A. K. Calais.

Senior Assistant of the Siberian Technological Institute.

The Reaction Rate in a Heterogeneous System and the Order of the Rate of Certain Reactions in This System, as Determined by the Electrical-Conductivity Method

As is known from chemical statics, the rate of a given process is the smaller, the closer the given state is to the state of equilibrium; in exactly the same way, in the dissolution of a solid body, the rate of dissolution of the body is greatest in a pure solvent and approaches zero upon obtaining a saturated solution, when one may assume, just as for homogeneous equilibrium, that as much substance dissolves as precipitates from the solution. Let us first consider the process of dissolution in water of an indifferent substance (i.e., one that does not form compounds with the solvent).

This question was first examined by Stefan (1), and after him by Noyes and Whitney (2); they came to the conclusion that the rate of dissolution of a solid body is proportional to the difference between the concentrations of the saturated solution and that present at the given moment. In other words, the farther the given system (liquid, solid body) is from the state of equilibrium, the greater the rate of dissolution. This fundamental proposition may also be derived from the assumption that, in the immediate vicinity of the surface of the solid body, a layer of saturated solution is formed; according to Fick’s law (3), which had already been anticipated by Berthollet (4), the rate at which the dissolved substance is distributed by diffusion into the rest of the solution is proportional to the fall in concentration and, consequently, is entirely analogous to the rate of dissolution. Here, of course, it is assumed that the surface of the solid body remains constant; in general, however, the rate of dissolution is proportional to the surface. Thus, this regularity is entirely analogous to the so-called Newton’s law concerning the rate of transfer of heat from a hotter body to a colder body: this rate of transfer is directly proportional to the distance of the given system from the state of equilibrium, i.e., to the uniform heating of the whole system, since the more highly one body is heated than the other, the faster the transfer takes place; with respect to the quantity of heat transferred per unit time, here too the regularity of the rate of dissolution is preserved: it is proportional to the surface of the cooling body.

Noyes and Whitney (2) express the regularity in dissolution by the following formula

\[ \frac{dx}{dt}=kF(S-c). \qquad\qquad\qquad\qquad\qquad\qquad\qquad (1). \]

Here, \(x\) — the quantity of dissolved substance,
\(t\) — time from the beginning of dissolution,
\(k\) — a constant,
\(F\) — the surface of the body,
\(S\) — the concentration of the saturated solution,
\(c\) — „ solution at the given moment.

We can replace \(c=\dfrac{x}{v}\), where \(v\) is the volume of the solution; then

\[ \frac{dc}{dt}=\frac{kF}{v}(S-c)\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (II). \]

\[ \int \frac{dc}{S-c}=\int \frac{kF\,dt}{v}=\frac{kF}{v}\,t+\text{const.} \]

\[ -\ln(S-c)=\frac{kFt}{v}+\text{const};\quad \text{when } t=0,\ c=0;\ \text{const}=-\ln S \]

\[ -\ln(S-c)=\frac{kFt}{v}-\ln s;\quad \text{or}\quad \frac{kFt}{v}=\ln\frac{S}{S-c}\ldots\ldots\ldots\ldots\ldots (III). \]

\[ \text{i.e.}\quad k=\frac{v}{Ft}\ln\frac{S}{S-c}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (IV). \]

As we see from Eq. (IV), the rate constant is analogous to the rate constant of a first-order reaction (except that it includes the factor \(\dfrac{v}{F}\)). Nernst and Brunner (5) also use the following formula:

\[ \frac{dx}{dt}=\frac{DF}{\delta}(S-c),\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (V), \]

leaving the former notation and introducing instead

\[ k=\frac{D}{\delta}, \]

where \(D\) is the diffusion coefficient, and \(d\) is the thickness of the saturated layer near the surface.

On the basis of general considerations, M. Wildermann (6) was the first to give a theoretical derivation, and in essence all his considerations entered into the subsequent formulas, with the exception of the work of Nernst and Brunner (5) in the part where they replace \(k\) by \(\dfrac{D}{\delta}\).

