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STRUCTURE OF WATER AND IONIC SOLUTIONS
J. Bernal and R. Fowler (Cambridge)
Brief Summary
On the basis of a model of the water molecule, derived from spectral and X-ray data, and the internal structure of water proposed by us, the following properties of water and ionic solutions were derived, in quantitative agreement with experimental data:
- the crystalline structure of ice;
- the X-ray diffraction curve for water;
- the total energy of water and ice;
- the degree of hydration of positive and negative ions in water;
- the heats of solution of ions;
- the mobility of hydrogen and hydroxyl ions in water.
In addition, qualitative explanations were given for the following properties:
- the density of water and its changes;
- an explanation of the exceptional position of water among other molecular liquids;
- the dielectric properties of water and ice;
- the viscosity of dilute ionic solutions;
- the viscosity of concentrated acids.
Part I
1
Consideration of the spectroscopic model of the molecule \( \mathrm{H_2O}, \mathrm{O}<\begin{matrix}\mathrm{H}\\ \mathrm{H}'\end{matrix} \), the crystalline structure of ice, and the density of water leads to the assumption of an irregular structure of water with fourfold coordination. This structure was tested and proved capable of explaining the positions of the maxima of the X-ray diffraction of water, as well as the variation of this arrangement with temperature, markedly different from that for a simple liquid, such as, for example, mercury. Three types of arrangement of water molecules at different temperatures have been proposed: 1) of the ice—tridymite type (with fourfold coordination), at low temperatures below \(4^\circ\mathrm{C}\); 2) of the quartz type (with fourfold coordination), approximately between \(4\) and \(200^\circ\mathrm{C}\); 3) of the ammonia type, closely packed, between \(200\) and \(340^\circ\mathrm{C}\). These forms are not sharply separated and pass continuously into one another. This hypothesis explains the anomalous changes in the density of water. Its correctness is tested by observing the behavior of water under pressure, as well as the Raman spectra of water at different temperatures.
2
The high dielectric constant of water and ice and its changes with changes in temperature and frequency are considered, and the conclusion is drawn
* The Journal of Chemical Physics 1, 8, August 1933. Translated by N. Ya. Rabinovich.
that an explanation of the latter on the basis of existing theories, founded on the rotation of molecules, is possible only under the assumption that the corresponding number of water molecules does not possess free orientation under the influence of the external electric field. At the same time a new hypothesis is proposed on the structure of ice, based on electrostatic data and giving an explanation of the positions of the hydrogen atoms.
3
The physical properties of water are compared with the properties of analogous compounds, in particular with $\mathrm{NH_3}$, $\mathrm{HF}$, $\mathrm{H_2S}$, and $\mathrm{CH_3OH}$. It turns out that the exceptional position of water is explained not only by its dipolar character but, to a considerably greater degree, by the geometrical structure of its molecule, which is the simplest of the structures capable of forming extended lattices with tetrahedral coordination.
Proceeding from this model, it proved possible to calculate the total energy of ice and water, determined chiefly by the electrostatic potential of neighboring molecules. The agreement of the results of these calculations with experimental data is also a confirmation of the correctness of this hypothesis.
4
Turning to ionic solutions, we consider the nature of ionic hydration. A method has been developed for determining the degree of hydration from the specific gravity of solutions; the results obtained were compared with results calculated theoretically on the basis of the molecular model. We came to the conclusion that all strongly polarizing ions $\mathrm{H^+}$, $\mathrm{Li^+}$, $\mathrm{Na^+}$, as well as all divalent and trivalent positive ions, equally with $(\mathrm{OH})^-$ and $\mathrm{F^-}$, are hydrated, whereas $(\mathrm{NH_4})^+$, $\mathrm{Rb^+}$, $\mathrm{Cs^+}$, and the majority of negative ions are not hydrated. The degree of hydration depends chiefly on the ionic radius and is the same in solutions and in crystals, namely: $\mathrm{Be4H_2O}$, $\mathrm{Mg6H_2O}$.
Further, it proved possible to calculate the total heat of solution of any atomic ion. In the first approximation the latter may be expressed in the form $a + bP$, where $P$ is the mutual potential of the ion and the water molecule, depending on the radius and charge of the ion, whereas $a$ and $b$ depend exclusively on the charge of the ion. This formula agrees excellently with the experimental data.
Then the action of ions on the water in which they are dissolved is considered, and from the study of various properties, in particular viscosity, the conclusion is drawn that this action consists in increasing or decreasing the coupling between the molecules and the regularity of their arrangement. The concept of structural temperature is introduced, which is increased by large ions and decreased by small ions. From the point of view of these properties the ions $\mathrm{H^+}$ and $\mathrm{OH^-}$ appear completely anomalous. The anomalous hydration and mobilities of these ions require explanation from the standpoint of a new theory.
Part II
5
The ion $\mathrm{H^+}$ in solution must exist in the form $(\mathrm{OH_3})^+$, which considerably increases the difficulty of explaining the anomalous mobility of $\mathrm{H^+}$. We break with the existing theory. The positive hydrogen ion $(\mathrm{OH_3})^+$ moves in water under the influence of a potential gradient equal to $1\ \mathrm{V/cm}$, with a velocity of $32.5 \cdot 10^{-4}\ \mathrm{cm/sec}$ at room temperature. The corresponding velocity for $(\mathrm{OH})^-$ is $17.8 \cdot 10^{-4}$, and for all other ions is considerably lower (for example, $\mathrm{K^+}$, $(\mathrm{NH_4})^+$, $\mathrm{Cl^-}$, $6.7 \cdot 10^{-4}$). This discrepancy has been investigated, and we assert that the excess velocities of these ions, exceeding $6.7 \cdot 10^{-4}\ \mathrm{cm/sec}$, must be explained by a mechanism fundamentally different from the transfer of matter through the solution.
The idea is put forward that such a mechanism may be the transfer by means of hopping of a single proton from one water molecule to another under favorable configurations. An analogous idea was expressed by Hückel, but developed by him in a quite different way. This transfer is investigated quantum-mechanically on simple models, and it is shown that the result of applying an EMF should be a deflection in the direction of transfer along the direction of the electric field; moreover, the magnitude of this deflection, at average distances and heights of the potential barriers, is precisely such as to explain very well the exceptional (anomalous) velocities of these ions in water. The hydrogen isotope \(H^2\) should practically not possess an increased (anomalous) velocity, and therefore its velocity should be approximately equal to \(1/7\) of the velocity of \(H^1\). This explanation, naturally, does not affect the ordinary theory of the mobility of foreign ions in water. The theory requires a significant degree of organization in the structure of water, in full agreement with the arguments of the first part.
Introduction
§ 1. Anomalous mobilities of \(H^+\) and \((OH)^-\) in water
Modern theories of ionic mobility are based on the conception of water as a homogeneous liquid possessing a definite viscosity and dielectric constant, varying with temperature, and on the conception of ions as spherical charged particles subject to resistance forces proportional to their velocity. This theory gives a satisfactory explanation of the velocity of large ions \(K^+\), \(Cl^-\), etc., and also of the smaller mobilities of small ions, proceeding from the hypothesis of a more or less indeterminate degree of hydration. More exact agreement can be obtained for concentrated solutions if the electrostatic interaction of ions is taken into account, as in the Debye–Hückel–Onsager theory.
However, all these theories cannot explain the mobilities of the ions \(H^+\) and \((OH)^-\) in water. The equivalent radii for \(H^+\) and \((OH)^-\), calculated from Stokes’ law from their mobilities*, are \(2.6 \cdot 10^{-9}\) and \(4.8 \cdot 10^{-9}\) cm, i.e. far too small and entirely incompatible with physical conceptions (moreover, hydration and the interaction of ions can only reduce mobility). These anomalous mobilities of the ions \(H^+\) and \((OH)^-\) are limited only to water and to solvents nearest to it in their properties, such as, for example, methyl alcohol. In order to explain them, it is obviously necessary to subject the molecular structure of water to detailed investigation, so as to find out precisely where the hypothesis of a simple viscous dielectric liquid proves insufficient for explaining the experimental facts. Thus our principal theme concerning the anomalous mobility of the ion \(H^+\) passes over into the more general theme of the nature of water, to which the first part of our work is devoted. We return to the principal theme in the second part.
* By direct calculation from \(6\pi a\eta v = F\) (\(a\) — radius, \(\eta\) — viscosity), where \(v = 32.5;\ 17.8 \cdot 10^{-4}\) cm/sec for the field \(F\), equal to \(1\) V/cm.
Part 1. Structure of Water and Ionic Aqueous Solutions
§ 2. The Nature of the Water Molecule
Pure water, if one does not count a slight natural ionization, consists of H₂O molecules. There is no basis for supposing that these molecules, except for insignificant mutual deformations, differ from H₂O molecules in the vapor. This is confirmed by the Raman spectra and the infrared absorption spectra of water. The H₂O molecule in the vapor, according to the latest data of Mecke¹, based on the study of band spectra, consists of three nuclei arranged in the form of the letter V, the O—H distances being equal to 0.96 Å, and the HÔH angle being equal to 103–106°, i.e., very close to the tetrahedral angle—109°*.
Fig. 1. Distribution of electrons in the water molecule.
It is more difficult to imagine the distribution of the electrons. If it were simply spherical with its center at the O nucleus, then the dipole moment of the molecule would be equal to \(5.6 \cdot 10^{-18}\) electrostatic units, and not \(1.87 \cdot 10^{-18}\), as experiment shows. Consequently, we have a considerable screening effect, owing to the concentration of negative charge around the protons (see below, p. 607).
The wave-mechanical theory of the water molecule gives us some qualitative data on this question. According to Mulliken², the 10 electron wave functions of the water molecule may be defined as
\[ (1s)^2[2s_1]^2[2p_z]^2[2p_y]^2[2p_x]^2, \]
where \(x\), \(y\), and \(z\) are oriented as shown in Fig. 1. Of these, \([2s_1]^2[2p_z]^2[2p_y]^2\) bind the protons and jointly create a certain density of electrons surrounding the positions of the protons, whereas \([2p_x]^2\) is not a bonding wave function and gives a concentration of negative electricity in two regions situated at right angles to the plane
\[ \begin{array}{c} O\\[-2pt] / \ \backslash\\[-2pt] H \quad H \end{array} \]
. Therefore, the purely distributed—
* These results cannot be regarded as entirely exact, since up to the present time no analysis has been made of the fine structure of the rotational spectrum of the asymmetric top. However, most spectroscopists agree that the molecule forms an isosceles triangle, that the angle at the vertex lies between 90 and 120°, and that the O—H distance lies between 0.9 and 1.1 Å.
the distribution of electron density will resemble a tetrahedron with two corners of positive and two of negative charge. The “radius” of the water molecule is determined more easily. The smallest distance between water molecules in ice is 2.76 Å; very close values are also given by the most reliable measurements of crystallization water³. This gives us the radius of the water molecule, equal to 1.38 Å, i.e. somewhat larger than the radius of the ions isoelectronic with it, O\(^{-}\) (1.35), F\(^{-}\) (1.33), and (OH\(^{-}\)) (1.33), but considerably smaller than the radii of A (1.9), CH\(_4\) (2.08), or NH\(_3\) (1.80), likewise measured in the solid state.
§ 3. Arrangement of Molecules in Water
If we attempt to explain the structure of water on the basis of such a molecule and of our knowledge concerning the structure of a simple monatomic liquid, such as, for example, mercury, then from the very beginning we encounter difficulties in explaining its density. A simple, randomly close-packed assemblage of water molecules with a radius of 1.4 Å should have a density of 1.84, or conversely—for a density of 1.00 the equivalent radius would be 1.72 Å. Consequently, we must assume either that water is a simple close-packed liquid in which the effective radius changes from 1.4 Å in the solid state to 1.72 Å in the liquid, or that the radius continues to remain approximately equal to 1.4 Å, but that the mutual arrangement of the molecules differs considerably from that in a simple liquid. The already well-known anomalous properties of water and the extremely asymmetric and electropolar properties of the H\(_2\)O molecule argue in favor of the second alternative; but, fortunately, there exist independent data concerning the internal structure of water, obtained by means of X-ray diffraction. The most complete and nearly concordant results of investigations of X-ray diffraction, with corrections for absorption, the Compton effect, incoherent scattering, etc., are found in the works of Meyer⁴, Stewart⁵, and Amaldi⁶. The diffraction curve for water obtained by them is given in Fig. 2, experimental curve 2. Comparing this curve with the curve for mercury, which corresponds to curve 1 in Fig. 2, i.e. the theoretical curve for a close-packed arrangement according to the law of chance, we see that the association of molecules in water differs from that in a simple liquid, since instead of an alternation of maxima in the theoretical sequence with equivalent distances of 2.6, 1.65, and 1.1 Å, we have a very considerable maximum at 3.27 Å, followed by much weaker maxima at 2.3 and 1.4 Å. It is difficult to determine wherein the deviation from the distribution in a simple liquid consists. The best method is to construct models of the distribution functions of the molecules [i.e. the function \(g(r)\), giving the prob-
ity of finding the center of any molecule between distances \(r\) and \(r+dr\) from the center of the given molecule] according to various hypotheses,
Fig. 2. X-ray scattering curves for water. Curve 1—the theoretical curve for a disordered close packing; curve 2—the experimental curve; curve 3—the theoretical curve for a quartz-type distribution; curve 4—the theoretical curve for a tridymite-ice type distribution.
construction from them of the theoretical X-ray diffraction curve according to the formula:
\[ I_\theta=\int_{0}^{\infty}4\pi r^2\{g(r)-\rho\}\frac{\sin sr}{sr}\,dr, \tag{1} \]
where
\[ s=\frac{4\pi\sin\frac{\theta}{2}}{\lambda}, \tag{2} \]
and comparison of the curve with the experimental data. The various functions \(g(r)\) are given in Fig. 3, and the theoretically derived X-ray scattering curves in Figs. 2 and 5.
Curve \(a\) in Fig. 3 shows \(g(r)\) for the arrangement of closely packed molecules of a simple liquid; curve \(b\)—for the arrange-
tion of the ice type, and \(c\)—for an arrangement somewhat similar to quartz. We see that only \(b\) and \(c\) give diffraction curves in any way similar to the observed curve, with the greater similarity being found in the case of curve \(c\).
Distribution \(a\) was constructed for molecules having a radius of \(1.4\ \text{Å}\). Distributions \(b\) and \(c\) were constructed on the basis of the \(\mathrm{H_2O}\) molecule with a radius of \(1.4\ \text{Å}\), each molecule being surrounded by four others forming a more or less regular tetrahedron (Fig. 4). This distribution occurs in ice and follows necessarily from the quasi-tetrahedral angle of the \(\mathrm{H_2O}\) molecule. A proton near the surface of one molecule is situated opposite an empty place in the neighboring molecule, i.e., the place where a proton would be found in a molecule like \(\mathrm{CH_4}\). These are the positions of concentration of negative elasticity caused by the wave function \([2p_x]^2\), in Fig. 1.
However, such fourfold coordination of the molecules is not sufficient to determine the entire arrangement. In the case of silica, for which a similar fourfold coordination is valid, we have three principal crystalline forms—cristobalite, tridymite, and quartz, not counting amorphous quartz glass. The structure of tridymite corresponds to the structure of ordinary ice, and it would seem natural to imagine water simply as an irregular variant of such a structure (distribution \(b\)). In this case, however, it is difficult to explain the considerable decrease in volume on passing from ice to water. One might expect that, on passing from a regular crystalline arrangement to an irregular one, the volume should increase. This leads us to the idea that the arrangement of molecules in water corresponds not to the ice type, but to the quartz type. In such a lattice type the distance between the closest neighboring molecules remains equal to \(2.8\ \text{Å}\), whereas the distance to the molecule next after the nearest one, which in an arrangement of the ice—tridymite type is \(4.5\ \text{Å}\), falls to \(4.2\ \text{Å}\), which gives a decrease in volume of \(17\%\). Taking into account that the density of ice, and consequently also of structures of the ice—tridymite type, is \(0.91\), we obtain for a distribution of the quartz type a density equal to \(1.08\). There are grounds for supposing that water may have a lower density owing to certain irregularities. A more detailed examination of the theoretical curve will allow us to go still further. The chief difference between curve 3 in Fig. 2 for the case of a quartz-type distribution and the observed curve is that the principal maxima of the former occur at distances \(3.5\) and \(2.2\), whereas for the latter they occur at distances \(3.3\) and \(2.3\), and that the intermediate minimum at \(2.5\) is relatively much deeper: it amounts to not \(72\%\) but \(42\%\) of the principal maximum. Obviously, this may be ...
called the tendency toward a close-packed structure of water (curve 1), whose maximum is located precisely in this region. From the distribution curves for \(g(r)\) it is seen that above \(r = 2.7\ \text{Å}\) the density of the quartz and close-packed structures diverges for the same radii, and that therefore the effect of superposing distributions \(a\) and \(b\) reduces to smoothing out the difference from a homogeneous liquid, i.e., physically speaking, to decreasing the regularity of the packing, especially at distances exceeding \(4\ \text{Å}\) from any molecule. The scattering curve constructed on the basis of such a modified distribution function (\(d\) in Fig. 3) is shown as curve 5 in Fig. 5. It almost completely coincides with the experimental curve in the sense of the positions of the maxima and minima, but continues to differ from it in intensity, since the second maximum is undoubtedly lower than the first. The physical basis of the difference between the ideal quartz-type structure and the actual structure of water is the thermal motion of the molecules. At higher temperatures this leads to destruction of the quartz structure, brings it ever closer to a close-packed structure, and appears in X-ray diagrams as a convergence of the first and second maxima, with both shifting toward the mean position (\(2.7\ \text{Å}\)) and filling in of the minimum lying between them. Thus, between 2 and 98° the principal maximum shifts from 3.27 to 3.1, while the second shifts from 2.2 to 2.4 at 40° and then disappears altogether.
