Activity coefficient exponent

The activity coefficient exponent, v- 1 -i- Ioiy, for the solute ion, Y y, influences the solute activity coefficient through the solution properties, kay, and the ion valence, Vy. These effects are the familiar Debye-Huckel behavior resulting from screening of the electrostatic potential of a solute ion by the diffuse counter ion layer. Since Z < 1.0, will vary inversely with the activity coefficient due to solute ion screening by neighboring solute ions. To understand this behavior better it is instructive to examine electrostatic potential screening of solute ions. 

Activity coefficients for surface sites and protons have been written in terms of two variables (1) a base coefficient, Z, which is dependent upon the solution ionic strength, solvent type, and temperature and (2) an exponent, e, dependent mainly upon ion properties such as valence and effective diameter. The base activity coefficient, Z, appearing in all activity coefficient equations, (131)-(134), decreases from unity as the solution ionic strength increases. The activity coefficients exponents, E, are all positive. As discussed previously, the linearized Poisson-Boltzmann equation restricts the range of the primitive interfacial model activity coefficients from zero to unity while experimentally determined activity coefficients can be greater flian unity in concentrated solutions. 

The activity coefficient exponent, (1 + s ) /l + for the anionic surface site, SO, influences tire surface site activity coefficient through both interfacial (the... 

Equation (132) can be seen to predict the screening behavior described above and shown in Fig. 7. As s increases (becomes more positive), the activity coefficient exponent, e Q, increases, which decreases the activity coefficient. This result is reasonable because a more positive environment around an anionic surface site ion, SO will increase screening, q. This behavior can be clearly seen in a few numerical examples in Table 1, where a value of Z = 0.8 was chosen arbitrarily so that numerical values eould be ealeulated. In addition, polyvalent electrolytes enhance this effect as seen in Fig. 8. 

The activity coefficient exponent, (1 -i- s)Vl + ka, for the cationic surface site, SOH2 can be seen to decrease as the mean surface site electrostatic charge,... 

Equation (134) can be seen to prediet the screening behavior described above and shown in Fig. 7. As s increases (beeomes more positive), the activity coefficient exponent, f gOH deereases, which increases the activity coefficient,/gQjj. Note... 

In adsorption equations the product of the two proton mass action adsorption constants is used so it will be useful to determine the activity coefficient exponent... 

Note that a number of complicating factors have been left out for clarity For instance, in the EMF equation, activities instead of concentrations should be used. Activities are related to concentrations by a multiplicative activity coefficient that itself is sensitive to the concentrations of all ions in the solution. The reference electrode necessary to close the circuit also generates a (diffusion) potential that is a complex function of activities and ion mobilities. Furthermore, the slope S of the electrode function is an experimentally determined parameter subject to error. The essential point, though, is that the DVM-clipped voltages appear in the exponent and that cheap equipment extracts a heavy price in terms of accuracy and precision (viz. quantization noise such an instrument typically displays the result in a 1 mV, 0.1 mV, 0.01 mV, or 0.001 mV format a two-decimal instrument clips a 345.678. .. mV result to 345.67 mV, that is it does not round up ... 78 to ... 8 ). 

All quantities in Eq. (12.6) are measurable The concentrations can be determined by titration, and the combination of chemical potentials in the exponent is the standard Gibbs energy of transfer of the salt, which is measurable, just like the mean ionic activity coefficients, because they refer to an uncharged species. In contrast, the difference in the inner potential is not measurable, and neither are the individual ionic chemical potentials and activity coefficients that appear on the right-hand side of Eq. (12.3). 

I Solution Exponents of activity coefficients are the same as exponents of concentrations ... 

Experiment Type of Temp, diffusion range of coefficient experim. (°C) Diffusion Parameters Ref. Diffusion Pre-expon. Activation coefficient coefficient energy (23 °C) lg D0 Ed Dexp (cm2/s) x 10-8 - (kJ/mol) ... 

A number of other proton transfer reactions from carbon which have been studied using this approach are shown in Table 8. The results should be treated with reserve as it has not yet been established fully that the derived Bronsted exponents correspond exactly with those determined in the conventional way. One problem concerns the assumption that the activity coefficient ratios cancel, but doubts have also been raised by one of the originators of the method that, unless solvent effects on the transition state are intermediate between those on the reactants and products, anomalous Bronsted exponents will be obtained [172(c)]. The Bronsted exponents determined for menthone and the other ketones in Table 8 are roughly those expected by comparison with the values obtained for ketones using the conventional procedure (Table 2). For nitroethane the two values j3 = 0.72 and 0.65 which are shown in Table 8 result from the use of different H functions determined with amine and carbon acid indicators, respectively. Both values are roughly similar to the values (0.50 [103], 0.65 [104]), obtained by varying the base catalyst in aqueous solution. The result for 2-methyl-3-phenylpropionitrile fits in well with the exponents determined for malononitriles by general base catalysis but differs from the value j3 0.71 shown for l,4-dicyano-2-butene in Table 8. This latter result is also different from the values j3 = 0.94 and 0.98 determined for l,4-dicyano-2-butene in aqueous solution with phenolate ions and amines, respectively. However, the different results for l,4-dicyano-2-butene are to be expected, since hydroxide ion is the base catalyst used in the acidity function procedure and this does not fit the Bronsted plot observed for phenolate ions and amines. The primary kinetic isotope effects [114] also show that there are differences between the hydroxide ion catalysed reaction (feH/feD = 3.5) and the reaction catalysed by phenolate ions (kH /kP = 1.4). The result for chloroform, (3 = 0.98 shown in Table 8, fits in satisfactorily with the most recent results for amine catalysed detritiation [171(a)] from which a value 3 = 1.15 0.07 was obtained. 

