Encyclopaedia Britannica, 11th Edition, "Conduction, Electric"
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Title: Encyclopaedia Britannica, 11th Edition, "Conduction, Electric"
Author: Various
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Transcriber's notes:
(1) Numbers following letters (without space) like C2 were originally
printed in subscript. When letters are subscripted, they are
preceded by an underscore, like C_n.
(2) Characters following a carat (^) were originally printed in
superscript.
(3) Letters topped by Macron are represented as [=x].
(4) [oo] stands for infinity; [int] for integral; [alpha], [beta], etc.
for greek letters.
(5) The following typographical errors have been corrected:
Page 871: "an equation which is very useful, because it enables us,
if we know the distribution." 'because' amended from 'becaus'.
Page 890: "The colour of the luminosity due to positive rays is not
in general the same as that due to anode rays." 'positive' amended
from 'postive'.
Page 890: "Anode Rays.--Gehrcke and Reichenhein (Ann. der Phys. 25,
p. 861) have found that when the anode consists of a mixture of
sodium and lithium chloride." 'mixture' amended from 'mixure'.
ENCYCLOPAEDIA BRITANNICA
A DICTIONARY OF ARTS, SCIENCES, LITERATURE
AND GENERAL INFORMATION
ELEVENTH EDITION
VOLUME VI, SLICE VIII
Conduction, Electric
Article in This Slice:
CONDUCTION, ELECTRIC
CONDUCTION, ELECTRIC. The electric conductivity of a substance is that
property in virtue of which all its parts come spontaneously to the same
electric potential if the substance is kept free from the operation of
electric force. Accordingly, the reciprocal quality, electric
resistivity, may be defined as a quality of a substance in virtue of
which a difference of potential can exist between different portions of
the body when these are in contact with some constant source of
electromotive force, in such a manner as to form part of an electric
circuit.
All material substances possess in some degree, large or small, electric
conductivity, and may for the sake of convenience be broadly divided
into five classes in this respect. Between these, however, there is no
sharply-marked dividing line, and the classification must therefore be
accepted as a more or less arbitrary one. These divisions are: (1)
metallic conductors, (2) non-metallic conductors, (3) dielectric
conductors, (4) electrolytic conductors, (5) gaseous conductors. The
first class comprises all metallic substances, and those mixtures or
combinations of metallic substances known as alloys. The second includes
such non-metallic bodies as carbon, silicon, many of the oxides and
peroxides of the metals, and probably also some oxides of the
non-metals, sulphides and selenides. Many of these substances, for
instance carbon and silicon, are well-known to have the property of
existing in several allotropic forms, and in some of these conditions,
so far from being fairly good conductors, they may be almost perfect
non-conductors. An example of this is seen in the case of carbon in its
three allotropic conditions--charcoal, graphite and diamond. As charcoal
it possesses a fairly well-marked but not very high conductivity in
comparison with metals; as graphite, a conductivity about
one-four-hundredth of that of iron; but as diamond so little
conductivity that the substance is included amongst insulators or
non-conductors. The third class includes those substances which are
generally called insulators or non-conductors, but which are better
denominated dielectric conductors; it comprises such solid substances as
mica, ebonite, shellac, india-rubber, gutta-percha, paraffin, and a
large number of liquids, chiefly hydrocarbons. These substances differ
greatly in insulating power, and according as the conductivity is more
or less marked, they are spoken of as bad or good insulators. Amongst
the latter many of the liquid gases hold a high position. Thus, liquid
oxygen and liquid air have been shown by Sir James Dewar to be almost
perfect non-conductors of electricity.
The behaviour of substances which fall into these three classes is
discussed below in section I., dealing with metallic conduction.
The fourth class, namely the electrolytic conductors comprises all those
substances which undergo chemical decomposition when they form part of
an electric circuit traversed by an electric current. They are discussed
in section II., dealing with electrolytic conduction.
The fifth and last class of conductors includes the gases. The
conditions under which this class of substance becomes possessed of
electric conductivity are considered in section III., on conduction in
gases.
In connexion with metallic conductors, it is a fact of great interest
and considerable practical importance, that, although the majority of
metals when in a finely divided or powdered condition are practically
non-conductors, a mass of metallic powder or filings may be made to pass
suddenly into a conductive condition by being exposed to the influence
of an electric wave. The same is true of the loose contact of two
metallic conductors. Thus if a steel point, such as a needle, presses
very lightly against a metallic plate, say of aluminium, it is found
that this metallic contact, if carefully adjusted, is non-conductive,
but that if an electric wave is created anywhere in the neighbourhood,
this non-conducting contact passes into a conductive state. This fact,
investigated and discovered independently by D. E. Hughes, C. Onesti, E.
Branly, O. J. Lodge and others, is applied in the construction of the
"coherer," or sensitive tube employed as a detector or receiver in that
form of "wireless telegraphy" chiefly developed by Marconi. Further
references to it are made in the articles ELECTRIC WAVES and TELEGRAPHY:
_Wireless_.
_International Ohm._--The practical unit of electrical resistance was
legally defined in Great Britain by the authority of the queen in
council in 1894, as the "resistance offered to an invariable electric
current by a column of mercury at the temperature of melting ice,
14.4521 grammes in mass, of a constant cross-sectional area, and a
length 106.3 centimetres." The same unit has been also legalized as a
standard in France, Germany and the United States, and is denominated
the "International or Standard Ohm." It is intended to represent as
nearly as possible a resistance equal to 10° absolute C.G.S. units of
electric resistance. Convenient multiples and subdivisions of the ohm
are the microhm and the megohm, the former being a millionth part of
an ohm, and the latter a million ohms. The resistivity of substances
is then numerically expressed by stating the resistance of one cubic
centimetre of the substance taken between opposed faces, and expressed
in ohms, microhms or megohms, as may be most convenient. The
reciprocal of the ohm is called the mho, which is the unit of
conductivity, and is defined as the conductivity of a substance whose
resistance is one ohm. The absolute unit of conductivity is the
conductivity of a substance whose resistivity is one absolute C.G.S.
unit, or one-thousandth-millionth part of an ohm. Resistivity is a
quality in which material substances differ very widely. The metals
and alloys, broadly speaking, are good conductors, and their
resistivity is conveniently expressed in microhms per cubic
centimetre, or in absolute C.G.S. units. Very small differences in
density and in chemical purity make, however, immense differences in
electric resistivity; hence the values given by different
experimentalists for the resistivity of known metals differ to a
considerable extent.
I. CONDUCTION IN SOLIDS
It is found convenient to express the resistivity of metals in two
different ways: (1) We may state the resistivity of one cubic centimetre
of the material in microhms or absolute units taken between opposed
faces. This is called the _volume-resistivity_; (2) we may express the
resistivity by stating the resistance in ohms offered by a wire of the
material in question of uniform cross-section one metre in length, and
one gramme in weight. This numerical measure of the resistivity is
called the _mass-resistivity_. The mass-resistivity of a body is
connected with its volume-resistivity and the density of the material in
the following manner:--The mass-resistivity, expressed in microhms per
metre-gramme, divided by 10 times the density is numerically equal to
the volume-resistivity per centimetre-cube in absolute C.G.S. units. The
mass-resistivity per metre-gramme can always be obtained by measuring
the resistance and the mass of any wire of uniform cross-section of
which the length is known, and if the density of the substance is then
measured, the volume-resistivity can be immediately calculated.
If R is the resistance in ohms of a wire of length l, uniform
cross-section s, and density d, then taking [rho] for the
volume-resistivity we have 10^9R = [rho]l/s; but lsd = M, where M is
the mass of the wire. Hence 10^9R = [rho]dl²/M. If l = 100 and M = 1,
then R = [rho]'= resistivity in ohms per metre-gramme, and 10^9[rho]'
= 10,000d[rho], or [rho] = 10^5[rho]'/d, and [rho]' = 10,000MR/l².
The following rules, therefore, are useful in connexion with these
measurements. To obtain the mass-resistivity per metre-gramme of a
substance in the form of a uniform metallic wire:--Multiply together
10,000 times the mass in grammes and the total resistance in ohms, and
then divide by the square of the length in centimetres. Again, to
obtain the volume-resistivity in C.G.S. units per centimetre-cube, the
rule is to multiply the mass-resistivity in ohms by 100,000 and divide
by the density. These rules, of course, apply only to wires of uniform
cross-section. In the following Tables I., II. and III. are given the
mass and volume resistivity of ordinary metals and certain alloys
expressed in terms of the international ohm or the absolute C.G.S.
unit of resistance, the values being calculated from the experiments
of A. Matthiessen (1831-1870) between 1860 and 1865, and from later
results obtained by J. A. Fleming and Sir James Dewar in 1893.
TABLE I.--_Electric Mass-Resistivity of Various Metals at 0° C., or
Resistance per Metre-gramme in International Ohms at 0° C._
(Matthiessen.)
+---------------------+---------------------+-------------+
| | Resistance at 0° C. | |
| | in International | Approximate |
| Metal. | Ohms of a Wire | Temperature |
| | 1 Metre long and | Coefficient |
| | Weighing | near 20° C. |
| | 1 Gramme. | |
+---------------------+---------------------+-------------+
|Silver (annealed) | .1523 | 0.00377 |
|Silver (hard-drawn) | .1657 | .. |
|Copper (annealed) | .1421 | 0.00388 |
|Copper (hard-drawn) | .1449 (Matthiessen's Standard) |
|Gold (annealed) | .4025 | 0.00365 |
|Gold (hard-drawn) | .4094 | .. |
|Aluminium (annealed) | .0757 | .. |
|Zinc (pressed) | .4013 | .. |
|Platinum (annealed) | 1.9337 | .. |
|Iron (annealed) | .765 | .. |
|Nickel (annealed) | 1.058[1] | .. |
|Tin (pressed) | .9618 | 0.00365 |
|Lead (pressed) | 2.2268 | 0.00387 |
|Antimony (pressed) | 2.3787 | 0.00389 |
|Bismuth (pressed) | 12.8554[1] | 0.00354 |
|Mercury (liquid) | 12.885[2] | 0.00072 |
+---------------------+---------------------+-------------+
The data commonly used for calculating metallic resistivities were
obtained by A. Matthiessen, and his results are set out in the Table
II. which is taken from Cantor lectures given by Fleeming Jenkin in
1866 at or about the date when the researches were made. The figures
given by Jenkin have, however, been reduced to international ohms and
C.G.S. units by multiplying by ([pi]/4) × 0.9866 × 10^5 = 77,485.
Subsequently numerous determinations of the resistivity of various
pure metals were made by Fleming and Dewar, whose results are set out
in Table III.
TABLE II.--_Electric Volume-Resistivity of Various Metals at 0° C.,
or Resistance per Centimetre-cube in C.G.S. Units at 0° C._
+---------------------+---------------------+
| | Volume-Resistivity. |
| Metal. | at 0° C. in C.G.S. |
| | Units |
+---------------------+---------------------+
|Silver (annealed) | 1,502 |
|Silver (hard-drawn) | 1,629 |
|Copper (annealed) | 1,594 |
|Copper (hard-drawn) | 1,630[3] |
|Gold (annealed) | 2,052 |
|Gold (hard-drawn) | 2,090 |
|Aluminium (annealed) | 3,006 |
|Zinc (pressed) | 5,621 |
|Platinum (annealed) | 9,035 |
|Iron (annealed) | 10,568 |
|Nickel (annealed) | 12,429[4] |
|Tin (pressed) | 13,178 |
|Lead (pressed) | 19,580 |
|Antimony (pressed) | 35,418 |
|Bismuth (pressed) | 130,872 |
|Mercury (liquid) | 94,896[5] |
+---------------------+---------------------+
TABLE III.--_Electric Volume-Resistivity of Various Metals at 0° C.,
or Resistance per Centimetre-cube at 0° C. in C.G.S. Units._
(Fleming and Dewar, _Phil. Mag._, September 1893.)
+--------------------------+------------------+------------------+
| | Resistance | Mean Temperature |
| Metal. | at 0° C. per | Coefficient |
| | Centimetre-cube | between 0° C. |
| | in C.G.S. Units. | and 100° C. |
+--------------------------+------------------+------------------+
|Silver (electrolytic and | | |
| well annealed)[6] | 1,468 | 0.00400 |
|Copper (electrolytic and | | |
| well annealed)[6] | 1,561 | 0.00428 |
|Gold (annealed) | 2,197 | 0.00377 |
|Aluminium (annealed) | 2,665 | 0.00435 |
|Magnesium (pressed) | 4,355 | 0.00381 |
|Zinc | 5,751 | 0.00406 |
|Nickel (electrolytic)[6] | 6,935 | 0.00618 |
|Iron (annealed) | 9,065 | 0.00625 |
|Cadmium | 10,023 | 0.00419 |
|Palladium | 10,219 | 0.00354 |
|Platinum (annealed) | 10,917 | 0.003669 |
|Tin (pressed) | 13,048 | 0.00440 |
|Thallium (pressed) | 17,633 | 0.00398 |
|Lead (pressed) | 20,380 | 0.00411 |
|Bismuth (electrolytic)[7] | 110,000 | 0.00433 |
+--------------------------+------------------+------------------+
_Resistivity of Mercury._--The volume-resistivity of pure mercury is a
very important electric constant, and since 1880 many of the most
competent experimentalists have directed their attention to the
determination of its value. The experimental process has usually been
to fill a glass tube of known dimensions, having large cup-like
extensions at the ends, with pure mercury, and determine the absolute
resistance of this column of metal. For the practical details of this
method the following references may be consulted:--"The Specific
Resistance of Mercury," Lord Rayleigh and Mrs Sidgwick, _Phil.
Trans._, 1883, part i. p. 173, and R. T. Glazebrook, _Phil. Mag._,
1885, p. 20; "On the Specific Resistance of Mercury," R. T. Glazebrook
and T. C. Fitzpatrick, _Phil. Trans._, 1888, p. 179, or _Proc. Roy.
Soc._, 1888, p. 44, or _Electrician_, 1888, 21, p. 538; "Recent
Determinations of the Absolute Resistance of Mercury," R. T.
Glazebrook, _Electrician_, 1890, 25, pp. 543 and 588. Also see J. V.
Jones, "On the Determination of the Specific Resistance of Mercury in
Absolute Measure," _Phil. Trans._, 1891, A, p. 2. Table IV. gives the
values of the volume-resistivity of mercury as determined by various
observers, the constant being expressed (a) in terms of the resistance
in ohms of a column of mercury one millimetre in cross-section and 100
centimetres in length, taken at 0° C.; and (b) in terms of the length
in centimetres of a column of mercury one square millimetre in
cross-section taken at 0° C. The result of all the most careful
determinations has been to show that the resistivity of pure mercury
at 0° C. is about 94,070 C.G.S. electromagnetic units of resistance,
and that a column of mercury 106.3 centimetres in length having a
cross-sectional area of one square millimetre would have a resistance
at 0° C. of one international ohm. These values have accordingly been
accepted as the official and recognized values for the specific
resistance of mercury, and the definition of the ohm. The table also
states the methods which have been adopted by the different observers
for obtaining the absolute value of the resistance of a known column
of mercury, or of a resistance coil afterwards compared with a known
column of mercury. A column of figures is added showing the value in
fractions of an international ohm of the British Association Unit
(B.A.U.), formerly supposed to represent the true ohm. The real value
of the B.A.U. is now taken as .9866 of an international ohm.
TABLE IV.--_Determinations of the Absolute Value of the
Volume-Resistivity of Mercury and the Mercury Equivalent of the
Ohm._
+-----------------+------+----------------------+--------+----------+---------+
| | | | |Value of |Value of |
| | | |Value of|100 Centi-|Ohm in |
| Observer. | Date.| Method. |B.A.U. |metres of |Centi- |
| | | |in Ohms.|Mercury |metres of|
| | | | |in Ohms. |Mercury. |
+-----------------+------+----------------------+--------+----------+---------+
|Lord Rayleigh | 1882 | Rotating coil | .98651 | .94133 | 106.31 |
|Lord Rayleigh | 1883 | Lorenz method | .98677 | .. | 106.27 |
|G. Wiedemann | 1884 | Rotation through 180°| .. | .. | 106.19 |
|E. E. N. Mascart | 1884 | Induced current | .98611 | .94096 | 106.33 |
|H. A. Rowland | 1887 | Mean of several | .98644 | .94071 | 106.32 |
| | | methods | | | |
|F. Kohlrausch | 1887 | Damping of magnets | .98660 | .94061 | 106.32 |
|R. T. Glazebrook | 1882 | Induced currents | .98665 | .94074 | 106.29 |
| | 1888 | | | | |
|Wuilleumeier | 1890 | | .98686 | .94077 | 106.31 |
|Duncan and Wilkes| 1890 | Lorenz | .98634 | .94067 | 106.34 |
|J. V. Jones | 1891 | Lorenz | .. | .94067 | 106.31 |
| | | +--------+----------+---------+
| | | Mean value .98653 | | |
| | | | | |
|Streker | 1885 | An absolute determin-| | .94056 | 106.32 |
|Hutchinson | 1888 | ation of resistance| | .94074 | 106.30 |
|E. Salvioni | 1890 | was not made. The | | .94054 | 106.33 |
|E. Salvioni | .. | value .98656 has | | .94076 | 106.30 |
| | | been used | | | |
| | | | +----------+---------+
| | | Mean value .94076 | 106.31 |
| | | | |
|H. F. Weber | 1884 | Induced current | Absolute measure- | 105.37 |
|H. F. Weber | .. | Rotating coil | ments compared | 106.16 |
|A. Roiti | 1884 | Mean effect of | with German silver| 105.89 |
| | | induced current | wire coils issued | |
|F. Himstedt | 1885 | | by Siemens and | 105.98 |
| | | | Streker | |
|F. E. Dorn | 1889 | Damping of a magnet | | 106.24 |
|Wild | 1883 | Damping of a magnet | | 106.03 |
|L. V. Lorenz | 1885 | Lorenz method | | 105.93 |
+-----------------+------+----------------------+-------------------+---------+
For a critical discussion of the methods which have been adopted in
the absolute determination of the resistivity of mercury, and the
value of the British Association unit of resistance, the reader may be
referred to the _British Association Reports_ for 1890 and 1892
(_Report of Electrical Standards Committee_), and to the
_Electrician_, 25, p. 456, and 29, p. 462. A discussion of the
relative value of the results obtained between 1882 and 1890 was given
by R. T. Glazebrook in a paper presented to the British Association at
Leeds, 1890.
