Svante Arrhenius

In 1887 a young Swedish physicist, Svante Arrhenius, advanced the theory of solutions.[23]

The ideas of Williamson and Clausius on the spontaneous dissociation of electrolytes led to the properties discovered by Kohlrausch[24].

Arrhenius inferred:

• that in very dilute solutions the electrolyte is completely dissociated into ions, but that in more concentrated solutions the salt is less completely dissociated; and - that as in all solutions the transport of electricity in the solution is effected solely by the movement of ions, the equivalent conductivity[25] must be proportional to the fraction which expresses the degree of ionization.

It thus became possible to:

• estimate the dissociation quantitatively, and
• construct a general theory of electrolytes.

Contemporary physicists and chemists found it difficult at first to believe that a salt exists in dilute solution only in the form of ions. For example, sodium and chlorine exist separately and independently in a solution of common salt.

But there is chemical evidence in favour of Arrhenius’ conception.

For instance, the tests in chemical analysis are really tests for the ions.

Iron as ferrocyanide, and chlorine as chlorate, do not respond to the characteristic tests for iron and chlorine respectively. These are really the tests for iron and chlorine ions.

The general acceptance of Arrhenius’ views was hastened by the advocacy of Ostwald, who brought to light further evidence in their favour.

For instance, all permanganates in dilute solution show the same purple colour;

Ostwald considered their absorption-spectra to be identical[26].

This identity is easily accounted for on Arrhenius’ theory, by supposing that the spectrum in question is that of the anion which corresponds to the acid radicle.

The blue colour which is observed in dilute solutions of copper salts, even when the strong solution is not bine, may in the same way be ascribed to a blue copper cation. A striking instance of the same kind is afforded by ferric sulphocyanide; here the strong solution shows a deep red colour, due to the salt itself; but on dilution the colour disappears, the ions being colourless.

If it be granted that ions can have any kind of permanent existence in a salt solution, it may be shown from thermodynamical considerations that the degree of dissociation must increase as te dilution increases, and that at infinite dilution there must be complete dissociation.

For the available energy of a dilute solution of volume V, containing n1, gramme-molecules of one substance, n2 gramme-molecules of another, and so on, is (as may be shown by an obvious extension of the reasoning already employed in connexion with concentration-cells)[27]

the available energy { the available energy}

possessed by the solvent before the introduction of the solutes, where φr(T) depends on T and on the nature of the rth solute, but not on V, and R denotes the constant which occurs in the equation of state of perfect gases.

When the system is in equilibrium, the proportions of the reacting substances will be so adjusted that the available energy has a stationary value for small virtual alterations ∂n1, ∂n2, …… of the proportions; and therefore

Applying this to the case of an electrolyte in which the disappearance of one molecule of salt indicated by the suffix 1) gives rise to one cation (indicated by the suffix 2) and one anion (indicated by the suffix 3), we have ∂n1 = -∂n2 = -∂n3; so the equation becomes

or

… a function of

… only

…

Since in a neutral solution the number of anions is equal to the number of cations, this equation may be written

…

a function of

…

only

…

times

it shows that when V is very large (so that the solution is very dilute), n2 is very large compared with n1; that is to say, the salt tends towards a state of complete dissociation.

The ideas of Arrhenius contributed to the success of Walther Nernst[28] in perfecting Helmholtz theory of concentration-cells, and representing their mechanism in a much more definite fashion than had been done heretofore.

In an electrolytic solution let the drift-velocity of the cations under unit electric force be u, and that of the anions be v, so that the fraction u/(u + v) of the current is transported by the cations, and the fraction v/(u + v) by the anions.