Fig. 3. Distribution function \(g(r)\) for molecules situated around a water molecule (nearest distance \(2.76\ \text{Å}\)), for various types of arrangement.
Labels in Fig. 3:
- \(a)\) close-packed
- \(b)\) Bernal type of water
- \(c)\) quartz type
- \(d)\) modified quartz type
Fig. 4. Tetrahedral coordination of water molecules. Four molecules surrounding a water molecule are shown. Of these, two lie in the plane of the paper, one above and one below it.
The displacement of the principal maximum toward the middle causes the appearance of a visible contraction when water is heated, which, as we now see, is apparent. At the lowest temperatures, \(2^\circ\mathrm{C}\), the deepening of the first minimum suggests the appearance of a lattice of the ice—tridymite type. It may be supposed that such a structure is usual for supercooled water and that its disappearance is the cause of the anomalous contraction of water when heated above \(0^\circ\mathrm{C}\). However, these changes in volume are very small in comparison with the change on passing from ice to water. Therefore we can only say that cold water has, in general, a structure of the quartz type, with a noticeable tendency toward the ice—tridymite type. Thus it may be considered that there exist three principal forms of arrangement of \(\mathrm{H_2O}\) molecules in water: water I, of the ice—tridymite type, occurring comparatively rarely and present in some amount at low temperatures below \(4^\circ\mathrm{C}\); water II, of the quartz type, predominating at ordinary temperatures; water III, a densely packed, ideal liquid, of the ammonia type, predominating at high temperatures in some interval below the critical point \((374^\circ\mathrm{C})\). With a change in temperature these forms pass continuously one into another.
Fig. 5. Scattering curves of X-rays for water. Curve 2—experimental curve; curve 5—theoretical curve for water of the quartz type, modified toward close packing.
| max. | min. | max. | min. | max. | min. | ||
|---|---|---|---|---|---|---|---|
| Observed | 3.27 | 2.37 | 2.27 | 1.60 | 1.45 | 1.1 | |
| Calculated | 3.33 | 2.4 | 2.2 | (bend) | 1.55 | 1.40 | 1.1 |
However, as Stewart points out, here there is no mixing of volumes of different structure; at all temperatures the liquid remains homogeneous, and only the average mutual arrangements of the molecules resemble, to a greater or lesser degree, water I, II, or III. The transition from water I to II and III is accompanied by an increase in the rotational and translational motion of the molecules and by a corresponding decrease in the dipole cohesive forces of the liquid and a relative increase in the van der Waals component. This is a consequence of the increase in fluidity, but not of an increase in volume. The immediate result of the destruction of the relatively loose structure of the ice type—water I—is a decrease in volume on transition to water II, accompanied by a more normal increase on transition to water III, where the increase in the mean distance between neighboring molecules,
caused by thermal motion, more than compensates for the geometrical compression in the transition from a quartz-type structure to close-packed structures. Chemists are familiar with the theory of polymorphic forms of water chiefly thanks to the work of Armstrong. If one adopts the point of view set forth above, it becomes obvious that these theories are closely connected with the facts, but are too much sustained in the spirit of molecular chemistry.
Trihydrol \((\mathrm{H_2O})_3\), dihydrol \((\mathrm{H_2O})_2\), and hydrol \((\mathrm{H_2O})\) have no immediate direct structural analogy with water I, water II, and water III; yet, given the geometrical internal structure of the liquid, the latter representations provide an explanation of physical properties which were first attempted to be explained by admitting hypothetical molecules.
§ 4. Effects of High Pressures
These considerations may be supported by examining the behavior of water under pressure.
TABLE 1
Equivalent volumes of water and various types of ice
(Bridgman)
| Pressure in atmospheres | Form of ice | Volume of ice (observed) | Equivalent volume of water (observed) | Geometrical equivalent volume of ice | Relative volume of a water molecule | Geometrical equivalent volume of water |
|---|---|---|---|---|---|---|
| 1 | I | 1.09000 | 1.0000 | 1.090 | 1.000 | 1.000 |
| 2 000 | I | 1.0571 | 0.9253 | 1.090 | 0.970 | 0.955 |
| 2 000 | III | 0.8774 | 0.9253 | 0.905 | 0.970 | 0.955 |
| 3 500 | II | 0.8636 | 0.8815 | 0.905 | 0.956 | 0.922 |
| 3 500 | V | 0.8085 | 0.8815 | 0.847 | 0.956 | 0.922 |
| 6 000 | V | 0.7929 | 0.8472 | 0.847 | 0.937 | 0.904 |
| 6 000 | VI | 0.7636 | 0.8472 | 0.816 | 0.937 | 0.904 |
| 10 000 | VI | 0.7385 | 0.8055 | 0.816 | 0.906 | 0.890 |
| Observed volumes with correction for the magnitude of the water molecule: column 3 / column 6 | From the change in volume of the preceding ice | column 4 / column 6 |
The equilibrium diagram of water was given by Bridgman 8. It was found that at various temperatures four kinds of ice are in equilibrium with water: I, III, V, VI. Their relative volumes
(normal cold water 1.000) and the volumes of water in equilibrium with them are given in Table 1. Modifications of ice III, V, and VI have densities greater than the density of water, which indicates that the latter does not possess a close-packed structure.
In considering these equivalent volumes it is necessary to take into account the effect of the compressibility of the water molecules themselves. This can be achieved approximately if it is assumed that the change in volume of each ice modification at different temperatures within its region of stability is caused by compression of the atoms themselves (this is absolutely true only for a crystalline structure without variable parameters). Thus, by the volume ratio of each ice modification in the region of maximum pressure in Table 1 we understand the volume ratio of water molecules in the same pressure region. It then becomes possible to determine what part of the decrease in the volume of water at high pressures is caused by compression of the molecules (column 6, relative volume of the water molecule) and what part is caused by destruction of the organized structure of the liquid (column 7, geometrically equivalent volume of the liquid). We shall see further that at the highest pressures (10,000 atm) this geometrical crushing reaches 11%, although the volume still amounts to 44% of the volume of a close-packed structure of water molecules with radius equal to 1.4 Å. The geometrical factor of compression is much more significant for ice modifications III, V, and VI than for water. The significance of this would be fully clarified only if the crystalline structures of these forms could be determined by means of X-rays. The isomorphism observed between tridymite and ice I leads one to suppose that at least one of the modifications of ice III, V, and VI must have the structure of quartz. In any case, the considerable decrease in volume (5, 9, and 11%) on going from water to ice III, V, and VI confirms the X-ray-analysis data that the structure of water is not a close-packed structure.
§ 5. Data obtained from Raman spectra
The nature of this difference appears still more clearly in the study of the interaction of water with electromagnetic waves of greater wavelength. The Raman spectrum of water has been studied many times. It consists of three bands at \(\Delta \nu = 3216,\ 3435\) and \(3582\), which evidently correspond to the fundamental frequency \(\nu = 3700\) of the \(\mathrm{H_2O}\) molecule, found from the infrared absorption of water vapor. The bands of water differ from the bands of other liquids in their breadth, as a result of which they almost completely overlap. This indicates the effect of perturbation of the energy levels by local dipoles and in itself is evidence of an internal structure different from the structure of normal liquids. The bands of water in the Raman spectrum of ice correspond to the lines \(\Delta \nu = 3193,\ 3391\) and \(3549\). This shows
shows that, in a first approximation, the influence of neighboring molecules on any molecule is the same in both phases and that, in all probability, water, like ice, has a tetrahedral structure, with fourfold coordination. The relative intensities of the lines show that this coordination is not exact and has a tendency to break down at higher temperatures. The predominant line for ice is the 3200 line, for cold water—the 3400 line, and for hot water—the 3600 line. Therefore the weakening of the 3200 line in the Raman spectrum is an indication, parallel to the maximum at 2.2 Å in X-ray diffraction, and signifies the beginning of a transition to a close-packed structure. It would be very tempting to regard the bands 3200, 3400, and 3600 as corresponding to the above-mentioned structures of water I, II, and III; however, this is impossible until the nature of the transitions corresponding to these bands has been studied in detail.
There is also observed a shift toward higher frequencies, indicating a weakening of the mutual polarization forces acting between molecules in water, as compared with ice.
§ 6a. Structure of Ice
Before proceeding to a discussion of the dielectric constants of water and ice, it is necessary to consider in detail the structure of ice. In the normal structure of ice the positions of the H nuclei, and consequently the orientation of the molecules, are fixed. This follows from the determination of the positions of the O atoms by means of X-rays, and also from our knowledge of the structure of the molecule obtained from the study of band spectra. Each molecule is surrounded by four others, forming a tetrahedron, and this arrangement has the lowest energy in the case when the H nuclei, situated at an angle of approximately 109° with the vertex at the center of the molecule,* lie opposite two neighboring molecules, while one H nucleus of each of the two other neighboring molecules lies opposite the negative angle of the original molecule (Fig. 4). Such an arrangement is, of course, devoid of trigonal symmetry and at first sight cannot be reconciled with the structure of ice known to us from X-ray-analysis data⁹). But in deriving the latter, only the influences of the O atoms could be taken into account, and thus no structure leaving the positions of the latter unchanged can be rejected on the basis of X-ray data. The simplest of the physically possible structures is shown in Fig. 6. It requires a unit cell three times as large as the cell proposed by Barnes (Barnes), but possesses trigonal—
* The angle of 103–106°, indicated above on p. 589, refers to the angle between the H nuclei and O. In the present case the center of the molecule is not its center of mass near the O nucleus, but its electronic center, which is displaced toward the pair of H nuclei (see below, p. 607).
symmetry. Moreover, it belongs to the polar class, whereas the Barnes structure belongs to the holohedral class.
The crystallographic data are contradictory, but the careful investigations of Adams indicate that the polar class is the correct one, although it is often masked by the formation of twins on the basal plane. However, Wooster’s (W. A. Wooster) attempt to prove the polar character of ice by means of its pyroelectric properties, using Martin’s method,^10 gave a negative result. This may be explained by the formation of twins.
Undoubtedly, it is quite possible to construct models of the correct structure of ice that are devoid of polar properties. However, for reasons of geometrical order these structures must be extremely complicated and have a very large cell. The simplest of them, with symmetry \(C_3^2 — C3\), geometrically but not physically polar, contains 96 molecules in the unit cell. The formation of such a complex arrangement under ordinary conditions is highly improbable. Fig. 6 gives the simplest correct structure of ice. It is therefore quite possible, and even probable, that at temperatures below the melting point the distribution of the molecules remains partly (and even for the most part) irregular, although at every point it preserves tetrahedral coordination and balanced dipoles. In that case ice would be crystalline only with respect to the positions of its molecules, but glass-like with respect to their orientation. Such a hypothesis may still be necessary to explain the dielectric constant and the absence of pyroelectricity.
§ 6b. Theoretical discussion of the dielectric constant of water and ice
One of the most characteristic properties of water is its high dielectric constant, equal to 88 at \(0^\circ\)C, for frequencies up to \(10^8\) cycles per second. The dielectric constant of ice is sometimes considered low. However, the difference between them depends to a large extent on the frequencies at which the dielectric constant is measured. The changes in the dielectric constant of water and ice at various temperatures as a function of changing frequency are shown in Fig. 7.
The nature of the dielectric constant as a function of frequency is approximately the same at all temperatures, apparently both for water and for ice; important differences are observed only in the frequency scale for the different curves. At sufficiently low frequencies the dielectric constant even of very cold ice may reach large values, possibly even exceeding the usual values for water. At sufficiently high frequencies the dielectric constant of water falls below its constant value for low frequencies, but only at higher frequencies for higher temperatures. The value of the dielectric constant at ...
Upper layer. Lower layer.
Fig. 6. Structure of ice, considered along the hexagonal axes. Explanation: the structure* is shown in two layers; — denotes the boundaries of the true cell, \(a=7.81\ \text{Å}\); — denotes the boundaries of the Barnes cell, \(a=4.51\ \text{Å}\),
○ Indicates, in molecules of the upper layer \((5/8)\), \(c=4.61\ \text{Å}\) above the base of the cell.
○ Indicates, in molecules of the lower layer \((1/8)\), \(c=0.92\ \text{Å}\) above the base of the cell. Both H lie above O, and the directions of OH are inclined at an angle of \(10^\circ 16'\) to the \(c\)-axis.
○ Indicates, in molecules of the upper layer \((7/8)\), \(c=6.45\ \text{Å}\) above the base of the cell.
○ Indicates, in molecules of the lower layer \((1/8)\), \(c=0.92\ \text{Å}\) above the base of the cell. One H lies above O along the \(c\)-axis, the other under O at an angle of \(70^\circ 16'\) to the \(c\)-axis; in both cases the dipole axis is directed with its positive end upward at an angle of \(54^\circ\) to the \(c\)-axis.
\[ \begin{aligned} 6\mathrm{O}_{\mathrm{I}}\quad \text{at}\quad &\left\{ \begin{array}{cccccccc} \frac{1}{3} & 0\,\frac{1}{16} & \frac{2}{3} & \frac{2}{3} & \frac{1}{16} & 0\,\frac{1}{3} & \frac{1}{16} \\ \frac{2}{3} & 0\,\frac{9}{16} & \frac{1}{3} & \frac{1}{3} & \frac{9}{16} & 0\,\frac{2}{3} & \frac{9}{16} \end{array} \right. \\[1em] 6\mathrm{O}_{\mathrm{II}}\quad \text{at}\quad &\left\{ \begin{array}{cccccccc} \frac{2}{3} & 0\,\frac{15}{16} & \frac{1}{3} & \frac{1}{3} & \frac{15}{16} & 0\,\frac{2}{3} & \frac{15}{16} \\ \frac{1}{3} & 0\,\frac{7}{16} & \frac{2}{3} & \frac{2}{3} & \frac{7}{16} & 0\,\frac{1}{3} & \frac{7}{16} \end{array} \right. \\[1em] 6\mathrm{H}_{\mathrm{I}}\quad \text{at}\quad &\left\{ \begin{array}{cccccccc} \frac{1}{3} & 0\,\frac{1}{16}+3z & \frac{2}{3} & \frac{2}{3} & \frac{1}{16}+3z & 0\,\frac{1}{3} & \frac{1}{16}+3z \\ \frac{2}{3} & 0\,\frac{9}{16}+3z & \frac{1}{3} & \frac{1}{3} & \frac{9}{16}+3z & 0\,\frac{2}{3} & \frac{9}{16}+3z \end{array} \right. \\[1em] 6\mathrm{H}_{\mathrm{I}}\quad \text{at}\quad &\left\{ \begin{array}{cccccccc} \frac{1}{3}+y & 0\,\frac{1}{16}-z & \frac{2}{3}-y & \frac{2}{3}-y & \frac{1}{16}-z & 0\,\frac{1}{3}+y & \frac{1}{16}-z \\ \frac{2}{3}-y & 0\,\frac{9}{16}-z & \frac{1}{3}+y & \frac{1}{3}+y & \frac{9}{16}-z & 0\,\frac{2}{3}-y & \frac{1}{16}-z \end{array} \right. \\[1em] 12\mathrm{H}_{\mathrm{II}}\quad &\left\{ \begin{array}{cccccccc} \frac{2}{3}+y & y\,\frac{15}{16}+z & \frac{1}{3} & \frac{1}{3}-y & \frac{15}{16}+z & y\,\frac{2}{3}+y & \frac{15}{16}+z \\ \frac{2}{3} & -y\,\frac{15}{16}+z & \frac{1}{3}-y & \frac{1}{3} & \frac{15}{16}+z-y & \frac{2}{3} & \frac{15}{16}+z \\ \frac{1}{3} & y\,\frac{7}{16}+z & \frac{2}{3} & \frac{2}{3}+y & \frac{7}{16}+z & y\,\frac{1}{3} & \frac{7}{16}+z \\ \frac{1}{3}-y & -y\,\frac{7}{16}+z & \frac{2}{3}+y & \frac{2}{3} & \frac{7}{16}+z-y & \frac{1}{3}-y & \frac{7}{16}+z \end{array} \right. \end{aligned} \]
\[ y=0.105,\qquad z=0.037 \]
* Complete description of the structure: hexagonal with \(a=7.82,\ c=7.36\), two molecules of \(\mathrm{H_2O}\) in the cell; space group \(C_{6v}^{3} - C6mc\); atomic positions somewhat idealized.
at low frequencies decreases for water with increasing temperature, and for ice, probably, with decreasing temperature. The greatest values are attained for cold water (or, perhaps, for warm ice). In Fig. 7 the most recent values are given, as well as some of the earlier, older values at low temperatures.
Fig. 7. Change of the dielectric constant of water and ice with change in frequency (according to International Critical Tables).
