Activity coefficient equilibrium constant derivation

These equations, and in particular eqn [6], could then be used to determine the rate coefficients such as the activation-deactivation equilibrium constant. Nevertheless, this was experimentally difficult to obtain when high conversions are reached. This is based on the inaccurate assumption that the initiator concentration (RY or dormant species in polymerization) does not change as the reaction proceeds. Tang et then derived new equations to take into accoimt the consumption of the initiator. 

By examining the compositional dependence of the equilibrium constant, the provisional thermodynamic properties of the solid solutions can be determined. Activity coefficients for solid phase components may be derived from an application of the Gibbs-Duhem equation to the measured compositional dependence of the equilibrium constant in binary solid solutions (10). 

Finally, it is not appropriate to derive thermodynamic properties of solid solutions from experimental distribution coefficients unless it can be shown independently that equilibrium has been established. One possible exception applies to trace substitution where the assumptions of stoichiometric saturation and unit activity for the predominant component allow close approximation of equilibrium behavior for the trace components (9). The method of Thorstenson and Plummer (10) based on the compositional dependence of the equilibrium constant, as used in this study, is well suited to testing equilibrium for all solid solution compositions. However, because equilibrium has not been found, the thermodynamic properties of the KCl-KBr solid solutions remain provisional until the observed compositional dependence of the equilibrium constant can be verified. One means of verification is the demonstration that recrystallization in the KCl-KBr-H20 system occurs at stoichiometric saturation. 

The substituent constant of the Hammett equation has been related successfully to the logarithm of the activity coefficient ratio at infinite dilution for a series of meta and para isomers of phenol. Hammett stated that a free energy relationship should exist between the equilibrium or rate behavior of a benzene derivative and a series of corresponding meta and para monosubstitut-ed benzene derivatives. The Hammett equation may be written... 

Many additional consistency tests can be derived from phase equilibrium constraints. From thermodynamics, the activity coefficient is known to be the fundamental basis of many properties and parameters of engineering interest. Therefore, data for such quantities as Henry s constant, octanol—water partition coefficient, aqueous solubility, and solubility of water in chemicals are related to solution activity coefficients and other properties through fundamental equilibrium relationships (10,23,24). Accurate, consistent data should be expected to satisfy these and other thermodynamic requirements. Furthermore, equilibrium models may permit a missing property value to be calculated from those values that are known (2). 

The derivation of the law of mass action from the second law of thermodynamics defines equilibrium constants K° in terms of activities. For dilute solutions and low ionic strengths, the numerical values of the molar concentration quotients of the solutes, if necessary amended by activity coefficients, are acceptable approximations to K° [Equation (3)]. However, there exists no justification for using the numerical value of a solvent s molar concentration as an approximation for the pure solvent s activity, which is unity by definition.76,77... 

As a rough approximation (neglecting the activity coefficients), the distribution ratio of a given trivalent metallic cation (DM) can be derived from the logarithm expression of the concentration equilibrium constant Kex ... 

In theory, it should be possible to deal with all carbonate geochemistry in seawater simply by knowing what the appropriate activity coefficients are and how salinity, temperature, and pressure affect them. In practice, we are only now beginning to approach the treatment of activity coefficients under this varying set of conditions with sufficient accuracy to be useful for most problems of interest. That is why "apparent" and stoichiometric equilibrium constants, which do not involve the use of activity coefficients, have been in widespread use in the study of marine carbonate chemistry for over 20 years. The stoichiometric constants, usually designated as K. involve only the use of concentrations, whereas expressions for apparent equilibrium constants contain both concentrations and aH+ derived from "apparent pH". These constants are usually designated as K Examples of these different types of constants are ... 

ACTIVITY AND ACTIVITY COEFFICIENTS In our deduction of the law of mass action we used the concentrations of species as variables, and deduced that the value of the equilibrium constant is independent of the concentrations themselves. More thorough investigations however showed that this statement is only approximately true for dilute solutions (the approximation being the better, the more dilute are the solutions), and in more concentrated solutions it is not correct at all. Similar discrepancies arise when other thermodynamic quantities, notably electrode potentials or chemical free energies are dealt with. To overcome these difficulties, and still to retain the simple expressions derived for such quantities, G. N. Lewis introduced a new thermodynamic quantity, termed activity, which when applied instead of concentrations in these thermodynamic functions, provides an exact fit with experimental results. This quantity has the same dimensions as concentration. The activity, aA, of a species A is proportional to its actual concentration [A], and can be expressed as... 