_Resistivity of Copper._--In connexion with electro-technical work the
determination of the conductivity or resistivity values of annealed
and hard-drawn copper wire at standard temperatures is a very
important matter. Matthiessen devoted considerable attention to this
subject between the years 1860 and 1864 (see _Phil. Trans._, 1860, p.
150), and since that time much additional work has been carried out.
Matthiessen's value, known as _Matthiessen's Standard_, for the
mass-resistivity of pure hard-drawn copper wire, is the resistance of
a wire of pure hard-drawn copper one metre long and weighing one
gramme, and this is equal to 0.14493 international ohms at 0° C. For
many purposes it is more convenient to express temperature in
Fahrenheit degrees, and the recommendation of the 1899 committee on
copper conductors[8] is as follows:--"Matthiessen's standard for
hard-drawn conductivity commercial copper shall be considered to be
the resistance of a wire of pure hard-drawn copper one metre long,
weighing one gramme which at 60° F. is 0.153858 international ohms."
Matthiessen also measured the mass-resistivity of annealed copper, and
found that its conductivity is greater than that of hard-drawn copper
by about 2.25% to 2.5% As annealed copper may vary considerably in its
state of annealing, and is always somewhat hardened by bending and
winding, it is found in practice that the resistivity of commercial
annealed copper is about 1¼% less than that of hard-drawn copper. The
standard now accepted for such copper, on the recommendation of the
1899 Committee, is a wire of pure annealed copper one metre long,
weighing one gramme, whose resistance at 0° C. is 0.1421 international
ohms, or at 60° F., 0.150822 international ohms. The specific gravity
of copper varies from about 8.89 to 8.95, and the standard value
accepted for high conductivity commercial copper is 8.912,
corresponding to a weight of 555 lb per cubic foot at 60° F. Hence the
volume-resistivity of pure annealed copper at 0° C. is 1.594 microhms
per c.c., or 1594 C.G.S. units, and that of pure hard-drawn copper at
0° C. is 1.626 microhms per c.c., or 1626 C.G.S. units. Since
Matthiessen's researches, the most careful scientific investigation on
the conductivity of copper is that of T. C. Fitzpatrick, carried out
in 1890. (_Brit. Assoc. Report_, 1890, Appendix 3, p. 120.)
Fitzpatrick confirmed Matthiessen's chief result, and obtained values
for the resistivity of hard-drawn copper which, when corrected for
temperature variation, are in entire agreement with those of
Matthiessen at the same temperature.
The volume resistivity of alloys is, generally speaking, much higher
than that of pure metals. Table V. shows the volume resistivity at 0° C.
of a number of well-known alloys, with their chemical composition.
TABLE V.--_Volume-Resistivity of Alloys of known Composition at 0°
C. in C.G.S. Units per Centimetre-cube. Mean Temperature
Coefficients taken at 15° C._ (Fleming and Dewar.)
+-----------------------+-----------+-----------+-------------------+
| |Resistivity|Temperature|Composition in per |
| Alloys. |at 0° C. |Coefficient| cents. |
| | |at 15° C. | |
+-----------------------+-----------+-----------+-------------------+
|Platinum-silver | 31,582 | .000243 |Pt 33%, Ag 66% |
|Platinum-iridium | 30,896 | .000822 |Pt 80%, Ir 20% |
|Platinum-rhodium | 21,142 | .00143 |Pt 90%, Rd 10% |
|Gold-silver | 6,280 | .00124 |Au 90%, Ag 10% |
|Manganese-steel | 67,148 | .00127 |Mn 12%, Fe 78% |
|Nickel-steel | 29,452 | .00201 |Ni 4.35%, remaining|
| | | | percentage |
| | | | chiefly iron, but|
| | | | uncertain |
|German silver | 29,982 | .000273 |Cu5Zn3Ni2 |
|Platinoid[9] | 41,731 | .00031 | |
|Manganin | 46,678 | .0000 |Cu 84%, Mn 12%, |
| | | |Ni 4% |
|Aluminium-silver | 4,641 | .00238 |Al 94%, Ag 6% |
|Aluminium-copper | 2,904 | .00381 |Al 94%, Cu 6% |
|Copper-aluminium | 8,847 | .000897 |Cu 97%, Al 3% |
|Copper-nickel-aluminium| 14,912 | .000643 |Cu 87%, Ni 6.5%, |
| | | | Al 6.5% |
|Titanium-aluminium | 3,887 | .00290 | |
+-----------------------+-----------+-----------+-------------------+
Generally speaking, an alloy having high resistivity has poor
mechanical qualities, that is to say, its tensile strength and
ductility are small. It is possible to form alloys having a
resistivity as high as 100 microhms per cubic centimetre; but, on the
other hand, the value of an alloy for electro-technical purposes is
judged not merely by its resistivity, but also by the degree to which
its resistivity varies with temperature, and by its capability of
being easily drawn into fine wire of not very small tensile strength.
Some pure metals when alloyed with a small proportion of another metal
do not suffer much change in resistivity, but in other cases the
resultant alloy has a much higher resistivity. Thus an alloy of pure
copper with 3% of aluminium has a resistivity about 5½ times that of
copper; but if pure aluminium is alloyed with 6% of copper, the
resistivity of the product is not more than 20% greater than that of
pure aluminium. The presence of a very small proportion of a
non-metallic element in a metallic mass, such as oxygen, sulphur or
phosphorus, has a very great effect in increasing the resistivity.
Certain metallic elements also have the same power; thus platinoid has
a resistivity 30% greater than German silver, though it differs from
it merely in containing a trace of tungsten.
The resistivity of non-metallic conductors is in all cases higher than
that of any pure metal. The resistivity of carbon, for instance, in the
forms of charcoal or carbonized organic material and graphite, varies
from 600 to 6000 microhms per cubic centimetre, as shown in Table VI.:--
TABLE VI.--_Electric Volume-Resistivity in Microhms per
Centimetre-cube of Various Forms of Carbon at 15° C._
+-----------------------------------------+---------------+
| Substance. | Resistivity. |
+-----------------------------------------+---------------+
| Arc lamp carbon rod | 8000 |
| Jablochkoff candle carbon | 4000 |
| Carré carbon | 3400 |
| Carbonized bamboo | 6000 |
| Carbonized parchmentized thread | 4000 to 5000 |
| Ordinary carbon filament from glow-lamp | |
| "treated" or flashed | 2400 to 2500 |
| Deposited or secondary carbon | 600 to 900 |
| Graphite | 400 to 500 |
+-----------------------------------------+---------------+
The resistivity of liquids is, generally speaking, much higher than that
of any metals, metallic alloys or non-metallic conductors. Thus fused
lead chloride, one of the best conducting liquids, has a resistivity in
its fused condition of 0.376 ohm per centimetre-cube, or 376,000
microhms per centimetre-cube, whereas that of metallic alloys only in
few cases exceeds 100 microhms per centimetre-cube. The resistivity of
solutions of metallic salts also varies very largely with the proportion
of the diluent or solvent, and in some instances, as in the aqueous
solutions of mineral acids; there is a maximum conductivity
corresponding to a certain dilution. The resistivity of many liquids,
such as alcohol, ether, benzene and pure water, is so high, in other
words, their conductivity is so small, that they are practically
insulators, and the resistivity can only be appropriately expressed in
megohms per centimetre-cube.
In Table VII. are given the names of a few of these badly-conducting
liquids, with the values of their volume-resistivity in megohms per
centimetre-cube:--
TABLE VII.--_Electric Volume-Resistivity of Various Badly-Conducting
Liquids in Megohms per Centimetre-cube._
+---------------------------+----------------+---------------------+
| | Resistivity | |
| Substance. | in Megohms | Observer. |
| | per c.c. | |
+---------------------------+----------------+---------------------+
| Ethyl alcohol | 0.5 | Pfeiffer. |
| Ethyl ether | 1.175 to 3.760 | W. Kohlrausch. |
| Benzene | 4.700 | |
| Absolutely pure water | 25.0 at 18° C. | Value estimated |
| approximates probably to | | by F. Kohlrausch |
| | | and A. Heydweiler. |
| All very dilute aqueous | 1.00 at 18° C. | From results by |
| salt solutions having a | | F. Kohlrausch |
| concentration of about | | and others. |
| 0.00001 of an equivalent | | |
| gramme molecule[10] per | | |
| litre approximate to | | |
+---------------------------+----------------+---------------------+
The resistivity of all those substances which are generally called
dielectrics or insulators is also so high that it can only be
appropriately expressed in millions of megohms per centimetre-cube, or
in megohms per quadrant-cube, the quadrant being a cube the side of
which is 10^9 cms. (see Table VIII.).
Effects of Heat.--Temperature affects the resistivity of these different
classes of conductors in different ways. In all cases, so far as is yet
known, the resistivity of a pure metal is increased if its temperature is
raised, and decreased if the temperature is lowered, so that if it could
be brought to the absolute zero of temperature (-273° C.) its resistivity
would be reduced to a very small fraction of its resistance at ordinary
temperatures. With metallic alloys, however, rise of temperature does not
always increase resistivity: it sometimes diminishes it, so that many
alloys are known which have a maximum resistivity corresponding to a
certain temperature, and at or near this point they vary very little in
resistance with temperature. Such alloys have, therefore, a negative
temperature-variation of resistance at and above fixed temperatures.
Prominent amongst these metallic compounds are alloys of iron, manganese,
nickel and copper, some of which were discovered by Edward Weston, in the
United States. One well-known alloy of copper, manganese and nickel, now
called manganin, which was brought to the notice of electricians by the
careful investigations made at the Berlin Physikalisch-Technische
Reichsanstalt, is characterized by having a zero temperature coefficient
at or about a certain temperature in the neighbourhood of 15° C. Hence
within a certain range of temperature on either side of this critical
value the resistivity of manganin is hardly affected at all by
temperature. Similar alloys can be produced from copper and
ferro-manganese. An alloy formed of 80% copper and 20% manganese in an
annealed condition has a nearly zero temperature-variation of resistance
between 20° C. and 100° C. In the case of non-metals the action of
temperature is generally to diminish the resistivity as temperature
rises, though this is not universally so. The interesting observation has
been recorded by J. W. Howell, that "treated" carbon filaments and
graphite are substances which have a minimum resistance corresponding to
a certain temperature approaching red heat (_Electrician_, vol. xxxviii.
p. 835). At and beyond this temperature increased heating appears to
increase their resistivity; this phenomenon may, however, be accompanied
by a molecular change and not be a true temperature variation. In the
case of dielectric conductors and of electrolytes, the action of rising
temperature is to reduce resistivity. Many of the so-called insulators,
such as mica, ebonite, indiarubber, and the insulating oils, paraffin,
&c., decrease in resistivity with great rapidity as the temperature
rises. With guttapercha a rise in temperature from 0° C. to 24° C. is
sufficient to reduce the resistivity of one-twentieth part of its value
at 0° C., and the resistivity of flint glass at 140° C. is only
one-hundredth of what it is at 60° C.
TABLE VIII.--_Electric Volume-Resistivity of Dielectrics reckoned in
Millions of Megohms (Mega-megohms) per Centimetre-cube, and in Megohms
per Quadrant-cube, i.e. a Cube whose Side is 10^9 cms._
+------------------------+---------------------------+----------+
| | Resistivity. | |
| +------------+--------------+ Tempera- |
| Substance. |Mega-megohms| Megohms per | tura |
| | per c.c. |Quadrant-cube.| Cent. |
+------------------------+------------+--------------+----------+
|Bohemian glass | 61 | .061 | 60° |
|Mica | 84 | .084 | 20° |
|Gutta-percha | 450 | .45 | 24° |
|Flint glass | 1,020 | 1.02 | 60° |
|Glover's vulcanized | | | |
| indiarubber | 1,630 | 1.63 | 15° |
|Siemens' ordinary pure | | | |
| vulcanized indiarubber | 2,280 | 2.28 | 15° |
|Shellac | 9,000 | 9.0 | 28° |
|Indiarubber | 10,900 | 10.9 | 24° |
|Siemens' high-insulating| | | |
| fibrous material | 11,900 | 11.9 | 15° |
|Siemens' special | | | |
| high-insulating | | | |
| indiarubber. | 16,170 | 16.17 | 15° |
|Flint glass | 20,000 | 20.0 | 20° |
|Ebonite | 28,000 | 28. | 46° |
|Paraffin | 34,000 | 34. | 46° |
+------------------------+------------+--------------+----------+
A definition may here be given of the meaning of the term _Temperature
Coefficient_. If, in the first place, we suppose that the resistivity
([rho]t) at any temperature (t) is a simple linear function of the
resistivity ([rho]0) at 0° C., then we can write [rho]t = [rho]0(1 +
[alpha]t), or [alpha] = ([rho]t - [rho]0)/[rho]0t.
The quantity [alpha] is then called the temperature-coefficient, and
its reciprocal is the temperature at which the resistivity would
become zero. By an extension of this notion we can call the quantity
d[rho]/[rho]dt the temperature coefficient corresponding to any
temperature t at which the resistivity is [rho]. In all cases the
relation between the resistivity of a substance and the temperature is
best set out in the form of a curve called a temperature-resistance
curve. If a series of such curves are drawn for various pure metals,
temperature being taken as abscissa and resistance as ordinate, and if
the temperature range extends from the absolute zero of temperature
upwards, then it is found that these temperature-resistance lines are
curved lines having their convexity either upwards or downwards. In
other words, the second differential coefficient of resistance with
respect to temperature is either a positive or negative quantity. An
extensive series of observations concerning the form of the
resistivity curves for various pure metals over a range of temperature
extending from -200° C. to +200° C. was carried out in 1892 and 1893
by Fleming and Dewar (_Phil. Mag._ Oct. 1892 and Sept. 1893). The
resistance observations were taken with resistance coils constructed
with wires of various metals obtained in a state of great chemical
purity. The lengths and mean diameters of the wires were carefully
measured, and their resistance was then taken at certain known
temperatures obtained by immersing the coils in boiling aniline,
boiling water, melting ice, melting carbonic acid in ether, and
boiling liquid oxygen, the temperatures thus given being +184°.5 C.,
+100° C., 0° C., -78°.2 C. and -182°.5 C. The resistivities of the
various metals were then calculated and set out in terms of the
temperature. From these data a chart was prepared showing the
temperature-resistance curves of these metals throughout a range of
400 degrees. The exact form of these curves through the region of
temperature lying between -200° C. and -273° C. is not yet known. As
shown on the chart, the curves evidently do not converge to precisely
the same point. It is, however, much less probable that the resistance
of any metal should vanish at a temperature above the absolute zero
than at the absolute zero itself, and the precise path of these curves
at their lower ends cannot be delineated until means are found for
fixing independently the temperature of some regions in which the
resistance of metallic wires can be measured. Sir J. Dewar
subsequently showed that for certain pure metals it is clear that the
resistance would not vanish at the absolute zero but would be reduced
to a finite but small value (see "Electric Resistance Thermometry at
the Temperature of Boiling Hydrogen," _Proc. Roy. Soc._ 1904, 73, p.
244).
The resistivity curves of the magnetic metals are also remarkable for
the change of curvature they exhibit at the magnetic critical
temperature. Thus J. Hopkinson and D. K. Morris (_Phil. Mag._
September 1897, p. 213) observed the remarkable alteration that takes
place in the iron resistance temperature curve in the neighbourhood of
780° C. At that temperature the direction of the curvature of the
curve changes so that it becomes convex upwards instead of convex
downwards, and in addition the value of the temperature coefficient
undergoes a great reduction. The mean temperature coefficient of iron
in the neighbourhood of 0° C. is 0.0057; at 765° C. it rises to a
maximum value 0.0204; but at 1000° C. it falls again to a lower value,
0.00244. A similar rise to a maximum value and subsequent fall are
also noted in the case of the specific heat of iron. The changes in
the curvature of the resistivity curves are undoubtedly connected with
the molecular changes that occur in the magnetic metals at their
critical temperatures.
A fact of considerable interest in connexion with resistivity is the
influence exerted by a strong magnetic field in the case of some
metals, notably bismuth. It was discovered by A. Righi and confirmed
by S. A. Leduc (_Journ. de Phys._ 1886, 5, p. 116, and 1887, 6, p.
189) that if a pure bismuth wire is placed in a magnetic field
transversely to the direction of the magnetic field, its resistance is
considerably increased. This increase is greatly affected by the
temperature of the metal (Dewar and Fleming, _Proc. Roy. Soc._ 1897,
60, p. 427). The temperature coefficient of pure copper is an
important constant, and its value as determined by Messrs Clark, Forde
and Taylor in terms of Fahrenheit temperature is
[rho]t = [rho]32 {1 + 0.0023708(t - 32) + 0.0000034548(t - 32)²}.
