Abstract
This article aims to examine the principal aspects of the modern electrical theory of solutions. The necessity of the division adopted below into two chapters, the first of which considers thermodynamic phenomena and the second electrical conductivity, is dictated by the consideration that the passage of an electric current through a solution is an example of an irreversible process; consequently, the methods developed to explain the “thermodynamic” behavior of solutions prove inapplicable in the study of electrical conductivity.
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ELECTRICAL THEORY OF SOLUTIONS OF STRONG ELECTROLYTES.
B. N. Finkelstein, Leningrad.
The theory of dilute solutions, created by van’t Hoff, Arrhenius, and Planck, suffered an unexpected fiasco in explaining the “anomalous” behavior of the so-called strong electrolytes, i.e., those salts, strong bases, and acids which, from the point of view of this theory, are characterized by a large degree of electrolytic dissociation. After all the numerous attempts to cope with the difficulties that arose, without encroaching upon the foundations of the old conceptions, proved unsuccessful1, alongside the harmonious edifice of the old theory, which had rightly earned the name “classical,” the outlines of a new conception began to emerge, now known as the “electrical theory of solutions.”
The pioneer in this field was the Australian physicist W. Sutherland2, who as early as 1907 expressed a number of considerations forming the basis of the modern electrical theory of solutions. Advancing the hypothesis of the complete dissociation of electrolytes3, Sutherland assumed—and herein lay the original aspect of his views—that in solutions of fully dissociated electrolytes the principal role belongs to the electrical forces with which the ions of the dissolved substance act upon one another and upon the molecules of the solvent. In this way he anticipated the later ideas of Bjerrum, Kjellin, P. Hertz, Milner, I. Ch. Ghosh, and Debye.
The works of Kjellin, P. Hertz, and Ghosh are at present of historical interest only, and therefore we shall not dwell on them. With regard to Ghosh’s theory, let us note—
...that it had a wholly undeserved success, which can be explained only by the circumstance that, without making use of a complex mathematical apparatus and proceeding from a simple (but incorrect) physical hypothesis, this theory, quite by chance, made it possible under certain circumstances to obtain satisfactory agreement with experiment.
Niels Bjerrum (1909), on the basis of the results of investigations of the optical properties of solutions, limited the hypothesis of Sutherland by the assertion that only strong electrolytes dissolved in media with a high dielectric constant are completely dissociated even at high concentrations. Further, he was also the first to place on a firm footing a quantitative description of the deviations in the behavior of solutions of strong electrolytes from the laws established by the classical theory. The three coefficients introduced by him for this purpose—the osmotic coefficient \(g\), the conductivity coefficient \(f\), and the activity coefficient \(f\)—make possible a complete quantitative description, independent of any special molecular theory, of the effects caused by the electrical forces of interaction between ions. Thus, for example, the osmotic coefficient \(g\) is defined by means of the following relation:
\[ g = \frac{P}{\bar P}. \]
\(P\) denotes the true (measured) osmotic pressure of the solution; \(\bar P\) is the osmotic pressure calculated according to the van ’t Hoff formula under the assumption of complete dissociation of the electrolyte (degree of dissociation \(\alpha = 1\)).
The task of the electrical theory of solutions, in Bjerrum’s opinion, consists in revealing the functional dependence of these quantities on temperature, the dielectric properties of the solvent, the concentration of the electrolyte, valence, and other individual properties of the ions, etc.
The new theory owes its first success to Milner,\(^1\) who theoretically calculated the influence of the electric charges of ions on the osmotic pressure and the thermodynamically related lowering of the freezing point of solutions of binary uni-univalent electrolytes; in doing so, for highly dilute solutions good agreement with experiment was obtained.
Milner’s method, irreproachable in theoretical respects, presents great mathematical difficulties, which completely preclude the possibility of taking into account, with its aid, individual peculiarities—
\(^1\) Phil. Mag. (6), 23, 551, 1912; 25, 742, 1913; 35, 214, 352, 1918.
B. N. FINKELSTEIN
properties (for example, dimensions) of the ions; thus the field of application of this method remains very limited.
With the work of Debye and Hückel (E. Hückel), which appeared in 1923, a new period begins in the development of the electrical theory of solutions. Basing himself on the hypothesis of complete dissociation of strong electrolytes and assuming that, in the interaction of ions, the predominant role is played by the Coulomb forces of electrostatic attraction and repulsion, Debye developed a new method for the theoretical calculation of the above-mentioned Bjerrum coefficients, quite sufficient for an exhaustive description of the behavior of solutions of strong electrolytes.
The present article sets itself the task of considering the principal aspects of the modern electrical theory of solutions.
The necessity of the division adopted below into two chapters, of which the first considers thermodynamic phenomena and the second—electrical conductivity, is dictated by the consideration that the passage of an electric current through a solution is an example of an irreversible process; as a consequence, the methods worked out for explaining the “thermodynamic” behavior of solutions prove inapplicable in the study of electrical conductivity.
Finally, we note that in the present article questions connected with surface phenomena in solutions of strong electrolytes (Frumkin, Wagner, Wessel), as well as the Born theory of the mobilities of electrolytic ions (Born, Schwick), are not touched upon at all.
CHAPTER ONE.
THERMODYNAMIC PHENOMENA.
I. GENERAL THERMODYNAMIC THEORY.
§ 1. Ideal and real solutions. The thermodynamic potential of a real solution. Activity coefficients and activity potentials.
Before proceeding to the exposition of the foundations of the specifically “electrical” theory and to the theoretical calculation of the fundamental quantities characterizing the thermodynamic behavior of dilute solutions of strong electrolytes, let us first consider general thermodynamic relations that do not depend on any special molecular theory, and derive formulas, convenient in practical respects, for quantities that are subject to direct and exact measurement.
Let us call, following Debye1, an “ideal solution” such an imaginable system which, with respect to all its thermodynamic—
properties—as “osmotic” ones (osmotic pressure, lowering of the freezing point, lowering of the vapor pressure, etc.), as well as “chemical” ones (solubility, electromotive forces, etc.)—obeys the laws of the classical theory, with the sole difference from the latter that the number of particles corresponds to the state of complete dissociation of the dissolved substance. From the point of view of the thermodynamic theory of dilute solutions, the definition of an ideal solution given here is equivalent to the assertion that in the latter there exists no other interaction except the interaction of the molecules of the solvent with one another and with the particles of the dissolved substance1. Thus, a real solution could be transformed into an “ideal” one if it were possible in some way to destroy the actions caused by the electric charges of the ions, i.e., in other words, to neutralize the latter; in that case all three Bjerrum coefficients would take values equal to unity.
Choosing as independent variables the external pressure \(p\) and the absolute temperature \(T\), to describe the thermodynamic behavior of the solution we shall use the thermodynamic potential \(\Phi\) of Gibbs (W. Gibbs):
\[ \Phi = U - TS + pV \tag{1} \]
\[ d\Phi = -S\,dT + V\,dp \tag{1'} \]
(\(U\)—energy of the system, \(S\)—entropy, \(V\)—volume).