In principle, it would be possible to determine the outcome of any chemical reaction if (a) The reaction mechanisms were known in detail, i.e. if all equilibrium constants and all rate constants of intermediary steps were known and (b) the initial concentrations of the reactants and the activity coefficients of all species involved were perfectly known. However, this is never the case in practice. It would be impossible to derive such a model by deduction from physical chemical theory without introducing drastic assumptions and simplifications. A consequence of this is, that the precision of any detailed prediction from such hard models will be low. In addition to this, physical chemical models rarely take interaction effects between experimental variables into account, which means that, in practice, such models will not be very useful for analysing the influence of experimental variables on synthetic operations. 

It is useful to construct a graph relating carbonate mineral solubilities to CO2 pressure. This can be done for calcite starting with equilibrium constant expression (6.2) above. If done rigorously, the derivation accounts for the effects of ion activity coefficients and the presence of CaHCOI and CaCOf ion pairs and of CaOH. Considering all complexation, the exact charge-balance equation for a pure water in which calcite is dissolving is... 

Summary We have derived a deceptively simple equation, AG°= -Rl n K, which enables us to calculate equilibrium constants from AG° data. Having derived and discussed the concept of activity, we can easily take into account the non-ideal behaviour of solutions, and of gases at significant pressures. However, in cases where solutions are dilute, or gas pressures low, calculations for the ideal case are also easily made, by putting all activity coefficients equal to unity. 

The subscripts ale, e and w refer to alcohol, ethylene and water, respectively. The term in pressures is usually written Kp, and the partial pressures are derived from the total pressure multiplied by mole fraction. The expression in activity coefficients, because of its similarity in form to Kp, is often abbreviated Ky it is not of course an equilibrium constant. In short, we may write K=KpKr The activity coefficients are determined from the generalized chart shown as Fig. 7.7, from known or estimated pressures. However, to start with we know only K and the total pressure, and must estimate starting values of each pressure in order to find values for y in each case. We can then estimate Kp as K/ffi and calculate equilibrium pressures once more. If the new values for y are appreciably different, then another iteration (calculation) is called for. The approach is similar to that of a golfer approaching his hole he arrives by successive approximation. Let us see how this works out in practice. 

Summary This chapter has presented a set of variations on the theme of free energy changes. We have seen how accurate values of AG° may be derived at any temperature, how this data may be summarized either as free energy functions or in terms of Ellingham diagrams, and how the data may be applied in a few instances. In virtually all cases, we have seen that activity can simplify the calculations of equilibrium constants and that allowances can always be made for non-ideal behaviour, assuming that activity coefficient data are available. Complete thermodynamic data have been published for relatively few compounds, however, and there are for example many common organic compounds for which only an enthalpy of formation has been determined. As more complete information is circulated, the number of applications of chemical... 

This is necessary if an elementary discussion of the principles of equilibrium is given before the full thermodynamic derivation of the algebraic form of the equifibrium constant. Much can be said about equilibrium without a thermodynamic approach, and without inclusion of the effects of non-ideality, i.e. without involving activity coefficients. 

Note that both Eqs. (5.142) and (5.137) are derived under the same assumptions the activity coefficients are neglected and Kc is equal to the equilibrium constant. Equation (5.142) can be rearranged further under the assumptions that ai = 0 and ai = 1... 

Any real model calculations require more or less simplification. As only a very few measurements were made at different temperatures, only equilibrium constants or the corresponding AG (7) values can be derived. Activity coefficients are often disregarded by measuring at nearly constant ionic strength obtained by the addition of a sufficient number of background ions. Here the equilibrium constants will contain an "activity factor." Further simphfications will depend on the particular system. 

Distribution coefficient (a) is the ratio of the concentrations of a solute in two immiscible solvents in equilibrium with each other. As an equilibrium constant, it can be substituted for K in Equation (4) and is therefore logarithmically related to free energy. Most of the work on correlating distribution coefficients with biological activities has been based on the system 1-octanol water, and the distribution coefficients quoted in a dimensionless form n. it is defined by Equation (40), where the suffix H represents the unsubstituted parent compound and X the derivative in which hydrogen has been replaced by the group X. A review by Hansch [47 ] summarises much of this work. 

There is, however, a difficulty with this derivation. The fact that the equilibrium equation (4) involves activity coefficients requires that the rate constants k2 and k j for reaction in forward and reverse directions also involve activity coefficients, since K = k /k-i. But if the rate constant k-i for the reverse reaction involves activity coefficients, how can one say that v = k [s] It is unreasonable to suppose that the intermediate S follows a... 

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