Let us consider in more detail the diffusion theory of dissolution given in the works of Nernst and Brunner (2). Nernst, in his article, which is as it were introductory to Brunner’s work, points out that many facts lead to the supposition that equilibrium at the boundary of two different phases is established extremely rapidly, since at the surface of two phases there are “infinitely great forces” causing the process to occur with enormous speed, so that at the surface of a solid body in contact with a liquid, equilibrium is established “infinitely rapidly.” He further says that if we are dealing not with a pure process of dissolution, but with a chemical reaction in a heterogeneous system, as, for example, with the solubility of MgO in acids, then here too, in infinite proximity to the surface of MgO, the solution becomes saturated and even “weakly alkaline”; the diffusing acid HCl must be completely neutralized at the interface, and the rate of dissolution of MgO will depend substantially on the rate at which HCl penetrates to the interface between MgO and the solution. If, in this case, the solution is stirred vigorously, then the composition of the solution may be taken as constant, and the “Diffusionsgefälle,” the diffusion gradient, slope, will be limited to a thin layer, of thickness \(\delta\), adjoining the interface. And with respect to various electrochemical reactions, Nernst likewise considers only this diffusion theory of his to be applicable, in contrast to van’t Hoff’s theory, which regards it as possible to apply the determination of the order of a reaction by the number of reacting molecules also in a heterogeneous system; this latter theory Nernst considers “entirely un-

“motivated” and even “untenable,” since in a heterogeneous system the reaction rate will depend wholly or partly on the rate of diffusion, which has no connection whatever with the order of the reaction.

Meyer Wildermann (7) opposed this diffusion theory with many weighty objections. He does not recognize any layer with a saturated solution around the solid body and maintains, in accordance with the general law, that the rate at which any form of energy, removed from the state of equilibrium, tends toward this state is directly proportional to the departure of this form from the state of equilibrium, or proportional to the difference of the potentials of the given point and of the state of equilibrium.

He considers that all equations mathematically expressed in the form

\[ -\frac{dc}{dt} = \]

\[ = kF(S-c) \]

which determine the reaction rate in a heterogeneous system and which were used by Noyes and Whitney (2), Brunner and Tolloczko (8), are nothing other than the formulation given by him as early as 1896 (see 6).

Nernst and Brunner (5), instead of \(k\), substitute \(\frac{D}{\delta}\), where \(D\) is the rate of diffusion, and \(\delta\) is the thickness of the layer adjacent to the solid body; thus to attach a meaning to the constant \(k\) simply means to give it its explanation and nothing more; for this reason he lays claim to priority in the clarification of this question. However, it seems to me that priority most probably belongs to Boguskii (9), who worked in D. I. Mendeleev’s laboratory, since as early as 1876 he used, in his work in which he determined the rate of dissolution of Carrara marble, the formula

\[ \frac{dx}{dt}=kF(C-x) \]

Among the further inaccuracies noted by Wildermann in the work of Nernst and Brunner, the following must also be mentioned:

1) The material used was not chemically pure; for example, \(Mg(OH)_2\), by their method of preparation, always contained \(MgCO_3\), since without \(MgCO_3\) \(Mg(OH)_2\) alone would not have yielded plates that did not fall apart.

2) When Nernst and Brunner (5), in order to obtain dense plates of “\(MgO\),” bound it with 10% gum arabic, they naturally introduced K and Ca salts of arabic acid. On this point Wildermann exclaims: “And on the basis of such mixtures Nernst and Brunner give their quantitative determinations of the rate of dissolution, their concordant results, their important proofs in favor of the diffusion theory.”

3) Against the diffusion theory of dissolution there also speaks the circumstance that the change of the rate constant with temperature obeys van’t Hoff’s law for molecular reactions in a homogeneous medium, as he showed in a number of examples.

4) A very strong refutation of Nernst’s diffusion theory is also the unequal rate of dissolution of different faces of crystals. Thus, according to Nernst’s theory, \(CaSO_4.2H_2O\) should, regardless of which face of the crystal we take, dissolve at the same rate; \(k\), ceteris paribus, should be one and the same, since, first, the reaction rate at the surface is infinitely large, and second, the saturated solution in all cases should be a solution of \(CaSO_4\). But, as has been shown by many authors, \(k\), the rate of dissolution, both of the different faces of \(CaSO_4.2H_2O\) and of different varieties of \(CaSO_4.2H_2O\), such as selenite, alabaster, and fibrous gypsum, despite identical conditions ...

of dissolution are different; for example, the ratios for the rates of dissolution of selenite : fibrous gypsum : alabaster = 1 : 3.5 : 5.4.