Denoting further by \(N_0\) the number of solvent molecules, and by \(N_i\) \((i = 1,2,\ldots,s)\) the number of dissolved particles of the \(i\)-th kind2, for the thermodynamic potential \(\overline{\Phi}\) of an ideal solution we obtain the following well-known expression:
\[ \overline{\Phi} = \sum_{i=0}^{s} N_i \bigl(\varphi_i + kT \log c_i\bigr) \tag{2} \]
The functions \(\varphi_i = \varphi_i(p,T)\) depend only on the temperature, external pressure, and the properties of the particles present in the solution, but not on their number;
\(k = 1.37 \times 10^{-16}\ \mathrm{erg}\cdot \mathrm{grad}^{-1}\) is the Boltzmann constant (Boltzmann),
\[ c_i = \frac{N_i}{\displaystyle\sum_{i=0}^{s} N_i} \]
is the concentration of particles of the \(i\)-th kind.
Deviations in the behavior of real solutions from ideal ones—in particular, the “anomalies” of strong electrolytes—in the first approxima-
...can be explained by the forces of interaction between the particles of the dissolved substance, due to the electric charges of the latter.
In view of the fact that the thermodynamic potential is a homogeneous function of the first degree in the variables expressing the number of molecules of the \(i\)-th kind (i.e., in all \(s\) quantities \(N\)¹), it may be represented by the formula:
\[ \Phi=\sum_{i=0}^{s} N_i \lambda_i, \]
\[ \lambda_i=\lambda_i(p,T,c_1,\ldots,c_s). \]
Thus, without entering at all into consideration of the nature of the forces of interaction between the ions, we may express the thermodynamic potential of a real solution by the formula:
\[ \Phi=\overline{\Phi}+\sum_{i=0}^{s} N_i w_i \tag{3} \]
or, referring to (2):
\[ \Phi=\sum_{i=0}^{s} N_i\left(\varphi_i+kT\log c_i+w_i\right), \tag{3'} \]
where \(w_i\) are as yet unknown functions of the external pressure, temperature, concentrations, and properties of the ions.
The equilibrium condition for a real solution when the number of molecules is changed will be written in the following form:
\[ (\delta\Phi)_{p,T}=0 \]
\[ (\delta\Phi)_{p,T} = \sum_{i=0}^{s}\frac{\partial\Phi}{\partial N_i}\delta N_i = \sum_{i=0}^{s} \left[ \varphi_i+kT\log c_i+w_i+ \sum_j N_j\frac{\partial w_j}{\partial N_i} \right]\delta N_i \]
(note that
\[ \sum_j N_j\frac{\partial\log c_j}{\partial N_i}=0 \]
by virtue of the obvious relation
\[ \sum_j c_j=1 \]
).
We shall call the activity coefficient of particles of the \(i\)-th kind the quantity defined by the formula:
\[ \log f_i=\left(w_i+\sum_j N_j\frac{\partial w_j}{\partial N_i}\right)\frac{1}{kT}; \]
¹ M. Planck. loc. cit.
further, without limiting the generality of our reasoning, let us put:
\[ w_i = kT \log h_i, \tag{4} \]
whence it follows:
\[ \Phi = \sum_{i=0}^{s} N_i \left[ \varphi_i + kT \log (c_i h_i) \right] \tag{3''} \]
\[ (\delta \Phi)_{p,T} = \sum_{i=0}^{s} \left[ \varphi_i + kT \log (c_i f_i) \right]\delta N_i \tag{5} \]
\[ \log f_i = \log h_i + \sum_j N_j \frac{\partial \log h_j}{\partial N_i} \tag{6} \]
(the quantities \(h_i\) are called “activity potentials”).
Finally, the condition of equilibrium may be written in the following form:
\[ (\delta \Phi)_{p,T} = \sum \mu_i \delta N_i, \tag{7} \]
where \(\mu_i = \varphi_i + kT \log (c_i f_i)\) is called the chemical potential of particles of the \(i\)-th kind (Gibbs). It differs from the corresponding quantity for an ideal solution only in that, instead of the true concentration \(c_i\), the “activity” \(c_i^{*} = c_i f_i\) appears here.
Let us also note that, in the case where each of the \(N\) molecules of the electrolyte dissociates into \(\nu_i\) ions of the \(i\)-th kind \((i = 1, 2, \ldots, s)\), it is expedient, along with the activity coefficients of the individual kinds of ions, to introduce the activity coefficient of the electrolyte \(f\), defined by the formula:
\[ \nu \log f = \sum_{1}^{s} \nu_i \log f_i, \tag{8} \]
where \(\nu = \sum_{1}^{s} \nu_i\).
§ 2. Osmotic properties of solutions (osmotic pressure, lowering of vapor pressure) and activity coefficients.
The reasoning of the preceding section is applicable to the case of equilibrium of two contacting phases. Let us begin with the consideration of the group of so-called “osmotic” phenomena, in which only the solvent molecules pass from one phase of the system under study into the other.
a) Osmotic pressure.
Suppose that the pure solvent is separated from the solution by a semipermeable membrane. For both parts (phases) of our system one may write the following expression:
\[ \text{(solution)}\quad \Phi=N_o\varphi_o+\sum_{1}^{s}N_i\varphi_i +kT\left\{N_o\log c_oh_o+\sum_{1}^{s}N_i\log c_ih_i\right\} \tag{9} \]
\[ \text{(pure solvent)}\quad \Phi'=N_o'\varphi_o' \tag{9'} \]
The condition of equilibrium
\[ \delta(\Phi+\Phi')_{p,T}=0 \tag{10} \]
must hold for all transfers of solvent molecules from one phase to the other satisfying the relation
\[ \delta N_o=-\delta N_o', \tag{10'} \]
whence it follows:
\[ \delta\Phi=\varphi_o\delta N_o +kT\left\{\log c_oh_o +N_o\frac{\partial\log(c_oh_o)}{\partial N_o} +\sum_{1}^{s}N_i\frac{\partial\log(c_ih_i)}{\partial N_i}\right\}\delta N_o \tag{11} \]
\[ \delta\Phi'=\varphi_o'\delta N_o'=-\varphi_o'\delta N_o \tag{11'} \]
By the definition of the activity coefficient we have:
\[ \log f_o=\log h_o+\sum_{o}^{s}N_i\frac{\partial\log h_i}{\partial N_o}; \]
therefore, taking into account that
\[ \sum_{o}^{s}N_i\frac{\partial\log c_i}{\partial N_i}=0, \]
we obtain:
\[ \varphi-\varphi_o'=-kT\left\{\log c_o+\log h_o+\sum_{o}^{s}N_i\frac{\partial\log h_i}{\partial N_i}\right\} =-kT\log c_of_o \tag{12} \]
The difference between the potentials of the solution and of the pure solvent, separated by a semipermeable membrane and under identical external conditions, is due to the presence of osmotic pressure:
\[ P=p-p'. \]
Thus the quantity \(\Delta\varphi=\varphi_o-\varphi_o'\) is a function of \(P\) and may be expanded in a Taylor series:
\[ \varphi_o=\varphi_o' + \left(\frac{\partial\varphi_o'}{\partial p}\right)_T P + O(P^2), \]
limiting ourselves in this expansion to the first approximation and taking (12) into account, we obtain:
\[ P\left(\frac{\partial\varphi_o'}{\partial p}\right)_T=-kT\log c_o f_o. \]
On the other hand, from relation \((9')\) it follows that:
\[ \varphi_o'=\frac{\Phi'}{N_o'};\quad \left(\frac{\partial\varphi_o'}{\partial p}\right)_T = \frac{1}{N_o'}\left(\frac{\partial\Phi'}{\partial p}\right)_T = \frac{V}{N_o'}=v_o \]
(the volume of one molecule of solvent).
