Co-sphere

The picture of a very dilute solution that we must adopt is shown schematically in Fig 2 Each ion is enclosed in its own co-sphere, while the remainder of the solvent between the ions docs not differ in any way from ordinary pure solvent. As a result of recent progress in atomic physics, we now know in great detail the structure and properties of different species of atomic ions in a vacuum and at the same time we know the physical properties of the pure solvent. In order to understand the properties of a very dilute solution, we need to discuss the portions of solvent that lie in the co-spheres of the ions. 

The term Brownian motion was originally introduced to refer to the random thermal motion of visible particles. There is no reason why we should not extend its use to the random motion of the molecules and ions themselves. Even if the ion itself were stationary, the solvent molecules in the outer regions of the co-sphere would be continually changing furthermore, the ion itself executes a Brownian motion. We must use the term co-sphere to refer to the molecules which at any time are momentarily in that region of solvent which is appreciably modified by the ion. In this book we are primarily interested in solutions that are so dilute that the co-spheres of the ions do not overlap, and we are little concerned with the size of the co-spheres. In studying any property... 

After discussing in Chapter 1 a charged sphere in a dielectric, we saw in Sec. 14 that, when any pair of ions is added to a solvent, there will be a change of entropy in the co-sphere of each ion. If, for example, we knew the value of this change for the ion pair (Ag+ + Cl ) and likewise for the ion pair (Ag+ + I ), any difference between the two quantities could at once be ascribed to the difference between the co-spheres of the chloride ion Cl and the iodode ion I-, since the contribution from the Ag+ ion and its co-sphere would be the same in the two cases. 

We can now answer the question under discussion. Since 29.1 e.u. is considerably smaller than the total difference (116.8 — 75.8) = 41.0 e.u., mentioned above, we conclude that the unitary term for the ion pair (Ag+ + I") is greater than for (Ag+ + Cl"). The contribution from the Ag+ ion and its co-sphere must be, within the experimental error, the same in the two cases. Finally then, we reach the conclusion that in water at 25°C the entropy associated with the iodide ion I" and its co-sphere is greater than that associated with the ion Cl" and its co-sphere in fact, the excess lies in the neighborhood of (41 — 29) = 12 e.u. 

Referring to Fig. 9, it will be recalled that the curve of Fig. 96 differs from the curve of Fig. 9a because, when the ion is introduced into the solvent, the molecular dipoles in the co-sphere of the ion lose a certain amount of free energy. The co-sphere of each ion thereby makes a contribution (positive or negative) to the e.m.f. of the cell. For-each ion added or removed, the cratic term likewise makes a contribution to the e.m.f. fi of the cell. 

When an ion is in a solvent, the energy associated with its ionic field, being sensitive to the environment, is sensitive to the temperature of the solvent, as we saw in (19). On the other hand, quantum-mechanical forces will be relatively insensitive to the temperature of the solvent. The electrical analogue of magnetic heating and cooling arises entirely, or almost entirely, from the interaction between the ion and the solvent in its co-sphere. 

In a very dilute solution, between the co-spheres of the ions the interstitial solvent is unmodified and has the same properties as in the pure. solvent,. The co-sphere of each positive ion and the co-sphere of each negative ion, however, may contribute toward a change in the viscosity. We should expect to find, in a very dilute solution, for each species of ion present, a total contribution proportional to the number of ions of that species present in unit volume. At the same time, we may anticipate that the electrostatic forces between the positively and the negatively charged ions must be taken into account. 

If the B-coefficients represent the contributions from the co-spheres of the ions, we should expect that in very dilute solution the contributions from the positive and the negative ions would be independent, and... 

It will be recalled that in Fig. 28 we found that for the most mobile ions the mobility has the smallest temperature coefficient. If any species of ion in aqueous solution at room temperature causes a local loosening of the water structure, the solvent in the co-sphere of each ion will have a viscosity smaller than that of the normal solvent. A solute in which both anions and cations are of this type will have in (160) a negative viscosity //-coefficient. At the same time the local loosening of the water structure will permit a more lively Brownian motion than the ion would otherwise have at this temperature. Normally a certain rise of temperature would be needed to produce an equal loosening of the water structure. If, in the co-sphere of any species of ion, there exists already at a low temperature a certain loosening of the water structure, the mobility of this ion is likely to have an abnormally small temperature coefficient, as pointed out in Sec. 34. 

The Acetate Ion. For the B-coefficients of lithium acetate and potassium acetate, which are of course completely dissociated in aqueous solution, Cox and Wolfenden obtained at 25°C the values +0.397 and +0.238. These large values could be due entirely to the large size of the molecular ion or could be due partly to the fact that the anion produces order in its co-sphere. To test this, Laurence and Wolfenden measured the B-coefficient of acetic acid in aqueous solution at 25°C. 