Thus Wildermann believes that Nernst’s assertion that van’t Hoff’s theory of the homogeneous system “has nothing to do” with a heterogeneous system is incorrect; that, on the contrary, van’t Hoff’s theory contains principles applicable also to a heterogeneous system, with the addition, of course, of new factors, and that the sooner we rid ourselves of the numerous and unnatural assumptions of the “diffusion theory,” the better it will be for this field of research.

Without dwelling on the works of other authors, among whom should be mentioned Drucker (10), Danneel (11), Andreev (12), Marck (13), Denham (14), Roth (15), Yamasaki (16), Name (17), among whom there were quite a few adherents of Nernst’s diffusion theory, I shall in what follows discuss in greater detail only the work of Centnerszwer (19).

In view of the fact that in the works of the authors mentioned considerable fluctuations were often observed in the value \(k\)—the reaction-rate constant—and that all calculations of this value were made, in essence, by the formulae for monomolecular reactions, I decided to devise a method and to set up a series of experiments that would make it possible, with sufficient speed and accuracy, at any moment from the beginning of the reaction, to determine the amount of one of the reaction products formed and to ascertain for which order of reaction the constancy of the value \(k\) is best preserved.

As such a method I used the determination of the amount of the substance formed from the electrical conductivity of the solutions.

It goes without saying that this method could be applied only to such reactions in which all the reaction products, except one, possess negligibly small electrical conductivity; a characteristic reaction of this kind is the action of metals on mercuric chloride, the solutions of which are very weakly dissociated.

Indeed, for example, the electrical conductivity is

\[ \begin{array}{rll} \text{for a } 2.5\% \text{ solution of } \mathrm{ZnCl_2} & K_{18}\,10^4 &= 276,\\ \text{for a } 5.08\% \text{ solution of } \mathrm{HgCl_2} & & = 4.21,\\ \text{for a } 1.013\% \text{ solution of } \mathrm{HgCl_2} & & = 1.14,\\ \text{for a saturated solution of } \mathrm{Hg_2Cl_2} & & = 0.012. \end{array} \]

Thus the change in the electrical conductivity of a solution of \(\mathrm{HgCl_2}\) into which, for example, Zn is immersed may practically be attributed only to the increasing amount of \(\mathrm{ZnCl_2}\) and only to a small extent to the decrease in the amount of decomposing \(\mathrm{HgCl_2}\).

The procedure was as follows:

1) On a platinum plate (\(8\ \mathrm{cm^2}\)) a layer of the metal under investigation was deposited electrolytically.

2) A definite solution of \(\mathrm{HgCl_2}\) was prepared, the concentration of which was checked by the specific electrical conductivity.

3) The platinum plate with the metal layer deposited on it was attached to the axis of a vertically mounted electric motor and introduced into the \(\mathrm{HgCl_2}\) solution, where electrodes for determining the electrical conductivity had already been placed beforehand.

I shall present, from a series of experiments carried out by this method, some with detailed numerical data.

The experiment given in Table No. 1 was carried out with a \(1.125\%\) solution of \(\mathrm{HgCl_2}\). The logarithm of the capacity of the vessel for determining the electrical conductivity = \(1.16406\). Having attempted to apply to the numerical data obtained

According to the formulas of the first, second, and even third order, I obtained a strongly varying value of \(k\). The latter proved constant only when I referred it to \(2/3\) molecular, i.e. the reaction corresponded to the equation

\[ \frac{dx}{dt}=k(a-x)^{2/3} \]

Table No. 1.