The dielectric properties of water and ice were carefully studied by Debye[^11]. For the water molecule H₂O he uses a model consisting of spherical carriers of dipoles, which can rotate freely under the combined action of the effective electric field and the internal friction of water or ice. For ice this internal friction must be understood in a generalized sense. Debye shows that the form of the curves as functions of frequency agrees surprisingly well with this very crude theory. The internal friction, the relaxation time, and consequently also the wavelength of oscillations at which the large change in the dielectric constant occurs, apparently increase (by \(10^5\) times) upon freezing by a sharp jump (which is quite natural), but otherwise the transition from the properties of water to the properties of ice takes place smoothly and continuously.
Fig. 7a. Dielectric constant of water and ice at low frequencies. Circles: Lattey, 1931; triangles: Fleming and Dwar, 1897; crosses: Wintsch, 1932 (Helv. Phys. Acta V, 126, 1932).
To this part of Debye’s reasoning we shall add nothing, since there is no need to do so. But Debye does not analyze the absolute values of the dielectric constants at low frequencies as functions of temperature; we, however, think that much can be achieved by means of such an analysis carried out on the same model[^12]. In doing so it is important to remember ...
that this analysis is based on the hypothesis of Mosotti, according to which the effective internal field strength at each point, acting on each of our dipole carriers, is equal to \(E+\dfrac{4}{3}\pi P\), where \(E\) is the electric force (or field strength), and \(P\) is the polarization of the medium. The applicability of the Mossotti hypothesis may be called into question, as Debye cautiously points out. Strictly speaking, its applicability is rigorously guaranteed by the Lorentz lemma \(^{13}\) only for a completely disordered distribution of dipoles around any chosen dipole, or for distributions with cubic symmetry. The latter include distributions in which all neighbors can be grouped four at a time at the vertices of regular tetrahedra whose centers lie in the selected molecule. To this model of rotating dipoles the Weiss–Langevin ferromagnetic theory is applicable in its simplest form, so that
\[ P=N\mu L\left(\frac{\mu\left|E+\frac{4}{3}\pi P\right|}{kT}\right), \tag{3} \]
where \(N\) is the number of orienting dipoles, with moment \(\mu\), per unit volume, and \(L(x)\) is the Langevin function:
\[ L(x)=\operatorname{ctg} x-\frac{1}{x}. \tag{4} \]
From equations (3) and (4) it follows for small fields:
\[ P\left(1-\frac{4\pi N\mu^2}{9kT}\right)=\frac{N\mu^2}{3kT}E, \]
or
\[ P=\frac{N\mu^2}{3k(T-T_c)}E \qquad \left(T_c=\frac{4\pi N\mu^2}{9k}\right). \tag{5} \]
The dielectric constant \(\eta\) is given by the equation:
\[ \eta-1=\frac{4\pi P}{E}, \]
so that
\[ \eta=1+\frac{4\pi N\mu^2}{3k(T-T_c)} =1+\frac{3T_c}{T-T_c}. \tag{6} \]
If all dipoles of water are freely orientable, then
\[ N=\frac{6.06\cdot 10^{23}}{18},\qquad \mu=1.87\cdot 10^{-18},\qquad T_c=1200. \]
This, of course, is impossible, since it would correspond to a state of permanent polarization of the ferromagnetic type. If we wish to explain the observed values of \(\eta(T)\) by means of the theory of orienting dipoles, then we must determine \(T_c\) from equation (6), the observed \(\eta(T)\), and \(N\) from equation (5). This means that we must use a variable number of effective freely rotating dipoles, which a priori is entirely permissible.
conclusion. The actual effective numbers of freely rotating dipoles, expressed as fractions of \(N\), required to obtain the values (Fig. 7a), are given in Table 2. In these approximate calculations the part of the dielectric constant explained by orientation is taken to be \(\eta(T)-2\).
TABLE 2
Fraction \(f\) of molecules capable of orientation required to explain \(\eta(T)\)
| Water | Water | Ice | Ice |
|---|---|---|---|
| Temperature in °K | \(f\) | Temperature in °K | \(f\) |
| 273 | 0.220 | 270 | 0.216 |
| 293 | 0.235 | 250 | 0.200 |
| 325 | 0.26 | 225 | 0.178 |
| 350 | 0.28 | 200 | 0.154 |
| — | — | 175 | 0.118 |
These deviations are in themselves quite probable, even if we cannot provide a special theory for them. They force us to acknowledge that from \(1/5\) to \(1/4\) of the water molecules, or from \(1/10\) to \(1/5\) of the ice molecules, must be regarded as free rotators. At the same time, a large part of the molecules of ice or water must be definitively recognized as incapable of free rotation. This does not contradict the theory of the structure of water that we are developing, but the number of non-rotating molecules may perhaps differ considerably from unity, especially for crystalline ice. Usually, on the basis of Debye’s calculations\(^{14}\), it is assumed that in fields of ordinary intensity such an insignificant number of ice molecules should rotate (1 in 5,000,000 at a field strength equal to 1 V/cm) that this cannot disturb the ice lattice. We, however, assert something rather different: our calculations give not the number of rotating dipoles, but the number of those possessing freedom of rotation, and it is possible that this latter number is of greater importance for the question of lattice stability than Debye’s number.
Summarizing the foregoing, we may state: 1) the theory of rotating dipoles may be accepted for explaining the dielectric constant of water and ice without contradicting the theory of the structure of water that we propose, but 2) the theory of rotating dipoles has inherent difficulties which compel us to accept it only with certain reservations. These difficulties become more evident after, in the subsequent exposition (p. [[unclear: page number]]), we show that the total energy of ice or water is composed, over the entire
probabilities, chiefly from the interaction of dipoles incapable of free rotation (with the exception of the insignificant form of rotation about the dipole axis). But if the theory of free dipoles must be abandoned or modified, then it seems quite probable that the dielectric constant can be derived from the change in orientation (reorientation) of groups (blocks) of polar crystals under an applied field. In water these groups will be rather indefinite liquid crystals, very readily passing one into another; for ice the transition time will, of course, be longer and will increase as the temperature falls. For the present we see no necessity for a detailed development of this theory, but in this way one could remove what we consider the most important objection to the theory of free dipoles, namely the notion that very small changes in \(f\) correspond to enormous changes in the dielectric constant, so that the factors determining \(f\) must in one form or another explicitly enter into the theory.
§ 7. Comparison of water with other liquids
The physical properties of water and ice seem still more anomalous when compared with other compounds of analogous electronic structure. The molecule \(\mathrm{H_2O}\), like the molecules \(\mathrm{CH_4}\), \(\mathrm{NH_3}\), and \(\mathrm{HF}\), belongs to the neon type, possessing ten electrons. The most important properties of these substances are given in Table 3. For comparison, the same properties are given for the hydrogen compounds of the elements of the second row, \(\mathrm{PH_3}\), \(\mathrm{SH_2}\), and \(\mathrm{HCl}\), and also for certain hydrogen compounds of diatomic complexes, such as, for example, \(\mathrm{CC}\), \(\mathrm{CN}\), \(\mathrm{CO}\), \(\mathrm{CF}\), \(\mathrm{NN}\), \(\mathrm{NO}\), \(\mathrm{OO}\). At first sight it is clear that these substances can be divided into three classes. The majority are typically molecular substances with low melting and boiling points and a narrow range of the liquid state. Others, such as nitrous acid \(\mathrm{HNO}\), formaldehyde \(\mathrm{H_2CO}\), and acetylene \(\mathrm{HCCH}\), have a tendency to polymerize. The remaining \(\mathrm{H_2O}\), \(\mathrm{HF}\), \(\mathrm{H_3COH}\), \(\mathrm{H_2NNH_2}\), \(\mathrm{H_2NOH}\), \(\mathrm{H_2O_2}\), and \(\mathrm{HCN}\) form a group of typical associated liquids with relatively high critical temperatures and boiling points and a wide range of the liquid state. All these molecules, with the exception of \(\mathrm{H_2NNH_2}\), possess one highly polar group \(\mathrm{OH}\) (or \(\mathrm{FH}\), \(\mathrm{C-H}\)), and it is quite obvious that the attraction between the positive end of these dipoles and the negatively charged remaining part of the molecule determines the exceptional cohesive forces observed in these cases. In the case of \(\mathrm{HF}\) it would be more correct to suppose that cohesion is caused not by dipoles, but by a “hydrogen bond.” But even in this group water occupies a special position. It has the highest van der Waals constant \(a\), corresponding to the greatest attraction between molecules, and the lowest \(b\), corresponding to the smallest distan-
TABLE 3
Properties of substances with simple molecules
| Substance | Crystal structure | Molecular volume in liquid state, ų | Melting temperature, \(T^\circ\) abs | Boiling temperature, \(T^\circ\) abs | Critical temperature, \(T^\circ\) abs | Constants \(a\) | van der Waals \(b\) | Latent heat of fusion, Cal/g-mol | Latent heat of vaporization, Cal/g-mol | Viscosity at temperature \(T^\circ\) abs | Dielectric constant | Dipole moment |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| CH₄ | CP | 63 | 89 | 109 | 191 | 0,0036 | 0,00162 | 0 023 | — | 0,0025 at 34 | — | 0 |
| NH₃ | CP | 45 | 196 | 234 | 405 | 0,0080 | 0,00161 | 1,84 | 5,6 | 0,0179 at 0 | 25,4 | 1,5 |
| OH | \(I\) | 30 | 273 | 373 | 647 | 0,0118 | 0,00150 | 1,41 | 9,72 | Mobile | 80 | 1,87 |
| FH | Pol | 33 | 190 | 293 | — | — | — | 1,09 | 6,02 | — | 83,5 | — |
| PH | CP | 76 | 140 | 188 | 327 | 0,0094 | 0,00233 | — | — | — | — | 0,25 |
| SH | CP | 63 | 187 | 211 | 373 | 0,0089 | 0,00193 | — | 4,23 | 0,00435 63 | 5,03 | 1,10 |
| ClH | CP | 65 | 160 | 190 | 325 | — | — | 0 050 | 3 54 | 0,00155—45 | 8,85 | 1 03 |
| H₃CCH₃ | HCP | 110 | 102 | 184 | 305 | 0,0119 | 0,00312 | — | — | — | — | 0 |
| H₃CNH₂ | — | 80 | — | 267 | 430 | 0,0144 | 0,00272 | — | — | 0 00236 0 | 10,5 | 1 31 |
| H₃COH | — | 65 | 175 | 338 | 513 | 0,0199 | 0,00318 | 0,052 | 11,0 | 0,0055, 25 | 34 | 1 73 |
| H₃CF | — | — | — | 195 | 316 | 0,0092 | 0,00235 | — | — | — | — | — |
| H₂NNH₂ | — | 52,5 | 274 | 386 | 653 | — | — | — | — | Viscous | — | — |
| H₂NOH | — | 44,5 | 306 | 340 | — | — | — | — | — | — | — | — |
| HOOH | — | 38 | 271 | 357** | — | — | — | 0 25 | 12,3 | Viscous | 92,8 | — |
| H₂CCH₂ | — | 76 | 103 | 170 | 282 | 0,0089 | 0 00255 | — | — | — | — | 0 |
| H₂CO | — | 61 | 181 | 252 | — | — | — | — | — | — | — | — |
| HNO | Pol | — | Solid at 290 | — | — | — | — | — | — | — | — | — |
| HCCH | Pol | 69 | — | 191 | 308 | 0,0087 | 0,00239 | — | — | — | — | — |
| HCN | Pol | 64 | 259 | 300 | 457 | — | — | — | 5,6 | 0,0020—7 | — | — |
CP — cubic close packing
CP — polar modified cubic close packing
HCP — hexagonal close packing
\(I\) — structure of tridymite
Pol — polymerizes
* At 60 mm and unstable at higher temperatures
** At 68 mm and unstable at higher temperatures
...between the centers of molecules. This confirms what we attempted to show above, namely, that the peculiar cohesion of water is explained not only by the presence of dipoles, but also by the geometrical possibility of bringing the molecules together. A tetrahedral arrangement of the molecules is possible only in water. The presence in each molecule of two, and only two, hydrogen atoms makes it possible to attach two molecules in this way, and two others by means of their own hydrogen atoms, imitating the ionic structure of quartz; moreover, instead of one oxygen atom between two silicon atoms, in water we have one hydrogen atom between two oxygen atoms (Fig. 8).
Fig. 8. “Quartz” structure of water.
Molecules are arranged in three layers: \((1/3)c\), \(0\), \((1/3)c\) from the base of the cell. The directions OH upward are shown by ——H. The directions OH downward are shown by ---H. The central atom shows a distorted tetrahedral coordination.
Such a three-dimensional, infinitely repeating structure is impossible for ammonia, which has three hydrogen atoms and only one place for one atom, just as it is for hydrofluoric acid, which has only one hydrogen atom. It is natural to suppose that in the case of HF we are dealing only with a linear dipole, that the tetrahedral framework is not developed, and that the only possible point of attachment for another hydrogen atom lies approximately opposite the first H. In this case we shall have only linear coordination; HF must form rings or chains:
\[ \begin{array}{ccccccc} & & F & & & & \\ & H & & H & & & \\ F & & & & F & & \\ H & & & & & H; & \quad FHFHFHFH\ldots \\ F & & & & F & & \\ & H & & H & & & \\ & & F & & & & \end{array} \]
If we even assume, in the case of \(\mathrm{NH_3}\) or HF, the existence of latent tetrahedra, we still obtain only separate rings or chains. For example, for \(\mathrm{NH_3}\) we can obtain the structure shown in Fig. 9, where the sign \(+\) indicates the position of hydrogen nuclei, and the sign \(\odot\) indicates vacant sites. If we wish to add extra molecules, this can be achieved in the manner indicated in the figure by dotted circles; however, these chains
will never be able to come together again and must tend toward a weakening of the mutual bonds. If the \(+\) signs are replaced by \(\odot\) signs and conversely, then this same drawing may serve as an illustration of the structure of HF. The internal structure of \(\mathrm{H_2S}\) is similar to the structure of water, but its hydrogen atoms are situated too deeply and cannot play the role of dipoles in thermal rotation. Of all liquids, methyl alcohol comes closest to water, but the nonpolar methyl group gives it a decidedly more molecular character. However, the coherent nature of liquid hydrazine \((\mathrm{H_2NNH_2})\) cannot be explained by any of the methods listed above. Its properties deserve more careful study. Two possibilities present themselves: hydrogen bonds may be formed between neighboring molecules, although this has never been observed in \(\mathrm{N—H}\) compounds, or else the molecules in the liquid may be transformed into amphoteric ions \((\mathrm{H_2N})^{+}(\mathrm{NH})^{-}\), held together by electrostatic attraction.
Fig. 9. Rings or chains in \(\mathrm{NH_3}\).
On the basis of the foregoing, we arrive at the conclusion that the exceptional properties of water are explained by the structure of its molecule, which allows it to form, in the solid and liquid phases, an extended electropolar complex characterized by tetrahedral (fourfold) coordination.
§ 8. Total Internal Energy of Water
This conception of the structure of water can be subjected to an approximate quantitative check. In this model the total energy of water is determined chiefly by the mutual potential between neighboring molecules, caused by their electropolar character. If we knew the distribution of charges in the \(\mathrm{H_2O}\) molecule, then the total energy could be calculated exactly; at present, however, we can obtain approximate values by combining spectral data on the relative positions of the centers of mass in the molecule with the value of the dipole moment of the molecule and the distance between molecules obtained from X-ray data.
If, in the spectrographic model (Fig. 10), a charge \(e\) is placed in the positions of H, and a charge \(-2e\) in the O nucleus, then we obtain a molecule with a dipole moment \(5.6 \cdot 10^{-18}\) instead of the observed dipole moment \(1.87 \cdot 10^{-18}\). Obviously, this discrepancy arises because we do not take into account the influence of the two \(\mathrm{H}^{+}\) on the \(\mathrm{O}^{--}\) ion. This effect may be twofold. The center of the negative
of the electronic charge will move from the oxygen nucleus to a position intermediate between the latter and the hydrogen nucleus, and some concentration of negative charge will screen the hydrogen charges. The simplest assumption is that effective charges \(e'\) (smaller than \(e\)) are located at the positions of hydrogen, and the charge \(-2e'\) is located at a distance \(x\) Å from the nucleus O on the bisector of the HOH angle. In the presence of an electric moment equal to \(2\cdot 10^{-18}\), the latter are related by the relation:
\[ (0.58-x)e' = 0.21e. \]
This leaves the choice of \(x\) free only between 0 and \(0.37\) Å. Obviously, \(x\) will be closer to the first value because, for six out of ten electrons of the system, the negative center must be located very close to the oxygen nucleus. A value of \(0.15\) is quite acceptable for \(x\), giving \(e' = 0.49e = 2.33\cdot 10^{-10}\), which has the advantage that in the calculations we obtain a tetrahedral arrangement of positive and negative charges*. Fortunately, we have an independent way of checking this value by using the results of measurements of the dipole moment of hydroquinone. In this dihydroxy compound \(\mathrm{HOC_6H_4OH}\), the hydroxyl groups rotate completely independently, and since they are arranged symmetrically, only the transverse electric moments \(\mu_t\) are effective (Fig. 11). According to Fuchs’ theory\({}^{15}\), the observed moments \(\mu\) are the time averages of the moments of the end groups, neutralizing one another in the trans-positions and additive in the cis-positions, i.e. \(\mu = \sqrt{2}\mu_t\).
