Partition equilibrium constants

Firstly, we need to note that stating a value of (partition) is useless unless we know how the equilibrium constant (partition) was written, i.e. which of the phases T and 2 in Equation (5.8) is the air and which is the liquid ... 

Figure 5.17 The solubility s of a partially soluble salt is related to the equilibrium constant (partition) and obeys the van t Hoff isochore, so a plot of In s (as y ) against 1 IT (as x ) should be linear, with a slope of AF lution -t- R . Note how the temperature is expressed in kelvin a graph drawn with temperatures expressed in Celsius would have produced a curved plot. The label KIT on the x-axis comes from l/T -t- 1/K...

Magnitude of the flux vector Equilibrium constant partition coefficient Apparent equilibrium constant... 

K Binding constant/equilibrium constant/partitioning coefficient... 

Because of their definition as equilibrium constants, partition coefficients do not refer to the total amount of chemicals in the system but rather indicate how large a fraction of the release tends to transfer into the different phases. Partition coefficients may be used as an extrathermodynamic reference scale for characterizing lipophilicity as one of the most important properties of small molecules acting on macromolecular systems in aqueous solutions for a comprehensive review of hydrophobicity see, for example, Taylor (1990). In QSAR analyses, the log P term is frequently used to account for two major processes related to partitioning ... 

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. 

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... 

It should finally be noted that the amount of the neutral and zweitterionic forms of a compound in solution is determined by its tatuomeric equilibrium constant, defined as Kz = cz/cn. Therefore, the neutral species and the zwitterion coexist around the isoelectric pH, and membrane permeation is conditioned by Kx and by the partition coefficient values of both tautomers. 

As mentioned in Chapater 3, the law of mass action sets the concentration relations of the reactants and products. So, the equilibrium constants, termed the partition coefficients, are the quotients... 

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