_Time Effects._--In the case of dielectric conductors, commonly called
insulators, such as indiarubber, guttapercha, glass and mica, the
electric resistivity is not only a function of the temperature but also
of the time during which the electromotive force employed to measure it
is imposed. Thus if an indiarubber-covered cable is immersed in water
and the resistance of the dielectric between the copper conductor and
the water measured by ascertaining the current which can be caused to
flow through it by an electromotive force, this current is found to vary
very rapidly with the time during which the electromotive force is
applied. Apart from the small initial effect due to the electrostatic
capacity of the cable, the application of an electromotive force to the
dielectric produces a current through it which rapidly falls in value,
as if the electric resistance of the dielectric were increasing. The
current, however, does not fall continuously but tends to a limiting
value, and it appears that if the electromotive force is kept applied
to the cable for a prolonged time, a small and nearly constant current
will ultimately be found flowing through it. It is customary in
electro-technical work to consider the resistivity of the dielectric as
the value it has after the electromotive force has been applied for one
minute, the standard temperature being 75° F. This, however, is a purely
conventional proceeding, and the number so obtained does not necessarily
represent the true or ohmic resistance of the dielectric. If the
electromotive force is increased, in the case of a large number of
ordinary dielectrics the apparent resistance at the end of one minute's
electrification decreases as the electromotive force increases.
_Practical Standards._--The practical measurement of resistivity
involves many processes and instruments (see WHEATSTONE'S BRIDGE and
OHMMETER). Broadly speaking, the processes are divided into _Comparison
Methods_ and _Absolute Methods_. In the former a comparison is effected
between the resistance of a material in a known form and some standard
resistance. In the _Absolute Methods_ the resistivity is determined
without reference to any other substance, but with reference only to the
fundamental standards of length, mass and time. Immense labour has been
expended in investigations concerned with the production of a standard
of resistance and its evaluation in absolute measure. In some cases the
absolute standard is constructed by filling a carefully-calibrated tube
of glass with mercury, in order to realize in a material form the
official definition of the ohm; in this manner most of the principal
national physical laboratories have been provided with standard mercury
ohms. (For a full description of the standard mercury ohm of the Berlin
Physikalisch-Technische Reichsanstalt, see the _Electrician_, xxxvii.
569.) For practical purposes it is more convenient to employ a standard
of resistance made of wire.
Opinion is not yet perfectly settled on the question whether a wire
made of any alloy can be considered to be a perfectly unalterable
standard of resistance, but experience has shown that a platinum
silver alloy (66% silver, 33% platinum), and also the alloy called
manganin, seem to possess the qualities of permanence essential for a
wire-resistance standard. A comparison made in 1892 and 1894 of all
the manganin wire copies of the ohm made at the Reichsanstalt in
Berlin, showed that these standards had remained constant for two
years to within one or two parts in 100,000. It appears, however, that
in order that manganin may remain constant in resistivity when used in
the manufacture of a resistance coil, it is necessary that the alloy
should be _aged_ by heating it to a temperature of 140° C. for ten
hours; and to prevent subsequent changes in resistivity, solders
containing zinc must be avoided, and a silver solder containing 75% of
silver employed in soldering the manganin wire to its connexions.
The authorities of the Berlin Reichsanstalt have devoted considerable
attention to the question of the best form for a wire standard of
electric resistance. In that now adopted the resistance wire is
carefully insulated and wound on a brass cylinder, being doubled on
itself to annul inductance as much as possible. In the coil two wires
are wound on in parallel, one being much finer than the other, and the
final adjustment of the coil to an exact value is made by shortening the
finer of the two. A standard of resistance for use in a laboratory now
generally consists of a wire of manganin or platinum-silver carefully
insulated and enclosed in a brass case. Thick copper rods are connected
to the terminals of the wire in the interior of the case, and brought to
the outside, being carefully insulated at the same time from one another
and from the case. The coil so constructed can be placed under water or
paraffin oil, the temperature of which can be exactly observed during
the process of taking a resistance measurement. Equalization of the
temperature of the surrounding medium is effected by the employment of a
stirrer, worked by hand or by a small electric motor. The construction
of a standard of electrical resistance consisting of mercury in a glass
tube is an operation requiring considerable precautions, and only to be
undertaken by those experienced in the matter. Opinions are divided on
the question whether greater permanence in resistance can be secured by
mercury-in-glass standards of resistance or by wire standards, but the
latter are at least more portable and less fragile.
A full description of the construction of a standard wire-resistance
coil on the plan adopted by the Berlin Physikalisch-Technische
Reichsanstalt is given in the Report of the British Association
Committee on Electrical Standards, presented at the Edinburgh Meeting
in 1892. For the design and construction of standards of electric
resistances adapted for employment in the comparison and measurement
of very low or very high resistances, the reader may be referred to
standard treatises on electric measurements.
BIBLIOGRAPHY.--See also J. A. Fleming, _A Handbook for the Electrical
Laboratory and Testing Room_, vol. i. (London, 1901); _Reports of the
British Association Committee on Electrical Standards_, edited by
Fleeming Jenkin (London, 1873); A. Matthiessen and C. Vogt, "On the
Influence of Temperature on the Conducting Power of Alloys," _Phil.
Trans._, 1864, 154, p. 167, and _Phil. Mag._, 1865, 29, p. 363; A.
Matthiessen and M. Holtzmann, "On the Effect of the Presence of Metals
and Metalloids upon the Electric Conducting Power of Pure Copper,"
_Phil. Trans._, 1860, 150, p. 85; T. C. Fitzpatrick, "On the Specific
Resistance of Copper," _Brit. Assoc. Report_, 1890, p. 120, or
_Electrician_, 1890, 25, p. 608; R. Appleyard, _The Conductometer and
Electrical Conductivity_; Clark, Forde and Taylor, _Temperature
Coefficients of Copper_ (London, 1901). (J. A. F.)
II. CONDUCTION IN LIQUIDS
Through liquid metals, such as mercury at ordinary temperatures and
other metals at temperatures above their melting points, the electric
current flows as in solid metals without changing the state of the
conductor, except in so far as heat is developed by the electric
resistance. But another class of liquid conductors exists, and in them
the phenomena are quite different. The conductivity of fused salts, and
of solutions of salts and acids, although less than that of metals, is
very great compared with the traces of conductivity found in so-called
non-conductors. In fused salts and conducting solutions the passage of
the current is always accompanied by definite chemical changes; the
substance of the conductor or electrolyte is decomposed, and the
products of the decomposition appear at the electrodes, i.e. the
metallic plates by means of which the current is led into and out of the
solution. The chemical phenomena are considered in the article
ELECTROLYSIS; we are here concerned solely with the mechanism of this
_electrolytic_ conduction of the current.
To explain the appearance of the products of decomposition at the
electrodes only, while the intervening solution is unaltered, we suppose
that, under the action of the electric forces, the opposite parts of the
electrolyte move in opposite directions through the liquid. These
opposite parts, named ions by Faraday, must therefore be associated with
electric charges, and it is the convective movement of the opposite
streams of ions carrying their charges with them that, on this view,
constitutes the electric current.
In metallic conduction it is found that the current is proportional to
the applied electromotive force--a relation known by the name of Ohm's
law. If we place in a circuit with a small electromotive force an
electrolytic cell consisting of two platinum electrodes and a solution,
the initial current soon dies away, and we shall find that a certain
minimum electromotive force must be applied to the circuit before any
considerable permanent current passes. The chemical changes which are
initiated on the surfaces of the electrodes set up a reverse
electromotive force of polarization, and, until this is overcome, only a
minute current, probably due to the slow but steady removal of the
products of decomposition from the electrodes by a process of diffusion,
will pass through the cell. Thus it is evident that, considering the
electrolytic cell as a whole, the passage of the current through it
cannot conform to Ohm's law. But the polarization is due to chemical
changes, which are confined to the surfaces of the electrodes; and it is
necessary to inquire whether, if the polarization at the electrodes be
eliminated, the passage of the current through the bulk of the solution
itself is proportional to the electromotive force actually applied to
that solution. Rough experiment shows that the current is proportional
to the excess of the electromotive force over a constant value, and thus
verifies the law approximately, the constant electromotive force to be
overcome being a measure of the polarization. A more satisfactory
examination of the question was made by F. Kohlrausch in the years 1873
to 1876. Ohm's law states that the current C is proportional to the
electromotive force E, or C = kR, where k is a constant called the
conductivity of the circuit. The equation may also be written as C =
E/R, where R is a constant, the reciprocal of k, known as the resistance
of the circuit. The essence of the law is the proportionality between C
and E, which means that the ratio E/C is a constant. But E/C = R, and
thus the law may be tested by examining the constancy of the measured
resistance of a conductor when different currents are passing through
it. In this way Ohm's law has been confirmed in the case of metallic
conduction to a very high degree of accuracy. A similar principle was
applied by Kohlrausch to the case of electrolytes, and he was the first
to show that an electrolyte possesses a definite resistance which has a
constant value when measured with different currents and by different
experimental methods.
_Measurement of the Resistance of Electrolytes._--There are two effects
of the passage of an electric current which prevent the possibility of
measuring electrolytic resistance by the ordinary methods with the
direct currents which are used in the case of metals. The products of
the chemical decomposition of the electrolyte appear at the electrodes
and set up the opposing electromotive force of polarization, and unequal
dilution of the solution may occur in the neighbourhood of the two
electrodes. The chemical and electrolytic aspects of these phenomena are
treated in the article ELECTROLYSIS, but from our present point of view
also it is evident that they are again of fundamental importance. The
polarization at the surface of the electrodes will set up an opposing
electromotive force, and the unequal dilution of the solution will turn
the electrolyte into a concentration cell and produce a subsidiary
electromotive force either in the same direction as that applied or in
the reverse according as the anode or the cathode solution becomes the
more dilute. Both effects thus involve internal electromotive forces,
and prevent the application of Ohm's law to the electrolytic cell as a
whole. But the existence of a definite measurable resistance as a
characteristic property of the system depends on the conformity of the
system to Ohm's law, and it is therefore necessary to eliminate both
these effects before attempting to measure the resistance.
The usual and most satisfactory method of measuring the resistance of
electrolytes consists in eliminating the effects of polarization by the
use of alternating currents, that is, currents that are reversed in
direction many times a second.[11] The chemical action produced by the
first current is thus reversed by the second current in the opposite
direction, and the polarization caused by the first current on the
surface of the electrodes is destroyed before it rises to an appreciable
value. The polarization is also diminished in another way. The
electromotive force of polarization is due to the deposition of films of
the products of chemical decomposition on the surface of the electrodes,
and only reaches its full value when a continuous film is formed. If the
current be stopped before such a film is completed, the reverse
electromotive force is less than its full value. A given current flowing
for a given time deposits a definite amount of substance on the
electrodes, and therefore the amount per unit area is inversely
proportional to the area of the electrodes--to the area of contact, that
is, between the electrode and the liquid. Thus, by increasing the area
of the electrodes, the polarization due to a given current is decreased.
Now the area of free surface of a platinum plate can be increased
enormously by coating the plate with platinum black, which is metallic
platinum in a spongy state, and with such a plate as electrode the
effects of polarization are diminished to a very marked extent. The
coating is effected by passing an electric current first one way and
then the other between two platinum plates immersed in a 3% solution of
platinum chloride to which a trace of lead acetate is sometimes added.
The platinized plates thus obtained are quite satisfactory for the
investigation of strong solutions. They have the power, however, of
absorbing a certain amount of salt from the solutions and of giving it
up again when water or more dilute solution is placed in contact with
them. The measurement of very dilute solutions is thus made difficult,
but, if the plates be heated to redness after being platinized, a grey
surface is obtained which possesses sufficient area for use with dilute
solutions and yet does not absorb an appreciable quantity of salt.
Any convenient source of alternating current may be used. The currents
from the secondary circuit of a small induction coil are satisfactory,
or the currents of an alternating electric light supply may be
transformed down to an electromotive force of one or two volts. With
such currents it is necessary to consider the effects of self-induction
in the circuit and of electrostatic capacity. In balancing the
resistance of the electrolyte, resistance coils may be used in which
self-induction and the capacity are reduced to a minimum by winding the
wire of the coil backwards and forwards in alternate layers.
[Illustration: FIG. 1.]
With these arrangements the usual method of measuring resistance by
means of Wheatstone's bridge may be adapted to the case of electrolytes.
With alternating currents, however, it is impossible to use a
galvanometer in the usual way. The galvanometer was therefore replaced
by Kohlrausch by a telephone, which gives a sound when an alternating
current passes through it. The most common plan of the apparatus is
shown diagrammatically in fig. 1. The electrolytic cell and a resistance
box form two arms of the bridge, and the sliding contact is moved along
the metre wire which forms the other two arms till no sound is heard in
the telephone. The resistance of the electrolyte is to that of the box
as that of the right-hand end of the wire is to that of the left-hand
end. A more accurate method of using alternating currents, and one more
pleasant to use, gets rid of the telephone (_Phil. Trans._, 1900, 194,
p. 321). The current from one or two voltaic cells is led to an ebonite
drum turned by a motor or a hand-wheel and cord. On the drum are fixed
brass strips with wire brushes touching them in such a manner that the
current from the brushes is reversed several times in each revolution of
the drum. The wires from the brushes are connected with the Wheatstone's
bridge. A moving coil galvanometer is used as indicator, its connexions
being reversed in time with those of the battery by a slightly narrower
set of brass strips fixed on the other end of the ebonite commutator.
Thus any residual current through the galvanometer is direct and not
alternating. The high moment of inertia of the coil makes the period of
swing slow compared with the period of alternation of the current, and
the slight periodic disturbances are thus prevented from affecting the
galvanometer. When the measured resistance is not altered by increasing
the speed of the commutator or changing the ratio of the arms of the
bridge, the disturbing effects may be considered to be eliminated.
[Illustration: FIG. 2.]
[Illustration: FIG. 3.]
The form of vessel chosen to contain the electrolyte depends on the
order of resistance to be measured. For dilute solutions the shape of
cell shown in fig. 2 will be found convenient, while for more
concentrated solutions, that indicated in fig. 3 is suitable. The
absolute resistances of certain solutions have been determined by
Kohlrausch by comparison with mercury, and, by using one of these
solutions in any cell, the constant of that cell may be found once for
all. From the observed resistance of any given solution in the cell the
resistance of a centimetre cube--the so-called specific resistance--may
be calculated. The reciprocal of this, or the conductivity, is a more
generally useful constant; it is conveniently expressed in terms of a
unit equal to the reciprocal of an ohm. Thus Kohlrausch found that a
solution of potassium chloride, containing one-tenth of a gram
equivalent (7.46 grams) per litre, has at 18° C. a specific resistance
of 89.37 ohms per centimetre cube, or a conductivity of 1.119 × 10^-2
mhos or 1.119 × 10^-11 C.G.S. units. As the temperature variation of
conductivity is large, usually about 2% per degree, it is necessary to
place the resistance cell in a paraffin or water bath, and to observe
its temperature with some accuracy.
Another way of eliminating the effects of polarization and of dilution
has been used by W. Stroud and J. B. Henderson (_Phil. Mag._, 1897 [5],
43, p. 19). Two of the arms of a Wheatstone's bridge are composed of
narrow tubes filled with the solution, the tubes being of equal diameter
but of different length. The other two arms are made of coils of wire of
equal resistance, and metallic resistance is added to the shorter tube
till the bridge is balanced. Direct currents of somewhat high
electromotive force are used to work the bridge. Equal currents then
flow through the two tubes; the effects of polarization and dilution
must be the same in each, and the resistance added to the shorter tube
must be equal to the resistance of a column of liquid the length of
which is equal to the difference in length of the two tubes.
A somewhat different principle was adopted by E. Bouty in 1884. If a
current be passed through two resistances in series by means of an
applied electromotive force, the electric potential falls from one end
of the resistances to the other, and, if we apply Ohm's law to each
resistance in succession, we see that, since for each of them E = CR,
and C the current is the same through both, E the electromotive force or
fall of potential between the ends of each resistance must be
proportional to the resistance between them. Thus by measuring the
potential difference between the ends of the two resistances
successively, we may compare their resistances. If, on the other hand,
we can measure the potential difference in some known units, and
similarly measure the current flowing, we can determine the resistance
of a single electrolyte. The details of the apparatus may vary, but its
principle is illustrated in the following description. A narrow glass
tube is fixed horizontally into side openings in two glass vessels, and
an electric current passed through it by means of platinum electrodes
and a battery of considerable electromotive force. In this way a steady
fall of electric potential is set up along the length of the tube. To
measure the potential difference between the ends of the tube, tapping
electrodes are constructed, e.g. by placing zinc rods in vessels with
zinc sulphate solution and connecting these vessels (by means of thin
siphon tubes also filled with solution) with the vessels at the ends of
the long tube which contains the electrolyte to be examined. Whatever be
the contact potential difference between zinc and its solution, it is
the same at both ends, and thus the potential difference between the
zinc rods is equal to that between the liquid at the two ends of the
tube. This potential difference may be measured without passing any
appreciable current through the tapping electrodes, and thus the
resistance of the liquid deduced.
_Equivalent Conductivity of Solutions._--As is the case in the other
properties of solutions, the phenomena are much more simple when the
concentration is small than when it is great, and a study of dilute
solutions is therefore the best way of getting an insight into the
essential principles of the subject. The foundation of our knowledge was
laid by Kohlrausch when he had developed the method of measuring
electrolyte resistance described above. He expressed his results in
terms of "equivalent conductivity," that is, the conductivity (k) of the
solution divided by the number (m) of gram-equivalents of electrolyte
per litre. He finds that, as the concentration diminishes, the value of
k/m approaches a limit, and eventually becomes constant, that is to say,
at great dilution the conductivity is proportional to the concentration.
Kohlrausch first prepared very pure water by repeated distillation and
found that its resistance continually increased as the process of
purification proceeded. The conductivity of the water, and of the slight
impurities which must always remain, was subtracted from that of the
solution made with it, and the result, divided by m, gave the equivalent
conductivity of the substance dissolved. This procedure appears
justifiable, for as long as conductivity is proportional to
concentration it is evident that each part of the dissolved matter
produces its own independent effect, so that the total conductivity is
the sum of the conductivities of the parts; 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 [root 3]m as abscissae, [root 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.
