Hydration numbers from activity coefficients

A nonspeetroseopie method that has been used to obtain hydration numbers at high concentrations will be described here only in qualitative outline. Understanding it quantitatively requires a knowledge of ion-ion interactions, which will be developed in Chapter 3. Here, therefore, arejust a few words of introduction. 

The basic point is that the mass action laws of chemistry ([A][B]/[AB] = constant) do not work for ions in solution. The reason they do not work puzzled ehemists for 40 years before an acceptable theory was found. The answer is based on the effects of electrostatic interaction forces between the ions. The mass aetion laws (in terms of concentrations) work when there are no charges on the partieles and hence no long-range attraction between them. When the particles are charged. Coulomb s law applies and attractive and repulsive forces (dependent on 1/r where r is the distanee between the ions) come in. Now the particles are no longer independent but puU on each other and this impairs the mass action law, the silent assumption of which is that ions are free to act alone. 

There are several ways of taking the interionic attraction into account. One can 

From a very dilute solution (10 mol dm ) to about 10 mol the ratio of the activity to the concentration (a/c,-or the activity coefficient, y,-) keeps on getting smaller (the deviations from the independent state increase with increasing eoneen-tration). Then, somewha-e between lO and 10 M solutions of electrolytes sueh as NaCl, the activity coefficient (the arbiter of the deviations) starts to hesitate as to which direction to change with increasing concentration above 1 mol dm , it turns around and increases with increasing concentration. This can be seen schematically in Fig. 2.18. 

Why does it do this Tha-e may be more than one reason. A reason that was suggested long ago by Bjerrum and developed intensively by Stokes and Robinson in the 1950s is concerned with solvation and has more than historical interest. This is how they argued. 

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Ion Observed diffusion coefficient at infinite dilution" Hydration numbers from diffusion at infinite dilution Hydration numbers from diffusion s concentration dependence Hydration numbers from activity coefficients" Hydration numbers from transference methods"... 

Table 6 Comparison of ion hydration numbers he calculated from Glueckauf s treatment of static dielectric constant with values calculated from activity coefficients (Ao) and entropies (A,)...

Effective hydration number, taken as the number of water molecules whose strength of interaction with the cation is large compared to kT, as estimated from activity coefficients of... 

In this equation, h is the maximum number of waters in the hydration shell (at low ionic strength) and k is the distribution coefficient for water between solvent and hydration shell. Fitting activity coefficient data then results in values of and kh. For many chloride 1 1 and 2 1 electrolytes, the fit is identical to that from Equation (15.29) at low ionic strength, but remains good to much higher ionic strengths. Wolery and Jackson (1990) have proposed other modifications to the hydration approach. 

The correlation between entropy and hydration number is illustrated by the ratio of AShyd between La and Lu. Bertha and Choppin estimated this ratio to be 1.2. From activity coefficient measurements, Glueckauf (46) reported outer sphere hydration numbers of 7.5 and 8.7 for La and Lu, respectively, and also a La/Lu ratio of 1.2. Choppin and Graffeo (47) calculated hydration numbers from conductance data and reported a La/Lu hydration ratio of 1.1, while Padova measured molar volumes to obtain a La/Lu hydration ratio of 1.2. The consistency in this ratio calculated by four different methods supports the correlation between the total hydration numbers, the S hyd values, and the atomic number. 

These ideas are frequently qualitatively useful, but they are rarely quantitatively applicable. The data in Table 6.2-1 illustrate this by comparing hydration numbers found from diffusion, from activity coefficients, and from transference methods. Qualitatively, these values supply insights. For example, the diffusion of lithium is slower than that of sodium, which is slower than that of potassium, etc. This suggests that the radii of the diffusing solutes are in the order Li >Na >K >Cs, exactly the reverse of the ionic radii found in the solid state. Such inverted behavior seems to be the result of hydration. 

Raji Heyrovska [18] has developed a model based on incomplete dissociation, Bjermm s theory of ion-pair formation, and hydration numbers that she has found fits the data for NaCl solutions from infinite dilution to saturation, as well as several other strong electrolytes. She describes the use of activity coefficients and extensions of the Debye-Hiickel theory as best-fitting parameters rather than as explaining the significance of the observed results. ... 

After de Forcrand s Clapeyron, and Handa s methods, a third method for the determination of hydrate number, proposed by Miller and Strong (1946), was determined to be applicable when simple hydrates were formed from a solution with an inhibitor, such as a salt. They proposed that a thermodynamic equilibrium constant K be written for the physical reaction of Equation 4.14 to produce 1 mol of guest M, and n mol of water from 1 mol of hydrate. Writing the equilibrium constant K as multiple of the activity of each product over the activity of the reactant, each raised to its stoichiometric coefficient, one obtains ... 

Use the data of Table 2.8 to calculate the mean activity coefficient of a 5 MNaCI solution, assuming the total hydration number at this high concentration is <3. Values for A and B of the Debye-Hiickel equation can be recovaed from the text. 

Plot the values for the activity coefficient of the electrolyte as calculated in Problem 3 against the ionic strength. Then see what degree of match you can obtain from the Debye-Htickel law (one-parameter equation). Do a similar calculation with the equation in the text which brings in the distance of closest approach (a) and allows for the removal of water from the solution (two-parameter equation). Describe which values ofa,andn (the hydration number) fit best. Discuss the degree to which the values you had to use were physically sensible. 