Fig. 10. Model of the water molecule. \(H^+H^+\) indicate the hydrogen nuclei; O—the oxygen nucleus; \(\bar O\)—the center of the molecule and of the negative charge.
But \(\mu = 2.34\cdot 10^{-18}\) and, consequently, \(\mu_t = 1.66\cdot 10^{-18}\). In our model of the water molecule, \(\mu_t = 0.75e' = 1.75\cdot 10^{-18}\). Thus the transverse moment calculated for the water model is 5% greater than that obtained from the hydroquinone molecule. This difference of 5% is not unexpected for us, since above we assumed an incre—
* In some subsequent calculations this value is taken, for simplicity, to be equal to 0.5.
a 5% increase in the direct moment—from 1.8 to \(2.0\cdot 10^{-18}\)—to take account of the polarizing action of neighboring water molecules.
Thus we may suppose that the model of the water molecule is correct to a first approximation. It could have been derived independently of spectroscopic data, exclusively from measurements of the dipole moment.
The mutual potential energy of two such molecules situated relative to each other as we observe for ice (Fig. 6), i.e., when their centers are at a distance \(2.72\cdot 10^{-8}\)* and their effective OH directions form tetrahedral angles with one another, can be calculated electrostatically. (In most cases these calculations were carried out graphically.) For the above-mentioned model it reaches, on the average, \(0.533\cdot 10^{-12}\) ergs. In fact, in ice we find three positions satisfying these conditions: they may be called the cis, trans, and screw arrangements, which differ from one another only in the azimuthal angular position of the two water tetrahedra relative to the line joining them. The corresponding mutual potential energies are \(0.504\), \(0.527\), \(0.551\cdot 10^{-12}\) ergs for each pair of molecules. In any structure of ice these types must occur in the ratio \(1:1:2\), and the mean value, weighted in this way, is \(0.533\cdot 10^{-12}\) ergs for each pair of molecules. Each molecule has four neighbors; thus the potential energy per molecule is \(1.066\cdot 10^{-12}\) ergs, with each molecule counted only once, which is equivalent to \(-15.3\) cal/g-molecule.
Fig. 11. Hydroquinone.
In this calculation three factors were omitted.
- The mutual potential energy of non-contacting molecules. The latter is very small. The mutual potential energy of two dipoles decreases in proportion to the cube of the distance and, because of the mutual orientation, comprises approximately equal amounts of positive and negative terms. Of interest to us are only the molecules next after the nearest ones. By an approximate count the potential energy caused by the thirteen molecules following after
* The value \(2.76\ \text{Å}\) at \(0^\circ\) C, as a result of thermal contraction, falls to \(2.72\ \text{Å}\) at absolute zero.
nearest to the ice molecule, is equal to \(0.154\cdot 10^{-12}\) ergs, which corresponds to \(1.1\) cal/g.
-
Van der Waals attraction. According to London’s theory\(^ {16}\), the van der Waals attraction between two identical molecules located at a distance \(r\) from one another is given by the expression \(\left(\dfrac{3}{4}\right) I \alpha^2 r^{-6}\), where \(I\) is the ionization potential and \(\alpha\) the polarizability. Born\(^ {17}\) considers this quantity too small and multiplies it by the factor \(3/2\). Taking this correction and substituting \(13\ \mathrm{V}\) for \(I\), and \(1.5\cdot 10^{-24}\) for \(\alpha\), we find for the van der Waals energy of a pair of water molecules \(-0.143\cdot 10^{-12}\) ergs, or \(-4.1\) cal per gram molecule of ice. This value is almost certainly too small, since the application of London’s theory to such an asymmetric molecule as the water molecule is very artificial, but it should be correct as regards the order of magnitude.
-
Repulsive forces are much more difficult to calculate, but since they must constitute a large part of the total energy, we shall try to obtain approximate values of this quantity. For our purpose it is sufficient to restrict ourselves to considering only neighboring molecules. If the total potential is equal to
\[ V=-F(r)-\frac{C}{r^6}+\frac{B}{r^n}, \]
we can obtain \(B\), if \(n\) is known to us, from the equilibrium condition:
\[ -F'(r_0)+\frac{6}{r_0}\frac{C}{r_0^6} = \frac{n}{r_0}\frac{B}{r_0^n}. \]
\(F'(r)\)—the electrostatic force acting between molecules—can be obtained graphically from the model. It turns out to be equal to \(0.731\cdot 10^{-4}\) dynes for each pair of molecules. \(0.316\cdot 10^{-4}\) dynes falls to the share of the van der Waals force, so that
\[ \frac{n}{r_0}\frac{B}{r_0^n} = 1.047\cdot 10^{-4}\ \text{dynes}. \]
And since \(n\) cannot differ very markedly from its value for rare gases\(^ {18}\), \(\simeq 12\), we have:
\[ \frac{B}{r_0^n} = \frac{1.047\cdot 2.72\cdot 10^{-12}}{12} = 0.238\cdot 10^{-12}\ \text{ergs} \]
for each pair of molecules. Consequently, the repulsive forces account for \(0.475\cdot 10^{-12}\) ergs per molecule, or \(6.8\) cal/g-mol.
Introducing these corrections, we obtain an a priori theoretical value for the sublimation energy of ice at \(0^\circ K\) equal to
\[ 15.3-1.1+4.1-6.8=11.5\ \text{cal/g-mol}. \]
For comparison, we have the experimental value \(10.70\) cal/g-mol for the heat of vaporization of water at \(273^\circ K\). To this must be added: 1) the latent heat of melt-
of ice, \(1.44\ \mathrm{Cal}/\mathrm{g\text{-}mol}\), 2)
\[ \int_{0}^{273} C_p\;(\text{ice})\, dT = 1.28\ \mathrm{Cal}/\mathrm{g\text{-}mol}, \]
3)
\[ -\int_{0}^{273} C_v\;(\text{water vapor})\, dT = 1.61\ \mathrm{Cal}/\mathrm{g\text{-}mol}. \]
In this we do not take into account the energies of the zero point or of rotation in the solid. This gives for the heat of sublimation of ice at \(0^\circ K\) an experimental value with a theoretical value of \(11.5\ \mathrm{Cal}/\mathrm{g\text{-}mol}\). This agreement is a fortunate accident, since a change in the position of the effective center of the molecule by even \(0.05\ \text{\AA}\) changes the theoretical value by \(2.5\ \mathrm{Cal}/\mathrm{g\text{-}mol}\); a small change in the exponent of the repulsive forces* causes an even larger change. Nevertheless, the agreement is quite sufficiently real and undoubtedly shows that the overwhelming greater part of the cohesion of ice, and consequently also of water at low temperatures, is due to the electrical interaction of tripolar molecules in a relatively immobile tetrahedral configuration.
This calculation has been carried out for a regular structure devoid of thermal motion, and therefore applies exclusively to ice at absolute-zero temperature. The energy of water is naturally greater than this value, partly because of the motion of the molecules, and partly because of the destruction of the regular structure with fourfold coordination, which increases with rising temperature, so that at the critical point the state of water differs little from the state of \(\mathrm{NH_3}\) or \(\mathrm{CH_4}\) at the corresponding temperatures. The high values of the latent heat of fusion of ice and of the heat capacity of water can be explained by the gradual destruction of the structure with fourfold coordination. The breaking of one \(\mathrm{H_2O—H_2O}\) bond out of eight can explain the latent heat of ice; to explain the heat capacity of water it must be assumed that at \(100^\circ C\) one bond out of four is broken, and at \(250^\circ C\)—half of all the bonds. The sharpness of the transition of ice into water still remains unexplained. However, it is quite evident that the melting of ice is not a simple transition from a regular arrangement of atoms or molecules to an irregular one without a change in coordination, as in the cases of rare gases, paraffins
* If, instead of the older form of the theory, the repulsive term is taken in the form
\[ ae^{-r/\rho}, \]
where \(\rho = 0.34\ \text{\AA}\), obtained from the study of the halide salts of the alkali metals (Born und Meyer, Ztschr. f. Physik. 75, 1, 1932), the repulsive term will be approximately equal to \(10\ \mathrm{Cal}/\mathrm{g}\). This will lower the theoretically calculated energy to \(8\ \mathrm{Cal}\) instead of \(11.5\). This value is undoubtedly too low, but we are not sure that this does not apply also to the new salts, which have much larger terms expressing the action of the attractive forces. However, if this value proves correct, it will become necessary to find the correct value of the van der Waals term, which can easily restore the equilibrium.
or metals with a close-packed lattice. The melting of such substances, with its small latent heat and positive change in volume, may be called isomorphous melting. In the case of ice, however, as in the case of bismuth and many organic solids (salol, azoxyanisole), melting is equivalent to the destruction of the ordered arrangement of molecules plus a change in coordination, i.e., it corresponds to a change of crystalline state and may be called morphotropic melting. Associated with this type of melting are: a considerable latent heat, a considerable change in volume, usually negative, and the possibility of considerable supercooling, owing to the fact that the internal structure of the liquid does not correspond to the structure of the solid crystallizing from it. Solid form of water II is, in all probability, ice II, melting at \(-20^\circ\). This modification of ice can never be formed at ordinary pressure, since under such conditions water II instantaneously passes into water I of the ice type, even in those cases when, owing to the absence of nuclei, ice does not crystallize out. On the basis of the assumptions we have made in this chapter, it is possible to calculate the energy of the presumed solid water II. It proved to be equal to \(11.0\ \mathrm{Cal}/\mathrm{g\text{-}mol}\). The difference between this value and the theoretical value \(11.5\ \mathrm{Cal}/\mathrm{g\text{-}mol}\) is explained chiefly by the introduction of additional terms expressing the action of repulsive forces, introduced as a result of the mutual approach of molecules following after the nearest neighbors. The instability of water II in the solid state is compensated in the liquid state by the greater possibility of rotation of molecules and exchange with neighbors at higher temperatures. However, we must emphasize that the methods of calculating the internal energy of water are in all respects less reliable than those for ice, and that the numerical coincidences may be regarded as accidental. In particular, we still cannot give a quantitative explanation of the high latent heat of fusion of ice.
Nevertheless, the results of the investigation of the thermal properties of water confirm the data obtained from the X-ray study of changes in the structure of water with temperature, in particular in the transition from water II (of the quartz type) to water III (of the ammonia type).
§ 9. Properties of ionic solutions: density of the solution and hydration of ions
The present investigation of the structure of water was undertaken with the aim of elucidating the properties of ionic solutions, especially those containing hydrogen and hydroxyl ions. We may now proceed to consider the molecular picture of an ionic solution. Since the internal field of water is determined by the electrostatic field of dipoles, it is obvious that it will change strongly as a result of the introduction of charged ions. It is clear that the effect will be approximately proportional to the polarizing power of the ion, i.e., to the charge
of the latter divided by its radius. The effect of large monovalent ions will be the smallest, the effect of small ions possessing a large charge the greatest. This agrees with the long-known hypothesis of ion hydration, advanced to explain the seemingly anomalous fact that the mobilities of the large ions \(K^+\), \(Rb^+\), \(Cs^+\), \(Cl^-\), \(Br^-\), and \(J^-\) are all approximately the same, whereas small ions, such as, for example, \(Li^+\) or \(Mg^{++}\), move much more slowly. Several different experimental methods have been proposed for determining the degree of hydration of ions, but they all lead to different results and are, evidently, theoretically unsatisfactory. Methods based on the transport of water by ions in nonaqueous solutions are not consistent, since in this case the water molecules are held by the ions not against the attraction of other water molecules, but against the attraction of less polar molecules, and tend to give excessively high values. Methods based on mobilities interpret Stokes’ law too broadly, the interpretation being extended to regions where application of this law cannot be justified; but even in these cases what is measured is not so much the actual water molecules bound to a given ion as the mass of water carried by the ion hydrodynamically during motion through the liquid, and, thus, the values obtained always prove to be too high.
TABLE 4
Additive volume of one–monovalent salts in dilute aqueous solution, in \(\text{Å}^3\) per pair of ions
| \(OH^-\) | Difference \(F{-}OH\) | \(F^-\) | Difference \(Cl{-}F\) | \(Cl^-\) | Difference \(Br{-}Cl\) | \(Br^-\) | Difference \(J{-}Br\) | \(J\) | |
|---|---|---|---|---|---|---|---|---|---|
| \(H^+\) | 29.3 | 9.5 | 38.8 | 20.3 | 59.1 | ||||
| Difference \(Li{-}H\) | −8.3 | −0.8 | +1.0 | +2 | |||||
| \(Li^+\) | 29.0 | 10.8 | 39.8 | 16.6 | 56 | ||||
| Difference \(Na{-}Li\) | −0.9 | −0.9 | −0.4 | +0 | |||||
| \(Na^+\) | −9.2 | +5.0 | −4.2 | 32.3 | 28.1 | 10.3 | 38.4 | 18.2 | 57 |
| Difference \(K{-}Na\) | +16.4 | +17.1 | +17 | ||||||
| \(K^+\) | +17.6 | 31.1 | 44.5 | 11.0 | 55.5 | 18.6 | 74.1 | ||
| Difference \(Rb{-}K\) | +7.9 | 13.4 | +8.0 | +7.5 | +7.7 | ||||
| \(Rb^+\) | +14.7 | +7.2 | 31.9 | 52.5 | 10.5 | 63.0 | 18.8 | 81.8 | |
| Difference \(Cs{-}Rb\) | +5.5 | 20.6 | +11.9 | +11.8 | +11.2 | ||||
| \(Cs^+\) | 31.8 | 29.1 | 64.4 | 10.4 | 74.8 | 18.2 | 93.0 | ||
| \(NH_4^+\) | 31.8 | 29.1 | 60.9 | 2.1 | 63 | 13.5 | 76.5 |
The simplest and, possibly, the most accurate method for determining the actual degree of hydration of ions consists in measuring the densities of ionic solutions. In a sufficiently dilute solution the change in volume per each pair of ions is taken as constant. The values of this quantity for certain salts, calculated as additional volumes in cubic angstroms, are given in Table 4. From the constancy of the differences between rows and columns it is evident that the change in volume for ions is additive. It is also evident that the added volume is approximately proportional to the size of the ion. Proceeding from this assumption and dividing the share contributed by cesium chloride (for which the magnitudes of the positive and negative ions are approximately the same) proportionally to the volumes of these ions in the solid salt, we obtain the apparent volumes of ions in water given in Table 5. Taking the values 27.4 for $\mathrm{Cs}^{+}$ and 37 for $\mathrm{Cl}^{-}$, one can find the apparent ionic volumes of other ions. In the event of an incorrect distribution of the volume between $\mathrm{Cs}^{+}$ and $\mathrm{Cl}^{-}$, all volumes for positive ions will be too large and those for negative ions too small, or conversely, but the relative values will in both cases be quite correct. The table gives a normal relation between the volume in solution and in the solid state among positive ions only for $\mathrm{Rb}^{+}$ and $\mathrm{Cs}^{+}$, and among negative ions—for all, with the exception of $(\mathrm{OH})^{-}$ and $\mathrm{F}^{-}$. In all other cases the apparent volumes are either much smaller than in the solid state, or negative. The conclusion naturally suggests itself that in the first case we are dealing with non-hydrated ions, while in the second case the ions are hydrated to a greater or lesser degree. The loose structure of water postulated above evidently leads to a decrease in volume, being disrupted by the permanent aggregation of several molecules around a single ion. The number of water molecules around a fully hydrated ion is determined by Goldschmidt’s coordination number, i.e., the maximum hydration will be limited by the number of water molecules that can be arranged around the ion so that they are in contact with one another. The latter depends exclusively on the radius of the ions. The theoretical coordination number of various ions with water is given in the second column of Table 6. However, this coordination number is not only theoretical, but is also observed empirically for water of crystallization in known crystalline structures. These conclusions are confirmed by considering water of crystallization. Water occurs in crystals in three forms: 1) structural water, water located in voids or cracks in a rigid ionic structure and which can be removed by heating without destroying the structure, for example in zeolites; 2) ice in polyhydrates, such as, for example, $\mathrm{Na}_{2}\mathrm{CO}_{3}\,10\mathrm{H}_{2}\mathrm{O}$, in which the ions are the dispersed phase. Such crystals are said to dissolve in their own water of crystallization; 3) coordination water, in which mo-
TABLE 5
Apparent ionic volumes in dilute aqueous solution
| Ion | Apparent volume in solution, observed in Å\(^3\) per ion | Volume in solution, calculated from volume in the solid | Ion volume, cm\(^3\) H\(_2\)O per observed volume \((n)\) | Ion volume, cm\(^3\) H\(_2\)O calculated from crystallization data | Ion | Apparent volume in solution | Ion | Apparent volume in solution |
|---|---|---|---|---|---|---|---|---|
| H\(^+\) | −8 | 0.0 | 51(2) | 39 | (HS)\(^-\) | 40 | ||
| Li\(^+\) | −8 | 3 | 170(6) | 150 | (CN)\(^-\) | 41 | ||
| Na\(^+\) | −8.5 | 6 | 225(8) | 180 | (N\(_3\))\(^-\) | 52 | ||
| K\(^+\) | 8 | 15 | (CNS)\(^-\) | 72 | ||||
| Rb\(^+\) | 15.5 | 21 | (NO\(_3\))\(^-\) | 38 | ||||
| Cs\(^+\) | 27 | 28 | (CO\(_3\))\(^{--}\) | 7* | ||||
| NH\(_4\) | 12? | 18 | (HCO\(_3\))\(^-\) | 45 | ||||
| Tl | 21 | (ClO\(_3\))\(^-\) | 61 | |||||
| Be\(^{++}\) | −42.4 | 0.2 | 77(4) | 74 | Mn\(^{++}\) | −41 | (BrO\(_3\))\(^-\) | 64 |
| Mg\(^{++}\) | −48 | 3 | 130(6) | 150 | Fe\(^{++}\) | −39 | (JO\(_3\))\(^-\) | 42 |
| Ca\(^{++}\) | −45 | 7 | 133(6) | 180 | Co\(^{++}\) | −48 | (SO\(_3\))\(^{--}\) | 28 |
| Sr\(^{++}\) | −44? | 13 | 134(6) | 230 | Ni\(^{++}\) | −66 | (SO\(_4\))\(^{--}\) | 39 |
| Ba\(^{++}\) | −33 | 18 | 145(6) | 270 | Cu\(^{++}\) | −50 | (SeO\(_4\))\(^{--}\) | 46 |
| Al\(^{+++}\) | −87 | 1.1 | 91(6) | 110 | Zn\(^{++}\) | −56 | (H\(_2\)PO\(_4\))\(^-\) | 55 |
| Fe\(^{+++}\) | −77 | (MnO\(_4\))\(^-\) | 80 | |||||
| Cr\(^{+++}\) | −93 | (CrO\(_4\))\(^{--}\) | 45 | |||||
| (OH)\(^-\) | −2 | 15 | 28(1) | 34 | (MoO\(_4\))\(^{--}\) | 59 | ||
| F\(^-\) | 6 | 15 | 36 | 34 | (WO\(_4\))\(^{--}\) | 54 | ||
| Cl\(^-\) | 37 | 37 | (S\(_2\)O\(_3\))\(^{--}\) | 85 | ||||
| Br\(^-\) | 47.5 | 47 | (Cr\(_2\)O\(_7\))\(^{--}\) | 139 | ||||
| J\(^-\) | 66 | 66 | (SiO\(_3\))\(^{--}\) | 9* | ||||
| (VO\(_3\))\(^{--}\) | 53 |
The data are taken from the Landolt–Börnstein tables and reduced to 20°; most values are approximate.