[Illustration: 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 Faraday's 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 [eta]C,
where [eta] denotes the electro-chemical equivalent of hydrogen in
C.G.S. units. Thus we get
n(u + v) = [eta]C = [eta]ne(u + v),
and it follows that the charge, e, on 1 gram-equivalent of each kind
of ion is equal to 1/[eta]. We know that Ohm's Law holds good for
electrolytes, so that the current C is also given by k·dP/dx, where k
denotes the conductivity of the solution, and dP/dx the potential
gradient, i.e. the change in potential per unit length along the lines
of current flow. Thus
n
-----(u + v) = kdP/dx;
[eta]
therefore
k dP
u + v = [eta] - --.
n dx
Now [eta] 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
k dP
u + v = 1.036 × 10^-1 - -- --.
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, for simple salts when no complex ions are present, and
many further experiments have been made on the subject (see _Das
Leitvermögen der Elektrolyte_).
By combining the results thus obtained with the sum of the velocities,
as determined from the conductivities, Kohlrausch calculated the
absolute velocities of different ions under stated conditions. Thus,
in the case of the solution of potassium chloride considered above,
Hittorf's experiments show us that the ratio of the velocity of the
anion to that of the cation in this solution is .51 : .49. The
absolute velocity of the potassium ion under unit potential gradient
is therefore 0.000567 cm. per sec., and that of the chlorine ion
0.000592 cm. per sec. Similar calculations can be made for solutions
of other concentrations, and of different substances.
Table IX. shows Kohlrausch's values for the ionic velocities of three
chlorides of alkali metals at 18° C, calculated for a potential gradient
of 1 volt per cm.; the numbers are in terms of a unit equal to 10^-6 cm.
per sec.:--
TABLE IX.
+--------------------------------------------+----------------+
| | KCl | NaCl | LiCl |
+---------+-----------------+----------------+----------------+
| m | u + v u v | u + v u v |u + v u v |
+---------+-----------------+----------------+----------------+
| 0 | 1350 660 690 | 1140 450 690 | 1050 360 690 |
| 0.0001 | 1335 654 681 | 1129 448 681 | 1037 356 681 |
| .001 | 1313 643 670 | 1110 440 670 | 1013 343 670 |
| .01 | 1263 619 644 | 1059 415 644 | 962 318 644 |
| .03 | 1218 597 621 | 1013 390 623 | 917 298 619 |
| .1 | 1153 564 589 | 952 360 592 | 853 259 594 |
| .3 | 1088 531 557 | 876 324 552 | 774 217 557 |
| 1.0 | 1011 491 520 | 765 278 487 | 651 169 482 |
| 3.0 | 911 442 469 | 582 206 376 | 463 115 348 |
| 5.0 | | 438 153 285 | 334 80 254 |
| 10.0 | | | 117 25 92 |
+---------+-----------------+----------------+----------------+
These numbers show clearly that there is an increase in ionic velocity
as the dilution proceeds. Moreover, if we compare the values for the
chlorine ion obtained from observations on these three different salts,
we see that as the concentrations diminish the velocity of the chlorine
ion becomes the same in all of them. A similar relation appears in other
cases, and, in general, we may say that at great dilution the velocity
of an ion is independent of the nature of the other ion present. This
introduces the conception of specific ionic velocities, for which some
values at 18° C. are given by Kohlrausch in Table X.:--
Table X.
+------------------------------------+
| K 66 × 10^-5 cms. per sec. |
| Na 45 " " |
| Li 36 " " |
| NH4 66 " " |
| H 320 " " |
| Ag 57 " " |
| Cl 69 " " |
| I 69 " " |
| NO3 64 " " |
| OH 162 " " |
| C2H3O2 36 " " |
| C3H5O2 33 " " |
+------------------------------------+
Having obtained these numbers we can deduce the conductivity of the
dilute solution of any salt, and the comparison of the calculated with
the observed values furnished the first confirmation of Kohlrausch's
theory. Some exceptions, however, are known. Thus acetic acid and
ammonia give solutions of much lower conductivity than is indicated by
the sum of the specific ionic velocities of their ions as determined
from other compounds. An attempt to find in Kohlrausch's theory some
explanation of this discrepancy shows that it could be due to one of two
causes. Either the velocities of the ions must be much less in these
solutions than in others, or else only a fractional part of the number
of molecules present can be actively concerned in conveying the current.
We shall return to this point later.
_Friction on the Ions._--It is interesting to calculate the magnitude
of the forces required to drive the ions with a certain velocity. If
we have a potential gradient of 1 volt per centimetre the electric
force is 10^8 in C.G.S. units. The charge of electricity on 1
gram-equivalent of any ion is 1/.0001036 = 9653 units, hence the
mechanical force acting on this mass is 9653 × 10^8 dynes. This, let
us say, produces a velocity u; then the force required to produce unit
velocity is PA = 9.653 × 10^11/u dynes = 9.84 × 10^5/u kilograms-weight.
If the ion have an equivalent weight A, the force producing unit
velocity when acting on 1 gram is P1 = 9.84 × 10^5/Au kilograms-weight.
Thus the aggregate force required to drive 1 gram of potassium ions
with a velocity of 1 centimetre per second through a very dilute
solution must be equal to the weight of 38 million kilograms.
Table XI.
+-------------------------------+
| Kilograms-weight. |
+-------------------------------+
| PA P1 |
|K 15 × 10^8 38 × 10^6 |
|Na 22 " 95 " |
|Li 27 " 390 " |
|NH4 15 " 83 " |
|H 3.1 " 310 " |
|Ag 17 " 16 " |
|Cl 14 " 40 " |
|I 14 " 11 " |
|NO3 15 " 25 " |
|OH 5.4 " 32 " |
|C2H8O2 27 " 46 " |
|C3H5O2 30 " 41 " |
+------------- -----------------+
Since the ions move with uniform velocity, the frictional resistances
brought into play must be equal and opposite to the driving forces,
and therefore these numbers also represent the ionic friction
coefficients in very dilute solutions at 18° C.
_Direct Measurement of Ionic Velocities._--Sir Oliver Lodge was the
first to directly measure the velocity of an ion (_B.A. Report_, 1886,
p. 389). In a horizontal glass tube connecting two vessels filled with
dilute sulphuric acid he placed a solution of sodium chloride in solid
agar-agar jelly. This solid solution was made alkaline with a trace of
caustic soda in order to bring out the red colour of a little
phenol-phthalein added as indicator. An electric current was then passed
from one vessel to the other. The hydrogen ions from the anode vessel of
acid were thus carried along the tube, and, as they travelled,
decolourized the phenol-phthalein. By this method the velocity of the
hydrogen ion through a jelly solution under a known potential gradient
was observed to about 0.0026 cm. per sec, a number of the same order as
that required by Kohlrausch's theory. Direct determinations of the
velocities of a few other ions have been made by W. C. D. Whetham
(_Phil. Trans._ vol. 184, A, p. 337; vol. 186, A, p. 507; _Phil. Mag._,
October 1894). Two solutions having one ion in common, of equivalent
concentrations, different densities, different colours, and nearly equal
specific resistances, were placed one over the other in a vertical glass
tube. In one case, for example, decinormal solutions of potassium
carbonate and potassium bichromate were used. The colour of the latter
is due to the presence of the bichromate group, Cr2O7. When a current
was passed across the junction, the anions CO3 and Cr2O7 travelled in
the direction opposite to that of the current, and their velocity could
be determined by measuring the rate at which the colour boundary moved.
Similar experiments were made with alcoholic solutions of cobalt salts,
in which the velocities of the ions were found to be much less than in
water. The behaviour of agar jelly was then investigated, and the
velocity of an ion through a solid jelly was shown to be very little
less than in an ordinary liquid solution. The velocities could therefore
be measured by tracing the change in colour of an indicator or the
formation of a precipitate. Thus decinormal jelly solutions of barium
chloride and sodium chloride, the latter containing a trace of sodium
sulphate, were placed in contact. Under the influence of an
electromotive force the barium ions moved up the tube, disclosing their
presence by the trace of insoluble barium sulphate formed. Again, a
measurement of the velocity of the hydrogen ion, when travelling through
the solution of an acetate, showed that its velocity was then only about
the one-fortieth part of that found during its passage through
chlorides. From this, as from the measurements on alcohol solutions, it
is clear that where the equivalent conductivities are very low the
effective velocities of the ions are reduced in the same proportion.
Another series of direct measurements has been made by Orme Masson
(_Phil. Trans._ vol. 192, A, p. 331). He placed the gelatine solution of
a salt, potassium chloride, for example, in a horizontal glass tube, and
found the rate of migration of the potassium and chlorine ions by
observing the speed at which they were replaced when a coloured anion,
say, the Cr2O7 from a solution of potassium bichromate, entered the tube
at one end, and a coloured cation, say, the Cu from copper sulphate, at
the other. The coloured ions are specifically slower than the colourless
ions which they follow, and in this case it follows that the coloured
solution has a higher resistance than the colourless. For the same
current, therefore, the potential gradient is higher in the coloured
solution and lower in the colourless one. Thus a coloured ion which gets
in front of the advancing boundary finds itself acted on by a smaller
force and falls back into line, while a straggling colourless ion is
pushed forward again. Hence a sharp boundary is preserved. B. D. Steele
has shown that with these sharp boundaries the use of coloured ions is
unnecessary, the junction line being visible owing to the difference in
the optical refractive indices of two colourless solutions. Once the
boundary is formed, too, no gelatine is necessary, and the motion can be
watched through liquid aqueous solutions (see R. B. Denison and B. D.
Steele, _Phil. Trans._, 1906).
All the direct measurements which have been made on simple binary
electrolytes agree with Kohlrausch's results within the limits of
experimental error. His theory, therefore, probably holds good in such
cases, whatever be the solvent, if the proper values are given to the
ionic velocities, i.e. the values expressing the velocities with which
the ions actually move in the solution of the strength taken, and under
the conditions of the experiment. If we know the specific velocity of
any one ion, we can deduce, from the conductivity of very dilute
solutions, the velocity of any other ion with which it may be
associated, a proceeding which does not involve the difficult task of
determining the migration constant of the compound. Thus, taking the
specific ionic velocity of hydrogen as 0.00032 cm. per second, we can
find, by determining the conductivity of dilute solutions of any acid,
the specific velocity of the acid radicle involved. Or again, since we
know the specific velocity of silver, we can find the velocities of a
series of acid radicles at great dilution by measuring the conductivity
of their silver salts.
By such methods W. Ostwald, G. Bredig and other observers have found
the specific velocities of many ions both of inorganic and organic
compounds, and examined the relation between constitution and ionic
velocity. The velocity of elementary ions is found to be a periodic
function of the atomic weight, similar elements lying on corresponding
portions of a curve drawn to express the relation between these two
properties. Such a curve much resembles that giving the relation
between atomic weight and viscosity in solution. For complex ions the
velocity is largely an additive property; to a continuous additive
change in the composition of the ion corresponds a continuous but
decreasing change in the velocity. The following table gives Ostwald's
results for the formic acid series:--
Table XII.
+----------------------+----------+---------------------+
| | Velocity.| Difference for CH2. |
+----------------------+----------+---------------------+
| Formic acid HCO2 | 51.2 | .. |
| Acetic " H3C2O2 | 38.3 | -12.9 |
| Propionic " H5C3O2 | 34.3 | - 4.0 |
| Butyric " H7C4O2 | 30.8 | - 3.5 |
| Valeric " H9C5O2 | 28.8 | - 2.0 |
| Caprionic " H11C6O2 | 27.4 | - 1.4 |
+----------------------+----------+---------------------+
_Nature of Electrolytes._--We have as yet said nothing about the
fundamental cause of electrolytic activity, nor considered why, for
example, a solution of potassium chloride is a good conductor, while a
solution of sugar allows practically no current to pass.
All the preceding account of the subject is, then, independent of any
view we may take of the nature of electrolytes, and stands on the basis
of direct experiment. Nevertheless, the facts considered point to a very
definite conclusion. The specific velocity of an ion is independent of
the nature of the opposite ion present, and this suggests that the ions
themselves, while travelling through the liquid, are dissociated from
each other. Further evidence, pointing in the same direction, is
furnished by the fact that since the conductivity is proportional to the
concentration at great dilution, the equivalent-conductivity, and
therefore the ionic velocity, is independent of it. The importance of
this relation will be seen by considering the alternative to the
dissociation hypothesis. If the ions are not permanently free from each
other their mobility as parts of the dissolved molecules must be secured
by continual interchanges. The velocity with which they work their way
through the liquid must then increase as such molecular rearrangements
become more frequent, and will therefore depend on the number of solute
molecules, i.e. on the concentration. On this supposition the observed
constancy of velocity would be impossible. We shall therefore adopt as a
wording hypothesis the theory, confirmed by other phenomena (see
ELECTROLYSIS), that an electrolyte consists of dissociated ions.
It will be noticed that neither the evidence in favour of the
dissociation theory which is here considered, nor that described in the
article ELECTROLYSIS, requires more than the effective dissociation of
the ions from each other. They may well be connected in some way with
solvent molecules, and there are several indications that an ion
consists of an electrified part of the molecule of the dissolved salt
with an attendant atmosphere of solvent round it. The conductivity of a
salt solution depends on two factors--(1) the fraction of the salt
ionized; (2) the velocity with which the ions, when free from each
other, move under the electric forces.[12] When a solution is heated,
both these factors may change. The coefficient of ionization usually,
though not always, decreases; the specific ionic velocities increase.
Now the rate of increase with temperature of these ionic velocities is
very nearly identical with the rate of decrease of the viscosity of the
liquid. If the curves obtained by observations at ordinary temperatures
be carried on they indicate a zero of fluidity and a zero of ionic
velocity about the same point, 38.5° C. below the freezing point of
water (Kohlrausch, _Sitz. preuss. Akad. Wiss._, 1901, 42, p. 1026). Such
relations suggest that the frictional resistance to the motion of an ion
is due to the ordinary viscosity of the liquid, and that the ion is
analogous to a body of some size urged through a viscous medium rather
than to a particle of molecular dimensions finding its way through a
crowd of molecules of similar magnitude. From this point of view W. K.
Bousfield has calculated the sizes of ions on the assumption that
Stokes's theory of the motion of a small sphere through a viscous medium
might be applied (_Zeits. phys. Chem._, 1905, 53, p. 257; _Phil. Trans._
A, 1906, 206, p. 101). The radius of the potassium or chlorine ion with
its envelope of water appears to be about 1.2 × 10^-8 centimetres.
For the bibliography of electrolytic conduction see ELECTROLYSIS. The
books which deal more especially with the particular subject of the
present article are _Das Leitvermögen der Elektrolyte_, by F.
Kohlrausch and L. Holborn (Leipzig, 1898), and _The Theory of Solution
and Electrolysis_, by W. C. D. Whetham (Cambridge, 1902).
(W. C. D. W.)
III. ELECTRIC CONDUCTION THROUGH GASES
A gas such as air when it is under normal conditions conducts
electricity to a small but only to a very small extent, however small
the electric force acting on the gas may be. The electrical conductivity
of gases not exposed to special conditions is so small that it was only
definitely established in the early years of the 20th century, although
it had engaged the attention of physicists for more than a hundred
years. It had been known for a long time that a body charged with
electricity slowly lost its charge even when insulated with the greatest
care, and though long ago some physicists believed that part of the leak
of electricity took place through the air, the general view seems to
have been that it was due to almost unavoidable defects in the
insulation or to dust in the air, which after striking the charged body
was repelled from it and went off with some of the charge. C. A.
Coulomb, who made some very careful experiments which were published in
1785 (_Mém. de l'Acad. des Sciences_, 1785, p. 612), came to the
conclusion that after allowing for the leakage along the threads which
supported the charged body there was a balance over, which he attributed
to leakage through the air. His view was that when the molecules of air
come into contact with a charged body some of the electricity goes on to
the molecules, which are then repelled from the body carrying their
charge with them. We shall see later that this explanation is not
tenable. C. Matteucci (_Ann. chim. phys._, 1850, 28, p. 390) in 1850
also came to the conclusion that the electricity from a charged body
passes through the air; he was the first to prove that the rate at
which electricity escapes is less when the pressure of the gas is low
than when it is high. He found that the rate was the same whether the
charged body was surrounded by air, carbonic acid or hydrogen.
Subsequent investigations have shown that the rate in hydrogen is in
general much less than in air. Thus in 1872 E. G. Warburg (_Pogg. Ann._,
1872, 145, p. 578) found that the leak through hydrogen was only about
one-half of that through air: he confirmed Matteucci's observations on
the effect of pressure on the rate of leak, and also found that it was
the same whether the gas was dry or damp. He was inclined to attribute
the leak to dust in the air, a view which was strengthened by an
experiment of J. W. Hittorf's (_Wied. Ann._, 1879, 7, p. 595), in which
a small carefully insulated electroscope, placed in a small vessel
filled with carefully filtered gas, retained its charge for several
days; we know now that this was due to the smallness of the vessel and
not to the absence of dust, as it has been proved that the rate of leak
in small vessels is less than in large ones.