Thus
\[ Pv_o=-kT\log c_o f_o \]
or
\[ P=-\frac{1}{v_o}kT\log c_o f_o = -\frac{1}{v_o}kT\log\left[\left(1-\sum_1^s c_i\right)f_o\right] = -\frac{1}{v_o}kT\log\left(1-\sum_1^s c_i\right) \]
\[ -\frac{1}{v_o}kT\log f_o \simeq \frac{1}{v_o}kT\left(\sum_1^s c_i-\log f_o\right). \]
For an ideal solution \(f_o=1\) and, consequently,
\[ \overline{\Phi} = -\frac{1}{v_o}kT\log c_o = -\frac{1}{v_o}kT\log\left(1-\sum_1^s c_i\right) \simeq \frac{1}{v_o}kT\sum_1^s c_i. \]
By the definition of the osmotic coefficient \(g\), we have:
\[ P=g\overline{P} \]
or
\[ g=1-\frac{1}{\sum_1^s c_i}\log f_o \tag{12} \]
b) Lowering of the elasticity of vapor over a solution\({}^{1}\).
Assuming that the molecular volume of the liquid may be neglected in comparison with the molecular volume of the vapor, and that for the vapor the equation of state of an ideal gas is valid, with the help of the cal—
\({}^{1}\) O. E. Frivold. Phys. ZS. 25, 465, 1924.
reasoning entirely analogous to that given above, we obtain for the relative lowering of the vapor pressure over the solution the following expression:
\[ \frac{\Delta p}{p}=1-c_0 f_0 . \]
On the other hand, the relative lowering of the vapor pressure over an ideal solution is expressed by the formula:
\[ \frac{\overline{\Delta p}}{p}=1-c_0 . \]
From comparison of the last two formulas with (12) we obtain:
\[ g=\frac{1-c_0 f_0}{1-c_0}=\frac{\Delta p}{\overline{\Delta p}}{}^{1)} \tag{12} \]
§ 3. Electrode potentials and their application to the determination of activity coefficients.
Of the group of phenomena characterized by a change in the concentration of the dissolved substance, we shall consider only the electromotive forces arising at the boundary between the electrolyte and the metal immersed in it. Let us note that it is precisely to this region that the most accurate measurements belong.
Suppose that into a solution of a completely dissociated electrolyte there is immersed an electrode \(I\), “reversible” \(^{2)}\) with respect to ions of the \(k\)-th kind present in the solution. The dependence of interest to us, on the concentration, of the electric potential \(\Psi_I\) of electrode \(I\) relative to the solution can readily be obtained from the condition of thermodynamic equilibrium between the electrode and the solution. In the case under consideration this condition will be expressed by the formula:
\[ \delta \Phi+\delta \Phi' + \delta A_e=0, \tag{14} \]
which differs from the analogous expression (10) only by the term \(\delta A_e=\varepsilon z_k \Psi_I\), representing the work expended in transferring one \(z_k\)-valent ion from the solution into the electrode. Further,
\[ \delta \Phi=-\frac{\partial \Phi}{\partial N_k}\delta N_k'=-(\varphi_k+kT\log c_k f_k)\delta N_k' \]
\(^{1)}\) It is easy to show (see, e.g., P. Debye und E. Hückel, Phys. ZS. 24, 185, 1923) that \(g=\frac{\Delta}{\overline{\Delta}}\) (\(\overline{\Delta}\) is the freezing-point lowering calculated by the classical formula under the assumption of complete dissociation of the electrolyte; \(\Delta\) is the true lowering) coincides with \(g\) from formula (13).
\(^{2)}\) O. D. Khvolson, Course of Physics, vol. IV, part I; Prof. V. A. Kistyakovsky, Electrochemistry, part II, issue I, p. 217, St. Petersburg, 1914.
ELECTRICAL THEORY OF SOLUTIONS
applies to the solution:
\[ \delta \Phi'=\frac{\partial \Phi'}{\partial N'_k}\,\delta N'_k=\mu'_k\,\delta N'_k, \]
where \(\mu'_k\) denotes the chemical potential of ions of the selected kind, located in the electrode. Thus from the equilibrium condition (14) it follows that:
\[ \varphi_k+kT\log c_k f_k=\mu'_k+\varepsilon z_k\Psi_I, \tag{15} \]
whence
\[ \Psi_I=\frac{\varphi_k-\mu'_k}{\varepsilon z_k} +\frac{kT}{\varepsilon z_k}\log c_k f_k . \tag{15'} \]
The quantity \(\mu'_k\), generally speaking, depends on the concentration in the electrode of molecules and ions of other kinds, in addition to the \(k\)-th kind under consideration; however, for example, in the case of metallic electrodes, the state of the latter is entirely determined by the external pressure and temperature, as a result of which the indicated dependence may be neglected and one may assume that \(\mu'_k=\mu'_k(p,T)\). Putting
\[ \frac{\varphi_k-\mu'_k}{\varepsilon z_k} =\Psi_I^{\,0}(p,T), \]
whence we obtain:
\[ \Psi_I=\Psi_I^{\,0} +\frac{kT}{\varepsilon z_k}\log c_k f_k .\ ^1 \tag{15''} \]
Of practical interest is the case when one more electrode \(II\), “reversible” with respect to other ions of the \(l\)-th kind, is immersed in the solution. The potential difference between the two electrodes (“the electromotive force of the cell”) will be expressed by the following formula:
\[ \Psi=\Psi_I-\Psi_{II} =\Psi_0+\frac{kT}{\varepsilon} \left\{ \frac{1}{z_k}\log c_k f_k -\frac{1}{z_e}\log c_e f_e \right\}, \tag{16} \]
where \(\Psi_0=\Psi_I^0-\Psi_{II}^0\).
Restricting ourselves to solutions of binary uni-univalent electrolytes at \(25^\circ\text{C}\), for the electromotive force \(E\), expressed in volts, we obtain:
\[ E=E_0+0.1183\log_{10}cf \tag{17} \]
\[ (c_k=c_e=c;\ z_k=-z_e=1;\ \varepsilon=4.77\times10^{-10};\ T=298;\ f=\sqrt{f_k f_e}. \]
II. ELECTRICAL THEORY OF SOLUTIONS OF STRONG ELECTROLYTES
II. DEBYE.
(General foundations; application to thermodynamic phenomena).
§ 1. Calculation of the electrical energy of ions in a solution of a completely dissociated electrolyte.