* The low values are due to (OH)\(^-\).
cules of water are grouped around individual ions. Here we are interested only in the latter type. Taking into account only crystals of known structure, we find the coordination numbers, namely those which are given in parentheses in column 4 of Table 5. Starting from this coordination number, we can calculate the apparent volume occupied in the liquid by the hydrated complex, by adding the values of column 1 to the number \(29.7\), the volume occupied by each natural water molecule, taken \(n\) times. This is shown in column 4. This can be compared with the volume of the coordination spheres around the same ions observed in crystals. This volume is shown in column 5. The agreement is quite good, except in the cases of the large divalent ions \(\mathrm{Sr}^{++}\), \(\mathrm{Ba}^{++}\), where the determination of the coordination sphere is extremely approximate and the actually occupied volume is exaggerated. In general, one may say that investigation of the specific gravity of solutions shows that some ions are fully hydrated, whereas others are not hydrated at all.
TABLE 6
Additional energy in kcal, derived from the coordination of water molecules around ions with allowance for the effect of the interaction of water molecules. These numbers do not represent experimentally obtained hydration energies
| Ion | \(n = 1\) | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \(\mathrm{Li}^{+}\) | 15 | 28 | 39 | 46 | 49 |
| \(\mathrm{Na}^{+}\) | 11 | 21 | 29 | 33 | 28 |
| \(\mathrm{K}^{+}\) | 4.0 | 7.2 | 8.8 | 9.2 | 2.0 |
| \(\mathrm{Rb}^{+}\) | 2.6 | 4.6 | 5.6 | 4.9 | \(< 0\) |
| \(\mathrm{F}^{-}\) | 4.1 | 7.4 | 9.0 | 9.7 | 3.0 |
| \(\mathrm{Cs}^{+}\mathrm{Cl}^{-}\mathrm{Br}^{-}\mathrm{J}^{-}\) | \(< 0\) | \(< 0\) | \(< 0\) | \(< 0\) | \(< 0\) |
One may try to determine the degree of hydration theoretically, starting from the model of water developed above. Hydration of an ion occurs when the potential energy of the water molecule forming part of the coordination shell around the ion is less than the energy of the molecule in free water. In free water each molecule has four neighbors, while an individual molecule coordinated with an ion has three—the ion and two water molecules on the other side. This follows from the fact that the charge of the ion must symmetrically attract either both hydrogen nuclei or both vacant sites in the water molecule and, from the point of view of further coordination, occupy both of them. Consequently, an indispensable condition for an ion to possess
the condition that the potential energy of the water molecule, as determined by the ion, should be less than that which depends on the other two water molecules.*
The potential energy \(P\) of a water molecule with respect to an ion depends only on the radius and charge of the latter. The values calculated for monovalent ions are given in Fig. 12. If the electrostatic potential energy due to two water molecules is taken to be \(15\) Cal/g-ion, then it can be shown that for monovalent ions with radius greater than \(1.6\) Å hydration cannot be expected, whereas for all monatomic polyvalent ions hydration will always take place. In cases where more than one water molecule is coordinated with the ions, a correction must be made for their interaction; moreover, the latter will always tend to lower the negative potential depending on the ion, and will be the greater, the larger the coordination number. The value of this correction can be calculated; it is given by the formula:
\[ D_n \frac{\mu^2}{(r+r_w)^3}, \]
where \(\mu\) is the dipole moment, and \(r\) and \(r_w\) are, respectively, the radii of the ion and of the water molecule; \(D_n\) is a purely geometrical factor depending only on the coordination number \(n\) and given in the table:
| \(n\) | 2 | 3 | 4 | 6 | 8 | 12 |
|---|---|---|---|---|---|---|
| \(D_n\) | 0.0625 | 0.334 | 0.573 | 1.188 | 1.99 | 3.89 |
This term is significant only for monovalent ions. Introducing it, we can calculate the excess energy:
\[ n\left[P-\frac{D_n\mu^2}{(r+r_w)^3}-15\right], \]
obtained as a result of the coordination of one ion with \(2, 3, 4, 6\ldots\) water molecules. As is seen from Table 4, for \(\mathrm{Na^+}\), \(\mathrm{K^+}\), and \(\mathrm{F^-}\) this energy is maximal at \(n=4\)**; but four is the coordination number of water itself; consequently, we may conclude that the larger ions \(\mathrm{Rb^+}\), \(\mathrm{Cs^+}\), \(\mathrm{Cl^-}\), \(\mathrm{Br^-}\), \(\mathrm{J^-}\), possessing no appreciable or negative coordination energy, in fact also have coordination number four, judging from the insignificant change in their volumes on dissolution. Hence
* For the case of negative ions, it follows from the considerations of § 9a that the coordinating water molecule can have three water molecules as neighbors; but for any negative ion except \(\mathrm{F^-}\) and \((\mathrm{OH})\), the potential energy of coordination is so small that true hydration is not observed.
** \(\mathrm{Li^+}\) gives a maximum at \(n=6\). However, crystallographic data and observations on heats of dissolution compel the assumption that its coordination number is 4.
we conclude that all monovalent ions in a single solution have coordination number four. In the further exposition we shall present the remaining data confirming this proposition.
The difference between permanently hydrated ions Li\(^+\), Na\(^+\), (K\(^+\), FF\(^-\)) and other ions is not geometrical, but physical: small ions carry through the water the hydration shell permanently bound to them, whereas larger ions, in moving, exchange with neighboring water molecules.
For divalent and, still more, polyvalent simple ions the coordination number will always be the highest of the sterically possible ones, namely 6 for all rare gases and most ions with eighteen-electron shells, with the exception of Sr\(^{++}\), Ba\(^{++}\), which may have coordination number 8. However, owing to the large effect of the aqueous shell for these ions, there is a very small energy gain in passing from coordination number 6 to coordination number 8, and they are for the most part found in the former state.
§ 9a. Calculation of the heat of hydration
When the potential energy of a water molecule is lowered as a result of association with an ion, this energy appears as an additional term in the heat of hydration of the ion.
If water were an ideal insulator with dielectric constant \(\eta\), and the ion a conductor having radius \(a\) and charge \(zl\), then the hydration energy would be given by the equation:
\[ \frac{1}{2}\cdot\frac{\eta-1}{\eta}\cdot\frac{z^2 l^2}{a}. \]
As Debye\({}^{19}\) showed, this is valid only in the case where the field near the ion is less than the saturation field of the dielectric, and, consequently, is not applicable to every monatomic ion in water. According to Debye’s calculation, the critical radius \(a\), within which water does not possess the true properties of a dielectric, is equal to 11 Å for monovalent ions and 31, 57, and 88 Å for divalent, trivalent, and tetravalent ions. The theory of the structure of water proposed by us advances the determination of another critical radius \(a\) as the radius within which polarization reaches the greatest possible limit while preserving the tetrahedral structure, namely:
\[ \frac{\pi \mu}{\sqrt{3}} \]
or \(136\cdot 10^2\) V/cm. Within this sphere the water molecules are oriented not by their mutual dipole attractions, but by the ion, and therefore it may also be called the coordination sphere of the ion. Consequently, one may expect that the heat of hydration of an ion depends on four terms:
1) on the energy of the coordination sphere, proportional to the number of coordinated water molecules, and on the mutual energy of the ion and the molecule, with some corrections, which see below,
2) the energy of the intermediate sphere,
3) the energy of the main mass of water outside the Debye sphere,
4) it is obvious that the mutual energy of water molecules reoriented in the neighborhood of the ion must be subtracted from the expression for the hydration energy.
For ions with known charge and radius the energy of the first and third terms can be calculated. Calculation of the second term requires unknown experimental data concerning the polarization of water in fields close to saturation, and cannot be carried out on the basis of the present theory. However, we can avoid this difficulty by virtually eliminating the intermediate sphere and assuming the existence of a sphere of sharp discontinuity with radius \(R_z\), inside which the energy is equal to the energy of ionic coordination, and outside which it is equal to the energy of ordinary water. \(R_z\), naturally, must lie between \(a\) and \(a'\) and depend only on the charge, not on the size of the ion; however, its actual value can be obtained only empirically. On the basis of these assumptions, we can write for the hydration energy of an ion \(U_h\) the expression:
\[ U_h=\frac{\eta-1}{2\eta}\frac{z^2e^2}{R_z}+nP_{\mathrm{eff}}(r)-u_w . \]
In this formula \(R_z\) is the radius of the virtual saturation sphere, depending only on \(z\), \(n\) is the coordination number of the ion, \(P_{\mathrm{eff}}(r)\) is—
Fig. 12. Relation of ionic radius and mutual potential energy for water molecules and monovalent ions. On the ordinate scale is plotted \(P\) (Cal/g-ion), on the abscissa scale \(r\) (Å). For polyvalent ions \(P\) must be multiplied by \(Z\).
the mutual potential energy of an ion and a water molecule in its hydration sphere, depending on \(r\) and \(z\), and \(u_w\) is the electrostatic energy of disoriented water molecules, a small term which, as above, may be taken equal to \(31\ \mathrm{Cal}/\mathrm{g\text{-}ion}\), i.e. to the electrostatic energy of one water molecule in water.
In the first approximation, \(P_{\mathrm{eff}}(r)\) is proportional to the potential energy \(P\) of a water molecule due to the charge \(ze\) at the distance \(r+r_w\), where \(r\) and \(r_w\) are, respectively, the radii of the ion and of the water molecule.
\(P\) can be calculated from a model of the water molecule. Its values as a function of \(r\) are shown graphically in Fig. 12. The curves for positive and negative ions, in all probability, do not coincide, since the water molecules are distributed as shown in Fig. 13, \(a\) and \(b\). This is explained by the fact that \(H^+\) in the water molecule is a definite center of positive charges, whereas it seems doubtful whether, in the absence of another \(H^+\), the molecule can be considered to have two definite centers of negative charges instead of a common negative charge on the side remote from the \(H^+\) nuclei. For the majority of positive ions the value of \(P\) is sufficiently close to the simple expression for a dipole:
Fig. 13 \(a\) and \(b\). Orientation of water molecules in relation to positive and negative ions. \(- \to +\) is the dipole of the water molecule.
\[ \frac{\mu ez}{(r+r_w)^2}. \]
According to our theory, the complete expression for \(P_{\mathrm{eff}}(r)\) is
\[ P_{\mathrm{eff}}(r)=P\left\{1-F(r+r_w)+\frac{C}{(r+r_w)^1}+\frac{Kez}{(r+r_w)^2}-\frac{D_n\mu}{(r+r_w)ez}\right\}. \]
The correction terms in the brackets are:
\(F(r+r_w)\), which takes into account the repulsion between the ion and the water molecule.
\[ \frac{C}{(r+r_w)^1} \]
takes into account the van der Waals attraction.
\[ \frac{Kez}{(r+r_w)^2} \]
takes into account the change of the dipole moment in the field surrounding the ion
\[ \left(K=\frac{d\mu}{dE}\right). \]
\[ \frac{D_n\mu}{(r+r_w)ez} \]
takes into account the mutual potential energy of the water molecules in the coordination sphere of the ion. \(D_n\) is a geometrical multiplier, a factor given in the table on p. 614.
As for these terms, the determination of \(F(r+r_w)\) is difficult, and the value of \(K\) is unknown, so that we shall not attempt to calculate \(P_{\mathrm{eff}}(r)\) exactly.
However, a qualitative consideration of the terms in the brackets shows that
that they tend to compensate one another for different values of \(r\). The ratio \(\dfrac{P_{\mathrm{eff}}(r)}{P}=p_z\) does not differ appreciably from unity and will be approximately constant for one and the same value of \(z\), increasing with increasing \(z\) owing to the predominance of the polarization term \(\dfrac{Kez}{(r+r_w)^2}\). The actual value of this ratio can best be determined empirically. For this purpose we use the simplified expression:
\[ U_h=\frac{(\eta-1)z^2 e^2}{2\eta R_z}+np_zP(r)-u_w. \]
Expressing \(U_h\) as a function of \(P(r)\), we give it the simple form:
\[ U_h=a+bP(r), \]
where \(a\) and \(b\) are functions only of \(z\). Verification of these theoretical arguments by comparison with experimental data is hampered by the fact that the absolute heats of hydration of ions are unknown and cannot be measured directly. We can observe only the sum of the heats of hydration of ions of opposite sign, as, for example, in the case of dissolving a salt in water. For example, for a halide salt of a metal \(M^+X^-\):
\[ U_h(M)+U_h(X)=U_{MX\,\mathrm{cryst}}+U_s=Q_{MX\,\mathrm{aq}}+D_X+S_M-E_X+I_M, \]
where \(U_{MX}\) is the lattice energy, and \(U_s\) is the heat of solution of the salt; \(Q_{MX\,\mathrm{aq}}\) is the heat of formation of the aqueous salt at infinite dilution from \([M]_{\mathrm{cryst}}\) and \(X_{2\,\mathrm{gas}}\); \(S_M\) is the heat of sublimation, and \(I_M\) is the ionization potential of the metal; \(D_X\) is the heat of dissociation, and \(E_X\) is the electron affinity of the halogen. For univalent salts we made extensive use of the calculations of Mayer\(^{20}\) and Sherman\(^{21}\).
Calculations for other cases were carried out on the basis of data given in the International Critical Tables. As in the case of determining the ionic radius, knowing the absolute heat of hydration of one ion, one can determine the absolute heat of hydration of all the others. Attempts to determine such quantities were made by Fajans\(^{22}\) and Webb\(^{23}\).
Webb’s calculation is much more mathematically rigorous than the one we employ. However, Webb’s theory, advanced before accurate values of atomic and molecular radii became known, treats water as a continuum with variable dielectric properties. It is interesting to note that, by a completely independent method, Webb arrived at the value 82 for the heat of solution of \(\mathrm{K}^+\) instead of 94, as found by us. If the correctness of our argument is accepted, then all of Webb’s values for positive ions must be regarded as too low, and for negative ions as too high.
