Conduction In Solutions And Gases, From Faraday To Thomson

Grothuss and Davy gave a hypothesis[1] to explain the decomposition of electrolytes

• It was open to serious objection in more than one respect.

The electric force was supposed first to dissociate the molecules of the electrolyte into ions, and afterwards to set them in motion toward the electrodes.

• The doubling of the electric force should:
  • double both the dissociation of the molecules and the velocity of the ions
  • quadruple the electrolysis—an inference

These were not verified by observation.

Grothuss’ theory suggests that:

• some definite magnitude of electromotive force would be requisite for the dissociation
• no electrolysis at all would take place when the electromotive force was below this value

These were again contrary to experience.

Alex. Williamson[2] in 1850 tried to solve this by suggesting that in compound liquids decompositions and recombinations of the molecules are continually taking place throughout the whole mass of the liquid, quite independently of the application of an external electric force.

An atom of one element in the compound is thus paired now with one and now with another atom of another element, and in the intervals between these alliances the atom may be regarded as entirely free.

In 1857 this idea was made by R. Clausius,[3] of Zurich, the basis of a theory of electrolysis.

According to it, the electromotive force emanating from the electrodes does not effect the dissociation of the electrolyte into ions, since a degree of dissociation sufficient for the purpose already exists in consequence of the perpetual mutability of the molecules of the electrolyte.

Clausius assumed that these ions are in opposite electric conditions. The applied electric force therefore causes a general drift of all the ions of one kind towards the anode, and of all the ions of the other kind towards the cathode.

These opposite motions of the two kinds of ions constitute the galvanic current in the liquid.

The merits of the Williamson-Clausius hypothesis were not fully recognized for many years; but it became the foundation of that theory of electrolysis which was generally accepted at the end of the century.

Meanwhile, another aspect of electrolysis was receiving attention.

It had long been known that the passage of a current through an electrolytic solution is attended not only by the appearance of the products of decomposition at the electrodes, but also by changes of relative strength in different parts of the solution itself.

Thus, in the electrolysis of a solution of copper sulphate, with copper electrodes, in which copper is dissolved off the anode and deposited on the cathode, the concentration of the solution diminishes near the cathode, and increases near the anode.

Faraday[4] made some experiments on this in 1835. In 1844, this was further investigated by Frederic Daniell and W. A. Miller[5]. They asserted that the cation and anion have not (as had previously been supposed) the same facility of moving to their respective electrodes. Instead, the cation appears to move but little, while the transport is effected chiefly by the anion.

This idea was adopted by W. Hittorf of Münster. In 1853 to 1859, he published[6] a series of memoirs on the migration of the ions.

Let the velocity of the anions in the solution be to the velocity of the cations in the ratio v:u. Then it is easily seen that if (u + v) molecules of the electrolyte are decomposed by the current, and yielded up as ions at the electrodes, v of these molecules will have been taken from the fluid on the side of the cathode, and u of them from the fluid on the side of the anode.

By measuring the concentration of the liquid round the electrodes after the passage of a current, Hittorf determined the ratio v/u in a large number of cases of electrolysis.[7]

The theory of ionic movements was advanced a further stage by F. W. Kohlrausch[8] (b. 1840, d. 1910), of Würzburg. Kohlrausch showed that although the ohmic specific conductivity k of a solution diminishes indefinitely as the strength of the solution is reduced, yet the ratio k/m, where m denotes the number of gramme-equivalents[9] of salt per unit volume, tends to a definite limit when the solution is indefinitely dilute.

This limiting value may be denoted by λ. He further showed that λ may be expressed as the sum of two parts, one of which depends on the cation, but is independent of the nature of the anion; while the other depends on the anion, but not on the cation—a fact which may be explained by supposing that, in very dilute solutions, the two ions move independently under the influence of the electric force.

Let u and v denote the velocities of the cation and anion respectively, when the potential difference per em. in the solution is unity: then the total current carried through a cube of unit volume is mE (u + v), where E denotes the electric charge carried by one grammeequivalent of ion.[10]

Thus mE (u + v) = total current = k = mλ, or λ = E (u + v). The determination of v/u by the method of Hittorf, and of (u + v) by the method of Kohlrausch, made it possible to calculate the absolute velocities of drift of the ions from experimental data.

Meanwhile, important advances in voltaic theory were being effected in connexion with a different class of investigations.

Suppose that two mercury electrodes are placed in a solution of acidulated water, and that a difference of potential, insufficient to produce continuous decomposition of the water, is set up between the electrodes by an external agency.

Initially a slight electric current—the polarizing current,[11] as it is called—is observed; but after a short time it ceases; and after its cessation the state of the system is one of electrical equilibrium.

The polarizing current must in some way have set up in the cell an electromotive force equal and opposite to the external difference of potential; and it is also evident that the seat of this electromotive force must be at the electrodes, which are now said to be polarized.