Let us suppose that in a volume \(V\) of a solution of a strong, completely dissociated electrolyte there are \(N_1, N_2,\ldots, N_i,\ldots, N_s\) ions with charges \(z_1\varepsilon, z_2\varepsilon,\ldots, z_i\varepsilon,\ldots, z_s\varepsilon\) (\(z_1, z_2,\ldots, z_i,\ldots,z_s\) are positive and negative integers denoting the valencies of the ions). Since the whole solution is electrically neutral, it is evident that
\[ \sum_{i=1}^{s} N_i z_i = 0,\quad \text{or}\quad \sum_{i=1}^{s} n_i z_i = 0, \tag{18} \]
where \(n_i=\dfrac{N_i}{V}\) denotes the (mean) number of ions of the \(i\)-th kind in \(1\ \mathrm{cm}^3\) of the solution.
As a first approximation let us assume that the ions are point electric charges interacting according to the well-known Coulomb law; we shall consider that the influence of the solvent on this interaction is determined by the dielectric constant of the pure solvent.
In order to elucidate the character of the distribution of ions in the solution, let us carry out the following imaginary experiment: having imagined an “ideal” solution of the same concentration, let us attach to each particle (“neutral ion”) of the latter the sign \(+\) or \(-\), depending on the sign of the charge of the corresponding ion of the real solution. Then, choosing in the ideal solution some particle (for example, a “positive” one), we shall see that at a certain point of space lying in the immediate vicinity of the chosen particle, ions of both signs will be encountered equally often. Let us turn again to the real solution. Since here ions of the same name repel one another, while those of unlike name attract one another, around any definite ion there will be established, on the average (in time), such a distribution that in its immediate vicinity ions of the opposite sign will be encountered more often than ions of the same sign as the ion under consideration. Let us denote by \(\psi\) the mean electrostatic potential established around the ion under consideration. It follows from Boltzmann’s principle that if, in the element of volume \(dv\), in the absence of any force field, there are \(n_i\,dv\) ions of the \(i\)-th kind, then, in the presence of the mean electrostatic potential \(\psi\) in the same element of volume, there are
\[ n_i e^{-\frac{z_i\varepsilon\psi}{kT}} \]
ions of the \(i\)-th kind. On the other hand,
the unknown potential \(\psi\) satisfies Poisson’s differential equation:
\[ \Delta \psi=-\frac{4\pi}{D}\rho , \tag{19} \]
where \(D\) denotes the dielectric constant of the solvent, and \(\rho\) is the amount of electricity per unit volume.
The volume density of electricity in the case under consideration is evidently equal to
\[ \rho=\varepsilon \sum_{i=1}^{s} z_i n_i e^{-\frac{z_i\varepsilon\psi}{kT}} . \tag{20} \]
Substituting (20) into (19), we obtain the following differential equation for determining the unknown electrostatic potential \(\psi\):
\[ \Delta\psi=-\frac{4\pi\varepsilon}{D}\sum_{i=1}^{s} z_i n_i e^{-\frac{\varepsilon z_i\psi}{kT}} . \tag{19′} \]
The right-hand side of this equation may be expanded in a series in powers of the quantity \(\frac{\varepsilon z_i\psi}{kT}\) and limited to the first two terms of the expansion. We therefore put
\[ \Delta\psi=-\frac{4\pi\varepsilon}{D}\sum_{i=1}^{s} z_i n_i +\frac{4\pi\varepsilon^2}{DkT}\sum_{i=1}^{s} n_i z_i^2\psi, \]
whence, taking into account the condition of neutrality of the solution, we obtain
\[ \Delta\psi=\frac{4\pi\varepsilon^2}{DkT}\sum_{i=1}^{s} n_i z_i^2\psi \tag{21} \]
or
\[ \Delta\psi=\varkappa^2\psi, \tag{21′} \]
where
\[ \varkappa^2=\frac{4\pi\varepsilon^2}{DkT}\sum_{i=1}^{s} n_i z_i^2 . \tag{22} \]
The latter quantity, having the dimension of the square of an inverse length, plays a very important role in the Debye theory of solutions. Below we shall attempt to elucidate its immediate physical meaning.
Let us now calculate the distribution of the potential and of the density of electricity around the chosen ion. Taking the center of the latter as coinciding with the center of ...
taking the origin of the spherical coordinate system and taking into account that, owing to the obvious spherical symmetry of the distribution sought with respect to the ion under consideration, \(\psi=\psi(r)\), we obtain
\[ \frac{d^{2}\psi(r)}{dr^{2}}+\frac{2}{r}\frac{d\psi(r)}{dr}=\chi^{2}\psi(r). \tag{23} \]
The latter equation has the following general solution
\[ \psi(r)=A\frac{e^{-\chi r}}{r}+A'\frac{e^{\chi r}}{r}. \tag{24} \]
Since at an infinitely large distance the potential \(\psi\) tends to zero, it is obvious that \(A'=0\). The density of electricity around the ion under consideration is expressed by the formula
\[ \rho(r)=-\frac{D\chi^{2}}{4\pi}\frac{e^{-\chi r}}{r}\cdot A. \]
Let us calculate the volume density of electricity at two points separated from one another by a distance equal to \(\frac{1}{\chi}\):
\[ \rho(r)=C\frac{e^{-\chi r}}{r}, \]
where
\[ C=-\frac{D\chi^{2}}{4\pi}A \]
\[ \rho\left(r+\frac{1}{\chi}\right) = C\chi\frac{e^{-\chi\left(r+\frac{1}{\chi}\right)}}{\chi r+1}, \]
whence
\[ \frac{\rho(r)}{\rho\left(r+\frac{1}{\chi}\right)} = \frac{\chi r+1}{\chi r}e = \left(1+\frac{1}{\chi r}\right)e. \]
Thus, for \(r\gg \frac{1}{\chi}\), the quantity \(\frac{1}{\chi}\) is the distance over which the density of electricity in the “ionic atmosphere” decreases approximately by a factor of \(e\). For aqueous solutions of one-one valent electrolytes at \(0^\circ C\),
\[ \frac{1}{\chi}=\frac{3.06}{\sqrt{\gamma}}\cdot 10^{-8}\ \text{cm} \]
(\(\gamma\) denotes the number of moles of electrolyte in one liter of solution). Thus, for a normal solution \((\gamma=1)\), the length \(\frac{1}{\chi}\) reaches molecular dimensions.
