Chapter XV: Part 15

2CH3.COO + H2O = 2CH3.COOH + O

2CH3.COOH + O = C2H6 + 2CO2 + H2O.

The hydrogen at the cathode is developed by the secondary action

2Na + 2H2O = 2NaOH + H2.

Many organic compounds can be prepared by taking advantage of
secondary actions at the electrodes, such as reduction by the cathodic
hydrogen, or oxidation at the anode (see ELECTROCHEMISTRY).

It is possible to distinguish between double salts and salts of
compound acids. Thus J.W. Hittorf showed that when a current was
passed through a solution of sodium platino-chloride, the platinum
appeared at the anode. The salt must therefore be derived from an
acid, chloroplatinic acid, H2PtCl6, and have the formula Na2PtCl6, the
ions being Na and PtCl6", for if it were a double salt it would
decompose as a mixture of sodium chloride and platinum chloride and
both metals would go to the cathode.

_Early Theories of Electrolysis._--The obvious phenomena to be explained by any theory of electrolysis are the liberation of the products of chemical decomposition at the two electrodes while the intervening liquid is unaltered. To explain these facts, Theodor Grotthus (1785-1822) in 1806 put forward an hypothesis which supposed that the opposite chemical constituents of an electrolyte interchanged partners all along the line between the electrodes when a current passed. Thus, if the molecule of a substance in solution is represented by AB, Grotthus considered a chain of AB molecules to exist from one electrode to the other. Under the influence of an applied electric force, he imagined that the B part of the first molecule was liberated at the anode, and that the A part thus isolated united with the B part of the second molecule, which, in its turn, passed on its A to the B of the third molecule. In this manner, the B part of the last molecule of the chain was seized by the A of the last molecule but one, and the A part of the last molecule liberated at the surface of the cathode.

Chemical phenomena throw further light on this question. If two solutions containing the salts AB and CD be mixed, double decomposition is found to occur, the salts AD and CB being formed till a certain part of the first pair of substances is transformed into an equivalent amount of the second pair. The proportions between the four salts AB, CD, AD and CB, which exist finally in solution, are found to be the same whether we begin with the pair AB and CD or with the pair AD and CB. To explain this result, chemists suppose that both changes can occur simultaneously, and that equilibrium results when the rate at which AB and CD are transformed into AD and CB is the same as the rate at which the reverse change goes on. A freedom of interchange is thus indicated between the opposite parts of the molecules of salts in solution, and it follows reasonably that with the solution of a single salt, say sodium chloride, continual interchanges go on between the sodium and chlorine parts of the different molecules.

These views were applied to the theory of electrolysis by R.J.E. Clausius. He pointed out that it followed that the electric forces did not cause the interchanges between the opposite parts of the dissolved molecules but only controlled their direction. Interchanges must be supposed to go on whether a current passes or not, the function of the electric forces in electrolysis being merely to determine in what direction the parts of the molecules shall work their way through the liquid and to effect actual separation of these parts (or their secondary products) at the electrodes. This conclusion is supported also by the evidence supplied by the phenomena of electrolytic conduction (see CONDUCTION, ELECTRIC, S II.). If we eliminate the reverse electromotive forces of polarization at the two electrodes, the conduction of electricity through electrolytes is found to conform to Ohm's law; that is, once the polarization is overcome, the current is proportional to the electromotive force applied to the bulk of the liquid. Hence there can be no reverse forces of polarization inside the liquid itself, such forces being confined to the surface of the electrodes. No work is done in separating the parts of the molecules from each other. This result again indicates that the parts of the molecules are effectively separate from each other, the function of the electric forces being merely directive.

_Migration of the Ions._--The opposite parts of an electrolyte, which work their way through the liquid under the action of the electric forces, were named by Faraday the ions--the travellers. The changes of concentration which occur in the solution near the two electrodes were referred by W. Hittorf (1853) to the unequal speeds with which he supposed the two opposite ions to travel. It is clear that, when two opposite streams of ions move past each other, equivalent quantities are liberated at the two ends of the system. If the ions move at equal rates, the salt which is decomposed to supply the ions liberated must be taken equally from the neighbourhood of the two electrodes. But if one ion, say the anion, travels faster through the liquid than the other, the end of the solution from which it comes will be more exhausted of salt than the end towards which it goes. If we assume that no other cause is at work, it is easy to prove that, with non-dissolvable electrodes, the ratio of salt lost at the anode to the salt lost at the cathode must be equal to the ratio of the velocity of the cation to the velocity of the anion. This result may be illustrated by fig. 2. The black circles represent one ion and the white circles the other. If the black ions move twice as fast as the white ones, the state of things after the passage of a current will be represented by the lower part of the figure. Here the middle part of the solution is unaltered and the number of ions liberated is the same at either end, but the amount of salt left at one end is less than that at the other. On the right, towards which the faster ion travels, five molecules of salt are left, being a loss of two from the original seven. On the left, towards which the slower ion moves, only three molecules remain--a loss of four. Thus, the ratio of the losses at the two ends is two to one--the same as the ratio of the assumed ionic velocities. It should be noted, however, that another cause would be competent to explain the unequal dilution of the two solutions. If either ion carried with it some of the unaltered salt or some of the solvent, concentration or dilution of the liquid would be produced where the ion was liberated. There is reason to believe that in certain cases such complex ions do exist, and interfere with the results of the differing ionic velocities.