The pH of a solution in equilibrium with CaC03 and atmospheric CO2 is 8.3. Equation 7.18 yields a value of pH 8.5 because activity coefficient corrections are ignored and because of rounding-off errors in the exponents of the equilibrium constants. 

N denotes the number of active (growing) nuclei. The time y represents the time the nucleus got activated. The exponent m gives the dimension of nuclei growth. The law of nucleation can be postulated in various ways, such as unimolecular decay law. The left-hand side of the equation origins from Avrami s treatment for the nuclei overly. It gives the relation between the extended rate of conversion and the true rate of conversion. The pre-exponential coefficient includes several constants grouped together. 

At low and moderate concentrations, the theory predicts that the logarithm of the activity coefficient will be negative, and inversely dependent on the prodnct of the temperature and the relative permittivity of the solvent. At the lowest concentrations the dependence is on but at moderate concentrations the exponent... 

It should be noted that the aetivity eoefficient quotients, Z, are present even in nonionic solute adsorption equations beeause such quotients can never be zero in a solution or at a S-MO interfaee, whieh ean have a net electrostatie surfaee site eharge, v,. However, as seen in Eqs (336)-(338), two of the three terms in the exponent of Z are nonzero, but when eombined into the exponent for the aetivity coeffieient quotient for the adsorption eomplexation reaetion, caneellation causes the exponents for all three adsorption reaetions in Eqs (330), (332), and (334) to go to zero. Therefore, the activity coefficient quotients based on Coulombie interactions in adsorption equations (349)-(356) for nonionies are unity and ean be ignored. 

It should also be noted that Eqs (336)-(338) for activity coefficient quotient exponents have been derived for Coulombie interactions only. If electrostatic interactions other than Coulombie were considered, these exponents in Eqs (336)-(338) would not reduce to zero, but would have equations that would give small finite values. These effects are discussed in Chapter 4. 

A central ion, Y y, wifli valence, Py surrounds itself with an ion cloud of net electrostatic charge, —Vy e. As a result of the negative exponent, the value of the activity coefficient calculated from Eq. (60) for a solute ion can never be greater than unity. However, experimentally determined activity coefficients can exceed unity. The source of this discrepancy results largely from die use of the linearized Poisson—Boltzmann equation to obtain the elecfrostatic potentials for solute ions. 

The activity coefficient equations for a solute ion in the interior of a solution, Eq. (60), and for a siuface site ion in the Stem layer of a primitive interface, Eq. (88), are quite similar in mathematical form. In particular, the Bjerram association length, is found in both activity coefficient equations. In addition, the activity coefficient can be conveniently expressed in terms of the base activity coefficient, Z, and its exponent, Eg. However, the significance of these two variables, comprising the activity coefficient, is somewhat different for siuface site ions vis-i-vis solute ions. 

The explicit variables comprising the entire exponent of the activity coefficient, (Vg - s)2A g/i + Aflg, are related to the two different electrostatic potentials... 

From Eq. (195) explicit equations for activity coefficient quotients, based on both the Debye—Hiickel and primitive interfacial models, can be seen to vary simply by the exponent Table 6 lists for activity coefficient quotients. 

The above result states that at high concentrations the rational activity coefficient exhibits quadratic concentration dependence rather than the linear one predicted by Bahe s purely electrostatic pseudolattice treatment. As can be seen in Table I, this dependence can be proved to be valid for several electrolytes (Varela et al., 1997), where fn/+ + Ac — Be is fitted to Dc. As can be seen in the table, Varela et al. obtained that, for several 1 1 and 1 2 electrolytes, the calculated exponent is x=2, within the limits of experimental uncertainty for all the analyzed data. 

When using the selectivity constant or coefficient (k) mentioned by ISE suppliers, one must be sure that if the ion under test and the interfering ion have different valence the exponent in the activity term according to Nikolski has been taken into account it has become common practice to mention the interferent concentration that results in a 10% error in the apparent ion concentration these data facilitate the proper choice of an ISE for a specific analytical problem. Often maximum levels for no interference are indicated. 

This equation contains selectivity coefficient for cation J with respect to anion B consequently, activity a j ( 1) has a negative exponent. The behaviour of the ISE in fig. 3.4 can be explained in this manner. 

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