CRITIQUE OF THE ELECTROSTATIC THEORY OF SOLUTIONS.

V. K. Semenchenko, Moscow.

I.

The theory of strong electrolytes, the foundations of which were laid by Debye and Hückel (4), has managed, in the five years of its existence, to develop into one of the most interesting theories of contemporary molecular physics. First, it was able to explain—if not quantitatively, then at least qualitatively, from a unified point of view—a wider range of phenomena than any of the theories of solutions that had existed up to that time. Second, it gave an excellent molecular-electric interpretation of the theory of activity, which is the foundation of the thermodynamics of nonideal systems developed by the American physical chemist Lewis (13). Only after the work of Debye and Hückel and their followers did the physical meaning of the theory of activity become entirely clear—this distinctive chemical principle of relativity, which encroached upon the absolute significance of the principal independent variable of classical physical chemistry: concentration. The empirical regularities of the theory of activity, discovered by Lewis, Randall, and Bjerrum, received a reliable theoretical interpretation and justification. However, already at the very beginning of its development the theory encountered a certain difficulty, consisting in the fact that the constants characterizing the individual properties of ions—which Debye and Hückel, by the meaning of the derivation of the equations containing them, regarded as the radii of these ions—proved in some cases to be negative.

Soon a second difficulty was added to this one: a number of authors, who attempted on the basis of the theory to calculate the heats of dilution of solutions, obtained results very far from agreement with experiment. New determinations were then undertaken of the heats of dilution of very weak solutions, but the data obtained did not prove favorable for the theory; moreover, the discrepancies were observed not only in the magnitude but also in the sign of the observed effect: the theory predicted a positive effect, whereas in experiment it turned out, in a whole series of cases, to be negative.

These failures of the theory gave rise to a number of new works, which, on the one hand, subjected its principal conclusions to critical revision and, on the other, attempted to supplement the theory with certain new propositions. Before proceeding, however, to the exposition of works of a critical character, we shall allow ourselves to recall those basic assumptions which Debye and Hückel make in deriving the equation that constitutes the mathematical foundation of the entire theory, dwelling in particular on the physical meaning of the mathematical simplifications adopted by them. The course of the reasoning is as follows. Let us suppose that we have a solution containing several different electrolytes, whose degree of dissociation is so great that we may assume that the number of ions is practically equal to the number of dissolved molecules multiplied by the number of ions obtained from a given molecule. Debye and Hückel accept the supposition, expressed even before them by many investigators, that the cause producing, in electrolyte solutions, the deviation from the laws established by the classical van ’t Hoff–Arrhenius theory of solutions is the electric field of the ions. The principal task which Debye and Hückel set themselves consists in determining the potential of this field, or, more correctly, its time average. As follows from the very character of the interaction of the ions, this average must not be equal to zero, because if we begin to count the ions passing near some definite ion, then over a sufficiently long interval of time the number of ions of the opposite sign that have passed by it must be greater than the number of ions of the same sign.

there will always be more ions of the same sign. These excess ions will chiefly create the potential. But if such a potential exists, then, according to one of the fundamental principles of the kinetic theory, the Boltzmann principle, the time-average number of ions of a given kind \(n_i\) in the volume element \(dV\) containing the ion under consideration must differ from the number of ions \(n_0\,dV\) that would be there if the ions were distributed uniformly throughout the whole volume of the solution. Approximately the same thing occurs also in the crystal lattices of heteropolar compounds built of ions: the nearest neighbors of a given ion always have the opposite sign. In a solution, however, where ions do not have definite positions, such “crystal-like” arrangements of ions nevertheless occur more often than completely disordered arrangements, or arrangements in which the neighbors nearest to an ion have the same charge as it. Knowing the number of ions of each sign and their charges, we can also determine the density of electricity \(\rho\) in the given volume element, which will be equal to:

\[ \rho = \varepsilon z_1 \nu_1 n_{01} e^{-\frac{\varepsilon z_1}{kT}\psi} + \varepsilon z_2 \nu_2 n_{02} e^{-\frac{\varepsilon z_2}{kT}\psi} +\cdots = \varepsilon z_i \nu_i n_{0i} e^{-\frac{\varepsilon z_i}{kT}\psi} \]

since, according to the Boltzmann principle,
\[ n_i = n_{0i} e^{-\frac{\varepsilon z_i}{kT}\psi}, \]
where \(\varepsilon\) is the elementary charge \(4.774\cdot 10^{-10}\) e.s.u.; \(z_i\) is the valency of the ion; \(k=1.37\cdot 10^{-16}\ \mathrm{erg/grad.}\) is Boltzmann’s constant; \(T\) is the absolute temperature.