Great light was thrown on this subject by some experiments on the rates
of leak from charged bodies in closed vessels made almost simultaneously
by H. Geitel (_Phys. Zeit._, 1900, 2, p. 116) and C. T. R. Wilson
(_Proc. Camb. Phil._ Soc., 1900, 11, p. 32). These observers established
that (1) the rate of escape of electricity in a closed vessel is much
smaller than in the open, and the larger the vessel the greater is the
rate of leak; and (2) the rate of leak does not increase in proportion
to the differences of potential between the charged body and the walls
of the vessel: the rate soon reaches a limit beyond which it does not
increase, however much the potential difference may be increased,
provided, of course, that this is not great enough to cause sparks to
pass from the charged body. On the assumption that the maximum leak is
proportional to the volume, Wilson's experiments, which were made in
vessels less than 1 litre in volume, showed that in dust-free air at
atmospheric pressure the maximum quantity of electricity which can
escape in one second from a charged body in a closed volume of V cubic
centimetres is about 10^-8V electrostatic units. E. Rutherford and S. T.
Allan (_Phys. Zeit._, 1902, 3, p. 225), working in Montreal, obtained
results in close agreement with this. Working between pressures of from
43 to 743 millimetres of mercury, Wilson showed that the maximum rate of
leak is very approximately proportional to the pressure; it is thus
exceedingly small when the pressure is low--a result illustrated in a
striking way by an experiment of Sir W. Crookes (_Proc. Roy. Soc._,
1879, 28, p. 347) in which a pair of gold leaves retained an electric
charge for several months in a very high vacuum. Subsequent experiments
have shown that it is only in very small vessels that the rate of leak
is proportional to the volume and to the pressure; in large vessels the
rate of leak per unit volume is considerably smaller than in small ones.
In small vessels the maximum rate of leak in different gases, is, with
the exception of hydrogen, approximately proportional to the density of
the gas. Wilson's results on this point are shown in the following table
(Proc. Roy. Soc., 1901, 60, p. 277):--
+---------+------------------------+-----------------+
| Gas. | Relative Rate of Leak. | _Rate of Leak._ |
| | | Sp. Gr. |
+---------+------------------------+-----------------+
| Air | 1.00 | 1 |
| H2 | .184 | 2.7 |
| CO2 | 1.69 | 1.10 |
| SO2 | 2.64 | 1.21 |
| CH3Cl | 4.7 | 1.09 |
| Ni(CO)4 | 5.1 | .867 |
+---------+------------------------+-----------------+
The rate of leak of electricity through gas contained in a closed vessel
depends to some extent on the material of which the walls of the vessel
are made; thus it is greater, other circumstances being the same, when
the vessel is made of lead than when it is made of aluminium. It also
varies, as Campbell and Wood (_Phil. Mag._ [6], 13, p. 265) have shown,
with the time of the day, having a well-marked minimum at about 3
o'clock in the morning: it also varies from month to month. Rutherford
(_Phys. Rev._, 1903, 16, p. 183), Cooke (_Phil. Mag._, 1903 [6], 6, p.
403) and M'Clennan and Burton (_Phys. Rev._, 1903, 16, p. 184) have
shown that the leak in a closed vessel can be reduced by about 30% by
surrounding the vessel with sheets of thick lead, but that the reduction
is not increased beyond this amount, however thick the lead sheets may
be. This result indicates that part of the leak is due to a very
penetrating kind of radiation, which can get through the thin walls of
the vessel but is stopped by the thick lead. A large part of the leak we
are describing is due to the presence of radioactive substances such as
radium and thorium in the earth's crust and in the walls of the vessel,
and to the gaseous radioactive emanations which diffuse from them into
the atmosphere. This explains the very interesting effect discovered by
J. Elster and H. Geitel (_Phys. Zeit._, 1901, 2, p. 560), that the rate
of leak in caves and cellars when the air is stagnant and only renewed
slowly is much greater than in the open air. In some cases the
difference is very marked; thus they found that in the cave called the
Baumannshöhle in the Harz mountains the electricity escaped at seven
times the rate it did in the air outside. In caves and cellars the
radioactive emanations from the walls can accumulate and are not blown
away as in the open air.
The electrical conductivity of gases in the normal state is, as we have
seen, exceedingly small, so small that the investigation of its
properties is a matter of considerable difficulty; there are, however,
many ways by which the electrical conductivity of a gas can be increased
so greatly that the investigation becomes comparatively easy. Among such
methods are raising the temperature of the gas above a certain point.
Gases drawn from the neighbourhood of flames, electric arcs and sparks,
or glowing pieces of metal or carbon are conductors, as are also gases
through which Röntgen or cathode rays or rays of positive electricity
are passing; the rays from the radioactive metals, radium, thorium,
polonium and actinium, produce the same effect, as does also
ultra-violet light of exceedingly short wave-length. The gas, after
being made a conductor of electricity by any of these means, is found to
possess certain properties; thus it retains its conductivity for some
little time after the agent which made it a conductor has ceased to act,
though the conductivity diminishes very rapidly and finally gets too
small to be appreciable.
[Illustration: FIG. 5.]
This and several other properties of conducting gas may readily be
proved by the aid of the apparatus represented in fig. 5. V is a testing
vessel in which an electroscope is placed. Two tubes A and C are fitted
into the vessel, A being connected with a water pump, while the far end
of C is in the region where the gas is exposed to the agent which makes
it a conductor of electricity. Let us suppose that the gas is made
conducting by Röntgen rays produced by a vacuum tube which is placed in
a box, covered except for a window at B with lead so as to protect the
electroscope from the direct action of the rays. If a slow current of
air is drawn by the water pump through the testing vessel, the charge on
the electroscope will gradually leak away. The leak, however, ceases
when the current of air is stopped. This result shows that the gas
retains its conductivity during the time taken by it to pass from one
end to the other of the tube C.
The gas loses its conductivity when filtered through a plug of
glass-wool, or when it is made to bubble through water. This can readily
be proved by inserting in the tube C a plug of glass-wool or a water
trap; then if by working the pump a little harder the same current of
air is produced as before, it will be found that the electroscope will
now retain its charge, showing that the conductivity can, as it were, be
filtered out of the gas. The conductivity can also be removed from the
gas by making the gas traverse a strong electric field. We can show this
by replacing the tube C by a metal tube with an insulated wire passing
down the axis of the tube. If there is no potential difference between
the wire and the tube then the electroscope will leak when a current of
air is drawn through the vessel, but the leak will stop if a
considerable difference of potential is maintained between the wire and
the tube: this shows that a strong electric field removes the
conductivity from the gas.
The fact that the conductivity of the gas is removed by filtering shows
that it is due to something mixed with the gas which is removed from it
by filtration, and since the conductivity is also removed by an electric
field, the cause of the conductivity must be charged with electricity so
as to be driven to the sides of the tube by the electric force. Since
the gas as a whole is not electrified either positively or negatively,
there must be both negative and positive charges in the gas, the amount
of electricity of one sign being equal to that of the other. We are thus
led to the conclusion that the conductivity of the gas is due to
electrified particles being mixed up with the gas, some of these
particles having charges of positive electricity, others of negative.
These electrified particles are called _ions_, and the process by which
the gas is made a conductor is called the ionization of the gas. We
shall show later that the charges and masses of the ions can be
determined, and that the gaseous ions are not identical with those met
with in the electrolysis of solutions.
[Illustration: FIG. 6.]
One very characteristic property of conduction of electricity through a
gas is the relation between the current through the gas and the electric
force which gave rise to it. This relation is not in general that
expressed by Ohm's law, which always, as far as our present knowledge
extends, expresses the relation for conduction through metals and
electrolytes. With gases, on the other hand, it is only when the current
is very small that Ohm's law is true. If we represent graphically by
means of a curve the relation between the current passing between two
parallel metal plates separated by ionized gas and the difference of
potential between the plates, the curve is of the character shown in
fig. 6 when the ordinates represent the current and the abscissae the
difference of potential between the plates. We see that when the
potential difference is very small, i.e. close to the origin, the curve
is approximately straight, but that soon the current increases much less
rapidly than the potential difference, and that a stage is reached when
no appreciable increase of current is produced when the potential
difference is increased; when this stage is reached the current is
constant, and this value of the current is called the "saturation"
value. When the potential difference approaches the value at which
sparks would pass through the gas, the current again increases with the
potential difference; thus the curve representing the relation between
the current and potential difference over very wide ranges of potential
difference has the shape shown in fig. 7; curves of this kind have been
obtained by von Schweidler (_Wien. Ber._, 1899, 108, p. 273), and J. E.
S. Townsend (_Phil. Mag._, 1901 [6], 1, p. 198). We shall discuss later
the causes of the rise in the current with large potential differences,
when we consider ionization by collision.
The general features of the earlier part of the curve are readily
explained on the ionization hypothesis. On this view the Röntgen rays
or other ionizing agent acting on the gas between the plates, produces
positive and negative ions at a definite rate. Let us suppose that q
positive and q negative ions are by this means produced per second
between the plates; these under the electric force will tend to move,
the positive ones to the negative plate, the negative ones to the
positive. Some of these ions will reach the plate, others before
reaching the plate will get so near one of the opposite sign that the
attraction between them will cause them to unite and form an
electrically neutral system; when they do this they end their
existence as ions. The current between the plates is proportional to
the number of ions which reach the plates per second. Now it is
evident that we cannot go on taking more ions out of the gas than are
produced; thus we cannot, when the current is steady, have more than q
positive ions driven to the negative plate per second, and the same
number of negative ions to the positive. If each of the positive ions
carries a charge of e units of positive electricity, and if there is
an equal and opposite charge on each negative ion, then the maximum
amount of electricity which can be given to the plates per second is
qe, and this is equal to the saturation current. Thus if we measure
the saturation current, we get a direct measure of the ionization, and
this does not require us to know the value of any quantity except the
constant charge on the ion. If we attempted to deduce the amount of
ionization by measurements of the current before it was saturated, we
should require to know in addition the velocity with which the ions
move under a given electric force, the time that elapses between the
liberation of an ion and its combination with one of the opposite
sign, and the potential difference between the plates. Thus if we wish
to measure the amount of ionization in a gas we should be careful to
see that the current is saturated.
[Illustration: FIG. 7.]
The difference between conduction through gases and through metals is
shown in a striking way when we use potential differences large enough
to produce the saturation current. Suppose we have got a potential
difference between the plates more than sufficient to produce the
saturation current, and let us increase the distance between the plates.
If the gas were to act like a metallic conductor this would diminish the
current, because the greater length would involve a greater resistance
in the circuit. In the case we are considering the separation of the
plates will _increase_ the current, because now there is a larger volume
of gas exposed to the rays; there are therefore more ions produced, and
as the saturation current is proportional to the number of ions the
saturation current is increased. If the potential difference between the
plates were much less than that required to saturate the current, then
increasing the distance would diminish the current; the gas for such
potential differences obeys Ohm's law and the behaviour of the gaseous
resistance is therefore similar to that of a metallic one.
In order to produce the saturation current the electric field must be
strong enough to drive each ion to the electrode before it has time to
enter into combination with one of the opposite sign. Thus when the
plates in the preceding example are far apart, it will take a larger
potential difference to produce this current than when the plates are
close together. The potential difference required to saturate the
current will increase as the square of the distance between the plates,
for if the ions are to be delivered in a given time to the plates their
speed must be proportional to the distance between the plates. But the
speed is proportional to the electric force acting on the ion; hence the
electric force must be proportional to the distance between the plates,
and as in a uniform field the potential difference is equal to the
electric force multiplied by the distance between the plates, the
potential difference will vary as the square of this distance.
The potential difference required to produce saturation will, other
circumstances being the same, increase with the amount of ionization,
for when the number of ions is large and they are crowded together, the
time which will elapse before a positive one combines with a negative
will be smaller than when the number of ions is small. The ions have
therefore to be removed more quickly from the gas when the ionization is
great than when it is small; thus they must move at a higher speed and
must therefore be acted upon by a larger force.
When the ions are not removed from the gas, they will increase until the
number of ions of one sign which combine with ions of the opposite sign
in any time is equal to the number produced by the ionizing agent in
that time. We can easily calculate the number of free ions at any time
after the ionizing agent has commenced to act.
Let q be the number of ions (positive or negative) produced in one
cubic centimetre of the gas per second by the ionizing agent, n1, n2,
the number of free positive and negative ions respectively per cubic
centimetre of the gas. The number of collisions between positive and
negative ions per second in one cubic centimetre of the gas is
proportional to n1n2. If a certain fraction of the collisions between
the positive and negative ions result in the formation of an
electrically neutral system, the number of ions which disappear per
second on a cubic centimetre will be equal to [alpha]n1 n2, where
[alpha] is a quantity which is independent of n1, n2; hence if t is
the time since the ionizing agent was applied to the gas, we have
dn1/dt = q - [alpha]n1 n2, dn2/dt = q - [alpha]n1 n2.
Thus n1 - n2 is constant, so if the gas is uncharged to begin with, n1
will always equal n2. Putting n1 = n2 = n we have
dn/dt = q - [alpha]n² (1),
the solution of which is, since n = 0 when t = 0,
k([epsilon]^{2k[alpha]t} - 1)
n = ---------------------------- (2)
[epsilon]^{2k[alpha]t} + 1
if k² = q/[alpha]. Now the number of ions when the gas has reached a
steady state is got by putting t equal to infinity in the preceding
equation, and is therefore given by the equation
n0 = k = [root](q/[alpha]).
We see from equation (1) that the gas will not approximate to its
steady state until 2k[alpha]t is large, that is until t is large
compared with 1/2k[alpha] or with 1/2[root](q[alpha]). We may thus
take 1/2[root](q[alpha]) as a measure of the time taken by the gas to
reach a steady state when exposed to an ionizing agent; as this time
varies inversely as [root]q we see that when the ionization is feeble
it may take a very considerable time for the gas to reach a steady
state. Thus in the case of our atmosphere where the production of ions
is only at the rate of about 30 per cubic centimetre per second, and
where, as we shall see, [alpha] is about 10^-6, it would take some
minutes for the ionization in the air to get into a steady state if
the ionizing agent were suddenly applied.
We may use equation (1) to determine the rate at which the ions
disappear when the ionizing agent is removed. Putting q=0 in that
equation we get dn/[alpha]t = -[alpha]n².
Hence n = n0/(1 + n0[alpha]t) (3),
where n0 is the number of ions when t = 0. Thus the number of ions
falls to one-half its initial value in the time 1/n0[alpha]. The
quantity [alpha] is called the _coefficient of recombination_, and its
value for different gases has been determined by Rutherford (_Phil.
Mag._ 1897 [5], 44, p. 422), Townsend (_Phil. Trans._, 1900, 193, p.
129), McClung (_Phil. Mag._, 1902 [6], 3, p. 283), Langevin (_Ann.
chim. phys._ [7], 28, p. 289), Retschinsky (_Ann. d. Phys._, 1905, 17,
p. 518), Hendred (_Phys. Rev._, 1905, 21, p. 314). The values of
[alpha]/e, e being the charge on an ion in electrostatic measure as
determined by these observers for different gases, is given in the
following table:--
+-----+----------+----------+----------+------------+----------+
| | Townsend.| McClung. | Langevin.|Retschinsky.| Hendred. |
+-----+----------+----------+----------+------------+----------+
| Air | 3420 | 3380 | 3200 | 4140 | 3500 |
| O2 | 3380 | | | | |
| CO2 | 3500 | 3490 | 3400 | | |
| H2 | 3020 | 2940 | | | |
+-----+----------+----------+----------+------------+----------+
The gases in these experiments were carefully dried and free from
dust; the apparent value of [alpha] is much increased when dust or
small drops of water are present in the gas, for then the ions get
caught by the dust particles, the mass of a particle is so great
compared with that of an ion that they are practically immovable under
the action of the electric field, and so the ions clinging to them
escape detection when electrical methods are used. Taking e as 3.5 ×
10^-10, we see that [alpha] is about 1.2 × 10^-6, so that the number
of recombinations in unit time between n positive and n negative ions
in unit volume is 1.2 × 10^-6n². The kinetic theory of gases shows
that if we have n molecules of air per cubic centimetre, the number of
collisions per second is 1.2 × 10^-10n² at a temperature of 0° C.
Thus we see that the number of recombinations between oppositely
charged ions is enormously greater than the number of collisions
between the same number of neutral molecules. We shall see that the
difference in size between the ion and the molecule is not nearly
sufficient to account for the difference between the collisions in the
two cases; the difference is due to the force between the oppositely
charged ions, which drags ions into collisions which but for this
force would have missed each other.
Several methods have been used to measure [alpha]. In one method air,
exposed to some ionizing agent at one end of a long tube, is slowly
sucked through the tube and the saturation current measured at
different points along the tube. These currents are proportional to
the values of n at the place of observation: if we know the distance
of this place from the end of the tube when the gas was ionized and
the velocity of the stream of gas, we can find t in equation (3), and
knowing the value of n we can deduce the value of [alpha] from the
equation
1/n1 - 1/n2 = [alpha](t1 - t2),
where n1, n2 are the values of n at the times t1, t2 respectively. In
this method the tubes ought to be so wide that the loss of ions by
diffusion to the sides of the tube is negligible. There are other
methods which involve the knowledge of the speed with which the ions
move under the action of known electric forces; we shall defer the
consideration of these methods until we have discussed the question of
these speeds.
In measuring the value of [alpha] it should be remembered that the
theory of the methods supposes that the ionization is uniform
throughout the gas. If the total ionization throughout a gas remains
constant, but instead of being uniformly distributed is concentrated
in patches, it is evident that the ions will recombine more quickly in
the second case than in the first, and that the value of [alpha] will
be different in the two cases. This probably explains the large values
of [alpha] obtained by Retschinsky, who ionized the gas by the [alpha]
rays from radium, a method which produces very patchy ionization.
_Variation of [alpha] with the Pressure of the Gas._--All observers
agree that there is little variation in [alpha] with the pressures for
pressures of between 5 and 1 atmospheres; at lower pressures, however,
the value of [alpha] seems to diminish with the pressure: thus
Langevin (_Ann. chim. phys._, 1903, 28, p. 287) found that at a
pressure of 1/5 of an atmosphere the value of [alpha] was about 1/5 of
its value at atmospheric pressure.
