Partition functions equilibria

In order to extend the above theory, we have to make use of the equilibrium partition concepts described in Part II. Here we generally define the equilibrium partition function between any phase B and A by ... 

As seen from our discussion in Chapter 3, which dealt with onedimensional problems, in many relevant cases one actually does not need the knowledge of the behavior of the system in real time to find the rate constant. As a matter of fact, the rate constant is expressible solely in terms of the equilibrium partition function imaginary-time path integrals. This approximation is closely related to the key assumptions of TST, and it is not always valid, as mentioned in Section 2.3. The general real-time description of a particle coupled to a heat bath is the Feynman-Vernon... 

Flavor partition coefficients. The equilibrium distribution of a particular flavor molecule between two phases (e.g., oil-water, air-water, or air-oil) is characterized by an equilibrium partition function. These partition coefficients determine the distribution of the flavor molecules between the oil, water, and head space phases of an emulsion. 

In the calculation of chemical equilibrium, partition functions per unit volume are desired, so... 

By this time, you may have guessed that a time correlation function is a far more difficult object to calculate than an ordinary equilibrium partition function, and insofar as the latter is virtually impossible to evaluate exactly for realistic liquids, one might well wonder where we go from here. 

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. 

Here Zint is the intramolecular partition function accounting for rotations and vibrations. However, in equilibrium, the chemical potential in the gas phase is equal to that in the adsorbate, fi, so that we can write the desorption rate in (I) as... 

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... 

Carbon tetrachloride-hydrogen sulfide-water ternary system, 49, 51, 52 Carboniuin ion polymerization, 158 Carboxylic groups initiator, 174 Catalyst clathrates equilibrium, 35 Cell partition function, in calculation of thermodynamic quantities of clathrates, 26... 

Several methods have been developed for the quantitative description of such systems. The partition function of the polymer is computed with the help of statistical thermodynamics which finally permits the computation of the degree of conversion 0. In the simplest case, it corresponds to the linear Ising model according to which only the nearest segments interact cooperatively149. The second possibility is to start from already known equilibrium relations and thus to compute the relevant degree of conversion 0. 

In classical statistics all equilibrium properties of the system are expressed in terms of the partition function... 

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 the following, the MO applications will be demonstrated with two selected equilibrium reactions, most important in radical chemistry disproportionation and dimerization. The examples presented will concern MO approaches of different levels of sophistication ab initio calculations with the evaluation of partition functions, semiempirical treatments, and simple procedures employing the HMO method or perturbation theory. 

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... 

Using the formalism of statistical mechanics, Giddings et al. [135] investigated the effects of molecular shape and pore shape on the equilibrium distribution of solutes in pores. The equilibrium partition coefficient is defined as the ratio of the partition function in the pore... 

The model [39] was developed using three assumptions the conformers are in thermodynamic equilibrium, the peak intensities of the T-shaped and linear features are proportional to the populations of the T-shaped and linear ground-state conformers, and the internal energy of the complexes is adequately represented by the monomer rotational temperature. By using these assumptions, the temperature dependence of the ratio of the intensities of the features were equated to the ratio of the quantum mechanical partition functions for the T-shaped and linear conformers (Eq. (7) of Ref. [39]). The ratio of the He l Cl T-shaped linear intensity ratios were observed to decay single exponentially. Fits of the decays yielded an approximate ground-state binding... 

This distribution follows automatically if we require that the entropy of a system with many members that is in equilibrium is at maximum. The denominator in the Boltzmann distribution ensures that the frequencies P are normalized and add up to unity, or 100%. This summation of states (Zustandssumme in German) is called a partition function ... 

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. 

Hence, we conclude that the translational partition function of a particle depends on its mass, the temperature, the dimensionality as 3vell as the dimensions of the space in vhich it moves. As a result, translational partition functions may be large numbers. The translational partition function is conveniently calculated per volume, which is the quantity used, for example, when the equilibrium conditions are determined, as we shall see later. The partition function can conveniently be written as... 

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... 

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