Here we may rely directly on our theory. For ions of opposite signs, but of the same valency, \(U_h\) can depend only on the radius \(r\). Two ions with identical
Fig. 14. Relations between the heat of solution of an ion \(U_h^m\) and the mutual potential energy with respect to a water molecule \(P\). Detailed data for ions with charges 1 and 2. (For the latter quantities the ordinates have been reduced by half.)
radii will have approximately the same heat of solution. Such a pair is represented by \(K^+\) and \(F^-\) with empirical radii, equal for both, of \(1.33\) Å. In this work we
everywhere we use Goldschmidt’s empirical radii, which are more directly applicable than the Pauling or Zachariasen radii. The values 17.5 and 18.0 Cal/g-ion correspond to \(P\). The value
\[ U_h(\mathrm{K}^+) + U_h(\mathrm{F}^-) \]
is equal to 191 Cal/g-ion pair. Therefore we may take \(U_h(\mathrm{K}^+)\) as equal to 94, and \(U_h(\mathrm{F}^-)\) as equal to 97. These values of \(U_h\) may be plotted as a function of \(P\). For all values of \(U_h(\mathrm{K}^+)\) or \(U_h(\mathrm{F}^-)\), the points of positive and negative
Fig. 14a. Relation between the heat of solution of the ion \(U_h\) and the mutual potential energy with respect to the water molecule \(P\). General scheme.
ions lie on parallel straight lines. The above values were chosen so that, as far as possible, all points lay close to one line (Fig. 14). The errors must be small and should appear for ions of the same charge in the form of a constant term. The experimental results presented in Figs. 14 and 14a fully confirm the theory developed above. For all ions of the noble-gas type with the same
charge, lie on one straight line, whose slope gives \(\eta p_z\), and whose intersection with the \(U_h\) axis gives:
\[ \frac{(\eta - 1)e^2 z^2}{2\eta R_z} - u_w . \]
The values of \(\eta\), \(p_z\), and \(R_z\) are given in Table 7, while the observed and calculated quantities for all ions of the noble-gas type, derived with the aid of these quantities, are given in Table 8. Discrepancies in the majority of cases do not exceed the experimental errors. The heats of evaporation of metals and, in some cases, ionization potentials constitute the most frequent source of errors in finding the experimental quantities. For determining the heat of evaporation of \(\mathrm{Be}^{2+}\), \(\mathrm{La}^{3+}\), \(\mathrm{Th}^{4+}\) we proceeded from the melting point, whereas the ionization energy for \(\mathrm{Fe}^{3+}\) and \(\mathrm{Th}^{4+}\) was calculated by analogy with similar atoms. Therefore the data concerning \(3+\) and \(4+\) ions are not reliable and were used by us only in order to show their agreement with the general scheme.
TABLE 7
Constants in the equation for the heats of dissolution of ions. \(R_z\) and \(a\) are given in angstroms
| \(z\) | 1 | 1 | 2 | 2 | 3 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| \(p\) | 4 | 4 | 6 | 66 | 6 | 8 | |
| \(p_z\) | 0.85 | 1.17 | 0.92 | 1.28 | (1.73) | (1.30) | |
| \(R_z\) | 2.9 | 3.6 | 4.6 | (5.7) | |||
| \(a\) | 11 | 31 | 57 | 88 |
\[ u_w = 31\ \text{Cal/g-ion}. \]
The only exception from the linear law for ions of one valence is \(\mathrm{Be}^{++}\). This may result from an incorrectly determined heat of evaporation, but most probably from the fact that \(\mathrm{Be}^{++}\) has coordination number 4, being the only representative of this kind among divalent ions.
As was to be expected, the eighteen-electron shell and the paramagnetic ions do not obey the simple laws for ions of the noble-gas type and have much higher heats of hydration. This agrees perfectly with Goldschmidt’s point of view, according to which such ions possess an enhanced polarizing action caused by the interaction of electrons, which is facilitated by low-lying excited states. It is extremely interesting to note that, as far as can be judged, the points for \(\mathrm{Mn}^{++}\), \(\mathrm{Fe}^{++}\), \(\mathrm{Co}^{++}\), \(\mathrm{Ni}^{++}\), and for \(\mathrm{Zn}^{++}\), \(\mathrm{Cd}^{++}\) (but not for \(\mathrm{Hg}^{++}\), which, in all probability, is not fully ionized) lie
TABLE 8
Heats of solution of ions in the Kalit ion
| Ion | Ionization energy* | \(U_h\) observed | \(U_h\) calculated |
|---|---|---|---|
| \(\mathrm{H}^{+}\) | 313 | 276 | — |
| \(\mathrm{Li}^{+}\) | 124 | 136 | 131 |
| \(\mathrm{Na}^{+}\) | 118 | 114 | 116 |
| \(\mathrm{K}^{+}\) | 99.6 | 94 | 92 |
| \(\mathrm{Rb}^{+}\) | 95.9 | 87 | 87 |
| \(\mathrm{Cs}^{+}\) | 89.4 | 80 | 79 |
| \(\mathrm{NH}_{4}^{+}\) | 87 | 87 | |
| \(\mathrm{OH}_{3}^{+}\) | 130 | ||
| \(\mathrm{OH}^{-}\) | 105 | ||
| \(\mathrm{F}^{-}\) | 98.5** | 97 | 94 |
| \(\mathrm{Cl}^{-}\) | 92.5** | 65 | 67 |
| \(\mathrm{Br}^{-}\) | 87.1** | 57 | 63 |
| \(\mathrm{J}^{-}\) | 79.2** | 47 | 49 |
| \(\mathrm{Be}^{++}\) | 633 | 608 | \(\approx 600\) |
| \(\mathrm{Mg}^{++}\) | 521 | 490 | 495 |
| \(\mathrm{Ca}^{++}\) | 413 | 410 | 410 |
| \(\mathrm{Sr}^{++}\) | 384 | 376 | 370 |
| \(\mathrm{Ba}^{++}\) | 349 | 346 | 350 |
| \(\mathrm{Al}^{+++}\) | 1220 | 1149 | 1149*** |
| \(\mathrm{Sc}^{+++}\) | 1020 | 980 | |
| \(\mathrm{V}^{+++}\) | 910 | 830 | |
| \(\mathrm{La}^{+++}\) | 833 | 768 | 768*** |
| \(\mathrm{Th}^{++++}\) | \(\approx 1600\) | \(\approx 1540\) | 1540*** |
| \(\mathrm{Ag}^{+}\) | 174 | 162 | |
| \(\mathrm{Tl}^{+}\) | 140 | 107 | |
| \(\mathrm{Mn}^{++}\) | 534 | 479 | |
| \(\mathrm{Fe}^{++}\) | 561 | 500 | |
| \(\mathrm{Co}^{++}\) | 580 | 504 | |
| \(\mathrm{Ni}^{++}\) | 594 | 516 | |
| \(\mathrm{Cu}^{++}\) | 645 | 536 | |
| \(\mathrm{Zn}^{++}\) | 626 | 528 | |
| \(\mathrm{Cd}^{++}\) | 596 | 462 | |
| \(\mathrm{Hg}^{++}\) | 672 | 480 | |
| \(\mathrm{Fe}^{+++}\) | \(\approx 1346\) | 1185 | |
| \(\mathrm{In}^{+++}\) | 1210 | 980 |
* Energy required for the successive removal of the corresponding number of electrons.
** Electron affinity.
*** Used for determining \(p_z\) and \(R_z\).
are arranged on two straight lines, almost parallel to the line for ions of the noble-gas type. They can be brought into agreement with the general formula by adding to \(P\) a constant term \(P_{\mathrm{mag}}\), \(4\ \mathrm{Cal}/g\text{-ion}\) for the first group and \(P_{18}=8\ \mathrm{Cal}/g\text{-ion}\) for the second group.
If we had more data concerning the heats of formation and solution, this might help reveal further regularities, especially for rare-earth ions.
A more general and almost obvious conclusion can be drawn from consideration of the heats of hydration of ions. The heat of hydration is never much less than the ionization potential of the ion. Physically this means that the function of the ionizing medium consists simply in returning to the ion its missing electrons. This is effected, first of all, by the coordination shell, which for all multivalent ions provides a large part of the hydration energy. We now understand the reason for the stability, in the liquid and solid states, of ionic hydrates such as \(\mathrm{Be(OH_2)_4^{++}}\) or \(\mathrm{Al(OH_2)_6^{+++}}\), which may be regarded as complex ions with the same justification as \(\mathrm{PtCl_4^{--}}\) or \(\mathrm{Fe(CN)_6^{4---}}\).
§ 10. Influence of dissolved ions on the structure of water
The influence of ion hydration on the properties of the water in which they are dissolved was shown in Fajans’ work on refraction in ionic solutions. He showed that the effect on refractive power is additive for ions, but that, while the refraction of some ions in solution is similar to the refraction of the free ion or of the solid crystal, other ions—namely those which, as we have shown, are hydrated (for example \(\mathrm{Li^+}\), \(\mathrm{Ca^{++}}\), \(\mathrm{Al^{+++}}\))—give negative values, which can be explained only by a decrease in the refractive index of the solvent water as a result of the coordination of water molecules around these ions.
The simplest method of investigating the action of ions on the dissolving water is connected with consideration of the degree of disordering of the water molecules. This becomes obvious when one considers the diffraction of X-rays and the Raman spectra of solutions \(^{24}\). In both cases the addition of such ions as \(\mathrm{Li^+}\) or \(\mathrm{H^+}\) causes an increase in the sharpness of the lines and shifts of intensity corresponding to a more regular arrangement, i.e. in the direction: water I \(\to\) II \(\to\) III. The observations are not sufficiently numerous or reliable to permit a quantitative calculation, but they give quite convincing indications that small ions increase the regularity of the arrangement of water molecules.
The same effect can be observed when considering the viscosity of ionic solutions. The viscosity of a solution is a complex effect, but its
its nature becomes clear when considering the viscosity–concentration curve of CsCl in Fig. 15. For dilute solutions at low temperatures we first observe a drop in viscosity with increasing concentration, subsequently replaced by an increase. At higher temperatures we do not notice this decrease, which is not observed at all temperatures for salts with small ions, such as, for example, LiCl and CaCl₂. One may imagine that the viscosity of ionic solutions is determined by three factors:
a) Ions, regarded as independent massive particles, transfer momentum from one part of the liquid to another by means of their Brownian motion. This term will be proportional to the concentration, since the mean free path does not depend on the concentration and, probably, is almost independent of the temperature, as in the case of gases.
Fig. 15. Viscosity of CsCl solutions.
b) However, the ions are not fully independent, and the lattice formed by them, according to the Debye–Hückel theory, offers a certain resistance to the shearing force, which is manifested in the form of viscosity. One may expect this effect to increase proportionally to a high power of the concentration, but to decrease exponentially with increasing temperature.
c) Ions act upon the water molecules located in their vicinity, weakening or strengthening the structure, as was shown above. A looser structure caused by unhydrated ions gives a viscosity below that of pure water; a denser structure caused by hydrated ions gives a higher viscosity.
One may suppose that this term is in linear dependence on concentration for low concentrations and is to a large extent independent of temperature.
On this basis, for the viscosity of an ionic solution of a salt \(AX\), one may write (η, denoting viscosity, should not be confused with the η used by us above):
$$ \eta(AX, C, T)=\eta(\mathrm{aq}, T)-f_{(a)}(M_A, M_X) C T_s - f_{(b)}(C,z)e^{-\frac{z}{r}} - $$
$$ - f_{(c)}(r_A,r_X,z) C, $$
where $\eta(AX, C, T)$ is the viscosity of the solution $AX$ at concentration $C$ and temperature $T$; $\eta(\mathrm{aq}, T)$ is the viscosity of water at the same temperature; $f_{(a)}(M_A, M_X)$ is the Brownian coefficient, depending mainly on the masses of the ions and always being a positive quantity; $f_{(b)}(C,z)$ is the Debye–Hückel function, depending on the charge of the ions and on the concentration, likewise always positive; $f_{(c)}(r_A,r_X,z)$ is a function of the “structural temperature,” depending on the radius and charge of the ions and capable of having both positive and negative values.
For low temperatures and weak concentrations the principal term will be $f_{(c)}$; for high concentrations, $f_{(b)}$, and for high temperatures, $f_{(a)}$. This qualitative hypothesis gives a complete explanation of the shape of the curve for CsCl. As regards $f_{(c)}$, the addition of a hydrated or nonhydrated ion is equivalent to a lowering or raising of the temperature. Developing this idea further, we arrive at the conception of the structural temperature of an ionic solution, which may be defined as the temperature at which pure water will have the same internal structure and the same viscosity (X-ray diffraction, Raman spectra, etc.). It goes without saying that this applies only to dilute solutions of strong electrolytes, in which one may neglect the interaction of the ions and the amount of motion transferred by means of their Brownian motion. Both these factors tend to increase the viscosity at high concentrations, while the latter is the predominant factor at high temperatures.
The change in the structural temperature of the solvent is most clearly seen from the action of ionic solutions on the temperature of the maximum density of water. The effect always consists in a lowering of the temperature of maximum density, i.e., in an increase of the structural temperature, but it is much smaller for the chlorides of small ions, $\mathrm{H}^+$, $\mathrm{Li}^+$, $\mathrm{Mg}^{++}$, than for the chlorides of large ions. Obviously, in this case the increase is caused by $\mathrm{Cl}^-$ ions, while the positive ions either do not participate at all or lower the structural temperature. Of course, the change in the temperature of maximum density cannot be quantitatively identified with a change in structural temperature, but we note with satisfaction that both change in the same direction.
This conception of structural temperature may prove important in many fields of physical chemistry, especially those related to colloid chemistry, and also for many biochemical problems. With the aid of this conception the reason becomes clear for the antagonistic action of such ions as $\mathrm{Ca}^{++}$ and $\mathrm{K}^+$, of which the former lowers and the latter raises the structural temperature. However, we do not attempt to give this a precise
of physical significance. Only as a first rough approximation can the effect of bringing an ion into a solution be reduced to a change in temperature. The actual effect is not uniform, but gradually weakens with increasing distance from the ion and requires a much more detailed theoretical treatment than that which can be based on the crude model we used above.
§ 11. More on ionic mobilities
The theory of an ionic solution sketched above, although it refines many of our ideas, nevertheless gives nothing essentially new for solving the problem of ion mobility. Starting from the calculated hydration energy or, still better, from the observed heats of solution, one can construct a hydrodynamic theory of the motion of an ion at infinite dilution. Such a theory would clarify the obvious qualitative relations between high hydration energy and low ionic mobility; however, in the present work we do not attempt to derive such a theory, since it cannot help us in solving the task before us.
The study of the molecular structure of water and ionic solutions has shown that there is no hope of explaining in the usual way the high mobilities of the ions \(\mathrm{H}^{+}\) and \((\mathrm{OH})^{-}\). If \(\mathrm{H}^{+}\) and \((\mathrm{OH})^{-}\) are treated empirically like other ions, then, judging from the densities of their solutions, they turn out to be hydrated (see Tables 4 and 5). On the basis of this hydration they may be assigned small mobilities, approaching the mobility of \(\mathrm{Li}^{++}\) ions. But other properties, especially viscosity, show that \(\mathrm{H}^{+}\) and \((\mathrm{OH})^{-}\) do not behave normally. The influence of \(\mathrm{H}^{+}\) and \((\mathrm{OH})^{-}\) ions on the structural temperature is very large and negative. This shows that here we are dealing with a mechanism more complex than simple hydration, allowing \(\mathrm{H}^{+}\) and \((\mathrm{OH})^{-}\) ions to move rapidly in solution and at the same time imparting greater cohesion to it.
The second part of this work is devoted to an attempt to describe such a mechanism and to explain with its help the latter observed anomalous mobilities. We think that the explanation proposed by us is the only possible one, but only in the case where water for the most part possesses the structure described in the first part.
Part II. Quantum-mechanical theory of the exceptional mobility of \(\mathrm{H}^{+}\) and \((\mathrm{OH})^{-}\) in water
§ 12. Refinement of the statement of the problem
After the considerations presented in the first part of this work, one cannot agree with the point of view which holds that positive hydrogen ions are present in aqueous solution in the form of
naked protons. The latter must be firmly bound, at least to one molecule of water, and we can continue our consideration on the assumption that the positive hydrogen ion is present in solution in the form \((\mathrm{OH}_3)^+\), the oxonium ion, which, like the \((\mathrm{OH})^-\) ion and other small ions, is more or less hydrated. The additional proton in \((\mathrm{OH}_3)^+\) is undoubtedly placed in one, or adjacent to one, of the two free tetrahedral positions among the electron orbitals, the whole structure very much resembling the structure of the ammonia molecule \(\mathrm{NH}_3\).
If we take this point of view, for the acceptance of which quite simple energetic considerations are sufficient, the exceptional mobility of these ions becomes still more striking. If the mechanism of transport of this ion through the solution were the same as for other ions, which necessarily entails the material transport of identical ions in the direction of the corresponding electrode, then it would be incredible for its mobility to differ significantly from the mobility of the ammonium ion \((\mathrm{NH}_4)^+\), to which it is so similar in its electronic structure. But the observed mobility of \((\mathrm{OH}_3)^+\) exceeds the mobility of \((\mathrm{NH}_4)^+\) fivefold. As above, we conclude that in the present case, both for \((\mathrm{OH}_3)^+\) and for \((\mathrm{OH})^-\) (ions of water in water), we are dealing with a special mechanism, distinctly different from material transport. This point of view, which will be developed below, leaves unchanged the former theories of transport of any extraneous ion through any solution. The question of whether special mobility should be expected from ions of other solvents in these solvents will be considered below. We shall see that special conditions must be satisfied, so that exceptional mobilities should not occur often and, in any case, must be limited to ions whose effective charges are caused by an additional or missing proton.
An interesting confirmation of our point of view is provided by the mobility of \((\mathrm{OH}_3)^+\) in hydrofluoric acid. Pure dry hydrofluoric acid does not conduct electricity. Water present as an impurity acts as an electrolyte, causing the formation of \((\mathrm{OH}_3)^+\) and \(\mathrm{F}^-\) ions, of course in a more or less solvated state. But \((\mathrm{OH}_3)^+\) in liquid \(\mathrm{HF}\) has a perfectly normal mobility.
In the present work we attempt to describe the actual mechanism and its relation to the structure of water, and we think that we have succeeded in doing this, at least from a qualitative point of view. The basic idea of the theory is taken from quantum mechanics.