Hittorf and many other observers have made experiments to determine the unequal dilution of a solution round the two electrodes when a current passes. Various forms of apparatus have been used, the principle of them all being to secure efficient separation of the two volumes of solution in which the changes occur. In some cases porous diaphragms have been employed; but such diaphragms introduce a new complication, for the liquid as a whole is pushed through them by the action of the current, the phenomenon being known as electric endosmose. Hence experiments without separating diaphragms are to be preferred, and the apparatus may be considered effective when a considerable bulk of intervening solution is left unaltered in composition. It is usual to express the results in terms of what is called the migration constant of the anion, that is, the ratio of the amount of salt lost by the anode vessel to the whole amount lost by both vessels. Thus the statement that the migration constant or transport number for a decinormal solution of copper sulphate is 0.632 implies that of every gramme of copper sulphate lost by a solution containing originally one-tenth of a gramme equivalent per litre when a current is passed through it between platinum electrodes, 0.632 gramme is taken from the cathode vessel and 0.368 gramme from the anode vessel. For certain concentrated solutions the transport number is found to be greater than unity; thus for a normal solution of cadmium iodide its value is 1.12. On the theory that the phenomena are wholly due to unequal ionic velocities this result would mean that the cation like the anion moved against the conventional direction of the current. That a body carrying a positive electric charge should move against the direction of the electric intensity is contrary to all our notions of electric forces, and we are compelled to seek some other explanation. An alternative hypothesis is given by the idea of complex ions. If some of the anions, instead of being simple iodine ions represented chemically by the symbol I, are complex structures formed by the union of iodine with unaltered cadmium iodide--structures represented by some such chemical formula as I(CdI2), the concentration of the solution round the anode would be increased by the passage of an electric current, and the phenomena observed would be explained. It is found that, in such cases as this, where it seems necessary to imagine the existence of complex ions, the transport number changes rapidly as the concentration of the original solution is changed. Thus, diminishing the concentration of the cadmium iodine solution from normal to one-twentieth normal changes the transport number from 1.12 to 0.64. Hence it is probable that in cases where the transport number keeps constant with changing concentration the hypothesis of complex ions is unnecessary, and we may suppose that the transport number is a true migration constant from which the relative velocities of the two ions may be calculated in the matter suggested by Hittorf and illustrated in fig. 2. This conclusion is confirmed by the results of the direct visual determination of ionic velocities (see CONDUCTION, ELECTRIC, S II.), which, in cases where the transport number remains constant, agree with the values calculated from those numbers. Many solutions in which the transport numbers vary at high concentration often become simple at greater dilution. For instance, to take the two solutions to which we have already referred, we have--

+----------------------------------+------+------+------+------+------+------+------+-----+-----------+
|Concentration | 2.0 | 1.5 | 1.0 | 0.5 | 0.2 | 0.1 | 0.05 | 0.02|0.01 normal|
|Copper sulphate transport numbers | 0.72 | 0.714| 0.696| 0.668| 0.643| 0.632| 0.626| 0.62| .. |
|Cadmium iodide " " | 1.22 | 1.18 | 1.12 | 1.00 | 0.83 | 0.71 | 0.64 | 0.59|0.56 |
+----------------------------------+------+------+------+------+------+------+------+-----+-----------+

It is probable that in both these solutions complex ions exist at fairly high concentrations, but gradually gets less in number and finally disappear as the dilution is increased. In such salts as potassium chloride the ions seem to be simple throughout a wide range of concentration since the transport numbers for the same series of concentrations as those used above run--

Potassium chloride--
0.515, 0.515, 0.514, 0.513, 0.509, 0.508, 0.507, 0.507, 0.506.

The next important step in the theory of the subject was made by F. Kohlrausch in 1879. Kohlrausch formulated a theory of electrolytic conduction based on the idea that, under the action of the electric forces, the oppositely charged ions moved in opposite directions through the liquid, carrying their charges with them. If we eliminate the polarization at the electrodes, it can be shown that an electrolyte possesses a definite electric resistance and therefore a definite conductivity. The conductivity gives us the amount of electricity conveyed per second under a definite electromotive force. On the view of the process of conduction described above, the amount of electricity conveyed per second is measured by the product of the number of ions, known from the concentration of the solution, the charge carried by each of them, and the velocity with which, on the average, they move through the liquid. The concentration is known, and the conductivity can be measured experimentally; thus the average velocity with which the ions move past each other under the existent electromotive force can be estimated. The velocity with which the ions move past each other is equal to the sum of their individual velocities, which can therefore be calculated. Now Hittorf's transport number, in the case of simple salts in moderately dilute solution, gives us the ratio between the two ionic velocities. Hence the absolute velocities of the two ions can be determined, and we can calculate the actual speed with which a certain ion moves through a given liquid under the action of a given potential gradient or electromotive force. The details of the calculation are given in the article CONDUCTION, ELECTRIC, S II., where also will be found an account of the methods which have been used to measure the velocities of many ions by direct visual observation. The results go to show that, where the existence of complex ions is not indicated by varying transport numbers, the observed velocities agree with those calculated on Kohlrausch's theory.