If the concentration of the solution be c1 at one electrode, and c2, at the other, it follows from the formula previously found for the available energy that one gramme-ion of cations, in moving from one electrode to the other, is capable of yielding up an amount[29] RT log (c2/c1) of energy; while one gramme-ion of anions going in the opposite direction must absorb the same amount of energy. The total quantity of work furnished when one gramme-molecule of salt is transferred from concentration c2; to concentration c1 is therefore

The quantity of electric charge which passes in the circuit when one gramne-molecule of the salt is transferred is proportional to the valency ν of the ions, and the work furnished is proportional to the product of this charge and the electromotive force E of the cell; so that in suitable units we have

…

A typical concentration-cell to which this formula may be applied may be constituted in the following way:

…

Let a quantity of zinc amalgam, in which the concentration of zinc is c1, be in contact with a dilute solution of zinc sulphate, and let this in turn be in contact with a quantity of zinc amalgam of concentration c2.

When the 2 masses of amalgam are connected by a conducting wire outside the cell, an electric current flows in the wire from the weak to the strong amalgam,[30] while zinc cations pass through the solution from the strong amalgam to the weak.

The electromotive force of such a cell, in which the current may be supposed to be carried solely by cations, is

…

Not content with the derivation of the electromotive force from considerations of energy, Nernst proceeded to supply a definite mechanical conception of the process of conduction in electrolytes.

The ions are impelled by the electric force associated with the gradient of potential in the electrolyte. But this is not the only force which acts on them; for, since their available energy decreases as the concentration decreases, there must be a force assisting every process by which the concentration is decreased.

The matter may be illustrated by the analogy of a gas compressed in a cylinder fitted with a piston; the available energy of the gas decreases as its degree of compression decreases; and therefore that movement of the piston which tends to decrease the compression is assisted by a force—the “pressure” of the gas on the piston.

Similarly, if a solution were contained within a cylinder fitted with a piston which is permeable to the pure solvent but not to the solute, and if the whole were immersed in pure solvent, the available energy of the system would be decreased if the piston were to move outwards so as to admit more solvent into the solution; and therefore this movement of the piston would be assisted by a force—the “osmotic pressure of the solution,” as it is called.[31]

Consider, then, the case of a single electrolyte supposed to be perfectly dissociated; its state will be supposed to be the same at all points of any plane at right angles to the axis of x. Let ν denote the valency of the ions, and V the electric potential at any point.

Since[32] the available energy of a given quantity of a substance in very dilute solution depends on the concentration in exactly the same way as the available energy of a given quantity of a perfect gas depends on its density, it follows that the osmotic pressure p for each ion is determined in terms of the concentration and temperature by the equation of state of perfect gases

…

where M denotes the molecular weight of the salt, and c the mass of salt per unit volume.

Consider the cations contained in a parallelepiped at the place x, whose cross-section is of unit area and whose length is dx.

The mechanical force acting on them due to the electric field is -(vc/M)dV/dx.dx, and the mechanical force on them due to the osmotic pressure is -dp/dx.dx. If u denote the velocity of drift of the cations in a field of unit electric force, the total amount of charge which would be transferred by cations across unit area in unit time under the influence of the electric forces alone would be -(uνe/M)dV/dx; so, under the influence of both forces, it is

…

Similarly, if v denote the velocity of drift of the anions in a unit electric field, the charge transferred across unit area in unit time by the anions is

…

We have therefore, if the total current be denoted by i,

…

The first term on the right evidently represents the product of the current into the ohmic resistance of the parallelepiped dx, while the second term represents the internal electromotive force of the parallelepiped. It follows that if r denote the specific resistance, we must have

in agreement with Kohlrausch’s equation;[33] while by integrating the expression for the internal electromotive force of the parallelepiped dx, we obtain for the electromotive force of a cell whose activity depends on the transference of electrolyte between the concentrations c1 and c2, the value

or

in agreement with the result already obtained.

Although the current arising from a concentration cell which is kept at a constant temperature is capable of performing work, yet this work is provided, not by any diminution in the total internal energy of the cell, but by the abstraction of thermal energy from neighbouring bodies.

This (as may be seen by reference to W. Thomson’s general equation of available energy)[34] must be the case with any system whose available energy is exactly proportional to the absolute temperature.