But the potential, as is known, is related to the density of electricity by the equation:

\[ \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} = \Delta\psi = -\frac{4\pi\rho}{D} = -\frac{4\pi}{D}\sum \varepsilon z_i n_{0i} e^{-\frac{\varepsilon z_i}{kT}\psi}. \tag{1} \]

It is impossible to find an exact solution of this equation; Debye and Hückel therefore expand the exponential functions in a series and, restricting themselves to terms containing the first power of \(\psi\), obtain:

\[ \Delta\psi = \left( \frac{4\pi \varepsilon^2}{DkT} \sum n_{0i}\nu_i z_i^2 \right)\cdot \psi \tag{1'} \]

(\(\nu_i\) is the number of ions obtained from one molecule). Denoting:

\[ \frac{4\pi \varepsilon^2}{DkT}\sum n_{0i}\nu_i^{\,2} z_i^{\,2}=\chi^2, \]

we obtain:

\[ \Delta\psi=\chi^2\psi. \tag{1''} \]

This equation can be simplified still further if the left-hand side is transformed to polar coordinates, noting at the same time that, since the field may be regarded as symmetric, the derivatives depending on the angular coordinates will be equal to zero. Finally we obtain:

\[ \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d\psi}{dr}\right)=\chi^2\psi; \tag{1'''} \]

the integral of this equation is found easily and has the form:

\[ \psi=A_1\frac{e^{-\chi r}}{r}+A_2\frac{e^{\chi r}}{r}. \tag{2} \]

Let us now dwell on the physical meaning of the assumptions introduced by Debye and Hückel in determining the constants of integration \(A_1\) and \(A_2\). The constant \(A_2\) is equal to 0, since \(\psi\) must be equal to zero at \(r=\infty\) (\(r\) is the distance of the point for which we calculate the potential from the center of the volume element). If the ions are regarded as point charges and the potential is calculated at a point located at a distance \(r\) from one of the ions contained in the volume element under consideration, then, as \(r\) decreases, the potential produced by the selected ion will increasingly exceed the potentials produced by the others, whence

\[ \lim_{r\to 0}\psi=\frac{\varepsilon z}{Dr}=A_1\frac{e^{-\chi r}}{r}, \]

from which \(A_1=\frac{\varepsilon z}{D}\), and since the entire potential is composed of the potential of the ion located at the origin of the coordinates and of the ions surrounding it, we have:

\[ \psi=\frac{z\varepsilon e^{-\chi r}}{Dr}-\frac{\varepsilon z}{Dr} =-\frac{1-e^{-\chi r}}{r}\cdot\frac{\varepsilon z}{D}; \]

V. K. SEMENCHENKO

but we are interested in the potential of the surrounding ions (the “ionic atmosphere,” in Debye’s expression), and for small \(r\), expanding \(e^{-xr}\) in a series, we find that it is equal to:

\[ \psi = -\frac{\varepsilon x}{D}. \]

Of greatest interest is the determination of the constant \(A\) for the case of ions having finite dimensions. Debye and Hückel represent such an ion in the form of a sphere having radius \(a\) and a dielectric constant equal to the dielectric constant of the pure solvent, at the center of which there is a charge. Such a model of the ion is suitable only for ions surrounded by a shell of solvent molecules, for those ions which are solvated in the usual terminology, and the radii calculated according to Debye’s theory are the radii of such solvated ions. Therefore these radii have nothing in common with radii calculated from other data, for example from the smallest distances of ions in crystal lattices. If the ions are not solvated, then the radii calculated according to Debye’s theory will always be smaller than the true radii, since according to Debye the force acting between two such ions is equal to

\[ \frac{\varepsilon^2}{D r^2}, \]

whereas in reality it lies within the limits from

\[ \frac{\varepsilon^2}{D r^2} \quad \text{to} \quad \frac{\varepsilon^2}{r^2}, \]

i.e. the ions are as though brought closer together than they actually are. Below we shall see that experimental and theoretical results can be indicated which confirm this view. The very determination of the potential of the “ionic atmosphere” consisting of ions of finite dimensions is carried out in the following manner: inside the ion the potential, as is known from electrostatics, is equal to:

\[ \psi_i = \frac{\varepsilon}{D r} + B, \]

\(\frac{\varepsilon}{D r}\) denotes the potential arising from the charge situated at its center, \(B\) is the constant potential caused by the presen-

outside the ion an “ionic atmosphere.” Outside the ion the potential, as we have already established, is equal to:

\[ \phi_a = A \frac{e^{-\chi r}}{r}. \tag{2'} \]

On the surface of the ion, both the potentials themselves and their first derivatives, i.e. the field strengths, must be equal; consequently:

\[ A \frac{e^{-\chi a}}{a} = \frac{\varepsilon z}{D a} + B, \]

\[ A e^{-\chi a}\cdot \frac{1+\chi a}{a^2} = \frac{\varepsilon z}{D a^2}; \tag{3} \]

we need to know only the potential at the center of the ion, which, according to (3), is equal to:

\[ B = -\frac{\varepsilon z}{D}\frac{\chi}{1+a\chi}. \tag{3'} \]

