Activity coefficients defined

Some authors" express the transfer free energy in the form of an activity coefficient defined by Eq. (8-59). 

Here/+ is the mean ionic activity coefficient defined, by analogy with the mean ionic activity, as... 

The problem at hand is the evaluation of the activity coefficient defined in Eq. (76). It will be assumed that only pairwise interactions between the defects need be considered at the low defect concentrations we have in mind. (The theory can be extended to include non-pairwise forces.23) Then the cluster function R(n) previously defined in Eq. (78) is the sum of all multiply connected diagrams, in which each bond represents an /-function, which can be drawn among the set of n vertices, the /-function being defined by Eqs. (66), (56), and (43). The Helmholtz free energy of interaction of two defects appearing in this definition can be written as... 

An important component of equilibrium calculations is the conversion between ion activities, which equilibrium constants refer to, and ion concentrations, which mass balance and electrical neutrality equations refer to. The conversion is made with activity coefficients defined by the relation ... 

The approach of specific interactions, developed primarily by Pitzer (1973) and Whitfield (1975a,b), considers all salts, from a purely formal point of view, as completely dissociated, and embodies the effects of specific interactions into particular activity coefficients, defined as total activity coefficients or stoichiometric activity coefficients, with symbol y. For instance, for ion /,... 

The terms fugacity, activity and activity coefficient defined ... 

Activity coefficients defined within the infinite dilution activity scale cannot be formulated theoretically for the ionic medium of seawater. Since the oceans contain an ionic medium of practically constant composition, the ionic medium activity scale might be used advantageously in studying acid-base and other equilibria in seawater (see also Appendix 6.2 in Chapter 6). 

Corresponding to each chemical potential there is an activity coefficient defined in terms of equation (20.4). By convention, the activity coefficients of electrolytes are always expressed in terms of the ideal dilute solution as standard reference state, cf. chap. XXI, 3. Thus in the case of an aqueous NaCl solution we may write... 

We employ y to denote the activity coefficient defined relative to the pure substance and y the activity coefficient defined relative to an infinitely dilute solution. The only other use of a superscript asterisk (except in the statistical mechanical discussion of intermolecular forces and liquids) is to distinguish the pressure p from the fugacity, p. ... 

This result is identical with Henryks law [equation (36.3)], and so on the basis of the present standard state, a2/N2 will be unity, or as will equal N2, for all solutions obeying Henry s law. The activity coefficient 7n, equal to U2/N2, of the solute in any solution may thus be used to indicate its obedience of Henry s law. It is for this reason that the activity coefficient defined in this manner has been called the rational activity coefficient. 

It is clearly observed that the activity coefficients in (6.33)-(6.35) differ fundamentally from the activity coefficient introduced in sections 6.1 and 6.2. To stress this difference, we have used the superscript DI to denote deviations from DI behavior. Furthermore, each of the activity coefficients defined in (6.33)-(6.35) depends on the thermodynamic variables, say T and pB, or Tand P, or T and pB. This has also been indicated in the notation. In practical applications, however, one usually knows which variables have been chosen, in which case one can drop the arguments in the notation for y 1. 

As naphthalene in hexane follows Henry s Law (but not Raoul t s Law) the activity coefficient defined on this basis would be approximately unity, as the equation = fxf + RT nxt holds for this system. Whether the pure-liquid or the Henry s Law standard state is used in any thermodynamic calculation is purely a matter of convenience. There is no fundamental thermodynamic objection to using any standard state. In the particular problems we are considering it is sometimes helpful to have the activity coefficient equal to unity so we can apply the equation pLt = juf + RTIn x( to the system. On the other hand, if we wish to quantify the varying behaviour of naphthalene in different solvents the activity coefficients based on the pure-liquid standard state give us a convenient measure. 

We now have activity, molality and activity coefficient, defined in terms of the geometric mean of the ionic values, with, as usual ... 

Figure 6.15 displays the experimental values for liquid activity coefficients defined above. The non-ideality is significant. Note also that the shape of curves seems to indicate some errors in the high concentration domain. 

The behaviour of real fluid mixture may be described through deviations from ideal mixture [129, 138, 152, 154, 156] expressed by the activity coefficient / defined by... 

To measure deviations from a reference-solvent, Henry s law ideal solution, we introduce another activity coefficient defined by... 

Develop a spreadsheet showing values of activity coefficients for mono-, di- and tri-valentions using a) the Debye-Hiickel Limiting Law (L.L.), b) the extended L.L., and c) the Davies equation in solutions whose ionic strength varies from 0.001 to 0.200. Plot the above as well as the error in activity coefficient (defined as the difference of the value from that obtained with the Davies Equation) when a) the L.L. and b) the extended L.L. is used over the specified range. 

To illustrate the Raoultian activity-activity coefficient relationship we use activity coefficients defined by Equations (8.31), shown in Figure 8.2. In real systems these are measured quantities with associated uncertainties, and the shape of the activity curve may not fit any simple function. Figure 8.2 shows... 

The activity coefficients defined for the two representations (4.176) and (4.177) are obtained from comparison with (4.168) as... 

Fig. 4-6. The chemical potential p > of the electron as a function of its concentration in an isotropic crystal. The effective mass is w, and is the activity coefficient defined by the equation p > pl- kTlny,N,...

The activity coefficient defined in terms of weight fraction may be written by starting from the relation between volume and weight fractions of solute ... 

The relationship between the activity coefficients defined in terms of Conventions II and III can be obtained as follows. Consider a solute whose mole fraction is x and whose molality is m. Its actual chemical potential is, of course, quite independent of the choice of scale and therefore /i /i + RThiyuX... 

The general relation between x and m is equation (9-9), and it follows from this equation, together with (9 20), that ym is not equal to yxi except for a very dilute solution. Thus, even though a solution of appreciable concentration may be ideal, in the sense of the mole fraction scale (i.e. yjj = 1), it would not have a value of ym equal to unity. The activity coefficients defined on the molality scide evidently do not give a satisfactory measure of the deviations from ideality, except for a very dilute solution. This is equivalent to the remarks made at the end of the previous section. 

For i = a -1-1,..., c Eq. (1,139) applies, and the components are identified as solvents. The components belonging to the indexing of I = 1,..., a are known as solutes. When the solutes are not in the same state as the solution, the second convention becomes useful. This is often the case for liquid solutions of noncondensable gases. Activity coefficients defined by convention II are unsymmetricai. The relationship between and y will be discussed later in Chapter 5. Note that as 0 the corresponding y which is shown by has a defined value as Xi 0, the corresponding y which is shown by y - 1, as defined above. 

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