_Dissociation Theory._--The verification of Kohlrausch's theory of ionic velocity verifies also the view of electrolysis which regards the electric current as due to streams of ions moving in opposite directions through the liquid and carrying their opposite electric charges with them. There remains the question how the necessary migratory freedom of the ions is secured. As we have seen, Grotthus imagined that it was the electric forces which sheared the ions past each other and loosened the chemical bonds holding the opposite parts of each dissolved molecule together. Clausius extended to electrolysis the chemical ideas which looked on the opposite parts of the molecule as always changing partners independently of any electric force, and regarded the function of the current as merely directive. Still, the necessary freedom was supposed to be secured by interchanges of ions between molecules at the instants of molecular collision only; during the rest of the life of the ions they were regarded as linked to each other to form electrically neutral molecules.

In 1887 Svante Arrhenius, professor of physics at Stockholm, put forward a new theory which supposed that the freedom of the opposite ions from each other was not a mere momentary freedom at the instants of molecular collision, but a more or less permanent freedom, the ions moving independently of each other through the liquid. The evidence which led Arrhenius to this conclusion was based on van 't Hoff's work on the osmotic pressure of solutions (see SOLUTION). If a solution, let us say of sugar, be confined in a closed vessel through the walls of which the solvent can pass but the solution cannot, the solvent will enter till a certain equilibrium pressure is reached. This equilibrium pressure is called the osmotic pressure of the solution, and thermodynamic theory shows that, in an ideal case of perfect separation between solvent and solute, it should have the same value as the pressure which a number of molecules equal to the number of solute molecules in the solution would exert if they could exist as a gas in a space equal to the volume of the solution, provided that the space was large enough (i.e. the solution dilute enough) for the intermolecular forces between the dissolved particles to be inappreciable. Van 't Hoff pointed out that measurements of osmotic pressure confirmed this value in the case of dilute solutions of cane sugar.

Thermodynamic theory also indicates a connexion between the osmotic pressure of a solution and the depression of its freezing point and its vapour pressure compared with those of the pure solvent. The freezing points and vapour pressures of solutions of sugar are also in conformity with the theoretical numbers. But when we pass to solutions of mineral salts and acids--to solutions of electrolytes in fact--we find that the observed values of the osmotic pressures and of the allied phenomena are greater than the normal values. Arrhenius pointed out that these exceptions would be brought into line if the ions of electrolytes were imagined to be separate entities each capable of producing its own pressure effects just as would an ordinary dissolved molecule.

Two relations are suggested by Arrhenius' theory. (1) In very dilute solutions of simple substances, where only one kind of dissociation is possible and the dissociation of the ions is complete, the number of pressure-producing particles necessary to produce the observed osmotic effects should be equal to the number of ions given by a molecule of the salt as shown by its electrical properties. Thus the osmotic pressure, or the depression of the freezing point of a solution of potassium chloride should, at extreme dilution, be twice the normal value, but of a solution of sulphuric acid three times that value, since the potassium salt contains two ions and the acid three. (2) As the concentration of the solutions increases, the ionization as measured electrically and the dissociation as measured osmotically might decrease more or less together, though, since the thermodynamic theory only holds when the solution is so dilute that the dissolved particles are beyond each other's sphere of action, there is much doubt whether this second relation is valid through any appreciable range of concentration.

At present, measurements of freezing point are more convenient and accurate than those of osmotic pressure, and we may test the validity of Arrhenius' relations by their means. The theoretical value for the depression of the freezing point of a dilute solution per gramme-equivalent of solute per litre is 1.857 deg. C. Completely ionized solutions of salts with two ions should give double this number or 3.714 deg., while electrolytes with three ions should have a value of 5.57 deg.

The following results are given by H.B. Loomis for the concentration of 0.01 gramme-molecule of salt to one thousand grammes of water. The salts tabulated are those of which the equivalent conductivity reaches a limiting value indicating that complete ionization is reached as dilution is increased. With such salts alone is a valid comparison possible.

_Molecular Depressions of the Freezing Point._

_Electrolytes with two Ions._

Potassium chloride 3.60
Sodium chloride 3.67
Potassium hydrate 3.71
Hydrochloric acid 3.61
Nitric acid 3.73
Potassium nitrate 3.46
Sodium nitrate 3.55
Ammonium nitrate 3.58

_Electrolytes with three Ions._

Sulphuric acid 4.49
Sodium sulphate 5.09
Calcium chloride 5.04
Magnesium chloride 5.08

At the concentration used by Loomis the electrical conductivity indicates that the ionization is not complete, particularly in the case of the salts with divalent ions in the second list. Allowing for incomplete ionization the general concordance of these numbers with the theoretical ones is very striking.

The measurements of freezing points of solutions at the extreme dilution necessary to secure complete ionization is a matter of great difficulty, and has been overcome only in a research initiated by E.H. Griffiths.[3] Results have been obtained for solutions of sugar, where the experimental number is 1.858, and for potassium chloride, which gives a depression of 3.720. These numbers agree with those indicated by theory, viz. 1.857 and 3.714, with astonishing exactitude. We may take Arrhenius' first relation as established for the case of potassium chloride.