Knowing the potential, we can now immediately determine, by multiplying it by the charge of the corresponding ion, the potential energy of this ion and of the whole solution. Knowledge of the potential also allows us to calculate the quantity characterizing the deviation of the given system from the laws of the ideal state, introduced for the first time by the American physical chemist Lewis and called the “activity coefficient” \(f_a\). The activity coefficient is defined by the equation:

\[ \mu = \mu_0 + kT \ln f_a c, \tag{4} \]

where \(\mu\) denotes the chemical potential of the given substance, \(c\) its concentration; \(\mu_0\) depends only on temperature and pressure, but not on concentration. The chemical potential of an ideal solution is equal, as is known, to:

\[ \bar{\mu} = \mu_0 + kT \ln c, \tag{5} \]

whence we obtain that

\[ kT \ln f_a = \mu - \bar{\mu} \tag{4'} \]

but the difference standing on the right-hand side of this equation,

is equal to the isothermal reversible work of transfer of one ion from an ideal, i.e. infinitely dilute, solution into the given one. For all the ions contained in a unit volume of the solution this work, according to Debye (5), is equal to:

\[ -W=\sum n_i\int_0^1 \varepsilon z_i\psi_i\lambda^2\,d\lambda, \tag{6} \]

whence for \(\ln f_{ai}\) we obtain the following equation:

\[ \ln f_{ai}=\frac{1}{kT}\frac{\partial W}{\partial n_i} = -\int_0^1 \varepsilon z_i\psi_i\lambda^2\,d\lambda -\sum n_i\int_0^1 \varepsilon z_i\frac{\partial\psi_i}{\partial n_i}\lambda^2\,d\lambda, \tag{7} \]

and from this it is seen that the activity coefficient depends on \(\psi\), and we shall obtain different expressions for it according as we restrict ourselves to the approximation used by Debye or calculate \(\psi\) more accurately.

The very approximate Debye solution will coincide, as La Mer (11), King and Mason first pointed out, depending on what electrolytes we are dealing with, now with the first and now with the second approximation. For binary electrolytes (La Mer calls them symmetric), irrespective of their valence, when the exponential functions standing on the right-hand side of equation (1) are expanded, all terms of even degree in the series

\[ \nu_+\sum_0^\infty \frac{1}{i!}\left(-\frac{\varepsilon z}{kT}\psi\right)^i - \nu_-\sum_0^\infty \frac{1}{i!}\left(\frac{\varepsilon z}{kT}\psi\right)^i = -\nu\cdot 1+\nu\cdot 1-\nu\frac{\varepsilon z}{kT}\psi \]
\[ -\nu\frac{\varepsilon z}{kT}\psi +\frac{1}{2!}\nu\left(\frac{\varepsilon z}{kT}\right)^2\psi^2 -\frac{1}{2!}\nu\left(\frac{\varepsilon z}{kT}\right)^2\psi^2 -\frac{1}{3!}\nu\left(\frac{\varepsilon z}{kT}\right)^3\psi^3 \]
\[ -\frac{1}{3!}\nu\left(\frac{\varepsilon z}{kT}\right)^3\psi^3-\ldots \]

cancel; therefore for them the first approximation is at the same time also the second. Conversely, for all electrolytes for which \(\nu_+\) and \(\nu_-\), i.e. the numbers of positive and negative ions obtained from one molecule, are not equal, no cancellations occur, and the Debye solution will no longer coincide with the second approximation.

Experimental investigations by La Mer, King, and Mason showed that good agreement with experiment is obtained only for symmetrical electrolytes (the subject of the investigation was the influence of salts on the solubility of salts that have no ion in common with the influencing salts). Taking the second approximation, i.e., solving the equation:

\[ \frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\psi}{dr}\right) = \frac{4\pi\varepsilon^{2}}{DkT}\sum n_i z_i^{2} \left(\psi-\frac{\varepsilon\psi^{2}\sum n_i z_i^{3}}{2kT\sum n_i z_i^{2}}\right) = \]

\[ = \chi^{2}\left(\psi-\frac{\varepsilon}{2kT}q\psi^{2}\right); \qquad q=\frac{\sum n_i z_i^{3}}{\sum n_i z_i^{2}}, \]

La Mer obtained qualitative, but not quantitative, agreement with experiment.