Whence

\[ \frac{dx}{(a-x)^{2/3}}=k\,dt \]

\[ \int \frac{dx}{(a-x)^{2/3}}=kt+\mathrm{const}, \quad \text{i.e. } -\frac{(a-x)^{\frac{1}{3}}}{\frac{1}{3}}=kt+\mathrm{const}, \text{ or} \]

\[ -3\sqrt[3]{a-x}=kt+\mathrm{const}; \quad \text{when } t=0,\ x=0,\ \text{whence } \mathrm{const}.=-3\sqrt[3]{a} \]

\[ \text{Consequently, } \quad k=\frac{3}{t}\left(\sqrt[3]{a}-\sqrt[3]{a-x}\right). \]

This is the value of \(k\) which I used in calculating the results from the experimental data.

In the following Table No. 2 an experiment is given, carried out with a stronger solution of \(HgCl_2\), namely 1.687%, and there is also included a column with figures for the amount of decomposed \(HgCl_2\), calculated by the formula for a \(2/3\)-molecular reaction, for comparison with those found experimentally. The discrepancy, as is evident from the table, does not exceed 0.3%, a magnitude quite possible as the error of the observation itself.

A reaction carried out in a similar manner with Ni, also electrolytically deposited on Pt, likewise gave a perfectly clear indication that here too we are dealing with a \(2/3\)-molecular reaction. The discrepancy between the values calculated by the formula and the experimental data determining the amount of decomposed \(HgCl_2\) here also does not exceed 0.3%. The \(HgCl_2\) solution was 1.685%.

Taking one of the experiments in Centnerszwer’s work, published in Zeit. f. ph. Chemie for 1915 (p. 226, experiment No. 14), where he determines the rate of dissolution of an alloy of Zn with 4.47% Cu in HCl from the volume of hydrogen formed and evolved in this reaction, and applying, for the calculation of the quantity \(k\), the formula for a \(^{2}/_{3}\)-molecular reaction, I obtained values of \(k\) that are more constant than those obtained by the monomolecular formula used by him. This is clearly shown by Table 4 below. I do not give a dozen other calculations that present the same picture.

This same method, applied to the metal Cd, proved unsuitable, since it gave a continuously decreasing value of \(k\), owing apparently to the formation of a complex salt \(HgCl_2\) with a low degree of dissociation.

Table No. 2.

Table No. 3.

Table 4.

Conclusions

1) A method has been developed for the precise and rapid study of reaction rates in which only one of the reaction products is strongly dissociated.

2) The reaction of interaction between a sulema solution and metallic Zn and Ni is of \(^{2}/_{3}\)-molecular order.

3) The reaction of interaction between alloys of zinc and copper and HCl proceeds according to a \(^{2}/_{3}\)-molecular formula.

4) \(\mathrm{CdCl_2}\) forms a complex compound with a solution of \(\mathrm{HgCl_2}\).

Tomsk, October 24—26.

Literature:

1. Stefan—Monatsh. f. Ch. 10, 201. (1889).
2. Noyes u. Whitney—Zeit. ph. Ch. 23, 689. (1897).
3. Fick—Pogg. Ann. 94, 59. (1855).
4. Berthollet—Essai de st. chim., Paris 1803 I Th. 4 k.
5. Nernst u. Brunner—Zeit. ph. Ch. 47, 52, 63. (1904).
6. M. Wildermann—Repor. of Br. Ass., 1894. u. Z. ph. 30, 398 (1899).
7. M. Wildermann Z. ph. Ch. 1909, LXVI, 445.
8. Brunner u. Tolloczko—Z. ph. Ch. 47, 52. (1904).
9. Boguski—Kosmos 1, 538. (1875).
10. Drucker—Z. anorg. Ch. 29, 459. (1902).
11. Danneel—Z. ph. Ch. 33, 415. (1910).
12. Andreeff—Z. f. Krist. 43, 9. (1907).
13. Marc—Z. ph. Ch. 51, 386. (1908).
14. Denham—Jnaug. Diss., Heidelb. 1909. 76.
15. Roth—Z. f. Elektr. 16, 714, (1910).
16. Jamasaki—Z. f. Elektr. 16,85.
17. Name—Z. f. Elektr 73,97.
18. Centnerszwer u. Sacks.—Z. ph. Ch. 87, 692. (1914).
19. Centnerszwer—Z. ph. Ch. 89, 226. (1915).