_Variation of [alpha] with the Temperature._--Erikson (_Phil. Mag._,
Aug. 1909) has shown that the value of [alpha] for air increases as
the temperature diminishes, and that at the temperature of liquid air
-180° C., it is more than twice as great as at +12° C.
Since, as we have seen, the recombination is due to the coming
together of the positive and negative ions under the influence of the
electrical attraction between them, it follows that a large electric
force sufficient to overcome this attraction would keep the ions apart
and hence diminish the coefficient of recombination. Simple
considerations, however, will show that it would require exceedingly
strong electric fields to produce an appreciable effect. The value of
[alpha] indicates that for two oppositely charged ions to unite they
must come within a distance of about 1.5 × 10^-6 centimetres; at this
distance the attraction between them is e² × 10^12/2.25, and if X is
the external electric force, the force tending to pull them apart
cannot be greater than Xe; if this is to be comparable with the
attraction, X must be comparable with e × 10^12/2.25, or putting e = 4
× 10^-10, with 1.8 × 10²; this is 54,000 volts per centimetre, a
force which could not be applied to gas at atmospheric pressure
without producing a spark.
_Diffusion of the Ions._--The ionized gas acts like a mixture of
gases, the ions corresponding to two different gases, the non-ionized
gas to a third. If the concentration of the ions is not uniform, they
will diffuse through the non-ionized gas in such a way as to produce a
more uniform distribution. A very valuable series of determinations of
the coefficient of diffusion of ions through various gases has been
made by Townsend (_Phil. Trans._, 1900, A, 193, p. 129). The method
used was to suck the ionized gas through narrow tubes; by measuring
the loss of both the positive and negative ions after the gases had
passed through a known length of tube, and allowing for the loss by
recombination, the loss by diffusion and hence the coefficient of
diffusion could be determined. The following tables give the values of
the coefficients of diffusion D on the C.G.S. system of units as
determined by Townsend:--
Table I.--_Coefficients of Diffusion (D) in Dry Gases._
+-----+------------+------------+----------+------------------+
|Gas. |D for +ions.|D for -ions.|Mean Value| Ratio of D for |
| | | | of D. | - to D for +ions.|
+-----+------------+------------+----------+------------------+
| Air | .028 | .043 | .0347 | 1.54 |
| O2 | .025 | .0396 | .0323 | 1.58 |
| CO2 | .023 | .026 | .0245 | 1.13 |
| H2 | .123 | .190 | .156 | 1.54 |
+-----+------------+------------+----------+------------------+
Table II.--Coefficients of Diffusion in Moist Gases.
+-----+------------+------------+----------+------------------+
|Gas. |D for +ions.|D for -ions.|Mean Value| Ratio of D for |
| | | | of D. | - to D for +ions.|
+-----+------------+------------+----------+------------------+
| Air | .032 | .037 | .0335 | 1.09 |
| O2 | .0288 | .0358 | .0323 | 1.24 |
| CO2 | .0245 | .0255 | .025 | 1.04 |
| H2 | .128 | .142 | .135 | 1.11 |
+-----+------------+------------+----------+------------------+
It is interesting to compare with these coefficients the values of D
when various gases diffuse through each other. D for hydrogen through
air is .634, for oxygen through air .177, for the vapour of isobutyl
amide through air .042. We thus see that the velocity of diffusion of
ions through air is much less than that of the simple gas, but that it
is quite comparable with that of the vapours of some complex organic
compounds.
The preceding tables show that the negative ions diffuse more rapidly
than the positive, especially in dry gases. The superior mobility of
the negative ions was observed first by Zeleny (_Phil. Mag._, 1898
[5], 46, p. 120), who showed that the velocity of the negative ions
under an electric force is greater than that of the positive. It will
be noticed that the difference between the mobility of the negative
and the positive ions is much more pronounced in dry gases than in
moist. The difference in the rates of diffusion of the positive and
negative ions is the reason why ionized gas, in which, to begin with,
the positive and negative charges were of equal amounts, sometimes
becomes electrified even although the gas is not acted upon by
electric forces. Thus, for example, if such gas be blown through
narrow tubes, it will be positively electrified when it comes out, for
since the negative ions diffuse more rapidly than the positive, the
gas in its passage through the tubes will lose by diffusion more
negative than positive ions and hence will emerge positively
electrified. Zeleny snowed that this effect does not occur when, as in
carbonic acid gas, the positive and negative ions diffuse at the same
rates. Townsend (loc. cit.) showed that the coefficient of diffusion
of the ions is the same whether the ionization is produced by Röntgen
rays, radioactive substances, ultra-violet light, or electric sparks.
The ions produced by chemical reactions and in flames are much less
mobile; thus, for example, Bloch (_Ann. chim. phys._, 1905 [8], 4, p.
25) found that for the ions produced by drawing air over phosphorus
the value of [alpha]/e was between 1 and 6 instead of over 3000, the
value when the air was ionized by Röntgen rays.
_Velocity of Ions in an Electric Field._--The velocity of ions in an
electric field, which is of fundamental importance in conduction, is
very closely related to the coefficient of diffusion. Measurements of
this velocity for ions produced by Röntgen rays have been made by
Rutherford (_Phil. Mag._ [5], 44, p. 422), Zeleny (_Phil. Mag._ [5], 46,
p. 120), Langevin (_Ann. Chim. Phys._, 1903, 28, p. 289), Phillips
(_Proc. Roy. Soc._ 78, A, p. 167), and Wellisch (_Phil. Trans._, 1909,
209, p. 249). The ions produced by radioactive substance have been
investigated by Rutherford (_Phil. Mag._ [5], 47, p. 109) and by Franck
and Pohl (_Verh. deutsch. phys. Gesell._, 1907, 9, p. 69), and the
negative ions produced when ultra-violet light falls on a metal plate by
Rutherford (_Proc. Camb. Phil. Soc._ 9, p. 401). H. A. Wilson (_Phil.
Trans._ 192, p. 4O9), Marx (_Ann. de Phys._ 11, p. 765), Moreau (_Journ.
de Phys._ 4, 11, p. 558; _Ann. Chim. Phys._ 7, 30, p. 5) and Gold
(_Proc. Roy. Soc._ 79, p. 43) have investigated the velocities of ions
produced by putting various salts into flames; McClelland (_Phil. Mag._
46, p. 29) the velocity of the ions in gases sucked from the
neighbourhood of flames and arcs; Townsend (_Proc. Camb. Phil. Soc._ 9,
p. 345) and Bloch (_loc. cit._) the velocity of ions produced by
chemical reaction; and Chattock (_Phil. Mag._ [5], 48, p. 401) the
velocity of the ions produced when electricity escapes from a sharp
needle point into a gas.
Several methods have been employed to determine these velocities. The
one most frequently employed is to find the electromotive intensity
required to force an ion against the stream of gas moving with a known
velocity parallel to the lines of electric force. Thus, of two
perforated plane electrodes vertically over each other, suppose the
lower to be positively, the upper negatively electrified, and suppose
that the gas is streaming vertically downwards with the velocity V; then
unless the upward velocity of the positive ion is greater than V, no
positive electricity will reach the upper plate. If we increase the
strength of the field between the plates, and hence the upward velocity
of the positive ion, until the positive ions just begin to reach the
upper plate, we know that with this strength of field the velocity of
the positive ion is equal to V. By this method, which has been used by
Rutherford, Zeleny and H. A. Wilson, the velocity of ions in fields of
various strengths has been determined.
The arrangement used by Zeleny is represented in fig. 8. P and Q are
square brass plates. They are bored through their centres, and to the
openings the tubes R and S are attached, the space between the plates
being covered in so as to form a closed box. K is a piece of wire
gauze completely covering the opening in Q; T is an insulated piece of
wire gauze nearly but not quite filling the opening in the plate P,
and connected with one pair of quadrants of an electrometer E. A plug
of glass wool G filters out the dust from a stream of gas which enters
the vessel by the tube D and leaves it by F; this plug also makes the
velocity of the flow of the gas uniform across the section of the
tube. The Röntgen rays to ionize the gas were produced by a bulb at
O, the bulb and coil being in a lead-covered box, with an aluminium
window through which the rays passed. Q is connected with one pole of
a battery of cells, P and the other pole of the battery are put to
earth. The changes in the potential of T are due to ions giving up
their charges to it. With a given velocity of air-blast the potential
of T was found not to change unless the difference of potential
between P and Q exceeded a critical value. The field corresponding to
this critical value thus made the ions move with the known velocity of
the blast.
[Illustration: FIG. 8.]
[Illustration: FIG. 9.]
Another method which has been employed by Rutherford and McClelland is
based on the action of an electric field in destroying the
conductivity of gas streaming through it. Suppose that BAB, DCD (fig.
9) are a system of parallel plates boxed in so that a stream of gas,
after flowing between BB, passes between DD without any loss of gas in
the interval. Suppose the plates DD are insulated, and connected with
one pair of quadrants of an electrometer, by charging up C to a
sufficiently high potential we can drive all the positive ions which
enter the system DCD against the plates D; this will cause a deflexion
of the electrometer, which in one second will be proportional to the
number of positive ions which have entered the system in that time. If
we charge A up to a high potential, B being put to earth, we shall
find that the deflexion of the electrometer connected with DD is less
than it was when A and B were at the same potential, because some of
the positive ions in their passage through BAB are driven against the
plates B. If u is the velocity along the lines of force in the uniform
electric field between A and B, and t the time it takes for the gas to
pass through BAB, then all the positive ions within a distance ut of
the plates B will be driven up against these plates, and thus if the
positive ions are equally distributed through the gas, the number of
positive ions which emerge from the system when the electric field is
on will bear to the number which emerge when the field is off the
ratio of 1 - ut/l to unity, where l is the distance between A and B.
This ratio is equal to the ratio of the deflexions in one second of
the electrometer attached to D, hence the observations of this
instrument give 1 - ut/l. If we know the velocity of the gas and the
length of the plates A and B, we can determine t, and since l can be
easily measured, we can find u, the velocity of the positive ion in a
field of given strength. By charging A and C negatively instead of
positively we can arrive at the velocity of the negative ion. In
practice it is more convenient to use cylindrical tubes with coaxial
wires instead of the systems of parallel plates, though in this case
the calculation of the velocity of the ions from the observations is a
little more complicated, inasmuch as the electric field is not uniform
between the tubes.
[Illustration: FIG. 10.]
A method which gives very accurate results, though it is only
applicable in certain cases, is the one used by Rutherford to measure
the velocity of the negative ions produced close to a metal plate by
the incidence on the plate of ultra-violet light. The principle of the
method is as follows:--AB (fig. 10) is an insulated horizontal plate
of well-polished zinc, which can be moved vertically up and down by
means of a screw; it is connected with one pair of quadrants of an
electrometer, the other pair of quadrants being put to earth. CD is a
base-plate with a hole EF in it; this hole is covered with fine wire
gauze, through which ultra-violet light passes and falls on the plate
AB. The plate CD is connected with an alternating current dynamo,
which produces a simply-periodic potential difference between AB and
CD, the other pole being put to earth. Suppose that at any instant the
plate CD is at a higher potential than AB, then the negative ions from
AB will move towards CD, and will continue to do so as long as the
potential of CD is higher than that of AB. If, however, the potential
difference changes sign before the negative ions reach CD, these ions
will go back to AB. Thus AB will not lose any negative charge unless
the distance between the plates AB and CD is less than the distance
traversed by the negative ion during the time the potential of CD is
higher than that of AB. By altering the distance between the plates
until CD just begins to lose a negative charge, we find the velocity
of the negative ion under unit electromotive intensity. For suppose
the difference of potential between AB and CD is equal to a sin pt,
then if d is the distance between the plates, the electric intensity
is equal to a sin pt/d; if we suppose the velocity of the ion is
proportional to the electric intensity, and if u is the velocity for
unit electric intensity, the velocity of the negative ion will be ua
sin pt/d. Hence if x represent the distance of the ion from AB
dx ua
--- = --- sin pt
dT d
ua
x = ----(1 - cos pt), if x = 0 when t = 0.
pd
Thus the greatest distance the ion can get from the plate is equal to
2au/pd, and if the distance between the plates is gradually reduced to
this value, the plate AB will begin to lose a negative charge; hence
when this happens
d = 2au/pd, or u = pd²/2a,
an equation by means of which we can find u.
In this form the method is not applicable when ions of both signs are
present. Franck and Pohl (_Verh. deutsch. physik. Gesell._ 1907, 9, p.
69) have by a slight modification removed this restriction. The
modification consists in confining the ionization to a layer of gas
below the gauze EF. If the velocity of the positive ions is to be
determined, these ions are forced through the gauze by applying to the
ionized gas a small constant electric force acting upwards; if
negative ions are required, the constant force is reversed. After
passing through the gauze the ions are acted upon by alternating
forces as in Rutherford's method.
Langevin (_Ann. chim. phys._, 1903, 28, p. 289) devised a method of
measuring the velocity of the ions which has been extensively used; it
has the advantage of not requiring the rate of ionization to remain
uniform. The general idea is as follows. Suppose that we expose the
gas between two parallel plates A, B to Röntgen rays or some other
ionizing agent, then stop the rays and apply a uniform electric field
to the region between the plates. If the force on the positive ion is
from A to B, the plate B will receive a positive charge of
electricity. After the electric force has acted for a time T reverse
it. B will now begin to receive negative electricity and will go on
doing so until the supply of negative ions is exhausted. Let us
consider how the quantity of positive electricity received by B will
vary with T. To fix our ideas, suppose the positive ions move more
slowly than the negative; let T2 and T1 be respectively the times
taken by the positive and negative ions to move under the electric
field through a distance equal to AB, the distance between the planes.
Then if T is greater than T2 all the ions will have been driven from
between the plates before the field is reversed, and therefore the
positive charge received by B will not depend upon T. Next let T be
less than T2 but greater than T1; then at the time when the field is
reversed all the negative ions will have been driven from between the
plates, so that the positive charge received by B will not be
neutralized by the arrival of fresh ions coming to it after the
reversal of the field. The number of positive ions driven against the
plate B will be proportional to T. Thus if we measure the value of the
positive charge on B for a series of values of T, each value being
less than the preceding, we shall find that until T reaches a certain
value the charge remains constant, but as soon as we reduce the time
below this value the charge diminishes. The value of T when the
diminution in the field begins is T2, the time taken for a positive
ion to cross from A to B under the electric field; thus from T2 we can
calculate the velocity of the positive ion in this field. If we still
further diminish T, we shall find that we reach a value when the
diminution of the positive charge on B with the time suddenly becomes
much more rapid; this change occurs when T falls below T1 the time
taken for the negative ions to go from one plate to the other, for now
when the field is reversed there are still some negative ions left
between the plates, and these will be driven against B and rob it of
some of the positive charge it had acquired before the field was
reversed. By observing the time when the increase in the rate of
diminution of the positive charge with the time suddenly sets in we
can determine T1, and hence the velocity of the negative ions.
The velocity of the ions produced by the discharge of electricity from
a fine point was determined by Chattock by an entirely different
method. In this case the electric field is so strong and the velocity
of the ion so great that the preceding methods are not applicable.
Suppose P represents a vertical needle discharging electricity into
air, consider the force acting on the ions included between two
horizontal planes A, B. If P is the density of the electrification,
and Z the vertical component of the electric intensity, F the
resultant force on the ions between A and B is vertical and equal to
_ _ _
/ / /
| | | Z[rho]dxdydz.
_/_/_/
Let us suppose that the velocity of the ion is proportional to the
electric intensity, so that if w is the vertical velocity of the ions,
which are supposed all to be of one sign, w = RZ.
Substituting this value of Z, the vertical force on the ions between A
and B is equal to
_ _ _
1 / / /
- | | | w[rho]dxdydz.
R _/_/_/
But [integral][integral]w[rho]dxdy = [iota], where [iota] is the
current streaming from the point. This current, which can be easily
measured by putting a galvanometer in series with the discharging
point, is independent of z, the vertical distance of a plane between A
and B below the charging point. Hence we have
_
[iota] / [iota]
F = ------ | dz = ------·z.
R _/ R
This force must be counterbalanced by the difference of gaseous
pressures over the planes A and B; hence if pB and pA denote
respectively the pressures over B and A, we have
[iota]
pB - pA = ------ z.
R
Hence by the measurement of these pressures we can determine R, and
hence the velocity with which an ion moves under a given electric
intensity.
There are other methods of determining the velocities of the ions, but
as these depend on the theory of the conduction of electricity through
a gas containing charged ions, we shall consider them in our
discussion of that theory.
By the use of these methods it has been shown that the velocities of
the ions in a given gas are the same whether the ionization is
produced by Röntgen rays, radioactive substances, ultra-violet light,
or by the discharge of electricity from points. When the ionization is
produced by chemical action the ions are very much less mobile, moving
in the same electric field with a velocity less than one-thousandth
part of the velocity of the first kind of ions. On the other hand, as
we shall see later, the velocity of the negative ions in flames is
enormously greater than that of even the first kind of ion under
similar electric fields and at the same pressure. But when these
negative ions get into the cold part of the flame, they move
sluggishly with velocities of the order of those possessed by the
second kind. The results of the various determinations of the
velocities of the ions are given in the following table. The
velocities are in centimetres per second under an electric force of
one volt per centimetre, the pressure of the gas being 1 atmosphere.
V+ denotes the velocity of the positive ion, V- that of the negative.
V is the mean velocity of the positive and negative ions.
