and H. A. Wilson has shown that a relation of this form represents the
results of his experiments on the conduction of electricity through
flames.
The expression for the fall of potential at the cathode is inversely
proportional to q^(3/2), q being the number of ions produced per cubic
centimetre per second close to the cathode; thus any increase in the
ionization at the cathode will diminish the potential fall at the
cathode, and as practically the whole potential difference between the
electrodes occurs at the cathode, a diminution in the potential fall
there will be much more important than a diminution in the electric
force in the uniform part of the discharge, when the force is
comparatively insignificant. This consideration explains a very
striking phenomenon discovered many years ago by Hittorf, who found
that if he put a wire carrying a bead of a volatile salt into the
flame, it produced little effect upon the current, unless it were
placed close to the cathode where it gave rise to an enormous increase
in the current, sometimes increasing the current more than a
hundredfold. The introduction of the salt increases very largely the
number of ions produced, so that q is much greater for a salted flame
than for a plain one. Thus Hittorf's result coincides with the
conclusions we have drawn from the theory of this class of conduction.
The fall of potential at the cathode is proportional to i - i0, where
i0 is the stream of negative electricity which comes from the cathode
itself, thus as i0 increases the fall of potential at the cathode
diminishes and the current sent by a given potential difference
through the gas increases. Now all metals give out negative particles
when heated, at a rate which increases very rapidly with the
temperature, but at the same temperature some metals give out more
than others. If the cathode is made of a metal which emits large
quantities of negative particles, (i - i0) will for a given value of i
be smaller than if the metal only emitted a small number of
particles; thus the cathode fall will be smaller for the metal with
the greater emissitivity, and the relation between the potential
difference and the current will be different in the two cases. These
considerations are confirmed by experience, for it has been found that
the current between electrodes immersed in a flame depends to a great
extent upon the metal of which the electrodes are made. Thus
Pettinelli (_Acc. dei Lincei_ [5], v. p. 118) found that, _ceteris
paribus_, the current between two carbon electrodes was about 500
times that between two iron ones. If one electrode was carbon and the
other iron, the current when the carbon was cathode and the iron anode
was more than 100 times the current when the electrodes were reversed.
The emission of negative particles by some metallic oxides, notably
those of calcium and barium, has been shown by Wehnelt (_Ann. der
Phys._ 11, p. 425) to be far greater than that of any known metal, and
the increase of current produced by coating the cathodes with these
oxides is exceedingly large; in some cases investigated by Tufts and
Stark (_Physik. Zeits._, 1908, 5, p. 248) the current was increased
many thousand times by coating the cathode with lime. No appreciable
effect is produced by putting lime on the anode.
_Conduction when all the Ions are of one Sign._--There are many
important cases in which the ions producing the current come from one
electrode or from a thin layer of gas close to the electrode, no
ionization occurring in the body of the gas or at the other electrode.
Among such cases may be mentioned those where one of the electrodes is
raised to incandescence while the other is cold, or when the negative
electrode is exposed to ultra-violet light. In such cases if the
electrode at which the ionization occurs is the positive electrode,
all the ions will be positively charged, while if it is the negative
electrode the ions will all be charged negatively. The theory of this
case is exceedingly simple. Suppose the electrodes are parallel planes
at right angles to the axis of x; let X be the electric force at a
distance x from the electrode where the ionization occurs, n the
number of ions (all of which are of one sign) at this place per cubic
centimetre, k the velocity of the ion under unit electric force, e the
charge on an ion, and i the current per unit area of the electrode.
Then we have dX/dx = 4[pi]ne, and if u is the velocity of the ion neu
= i. But u = kX, hence we have kX/4[pi] . dX/dx = i, and since the
right hand side of this equation does not depend upon x, we get
kX^2/8[pi] = ix + C, where C is a constant to be determined. If l is
the distance between the plates, and V the potential difference
between them,
_ _____ _ _
/ l 1 /8[pi] | |
V = | Xdx = --- / ----- | ( il + C )^3/2 - C^3/2 |.
_/0 i \/ k |_ _|
We shall show that when the current is far below the saturation value,
C is very small compared with il, so that the preceding equation
becomes
V^2 = 8[pi]l^3i/k (1).
To show that for small currents C is small compared with il, consider
the case when the ionization is confined to a thin layer, thickness d
close to the electrode, in that layer let n0 be the value of n, then
we have q = [alpha]n0^2 + i/ed. If X0 be the value of X when x = 0,
kX0n0e = i, and,
kX0^2 i^2 [alpha] i^2
C = ----- = ------------ = --------- . -------- (2).
8[pi] n0^2ke.8[pi] 8[pi]ke^2 q + i/ed
Since [alpha]/8[pi]ke is, as we have seen, less than unity, C will be
small compared with il, if i/(eq + i/d) is small compared with l. If
I0 is the saturation current, q = I0/ed, so that the former expression
= id/(I0 + i), if i is small compared with I0, this expression is
small compared with d, and therefore _a fortiori_ compared with l, so
that we are justified in this case in using equation (1).
