when this ceases to hold, the concentration of the solution has in
general become so great that the conductivity of the solvent may
be neglected. The general result of these experiments can be
represented graphically by plotting *k*/*m* as ordinates and ^{3}√*m*
as abscissae, ^{3}√*m* being a number proportional to the reciprocal
of the average distance between the molecules, to which it seems
likely that the molecular conductivity may be related. The
general types of curve for a simple neutral salt like potassium or
sodium chloride and for a caustic alkali or acid are shown in fig. 4.
The curve for the neutral salt comes to a limiting value; that for
the acid attains a maximum at a certain very small concentration,
and falls again when the dilution
is carried farther. It has usually
been considered that this destruction
of conductivity is due to
chemical action between the acid
and the residual impurities in the
water. At such great dilution these
impurities are present in quantities
comparable with the amount of acid
which they convert into a less
highly conducting neutral salt. In
the case of acids, then, the maximum
must be taken as the limiting
value. The decrease in equivalent conductivity at great dilution
is, however, so constant that this explanation seems insufficient.
The true cause of the phenomenon may perhaps be connected
with the fact that the bodies in which it occurs, acids and
alkalis, contain the ions, hydrogen in the one case, hydroxyl in
the other, which are present in the solvent, water, and have,
perhaps because of this relation, velocities higher than those of any
other ions. The values of the molecular conductivities of all
neutral salts are, at great dilution, of the same order of magnitude,
while those of acids at their maxima are about three times as
large. The influence of increasing concentration is greater in the
case of salts containing divalent ions, and greatest of all in such
cases as solutions of ammonia and acetic acid, which are substances
of very low conductivity.

Fig. 4. |

*Theory of Moving Ions.*—Kohlrausch found that, when the
polarization at the electrodes was eliminated, the resistance of a
solution was constant however determined, and thus established
Ohm’s Law for electrolytes. The law was confirmed in the case
of strong currents by G. F. Fitzgerald and F. T. Trouton (*B.A.*
*Report*, 1886, p. 312). Now, Ohm’s Law implies that no work is
done by the current in overcoming reversible electromotive
forces such as those of polarization. Thus the molecular interchange
of ions, which must occur in order that the products may
be able to work their way through the liquid and appear at the
electrodes, continues throughout the solution whether a current is
flowing or not. The influence of the current on the ions is
merely directive, and, when it flows, streams of electrified ions
travel in opposite directions, and, if the applied electromotive
force is enough to overcome the local polarization, give up their
charges to the electrodes. We may therefore represent the facts
by considering the process of electrolysis to be a kind of convection.
Faraday’s classical experiments proved that when a current
flows through an electrolyte the quantity of substance liberated
at each electrode is proportional to its chemical equivalent
weight, and to the total amount of electricity passed. Accurate
determinations have since shown that the mass of an ion deposited
by one electromagnetic unit of electricity, *i.e.* its electro-chemical
equivalent, is 1.036×10^{−4}×its chemical equivalent
weight. Thus the amount of electricity associated with one
gram-equivalent of any ion is 10^{4}/1.036 = 9653 units. Each
monovalent ion must therefore be associated with a certain
definite charge, which we may take to be a natural unit of
electricity; a divalent ion carries two such units, and so on.
A cation, *i.e.* an ion giving up its charge at the cathode, as the
electrode at which the current leaves the solution is called, carries
a positive charge of electricity; an anion, travelling in the
opposite direction, carries a negative charge. It will now be seen
that the quantity of electricity flowing per second, *i.e.* the current
through the solution, depends on (1) the number of the ions
concerned, (2) the charge on each ion, and (3) the velocity with
which the ions travel past each other. Now, the number of ions
is given by the concentration of the solution, for even if all the
ions are not actively engaged in carrying the current at the same
instant, they must, on any dynamical idea of chemical equilibrium,
be all active in turn. The charge on each, as we have
seen, can be expressed in absolute units, and therefore the
velocity with which they move past each other can be calculated.
This was first done by Kohlrausch (*Göttingen Nachrichten*, 1876,
p. 213, and *Das Leitvermögen der Elektrolyte*, Leipzig, 1898)
about 1879.