We now proceed to the determination of the integration constant \(A\). Suppose that the ion we have chosen belongs to the \(j\)-th species. The potential which it creates at a distance \(r\) in a medium with dielectric constant \(D\) is equal to \(\dfrac{\varepsilon}{D}\dfrac{z_j}{r}\). Assuming that in the immediate vicinity of the ion (for \(r \to 0\)) this expression coincides with (24), we obtain
\[ A=\frac{\varepsilon z_j}{D}. \]
For the required potential we finally obtain:
\[ \psi(r)=\frac{\varepsilon z_j}{D}\frac{e^{-\chi r}}{r} =\frac{\varepsilon z_j}{D}\frac{1}{r} -\frac{\varepsilon z_j}{D}\frac{1-e^{-\chi r}}{r} =\psi_1+\psi_2 \]
\(\psi_1=\dfrac{\varepsilon z_j}{D}\dfrac{1}{r}\) represents the potential of the ion itself; \(\psi_2\) is the potential of the “ionic atmosphere.” For small values of \(r\) the latter quantity may be represented by the formula \(-\dfrac{\varepsilon z_j}{D}\chi\). Consequently, the potential energy of an ion of the \(j\)-th species with respect to its surrounding “ionic atmosphere” is equal to
\[ u_j=-\frac{\varepsilon^2 z_j^2}{D}\chi. \tag{25} \]
The additional electrical energy created by the electric charges of all ions contained in the volume \(v\) is expressed by the formula
\[ U_e=-\frac{\varepsilon^2\chi}{2D}\sum_{i=1}^{s} N_i z_i^2 . \tag{26} \]
The considerations developed here are, of course, insufficient for their successful application to the study of the thermodynamic behavior of solutions of strong electrolytes.
The point is that the interaction between ions is not exhausted by Coulomb forces alone, inversely proportional to the square of the distance; since ions are complex systems of electric charges (in the dynamic respect), additional “polarization” forces of attraction and repulsion, rapidly decreasing with distance, are added to the Coulomb forces.\(^1\)
\(^1\) See the book by Ya. I. Frenkel, Electrical Theory of Solids, and also his articles in Uspekhi Fizicheskikh Nauk 4, issue 6, p. 394, and in Zhurnal Russkogo Fiziko-Khimicheskogo Obshchestva, physical section, 56, 281, 1925.
However, further specialization of our ideas about the nature of these additional forces proves useless for the method described here, which is based on the application of Poisson’s differential equation. The fact is that a non-Coulomb force field cannot be described by means of a differential equation analogous to Poisson’s equation; as is known, in this case there is no possibility of representing the sources of the field in the form of some continuous distribution. Thus the formal difficulty hindering the development of the electrical theory of solutions and liquids is the absence, due to the circumstance indicated above, of a statistical method that would allow one to treat jointly forces that decrease rapidly with distance and Coulomb forces. The real necessity, nevertheless, of taking these additional forces into account (which in the first approximation reduce to repulsive forces) compels one to ascribe finite sizes to the ions, determining the smallest distances to which the ions can approach one another. Let us recall that an analogous device is used in deriving the equation of state for gases that differ from ideal gases. However, the method by which Debye introduces the finite sizes of ions is hardly deserving of approval. He regards an ion of the \(j\)-th kind as a dielectric sphere (filled with pure solvent) of some finite—previously unspecified—radius \(a_j\), at the center of which the entire charge of the ion is concentrated. It is further assumed that the electrostatic potential created by the “ionic atmosphere” inside this sphere is a constant quantity. Thus we have the following formulas for the potential:
for all space outside the sphere of radius \(a_j\)
\[ B\frac{e^{-\varkappa r}}{r} \]
inside the sphere
\[ \frac{\varepsilon z_j}{D}\frac{1}{r}+C. \]
From the condition of continuity of the potential and of its first derivative on the surface of the sphere, we obtain:
\[ B\frac{e^{-\varkappa a_j}}{a_j}=\frac{\varepsilon z_j}{D}\frac{1}{a_j}+C \]
\[ B e^{-\varkappa a_j}\frac{1+\varkappa a_j}{a_j^2} =\frac{\varepsilon z_j}{D}\frac{1}{a_j^2}, \]
whence it follows that
\[ B=-\frac{\varepsilon z_j}{D}\frac{e^{\chi a_j}}{1+\chi a_j}; \]
\[ C=-\frac{\varepsilon z_j \chi}{D}\frac{1}{1+\chi a_j}. \]
Since the quantity \(C\), by its definition, represents the value of the potential created by the “ionic atmosphere” (i.e., by the ions surrounding the ion under consideration) inside (and at the boundary of) the ion, it is obvious that the potential energy of an ion of the \(j\)-th kind with respect to its surroundings is equal to
\[ u_j=-\frac{\varepsilon^2 z_j^2}{D}\chi\,\frac{1}{1+\chi a_j}. \tag{27} \]
and, consequently, for the additional “electric” energy of the whole solution we obtain the formula
\[ U_e=-\frac{\varepsilon^2\chi}{2D}\sum_{i=1}^{s} N_i z_i^2 \frac{1}{1+\chi a_i}. \tag{28} \]
Below we shall see that the assumptions indicated here by Debye, with the aid of which he takes into account the additional interaction forces between ions caused by the complex structure of the latter, lead, when applied consistently, in some cases to entirely erroneous results. In view of this, Ya. I. Frenkel, for a summary description of the additional forces (which in a first approximation reduce to repulsive forces), proposed introducing the sizes of the ions by means of an equation of state, which in form is analogous to the well-known van der Waals equation and which describes quite satisfactorily, in quantitative terms, the behavior of solutions of considerable concentration1.
§ 2. Calculation of the Additional Part of the Thermodynamic Potential and of the Activity Coefficients.
Let us now turn to the application of the considerations developed above to the study of the thermodynamic behavior of solutions of strong electrolytes.
The principal task of the electrical theory—as we formulated it above—reduces to the calculation of the thermodynamic potential
of the real solution. For this purpose we shall use the method first proposed by Debye1 and based on the conclusions of the preceding section.
From the standpoint of a theory which assumes that all forces of interaction between ions are due to the electric charges of the latter, an ideal solution is a system consisting of neutral particles. Let us suppose that, by means of an imaginary infinitely slow process occurring at constant volume and unchanged temperature, the particles of the ideal solution acquire charges, as a result of which the system is transformed into a real solution. The work performed in this process, as is known from thermodynamics, is equal to the difference of the free energies of the system in the initial and final states:
\[ [W]=F-\overline{F} \]
Here \(F\) denotes the free energy of the real solution; \(\overline{F}\), the same quantity for the ideal solution.