The question of the adequacy of the Debye approximation was investigated in greater detail by G. Müller (14). He drew attention to the fact that, when ions of small radius are present in the solution, one can no longer confine oneself, in solving equation (1), to the first term, since the ions may approach one another at a very small distance and the potential of the “ionic atmosphere” thereby becomes so considerable that it is impermissible to neglect its higher powers. Therefore Müller solved the basic equation (1) by the method of successive approximations and, substituting the obtained values of \(\psi\) into the equation determining the activity coefficient or, more correctly, its logarithm, in this way found more accurate values of the activity coefficient. The most interesting result of Müller’s work may be considered the deviation, established by him, of the curves \(\ln f_a=f(c)\), when the ion radius \(a\) is decreased, from the limiting curve of the Debye theory for \(a=0\).

Müller showed that there exists a certain “critical radius,” equal to \(1.76\) Å; for ions whose radii are smaller than this, the curves \(\ln f_a=f(c)\) already pass below the curve \(\ln f_{a=0}=f(c)\), whereas according to Debye, as the radius decreases, they should tend toward asymptotic coincidence. Müller’s curves, calculated by him for ions with radii of \(1.07\) and \(0.47\) Å, deviate so sharply

from Debye's limiting curve, that even if one doubts (for which there is some basis) the accuracy of Müller's numerical results, the qualitative discrepancy is obvious. In his first paper Müller expressed doubt even that the Debye law for \(a=0\) is valid at infinitely small concentrations, and as a limiting law proposed the formula:

\[ \lim_{c\to 0}\ln f_a=\left(a+\frac{\beta}{a^4}\right)\sqrt{c}. \tag{9} \]

In his second paper, however, he abandoned this view and acknowledged that at vanishingly small concentrations the law is valid for ions of any radius, which is also confirmed by the results of Kramers' work, about which we shall have occasion to speak later. In any case, it may be regarded as established that at finite concentrations the results of Debye's theory are inapplicable to ions whose radii are smaller than the critical radius. The question of the applicability of the limiting law cannot be considered finally resolved, and its mathematical investigation is highly desirable.

In his second paper Müller gave formulas for calculating activity coefficients with any desired degree of accuracy. The extent to which the values calculated by Müller differ from the values obtained according to Debye's theory is seen from the following table.

Table I.

\(-\log_{10} f_a\) for monovalent ions in water at \(18^\circ\)

To the discussion of the assumptions on the basis of which Berrum’s values were calculated we shall return below.

From an entirely different point of view, a critical investigation of the foundations of the theory was carried out by Kramers in a paper published in 1927 (10). Kramers checked the correctness of Debye’s theory by the method of finding the value of the free energy of a solution containing ions, by the method set forth by Gibbs in his statistical mechanics. According to Gibbs, the free energy \(F\) of any molecular system is equal to:

\[ F=-kT\ln \int \cdots \int e^{-\frac{E}{kT}}\,dq_1\cdots dp_{mN}; \tag{10} \]

\(m\) here denotes the number of degrees of freedom belonging to one molecule of the system, \(N\)—the total number of molecules, \(E\)—the total energy of the system, \(q_i\)—coordinates determining the position of the molecules (geometrical coordinates), \(p_i\)—momenta, i.e. derivatives of the energy with respect to the time derivatives of the geometrical coordinates (dynamical coordinates). Integration with respect to \(p\) is performed very easily, since the part of the energy of the system that depends on \(p\) is expressed by a sum of the form

\[ E_{\mathrm{kin}}=\sum \frac{1}{2m_i}P_i^2, \]

and we obtain a product of integrals of the form:

\[ \int_{-\infty}^{+\infty} e^{-\frac{1}{2m_i kT}P_i^2}\,dp. \]

The part of the energy depending on \(q_i\) is precisely that electrostatic energy, arising from the interaction of ions, which plays the primary role in the Debye–Hückel theory. This part of the energy is expressed by the sum:

\[ E_{\mathrm{pot}}=\frac{1}{D}\sum\sum \frac{z_k z_l \varepsilon^2}{r_{kl}}; \tag{11} \]

\(r_{kl}\) denotes the distance between the corresponding ions, depending only on their position, i.e. on \(q_i\). Kramers

ingeniously bypasses the mathematical difficulties that arise in finding integrals depending on \(q_i\), by means of an artificial, purely mathematical device, and finds the value of the free energy. The calculation is carried out in such a way that the radii of the ions are not taken into account, and Kramers comes to the conclusion that up to a certain “critical concentration” the distribution of ions, and consequently the potential and free energy depending on this distribution, do not depend on the radius of the ions. Therefore, contrary to Müller’s assertion, the limiting law of the theory is correct. The critical concentration, starting from which the limiting law according to Kramers loses its force, begins for aqueous solutions of univalent electrolytes at \(0.03\) mole/l, and for divalent ones at \(0.0005\).

