Approximation method, equilibrium constant calculations

Therefore K b approximates the equilibrium constant of the hemoglobin oxygenation reaction. The has an important practical use in estimation of the correction factor f, required for the calculation of oxygen consumption by the Hb02 method. 

Dunken and Fritzsche (6) have summarized all the equilibrium constant calculations from infrared data used by earlier workers. They have shown that within the limits of accuracy in measurement, agreement can be reached between the results of two methods of calculation which contradict each other in their assumptions. By making infrared measurements at different temperatures, they also show that these chance agreements can come about at one temperature and not at another. Dunken and Fritzsche consider that simplified treatments of the type enumerated by Lussan are only approximations to the truth, and one should always employ a general model of association in which all the associated species (up to a certain maximum size) are present. The authors discuss the evidence for the cyclic dimer form but do not refer specifically to it in their calculations nor do they suggest that any of the higher multimers are cyclic. 

From the third law of thermodynamics, the entiopy 5 = 0 at 0 K makes it possible to calculate S at any temperature from statistical thermodynamics within the hamionic oscillator approximation (Maczek, 1998). From this, A5 of formation can be found, leading to A/G and the equilibrium constant of any reaction at 298 K for which the algebraic sum of AyG for all of the constituents is known. A detailed knowledge of A5, which we already have, leads to /Gq at any temperature. Variation in pressure on a reacting system can also be handled by classical thermodynamic methods. 

These four equations are perfectly adequate for equilibrium calculations although they are nonsense with respect to mechanism. Table 7.2 has the data needed to calculate the four equilibrium constants at the standard state of 298.15 K and 1 bar. Table 7.1 has the necessary data to correct for temperature. The composition at equilibrium can be found using the reaction coordinate method or the method of false transients. The four chemical equations are not unique since various members of the set can be combined algebraically without reducing the dimensionality, M=4. Various equivalent sets can be derived, but none can even approximate a plausible mechanism since one of the starting materials, oxygen, has been assumed to be absent at equilibrium. Thermodynamics provides the destination but not the route. 

In principle, this system of 20 equations can be solved provided the equilibrium constants, activities, Henry-constants and fugacities are available. While some results for most of these properties are available, there exists no approved method for calculating activities in concentrated aqueous solutions of weak electrolytes therefore, several approximations were developed. ... 

Polarography is valuable not only for studies of reactions which take place in the bulk of the solution, but also for the determination of both equilibrium and rate constants of fast reactions that occur in the vicinity of the electrode. Nevertheless, the study of kinetics is practically restricted to the study of reversible reactions, whereas in bulk reactions irreversible processes can also be followed. The study of fast reactions is in principle a perturbation method the system is displaced from equilibrium by electrolysis and the re-establishment of equilibrium is followed. Methodologically, the approach is also different for rapidly established equilibria the shift of the half-wave potential is followed to obtain approximate information on the value of the equilibrium constant. The rate constants of reactions in the vicinity of the electrode surface can be determined for such reactions in which the re-establishment of the equilibria is fast and comparable with the drop-time (3 s) but not for extremely fast reactions. For the calculation, it is important to measure the value of the limiting current ( ) under conditions when the reestablishment of the equilibrium is not extremely fast, and to measure the diffusion current (id) under conditions when the chemical reaction is extremely fast finally, it is important to have access to a value of the equilibrium constant measured by an independent method. 

Because K, depends on concentrations and the product KyKx is concentration independent, Kx must also depend on concentration. This shows that the simple equilibrium calculations usually carried out in first courses in chemistry are approximations. Actually such calculations are often rather poor approximations when applied to solutions of ionic species, where deviations from ideality are quite large. We shall see that calculations using Eq. (47) can present some computational difficulties. Concentrations are needed in order to obtain activity coefficients, but activity coefficients are needed before an equilibrium constant for calculating concentrations can be obtained. Such problems are usually handled by the method of successive approximations, whereby concentrations are initially calculated assuming ideal behavior and these concentrations are used for a first estimate of activity coefficients, which are then used for a better estimate of concentrations, and so forth. A G is calculated with the standard state used to define the activity. If molality-based activity coefficients are used, the relevant equation is... 

Equation 3 assumes that the exchange reaction coexists with the silicate equilibrium, but that the equilibrium constants of the two reactions are independent. If this is true, then something is amiss with respect to current methods for approximating the standard free energy of formation of minerals having exchangeable ions, because all such methods require inclusion of the exchangeable ion in the calculation (13, 14, 15). For a... 

However, these physicochemical investigations were to a substantial extent approximate and incomplete, due to the lack of thermodynamic constants of many rock-forming minerals (amphiboles, micas, chlorites, etc.) to imprecision in the tabulated calculated constants (siderite, ferrosilite, fayalite, etc.) to the substantial discrepancies between theoretical and experimental equilibrium data and to the lack of data on the properties of the fluid at high pressures. Moreover, many thermodynamic calculations were made by approximate methods, which led to substantial errors. 

For resenroir engineering calculations various properties of the crude oil and its associated gas and water must be known. It will be shown that theoretically many of these properties could be calculated by the methods presented in previous chapters, provided the composition of the system is known and complete equilibrium constant data for all of the components are available. However, since this information is seldom at hand, values of the reservoir fluid characteristics are usually experimentally determined or approximated by methods that experience has shown to be sufliciently accurate for most engineering computations. 

There have been many attempts to calculate AH independent of the equilibrium constant. The difficulty of a complete theoretical treatment of the H bond unfortunately requires approximations. The uncertainties thus introduced deprive the calculations of predictive value. Briefly, the usual approximations are based on some sort of electrostatic model, with computation of electrostatic, dispersion, and repulsive contributions by the methods of classical physics. Of course, the calculations require knowledge or estimation of such quantities as molecular arrangement, charge distribution, potential function, etc. Only a few systems have been treated. Reference 1327, for HF dimers 25, for carboxylic acids and 1561b, for ice furnish illustrative examples. Many other references are listed in Section 8.3, where a more complete discussion of the theoretical treatments is given. 

Several features of these results will be highlighted. First, the consistent agreement between observed and calculated values indicates that both the experimental method and the thermodynamic data base are valid. The Eh s are not remarkably sensitive, however, to the value for any single equilibrium constant. Typically, a change of +/- 0.2 log units in any constant will change Eh by no more than +/- 10 mV, which is approximately the limit of the accuracy of our data. This is somewhat surprising in that most of the Fe(III) is complexed in all of our runs (see Table III). 

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