Bearing in mind that in what follows we shall not have to deal with the dependence of thermodynamic equilibria on pressure, we may—owing to the negligibly small compressibility of liquids—identify the change in free energy with the change in thermodynamic potential. Hence the relation follows:
\[ [W]=W, \tag{29} \]
where [see (3)]
\[ W=\sum_i N_i w_i . \]
Let us proceed to calculate the work \([W]\) expended in the process of “charging” the ideal solution. Suppose that at some definite moment of this process the ions have acquired the charges
\[ \lambda z_1 \varepsilon,\ldots,\lambda z_s \varepsilon \quad (1\geq \lambda \geq 0). \]
Our process will consist in an infinitely slow change of the quantity \(\lambda\) from 0 to 1. Denoting by \(\psi_i(\lambda)\) the mean electrostatic potential established around the \(i\)-th ion at the “moment” \(\lambda\),
we obtain that, to increase the charge from \(z_i e\lambda\) to \(z_i e(\lambda+d\lambda)\), i.e. by \(z_i e\,d\lambda\), it is necessary to expend work \(du_i\) equal to
\[ du_i=\varepsilon z_i \psi_i(\lambda)\,d\lambda, \]
whence, comparing with (29), we obtain
\[ W=\sum_{i=1}^{s} N_i u_i =\varepsilon\sum_{i=1}^{s} N_i z_i \int_{0}^{1}\psi_i(\lambda)\,d\lambda, \]
where
\[ \psi_i(\lambda)=-\frac{\varepsilon z_i \chi}{D}\lambda^2, \]
therefore,
\[ \log h_i=\frac{1}{kT}u_i =-\frac{\varepsilon^2 z_i^2\chi}{DkT}\int_{0}^{1}\lambda^2\,d\lambda =-\frac{\varepsilon^2 z_i^2\chi}{3DkT}. \tag{30} \]
Let us calculate the activity coefficient \(f_i\) of ions of some, for example, the \(i\)-th kind. By the definition of the activity coefficient, we have:
\[ \log f_i=\log h_i+\sum_{j=1}^{s}N_j\frac{\partial\log h_j}{\partial N_i}; \]
taking (30) into account, we obtain
\[ \log f_i=-\frac{\varepsilon^2 z_i^2\chi}{3DkT} -\frac{\varepsilon^2}{3DkT}\sum_{1}^{s}N_j z_j^2\frac{\partial\chi}{\partial N_i}; \qquad (i=1,2,\ldots,s), \]
further,
\[ \chi^2=\frac{4\pi\varepsilon^2}{DkT}\sum_{1}^{s} n_k z_k^2 =\frac{1}{V}\frac{4\pi\varepsilon^2}{DkT}\sum_{1}^{s} N_k z_k^2 \left(n_k=\frac{N_k}{V};\ V=\sum N_k v_k\right), \]
whence
\[ 2\chi\frac{\partial\chi}{\partial N_i} =\frac{4\pi\varepsilon^2}{DkT}\frac{z_i^2}{V} -\frac{4\pi\varepsilon^2}{DkT}\sum_{1}^{s}N_k z_k^2\frac{1}{V^2}\frac{\partial V}{\partial N_i} \]
or
\[ 2\chi\frac{\partial\chi}{\partial N_i} =\frac{4\pi\varepsilon^2}{DkT}\frac{z_i^2}{V} -\frac{4\pi\varepsilon^2}{DkT}\sum_{1}^{s}n_k z_k^2\frac{v_i}{V}. \]
Formula (29′) was derived under the assumption that the ions have no extension; consequently,
\[ 2\chi\frac{\partial\chi}{\partial N_i} =\frac{4\pi\varepsilon^2}{DkT}\frac{z_i^2}{V}, \]
whence
\[ \log f_i = -\frac{\varepsilon^2 z_i^2 \chi}{3DkT} -\frac{4\pi\varepsilon^2}{6DkT}\,\frac{1}{\chi} \sum_{1}^{s}\frac{N_k}{V}z_k^2\cdot z_i^2 = \]
\[ = -\frac{\varepsilon^2 z_i^2 \chi}{3DkT} -\frac{4\pi\varepsilon^2}{3DkT}\,\frac{1}{\chi} \sum_{1}^{s} n_k z_k^2\cdot z_i^2 \]
or, finally,
\[ \log f_i=-\frac{\varepsilon^2 z_i^2 \chi}{2DkT}\quad (i=1,2,\ldots,s). \tag{31} \]
Let us now determine the activity coefficient of the solvent, i.e. of the particles of the “zero species” (quantities pertaining to the solvent we shall furnish with the subscript zero):
\[ \log h_0=\log f_0+\sum_{1}^{s} N_j\,\frac{\partial \log h_j}{\partial N_0} \]
from formula (30) it follows that \(\log h_0=0\), and, consequently, the activity potential of the solvent \(h_0\) is equal to unity; this follows from the fact that the particles of the solvent are neutral, i.e. \(z_0=0\). Thus
\[ \log f_0=\sum_{j=1}^{s}N_j\,\frac{\partial\log h_j}{\partial N_0} = -\frac{\varepsilon^2}{3DkT}\sum N_j z_j^2\,\frac{\partial\chi}{\partial N_0}. \]
Further,
\[ 2\chi\,\frac{\partial\chi}{\partial N_0} = -\frac{4\pi\varepsilon^2}{DkT}\sum n_k z_k^2\,\frac{v_0}{V}. \]
\[ \frac{\partial\chi}{\partial N_0}=-\frac{\chi v_0}{2V}, \]
whence
\[ \log f_0=\frac{v_0\varepsilon^2\chi}{6DkT}\sum_{k=1}^{s} n_k z_k^2. \tag{32} \]
Taking into account that in an ideal solution the activity coefficients of the solvent and of the ions are equal to unity, we see that upon passing to a real solution the “activity” of the dissolved substance decreases \((f_i<1;\ i=1,2,\ldots,s)\), while the “activity” of the solvent increases \((\log f_0\) is a substantially positive quantity).
The last formula makes it possible to calculate the deviation of the osmotic coefficient from unity, caused by electrical
(Coulomb) forces of interaction between ions. According to (12) we have:
\[ 1-g=\frac{1}{\sum_1^s c_i}\log f_o . \]
Let us suppose that there are \(N\) molecules of some electrolyte in the solution, each of them dissociating into \(\nu_i\) \((i=1,\ldots,s)\) ions of the \(i\)-th kind; taking into account that in the case of strong dilution
\[ c_i=\frac{N_i}{N_o+\sum_1^s N_i}\simeq \frac{N_i}{N_o} =\frac{\nu_i N}{N_o}\simeq \frac{\nu_i n N_o v_o}{N_o}=\nu_i n v_o \]
we obtain
\[ 1-g=\frac{\varepsilon^2\varkappa}{\nu\,6DkT}\sum\frac{n_k}{n}z_i^2 =\frac{\varepsilon^2\varkappa}{6DkT}\frac{\sum \nu_k z_k^2}{\nu}, \]
where
\[ \nu=\sum_1^s \nu_k . \]
Further we have
\[ n=\frac{\gamma \mathfrak{N}}{1000}, \]
where \(\gamma\) denotes the number of moles of electrolyte per liter of solution,
\[ \varkappa= \sqrt{\frac{4\pi \varepsilon^2}{DkT}}\sqrt{n\sum \nu_k z_k^2} = \sqrt{\frac{4\pi \varepsilon^2}{DkT}\frac{\gamma\mathfrak{N}}{1000}} \sqrt{\sum \nu_k z_k^2}\sqrt{\gamma} \]
and, finally,
\[ 1-g= \frac{\varepsilon^2}{6DkT} \sqrt{\frac{4\pi\varepsilon^2}{DkT}\frac{\mathfrak{N}}{1000}} \left(\frac{\sum \nu_k z_k^2}{\nu}\right)^{\frac{3}{2}} \sqrt{\gamma} \tag{33} \]
or
\[ 1-g=AD^{-\frac{3}{2}}\,w\sqrt{\gamma}, \tag{33'} \]
where
\[ A=\frac{\varepsilon^2}{6kT} \sqrt{\frac{4\pi\varepsilon^2}{kT}\frac{\mathfrak{N}}{1000}}; \]
\[ w=\left(\frac{\sum \nu_k z_k^2}{\nu}\right)^{\frac{3}{2}} . \]
§ 3. Comparison of the Theory with Experiment.