The second relation, as we have seen, is not a strict consequence of theory, and experiments to examine it must be treated as an investigation of the limits within which solutions are dilute within the thermodynamic sense of the word, rather than as a test of the soundness of the theory. It is found that divergence has begun before the concentration has become great enough to enable freezing points to be measured with any ordinary apparatus. The freezing point curve usually lies below the electrical one, but approaches it as dilution is increased.[4]

Returning once more to the consideration of the first relation, which deals with the comparison between the number of ions and the number of pressure-producing particles in dilute solution, one caution is necessary. In simple substances like potassium chloride it seems evident that one kind of dissociation only is possible. The electrical phenomena show that there are two ions to the molecule, and that these ions are electrically charged. Corresponding with this result we find that the freezing point of dilute solutions indicates that two pressure-producing particles per molecule are present. But the converse relation does not necessarily follow. It would be possible for a body in solution to be dissociated into non-electrical parts, which would give osmotic pressure effects twice or three times the normal value, but, being uncharged, would not act as ions and impart electrical conductivity to the solution. L. Kahlenberg (_Jour. Phys. Chem._, 1901, v. 344, 1902, vi. 43) has found that solutions of diphenylamine in methyl cyanide possess an excess of pressure-producing particles and yet are non-conductors of electricity. It is possible that in complicated organic substances we might have two kinds of dissociation, electrical and non-electrical, occurring simultaneously, while the possibility of the association of molecules accompanied by the electrical dissociation of some of them into new parts should not be overlooked. It should be pointed out that no measurements on osmotic pressures or freezing points can do more than tell us that an excess of particles is present; such experiments can throw no light on the question whether or not those particles are electrically charged. That question can only be answered by examining whether or not the particles move in an electric field.

The dissociation theory was originally suggested by the osmotic pressure relations. But not only has it explained satisfactorily the electrical properties of solutions, but it seems to be the only known hypothesis which is consistent with the experimental relation between the concentration of a solution and its electrical conductivity (see CONDUCTION, ELECTRIC, S II., "Nature of Electrolytes"). It is probable that the electrical effects constitute the strongest arguments in favour of the theory. It is necessary to point out that the dissociated ions of such a body as potassium chloride are not in the same condition as potassium and chlorine in the free state. The ions are associated with very large electric charges, and, whatever their exact relations with those charges may be, it is certain that the energy of a system in such a state must be different from its energy when unelectrified. It is not unlikely, therefore, that even a compound as stable in the solid form as potassium chloride should be thus dissociated when dissolved. Again, water, the best electrolytic solvent known, is also the body of the highest specific inductive capacity (dielectric constant), and this property, to whatever cause it may be due, will reduce the forces between electric charges in the neighbourhood, and may therefore enable two ions to separate.

This view of the nature of electrolytic solutions at once explains many well-known phenomena. Other physical properties of these solutions, such as density, colour, optical rotatory power, &c., like the conductivities, are _additive_, i.e. can be calculated by adding together the corresponding properties of the parts. This again suggests that these parts are independent of each other. For instance, the colour of a salt solution is the colour obtained by the superposition of the colours of the ions and the colour of any undissociated salt that may be present. All copper salts in dilute solution are blue, which is therefore the colour of the copper ion. Solid copper chloride is brown or yellow, so that its concentrated solution, which contains both ions and undissociated molecules, is green, but changes to blue as water is added and the ionization becomes complete. A series of equivalent solutions all containing the same coloured ion have absorption spectra which, when photographed, show identical absorption bands of equal intensity.[5] The colour changes shown by many substances which are used as indicators (q.v.) of acids or alkalis can be explained in a similar way. Thus para-nitrophenol has colourless molecules, but an intensely yellow negative ion. In neutral, and still more in acid solutions, the dissociation of the indicator is practically nothing, and the liquid is colourless. If an alkali is added, however, a highly dissociated salt of para-nitrophenol is formed, and the yellow colour is at once evident. In other cases, such as that of litmus, both the ion and the undissociated molecule are coloured, but in different ways.

Electrolytes possess the power of coagulating solutions of colloids such as albumen and arsenious sulphide. The mean values of the relative coagulative powers of sulphates of mono-, di-, and tri-valent metals have been shown experimentally to be approximately in the ratios 1:35:1023. The dissociation theory refers this to the action of electric charges carried by the free ions. If a certain minimum charge must be collected in order to start coagulation, it will need the conjunction of 6n monovalent, or 3n divalent, to equal the effect of 2n tri-valent ions. The ratios of the coagulative powers can thus be calculated to be 1:x:x^2, and putting x = 32 we get 1:32:1024, a satisfactory agreement with the numbers observed.[6]

The question of the application of the dissociation theory to the case of fused salts remains. While it seems clear that the conduction in this case is carried on by ions similar to those of solutions, since Faraday's laws apply equally to both, it does not follow necessarily that semi-permanent dissociation is the only way to explain the phenomena. The evidence in favour of dissociation in the case of solutions does not apply to fused salts, and it is possible that, in their case, a series of molecular interchanges, somewhat like Grotthus's chain, may represent the mechanism of conduction.

An interesting relation appears when the electrolytic conductivity of solutions is compared with their chemical activity. The readiness and speed with which electrolytes react are in sharp contrast with the difficulty experienced in the case of non-electrolytes. Moreover, a study of the chemical relations of electrolytes indicates that it is always the electrolytic ions that are concerned in their reactions. The tests for a salt, potassium nitrate, for example, are the tests not for KNO3, but for its ions K and NO3, and in cases of double decomposition it is always these ions that are exchanged for those of other substances. If an element be present in a compound otherwise than as an ion, it is not interchangeable, and cannot be recognized by the usual tests. Thus neither a chlorate, which contains the ion ClO3, nor monochloracetic acid, shows the reactions of chlorine, though it is, of course, present in both substances; again, the sulphates do not answer to the usual tests which indicate the presence of sulphur as sulphide. The chemical activity of a substance is a quantity which may be measured by different methods. For some substances it has been shown to be independent of the particular reaction used. It is then possible to assign to each body a specific coefficient of affinity. Arrhenius has pointed out that the coefficient of affinity of an acid is proportional to its electrolytic ionization.

