Experimental Determination of Activity Coefficients

A reasonably informative account has now been given as to how—albeit in very low concentrations—theoretical developments concerning interionic interaction give rise to theoretical values of the quantity (activity coefficient) by which the real 

During this presentation, the experimental mean activity coefficients showing up in the deductions have been taken for granted—nothing has been said about how they have been obtained. Obviously, one must be sure when dealing with theory that the experimental values with which the theory is compared are soundly based. 

It is time then that some account be given about the means by which experimental values of activity coefficients are known. Only two methods will be presented because the material contains no new ideas and is only presented so the reader is assured that the ground is firm. 

Howto Obtain So/ufoActivities from Data on So/venfActivities 

A characteistic of an ionic solution is that any vapor pressure due to the dissolved electrolyte itself is effectively zero. The vapor pressure of the solvent in the solution therefore falls with increasing concentration of the electrolyte in the solution. Thus, the solvent vapor pressure in the solution will be less than the vapor pressure of the pure solvent because the nonvolatile ions block out part of the surface from which, in the pure solvent, solvent molecules would evaporate. 

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In the experimental determination of activity coefficients of strong electrolytes, by the methods described below, the molalities, etc., of the ions are taken as the stoichiometric values, that is, the total possible molality, etc., disregarding incomplete dissociation, For example, in the last problem, the molalities of the sodium and sulfate ions in the 0.5 molal solution of sodium sulfate were taken as exactly 1.0 and 0.5, respectively, without allowing for the possibility that the salt may be only partially dissociated at the specified concentration. The activity coefficients obtained in this manner are called stoichiometric activity coefficients they allow for all variations from the postulated ideal behavior, including that due to incomplete dissociation. If the treatment is based on the actiuil ionic molalities, etc., in the given solution, as in the Debye-Httckel theory (Chapter XVII), there is obtained the true (or actual) activity coefficient. TTie ratio... 

The activity coefficients given by the Debye-Hiickel treatment presumably represent deviations from the dilute solution behavior, i.e., from Henry s law, and are consequently based on the standard state which makes the activity of an ion equal to its mole fraction at infinite dilution ( 37b, III B). In the experimental determination of activity coefficients, however, it is almost invariably the practice to take the activity as equal to the molarity or the molality at infinite dilution. The requisite corrections can be made by means of equation (39.13), but this is unnecessary, for in solutions that are sufficiently dilute for the Debye-Hackel limiting law to be applicable, the difference between the various activity coefficients is negligible. The equations derived above may thus be regarded as being independent of the standard state chosen for the ions, provided only that the activity coefficients are defined as being unity at infinite dilution. 

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. 

In the case of nonaqueous electrolyte solutions, of course, much less data are known. This is because it is not easy to determine absolute values in solvents different from water, since many types of experimental setups require a standard reference, which is not common for nonaqueous systems. Second, the solubility of salts is low in many nonaqueous systems thus rendering any experimental determinations of activity coefficients difficult. Apart from work done by Russian scientists in the 1950s-70s, e.g., [12], mainly the Regensburg group of Barthel and Neueder published relevant data during the last decades, e.g., [13-16], A Pitzer parameter compilation for electrolytes in methanol, ethanol, 2-propanol, acetonitrile, acetmie, dimethoxyethane, and dimethyl carbonate is given in the appendix of [17]. 

Of all the techniques, it is those of Group 1 that are likely to give the most realistic data, simply because they measure transport of charged species only. They are not the easiest experimental techniques to perform on polymeric systems and this probably explains why so few studies have been undertaken. The experimental difficulties associated with the Tubandt-Hittorf method are in maintaining nonadherent thin-film compartments. One way is to use crosslinked films [79], while an alternative has been to use a redesigned Hittorf cell [80]. Although very succesful experimentally, the latter has analytical problems. Likewise, emf measurements can be performed with relative ease [81, 82] it is the necessary determination of activity coefficients that is difficult. 

FIGURE 7.3 Extrapolation of experimental data for the determination of activity coefficients. 

The influence of molar mass, charge density as well as chain branching was also determined in the presence of low molecular mass salt. As seen in Fig. 16, the differences between theory and experiment are more important to low molar masses. In Fig. 16 the concentration dependence of the activity of the low molecular salt has been taken into account when calculating fac=fexp/fo [H4, 126], where fac and fexp are calculated and experimentally determined counterion activity coefficients, respectively f0 is the activity coefficient of the added low molecular salt in aqueous solution without polyelectrolyte. 