A procedure like this has been adopted in the literature, except that it is the value for the hydrogen ion that has been set equal to zero. This involves a slightly more difficult concept for, when a proton is added to water, it converts an H20 molecule into an (HsO)+ ion. The entropy of the original water molecule is replaced by the entropy of the (HsO)+ ion and its co-sphere. When the partial molal entropy of HC1 in aqueous solution has been determined, the whole is assigned to the Cl- ion that is to say, the value for the hydrogen ion is set equal to zero, and the values for all other species of ions are expressed relative to this zero. 

Solutes in Aqueous Solution. As mentioned in See. 88, when we say that we expect to find a correlation between the /1-coefficients of viscosity of various species of ions, and their entropy of solution, this refers only to the unitary part of the entropy, the part associated with the ionic co-sphere. We are inclined to adopt the view that a negative //-coefficient for a pair of ions should be accompanied by a positive increment in entropy, while a positive //-coefficient should be accompanied by a decrease in entropy. The values of AS0, the conventional entropy of solution, to be found in the literature, do not, give a direct answer to this question, since they contain the cratic term, which in water at room temperature amounts to 16 e.u. This must be subtracted. 

Properties of Different Solvents. In discussing molecular dipoles in Sec. 25, we estimated the force of attraction between an atomic ion and a dipole having the most favorable orientation and found this attraction to be very strong. In any ionic co-sphere those molecular dipoles which have a favorable orientation will bo attracted, while those that have the opposite orientation will be repelled. Since the former are more numerous the solvent in the co-sphere is, on the whole, attracted toward the ion. Since the liquid is not incompressible, we must expect that this will lead to a contraction in each co-sphere. In any ionic solution the sum of the contractions that have taken place in the co-spheres of the positive and negative ions will be apparent if we measure accurately the volume of the solution. 

At the same time, since there has been, in each co-sphere, an increment in the density of the solvent, we must expect some modification in other properties of the solvent, such as its compressibility. In a very dilute solution it may be difficult to detect such a change by measuring the compressibility of the solution. At higher concentrations, however, when a sufficient fraction of the total solvent lies within the ionic co-spheres, the sum of these local modifications can be detected by measuring the compressibility of the whole solution. 

The conclusions are evidently relevant to the amount of entropy lost by ions in methanol solution—see Table 29. If, however, the expression (170) is used for an atomic ion, we know that it is applicable only for values of R that are large compared with the ionic radius—that is to say, it will give quantitative results only when applied to the solvent dipoles in the outer parts of the co-sphere. The extent to which it applies also to the dipoles in the inner parts of the co-sphere must depend on the degree to which the behavior of these molecules simulates that of the more distant molecules. This can be determined only by experiment. In Table 29 we have seen that for the ion pair (K+ + Br ) and for the ion pair (K+ + Cl-) in methanol the unitary part of ASa amounts to a loss of 26.8 e.u. and 30.5 e.u., respectively, in contrast to the values for the same ions in aqueous solution, where the loss of entropy in the outer parts of the co-sphere is more than counterbalanced by a gain in entropy that has been attributed to the disorder produced by the ionic field. 

In view of this result, it becomes almost imperative to inquire whether the same method might be given wider application—whether, in discussing the small differences between homologous organic acids, we should pay attention to the possible states of order-disorder in the co-spheres... 

In both processes the ions are supposed to be introduced into solutions where the interionic forces are negligible. When in (191) the COJ ion is formed, the solvent in the co-sphere of this ion loses a certain amount of entropy. Likewise, in (192) when the CO, ion is formed, the solvent in the co-sphere of the ion loses precisely the same amount of entropy. At the same time, the amount of entropy associated with the thermal energy of the COJ" ion in aqueous solution is, of course, the same in (192) as in (191). In the process (192) we shall be concerned with the unitary term for the two Li+ ions in contrast to (191) where two protons are added to two II20 molecules. 

Turning next to the unitary part of AS0, this is given in Table 36 under the heading — N(dL/dT). It was pointed out in Secs. 90 and 106 that, to obtain the unitary part of AS0 in aqueous solution, one must subtract 16.0 e.u. for a uni-univalent solute, and 24.0 e.u. for a uni-divalent solute. In methanol solution the corresponding quantities are 14.0 and 21.0 e.u. In Table 36 it will be seen that, except for the first two solutes KBr and KC1, the values are all negative, in both solvents. It will be recalled that for KBr and KC1 the B-coefficients in viscosity are negative, and we associate the positive values for the unitary part of the entropy, shown in Table 29, with the creation of disorder in the ionic co-spheres. In every solvent the dielectric constant decreases with rise of temperature and this leads us to expect that L will increase. For KBr and KC1 in methanol solution, we see from Table 36 that dL/dT has indeed a large positive value. On the other hand, when these crystals dissolve in water, these electrostatic considerations appear to be completely overbalanced by other factors. 

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