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These may be expressed by the symbols ld, and lar, and we have the simple relation
lo = 96600 U The conductivities are given in the table, p. 257. One example may serve to illustrate the mobilities; hydrogen is the most rapid ion, probably because it occupies the least space, and so threads its way more easily between water molecules. For hydrogen, loo 318 at 18° C. Hence U = 0·00329 cm. per second (for one volt per centimetre). Accordingly, under this potential gradient, a moderate one in practice, a hydrogen ion would, even if it were uncombined all the time, only travel the small distance of 12 cms. in an hour.
Temperature has an important influence on the conductivity of electrolytes. In nearly all cases the conductivity increases very fast with temperature, a fact closely related to the decrease in viscosity of water on heating. Certain empirical relations have lately been brought to light by systematic measurements of temperature coefficients, which are important, although their theoretical explanation is not clear. It is found that the conductivity can be represented by a parabolic formula
ke = k18[1 + alt – 18) +00177(a - 00177)(t – 18)] where K, is the conductivity at t.
K 18 a is a specific constant, which varies in the case of
dilute solutions from o‘0163 for HNO3 to o'0262 for Na,CO3. Since the last term in the formula depends on the square of t - 18, it is unimportant at temperatures near the standard temperature 18°; a may therefore be described as the coefficient at 18°; i.e.
increase in conductivity per 1°
conductivity at 18° when the range of temperature considered is not far from 18°. We see, then, that nitric acid changes 1'63 per cent., sodium carbonate 2.62 per cent. of its value, per degree.
The temperature coefficients of the ionic conductivities may 1 Kohlrausch, Berl. Sitzber., 24. 1026-1033 (1901); 26. 572-580 (1902).
be calculated from the same observations, and it is found that the coefficient regularly increases as the conductivity decreases, as the following table shows :
One consequence of this is that rise of temperature tends in the direction of making the mobilities of the ions all equal, and consequently tends to make the migration ratio of all electrolytes approach the value }.
§ 8. THE LAW OF DILUTION
The agreement observed between the degree of dissociation of electrolytes as measured by conductivity, and by osmotic pressure measurements, is not the only argument in favour of Arrhenius' theory. An important confirmation has been obtained by applying the law of mass action to electrolytes.
It has already been remarked that in an electrolyte there is a constant interchange of ions : molecules are broken up, and their dissociated parts recombine in different ways; and the state of equilibrium that exists at any time is of the statistical character familiar in the theory of gases. That is, a certain average proportion of molecules are dissociated, but no particular molecules are always so.
The main condition governing equilibrium of this character is the law of mass action (Guldberg and Waage's law), the mathematical consequences of which have been very fully worked out. We need at present only consider its application to the special case of chemical equilibrium known as dissociation; i.e. when a single molecule breaks up into two or more parts.
See Mellor, “Chemical Dynamics,” in this series.
The simplest, and at the same time the most important case of this is the dissociation of a binary electrolyte, i.e. when each molecule forms two ions. If no other electrolyte is present, the number of anions is necessarily equal to that of cations.
If there be n equivalents of the electrolyte altogether in one cubic centimetre (total concentration), and of these the fraction y be dissociated, yn is the number of equivalents of either anions or cations in one cubic centimetre (ionic concentration); while the concentration of the undissociated salt molecules is n(I - y).
Then it may be shown that the law of mass action leads to the relation
yn)? = Kn(I – y)
Iwhere K is a numerical quantity known as the dissociation constant of the electrolyte in question.
This equation may be looked upon in the first place as a relation between the dissociation of a binary electrolyte and its concentration; other applications will be dealt with later.
As an example of the meaning, we may take a series of measurements on monochloracetic acid by van't Hoff and Reicher. They found that when one mol was dissolved in 205 litres of water, the equivalent conductivity was 132 (at 14°1°), the limiting value being 311. Accordingly y = 132 • 311 = 0'423, when n= 205000. Therefore
Again, when one mol was dissolved in 4080 litres, the molecular conductivity was 274. Hence y = 274 = 311 = 0·881,
-- = 1.60 X 10-6
Oʻ119 X 4080000 or about the same.
The calculation just given somewhat exaggerates the discrepancy between different observations. The best test of the
? Zeitschr. phys. Chem., 2.781 (1888).
equation is to find, as above, K from each measurement, and from the mean of these, recalculate the values of
This is done in the following table :
[K = 1'585 x 10-6]
very satisfactory agreement with the law of mass action.
The application of the law of mass action to dissociation of electrolytes is often known as Ostwald's law of dilution, Ostwald 1 having applied it to his numerous measurements of conductivity. The measurements were all made in a thermostat at 25°, so that the equivalent conductivities are considerably greater than those given by Kohlrausch for 18o. They refer chiefly to organic acids; the law has also been shown by Bredig a to apply to many organic bases. The results obtained by these and other observers are given at length in Kohlrausch and Holborn's monograph, to which we must refer; they are, however, considered below (Chap. II.), on account of the important chemical meaning of the dissociation constant. This constant is, in fact, a measure of the “strength" of the acid, as a few examples may serve to show :
1:8 X 10 8
155'0 X 108 Dichloracetic acid .
5100'0 X 108 · Zeitschr. phys. Chem., 3. 170, 241, 363 (1889); Lehrbuch d. Allg. Chemie (1893).
? Zeitschr. phys. Chem., 13. 289 (1894).
The substitution of each chlorine atom increases the strength of the acid considerably. (See Chap. II.)
Mineral acids and the strongest organic acids do not follow Ostwald's law, so that it is not possible to calculate a dissociation constant for them, in the sense given above. (Trichloracetic acid belongs to this class.)
Among bases also it is only the weaker that follow the rule ; . thus for ammonia K = 2'3 X 10-8. But tetra-ethyl-ammonium hydroxide and similar bases, as well as the alkalis, deviate largely from the law of dilution.
Salts in all cases deviate from the rule.
It appears, then, that the law of dissociation in the form given by Ostwald is only applicable to weakly dissociated substances : strong electrolytes—i.e. salts and strong acids and bases, all of which are considerably dissociated even in moderately dilute solution-are subject to disturbing influences.
Many attempts at explaining the behaviour of strong electrolytes were made; but none of them are of value, except that of Jahn. The usual theory of solutions, due chiefly to van't Hoff, is based on the assumption that there is an action between the solvent and the dissolved substance, but none between the parts of the dissolved substance itself; it being supposed that the molecules of dissolved substance are too widely separated for any appreciable forces to exist between them. It is by the mathematical expression of these views that Ostwald's law of dilution is obtained.
Jahn's explanation of the divergence from Ostwald's law by strong electrolytes, is that mutual action between the parts of the dissolved substance is too large to neglect. If this is so, there are two additional terms to be taken into account, one relating to mutual action between the ions (i.e, forces of attraction and repulsion, whilst the ions remain separate in the chemical sense) the other to influence of neutral molecules on ions. The former action is to be expected, for electrical forces exerted by ordinary charged bodies such as can be experimented with, are sensible at great distances, and it is therefore to be presumed that charged ions exert forces on
1 Zeitschr. phys. Chem., 37. 490-503 (1901); 41. 257-301 (1902).