Bjerrum (1) attempted to interpret more clearly the causes that give rise to deviations from the laws of the Debye and Hückel theory. The interactions occurring between ions in solution may be of two kinds: first, one may think that ions, approaching one another under the influence of electrostatic forces, combine into some new complex system, and that in this process there occurs a change in their energy subject to quantum conditions, or, to use the language of chemistry, the formation of an undissociated molecule takes place. Secondly, one may imagine the matter otherwise, considering that ions of opposite sign, which meet one another in the case when the kinetic energy of one relative to the other is less than the potential energy of interaction, begin to rotate around a common center of gravity, fully preserving, however, their individuality. Proceeding from the second point of view, one can, as was shown by the author of the present article (18), derive a number of relations previously known as empirical. Bjerrum also approached the question from a similar point of view, reasoning as follows: the probability that the distance between two ions lies between \(r\) and \(dr\), according to kinetic theory, is equal to:

\[ dW = r_i e^{-\frac{\varphi}{kT}} \cdot 4\pi r^2 dr, \tag{12} \]

where \(\varphi=-\dfrac{Z_1Z_2\varepsilon^2}{Dr}\) is the mutual potential energy of these ions, and therefore

\[ dW=n_i e^{\frac{Z_1Z_2\varepsilon^2}{DrkT}}\cdot 4\pi r^2dr; \tag{12'} \]

\(n_i\) is the number of ions in a unit volume of solution. First of all it is easy to find that, for a definite \(r\), \(dW\) will have a minimum value. This value is equal to:

\[ r_{\min}=\frac{Z_1Z_2\varepsilon^2}{2DkT}=q=3.52\ \text{\AA} \tag{13} \]

(water at \(T=291\)). The probability, and consequently also the number of ions located at distances \(r<r_{\min}\), increases rapidly if the ions have opposite signs, as is seen from Fig. 1.

Bjerrum calls ions situated from one another at a distance smaller than the critical distance associated, and considers that they no longer affect the activity coefficient and the potential energy of the solution. Strictly speaking, this is not quite exact, since only those ions may be considered associated which, as we have already said, owing to the small magnitude of their relative kinetic energy, will rotate in closed orbits. In order to determine the number of bound, associated ions (since Bjerrum, as one of the first prophets of the doctrine of complete dissociation, prefers to speak not of the degree of dissocia—

Fig. 1.

tion, but about the degree of association), Bjerrum calculates the value of the integral:

\[ Q=\int_a^q e^{\frac{\varepsilon_1 Z_2 \varepsilon^2}{DrkT}} r^2\,dr, \tag{14} \]

transforming it by the substitution:

\[ Z_1=Z_2=1;\quad y=\frac{\varepsilon^2}{DrkT};\quad b=\frac{\varepsilon^2}{aDkT} \]

(\(a\) — radius of the ion)

into the integral:

\[ Q(b)=\int_2^b e^y y^{-4}\,dy. \tag{14'} \]

Then the degree of association \(\alpha\) is equal to:

\[ \alpha=c\,\frac{4\pi N}{1000}\left(\frac{\varepsilon^2}{DkT}\right)^3 Q(b). \tag{15} \]

Assuming that we have the right to apply to the equilibrium between free and associated ions the generalized law of mass action, in which activities enter instead of concentrations, Bjerrum finally arrives at the equation:

\[ \frac{\alpha}{(1-\alpha)^2}=(f_a)^2 c\,\frac{4\pi N}{1000}\left(\frac{\varepsilon^2}{DkT}\right)^2 Q(b). \tag{16} \]

Solving this equation by the method of successive approximations, he obtains the values of the activity coefficient \(f_a\), given in Table I. These values lie closer to those of Müller than to those of Debye. For water at \(18^\circ\) and univalent ions, the minimum distance \(r_{min}\) beyond which Bjerrum considers ions to be associated is \(3.52\ \text{Å}\); therefore all ions whose sum of radii exceeds this value must strictly obey the Debye theory. It should be noted that the radii of ions calculated by Bjerrum, taking association phenomena into account, are always larger than the radii calculated by the Debye–Hückel method, and never take zero or negative values.

tions. As an example, we give a small table of radii calculated by the one and the other method.

Table II.

Ion radii (mean), calculated from activity coefficients.

Although the Bjerrum correction does lead to the elimination of some shortcomings of the theory in its original form (negative and zero radii), nevertheless the doubts arising in the analysis of Debye’s determination of the integration constants (the use of the macroscopic dielectric constant for ions situated at distances of the order of molecular dimensions or even in contact) are applicable to it as well. Bjerrum’s theory emphasizes still more the complexity of the phenomena occurring in solutions and the difficulty of taking them fully into account.

II.

