Activity coefficient numerical example

It can be seen from Figures 16.2 and 16.3 that the numerical values of the activity and activity coefficient of the solute are different for the two choices of standard state. The scale of activities, for example, is necessarily different. The activity coefficient at mole fraction X2(i) is given by the ratio N/M in both figures. Thus, when the standard state is chosen on the basis of Henry s law, the activity coefficients are less than 1, whereas when the pure solute is chosen as standard state, the activity coefficients all are greater than 1. 

The data needed are the rate equation, energy of activation, heat of reaction, densities, heat capacities, thermal conductivity, diffusivity, heat transfer coefficients, and usually the stoichiometry of the process. Simplified numerical examples are given for some of these cases. Item 4 requires the solution of a system of partial differential equations that cannot be made understandable in concise form, but some suggestions as to the procedure are made. 

There are a number of kinetic and equipment parameters used in the numerical example. The size of the reactor, the overall heat transfer coefficient, the heats of reaction, the specific reaction rates, and the activation energies are among the most important. All of these were explored. 

Chemical speciation in soil solutions and other natural waters can be calculated routinely with a number of software products offered in a variety of computational media.27 30 Five examples of these products are listed in Table 2.5. They differ principally in the method of solving the chemical equilibrium problem numerically, or in the chemical species and equilibrium constants considered, or in the model used to estimate single-species activity coefficients. Irrespective of these differences, all the examples follow a similar algorithm ... 

A modern set of methods which can be used to estimate partition coefficients is the group contribution method. These methods were developed to allow chemical engineers to estimate activity coefficients in liquid and polymeric systems. Of the numerous methods developed, UNIFAC, the oldest and most thoroughly tested method, is probably the most universally applicable to a wide variety of substances and sytems despite its known weaknesses (Baner, 1999). The use of UNIFAC and example calculations will be described later in this chapter. 

There are several limitations which lead to the discrepancies in Tables IV-X. First of all, no model will be better than the assumptions upon which it is based. The models compiled in this survey are based on the ion association approach whose general reliability rests on several non-thermodynamic assumptions. For example, the use of activity coefficients to describe the non-ideal behavior of aqueous electrolytes reflects our uncertain knowledge of ionic interactions and as a consequence we must approximate activity coefficients with semi-empirical equations. In addition, the assumption of ion association may be a naive representation of the true interactions of "ions" in aqueous solutions. If a consistent and comprehensive theory of electrolyte solutions were available along with a consistent set of thermodynamic data then our aqueous models should be in excellent agreement for most systems. Until such a theory is provided we should expect the type of results shown in Tables IV-X. No degree of computational or numerical sophistication can improve upon the basic chemical model which is utilized. 

Since the early work of Hansch et al., numerous examples of the quantitative correlation of biological activity with chemical structure have been reported. The success of QSAR relies heavily on the use of partition coefficients (log P or n) in extending the linear free-energy relationship (LEER) from homogenous organic chemical systems (i.e., the Hammett-Taft type approach) to compartmentalized heterogenous biological systems. 

It thus seems appropriate to use partition coefficients between aqueous and organic phases to represent molecular properties related to the hydrophobic interactions between small molecules and the biophase. This choice is well established at the level of a second order approximation in the extrathermodynamic derivation (see section A. 3). Numerous examples (137, 138) illustrate the utility of the partition coefficient in correlations of biological activity with chemical structure. 

The splitting of redox reaetions into two half cell reactions by introducing the symbol e is highly useful. It should be noted that the e notation does not in any way refer to solvated electrons. When calculating the equilibrium composition of a chemical system, both e , and can be chosen as components and they can be treated numerically in a similar way equilibrium constants, mass balances, etc. may be defined for both. However, while represents the hydrated proton in aqueous solution, the above equations use only the activity of e , and never the concentration of e . Concentration to activity conversions (or activity coefficients) are never needed for the electron cf. Appendix B, Example B.3). 