The affinities of acids have been compared in several ways. W. Ostwald
(_Lehrbuch der allg. Chemie_, vol. ii., Leipzig, 1893) investigated
the relative affinities of acids for potash, soda and ammonia, and
proved them to be independent of the base used. The method employed
was to measure the changes in volume caused by the action. His results
are given in column I. of the following table, the affinity of
hydrochloric acid being taken as one hundred. Another method is to
allow an acid to act on an insoluble salt, and to measure the quantity
which goes into solution. Determinations have been made with calcium
oxalate, CaC2O4+H2O, which is easily decomposed by acids, oxalic acid
and a soluble calcium salt being formed. The affinities of acids
relative to that of oxalic acid are thus found, so that the acids can
be compared among themselves (column II.). If an aqueous solution of
methyl acetate be allowed to stand, a slow decomposition goes on. This
is much quickened by the presence of a little dilute acid, though the
acid itself remains unchanged. It is found that the influence of
different acids on this action is proportional to their specific
coefficients of affinity. The results of this method are given in
column III. Finally, in column IV. the electrical conductivities of
normal solutions of the acids have been tabulated. A better basis of
comparison would be the ratio of the actual to the limiting
conductivity, but since the conductivity of acids is chiefly due to
the mobility of the hydrogen ions, its limiting value is nearly the
same for all, and the general result of the comparison would be
unchanged.

+-----------------+---------+---------+---------+---------+
| Acid. | I. | II. | III. | IV. |
+-----------------+---------+---------+---------+---------+
| Hydrochloric | 100 | 100 | 100 | 100 |
| Nitric | 102 | 110 | 92 | 99.6 |
| Sulphuric | 68 | 67 | 74 | 65.1 |
| Formic | 4.0 | 2.5 | 1.3 | 1.7 |
| Acetic | 1.2 | 1.0 | 0.3 | 0.4 |
| Propionic | 1.1 | .. | 0.3 | 0.3 |
| Monochloracetic | 7.2 | 5.1 | 4.3 | 4.9 |
| Dichloracetic | 34 | 18 | 23.0 | 25.3 |
| Trichloracetic | 82 | 63 | 68.2 | 62.3 |
| Malic | 3.0 | 5.0 | 1.2 | 1.3 |
| Tartaric | 5.3 | 6.3 | 2.3 | 2.3 |
| Succinic | 0.1 | 0.2 | 0.5 | 0.6 |
+-----------------+---------+---------+---------+---------+

It must be remembered that, the solutions not being of quite the same
strength, these numbers are not strictly comparable, and that the
experimental difficulties involved in the chemical measurements are
considerable. Nevertheless, the remarkable general agreement of the
numbers in the four columns is quite enough to show the intimate
connexion between chemical activity and electrical conductivity. We
may take it, then, that only that portion of these bodies is
chemically active which is electrolytically active--that ionization is
necessary for such chemical activity as we are dealing with here, just
as it is necessary for electrolytic conductivity.

The ordinary laws of chemical equilibrium have been applied to the
case of the dissociation of a substance into its ions. Let x be the
number of molecules which dissociate per second when the number of
undissociated molecules in unit volume is unity, then in a dilute
solution where the molecules do not interfere with each other, xp is
the number when the concentration is p. Recombination can only occur
when two ions meet, and since the frequency with which this will
happen is, in dilute solution, proportional to the square of the ionic
concentration, we shall get for the number of molecules re-formed in
one second yq^2 where q is the number of dissociated molecules in one
cubic centimetre. When there is equilibrium, xp = yq^2. If [mu] be the
molecular conductivity, and [mu]_([oo]) its value at infinite
dilution, the fractional number of molecules dissociated is
[mu]/[mu]_([oo]), which we may write as [alpha]. The number of
undissociated molecules is then 1 - [alpha], so that if V be the
volume of the solution containing 1 gramme-molecule of the dissolved
substance, we get

q = [alpha]/V and p = (1 - [alpha])/V,

hence x(1 - [alpha])V = ya^2/V^2,

[alpha]^2 x
and -------------- = -- = constant = k.
V(1 - [alpha]) y

This constant k gives a numerical value for the chemical affinity, and
the equation should represent the effect of dilution on the molecular
conductivity of binary electrolytes.