EXPERIMENTAL DETERMINATION OF ACTIVITIES AND ACTIVITY COEFFICIENTS OF STRONG ELECTROLYTES... 

There are different experimental methods for determining the gas-liquid partition coefficients leading to the determination of activity coefficients at infinite dilution y . The most frequently used methods are dynamic and static headspace methods. 

Experimental Determination of Activities and Activity Coefficients of Strong Electrolytes... 

The introduction of competitive alkali metal flame reactions has allowed the experimental determination of activation energy differences for alkali metal flame reactions. The method involves the reaction of sodium or potassium with a pair of organic halides, one of which contains chlorine-36. Analysis of the solid halides produced provides a method of obtaining relative yields of the halides and thus relative rate coefficients. The use of a large temperature range (90—120°C) allows accurate measurements of activation energy differences and ratios of Arrhenius A factors. The values in Table 1 were so obtained. 

Hence, the total excess Gibbs energy ean be easily determined from experimental values of activity coefficients, as it will be shown in the next section. Because In /, is a partial property, the following relation may be written ... 

The surfactant aggregation in the non-aqueous phase usually begins at 10" — 10 M and is several orders lower than the CCM in the aqueous phase [126]. Therefore, the study of the aggregation and the determination of activity coefficients at such concentrations turn out to be a comphcated experimental task. The surface pressure isotherms at the boundary between water and surfactant solutions in nonpolar solvents and other methods can be employed for this purpose. (Surfactants dissolved in the organic phase significantly affect the water-air interfacial tension [129]). 

It is important to note that the coefficients fp, gp, and hs are always nonvanishing, for both achiral and chiral isotropic films. On the other hand, fs, gs, and hp can only be nonvanishing if the isotropic film is chiral (nonracemic) because they completely depend on the chiral susceptibility components. Note that gs is always equal to zero within the electric dipole approximation. The sign of the chiral expansion coefficients changes between enantiomers, while that of the achiral expansion coefficients stays the same. Experimental determination of all expansion coefficients fully characterizes the nonlinearity and nonlinear optical activity of the sample. Once all expansion coefficients are... 

Changes in solubility product are one means of experimentally determining a value of activity coefficient, because we can independently determine the concentrations (e.g. via a titration) and the values of all y will be one at zero ionic strength. 

Most determinations of activity and osmotic coefficients of an electrolyte solution are based on these experimental techniques ... 

The final step of the convolution analysis is the determination of the transfer coefficient a. This coefficient, sometimes called the symmetry factor, describes how variations in the reaction free energy affect the activation free energy (equation 26). The value of a does not depend on whether the reaction is a heterogeneous or a homogeneous ET (or even a different type of reaction such as a proton transfer, where a is better known as the Bronsted coefficient). Since the ET rate constant may be described by equation (4), the experimental determination of a is carried out by derivatization of the ln/Chet-AG° and thus of the experimental Inkhei- plots (AG° = F E — E°)) (equation 27). 

Activity coefficients and concentration equilibrium constants. Strictly speaking, Eq. 6-31 applies only to thermodynamic equilibrium constants -that is, to constants that employ activities rather than concentrations. The experimental determination of such constants requires measurements of the apparent equilibrium constant or concentration equilibrium constant21 Kc at a series of different concentrations and extrapolation to infinite dilution (Eq. 6-32). 

Diffusion-Controlled Reactions. The specific rates of many of the reactions of elq exceed 10 Af-1 sec.-1, and it has been shown that many of these rates are diffusion controlled (92, 113). The parameters used in these calculations, which were carried out according to Debye s theory (41), were a diffusion coefficient of 10-4 sec.-1 (78, 113) and an effective radius of 2.5-3.0 A. (77). The energies of activation observed in e aq reactions are also of the order encountered in diffusion-controlled processes (121). A very recent experimental determination of the diffusion coefficient of e aq by electrical conductivity yielded the value 4.7 0.7 X 10 -5 cm.2 sec.-1 (65). This new value would imply a larger effective cross-section for e aq and would increase the number of diffusion-controlled reactions. A quantitative examination of the rate data for diffusion-controlled processes (47) compared with that of eaq reactions reveals however that most of the latter reactions with specific rates of < 1010 Af-1 sec.-1 are not diffusion controlled. 

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