By (numerical or graphical) integration, a2 can now be derived. Figure 2.3 gives as an example the activities of sucrose solutions. It is seen that the activities greatly deviate from the mole fractions at higher concentration. For example, at x2 = 0.1, the activity coefficient of water x 0.85/0.90 = 0.94, that of sucrose x 0.26/0.1 = 2.6. For mixtures of more than two components, the activities cannot be derived in this way. 

These equations, especially Eqs. 9.3-20, 9.3-22, and 9.3-23, are useful when one wants to correct the numerical values of activity coefficients obtained at one temperature and pressure for use at other temperatures and pressures. For example, Eq. 

The first term describes the contribution of the ideal mixing itself, while the second one describes the excess energy due to interactions The variation can exhibit a particular shape, as Illustrated by a numerical example in the Figure 6.18, where the activity coefficients have been calculated by Margules model with A,2=2 and A2,=1.5. It may be observed that in the immiscibility region a-b the value, resulting by... 

A collection of numerical data covering a relatively large number of quantities used in physical chemistry and thermodynamics, mainly for inorganic species for example acidity constants including those found in non-aqueous solvents, solubility constants and complexation constants. Regarding electrochemistry, you can find the redox potentials for numerous couples, the molar conductivities for the main ions in aqueous solution, the activity coefficients for electrolytes, as well as a small number of kinetic features (exchange current density, and transfer coefficient, etc.). 

Fugacity coefficients and hence activity coefficients can be calculated with the help of appropriate equations of state (see Section IV). This is possible, however, only for the gas phase (van der Waals equation, Redlich-Kwong equation, virial equation) for condensed phases no useful general equations of state are available. Experimental determination of activity coefficients in condensed phases is based on the study of equilibria. There are numerous methods, but only typical examples will be given. 

We now take a look at how numerical values are assigned to the thermodynamic properties of single ions. There is more than one method, of course. Most physical chemistry courses for example will mention the Macinnes convention, which postulates that because the K+ and Cr ions have similar properties, their activity coefficients should be identical. This means that... 

A more quantitative prediction of activity coefficients can be done for the simplest cases [18]. However, for most electrolytes, beyond salt concentrations of 0.1 M, predictions are a tedious task and often still impossible, although numerous attempts have been made over the past decades [19-21]. This is true all the more when more than one salt is involved, as it is usually the case for practical applications. Ternary salt systems or even multicomponent systems with several salts, other solutes, and solvents are still out of the scope of present theory, at least, when true predictions without adjusted parameters are required. Only data fittings are possible with plausible models and with a certain number of adjustable parameters that do not always have a real physical sense [1, 5, 22-27]. It is also very difficult to calculate the activity coefficients of an electrolyte in the presence of other electrolytes and solutes. Even the definition is difficult, because electrolyte usually dissociate, so that extrathermodynamical ion activity coefficients must be defined. The problem is even more complex when salts are only partially dissociated or when complex equilibriums of gases, solutes, and salts are involved, for example, in the case of CO2 with acids and bases [28, 29]. 

Still another function which is used, especially in connexion with electrolyte solutions, is the osmotic coefficient of the solvent. This is simply a logarithmic function of the activity coefficient, as already defined, but it is useful whenever the activity coefficient of the solvent differs from unity by only a very small amount. For example, in the case of dilute electrolyte solutions, the activity coefficient of the solvent may differ from unity by less than one part in 10, whilst the activity coefficient of the solute may differ from unity by several p>er cent. In such cases it is desirable to use a function which results in a larger numerical measure of the departure of the solvent from ideality. 

The pure-component vapor pressures play a key role in the reduction of low-pressure VLE data. Since they appear in the equilibrium equations as normalizing factors for the activity coefficients, they directly affect the numerical values of through every data point. However, vapor pressures are highly sensitive to temperature and to sample purity. Thus, to guarantee Internal consistency with the rest of the data set, they should be measured with the same equipment and on the same lots of materials as are the mixture vapor pressures. This unfortunately is not always done, and one sometimes has to analyze an otherwise satisfactory set of data with foreign vapor pressures, computed for example from an Antoine equation extracted from the literature. 

---