From equation (2) we see that the current increases as the square of
the potential difference. Here an increase in the potential difference
produces a much greater percentage increase than in conduction through
metals, where the current is proportional to the potential difference.
When the ionization is distributed through the gas, we have seen that
the current is approximately proportional to the square root of the
potential, and so increases more slowly with the potential difference
than currents through metals. From equation (1) the current is
inversely proportional to the cube of the distance between the
electrodes, so that it falls off with great rapidity as this distance
is increased. We may note that for a given potential difference the
expression for the current does not involve q, the rate of production
of the ions at the electrode, in other words, if we vary the
ionization the current will not begin to be affected by the strength
of the ionization until this falls so low that the current is a
considerable fraction of the saturation current. For the same
potential difference the current is proportional to k, the velocity
under unit electric force of the ion which carries the current. As the
velocity of the negative ion is greater than that of the positive, the
current when the ionization is confined to the neighbourhood of one of
the electrodes will be greater when that electrode is made cathode
than when it is anode. Thus the current will appear to pass more
easily in one direction than in the opposite.
Since the ions which carry the current have to travel all the way from
one electrode to the other, any obstacle which is impervious to these
ions will, if placed between the electrodes, stop the current to the
electrode where there is no ionization. A plate of metal will be as
effectual as one made of a non-conductor, and thus we get the
remarkable result that by interposing a plate of an excellent
conductor like copper or silver between the electrode, we can entirely
stop the current. This experiment can easily be tried by using a hot
plate as the electrode at which the ionization takes place: then if
the other electrode is cold the current which passes when the hot
plate is cathode can be entirely stopped by interposing a cold metal
plate between the electrodes.
_Methods of counting the Number of Ions._--The detection of the ions and
the estimation of their number in a given volume is much facilitated by
the property they possess of promoting the condensation of water-drops
in dust-free air supersaturated with water vapour. If such air contains
no ions, then it requires about an eightfold supersaturation before any
water-drops are formed; if, however, ions are present C. T. R. Wilson
(_Phil. Trans._ 189, p. 265) has shown that a sixfold supersaturation is
sufficient to cause the water vapour to condense round the ions and to
fall down as raindrops. The absence of the drops when no ions are
present is due to the curvature of the drop combined with the surface
tension causing, as Lord Kelvin showed, the evaporation from a small
drop to be exceeding rapid, so that even if a drop of water were formed
the evaporation would be so great in its early stages that it would
rapidly evaporate and disappear. It has been shown, however (J. J.
Thomson, _Application of Dynamics to Physics and Chemistry_, p. 164;
_Conduction of Electricity through Gases_, 2nd ed. p. 179), that if a
drop of water is charged with electricity the effect of the charge is to
diminish the evaporation; if the drop is below a certain size the effect
the charge has in promoting condensation more than counterbalances the
effect of the surface tension in promoting evaporation. Thus the
electric charge protects the drop in the most critical period of its
growth. The effect is easily shown experimentally by taking a bulb
connected with a piston arranged so as to move with great rapidity. When
the piston moves so as to increase the volume of the air contained in
the bulb the air is cooled by expansion, and if it was saturated with
water vapour before it is supersaturated after the expansion. By
altering the throw of the piston the amount of supersaturation can be
adjusted within very wide limits. Let it be adjusted so that the
expansion produces about a sixfold supersaturation; then if the gas is
not exposed to any ionizing agents very few drops (and these probably
due to the small amount of ionization which we have seen is always
present in gases) are formed. If, however, the bulb is exposed to strong
Rontgen rays expansion produces a dense cloud which gradually falls down
and disappears. If the gas in the bulb at the time of its exposure to
the Rontgen rays is subject to a strong electric field hardly any cloud
is formed when the gas is suddenly expanded. The electric field removes
the charged ions from the gas as soon as they are formed so that the
number of ions present is greatly reduced. This experiment furnishes a
very direct proof that the drops of water which form the cloud are only
formed round the ions.
This method gives us an exceedingly delicate test for the presence of
ions, for there is no difficulty in detecting ten or so raindrops per
cubic centimetre; we are thus able to detect the presence of this number
of ions. This result illustrates the enormous difference between the
delicacy of the methods of detecting ions and those for detecting
uncharged molecules; we have seen that we can easily detect ten ions per
cubic centimetre, but there is no known method, spectroscopic or
chemical, which would enable us to detect a billion (10^12) times this
number of uncharged molecules. The formation of the water-drops round
the charged ions gives us a means of counting the number of ions present
in a cubic centimetre of gas; we cool the gas by sudden expansion until
the supersaturation produced by the cooling is sufficient to cause a
cloud to be formed round the ions, and the problem of finding the number
of ions per cubic centimetre of gas is thus reduced to that of finding
the number of drops per cubic centimetre in the cloud. Unless the drops
are very few and far between we cannot do this by direct counting; we
can, however, arrive at the result in the following way. From the amount
of expansion of the gas we can calculate the lowering produced in its
temperature and hence the total quantity of water precipitated. The
water is precipitated as drops, and if all the drops are the same size
the number per cubic centimetre will be equal to the volume of water
deposited per cubic centimetre, divided by the volume of one of the
drops. Hence we can calculate the number of drops if we know their size,
and this can be determined by measuring the velocity with which they
fall under gravity through the air.
