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the plates is, according to definition, the conductivity of the solution. Next, if v is the number of cubic centimetres containing one gram-equivalent of the dissolved substance, place V c.c. of solution between the plates; then the conductance is v times as great as before. It is now a measure of the equivalent conductivity.
The most important result of experiment can be summed up by saying that, in general, while the actual conductivity k falls off very much when the solution is diluted, the equivalent conductivity of a salt (acid or base) increases with its dilution, but tends towards a finite limit for indefinitely great dilution. The following example will serve to illustrate the rule; solutions of varying strengths being made up, and their conductivities measured, the values of the equivalent conductivities are
100 1,000 10,000 100,000 1,000,000 10,000,000
982 III'9 1225 12706 129'5
0'049 1°32 4:60 143 41'0 107'0
258 45.0 72.2 IOI6 1133
In each case the increase with dilution is obvious. The approach to a finite maximum is clearly marked in the first case, and the limiting value can be obtained by extrapolation with some certainty : Kohlrausch gives it as 131'2. In the second case the limit is fairly obvious, but not calculable with certainty. With acetic acid it is not obvious; but there is indirect evidence that if it were possible to measure still more dilute solutions, it would be found.
Since the laws of dependence of the equivalent conductivity on concentration are not at all simple, it is of interest to draw curves showing the relations between the two quantities. But as measurements have to be taken over a very wide
of concentration, such diagrams would not be convenient to draw
except in the case of strong solutions. Kohlrausch recommends the use of diagrams in which the cube root of the concentration is taken as abscissa, the equivalent conductivity as ordinate ; curves so drawn are often nearly linear (see Fig. 25). It is also often convenient to take the logarithm of the concentration as abscissa.
Since, according to the hypothesis we have used, the conductivity of a solution is the joint effect of convection by anions and cations in it, the next step is to analyse the equivalent conductivities into a part due to the anions and a part due to the cations. These are called the partial or ionic conductivities ; they will be designated by la and lc, so that, using the same symbols as before, we shall expect to have
le = 96600 yUc
la = 96600 yUA and
= A Hittorf's migration ratios afford a means of effecting this analysis, and it is found that the ionic conductivities so determined are consistent amongst themselves. The most interesting numbers are the limiting values for infinite dilution, to which we shall return later ; but at present, for a test of the theory, we will use a set of values referring to decinormal solution (v 10000).
The molecular conductivity of KCl (v = 10000) is 111'9; that of NaNO3, 87'4. The migration ratios of the anion in these salts have been found 0'508 and 0.615 respectively. From these data we calculate the ionic conductivities :
K: = 0'492 X III'9 = 55'I
NO,' = 0.615 X 87'4 = 538 Now, if these numbers really depend on the properties of the single ions, we should expect to be able to build up from them the conductivities of other salts. This can be done with fair success ; 1.8: —
KNO, = 55'1 + 53*8 = 108 9
NaCl = 33.6 + 55-8 = 894 whilst the experimental values are 104:4 and 92'5 respectively. By taking the mean of a large number of different combinations, more reliable values of the ionic conductivities can be calculated.
The ionic conductivities are closely related to ionic mobilities (p. 34). If the numbers given in Kohlrausch's table be divided by the charge per equivalent, 96600 coulombs, the quotient expresses a velocity in centimetres per second for unit potential gradient. Thus the conductivity of the Kin decinormal KCI, 55'1 yields 55'1 – 96600 = 0'00057 cms. per second; this expresses the mobility of the potassium ions in such a solution, but in a sense that requires careful note. According to the equation of p. 59.
Here U is the average velocity acquired by the ion when exposed to unit potential gradient, and has been defined as the mobility, or, as we will say for distinction, the actual mobility, while
γ is a numerical factor known as the degree of dissociation. There are two aspects to y. In the first place, it represents the fraction of all the salt molecules which at any instant exist in the state of dissociation. But dissociation is a kinetic pheno
It is not that some molecules are always dissociated and others not, but that each molecule is broken up from time to time by some especially violent collision with its neighbours; and, again, that detached ions recombine frequently. In this way a certain average state of dissociation is kept up with considerable uniformity, but any individual atom may combine and dissociate again hundreds, possibly millions, of times per second. Accordingly, if we consider the history of a single atom (or ion), and average it over a long time, we may look upon y as measuring the fraction of all the time during which that atom is uncombined. It is obvious that this fraction must be the same as that obtained by averaging over all the molecules at a given instant.
But an ion only travels under the electric field when it is free, for if it combines with an oppositely charged ion, it becomes neutral, and there is no electric force acting on it (or two equal and opposite forces which neutralise). If, then, the ion when it is free travels at the average speed U, but is only free for the fraction y of the whole time, its velocity averaged over the whole time, whether free or not, is yU. obtain a conception that may be defined as the effective mobility. It may be regarded either as the average velocity of a single ion over a long time, intervals of rest 1 included, or the average velocity of all the ions, those which are for the time being combined included.
Accordingly, conductivity measurements only give, directly, the effective mobility of the ions; the conduction in any
1 These remarks refer, of course, only to the uniform drift due to electric force; when the ions are not moving in this way they still possess their heat motions, but as these are indifferently in all directions they lead nowhere, and do not interfere with the electric mobility.
In this way we