We have already said that attempts to calculate the heat of dilution on the basis of the Debye–Hückel theory were unsuccessful. We shall now proceed to an account of the experimental and theoretical works that attempted to resolve this question. The expression itself for the heat of dilution is obtained in the following way. Since, when a certain volume of dilute solution is mixed with pure solvent, the change in volume, i.e. the difference between the sum of the volumes of the solution and the solvent and the volume of the resulting solution, is negligible, therefore in the expression for the heat of the process:

\[ \Delta Q=\Delta U+p\Delta V \tag{17} \]

we may neglect the second term, proportional (at constant pressure) to \(\Delta V\), and confine ourselves to considera-

V. K. SEMENCHENKO

... of a term expressing the change in internal energy \(\Delta U\). But according to the well-known Gibbs–Helmholtz formula, \(U\) is equal to:

\[ U=F-T\frac{dF}{dT} \tag{18} \]

(\(F\) is the free energy), therefore finally:

\[ \Delta Q=\Delta F-\Delta T\frac{dF}{dT}. \tag{17'} \]

For ions of infinitely small radius, \(F\), according to Debye, is equal to:

\[ F=\sum N_i\left(\varphi_{0i}+kT-\frac{\varepsilon^2 z_i^2}{3D}\chi\right). \tag{18'} \]

Whence \(\Delta Q\) is also easily found:

\[ \Delta Q=\sum N_i\left[\varphi'_{0i}-\varphi_{0i}+kT\ln\frac{c'_i}{C_i} -\frac{\varepsilon^2 z_i^2}{3D}(\chi'-\chi)\right]- \]

\[ -T\sum N_i\left[k\ln\frac{c'_i}{C_i} +\left(\frac{\varepsilon^2 z_i^2}{6DT} +\frac{\varepsilon^2 z_i^2}{2D^2}\frac{dD}{dT}\right)(\chi'-\chi)\right], \tag{17''} \]

but \(\varphi'_{0i}=\varphi_{0i}\), since \(\varphi_0\) does not depend on concentration (the prime refers to the initial concentration), and consequently:

\[ \Delta Q=-\sum N_i\frac{\varepsilon^2 z_i^2}{2D} \left(1+\frac{T}{D}\frac{dD}{dT}\right)(\chi'-\chi) \tag{17'''} \]

or, substituting:

\[ D=81.1;\qquad T=291;\qquad \varepsilon=4.77\cdot10^{-10}; \]

\[ k=1.37\cdot10^{-16};\qquad 1+\frac{T}{D}\frac{dD}{dT}=-0.315 \quad\text{(according to Drude’s experiments),} \]

we obtain:

\[ \Delta Q=418(\sqrt{c'}-\sqrt{c}). \tag{17''''} \]

If we take into account the finite sizes of the ions, we obtain:

\[ \Delta Q=418\left( \frac{\sqrt{c'}}{1+0.327a\sqrt{c'}} - \frac{\sqrt{c}}{1+0.327\sqrt{c}} \right) \tag{18''} \]

Gross and Halpery (7), as well as Bjerrum (2), attempted to calculate the heats of dilution by this formula and found that in most cases it gives results not agreeing with experiment, sometimes even contradicting it in sign. The experimental data were taken by them from the work of Richards and Rowe (17). Fig. 2 shows how great the disagreement is between theory and experiment; only two substances, \(\mathrm{LiNO_3}\) and \(\mathrm{KOH}\), fall within the region approximately obeying the theory. For \(\mathrm{HCl}\) there is obtained approximate agreement with the limiting law \((a = 0)\), but there are hardly physical grounds for believing that the chlorine ion has infinitely small dimensions. A priori, one might have expected better agreement between theory and experiment in the region of more dilute solutions. Proceeding from this, Nernst and Orthmann (16) carried out determinations of the heats of dilution at low concentrations. The results obtained by them can in no way be regarded as favorable to the theory. The measurements themselves were made by the method of differential calorimetry developed by Nernst and his pupils.

Fig. 2.

The temperature difference was measured by 100 iron-constantan thermoelements, the junctions of which were in small tubes filled with mercury, placed in both calorimetric vessels. The sensitivity was equal to 0.00318 calories per millimeter of galvanometer deflection. The heat liberated owing to friction when solutions of the same concentration were mixed (Leereffekt) was sufficiently constant and sufficiently small and therefore had no effect on the final results. We give in Table III the data obtained by Orthmann. The 1st column contains

name of the salt investigated, the 2nd—the concentrations before and after mixing, the 3rd—the measured thermal effect in small calories, the 4th—the number of separate measurements, the 5th—the thermal effect calculated by formula (17‴), the 6th and 7th—the measured and calculated thermal effect in calories per gram-molecule.

Table III.