The theory of the fall of a heavy drop of water through a viscous
fluid shows that v = (2/9)ga^2/[mu], where a is the radius of the drop,
g the acceleration due to gravity, and [mu] the coefficient of
viscosity of the gas through which the drop falls. Hence if we know v
we can deduce the value of a and hence the volume of each drop and the
number of drops.
_Charge on Ion._--By this method we can determine the number of ions
per unit volume of an ionized gas. Knowing this number we can proceed
to determine the charge on an ion. To do this let us apply an electric
force so as to send a current of electricity through the gas, taking
care that the current is only a small fraction of the saturating
current. Then if u is the sum of the velocities of the positive and
negative ions produced in the electric field applied to the gas, the
current through unit area of the gas is neu, where n is the number of
positive or negative ions per cubic centimetre, and e the charge on an
ion. We can easily measure the current through the gas and thus
determine neu; we can determine n by the method just described, and u,
the velocity of the ions under the given electric field, is known from
the experiments of Zeleny and others. Thus since the product neu, and
two of the factors n, u are known, we can determine the other factor
e, the charge on the ion. This method was used by J. J. Thomson, and
details of the method will be found in _Phil. Mag._ [5], 46, p. 528;
[5], 48, p. 547; [6], 5, p. 346. The result of these measurements
shows that the charge on the ion is the same whether the ionization is
by Rontgen rays or by the influence of ultra-violet light on a metal
plate. It is the same whether the gas ionized is hydrogen, air or
carbonic acid, and thus is presumably independent of the nature of the
gas. The value of e formed by this method was 3.4 X 10^-10
electrostatic units.
H. A. Wilson (_Phil. Mag._ [6], 5, p. 429) used another method. Drops
of water, as we have seen, condense more easily on negative than on
positive ions. It is possible, therefore, to adjust the expansion so
that a cloud is formed on the negative but not on the positive ions.
Wilson arranged the experiments so that such a cloud was formed
between two horizontal plates which could be maintained at different
potentials. The charged drops between the plates were acted upon by a
uniform vertical force which affected their rate of fall. Let X be the
vertical electric force, e the charge on the drop, v1 the rate of fall
of the drop when this force acts, and v the rate of fall due to
gravity alone. Then since the rate of fall is proportionate to the
force on the drop, if a is the radius of the drop, and [rho] its
density, then
Xe + (4/3)[pi][rho]ga^3 v1
---------------------- = ---,
4/3[pi][rho]ga^3 v
or Xe = (4/3)[pi][rho]ga^3(v1 - v)/v.
But v = 2/9ga^2[rho]/[mu],
so that
/ / [mu]^3 v^(3/2)(v1 - v)
Xe = \/ 2.9[pi] - / ------ . ---------------.
\/ g[rho] v
Thus if X, v, v1 are known e can be determined. Wilson by this method
found that e was 3.1 X 10^-10 electrostatic units. A few of the ions
carried charges 2e or 3e.
Townsend has used the following method to compare the charge carried
by a gaseous ion with that carried by an atom of hydrogen in the
electrolysis of solution. We have
u/D = Ne/[Pi],
where D is the coefficient of diffusion of the ions through the gas, u
the velocity of the ion in the same gas when acted on by unit electric
force, N the number of molecules in a cubic centimetre of the gas when
the pressure is [Pi] dynes per square centimetre, and e the charge in
electrostatic units. This relation is obtained on the hypothesis that
N ions in a cubic centimetre produce the same pressure as N uncharged
molecules.
We know the value of D from Townsend's experiments and the values of u
from those of Zeleny. We get the following values for Ne X 10^-10:--
+---------------+---------------------+---------------------+
| | Moist Gas. | Moist Gas. |
+---------------+----------+----------+----------+----------+
| Gas. | Positive | Negative | Positive | Negative |
| | Ions. | Ions. | Ions. | Ions. |
+---------------+----------+----------+----------+----------+
| Air | 1.28 | 1.29 | 1.46 | 1.31 |
| Oxygen | 1.34 | 1.27 | 1.63 | 1.36 |
| Carbonic acid | 1.01 | .87 | .99 | .93 |
| Hydrogen | 1.24 | 1.18 | 1.63 | 1.25 |
+---------------+----------+----------+----------+----------+
| Mean | 1.22 | 1.15 | 1.43 | 1.21 |
+---------------+----------+----------+----------+----------+
Since 1.22 cubic centimetres of hydrogen at the temperature 15 deg. C.
and pressure 760 mm. of mercury are liberated by the passage through
acidulated water of one electromagnetic unit of electricity or 3 X
10^10 electrostatic units, and since in one cubic centimetre of the
gas there are 2.46 N atoms of hydrogen, we have, if E is the charge in
electrostatic units, on the atom of hydrogen in the electrolysis of
solutions