Turning to the experimental data, let us consider the curves expressing the dependence on concentration of the quantity \(1-g\), obtained from exact cryoscopic measurements. Restricting ourselves only to binary uni-univalent electrolytes, we observe (Fig. 1) that even in the region of very small concentrations the influence of the individual properties of the ions is noticeably manifested. Meanwhile, the formulas of the preceding section lead to the conclusion that the behavior of the solution—including its osmotic properties—is exactly the same in the case of salts of the same type. Thus, in order to obtain agreement with experiment, it immediately becomes necessary to make use of the following approximation, permitted by the theory under consideration, i.e., in other words, to assign finite dimensions to the ions.
Fig. 1.
For simplification, in the original formula (28), instead of the quantities \(a_i\) characterizing the dimensions of the corresponding ions, we introduce a certain mean radius \(a\) for the ions of the given electrolyte. Repeating exactly the reasoning of the preceding paragraph, we obtain
\[ 1-g=AD^{-\frac{3}{2}} w\sqrt{\gamma}\,\sigma(\varkappa a), \tag{34} \]
where
\[ \sigma(\varkappa a)=3\left[ \frac{1}{(\varkappa a)^2} +\frac{1}{(\varkappa a)^2}\frac{1}{1+\varkappa a} -\frac{2}{(\varkappa a)^3}\log(1+\varkappa a) \right]. \tag{35} \]
Let us note that for \(a=0\) formula (34) passes into (33'). Indeed,
\[ \lim_{a\to 0}\left|\sigma(\varkappa a)\right| = \lim_{a\to 0}\left|1-\frac{3}{2}\varkappa a+\frac{9}{5}(\varkappa a)^2-2(\varkappa a)^3+\cdots\right| =1. \]
In Fig. 2 is presented the graph of the function \(\sigma(x)\), which plays so essential a role in the Debye theory.
Formula (34), with a proper choice of the “mean” radius \(a\), leads to good agreement with experiment for strongly dilute sol-
On the particular example of the salt La \((\mathrm{NO}_3)_3\), we shall show how the determination of the quantity \(a\) is carried out. Substituting into formula (34) the numerical values of the universal constants and referring it to an aqueous solution at \(0^\circ\mathrm{C}\), we obtain:
\[ 1-g=0.270\,w\sqrt{\nu\gamma\sigma(\varkappa a)} \tag{36} \]
\[ (\mathfrak{N}=6.06\cdot10^{23};\quad \varepsilon=4.77\cdot10^{-10};\quad k=1.346\cdot10^{-16};\quad D=88.23;\quad T=273). \]
Fig. 2.
For the concentration \(\gamma=0.17486\) (the highest of those for which comparison with experiment was made) there corresponds the measured value
\[ 1-g=0.2547; \]
at the same time,
\[ \sqrt{\nu\gamma}=0.836. \]
On the other hand, for the same concentration, calculation by means of the “limiting” formula (33′) gives
\[ 1-g=1.173; \]
therefore,
\[ \sigma(\varkappa a)=\frac{0.2547}{1.173}=0.216. \]
Next, from the graph of the function \(\sigma(\varkappa a)\) (Fig. 2) we obtain: to the value \(\sigma(\varkappa a)=0.216\) there corresponds the argument
\[ \varkappa a=1.67. \]
Since for \(\sqrt{\nu\gamma}=0.836\) the quantity
\[ \varkappa=0.336\times10^{+8}, \]
we finally have
\[ a=4.97\cdot10^{-8}\ \mathrm{cm}=4.97\ \text{Å}. \]
In an analogous manner the “mean” radii may also be determined for other salts; the results obtained in this way are given in Table 1.
Table 1.
| salt | \(a\,10^8\ \mathrm{cm}\) |
|---|---|
| KCl | 3.76 |
| \(\mathrm{K}_2\mathrm{SO}_4\) | 2.69 |
| La \((\mathrm{NO}_3)_3\) | 4.97 |
| Mg \(\mathrm{SO}_4\) | 3.35 |
A comparison of the results of measurements with the data of calculations by formula (34) for an aqueous solution of KCl is shown in Table 2.
Table 2.
| $\gamma$ | $1-g$ (observ.) | $1-g$ (calc.) |
|---|---|---|
| 0.0050 | 0.0214 | 0.0237 |
| 0.0097 | 0.0295 | 0.0313 |
| 0.0166 | 0.0375 | 0.0392 |
| 0.0317 | 0.0485 | 0.0499 |
| 0.0580 | 0.0613 | 0.0618 |
| 0.1170 | 0.0758 | used to determine $a$ |
The first systematic comparison of the theory set forth with experiment belongs to its authors, who used for this purpose numerous cryoscopic measurements in aqueous solutions of strong electrolytes. Subsequently, nonaqueous solutions also attracted the attention of investigators. Thus, for example, Frivold1 applied Debye’s theory to the elevation of the boiling point of very dilute alcoholic solutions. He found good agreement between theory and experiment up to $\gamma = 0.5$; moreover, it turned out that the experimental curve expressing the dependence of $1-g$ on $\sqrt{\gamma}$ has at the origin a tangent in good agreement with the theoretical one.
Fundamental works by Brönsted and La-Mer2, as well as by Schärer3, are devoted to solubility phenomena from the standpoint of Debye’s theory. Let us note that, in order to obtain agreement with the experimental data, Schärer in some cases had to ascribe negative sizes to ions. This circumstance serves as an additional illustration of the dubious value of the method by which the theory under consideration takes into account all non-Coulomb forces of interaction between ions. In conclusion, we shall point to a considerable number of works by American investigators (Scatchard, Hovorka and Rodebush, Harned, A. Noyes, and many others), containing comparisons favorable to the theory between it and experimental determinations of the activity coefficients of strong electrolytes, chiefly on the basis of very accurate measurements of electrode potentials.
§ 4. Extension of the Debye theory to the region of high concentrations.
In its further development, the electrical theory of solutions of strong electrolytes sets as its principal task the expansion of the range of concentrations encompassed by it. The first (and in principle far from irreproachable) step in this direction, made by Debye and Hückel and reducible to the introduction of finite ion sizes, makes it possible to explain the thermodynamic behavior of solutions only in a very narrow range of concentrations.
As experiment shows, the osmotic coefficient (see Fig. 1), with increasing concentration, at first decreases and then, having reached a minimum, begins to increase. At some definite concentration the osmotic pressure (or the vapor pressure over the solution) may exceed the value indicated by the classical theory. In the same way, the activity coefficient of the electrolyte may reach a value exceeding unity. Meanwhile, as the formulas given above show, the logarithm of the activity coefficient of the electrolyte, while remaining a substantially negative quantity as the concentration increases, nevertheless increases in its absolute value. Thus the theoretical activity coefficient of the electrolyte is a monotonic (and, moreover, decreasing) function of the concentration. The indicated divergence of theory from experiment testifies only to the fact that, in the region of concentrations where the individual peculiarities of ions begin to manifest themselves especially strongly, the conception of the latter as nondeformable dielectric spheres of finite diameter carrying an electric charge ceases to be even approximately valid.
