Equilibrium constant partition functions

This factor can be obtained from the vibration partition function which was omitted from the expression for the equilibrium constant stated above and is, for one degree of vibrational freedom where vq is the vibrational frequency in the lowest energy state. 

Equation (5-43) has the practical advantage over Eq. (5-40) that the partition functions in (5-40) are difficult or impossible to evaluate, whereas the presence of the equilibrium constant in (5-43) permits us to introduce the well-developed ideas of thermodynamics into the kinetic problem. We define the quantities AG, A//, and A5 as, respectively, the standard free energy of activation, enthalpy of activation, and entropy of activation from thermodynamics we now can write... 

One can write for Eq. (7-49) an expression for the equilibrium constant. Statistical thermodynamics allows its formulation in terms of partition functions ... 

One can define a special equilibrium constant K (special in that it lacks the partition function for the unique vibration), giving for the rate constant... 

In contrast to disproportionation, in the dimerization equilibrium at lower temperatures the reaction product is favored. The contributions coming from partition functions and from the exponential term are presented together with the temperatures and logarithms of equilibrium constants of the reaction (103) in Table XI. 

Data for the 2 CHj C H, Reaction Temperature, Partition Function Contributions, Exponential Term, Logarithm of the Equilibrium Constant... 

Partition functions are very important in estimating equilibrium constants and rate constants in elementary reaction steps. Therefore, we shall take a closer look at the partition functions of atoms and molecules. Motion, or translation, is the only degree of freedom that atoms have. Molecules also possess internal degrees of freedom, namely vibration and rotation. 

In Chapter 2 we discussed both chemical equilibrium and equilibrium constants. We shall now return to the chemical reactions and see how equilibrium constants can be determined directly from the partition functions of the molecules participating in the reaction. Consider the following reaction, which was described in Chapter 2 ... 

Thus, given sufEcient detailed knowledge of the internal energy levels of the molecules participating in a reaction, we can calculate the relevant partition functions, and then the equilibrium constant from Eq. (67). This approach is applicable in general Determine the partition function, then estimate the chemical potentials of the reacting species, and the equilibrium constant can be determined. A few examples will illustrate this approach. 

It is instructive to illustrate the relation between the partition function and the equilibrium constant with a simple, entirely hypothetical example. Consider the equilibrium between an ensemble of molecules A and B, each with energy levels as indicated in Fig. 3.5. The ground state of molecule A is the zero of energy, hence the partition function of A vnll be... 

We have in Eq. (77) expressed the equilibrium constant in terms of the relevant partition functions, which must be calculated ... 

We express the equilibrium constant in terms of the partition functions of both the reactant and the transition state, and we take the partition function of the reaction coordinate separately ... 

We now need an expression for the equilibrium constant between the gas phase and the transition state complex. The reaction coordinate is again the (very weak) vibration between the atom and the surface. There are no other vibrations parallel to the surface, because the atom is moving in freely in two dimensions. The relevant partition functions for the atoms in the gas phase and in the transition state are... 

Note how the partition function for the transition state vanishes as a result of the equilibrium assumption and that the equilibrium constant is determined, as it should be, by the initial and final states only. This result will prove to be useful when we consider more complex reactions. If several steps are in equilibrium, and we express the overall rate in terms of partition functions, many terms cancel. However, if there is no equilibrium, we can use the above approach to estimate the rate, provided we have sufficient knowledge of the energy levels in the activated complex to determine the relevant partition functions. 

For t vo systems in chemical equilibrium we can calculate the equilibrium constant from the ratio of partition functions by requiring the chemical potentials of the t vo systems to be equal. 

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. 

Exercise 3.8 Partition Function, Average Energy and Equilibrium Constant... 

Exercise 3.9 Equilibrium Constants From Partition Functions... 

In statistical mechanics the properties of a system in equilibrium are calculated from the partition function, which depending on the choice for the ensemble considered involves a sum over different states of the system. In the very popular canonical ensemble, that implies a constant number of particles N, volume V, and temperature T conditions, the quasiclassical partition function Q is... 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

Clark, E.C.W., Glew, D.N. (1966) Evaluation of thermodynamic functions from equilibrium constants. Trans. Farad. Soc. 62, 539-547. Cole, J.G., Mackay, D. (2000) Correlating environmental partitioning properties of organic compounds The three solubility approach . Environ. Toxicol. Chem. 19, 265-270. 

This equilibrium constant is often incorrectly called a partition function - which is in fact a term from statistical mechanics. 

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

Thus, rate constant for a reaction can be given by equation (4.34) in terms of equilibrium constant for the formation of activated complex K which can be expressed either in terms of partition or thermodynamic functions. 

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