An abrupt fall of electric potential at an interface between two media, such as the mercury and the solution in the present case, requires that there should be a field of electric force, of considerable intensity, within a thin stratum at the interface: and this must owe its existence to the presence of electric charges.

Since there is no electric field outside the thin stratum, there must be as much vitreous as resinous electricity present; but the vitreous charges must preponderate on one side of the stratum, and the resinous charges on the other side; so that the system as a whole resembles the two coatings of a condenser with the intervening dielectric.

In the case of the polarized mercury cathode in acidulated water, there must be on the electrode itself a negative charge: the surface of this electrode in the polarized state may be supposed to be either mercury, or mercury covered with a layer of hydrogen.

In the solution adjacent to the electrode, there must be an excess of cations and a deficiency of anions, so as to constitute the other layer of the condenser: these cations may be either mercury cations dissolved from the electrode, or the hydrogen cations of the solution.

It was shown in 1870 by Cromwell Fleetwood Varley[12] that a mercury cathode, thus polarized in acidulated water, shows i tendency to adopt a definite superficial form, as if the surfacetension at the interface between the mercury and the solution were in some way dependent on the electric conditions. The matter was more fully investigated in 1873 by a young French physicist, then preparing for his inaugural thesis, Gabriel Lippmann.[13]

In Lippmann’s instrumental disposition, which is called a capillary electrometer, mercury electrodes are immersed in acidulated water: the anode H0, has a large surface, while the cathode H has a variable surface S small in comparison.

When the external electromotive force is applied, it is easily seen that the fall of potential at the large electrode is only slightly affected, while the fall of potential at the small electrode is altered by polarization by an amount practically equal to the external electromotive force. Lippmann found that the constant of capillarity of the interface at the small electrode was a function of the external electromotive force, and therefore of the difference of potential between the mercury and the electrolyte.

Let V denote the external electromotive force: we may, without loss of generality, assume the potential of H0, to be zero, so that the potential of H is -V. The state of the system may be varied by altering either V or S; we assume that these alterations may be performed independently, reversibly, and isothermally, and that the state of the large electrode H0, is not altered thereby. Let de denote the quantity of electricity which passes through the cell from H0, to H, when the state of the system is thus varied: then if E denote the available energy of the system, and γ the surface-tension at H, we have

γ being measured by the work required to increase the surface when no electricity flows through the circuit.

In order that equilibrium may be re-established between the electrode and the solution when the fall of potential at the cathode is altered, it will be necessary not only that some hydrogen cations should come out of the solution and be deposited on the electrode, yielding up their charges, but also that there should be changes in the clustering of the charged ions of hydrogen, mercury, and sulphion in the layer of the solution immediately adjacent to the electrode.

Each of these circumstances necessitates a flow of electricity in the outer circuit: in the one case to neutralize the charges of the cations deposited, and in the other case to increase the surface-density of electric charge on the electrode, which forms the opposite sheet of the quasi-condenser. Let Sf (V) denote the total quantity of electricity which has thus flowed in the circuit when the external electromotive force has attained the value V. Then evidently

so

…

Since this expression must be an exact differential, we have

…

so that -dy/dV is equal to that flux of electricity per unit of new surface formed, which will maintain the surface in a constant condition (V being constant) when it is extended. Integrating the previous equation, we have

…

Lippmann found that when the external electromotive force was applied, the surface-tension increased at first, until, when the external electromotive force amounted to about one volt, the surface-tension attained a maximum value, after which it diminished. He found that d2γ/dV2 was sensibly independent of V, so that the curve which represents the relation between γ and V is a parabola.[14]

The theory so far is more or less independent of assumptions as to what actually takes place at the electrode: on this latter question many conflicting views have been put forward. In 1878 Josiah Willard Gibbs,[15] of Yale (b. 1839, d. 1903), discussed the problem on the supposition that the polarizing current is simply all ordinary electrolytic conduction-current, which causes a liberation of hydrogen from the ionic form at the cathode.

If this be so, the amount of electricity which passes through the cell in any displacement must be proportional to the quantity of hydrogen which is yielded up to the electrode in the displacement; so that dγ/dV must be proportional to the amount of hydrogen deposited per unit area of the electrode.[16]

A different view of the physical conditions at the polarized electrode was taken by Helmholtz,[17] who assumed that the ions of hydrogen which are brought to the cathode by the polarizing current do not give up their charges there, but remain in the vicinity of the electrode, and form one face of a quasi-condenser of which the other face is the electrode itself.[18] If σ denote the surface-density of electricity on either face of this quasicondenser, we have, therefore,

…

This equation shows that when dγ/dV is zero—i.e., when the surface-tension is a maximum—o must be zero; that is to say, there must be no difference of potential between the mercury and the electrolyte. The external electromotive force is then balanced entirely by the discontinuity of potential at the other electrode H0; and thus a method is suggested of measuring the latter discontinuity of potential.

