Equilibrium constants partitioning

We have determined the ion-pair formation-partition equilibrium constant with picrate anion for a number of primary, secondary, tertiary and quaternary ammonium ions 23>. In aqueous media of pH 5-6, the ammonium ions and picrate are considered to exist almost completely as unpaired counter ions. When the aqueous solution is mixed with an immiscible organic solvent, the ions are partitioned into the organic phase as the ion pair. We expected that the steric effect of N-substituents in the ion-pair formation-partition equilibrium could be analyzed by a procedure similar to Eq. 24, and derived Eq. 27 for the set of quaternary ions 23). 

As mentioned above, ion pairing in the organic phase complicates the equilibrium analysis, but it also adds information on interactions that involve the cation and that may influence extraction selectivity. Continuing the discussion of salt partitioning in three-component systems, the expression for the salt partitioning equilibrium constant for an associated system [Eq. (34) is... 

The apparent diffusion coefficient, Da in Eq. (11) is a mole fraction-weighted average of the probe diffusion coefficient in the continuous phase and the microemulsion (or micelle) diffusion coefficient. It replaces D in the current-concentration relationships where total probe concentration is used. Both the zero-kinetics and fast-kinetics expressions require knowledge of the partition coefficient and the continuous-phase diffusion coefficient for the probe. Texter et al. [57] showed that finite exchange kinetics for electroactive probes results in zero-kinetics estimates of partitioning equilibrium constants that are lower bounds to the actual equilibrium constants. The fast-kinetics limit and Eq. (11) have generally been considered as a consequence of a local equilibrium assumption. This use is more or less axiomatic, since existing analytical derivations of effective diffusion coefficients from reaction-diffusion equations are approximate. 

The solubilization of diverse solutes in micelles is most often examined in tenns of partitioning equilibria, where an equilibrium constant K defines the ratio of the mole fraction of solute in the micelle (X and the mole fraction of solute in the aqueous pseudophase. This ratio serves to define the free energy of solubilization -RT In K). 

An equilibrium constant describing the distribution of a solute between two phases only one form of the solute is used in defining the partition coefficient... 

In a simple liquid-liquid extraction the solute is partitioned between two immiscible phases. In most cases one of the phases is aqueous, and the other phase is an organic solvent such as diethyl ether or chloroform. Because the phases are immiscible, they form two layers, with the denser phase on the bottom. The solute is initially present in one phase, but after extraction it is present in both phases. The efficiency of a liquid-liquid extraction is determined by the equilibrium constant for the solute s partitioning between the two phases. Extraction efficiency is also influenced by any secondary reactions involving the solute. Examples of secondary reactions include acid-base and complexation equilibria. 

This distinction between Kd and D is important. The partition coefficient is an equilibrium constant and has a fixed value for the solute s partitioning between the two phases. The value of the distribution ratio, however, changes with solution conditions if the relative amounts of forms A and B change. If we know the equilibrium reactions taking place within each phase and between the phases, we can derive an algebraic relationship between Kd and D. 

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... 

Both the CTC and the bromonium tribromide species have their A-max at 272 nm, but with very different e. Furthermore, at the employed Br2 concentrations the counteranion of the bromonium ion is partitioned between Br3 and Br5 , the latter having its A-max batochromically shifted at 310 nm. The equilibrium constant between the tribromide and the pentabromide species of Scheme 4 is 22.3 M l, that is nearly coincident with that found for the tetrabutylammonium tribromide-pentabromide equilibrium (ref. 22). 

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. 

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