_Velocities of Ions.--Ions produced by Röntgen Rays._
+----------------------+------+------+------+-----------------+
| Gas. | V+. | V-. | V. | Observer. |
+----------------------+------+------+------+-----------------+
| Air | .. | .. | 1.6 | Rutherford |
| Air (dry) | 1.36 | 1.87 | .. | Zeleny |
| " | 1.60 | 1.70 | .. | Langevin |
| " | 1.39 | 1.78 | .. | Phillips |
| " | 1.54 | 1.78 | .. | Wellisch |
| Air (moist) | 1.37 | 1.81 | .. | Zeleny |
| Oxygen (dry) | 1.36 | 1.80 | .. | " |
| Oxygen (moist) | 1.29 | 1.52 | .. | " |
| Carbonic acid (dry) | 0.76 | 0.81 | .. | " |
| " | 0.86 | 0.90 | .. | Langevin |
| " | 0.81 | 0.85 | .. | Wellisch |
| Carbonic acid (moist)| 0.82 | 0.75 | .. | Zeleny |
| Hydrogen (dry) | 6.70 | 7.95 | .. | " |
| Nitrogen | .. | .. | 1.6 | Rutherford |
| Sulphur dioxide | 0.44 | 0.41 | .. | Wellisch |
| Hydrochloric acid | .. | .. | 1.27 | Rutherford |
| Chlorine | .. | .. | 1.0 | " |
| Helium (dry) | 5.09 | 6.31 | .. | Franck and Pohl |
| Carbon monoxide | 1.10 | 1.14 | .. | Wellisch |
| Nitrous oxide | 0.82 | 0.90 | .. | " |
| Ammonia | 0.74 | 0.80 | .. | " |
| Aldehyde | 0.31 | 0.30 | .. | " |
| Ethyl alcohol | 0.34 | 0.27 | .. | " |
| Acetone | 0.31 | 0.29 | .. | " |
| Ethyl chloride | 0.33 | 0.31 | .. | " |
| Pentane | 0.36 | 0.35 | .. | " |
| Methyl acetate | 0.33 | 0.36 | .. | " |
| Ethyl formate | 0.30 | 0.31 | .. | " |
| Ethyl ether | 0.29 | 0.31 | .. | " |
| Ethyl acetate | 0.31 | 0.28 | .. | " |
| Methyl bromide | 0.29 | 0.28 | .. | " |
| Methyl iodide | 0.21 | 0.22 | .. | " |
| Carbon tetrachloride | 0.30 | 0.31 | .. | " |
| Ethyl iodide | 0.17 | 0.16 | .. | " |
+----------------------+------+------+------+-----------------+
_Ions produced by Ultra-Violet Light._
Air 1.4 Rutherford
Hydrogen 3.9 Rutherford
Carbonic acid 0.78 Rutherford
_Ions in Gases sucked from Flames._
Velocities varying from .04 to .23 McClelland
_Ions in Flames containing Salts._
Negative ions 12.9 cm./sec. Gold
+ions for salts of Li, Na,
K, Rb, Cs 62 H. A. Wilson
" 200 Marx
" 80 Moreau
_Ions liberated by Chemical Action._
Velocities of the order of 0.0005 cm./sec. Bloch
_Ions from Point Discharge._
+---------------+------+-------+------+----------+
| Hydrogen | 5.4 | 7.43 | 6.41 | Chattock |
| Carbonic acid | 0.83 | 0.925 | 0.88 | Chattock |
| Air | 1.32 | 1.80 | 1.55 | Chattock |
| Oxygen | 1.30 | 1.85 | 1.57 | Chattock |
+---------------+------+-------+------+----------+
It will be seen from this table that the greater mobility of the
negative ions is very much more marked in the case of the lighter and
simpler gases than in that of the heavier and more complicated ones;
with the vapours of organic substances there seems but little
difference between the mobilities of the positive and negative ions,
indeed in one or two cases the positive one seems slightly but very
slightly the more mobile of the two. In the case of the simple gases
the difference is much greater when the gases are dry than when they
are moist. It has been shown by direct experiment that the velocities
are directly proportional to the electric force.
_Variation of Velocities with Pressure._--Until the pressure gets low
the velocities of the ions, negative as well as positive, vary
inversely as the pressure. Langevin (loc. cit.) was the first to show
that at very low pressures the velocity of the negative ions increases
more rapidly as the pressure is diminished than this law indicates. If
the nature of the ion did not change with the pressure, the kinetic
theory of gases indicates that the velocity would vary inversely as
the pressure, so that Langevin's results indicate a change in the
nature of the negative ion when the pressure is diminished below a
certain value. Langevin's results are given in the following table,
where p represents the pressure measured in centimetres of mercury, V+
and V- the velocities of the positive and negative ions in air under
unit electrostatic force, i.e. 300 volts per centimetre:--
+----------------------+----------------------+
| Negative Ions. | Positive Ions. |
+-------+------+-------+-------+------+-------+
| p. | V-. |pV-/76.| p. | V+. |pV+/76.|
+-------+------+-------+-------+------+-------+
| 7.5 | 6560 | 647 | 7.5 | 4430 | 437 |
| 20.0 | 2204 | 580 | 20.0 | 1634 | 430 |
| 41.5 | 994 | 530 | 41.5 | 782 | 427 |
| 76.0 | 510 | 510 | 76.0 | 480 | 420 |
| 142.0 | 270 | 505 | 142.0 | 225 | 425 |
+-------+------+-------+-------+------+-------+
The increase in the case of pV- indicates that the structure of the
negative ion gets simpler as the pressure is reduced. Wallisch in some
experiments made at the Cavendish Laboratory found that the diminution
in the value of pV- at low pressures is much more marked in some gases
than in others, and in some gases he failed to detect it; but it must
be remembered that it is difficult to get measurements at pressures of
only a few millimetres, as the amount of ionization is so exceedingly
small at such pressures that the quantities to be observed are hardly
large enough to admit of accurate measurements by the methods
available at higher pressures.
_Effect of Temperature on the Velocity of the Ions._--Phillips (_Proc.
Roy. Soc._, 1906, 78, p. 167) investigated, using Langevin's method,
the velocities of the + and - ions through air at atmospheric pressure
at temperatures ranging from that of boiling liquid air to 411° C.; R1
and R2 are the velocities of the + and - ions respectively when the
force is a volt per centimetre.
+-------+-------+---------------------+
| R1. | R2. |Temperature Absolute.|
+-------+-------+---------------------+
| 2.00 | 2.495 | 411° |
| 1.95 | 2.40 | 399° |
| 1.85 | 2.30 | 383° |
| 1.81 | 2.21 | 373° |
| 1.67 | 2.125 | 348° |
| 1.60 | 2.00 | 333° |
| 1.39 | 1.785 | 285° |
| 0.945 | 1.23 | 209° |
| 0.235 | 0.235 | 94° |
+-------+-------+---------------------+
We see that except in the case of the lowest temperature, that of
liquid air, where there is a great drop in the velocity, the
velocities of the ions are proportional to the absolute temperature.
On the hypothesis of an ion of constant size we should, from the
kinetic theory of gases, expect the velocity to be proportional to the
square root of the absolute temperature, if the charge on the ion did
not affect the number of collisions between the ion and the molecules
of the gas through which it is moving. If the collisions were brought
about by the electrical attraction between the ions and the molecules,
the velocity would be proportional to the absolute temperature. H. A.
Wilson (_Phil. Trans._ 192, p. 499), in his experiments on the
conduction of flames and hot gases into which salts had been put,
found that the velocity of the positive ions in flames at a
temperature of 2000° C. containing the salts of the alkali metals was
62 cm./sec. under an electric force of one volt per centimetre, while
the velocity of the positive ions in a stream of hot air at 1000° C.
containing the same salts was only 7 cm./sec. under the same force.
The great effect of temperature is also shown in some experiments of
McClelland (_Phil. Mag._ [5], 46, p. 29) on the velocities of the ions
in gases drawn from Bunsen flames and arcs; he found that these
depended upon the distance the gas had travelled from the flame. Thus,
the velocity of the ions at a distance of 5.5 cm. from the Bunsen
flame when the temperature was 230° C. was .23 cm./sec. for a volt per
centimetre; at a distance of 10 cm. from the flame when the
temperature was 160° C. the velocity was .21 cm./sec; while at a
distance of 14.5 cm. from the flame when the temperature was 105° C.
the velocity was only .04 cm./sec. If the temperature of the gas at
this distance from the flame was raised by external means, the
velocity of the ions increased.
We can derive some information as to the constitution of the ions by
calculating the velocity with which a molecule of the gas would move
in the electric field if it carried the same charge as the ion. From
the theory of the diffusion of gases, as developed by Maxwell, we know
that if the particles of a gas A are surrounded by a gas B, then, if
the partial pressure of A is small, the velocity u with which its
particles will move when acted upon by a force Xe is given by the
equation
Xe
u = ------- D,
(p1/N1)
where D represents the coefficient of inter-diffusion of A into B, and
N1 the number of particles of A per cubic centimetre when the pressure
due to A is p1. Let us calculate by this equation the velocity with
which a molecule of hydrogen would move through hydrogen if it carried
the charge carried by an ion, which we shall prove shortly to be equal
to the charge carried by an atom of hydrogen in the electrolysis of
solutions. Since p1/N1 is independent of the pressure, it is equal to
[Pi]/N, where [Pi] is the atmospheric pressure and N the number of
molecules in a cubic centimetre of gas at atmospheric pressure. Now Ne
= 1.22 × 10^10, if e is measured in electrostatic units; [Pi] = 10^6
and D in this case is the coefficient of diffusion of hydrogen into
itself, and is equal to 1.7. Substituting these values we find
u = 1.97 × 10^4X.
If the potential gradient is 1 volt per centimetre, X = 1/300.
Substituting this value for X, we find u = 66 cm./sec, for the
velocity of a hydrogen molecule. We have seen that the velocity of the
ion in hydrogen is only about 5 cm./sec, so that the ion moves more
slowly than it would if it were a single molecule. One way of
explaining this is to suppose that the ion is bigger than the
molecule, and is in fact an aggregation of molecules, the charged ion
acting as a nucleus around which molecules collect like dust round a
charged body. This view is supported by the effect produced by
moisture in diminishing the velocity of the negative ion, for, as C.
T. R. Wilson (_Phil. Trans._ 193, p. 289) has shown, moisture tends to
collect round the ions, and condenses more easily on the negative than
on the positive ion. In connexion with the velocities of ions in the
gases drawn from flames, we find other instances which suggest that
condensation takes place round the ions. An increase in the size of
the system is not, however, the only way by which the velocity might
fall below that calculated for the hydrogen molecule, for we must
remember that the hydrogen molecule, whose coefficient of diffusion is
1.7, is not charged, while the ion is. The forces exerted by the ion
on the other molecules of hydrogen are not the same as those which
would be exerted by a molecule of hydrogen, and as the coefficient of
diffusion depends on the forces between the molecules, the coefficient
of diffusion of a charged molecule into hydrogen might be very
different from that of an uncharged one.
Wellisch (_loc. cit._) has shown that the effect of the charge on the
ion is sufficient in many cases to explain the small velocity of the
ions, even if there were no aggregation.
_Mixture of Gases._--The ionization of a mixture of gases raises some
very interesting questions. If we ionize a mixture of two very
different gases, say hydrogen and carbonic acid, and investigate the
nature of the ions by measuring their velocities, the question arises,
shall we find two kinds of positive and two kinds of negative ions
moving with different velocities, as we should do if some of the
positive ions were positively charged hydrogen molecules, while others
were positively charged molecules of carbonic acid; or shall we find
only one velocity for the positive ions and one for the negative? Many
experiments have been made on the velocity of ions in mixtures of two
gases, but as yet no evidence has been found of the existence of two
different kinds of either positive or negative ions in such mixtures,
although some of the methods for determining the velocities of the
ions, especially Langevin's, ought to give evidence of this effect, if
it existed. The experiments seem to show that the positive (and the
same is true for the negative) ions in a mixture of gases are all of
the same kind. This conclusion is one of considerable importance, as
it would not be true if the ions consisted of single molecules of the
gas from which they are produced.
_Recombination._--Several methods enable us to deduce the coefficient
of recombination of the ions when we know their velocities. Perhaps
the simplest of these consists in determining the relation between the
current passing between two parallel plates immersed in ionized gas
and the potential difference between the plates. For let q be the
amount of ionization, i.e. the number of ions produced per second per
unit volume of the gas, A the area of one of the plates, and d the
distance between them; then if the ionization is constant through the
volume, the number of ions of one sign produced per second in the gas
is qAd. Now if i is the current per unit area of the plate, e the
charge on an ion, iA/e ions of each sign are driven out of the gas by
the current per second. In addition to this source of loss of ions
there is the loss due to the recombination; if n is the number of
positive or negative ions per unit volume, then the number which
recombine per second is [alpha]n² per cubic centimetre, and if n is
constant through the volume of the gas, as will approximately be the
case if the current through the gas is only a small fraction of the
saturation current, the number of ions which disappear per second
through recombination is [alpha]n²·Ad. Hence, since when the gas is
in a steady state the number of ions produced must be equal to the
number which disappear, we have
qAd = iA/e + [alpha]n²·Ad,
q = i/ed + [alpha]^n2.
If u1 and u2 are the velocities with which the positive and negative
ions move, nu1e and nu2e are respectively the quantities of positive
electricity passing in one direction through unit area of the gas per
second, and of negative in the opposite direction, hence
i = nu1e + nu2e.
If X is the electric force acting on the gas, k1 and k2 the velocities
of the positive and negative ions under unit force, u1 = k1X, u2 =
k2X; hence
n = i/(k1 + k2)Xe,
and we have
i [alpha]i²
q = -- + --------------.
ed (k1 + k2)²e²X²
But qed is the saturation current per unit area of the plate; calling
this I, we have
d[alpha]i²
I - i = -------------
e(k1 + k2)²X²
or
i²·d[alpha]
X² = ------------------.
e(I - i)(k1 + k2)²
Hence if we determine corresponding values of X and i we can deduce
the value of [alpha]/e if we also know (k1 + k2). The value of I is
easily determined, as it is the current when X is very large. The
preceding result only applies when i is small compared with I, as it
is only in this case that the values of n and X are uniform throughout
the volume of the gas. Another method which answers the same purpose
is due to Langevin (_Ann. Chim. Phys._, 1903, 28, p. 289); it is as
follows. Let A and B be two parallel planes immersed in a gas, and let
a slab of the gas bounded by the planes a, b parallel to A and B be
ionized by an instantaneous flash of Röntgen rays. If A and B are at
different electric potentials, then all the positive ions produced by
the rays will be attracted by the negative plate and all the negative
ions by the positive, if the electric field were exceedingly large
they would reach these plates before they had time to recombine, so
that each plate would receive N0 ions if the flash of Röntgen rays
produced N0 positive and N0 negative ions. With weaker fields the
number of ions received by the plates will be less as some of them
will recombine before they can reach the plates. We can find the
number of ions which reach the plates in this case in the following
way:--In consequence of the movement of the ions the slab of ionized
gas will broaden out and will consist of three portions, one in which
there are nothing but positive ions,--this is on the side of the
negative plate,--another on the side of the positive plate in which
there are nothing but negative ions, and a portion between these in
which there are both positive and negative ions; it is in this layer
that recombination takes place, and here if n is the number of
positive or negative ions at the time t after the flash of Röntgen
rays,
n = n0/(1 + [alpha]n0t).
With the same notation as before, the breadth of either of the outer
layers will in time dt increase by X(k1 + k2)dt, and the number of ions
in it by X(k1 + k2)ndt; these ions will reach the plate, the outer
layers will receive fresh ions until the middle one disappears, which
it will do after a time l/X(k1 + k2), where l is the thickness of the
slab ab of ionized gas; hence N, the number of ions reaching either
plate, is given by the equation
_
/ l/X(k1+k2) n0X(k1 + k2) X(k1 + k2) / n0[alpha]l \
N = | --------------dt = ---------- log( 1 + ---------- ).
_/ 0 1 + n0[alpha]t [alpha] \ X(k1 + k2) /
If Q is the charge received by the plate,
X / Q0[epsilon]\
Q = Ne = -------------- log ( 1 + ----------- ),
4[pi][epsilon] \ 4[pi]X /
where Q0 = n0le is the charge received by the plate when the electric
force is large enough to prevent recombination, and [epsilon] =
[alpha]4[pi]e(R1 + R2). We can from this result deduce the value of
[epsilon] and hence the value of [alpha] when R1+R2 is known.
_Distribution of Electric Force when a Current is passing through an
Ionized Gas._--Let the two plates be at right angles to the axis of x;
then we may suppose that between the plates the electric intensity X
is everywhere parallel to the axis of x. The velocities of both the
positive and negative ions are assumed to be proportional to X. Let
k1X, k2X represent these velocities respectively; let n1, n2 be
respectively the number of positive and negative ions per unit volume
at a point fixed by the co-ordinate x; let q be the number of positive
or negative ions produced in unit time per unit volume at this point;
and let the number of ions which recombine in unit volume in unit time
be [alpha]n1n2; then if e is the charge on the ion, the volume density
of the electrification is (n1 - n2)e, hence
dX
-- = 4[pi](n1 - n2)e (1).
dx
If I is the current through unit area of the gas and if we neglect any
diffusion except that caused by the electric field,
n1ek1X + n2ek2X = I (2).
From equations (1) and (2) we have
1 / I k2 dX \
n1e = ------- ( - + ----- -- ) (3),
k1 + k2 \ X 4[pi] dx /
1 / I k1 dX \
n2e = ------- ( - - ----- -- ) (4),
k1 + k2 \ X 4[pi] dx /
and from these equations we can, if we know the distribution of
electric intensity between the plates, calculate the number of
positive and negative ions.