In the case of substances like ammonia and acetic acid, where the
dissociation is very small, 1 - [alpha] is nearly equal to unity, and
only varies slowly with dilution. The equation then becomes
[alpha]^2/V = k, or [alpha] = [root](Vk), so that the molecular
conductivity is proportional to the square root of the dilution.
Ostwald has confirmed the equation by observation on an enormous
number of weak acids (_Zeits. physikal. Chemie_, 1888, ii. p. 278;
1889, iii. pp. 170, 241, 369). Thus in the case of cyanacetic acid,
while the volume V changed by doubling from 16 to 1024 litres, the
values of k were 0.00 (376, 373, 374, 361, 362, 361, 368). The mean
values of k for other common acids were--formic, 0.0000214; acetic,
0.0000180; monochloracetic, 0.00155; dichloracetic, 0.051;
trichloracetic, 1.21; propionic, 0.0000134. From these numbers we can,
by help of the equation, calculate the conductivity of the acids for
any dilution. The value of k, however, does not keep constant so
satisfactorily in the case of highly dissociated substances, and
empirical formulae have been constructed to represent the effect of
dilution on them. Thus the values of the expressions [alpha]^2/(1 -
[alpha][root]V) (Rudolphi, _Zeits. physikal. Chemie_, 1895, vol. xvii.
p. 385) and [alpha]^3/(1 - [alpha])^2V (van 't Hoff, ibid., 1895, vol.
xviii. p. 300) are found to keep constant as V changes. Van 't Hoff's
formula is equivalent to taking the frequency of dissociation as
proportional to the square of the concentration of the molecules, and
the frequency of recombination as proportional to the cube of the
concentration of the ions. An explanation of the failure of the usual
dilution law in these cases may be given if we remember that, while
the electric forces between bodies like undissociated molecules, each
associated with equal and opposite charges, will vary inversely as the
fourth power of the distance, the forces between dissociated ions,
each carrying one charge only, will be inversely proportional to the
square of the distance. The forces between the ions of a strongly
dissociated solution will thus be considerable at a dilution which
makes forces between undissociated molecules quite insensible, and at
the concentrations necessary to test Ostwald's formula an electrolyte
will be far from dilute in the thermodynamic sense of the term, which
implies no appreciable intermolecular or interionic forces.

When the solutions of two substances are mixed, similar considerations
to those given above enable us to calculate the resultant changes in
dissociation. (See Arrhenius, loc. cit.) The simplest and most
important case is that of two electrolytes having one ion in common,
such as two acids. It is evident that the undissociated part of each
acid must eventually be in equilibrium with the free hydrogen ions,
and, if the concentrations are not such as to secure this condition,
readjustment must occur. In order that there should be no change in
the states of dissociation on mixing, it is necessary, therefore, that
the concentration of the hydrogen ions should be the same in each
separate solution. Such solutions were called by Arrhenius
"isohydric." The two solutions, then, will so act on each other when
mixed that they become isohydric. Let us suppose that we have one very
active acid like hydrochloric, in which dissociation is nearly
complete, another like acetic, in which it is very small. In order
that the solutions of these should be isohydric and the concentrations
of the hydrogen ions the same, we must have a very large quantity of
the feebly dissociated acetic acid, and a very small quantity of the
strongly dissociated hydrochloric, and in such proportions alone will
equilibrium be possible. This explains the action of a strong acid on
the salt of a weak acid. Let us allow dilute sodium acetate to react
with dilute hydrochloric acid. Some acetic acid is formed, and this
process will go on till the solutions of the two acids are isohydric:
that is, till the dissociated hydrogen ions are in equilibrium with
both. In order that this should hold, we have seen that a considerable
quantity of acetic acid must be present, so that a corresponding
amount of the salt will be decomposed, the quantity being greater the
less the acid is dissociated. This "replacement" of a "weak" acid by a
"strong" one is a matter of common observation in the chemical
laboratory. Similar investigations applied to the general case of
chemical equilibrium lead to an expression of exactly the same form as
that given by C.M. Guldberg and P. Waage, which is universally
accepted as an accurate representation of the facts.

The temperature coefficient of conductivity has approximately the same value for most aqueous salt solutions. It decreases both as the temperature is raised and as the concentration is increased, ranging from about 3.5% per degree for extremely dilute solutions (i.e. practically pure water) at 0 deg. to about 1.5 for concentrated solutions at 18 deg. For acids its value is usually rather less than for salts at equivalent concentrations. The influence of temperature on the conductivity of solutions depends on (1) the ionization, and (2) the frictional resistance of the liquid to the passage of the ions, the reciprocal of which is called the ionic fluidity. At extreme dilution, when the ionization is complete, a variation in temperature cannot change its amount. The rise of conductivity with temperature, therefore, shows that the fluidity becomes greater when the solution is heated. As the concentration is increased and un-ionized molecules are formed, a change in temperature begins to affect the ionization as well as the fluidity. But the temperature coefficient of conductivity is now generally less than before; thus the effect of temperature on ionization must be of opposite sign to its effect on fluidity. The ionization of a solution, then, is usually diminished by raising the temperature, the rise in conductivity being due to the greater increase in fluidity. Nevertheless, in certain cases, the temperature coefficient of conductivity becomes negative at high temperatures, a solution of phosphoric acid, for example, reaching a maximum conductivity at 75 deg. C.

The dissociation theory gives an immediate explanation of the fact that, in general, no heat-change occurs when two neutral salt solutions are mixed. Since the salts, both before and after mixture, exist mainly as dissociated ions, it is obvious that large thermal effects can only appear when the state of dissociation of the products is very different from that of the reagents. Let us consider the case of the neutralization of a base by an acid in the light of the dissociation theory. In dilute solution such substances as hydrochloric acid and potash are almost completely dissociated, so that, instead of representing the reaction as

HCl + KOH = KCl + H2O,

we must write

+ - + - + -
H + Cl + K + OH = K + Cl + H2O.

