrease with increasing temperature. This counterintuitive situation can be explained via the transition state theory for reactions with no activation energy harriers that is, preexponential terms can exhibit negative temperature dependencies for polyatomic reactions as a consequence of partition function considerations (see, for example, Table 5.2 in Moore and Pearson, 1981). However, another plausible explanation involves the formation of a bound intermediate complex (Fontijn and Zellner, 1983 Mozurkewich and Benson, 1984). To... 

The enthalpy of formation for reactants and intermediates may be calculated from the expression for the partition function... 

At the actual reaction temperature, the molecular partition function for all intermediates are calculated from the properties of each intermediate. 

Calorimetric data provide a complete thermodynamic characterization as well as a direct experimental evaluation of the folding/ unfolding partition function and the population of intermediate states. This approach has been used in numerous calorimetric applications during the last decade and will not be reviewed here (the reader is referred to Freire and Biltonen, 1978b Privalov, 1982 Freire, 1989 Freire et al., 1990, for reviews in this area). 

Fig. 4. Schematic representation of the partition function [Eq. (1)] for protein folding/unfolding. Each state, from the native state (i = 0) to the unfolded state (i = n) and all intermediates (i = 1 to n - 1), is assigned a AG relative to the native state from which the statistical weights are obtained. The partition function, Q, is simply the sum of the statistical weights of all the states. Other important parameters, including the population of each state [Eq. (2)], the excess enthalpy [Eq. (3)], and the excess heat capacity [Eq. (4)], are determined from the partition function as described in the text.

It has been shown that under standard conditions most singledomain globular proteins exhibit a folding/unfolding behavior consistent with the two-state mechanism (Freire and Biltonen, 1978a Privalov, 1979). From a statistical thermodynamic standpoint the implication is that the population of partially folded intermediate states is negligible and the partition function reduces to two terms ... 

Fortunately, the number of states that ever become populated is relatively small, even under conditions that maximize the population of intermediates. It is apparent that the folding/unfolding partition function can be simplified so that it includes only those states that are relevant to the folding process. The approach that we have undertaken involves the use of the native conformation as a template to generate partially folded conformations, and to evaluate the Gibbs free energy of those conformations according to the rules described in Section III. 

An a-helix bundle may become a second-order cooperative folding unit if the interaction energy terms are such that the intermediate terms in the partition function become negligibly small [Eq. (14)] and the entire partition function reduces to a two-state partition function (i.e., a partition function of the form 1 + e G/RT). If such is the case, the a-helix bundle will be either completely folded or unfolded. Higher order cooperative folding units can be constructed from lower order ones following the same rules. The most immediate application of this approach is to proteins exhibiting pure a-helical structural motifs. 

As seen in Fig. 10, the model accurately predicts the presence, location, and area of the cold and heat denaturation peaks. Under these conditions, the hierarchical partition function predicts a heat denaturation peak centered at 58°C and a cold denaturation peak centered at 4°C. The enthalpy change for the heat denaturation peak is 59 kcal mol-1 and the ACp is equal to 2.45 kcal K-1 mol-1. The experimental values reported by Privalov et al. (1986) are 57.5 and 3°C for the heat and cold denaturation transition temperatures, 53 kcal mol-1 for the enthalpy change, and 2.5 kcal K-1 mol-1 for ACp. Analysis of the theoretical curve indicates that it corresponds to a two-state transition, in agreement with the experimental data. The population of partially folded intermediates is never greater than 10-5 during the heat denaturation transition. 

Another application of intermediate coupling calculations has been to use the calculated results to reevaluate dissociation energies derived using the third-law method and mass- spectral data. Balasubramanian and Pitzer have shown how this can be accomplished in their calculations on Sn2 and Pb2 (90). This method requires the molecular partition function, which can be written... 

The proper evaluation of the quantized energy levels within the SACM requires a separable reaction coordinate and thus numerical implementations have implicitly assumed a center-of-mass separation distance for the reaction coordinate, as in flexible RRKM theory. Under certain reasonable limits the underlying adiabatic channel approximation can be shown to be equivalent to the variational RRKM approximations. Thus, the key difference between flexible RRKM theory and the SACM is in the focus on the underlying potential energy surface in flexible RRKM theory as opposed to empirical interpolation schemes in the SACM. Forst s recent implementation of micro-variational RRKM theory [210], which is based on interpolations of product and reactant canonical partition functions, provides what might be considered as an intermediate between these two theories. 

In the path integral approach, the transition amplitude between two states of the system can be calculated by summing amplitudes for all possible paths between them. By inserting a sequence of sums over sets of intermediate states into the expression for the partition function, Eq. (48) becomes... 

Boltzmann constant, and g, is a statistical weight for the ith excited state. The summation over all possible states is the electronic partition function. If the flame temperature is constant throughout the analysis, the signal level will be subject only to the amount of sample in this region. Thus the intermediate zone is usually aligned with the optical path and is of most importance for analytical measurements. However, this alignment of the optical path should also be optimized for the particular element to be quantitated. 

Intermediates. Another comphcation is when an intermediate is formed that partitions to a significant extent both forward and backward. The observable KIEs will be a function of the intermediate partitioning." Again, it may be possible to... 

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