On the other hand, the use of the dielectric constant of the pure solvent is hardly permissible in calculating the influence exerted by the solvent on the interaction of ions in relatively strong solutions. It is quite obvious that, in the case of the presence in the solvent of a large number of electric charges, the latter undergoes strong changes in its “dielectric” respect.
The first attempt at a theoretical investigation of the influence exerted on the thermodynamic behavior of strong aqueous solutions of strong electrolytes by a change in the dielectric properties of the solvent (water) belongs to Hückel1.
The large value of the dielectric constant of water, as is known, is closely connected with the dipolar character of its molecules; thanks to this latter circumstance, water is a strongly “polarizable” medium; moreover, the polarization produced, for example, by a weak external
the electric field is manifested not so much in the deformation of the molecules as, chiefly, in their orientation in the direction of the external field1.
In his work, Hückel proceeds from the following purely phenomenological considerations:
1) in the immediate vicinity of an ion, the “ready-made” dipoles of the water molecules are strictly oriented (“fixed”) in the electrostatic field of this ion;
2) such “binding” of part of the solvent molecules (increasing with increasing ion concentration) is equivalent to a decrease in the dielectric constant of the solution in comparison with the same quantity for the pure solvent;
3) each kind of ion is characterized by its own dependence of the dielectric constant of the solution on concentration.
He expresses the dielectric constant \(D\) of the solution by the following linear formula:
\[ D = D_{0} - \sum_{1}^{s} \delta_i \gamma_i \]
or
\[ D = D_{0}\left(-1 \sum_{1}^{s} \frac{a_i}{D^{0}} n_i\right) = D_{0}(1-\beta), \]
where
\[ \beta = \sum_{1}^{s} \frac{a_i}{D_{0}} n_i;\qquad n_i=\frac{\mathcal{N}\gamma_i}{1000}; \]
\(\delta_i\) is the amount by which the dielectric constant of the pure solvent is decreased when \(\mathcal{N}\) ions of the \(i\)-th kind are added to one liter of solution.
This formula is evidently based on the assumption that (at least in some interval of concentrations) each ion “binds,” on the average, the same number of water molecules.
Assuming further that \(\beta\) is a small quantity and neglecting its higher powers, Hückel calculates the activity coefficients of the solvent and of the electrolyte. The formulae obtained by him contain the unknown constants \(\delta_i\) and \(a_i\) \((i=1,2,\ldots,s)\), which can be determined from experimental data in exactly the same way as was shown above. In their final form Hückel’s formulae give fairly good agreement with experiment over a very broad range of concentrations (from \(\gamma=0\) up to \(\gamma=5.0\)).
As an example, let us consider an aqueous solution of a binary univalent electrolyte, at the temperature \(t = 25^\circ\mathrm{C}\) (\(T = 298\,\mathrm{K}\)). In this case
\[ D = D_0 - (\delta_1 + \delta_2)\gamma . \]
Assuming, for simplicity, that \(a_1 = a_2 = a\), Hückel obtains the following formula for the activity coefficient of the electrolyte \(f\):
\[ \log f = \frac{1}{2}(\log f_1 + \log f_2) \]
\[ \log_{10} f = 0.354 \sqrt{2\gamma}\,\frac{1}{1+x_0 a} + \frac{0.0194 \times 10^{-8}}{a}\,2\delta\,2\gamma\,\frac{1}{1+xa} -0.00225\,\delta\,(2\gamma)^{\frac{3}{2}}\, \frac{1}{(1+x_0a)^2} + \]
\[ + \frac{0.000246 \cdot 10^{-8}}{a}\,3\delta\,(2\gamma)^2 \left[ \frac{3}{4}\frac{1}{1+x_0a} + \frac{1}{4}\frac{1}{(1+x_0a)^3} \right], \]
where
\[ \delta = \frac{1}{2}(\delta_1+\delta_2); \]
\[ x_0 = \sqrt{ \frac{4\pi e^2}{D_0 kT}\sum n_i z_i^2 } = 0.232 \cdot 10^8 \sqrt{2\gamma}; \qquad D_0 = 78.77. \]
Using the data of very precise measurements of the electromotive force in the cell
\[ \mathrm{HgNa_x}\;|\;\mathrm{NaCl}\;|\;\mathrm{HgCl}\mathrm{Hg} \]
for three different concentrations
\[ \gamma = 0.02;\quad 0.50;\quad 5.41, \]
Hückel obtained the following result for an aqueous solution of NaCl:
\[ a = 2.35 \cdot 10^{-8}\ \mathrm{cm}, \qquad \delta \simeq 4.5. \]
In an analogous way, for an aqueous solution of KCl one obtains \(\delta \simeq 3\). On the other hand, measurement of the dielectric constants of solutions of strong electrolytes, carried out by Walden, Ulich, and Werner\(^1\), leads in the case, for example, of KCl to a completely different result:
\[ \delta = 810. \]
\(^1\) P. Walden, H. Ulich und O. Werner, ZS f. phys. Chemie, 116, 261, 1925.
Without entering into a discussion of the reasons for so sharp a discrepancy between experiment and theory, we shall note that, whereas the measurement data do not arouse particular doubts as to their accuracy, the calculations are based on hypothetical, highly schematized representations. The latter circumstance permits one to suppose that Hückel’s conclusions, based on data obtained in the study of the thermodynamic behavior of concentrated aqueous solutions of strong electrolytes, should in fact be referred not to the ordinary “macroscopic” dielectric constant of the solution, but to a certain quantity that does not have such a simple and strictly defined physical meaning. Indeed, the dielectric constant characterizes the behavior of the entire system as a whole—i.e., in the present case, of the free water molecules and ions together with the “bound” water molecules—in a weak external field (in comparison with the fields in the immediate vicinity of the ions). The interaction of the ions, however, is determined chiefly by the polarization of the solvent in their immediate vicinity.
Thus, one must conclude that the success of Hückel’s theory is achieved by means of a method whose theoretical value cannot but arouse serious doubts. Nevertheless this theory deserves our attention, because it represents the first attempt to elucidate—from the standpoint of electrical theory—the influence exerted on the thermodynamic properties of aqueous solutions by the “binding” of part of the water molecules by the electrolyte ions, which in its main features constitutes the essence of that “hydration” to which it is customary to refer in explaining the behavior of concentrated solutions.
In conclusion, let us note that lack of space permits us only to make a brief mention of two recently published theoretical works by Zwicky¹) and Webb²), in which the electric polarization of the solvent, caused by the strong fields of the ions, is calculated directly, as is the compression of the solvent (electrostriction) arising as a result of the interaction of the electrolyte ions with those electric charges of which the solvent molecules are composed.
(To be continued.)
¹) F. Zwicky, Phys. ZS. 27, 271, 1926.
²) T. I. Webb, Journ. Amer. Chem. Soc. 48, 2589, 1926.