In order to develop Kohlrausch’s theory, let us take, as an example,
the case of an aqueous solution of potassium chloride, of concentration
n gram-equivalents per cubic centimetre. There will then
be n gram-equivalents of potassium ions and the same number of
chlorine ions in this volume. Let us suppose that on each gram-equivalent
of potassium there reside +*e* units of electricity, and on
each gram-equivalent of chlorine ions −*e* units. If *u* denotes the
average velocity of the potassium ion, the positive charge carried
per second across unit area normal to the flow is *n e u*. Similarly, if
*v* be the average velocity of the chlorine ions, the negative charge
carried in the opposite direction is *n e v*. But positive electricity
moving in one direction is equivalent to negative electricity moving
in the other, so that, before changes in concentration sensibly supervene,
the total current, C, is *ne*(*u* + *v*). Now let us consider the
amounts of potassium and chlorine liberated at the electrodes by
this current. At the cathode, if the chlorine ions were at rest, the
excess of potassium ions would be simply those arriving in one second,
namely, *nu*. But since the chlorine ions move also, a further separation
occurs, and *nv* potassium ions are left without partners. The
total number of gram-equivalents liberated is therefore *n*(*u* + *v*).
By Faradays law, the number of grams liberated is equal to the
product of the current and the electro-chemical equivalent of the
ion; the number of gram-equivalents therefore must be equal to
ηC, where η denotes the electro-chemical equivalent of hydrogen in
C.G.S. units. Thus we get

*n*(

*u*+

*v*) = ηC = η

*ne*(

*u' +*v

*),*

and it follows that the charge, *e*, on 1 gram-equivalent of each kind
of ion is equal to 1/η. We know that Ohm’s Law holds good for
electrolytes, so that the current C is also given by *k*·*d*P/*dx*, where
*k* denotes the conductivity of the solution, and *d*P/*dx* the potential
gradient, *i.e.* the change in potential per unit length along the lines of
current flow. Thus

*n*η(

*u*+

*v*) =

*kd*P/

*dx*;

therefore

u + v = η |
k | dP | . | |

n | dx |

Now η is 1.036×10^{−4}, and the concentration of a solution is usually
expressed in terms of the number, *m*, of gram-equivalents per litre
instead of per cubic centimetre. Therefore

u + v = 1.036×10^{−1} |
k | dP | . | |

m | dx |

When the potential gradient is one volt (10^{8} C.G.S. units) per
centimetre this becomes

*u*+

*v*= 1.036×10

^{−7}×

*k*/

*m*.

Thus by measuring the value of *k*/*m*, which is known as the
equivalent conductivity of the solution, we can find *u* + *v*, the
velocity of the ions relative to each other. For instance, the equivalent
conductivity of a solution of potassium chloride containing one-tenth
of a gram-equivalent per litre is 1119×10^{−13} C.G.S. units at
18° C. Therefore

*u*+

*v*= 1.036×10

^{7}×1119×10

^{−13}

= 1.159×10^{−3} = 0.001159 cm. per sec.

In order to obtain the absolute velocities *u* and *v*, we must find
some other relation between them. Let us resolve *u* into ½(*u* + *v*)
in one direction, say to the right, and ½(*u* − *v*) to the left. Similarly
*v* can be resolved into ½(*v* + *u*) to the left and ½(*v* − *u*) to the right.
On pairing these velocities we have a combined movement of the
ions to the right, with a speed of ½(*u* − *v*) and a drift right and left,
past each other, each ion travelling with a speed of ½(*u* + *v*), constituting
the electrolytic separation. If *u* is greater than *v*, the combined
movement involves a concentration of salt at the cathode, and a
corresponding dilution at the anode, and *vice versa*. The rate at
which salt is electrolysed, and thus removed from the solution at
each electrode, is ½(*u* + *v*). Thus the total loss of salt at the cathode
is ½(*u* + *v*) − ½(*u* − *v*) or *v*, and at the anode, ½(*v* + *u*) − ½(*v* − *u*), or *u*.
Therefore, as is explained in the article Electrolysis, by measuring
the dilution of the liquid round the electrodes when a current passed,
W. Hittorf (*Pogg. Ann.*, 1853–1859, 89, p. 177; 98, p. 1; 103, p. 1; 106,
pp. 337 and 513) was able to deduce the ratio of the two velocities,