The table shows that there can be no question of agreement between theory and experiment. In many cases, just as before, there is not even agreement in sign. Soon after the work of Nernst and Orthmann there appeared the work of Lange and Messner (12), devoted to the same question. Their differential calorimeter consisted (Fig. 3) of a Dewar vessel divided into two halves by an ebonite plate, on both sides of which sat piles of 1000 thermoelements. The vessel itself was placed in a large water bath. The solution being diluted was added from a large pipette, whereby the Leereffekt increased, but Lange and Messner compensated it by adding the same quantity of water from the other side. The sensitivity was equal to 0.0009 cal. per 1 mm.

...deflections of the galvanometer with an accuracy of 2 mm. The results of the experiments of Lange and Messner are given in Table IV; the 8th and 9th columns contain \(\Delta Q\), calculated by the theoretical formula, determining

\[ \frac{dD}{dT} \]

either from the data of Drude (6) or from the new and more accurate measurements of L. Kockel (9). The discrepancies between the theoretical and experimental values are smaller than in Nernst and Orttmann, but nevertheless still considerable. The authors themselves believe that the theoretical formula is correct as regards the sign and the dependence of \(\Delta Q\) on the dielectric constant, temperature, and valence of the electrolyte, but that, for complete agreement, it is necessary to introduce a constant factor whose physical meaning is unclear. Moreover, in their opinion, the values

\[ \frac{dD}{dT} \]

are correct to no more than 10–20%, and they suppose that it would be more proper to use not the dielectric constant of the solvent, but the dielectric constant of the solution, for which the temperature dependence is completely unknown (see Table IV, p. 664).

Fig. 3.

Nernst (16) attempted to clarify theoretically the cause of the discrepancy between the experimental data and the formulas of Debye–Hückel. In his opinion the formulas themselves are entirely exact; the discrepancy between theory and experiment arises because the dissociation of electrolyte molecules is by no means complete, and upon dilution of the solution part of the undissociated molecules dissociates. The heat of dissociation is always negative; therefore the experimental values of the heats of dilution are always less than the theoretical ones. Thus Nernst returns to the viewpoint of the classical theory of electrolytic dissociation. The total heat effect of dilution, in his opinion, is composed of two parts: the negative heat of dissociation and the posi-

Table IV*.

positive (according to Debye) heat of dilution. For dilution from some concentration \(c\) to 0:

\[ \Delta Q=-Q(1-a)+B\sqrt{c}. \tag{19} \]

For large \(a\), \((1-a)\), by the dilution law

\[ \frac{a^2}{1-a}c=K \]

may be regarded as proportional to the concentration, and \(\Delta Q\) will be

fall into two parts: one proportional to the concentration, and the other proportional to the square root of the concentration. Since, according to Debye’s theory, \(B\) is known, the product \(Q(1-a)\) is readily calculated; in order to compute \(Q\) and \(1-a\) separately, Nernst used the results of measurements of the heats of solution at various temperatures, carried out by Naude (15) in his laboratory. Integrating the known van ’t Hoff equation

\[ \frac{d}{dT}\ln K=\frac{Q}{RT^2}, \]

we obtain (\(K\) is Ostwald’s dilution constant):

\[ \ln\frac{K_1}{K_2} =\ln\frac{a_1^2}{a_2^2}\cdot\frac{1-a_2}{1-a_1} =\frac{Q}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right). \tag{20} \]

From this equation and from the values \(Q(1-a_1)\) and \(Q(1-a_2)\), Nernst obtains \(a_1\) and \(a_2\) by the method of successive approximations. However, this method of calculation cannot be regarded as reliable, since Nernst uses Ostwald’s law in its classical form, taking the activity coefficient to be equal to unity. But

\[ \ln f_a=-\frac{\varepsilon^2 z^2}{2DkT}\chi \tag{7'} \]

and it is clear that \(f_a\) is equal to 1 only when \(\chi=0\), since all the other quantities can equal neither zero nor infinity. For values of \(\chi\) close to zero, the “Debye effect” on the basis of which the product \(Q(1-a)\) is calculated will also be equal to zero. In order to bring formula (19) into agreement with experiment, Nernst had to change also the value of the coefficient \(B\) at \(\sqrt{c}\), and he obtained the formula:

\[ U-\Delta Q_{c\to\infty}=320\sqrt{c}-4000(1-a), \tag{21} \]

by means of which, as Table V shows, one can obtain good agreement with the experimental results. It follows from Naude’s measurements that the coefficient of \(\sqrt{c}\) is almost independent of temperature.

Nernst’s theory cannot be considered rigorous, and formula (21) must therefore be regarded as empirical.

Therefore, the degree of dissociation determined on the basis of this formula is hardly actually such as Nernst thinks. The value of the works of Nernst, Ortmann, and Naudé lies in establishing the inapplicability of the theoretical formulas of the electrostatic theory of solutions even at low concentrations, and not at all in the theoretical interpretation of their own experiments.