If one recognizes the existence of \((\mathrm{OH}_3)^+\) structures in solution, then from the point of view of quantum mechanics it will immediately become clear that the ion
\((\mathrm{OH}_3)^+\), being in sufficiently close contact with an \(\mathrm{OH}_2\) molecule, may fail to retain its excess proton and may transfer it to the other molecule. The proton jumps back and forth from one molecule to the other in the case of a favorable configuration of both molecules, and we can show that this leads to a noticeable displacement in the direction of the field.
In formulating this problem, we are merely repeating ideas which at the present time have become commonplace for everyone studying chemical physics. Presenting our solution of this problem in its quantum-mechanical garb, we are only trying to express in a more exact form ideas familiar to everyone in a more general form. These ideas in fact are of very respectable age, since in essence we are reviving, in a more modern form, the idea of Grotthuss chains. Not long ago Hückel \({}^{25}\), in a work with which we were not acquainted when we began the present investigation, expressed the same idea of proton jumps. However, he did not attempt to calculate the frequency of these jumps and developed his argument in such a way that it coincides with ours only to a slight extent. On the quantitative side we have not advanced very far; a quantitative explanation, perhaps, is not yet possible at present, if one takes into account the probable structure of water itself. But our arguments, like Hückel’s, allow one to draw a conclusion as to the correctness of the orders of magnitude and to recognize that proton jumps are the basis of the exceptional mobility. The other exceptional mobilities known to us apparently do not contradict this explanation.
§ 13. Mechanism of the transfer of \((\mathrm{OH}_3)\) and \((\mathrm{OH})^{-}\)
We have already mentioned that, according to quantum mechanics, an \((\mathrm{OH}_3)^+\) ion in sufficiently close contact with a water molecule oriented in a suitable way should not retain its excess proton. The reason for this consists, of course, in the existence of another possible configuration, possessing equal energy, in which the extra proton has exchanged its molecule for another. Then one may assume that the three protons in \((\mathrm{OH}_3)\) are in equivalent states, or that there is exchange degeneracy among them, so that each of them is capable, at least in turn, of playing the role of the jumping proton. If the two systems are situated far from one another or are in an unsuitable relative orientation, then the proton must pass through a considerable barrier by means of the tunnel effect, which occurs very rarely. The large mass of the proton makes it incapable of passing through barriers through which electrons pass freely. But when both systems are in close contact, as neighbors in water, and if they are oriented in such a way that one of the protons (or the jumping proton) of the \((\mathrm{OH}\)
if it is located approximately opposite a certain position in \(\mathrm{OH}_2\) into which it must fall when changing partners, the distortion of the electronic orbits may be such that only a small barrier remains, or the latter may almost completely disappear; in both cases the proton can move fairly freely from one system to the other during the period of this close contact.
In the absence of any external field we may conclude from symmetry considerations that, if the favorable mutual arrangement ceases, the excess proton may be found with equal probability in either of the two initial systems. Such an exchange will lead to a random migration of free protons through the solution, and consequently also of the apparent system \((\mathrm{OH}_3)^+\), with no pure jump being observed. From this point of view, of course, it will not be the very same proton that moves, and still less the same system \((\mathrm{OH}_3)^+\). But if we have an applied external field \(F\), then the probability that, after separation, the proton will prove to be bound to one of the two \(\mathrm{OH}_2\) systems changes slightly by an amount proportional to \(F\) (since the effect is not large), preference being given to the \(\mathrm{OH}_2\) system lying in the region of lower potential energy for the proton. We shall prove this rigorously for the simple model described in § 14. Then we shall have a stationary flux of apparent \((\mathrm{OH}_3)^+\) ions along the potential gradient with a velocity proportional to \(F\), which is precisely their observed excess mobility.
A very similar mechanism should also hold for \((\mathrm{OH})^-\). With a favorable position of \((\mathrm{OH})^-\) and another \(\mathrm{OH}_2\), the proton can again separate, producing an exchange between the ion and the neutral system. It is evident that the two hydrogens in \(\mathrm{OH}_2\) will lie somewhat deeper in the electronic system than the three hydrogens in \((\mathrm{OH}_3)^+\), and therefore will have a somewhat smaller possibility of exchange. Hence the lower mobility.
Apparently it is impossible to prove a priori that this proton transition must occur with the speed required by our theory in the case of a favorable configuration, but only that under these conditions such a transition in general can take place. However, it seems to us that in this case we may reason in reverse and assert, first, that the proposed transfer mechanism is an indisputably convenient working hypothesis, and, second, that apart from the material transport of the \((\mathrm{OH}_3)^+\) complex through the medium there is no other mechanism capable of imparting a noticeable mobility to the apparent hydronium ions; and, as is known, such material transport is too slow. Therefore we may conclude that the observed velocities establish a sufficient rate of transfer.
§ 14. Approximate calculation
The form of the approximate calculations necessary for calculating the transitions of a proton from an \((\mathrm{OH}_3)^+\)-ion to a neighboring \(\mathrm{OH}_2\) molecule was given several years ago by McCree \(^{26}\) in an attempt at a theoretical calculation of electronic conductivity. His work may be adopted with very slight corrections and represents just the type of calculation suitable for the process considered by us.
Two water molecules, situated in a suitable configuration, represent two adjacent possible sites for the proton. Let the wave equation for the proton in one of these positions in the isolated state be
\[ \nabla^2 \chi+\kappa^2\{E-U(r_1)\}\chi=0 \qquad \left(\kappa^2=8\pi^2\frac{m_{\mathrm H}}{h^2}\right) \tag{7} \]
with the normalized solution \(\chi=f(r_1)=\psi,\ E=E_0\) for the lowest energy state. We shall neglect the effect of excited states, which may be justified a posteriori. When we have two positions on neighboring molecules, we shall assume that the wave equation is given, with a sufficient degree of approximation, by the expression:
\[ \nabla^2\chi+\kappa^2\{E-U(r_1)-U(r_2)\}\chi=0 \tag{8} \]
with two solutions (in the zeroth approximation):
\[ \chi=f(r_1)=\psi;\qquad \chi=-f(r_2)=-\varphi;\qquad (E=E_0). \]
The solution \(\psi\) means that the proton is on molecule 1, and the solution \(-\varphi\) that it is on molecule 2. With the aid of the usual proofs, the two lowest stationary states of this system may be determined approximately by means of the wave functions:
\[ \chi_{\pm}=\frac{\psi\pm\varphi}{q_{\pm}},\qquad E=E_0-e_{\pm}, \tag{9} \]
where
\[ q_{\pm}=\sqrt{2(1\pm\varphi\psi)},\qquad e_{\pm}=\frac{U\pm U^*}{1\pm\varphi\psi} \tag{10} \]
\[ (\varphi\psi)=\int \varphi\psi\,d\tau,\qquad U=\int U(r_2)\psi^2\,d\tau,\qquad U^*=\int U(r_2)\varphi\psi\,d\tau. \tag{11} \]
\(\int \ldots d\tau\) denotes integration over the entire region of configuration of the proton. From the symmetrical form of these functions one may conclude that, for the proton in both stationary states, both positions are equally probable.
Let the electromotive force \(F\) act on the proton in the direction of increasing \(x\), which for the time being we shall identify with the direction from the position \(\psi\) to the position \(\varphi\). Assuming that the origin of \(x\) lies midway between these two positions, we may represent the perturbing potential energy for the proton as \(-eFx\). It may be assumed that this perturbation is insignificant in comparison with the perturbation arising from the interaction of the two positions, and therefore it may be applied as a new perturbation to both states analyzed in (9)—(11). But the perturbation caused by the interaction is itself so small that, in calculating the new perturbation, it is sufficient to take into account only these two nearby states. Then the perturbed wave equation will take the form:
\[ \nabla^2\chi+\chi^2\{E-U(r_1)-U(r_2)+eFx\}\chi=0, \tag{12} \]
and the perturbed solutions:
\[ \begin{aligned} \chi_+^*&=\chi_+-\zeta\chi_- \qquad (E=E_0+e_+),\\ \chi_-^*&=\chi_- - \zeta\chi_+ \qquad (E=E_0+e_-); \end{aligned} \tag{13} \]
where
\[ \zeta=eF\int \frac{x\chi_+\chi_-\,d\tau}{e_- - e_+}. \tag{14} \]
There are no first-order changes in the energies. We may proceed further in the calculation of \(\zeta\), since
\[ \int x\chi_+\chi_-\,d\tau = \int \frac{x(\psi^2-\varphi^2)\,d\tau}{q_+q_-}. \]
From the assumed symmetry of the field \(U(r)\), and hence also of the wave function \(\psi\) (or \(\varphi\)) of the principal fundamental mode, it follows that
\[ \int x\psi^2\,d\tau=-\frac{1}{2}R, \qquad \int x\varphi^2\,d\tau=\frac{1}{2}R, \]
if \(R\) is the distance between the centers of the two positions. Thus:
\[ \zeta=-\frac{eFR}{q_+q_-(e_- - e_+)}. \tag{15} \]
Consequently, the general wave function for the perturbed system with time factors will be:
\[ \chi = \alpha(\chi_+-\zeta\chi_-) e^{-\frac{2\pi i(E_0+e_+)t}{h}} + \beta(\chi_-+\zeta\chi_+) e^{-\frac{2\pi i(E_0+e_-)t}{h}}. \tag{16} \]
…be in the position \(\varphi\). These [[unclear: remainder cut off]]
where \(\alpha\) and \(\beta\) are constants. This represents the change of the system in time. If \(\alpha\) and \(\beta\) are chosen in such a way that \(\chi=\psi\) for \(\tau=0\) represents a proton initially situated in the position \(\psi\), then equation (16) shows how the latter passes into the position \(\varphi\), and in what manner it is, on the average, distributed between these two positions. We may apply either of the two methods. We shall compute the mean distribution starting from a proton in \(\psi\) (or \(\varphi\)), and then derive from this the probability of the transition from \(\psi \to \varphi\) (or \(\varphi \to \psi\)) found after separation. This is sufficiently accurate, provided that the state may be assumed to be sufficiently long-lived, i.e. over a time interval comparable with half an oscillation, expressed by equation (16).
For this purpose we express equation (16) in terms of \(\psi\) and \(\varphi\) and compute \(\overline{|\chi|^2}\), the time-mean value of \(|\chi|^2\), obtaining the equation:
\[ \overline{|\chi|^2} = \alpha^2 \left\{ \psi\left(\frac{1}{q_+}-\frac{\zeta}{q_-}\right) + \varphi\left(\frac{1}{q_+}+\frac{\zeta}{q_-}\right) \right\}^2 + \beta^2 \left\{ \psi\left(\frac{1}{q_-}+\frac{\zeta}{q_+}\right) + \varphi\left(\frac{1}{q_-}-\frac{\zeta}{q_+}\right) \right\}^2 . \tag{17} \]
We retain only terms up to the first order in \(\zeta\). In order that \(\chi\) may be reduced to \(\psi\) at \(t=0\), we must have:
\[ \alpha=\frac{1}{2}\left(q_+-\zeta q_-\right), \qquad \beta=\frac{1}{2}\left(q_-+\zeta q_+\right). \]
With these values of \(\alpha\) and \(\beta\),
\[ \overline{|\chi|^2} = \frac{\psi^2}{2} + \frac{\varphi^2}{2} \frac{1+2\zeta\left(q_+^2-q_-^2\right)}{q_+q_-} - \frac{\varphi\psi\zeta}{q_+q_-} \left(q_+^2+q_-^2\right). \tag{18} \]
The term \(\varphi\psi\) is symmetric and represents an equal probability of finding the proton in either of the two positions. Thus we have a probability equal to \(\dfrac{1}{2}-\vartheta\) that the proton may be in the position \(\psi\), and equal to
\[ \frac{1}{2}-\vartheta+\frac{\zeta\left(q_+^2-q_-^2\right)}{q_+q_-}, \]
that it may be found [[unclear: continuation cut off at bottom of page]].
to be in the position \(\varphi\). These probabilities, when added, give 1, so that
\[ \psi = \frac{\dfrac{1}{2}\,\xi\left(q_+^2-q_-^2\right)} {q_+q_-} \]
which can be checked by direct calculation. Thus we have the probability that the proton remains at \(\psi\), equal to
\[ \frac{1}{2}\left(1-\frac{\xi\left(q_+^2-q_-^2\right)}{q_+q_-}\right), \]
and at \(\varphi\), equal to
\[ \frac{1}{2}\left(1+\frac{\xi\left(q_+^2-q_-^2\right)}{q_+q_-}\right). \]
Consequently, we have a probability equal to
\[ \frac{1}{2}\left(1+\frac{\xi\left(q_+^2-q_-^2\right)}{q_+q_-}\right), \tag{19} \]
that, upon destruction of the configuration, the proton will have made a jump downward along the applied field, if it emerges from a molecule located higher in the field, and a probability equal to
\[ \frac{1}{2}\left(1-\frac{\xi\left(q_+^2-q_-^2\right)}{q_+q_-}\right), \tag{20} \]
that it will have jumped upward along the field, in the case where it emerges from a molecule located lower in the field.
Each such jump is equivalent to a displacement by a distance \(\Delta\), where \(\Delta\) is the mean distance between neighboring \(\mathrm{H_2O}\) molecules in water*. For a large number \(N\) of such jumps, half will begin with the upper position of the proton and half with the lower position of the proton in the field. As a result of these \(N\) events the oxonium ion will have effectively moved in water downward along the field by a distance \(D\), given by the equation:
* It must be admitted that, in this simple jump, the effective charge advances not by the distance \(\Delta\), but, of course, only by the distance \(R\); motion to \(A\) would be consolidated only if dipole regrouping by thermal motion occurred sufficiently often. As was pointed out to us in discussion by Ya. I. Frenkel, this thermal reorientation, which we have so far not mentioned for the sake of simplicity, must be an inalienable part of the complete description of the phenomenon. We return to this aspect of a more exact theory at the end of § 18.
\[ D=\frac{1}{2}N\frac{1}{2}\left(1+\frac{\xi(q_+^2-q_-^2)}{q_++q_-}\right)\Delta -\frac{1}{2}N\frac{1}{2}\left(1-\frac{\xi(q_+^2-q_-^2)}{q_++q_-}\right)\Delta = \]
\[ = N\Delta \xi\,\frac{q_+^2-q_-^2}{q_++q_-}. \tag{21} \]
If this is written out in full and \((\varphi\psi)^2\) is neglected in comparison with unity, this equation assumes the form:
\[ D=\frac{N\Delta}{4}\, \frac{eFR\int \varphi\psi\,d\tau} {-\int \varphi\psi\,U\,d\tau+(\int \varphi\psi\,d\tau)\int U(r_2)\psi^2\,d\tau}. \tag{22} \]
With the aid of this formula one can find the velocity of motion of the effective hydrogen ion in water, in cm/sec, under the influence of a potential gradient equal to unity, equivalent to its mobility.
§ 15. Numerical values
The value of the migration velocity for the effective hydrogen ion under the influence of a potential gradient1 equal to \(F^*\) V/cm is \(32.5\cdot10^{-4}\) cm/sec. The value for the hydroxyl ion is \(17.8\cdot10^{-4}F^*\). The value for the ammonia ion is the same as for the potassium ion, namely \(6.7\cdot10^{-4}F^*\). It may be expected that the oxonium ion and the hydroxyl ion possess an ordinary velocity of this order, \(6.7\cdot10^{-4}F^*\), caused by material transport through the liquid in the ordinary way. Consequently, they possess an additional velocity caused by the interaction mechanism:
\[ (\mathrm{OH}_3)^+:\;25.8\cdot10^{-4}\;F^*\,\mathrm{cm/sec}, \]
\[ (\mathrm{OH})^-:\;11.1\cdot10^{-4}\;F^*\,\mathrm{cm/sec}. \]
The exact numerical values are not important. They refer to \(291^\circ\) K. It remains to consider whether equation (22) can give comparable values when brought into correspondence with the conditions in solution.
-
The value \(\Delta\). \(\Delta\) represents the mean distance between \(\mathrm{H_2O}\) molecules in water in the case when they are capable of interacting in the indicated manner, one of them being the carrier of the additional proton. For this distance we should, in all probability, have taken the mean distance between coordinated neighboring molecules (see Part I), namely \(28\cdot10^{-18}\) cm.
-
The value \(F\). The effective value of \(F\) should have been calculated taking into account the dielectric constant of water. In constructing a theory of transport for ordinary ions, we must of course bear in mind that only the electric force is effective. We may imagine the motion of ions as taking place in a cavity having the shape of a needle, from which the polarizable material has been removed and in which the effective force is the electric
intensity corresponding to the gradient, in V/cm, applied between the electrodes. But it is obvious that in the transfers considered in §14, in which the entire ionic system as a whole remains immobile, the effective force acting on the hop (and this action has the nature of a polarization effect), according to Lorentz’s reasoning, is the force at the center of the spherical cavity surrounding the system, since the symmetry of the distribution around the immobile system leads to the fact that the matter inside the sphere adds nothing to it. From this it is not difficult to conclude that the effective value of \(F\) in equation (16) must be equal to:
\[ \frac{1}{3}\frac{(\eta+2)F^*}{300}, \]
where \(F^*\) is given in V/cm between the electrodes, and \(\eta\) is the dielectric constant of water. It goes without saying that the factor \(1/300\) converts volts to absolute electrostatic units.