We shall now discuss the depression of the static permittivity of water by the addition of eiectrolyte solutes, which is a phenomenon of some importance in the understanding of the hydration sheath of the ions. It is essentially a dielectric saturation phenomenon the strong electric fields in the neighbourhood of the ions produce a non-linear polarization, which renders the local water moleodes ineffective as regards orientation in the applied field. It is possible to make estimates of the extent of hydration, or hydration number , of water molecules considered to be bound irrotationally to the average ion these estimates are in reasonable agreement with hydration numbers estimated on the basis of activity coefficients, entropies, mobilities, and viscosities. The hydration number must be distinguished from the number of water molecules actually adjacent to the ion in the first or second layers of hydration (the hydration sheath) it does not follow that all of these molecules can be considered to be attached to the ion as it moves in the solution. 

This has always been an important, though very indirect, source of hydration numbers. Colligative properties and emfs give high precision experimental data from which the activity of the solute can be calculated and stoichiometric mean ionic activity coefficients found. If it is assumed that solvent molecules are bound to the ions, a relation between... 

Hydration theory deals with aqueous solutions in terms of two parallel definitions of the set of components, the usual one in which the solutes are considered formally unhydrated and the amount of solvent is the nominal amount, and a second in which the solutes are formally hydrated and the amount of water is reduced from the nominal amount. The objective is to correct for hydration effects and obtain a model giving activity coefficients in terms of the usual set of components. Quantities pertaining to the second set of components will be denoted by an asterisk. The number of moles of water in the first set is given by... 

The presence of an ion of this composition in solution was first deduced from the specific heats of aqueous acids, though the argument was based upon a rather artificial picture of water as a mixture of polymers. Supporting evidence comes from the thermodynamic properties of concentrated acid solutions. The rapid rise in the activity coefficients of electrolytes in concentrated solutions can be attributed largely to the removal of water by ionic hydration, with a consequent increase in the true mole fraction of the solute. A quantitative treatment in terms of a reasonable model " yields the mean hydration numbers of the ions, and a value close to 4 is found for the hydrogen ion. A similar deduction can be made from the indicator equilibria used in determining the acidity of concentrated... 

The degrees of dissociation and hydration numbers calculated from vapor pressures correlate quantitatively with the properties of dilute as well as concentrated solutions of strong electrolytes. Simple mathematical relations have been provided for the concentration dependences of vapor pressure, e.m.f. of concentration cells, solution density, equivalent conductivity and diffusion coefficient. Non-ideality has thus been shown to be mainly due to solvation and incomplete dissociation. The activity coefficient corrections are, therefore, no longer necessary in physico-chemical thermodynamics and analytical chemistry. 

Hydration studies for higher oxidation state actinides are reported for some AnOj salts (UO2 or NpO ) but not for any AnOj systems. The Raman spectra of aqueous uranyl solutions were interpreted to show the presence of six water molecules coordinated in the primary plane perpendicular to the O-U-O axis (Sutton 1952). However, similar hydration numbers have been obtained by methods in which the secondary hydration shell may also contribute to the value. For example, the activity coefficient measurements suggest a hydration number of 7.4 [relative to an assumed hydration number of zero for Cs(I) (Hinton and Amis 1971)]. Similarly, a hydration number of seven has been derived from Gusev s conductivity method (Gusev 1971, 1972,1973). 

The eventual upturn of the mean ionic activity coefficients as the electrolyte concentration increases, described in Equation 7.11 by the linear term Cm and in Equations 7.6 and 7.7 by higher powers of the concentration, can be interpreted in several ways. A concept that differs from that involving hydration numbers. Equations 7.14 and 7.15, is in terms of the association of ions of opposite charges. In fact, ion association competes with the solvation of the ions and in certain cases an ion of opposite charge may replace some of the solvent in the solvation shell of a given ion. 

The MC simulations also provided details on the structural changes that accompany the reaction. The key observation is that the hydration-induced activation barrier is caused by a reduction in strength rather than in the number of solute-water hydrogen bonds that accompany charge delocalization in proceeding to the transition state. The recent dynamics results for the reaction in water are also nicely complementary a transmission coefficient near 0.5 was obtained and carefully analyzed. The deviation from TST for such a sharp barrier attests to the strong solute-solvent coupling. 

Figure 2 shows the initial activity of the catalytic composites for camphene hydration (A), expressed as the initial reaction rate calculated finm the slope of the camphene kinetic curve and the initial activity for isobomeol oxidation (B), expressed as the initial reaction rate calculated from the slope of the isobomeol kinetic curve. It was observed that the initial activity regarding camphene hydration, decreases when the crosslinking degree increases, for the catalysts Co(acac)2/PVAx (fig. 2 bar Cl > bar C2), in spite of the increase in the number of acid sites. This result, which is also in contradiction with those observed for the oxidation of isobomeol, is likely to be due to the expectable decrease in the sorption coefficient of camphene caused by the increase of the number of sulfonic groups. The same explanation may be given for the decrease in activity observed when the load of Co(acac)2trien NaY in the polymeric matrix... 

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