Berrum (3) indicated other ways of solving this same question. As we have already pointed out in § 1, the physical meaning of determining the integration constant in the case of ions of finite size reduces to the assumption that they possess a shell of solvent molecules, more or less firmly bound to them.

With increasing temperature these shells will be destroyed, and therefore the radius of the ion will decrease. Hence, in the expression for the heat of solution there must enter a term connected with the dependence of the radius on temperature. This term, as Berrum shows, must have the form:

\[ \Delta Q_a=-217\,\frac{da}{d\ln T}\cdot c \tag{22} \]

Table V.

\[ \mathrm{KNO}_3 \]

Thus, from entirely different considerations, Berrum also arrives at the conclusion that in the formula for the heat of dilution there must be a term proportional to the concentration. Calculating \(\Delta Q_a\) on the basis of the data

Richards and Rowe, Bjerrum determines the sign and magnitude of \(\dfrac{da}{d\ln T}\) from the equation:

\[ \frac{\Delta Q(0{,}555-0{,}139)}{0{,}555-0{,}139}=-217\frac{da}{d\ln T}; \tag{23} \]

\(\Delta Q(0{,}555-0{,}139)\) denotes the heat of dilution from a concentration of \(0{,}555\) to \(0{,}139\) mole/l. It is interesting to note that for the ions H and Li, which may most likely be regarded as hydrated, the radius does indeed decrease with increasing temperature; for other ions, which, as for example \(KNO_3\) and \(CsNO_3\), obey the theoretical formula (18) worst of all, the radii increase with increasing temperature. In Table VI we give the results of Bjerrum’s calculations:

Table VI.

There have also been suggestions (17) concerning the possibility of interpreting the observed deviations of the dissolution temperature from those predicted theoretically from the standpoint of the theory of concentrated solutions developed by Hückel (8, cf. also 21, pp. 373—375). We shall not, however, dwell on them, since in the formulas obtained on the basis of these assumptions there is a temperature coefficient of the coefficient characterizing the lowering of the dielectric constant by the action of the electrolyte (as is known, the hypothesis underlying Hückel’s theory is the hypothesis of a lowering of the dielectric constant of the solution propor-

tional to the concentration of the added electrolyte). However, our information not only about the dependence of the influence of electrolytes on the \(DK\) of the solution on temperature, but even about the influence itself, is so inexact that theoretical constructions based on it can hardly be regarded as clarifying anything. Thus the question of the energy quantities in Debye’s theory also cannot be considered solved, and its present state speaks rather not in favor of Nernst, Bjerrum, and Debye himself; attempts to explain theoretically the appearance of an effect proportional to concentration cannot be regarded as clarifying the essence of the matter; perhaps the solution of the questions connected with the foundations of the theory, of which we spoke at the beginning of our review, will help to resolve this question as well.

LITERATURE

1. Bjerrum. Det. Kyl. danske vidensk. selskab. Manth. fys. Meddelelser, VII, H. 9. 1926.
2. — ZS. f. phys. Chem., 119, 146. 1926.
3. — Trans. of the Faraday Soc. Vol. XXIII, 445. 1927.
4. Debye u. Hückel. Phys. ZS., 24, 185. 1923.
5. Debye. Phys. ZS., 23, 97. 1924.
6. Drude. Wied. Ann., 59, 48. 1896.
7. Gross u. Calpern. Phys. ZS. 26, 403. 1925.
8. Hückel. Phys. ZS., 26, 93. 1925.
9. Kockel. Ann. d. Phys., 77, 430. 1926.
10. Kramers. Kyl. akad. van wet nsch. te Amsterdam. XXX, 145. 1927.
11. La Mer, King a. Mason. Journ. Amer. Chem. Soc., 49, 419. 1927.
12. Lange u. Messner. ZS. f. Elektrochem., 33, 431. 1927; Naturwiss., 15, 521. 1927.
13. Lewis a. Randall. Thermodynamics and the free Energy of chemical Substances. Hill Company, 1923. Chapt. XXII, XXIII, XXVI, XXVII and XXVIII.
14. Müller. Phys. ZS., 28, 324. 1927; 29, 78. 1928.
15. Naude. ZS. f. Elektrochem., 33, 532. 1927.
16. Nernst u. Orthmann. Berl. Ber. 51, 1926: 136, 1927.
17. Orthmann. Erg. d. exakt. Naturwiss., Bd. VI, 155. 1927.
18. Richards a. Rowe. Journ. of the Amer. Chem. Soc., 42, 1621, 1920; 43, 770, 1921.
19. Sementschenko. ZS. f. phys. Chem., 112, 128. 1924.
20. Semenchenko. Zh. RFKhO, phys. part, 56, 541. 1925.
21. Finkelstein. UFN, 7, issue 5 and 6. 1927.