The ions K and Cl suffer no change, but the hydrogen of the acid and the hydroxyl (OH) of the potash unite to form water, which is only very slightly dissociated. The heat liberated, then, is almost exclusively that produced by the formation of water from its ions. An exactly similar process occurs when any strongly dissociated acid acts on any strongly dissociated base, so that in all such cases the heat evolution should be approximately the same. This is fully borne out by the experiments of Julius Thomsen, who found that the heat of neutralization of one gramme-molecule of a strong base by an equivalent quantity of a strong acid was nearly constant, and equal to 13,700 or 13,800 calories. In the case of weaker acids, the dissociation of which is less complete, divergences from this constant value will occur, for some of the molecules have to be separated into their ions. For instance, sulphuric acid, which in the fairly strong solutions used by Thomsen is only about half dissociated, gives a higher value for the heat of neutralization, so that heat must be evolved when it is ionized. The heat of formation of a substance from its ions is, of course, very different from that evolved when it is formed from its elements in the usual way, since the energy associated with an ion is different from that possessed by the atoms of the element in their normal state. We can calculate the heat of formation from its ions for any substance dissolved in a given liquid, from a knowledge of the temperature coefficient of ionization, by means of an application of the well-known thermodynamical process, which also gives the latent heat of evaporation of a liquid when the temperature coefficient of its vapour pressure is known. The heats of formation thus obtained may be either positive or negative, and by using them to supplement the heat of formation of water, Arrhenius calculated the total heats of neutralization of soda by different acids, some of them only slightly dissociated, and found values agreeing well with observation (_Zeits. physikal. Chemie_, 1889, 4, p. 96; and 1892, 9, p. 339).

_Voltaic Cells._--When two metallic conductors are placed in an electrolyte, a current will flow through a wire connecting them provided that a difference of any kind exists between the two conductors in the nature either of the metals or of the portions of the electrolyte which surround them. A current can be obtained by the combination of two metals in the same electrolyte, of two metals in different electrolytes, of the same metal in different electrolytes, or of the same metal in solutions of the same electrolyte at different concentrations. In accordance with the principles of energetics (q.v.), any change which involves a decrease in the total available energy of the system will tend to occur, and thus the necessary and sufficient condition for the production of electromotive force is that the available energy of the system should decrease when the current flows.

In order that the current should be maintained, and the electromotive force of the cell remain constant during action, it is necessary to ensure that the changes in the cell, chemical or other, which produce the current, should neither destroy the difference between the electrodes, nor coat either electrode with a non-conducting layer through which the current cannot pass. As an example of a fairly constant cell we may take that of Daniell, which consists of the electrical arrangement--zinc | zinc sulphate solution | copper sulphate solution | copper,--the two solutions being usually separated by a pot of porous earthenware. When the zinc and copper plates are connected through a wire, a current flows, the conventionally positive electricity passing from copper to zinc in the wire and from zinc to copper in the cell. Zinc dissolves at the anode, an equal amount of zinc replaces an equivalent amount of copper on the other side of the porous partition, and the same amount of copper is deposited on the cathode. This process involves a decrease in the available energy of the system, for the dissolution of zinc gives out more energy than the separation of copper absorbs. But the internal rearrangements which accompany the production of a current do not cause any change in the original nature of the electrodes, fresh zinc being exposed at the anode, and copper being deposited on copper at the cathode. Thus as long as a moderate current flows, the only variation in the cell is the appearance of zinc sulphate in the liquid on the copper side of the porous wall. In spite of this appearance, however, while the supply of copper is maintained, copper, being more easily separated from the solution than zinc, is deposited alone at the cathode, and the cell remains constant.

It is necessary to observe that the condition for change in a system is that the total available energy of the whole system should be decreased by the change. We must consider what change is allowed by the mechanism of the system, and deal with the sum of all the alterations in energy. Thus in the Daniell cell the dissolution of copper as well as of zinc would increase the loss in available energy. But when zinc dissolves, the zinc ions carry their electric charges with them, and the liquid tends to become positively electrified. The electric forces then soon stop further action unless an equivalent quantity of positive ions are removed from the solution. Hence zinc can only dissolve when some more easily separable substance is present in solution to be removed pari passu with the dissolution of zinc. The mechanism of such systems is well illustrated by an experiment devised by W. Ostwald. Plates of platinum and pure or amalgamated zinc are separated by a porous pot, and each surrounded by some of the same solution of a salt of a metal more oxidizable than zinc, such as potassium. When the plates are connected together by means of a wire, no current flows, and no appreciable amount of zinc dissolves, for the dissolution of zinc would involve the separation of potassium and a gain in available energy. If sulphuric acid be added to the vessel containing the zinc, these conditions are unaltered and still no zinc is dissolved. But, on the other hand, if a few drops of acid be placed in the vessel with the platinum, bubbles of hydrogen appear, and a current flows, zinc dissolving at the anode, and hydrogen being liberated at the cathode. In order that positively electrified ions may enter a solution, an equivalent amount of other positive ions must be removed or negative ions be added, and, for the process to occur spontaneously, the possible action at the two electrodes must involve a decrease in the total available energy of the system.