In a steady state the number of positive and negative ions in unit
volume at a given place remains constant, hence neglecting the loss by
diffusion, we have
d
--(k1n1X) = q - [alpha]n1n2 (5).
dx
d
- --(k2n2X) = q - [alpha]n1n2 (6).
dx
If k1 and K2 are constant, we have from (1), (5) and (6)
d²X² / 1 1 \
---- = 8[pi]e(q - [alpha]n1n2)( --- + --- ) (7),
dx² \ k1 k2 /
an equation which is very useful, because it enables us, if we know
the distribution of X², to find whether at any point in the gas the
ionization is greater or less than the recombination of the ions. We
see that q - [alpha]n1n2, which is the excess of ionization over
recombination, is proportional to d²X²/dx². Thus when the ionization
exceeds the recombination, i.e. when q - [alpha]n1n2 is positive, the
curve for X² is convex to the axis of x, while when the recombination
exceeds the ionization the curve for X² will be concave to the axis of
x. Thus, for example, fig. 11 represents the curve for X² observed by
Graham (_Wied. Ann._ 64, p. 49) in a tube through which a steady
current is passing. Interpreting it by equation (7), we infer that
ionization was much in excess of recombination at A and B, slightly so
along C, while along D the recombination exceeded the ionization.
Substituting in equation (7) the values of n1, n2 given in (3), (4),
we get
_ _
d²X² | [alpha] / k² dX²\ / k2 dX²\ | / 1 1 \
---- = 8[pi]e |q - -------------- (1 + ----- --- ) (1 - ----- --- )| ( --- + --- ) (8).
dx² |_ e²X²(k1 + k2)² \ 8[pi] dx / \ 8[pi] dx /_| \k1 k2 /
[Illustration: Fig. 11.]
This equation can be solved (see Thomson, _Phil. Mag._ xlvii. P. 253),
when q is constant and k1 = k2. From the solution it appears that if
X1 be the value of x close to one of the plates, and X0 the value
midway between them,
1
X1/X0 = ------------------
[beta]² - 2/[beta]
where [beta] = 8[pi]ek1/[alpha].
Since e = 4 × 10^-10, [alpha] = 2 × 10^-6, and k1 for air at
atmospheric pressure = 450, [beta] is about 2.3 for air at atmospheric
pressure and it becomes much greater at lower pressures.
Thus X1/X0 is always greater than unity, and the value of the ratio
increases from unity to infinity as [beta] increases from zero to
infinity. As [beta] does not involve either q or I, the ratio of X1 to
X0 is independent of the strength of the current and of the intensity
of the ionization.
No general solution of equation (8) has been found when k1 is not
equal to k2, but we can get an approximation to the solution when q is
constant. The equations (1), (2), (3), (4) are satisfied by the
values--
n1 = n2 = (q / [alpha])^½
k1
k1n1Xe = ------- I,
k1 + k2
k2
k2n2Xe = ------- I,
k1 + k2
/[alpha]\^½ I
X = ( ------- ) ----------.
\ q / e(k1 + k2)
These solutions cannot, however, hold right up to the surface of the
plates, for across each unit of area, at a point P, k1I/(k1+k2)e
positive ions pass in unit time, and these must all come from the
region between P and the positive plate. If [lambda] is the distance
of P from this plate, this region cannot furnish more than q[lambda]
positive ions, and only this number if there are no recombinations.
Hence the solution cannot hold when q[lambda] is less than k1I/(k1 +
k2)e, or where [lambda] is less than k1I/(k1 + k2)qe.
Similarly the solution cannot hold nearer to the negative plate than
the distance k2I/(k1 + k2)qe.
[Illustration: FIG. 12.]
The force in these layers will be greater than that in the middle of
the gas, and so the loss of ions by recombination will be smaller in
comparison with the loss due to the removal of the ions by the
current. If we assume that in these layers the loss of ions by
recombination can be neglected, we can by the method of the next
article find an expression for the value of the electric force at any
point in the layer. This, in conjunction with the value
/[alpha]\^½ I
X0 = ( ------- ) ----------
\ q / e(k1 + k2)
for the gas outside the layer, will give the value of X at any point
between the plates. It follows from this investigation that if X1 and
X2 are the values of X at the positive and negative plates
respectively, and X0 the value of X outside the layer,
/ k1 I \^½ / k2 I \^½
X1 = X0 ( I + -- --------- ) , X2 = X0 ( I + -- --------- ) ,
\ k2 [epsilon]/ \ k1 [epsilon]/
where [epsilon] = [alpha]/4[pi]e(k1 + k2). Langevin found that for air
at a pressure of 152 mm. [epsilon] = 0.01, at 375 mm. [epsilon] =
0.06, and at 760 mm. [epsilon] = 0.27. Thus at fairly low pressures
1/[epsilon] is large, and we have approximately
/k1\^½ I /k2\^½ I
X1 = X0 ( -- ) ---------------, X2 = X0 ( -- ) ---------------.
\k2/ [root][epsilon] \k1/ [root][epsilon]
Therefore X1/X2 = k1/k2,
or the force at the positive plate is to that at the negative plate as
the velocity of the positive ion is to that of the negative ion. Thus
the force at the negative plate is greater than that at the positive.
The falls of potential V1, V2 at the two layers when 1/[epsilon] is
large can be shown to be given by the equations
/[epsilon]\^3/2 /k1\^1/2
V1 = 8[pi]²( --------- ) k1 ( -- ) i²,
\q [alpha]/ \k2/
/[epsilon]\^3/2 /k2\^1/2
V2 = 8[pi]²( --------- ) k2 ( -- ) i²,
\q [alpha]/ \k1/
hence V1/V2 = k1²/k2²,
so that the potential falls at the electrodes are proportional to the
squares of the velocities of the ions. The change in potential across
the layers is proportional to the square of the current, while the
potential change between the layers is proportional to the current,
the total potential difference between the plates is the sum of these
changes, hence the relation between V and i will be of the form
V = Ai + Bi².
Mie (_Ann. der. Phys._, 1904, 13, P. 857) has by the method of
successive approximations obtained solutions of equation (8) (i.) when
the current is only a small fraction of the saturation current, (ii.)
when the current is nearly saturated. The results of his
investigations are represented in fig. 12, which represents the
distribution of electric force along the path of the current for
various values of the current expressed as fractions of the saturation
current. It will be seen that until the current amounts to about
one-fifth of the maximum current, the type of solution is the one just
indicated, i.e. the electric force is constant except in the
neighbourhood of the electrodes when it increases rapidly.
Though we are unable to obtain a general solution of the equation (8),
there are some very important special cases in which that equation can
be solved without difficulty. We shall consider two of these, the
first being that when the current is saturated. In this case there is
no loss of ions by recombination, so that using the same notation as
before we have
d
--(n1k1X) = q,
dx
d
--(n2k2X) = -q.
dx
The solutions of which if q is constant are
n1k1X = qx,
n2k2X = I/e - qx = q(l - x),
if l is the distance between the plates, and x = 0 at the positive
electrode. Since
dX/dx = 4[pi](n1 - n2)e,
we get
1 dX² / 1 1 \ l
----- --- = qx ( -- + -- ) - q --,
8[pi] d²x \k1 k2 / k2
or
X² x² / 1 1 \ lx
----- = q -- ( -- + -- ) - q -- + C,
8[pi] 2 \k1 k2 / k2
where C is a quantity to be determined by the condition that
_
/ l
| Xdx = V,
_/0
where V is the given potential difference between the plates. When the
force is a minimum dX/dx = 0, hence at this point
lk1 lk2
x = -------, l - x = -------.
k1 + k2 k1 + k2
Hence the ratio of the distances of this point from the positive and
negative plates respectively is equal to the ratio of the velocities
of the positive and negative ions.
The other case we shall consider is the very important one in which
the velocity of the negative ion is exceedingly large compared with
the positive; this is the case in flames where, as Gold (_Proc. Roy.
Soc._ 97, p. 43) has shown, the velocity of the negative ion is many
thousand times the velocity of the positive; it is also very probably
the case in all gases when the pressure is low. We may get the
solution of this case either by putting k1/k2 = 0 in equation (8), or
independently as follows:--Using the same notation as before, we have
i = n1k1Xe + n2k2Xe,
d
--(n2k2X) = q - [alpha]n1n2,
dx
dX
-- = 4[pi](n1 - n2)e.
dx
In this case practically all the current is carried by the negative
ions so that i = n2k2Xe, and therefore q = [alpha]n1n2.
Thus
n2 = i/k2Xe, n1 = qk2Xe/[alpha]i.
Thus
dX 4[pi]e²k2qX 4[pi]i
-- = ----------- - ------,
dx [alpha]i k2X
or
dX² 8[pi]e²k2qX² 8[pi]i
-- - ------------ = - ------.
dx [alpha]i k2
The solution of this equation is
[alpha] i²
X² = ------- ----- + C[epsilon]^(8[pi]e²k2qx/[alpha]i)
q k2²e²
Here x is measured from the positive electrode; it is more convenient
in this case, however, to measure it from the negative electrode. If x
be the distance from the negative electrode at which the electric
force is X, we have from equation (7)
[alpha] i²
X² = ------- ----- + C¹[epsilon]^(8[pi]e²k2qx/[alpha]i)
q k2²e²
To find the value of C¹ we see by equation (7) that
d²X² k1k2 1
--- ------- ------ = q - [alpha]n1n2;
dX² k1 + k2 8[pi]e
hence
_ _ _
| dX² k1k2 1 |^x1 / x1
| --- ------- ------ | = | (q - [alpha]n1n2)dx.
|_dX k1 + k2 8[pi]e_| _/0
The right hand side of this equation is the excess of ionization over
recombination in the region extending from the cathode to x1; it must
therefore, when things are in a steady state, equal the excess of the
number of negative ions which leave this region over those which enter
it. The number which leave is i/e and the number which enter is i0/e,
if it is the current of negative ions coming from unit area of the
cathode, as hot metal cathodes emit large quantities of negative
electricity i0 may in some cases be considerable, thus the right hand
side of equation is (i - i0)/e. When x1 is large dX²/dx = 0; hence we
have from equation
[alpha]i(i - i0) k1 + k2
C¹ = ---------------- -------,
qk1k2e² k2
and since k1 is small compared with k2, we have
[alpha]i² / k2 i - i0 \
X² = --------- (1 + -- ------ [epsilon]^{-8[pi]e²k2·qx/[alpha]·i} ).
qk2²e² \ k1 i /
From the values which have been found for k2 and [alpha], we know that
8[pi]ek2/[alpha] is a large quantity, hence the second term inside the
bracket will be very small when eqx is equal to or greater than i;
thus this term will be very small outside a layer of gas next the
cathode of such thickness that the number of ions produced on it would
be sufficient, if they were all utilized for the purpose, to carry the
current; in the case of flames this layer is exceedingly thin unless
the current is very large. The value of the electric force in the
uniform part of the field is equal to i/k2e·[root]([alpha]/q), while
when i0 = 0, the force at the cathode itself bears to the uniform
force the ratio of (k1 + k2)^½ to k1^½. As k1 is many thousand times
k2 the force increases with great rapidity as we approach the cathode;
this is a very characteristic feature of the passage of electricity
through flames and hot gases. Thus in an experiment made by H. A.
Wilson with a flame 18 cm. long, the drop of potential within 1
centimetre of the cathode was about five times the drop in the other
17 cm. of the tube. The relation between the current and the potential
difference when the velocity of the negative ion is much greater than
the positive is very easily obtained. Since the force is uniform and
equal to i/k2e·[root]([alpha]/q), until we get close to the cathode
the fall of potential in this part of the discharge will be very
approximately equal to i/k2e·[root]([alpha]l/q), where l is the
distance between the electrodes. Close to the cathode, the electric
force when i0 is not nearly equal to i is approximately given by the
equation
i /[alpha]\^½
X = --------- (---------) [epsilon]^{-4[pi]e²k2qx/[alpha]i},
e(k1k2)^½ \ q / ,
and the fall of potential at the cathode is equal approximately to
_[oo]
/
| X dx,
_/0
that is to
i /[alpha]\^½ [alpha] i
--------- (---------) ----------.
e(k1k2)^½ \ q / 4[pi]e²k2q
The potential difference between the plates is the sum of the fall of
potential in the uniform part of the discharge plus the fall at the
cathode, hence
/[alpha]\^½ i / i[alpha]² 1 \
V = (---------) --- ( il + --------- ------------ ).
\ q / ek2 \ 4[pi]e²q [root](k1k2)/
The fall of potential at the cathode is proportional to the square of
the current, while the fall in the rest of the circuit is directly
proportional to the current. In the case of flames or hot gases, the
fall of potential at the cathode is much greater than that in the rest
of the circuit, so that in such cases the current through the gas
varies nearly as the square root of the potential difference. The
equation we have just obtained is of the form
V = Ai + Bi²,
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
kX²/8[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
V² = 8[pi]l³i/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]n0² + i/ed. If X0 be the value of X when x = 0,
kX0n0e = i, and,
kX0² i² [alpha] i²
C = ----- = ----------- = -------- · -------- (2).
8[pi] n0²ke·8[pi] 8[pi]ke² 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
Röntgen 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 Röntgen 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)ga²/[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 Röntgen 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 × 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]ga³ v1
---------------------- = ---,
4/3[pi][rho]ga³ v
or Xe = (4/3)[pi][rho]ga³(v1 - v)/v.
But v = 2/9ga²[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 × 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 × 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° C. and
pressure 760 mm. of mercury are liberated by the passage through
acidulated water of one electromagnetic unit of electricity or 3 ×
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
2.46NE = 3 × 10^10,
or
NE = 1.22 × 10^10.
The mean of the values of Ne in the preceding table is 1.24 × 10^10.
Hence we may conclude that the charge of electricity carried by a
gaseous ion is equal to the charge carried by the hydrogen atom in the
electrolysis of solutions. The values of Ne for the different gases
differ more than we should have expected from the probable accuracy of
the determination of D and the velocity of the ions: Townsend (_Proc.
Roy. Soc._ 80, p. 207) has shown that when the ionization is produced
by Röntgen rays some of the positive ions carry a double charge and
that this accounts for the values of Ne being greater for the positive
than for the negative ions. Since we know the value of e, viz. 3.5 ×
10^-10, and, also Ne, = 1.24 × 10^10, we find N the number of
molecules in a cubic centimetre of gas at standard temperature and
pressure to be equal to 3.5 × 10^19. This method of obtaining N is the
only one which does not involve any assumption as to the shape of the
molecules and the forces acting between them.
Another method of determining the charge carried by an ion has been
employed by Rutherford (_Proc. Roy. Soc._ 81, pp. 141, 162), in which
the positively electrified particles emitted by radium are made use
of. The method consists of: (1) Counting the number of [alpha]
particles emitted by a given quantity of radium in a known time. (2)
Measuring the electric charge emitted by this quantity in the same
time. To count the number of the [alpha] particles the radium was so
arranged that it shot into an ionization chamber a small number of
[alpha] particles per minute; the interval between the emission of
individual particles was several seconds. When an [alpha] particle
passed into the vessel it ionized the gas inside and so greatly
increased its conductivity; thus, if the gas were kept exposed to an
electric field, the current through the gas would suddenly increase
when an [alpha] particle passed into the vessel. Although each [alpha]
particle produces about thirty thousand ions, this is hardly large
enough to produce the conductivity appreciable without the use of very
delicate apparatus; to increase the conductivity Rutherford took
advantage of the fact that ions, especially negative ones, when
exposed to a strong electric field, produce other ions by collision
against the molecules of the gas through which they are moving. By
suitably choosing the electric field and the pressure in the
ionization chamber, the 30,000 ions produced by each [alpha] particle
can be multiplied to such an extent that an appreciable current passes
through the ionization chamber on the arrival of each [alpha]
particle. An electrometer placed in series with this vessel will show
by its deflection when an [alpha] particle enters the chamber, and by
counting the number of deflections per minute we can determine the
number of [alpha] particles given out by the radium in that time.
Another method of counting this number is to let the particles fall on
a phosphorescent screen, and count the number of scintillations on the
screen in a certain time. Rutherford has shown that these two methods
give concordant results.
The charge of positive electricity given out by the radium was
measured by catching the [alpha] particles in a Faraday cylinder
placed in a very highly exhausted vessel, and measuring the charge per
minute received by this cylinder. In this way Rutherford showed that
the charge on the [alpha] particle was 9.4 × 10^-10 electrostatic
units. Now e/m for the [alpha] particle = 5 × 10³, and there is
evidence that the [alpha] particle is a charged atom of helium; since
the atomic weight of helium is 4 and e/m for hydrogen is 10^4, it
follows that the charge on the helium atom is twice that on the
hydrogen, so that the charge on the hydrogen atom is 4.7 × 10^-10
electrostatic units.
_Calculation of the Mass of the Ions at Low Pressures._--Although at
ordinary pressures the ion seems to have a very complex structure and to
be the aggregate of many molecules, yet we have evidence that at very
low pressures the structure of the ion, and especially of the negative
one, becomes very much simpler. This evidence is afforded by
determination of the mass of the atom. We can measure the ratio of the
mass of an ion to the charge on the ion by observing the deflections
produced by magnetic and electric forces on a moving ion. If an ion
carrying a charge e is moving with a velocity v, at a point where the
magnetic force is H, a mechanical force acts on the ion, whose direction
is at right angles both to the direction of motion of the ion and to the
magnetic force, and whose magnitude is evH sin [theta], where [theta] is
the angle between v and H. Suppose then that we have an ion moving
through a gas whose pressure is so low that the free path of the ion is
long compared with the distance through which it moves whilst we are
experimenting upon it; in this case the motion of the ion will be free,
and will not be affected by the presence of the gas.
Since the force is always at right angles to the direction of motion
of the ion, the speed of the ion will not be altered by the action of
this force; and if the ion is projected with a velocity v in a
direction at right angles to the magnetic f
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