All previous measurements of differences of potential had involved the employment of more than one interface; and it was not known how the measured difference of potential should be distributed among these interfaces; so that the suggestion of a means of measuring single differences of potential was a distinct advance, even though the hypotheses on which the method was based were somewhat insecure.

A further consequence deduced by Helmholtz from this theory leads to a second method of determining the difference of potential between mercury and an electrolyte. If a mercury surface is rapidly extending, and electricity is not rapidly transferred through the electrolyte, the electric surface-density in the double layer must rapidly decrease, since the same quantity of electricity is being distributed over an increasing area Thus it may be inferred that a rapidly extending mercury-surface in an electrolyte is at the same potential as the electrolyte.

This conception is realized in the dropping-electrode, in which a jet of mercury, falling from a reservoir into an electrolytic solution, is so adjusted that it breaks into drops when the jet touches the solution. According to Helmholtz’s conclusion there is no difference of potential between the drops and the electrolyte.

Therefore, the difference of potential between the electrolyte and a layer of mercury underlying it in the same vessel is equal to the difference of potential between this layer of mercury and the mercury in the upper reservoir, which difference is a measurable quantity.

It will be seen that according to the theories both of Gibbs and of Helmholtz, and indeed according to all other theories on the subject,[19] dγ/dV is zero for an electrode whose surface is rapidly increasing—e.g., a dropping electrode; that is to say, the difference of potential between an ordinary mercury electrode and the electrolyte, when the surface-tension has its maximum value, is equal to the difference of potential between a dropping-electrode and the same electrolyte.

This result has been experimentally verified by various investigators, who have shown that the applied electromotive force when the surface-tension has its maximum value in the capillary electrometer, is equal to the electromotive force of a cell having as electrodes a large mercury electrode and a dropping electrode.

Another memoir which belongs to the same period of Helmholtz’ career, and which has led to important developments, was concerned with a special class of voltaic cells. The most usual type of cell is that in which the positive electrode is composed of a different metal from the negative electrode, and the evolution of energy depends on the difference in the chemical affinities of these metals for the liquids in the cell.

But in the class of cells now considered[20] by Helmholtz, the two electrodes are composed of the same metal (say, copper); and the liquid (say, solution of copper sulphate) is more concentrated in the neighbourhood of one electrode than in the neighbourhood of the other.

When the cell is in operation, the salt passes from the places of high concentration to the places of low concentration, so as to equalize its distribution, and this process is accompanied by the flow of a current in the outer circuit between the electrodes. Such cells had been studied experimentally by James Moser a short time previously[21] to Helmholtz’ investigation.

The activity of the cell is due to the fact that the available energy of a solution depends on its concentration, the molecules

of salt, in passing from a high to a low concentration, are therefore capable of supplying energy, just as a compressed gas is capable of supplying energy when its degree of compression is reduced. To examine the matter quantitatively, let nf(n/V) denote the term in the available energy of a solution, which is due to the dissolution of n gramme-molecules of salt in a volume V of pure solvent; the function f will of course depend also on the temperature.

Then when dn gramme-molecules of solvent are evaporated from the solution, the decrease in the available energy of the system is evidently equal to the available energy of dn gramme-molecules of liquid solvent, less the available energy of dn gramme-molecules of the vapour of the solvent, together with nf(n/V) less nf{n/(V-vdn)}, where v denotes the volume of one gramme-molecule of the liquid. But this decrease in available energy must be equal to the mechanical work supplied to the external world, which is dn.p1(u′ – v), if p1, denote the vapour-pressure of the solution at the temperature in question, and {{Wikimath|v′ denote the volume of one gramme-molecule of vapour. We have therefore

…-available energy of dn gramme-molecules of solvent vapour +available energy of dn gramme-molecules of liquid solvent

…

Subtracting from this the equation obtained by making n zero, we have

…

where p0 denotes the vapour-pressure of the pure solvent at the temperature in question; so that

…

When a salt is dissolved in water, the vapour-pressure is lowered in proportion to the concentration of the salt – at any rate, when the concentration is small: in fact, by the law of Raoult, {{Wikimath|(p0-p1/p0, is approximately equal to nv/V; so that the previous equation becomes

Neglecting v in comparison with v′, and making use of the equation of state of perfect gases (namely,

…

where T denotes the absolute temperature, and R denotes the constant of the equation of state), we have

…

and therefore

…

Thus in the available energy of one gramme-molecule of a dissolved salt, the term which depends on the concentration is proportional to the logarithm of the concentration; and hence, if in a concentration-cell one gramme-molecule of the salt passes from a high concentration c2, at one electrode to a low concentration c1 at the other electrode, its available energy is thereby diminished by an amount proportional to log c2/c1.

The energy which thus disappears is given up by the system in the form of electrical work; and therefore the electromotive force of the concentration-cell must be proportional to log c2/c1. The theory of solutions and their vapour-pressure was not at the time sufficiently developed to enable Helmholtz to determine precisely the coefficient of log c2/c1 in the expression.[22]