Considered thermodynamically, voltaic cells must be divided into reversible and non-reversible systems. If the slow processes of diffusion be ignored, the Daniell cell already described may be taken as a type of a reversible cell. Let an electromotive force exactly equal to that of the cell be applied to it in the reverse direction. When the applied electromotive force is diminished by an infinitesimal amount, the cell produces a current in the usual direction, and the ordinary chemical changes occur. If the external electromotive force exceed that of the cell by ever so little, a current flows in the opposite direction, and all the former chemical changes are reversed, copper dissolving from the copper plate, while zinc is deposited on the zinc plate. The cell, together with this balancing electromotive force, is thus a reversible system in true equilibrium, and the thermodynamical reasoning applicable to such systems can be used to examine its properties.

Now a well-known relation connects the available energy of a reversible system with the corresponding change in its total internal energy.

The available energy A is the amount of external work obtainable by an
infinitesimal, reversible change in the system which occurs at a
constant temperature T. If I be the change in the internal energy, the
relation referred to gives us the equation

A = I + T(dA/dT),

where dA/dT denotes the rate of change of the available energy of the
system per degree change in temperature. During a small electric
transfer through the cell, the external work done is Ee, where E is
the electromotive force. If the chemical changes which occur in the
cell were allowed to take place in a closed vessel without the
performance of electrical or other work, the change in energy would be
measured by the heat evolved. Since the final state of the system
would be the same as in the actual processes of the cell, the same
amount of heat must give a measure of the change in internal energy
when the cell is in action. Thus, if L denote the heat corresponding
with the chemical changes associated with unit electric transfer, Le
will be the heat corresponding with an electric transfer e, and will
also be equal to the change in internal energy of the cell. Hence we
get the equation

Ee = Le + Te(dE/dT) or E = L + T(dE/dT),

as a particular case of the general thermodynamic equation of
available energy. This equation was obtained in different ways by J.
Willard Gibbs and H. von Helmholtz.

It will be noticed that when dE/dT is zero, that is, when the
electromotive force of the cell does not change with temperature, the
electromotive force is measured by the heat of reaction per unit of
electrochemical change. The earliest formulation of the subject, due
to Lord Kelvin, assumed that this relation was true in all cases, and,
calculated in this way, the electromotive force of Daniell's cell,
which happens to possess a very small temperature coefficient, was
found to agree with observation.

When one gramme of zinc is dissolved in dilute sulphuric acid, 1670
thermal units or calories are evolved. Hence for the electrochemical
unit of zinc or 0.003388 gramme, the thermal evolution is 5.66
calories. Similarly, the heat which accompanies the dissolution of one
electrochemical unit of copper is 3.00 calories. Thus, the thermal
equivalent of the unit of resultant electrochemical change in
Daniell's cell is 5.66 - 3.00 = 2.66 calories. The dynamical
equivalent of the calorie is 4.18 X 10^7 ergs or C.G.S. units of work,
and therefore the electromotive force of the cell should be 1.112 X
10^8 C.G.S. units or 1.112 volts--a close agreement with the
experimental result of about 1.08 volts. For cells in which the
electromotive force varies with temperature, the full equation given
by Gibbs and Helmholtz has also been confirmed experimentally.

As stated above, an electromotive force is set up whenever there is a difference of any kind at two electrodes immersed in electrolytes. In ordinary cells the difference is secured by using two dissimilar metals, but an electromotive force exists if two plates of the same metal are placed in solutions of different substances, or of the same substance at different concentrations. In the latter case, the tendency of the metal to dissolve in the more dilute solution is greater than its tendency to dissolve in the more concentrated solution, and thus there is a decrease in available energy when metal dissolves in the dilute solution and separates in equivalent quantity from the concentrated solution. An electromotive force is therefore set up in this direction, and, if we can calculate the change in available energy due to the processes of the cell, we can foretell the value of the electromotive force. Now the effective change produced by the action of the current is the concentration of the more dilute solution by the dissolution of metal in it, and the dilution of the originally stronger solution by the separation of metal from it. We may imagine these changes reversed in two ways. We may evaporate some of the solvent from the solution which has become weaker and thus reconcentrate it, condensing the vapour on the solution which had become stronger. By this reasoning Helmholtz showed how to obtain an expression for the work done. On the other hand, we may imagine the processes due to the electrical transfer to be reversed by an osmotic operation. Solvent may be supposed to be squeezed out from the solution which has become more dilute through a semi-permeable wall, and through another such wall allowed to mix with the solution which in the electrical operation had become more concentrated. Again, we may calculate the osmotic work done, and, if the whole cycle of operations be supposed to occur at the same temperature, the osmotic work must be equal and opposite to the electrical work of the first operation.

The result of the investigation shows that the electrical work Ee is
given by the equation
_
/ p2
Ee = | vdp,
_/ p1

where v is the volume of the solution used and p its osmotic pressure.
When the solutions may be taken as effectively dilute, so that the gas
laws apply to the osmotic pressure, this relation reduces to

nrRT c1
E = ---- log_[epsilon] --
ey c2

where n is the number of ions given by one molecule of the salt, r the
transport ratio of the anion, R the gas constant, T the absolute
temperature, y the total valency of the anions obtained from one
molecule, and c1 and c2 the concentrations of the two solutions.

If we take as an example a concentration cell in which silver plates
are placed in solutions of silver nitrate, one of which is ten times
as strong as the other, this equation gives

E = 0.060 X 10^8 C.G.S. units
= 0.060 volts.

W. Nernst, to whom this theory is due, determined the electromotive force of this cell experimentally, and found the value 0.055 volt.