- Values of the interaction integrals. The calculation of this last factor in equation (16) is more difficult and must first be carried out on a simple model. If we take a model in which each position is represented by a cavity with vertical walls and constant depth (outside this depression \(U=0\), inside the depression \(U=-B\)). Then, without calculating the wave function, we can see that approximately:
\[ \int \varphi\psi\,d\tau = 2\int_{\mathrm{cav}} \varphi\psi\,d\tau, \]
(actually this value is a lower limit), whereas
\[ \int U(r_2)\psi^2\,d\tau \]
is considerably smaller.
Thus for this model
\[ \frac{\int \varphi\psi\,d\tau}{-\int \varphi\psi\,U\,d\tau+\left(\int \varphi\psi\,d\tau\right)\int U(r_2)\psi^2\,d\tau} = \frac{2}{B}; \tag{23} \]
for any reasonably similar model we shall obtain a result of the same form, provided that for \(B\) a suitable mean depth of the cavity is taken. It is important to note that relation (23) does not depend to any appreciable degree on \(R\). As we shall soon see, the role of \(R\) lies not in its effect on the distribution of the proton between a pair of water molecules, but in its strong effect on the rate of establishment of the distribution.
- Transformation of equation (16) into three-dimensional form. Equation (22) refers to a one-dimensional world, in which hops can occur only upward or downward along the field. To transform it for the three-dimensional world, it is sufficient to assume that hops can occur in all directions, arranged
according to the laws of chance. For directions making an angle \(\theta\) with the applied field \(\Delta\) will be equal to \(\Delta \cos \theta\) and \(F^{\ddagger}\), \(F^{\ddagger}\cos \theta\). If \(N\) still denotes the rate at which occasions for jumps in all directions present themselves, then, in order to apply it to three-dimensional space, equation (16) must be multiplied by \(\overline{\cos^2 \theta}\), or by \(1/3\). Another result of the transition to three dimensions is that in three-dimensional space the action of an individual oxonium ion on the polarity of water molecules need not extend infinitely along the coordinated chain, but is broken by the formation of rings. This speaks in favor of the simple interpretation of the theory given here.
5. Values of \(R\) and \(B\). A pair of neighboring water molecules are at a distance of \(2.8 \cdot 10^{-8}\,\mathrm{cm}\) from one another, and in a free water molecule the protons are situated at a distance approximately equal to \(10^{-8}\,\mathrm{cm}\) from the oxygen nucleus. If we did not have a modification of neutral \(\mathrm{OH}_2\), the centers of both positions would be separated from one another by approximately \(0.8 \cdot 10^{-8}\). But such a modification is unavoidable, since the third proton will tend to increase the distance of all the protons from the center, while the polarization of the molecules by a proton close to their point of contact will promote a further displacement of the proton and a lowering of the barrier between the positions. Therefore the effective distance between the positions will be less than \(8 \cdot 10^{-9}\,\mathrm{cm}\) and may be taken approximately equal to \(6 \cdot 10^{-9}\,\mathrm{cm}\).
A satisfactory calculation of \(B\) presents difficulties. It is self-evident that the third proton is not bound as firmly as the second to a neutral \(\mathrm{OH}\) molecule in the formation of \(\mathrm{OH}_2\). But from consideration of the general magnitude of the energy participating in the process, it is clear that \(B\) will be rather of the order of magnitude of \(1\,\mathrm{V}\) than \(1/10\) or \(10\,\mathrm{V}\), and we shall take \(B = 0.75\), acknowledging that we may be allowing a considerable error. The choice of this order of magnitude is confirmed by some preliminary calculations based on the theory proposed by us.
Substituting these numerical values, we obtain:
\[ D = 1.04 \cdot 10^{-15} N F^{\ddagger}. \tag{24} \]
Let us note that this represents the observed value for \((\mathrm{OH}_3)+\) in the case \(N = 2.5 \cdot 10^{12}\), while the determination of the value of \(N\) presents difficulties.
6. Value of \(N\). \(N\) is the number of favorable configurations into which, as we may expect, an oxonium ion enters during one second, each of which gives the proton the possibility of jumping in some direction to a neighboring molecule within the limits imposed by the theory.
We shall begin by recalling that our calculations are based on the assumption that the duration of existence of a favorable configuration is, in the worst case, comparable with the duration
a proton jump between the two positions presented to it. It is therefore necessary to estimate this duration. For the model we have used, mentioned in paragraph 3 of this section, we obtained with sufficient accuracy:
\[ \varepsilon_- - \varepsilon_+ = 2B \int_{\mathrm{vol}} \psi\, d\tau . \]
For an order-of-magnitude estimate this may be taken to be equal to
\[ 2B e^{-\chi R'} = \bar E, \tag{25} \]
where in this model \(E_0\) \((<0)\) is the depth of the natural energy level below the zero energy outside the cavity, and \(R'\) is the width of the barrier. If we take cavities with radius equal to \(10^{-9}\ \mathrm{cm}\), then we obtain the approximate values \(E_0 = \dfrac{1}{2}\ \mathrm{V}\), and \(R' = 4 \cdot 10^{-9}\ \mathrm{cm}\).
From consideration of the time factors in equation (16) we see that the transition time \(\tau\) is equal to \(\dfrac{h}{2(\varepsilon_- - \varepsilon_+)}\). Consequently, from this:
\[ \tau \approx \frac{h}{4B} e^{\chi R' - E_0} = 1.3 \cdot 10^{-15} e^{7 \cdot 10^{?} R' - E_0} = \tag{26} \]
\[ = 1.3 \cdot 10^{-15 + 3.05 \cdot 10^{?} R' - E_0}. \tag{27} \]
Substituting the value of \(E_0\) for our model, we obtain:
\[ \tau \approx 6.5 \cdot 10^{-13}. \tag{28} \]
We see at once that this value is practically equal to \(\dfrac{1}{N}\), where \(N\) has the value required by equation (24). Hence we conclude that \(\tau\) and \(\dfrac{1}{N}\) are of the same order of magnitude. This allows us to continue our derivation further with greater accuracy than is possible by means of only an a priori quantitative determination of \(N\), since in reality we now see that the rate of migration is so great that it is determined by \(\tau\), and not by \(N\). The actual additional mobility will be precisely of the order of magnitude that may be expected in the case where the possibilities of jumping from one molecule to the next are infinitely frequent or, more exactly, if the possibilities for the jump of the ion \((\mathrm{OH}_3)^+\) present themselves at least as often as it can make use of them, while the time required for each jump,
will be of the order of magnitude of \(\tau\). Consequently, it remains only for us to consider how far this conclusion agrees with our conception of the structure of water.
§ 16. The value of \(N\) and the structure of water
Let us first consider what value \(N\) should have if water were a quasi-closely packed structure consisting of freely rotating molecules and ions. This is the most disordered arrangement possible in a liquid, and one corresponding fairly accurately to the actual structure of, say, liquid argon, and also to the actual structure of water in the case where the water molecules were not carriers of strong dipoles.
Such water molecules may, with a sufficient degree of accuracy, be regarded as classical isotropic rotators, whose moment of inertia \(I\) is approximately \(2\cdot 10^{-40}\ \mathrm{g\cdot cm^2}\). Their mean component of angular velocity \(\omega_x\) is given by the equation:
\[ \frac{1}{2} I\omega_x^2=\frac{1}{2}kT, \]
where \(k\) is Boltzmann’s constant. Therefore, at ordinary temperatures \(\omega_x=1.4\cdot 10^{13}\) rad/sec, while the resultant \(\omega\) is approximately \(2.5\cdot 10^{13}\) rad/sec, i.e. a velocity comparable with \(1/\tau\).
But each \((\mathrm{OH}_3)^+\) ion carries three protons, and each \(\mathrm{OH}_2\) molecule at least one vacant (for a proton) site, so that, in close packing, we should have twelve nearest neighbors around each ion. However, the sites are rather small, and the hits must be very accurate in order to be effective. Therefore, despite the large number of favorable possibilities during a single revolution, it is evident that successful orientation on each revolution is an exceedingly improbable event*.
* The number of favorable configurations per revolution can be approximately determined by the following argument.
During one complete revolution of an oxonium ion, each of the protons carried by the latter, \(P\) in number, describes small circles of radius \(\frac{1}{2}\Delta \sin\theta\), where \(\theta\) is the angle formed by the radius vector of the proton and the axis of rotation. The length of the path traversed, averaged over all axes of rotation, is \(\frac{1}{2}\pi\Delta\), where \(\Delta\), as before, is the diameter of the ion. Around each ion there are \(\chi\) nearest neighbors, and each of the \(\chi\) lines connecting the centers with these neighbors is a place at which a jump may occur if a proton on the ion and a vacant position on the molecule meet there simultaneously. The probability that during
Consequently, this conclusion is in extraordinarily satisfactory agreement with the views developed in the present article concerning the coordinated structure of water. In general, the protons are permanently arranged in such a way that jumps from one \(\mathrm{OH}_2\) carrier to the next can take place at a rate on the order of
\[ \frac{1}{\tau} \]
times per second, i.e., just often enough to explain the anomalous mobility. Our argument concerning an isolated pair of molecules applies, at least approximately, to any pair of molecules in a chain or in another coordinated structure in the presence of an additional proton.
§ 17. Hydroxyl Ion
The additional mobility of the ion \((\mathrm{OH})^{-}\) is approximately equal to one half of the additional mobility of the ion \((\mathrm{OH}_3)^{+}\). Obviously, it is necessary and possible to adopt the same general explanation of the special mobility for both ions. It remains only to determine whether a theoretical explanation can be given for the fact that the additional mobility of \((\mathrm{OH}_3)^{+}\) is greater than that of \((\mathrm{OH})^{-}\). Obviously, this follows from the theory, since both processes may be represented as follows: 1) transfer of \(\mathrm{H}^{+}\) from \((\mathrm{OH}_3)^{+}\) to \(\mathrm{OH}_2\); 2) transfer of \(\mathrm{H}^{+}\) from \(\mathrm{OH}_2\) to \((\mathrm{OH})^{-}\). However, the proton levels in \((\mathrm{OH}_3)^{+}\) will be located
during one revolution, one of the protons will be at a small distance \(\delta\) from one of the lines of centers, equal to the ratio of the area of a trajectory of width \(2\delta\), traversed by the proton, to the cross-sectional area of the sphere of oxonium, i.e.,
\[ \frac{1}{2}\,\frac{\pi \Delta 2\delta}{\pi \Delta^{2}}. \]
Hence the average number of such approaches per revolution is equal to:
\[ \frac{\chi P \delta}{\Delta}. \]
Such an approach can prove effective only if, at the required moment, we also have a free site. The probability that, at the required moment, a site will be located at a distance not greater than \(\delta'\) from the line of centers is equal to:
\[ \frac{\pi \delta'^{2}}{\pi \Delta^{2}}. \]
Combining this probability with the preceding calculation, we obtain that the average number of double approaches per revolution is equal to
\[ \frac{\chi P \delta \delta'^{2}}{\Delta^{3}}. \]
This, of course, is only a rough outline; exact calculations would have to correspond to such calculations for the triple collision in a gas. But the order of magnitude should be correct for the number of intimate approaches per revolution in which a transition can occur. If we take \(P = 3\), \(\chi = 12\), \(\delta = \delta' = 2.5 \cdot 10^{-9}\), \(\Delta = 2.8 \cdot 10^{-8}\ \mathrm{cm}\), then their number per revolution will be equal to \(1/40\).
are situated somewhat closer to the periphery of the molecule and, perhaps, are bound somewhat less tightly than the levels of the proton in \(\mathrm{OH}_2\), owing to the repulsive action of the additional proton. Turning to formulas (22) and (26) for \(D\) and \(\tau\), we see that small changes in \(R\) and \(B\) are possible which can have no effect on \(D\), except through their effect on \(\tau\). But small increases of \(R\) and \(-E_0\) increase \(\tau\) and, consequently, decrease \(N\); and this must be the predominating effect, since it occurs in the exponent of the mean value. Consequently, on the basis of the theory we should expect a smaller mobility for \((\mathrm{OH})^{-}\), which is also confirmed by observation.
§ 18. Effect of temperature
In the present work we do not intend to enter into a discussion of the theory of the temperature coefficient \(D\). We shall confine ourselves to indicating the general way in which the influence of temperature may show itself. First of all, in our theory in its present state the only factor directly dependent on temperature is \(\eta\)—the dielectric constant. \(\eta\) falls with increasing temperature, and the special mobility must likewise fall. (We assume that the structure of water continues to remain so organized as to present a large number of possibilities for proton jumps.) The special mobility is anomalous and does not increase as rapidly as the ordinary mobility, which is determined by the viscosity of water. But apparently it does not fall, and at ordinary temperatures increases with increase of the latter, which may be explained, on the basis of this simple interpretation of the theory, as being caused by a decrease of \(\tau\) for some fraction of the protons. This is possible because some protons are thermally excited into higher states and thus can make more rapid transitions. On the other hand, the thermal activation of motion may contribute to bringing some of the water molecules into unusually close contact with one another. Our purpose was only to show that, even for the largest possible values of \(\tau\), the transition may be sufficiently rapid to produce the special mobility in suitably coordinated water. It is also possible that the actual \(\tau\) is smaller and that the mobility depends partly on \(\tau\), and partly on the rate of thermal rearrangement, so that the observed temperature effect may appear at this point.
§ 19. Mobility of the isotope \(\mathrm{H}^2\)
We do not consider that the theory developed by us, or the details of the model proposed by us, should be regarded as entirely reliable. Indeed, the details of the model on which our arguments concerning mobility are based have been chosen rather unfavorably for
theory, in order to show that quantum-mechanical jumps must, in the worst case, be capable of giving the required additional mobility, provided there is a sufficient amount of permanent coordination in the structure of water. However, under the condition that the details are only approximately correct, we can show that the effective mobility of the isotope \((\mathrm{H}^2)\) must be much less than the mobility of ordinary \((\mathrm{H}^1)\). The only significant effect of the change in mass is manifested in \(\tau\). In \(\tau\) we have \(\chi'=\sqrt{2\chi}\), and also a small increase in \(-E_0\). This will turn the second part of the exponent in equation (27) into \(4.31\cdot 10^6 \sqrt{-E_0}\), so that \(\tau \approx 10^{-11}\) will become approximately 20 times larger. Therefore the additional mobility of \((\mathrm{H}^2)\) will be approximately 20 times smaller than that of \((\mathrm{H}^1)\), and hence may be completely omitted in comparison with the mobility by material transfer, even if the factor 20 proves to be somewhat exaggerated.
We arrive at the conclusion that the effective mobility of \((\mathrm{H}^1)\) is approximately five times greater than the mobility of \((\mathrm{H}^2)\) in an aqueous solution. This conclusion may be of significance in the theory of the separation of hydrogen isotopes by electrolysis. However, it should not be assumed that this is necessarily the predominant factor. Electrode processes, which will also proceed while favoring the more rapid removal of \((\mathrm{H}^1)\) from the solution, may prove to be more important.
With this we conclude the present article. It is clear that the ideas developed by us, if they are at all correct, will find many new applications in the same direction, especially in the field of physical and chemical applications of the hydrogen bond. However, the recent successes of Lewis and others in the field of concentrating \(\mathrm{H}^2\) in considerable quantities open up such broad possibilities for new experiments in this field that at present we consider it undesirable to publish further unverified speculations.
REFERENCES
- Mecke und Baumann, Phys. Ztschr. 33, 333, 1932.
- Mulliken, Phys. Rev. 41, 756. 1932.
- C. Beevers und H. Lipson. Ztschr. Kryst. 82, 297, 1932; 83, 123, 1932.
- A. M. Meyer, Ann. d. Phys. 5, 701, 1931.
- Steward, Phys. Rev. 35, 1426. 1930; 37, 9. 1931.
- Amaldi, Phys. Ztschr. 32, 914, 1931.
- Debye und Mecke, Phys. Ztschr. 31, 799, 1930; Prins. Ztschr. Physik, 56, 617, 1929.
- Bridgman, Proc. Nat. Ac. Sci. 47, 441, 1912.
- Adams, Proc. Roy. Soc. A. 128, 588, 1930.
- Martin, Miner. Mag. 21, 519, 1931.
- Debye, Polare Molekeln, ch. V, 1929.
- Cf. also Zwicky, Phys. Ztschr. 27, 271, 1926.
- Lorentz, Theory of Electrons, notes 54, 55.
- Debye, loc. cit., note 12, p. 122.
- Fuchs, Ztschr. f. physik. Chemie, vol. 14, 339, 1931.
- London, Ztschr. physikal. Chem., vol. 11, 222, 1930.
- Born, Ztschr. Physik. 75, 1, 1932.
- Lennard Jones, Proc. Roy. Soc. A. 112, 214, 1926.
- Debye, loc. cit., note 12, p. 7.
- Meyer and Helmholtz, Ztschr. Physik. 75, 19, 1932.
- J. Sherman, Chem. Rev. 11, 93, 1932.
- Fajans, Ztschr. Elektrochem. 34, 502, 1928; 34, 546, 1928.
- Webb, Journ. Am. Chem. Soc. 48, 2589, 1926.
- Canes and Vonkatenwaren, Ind. J. Phys. 4, 125, 1929.
- Hückel, Ztschr. Elektrochem. 34, 546, 1928.
- Mc. Crea, Proc. Camb. Phil. Soc. 24, 438, 1928.
- Lewis, System of Physical Chemistry 1, 212, 1918.
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