Chapter XII: ). The constants may be expressed, as in the text,

in terms of (molar) ‹concentrations› of the ions, or in terms
of the ‹osmotic pressures› of the ions, a molar solution at 0°
producing an osmotic pressure of 22.4 atmospheres. Where osmotic
pressure and concentration are not strictly proportional (‹e.g.›
for concentrated solutions), the osmotic pressure, rather than the
concentration, is the determining factor and, when known, is used
in exact calculations. The plan, pursued in the text, is adopted
in order to express these constants in the terms used for all the
other equilibrium constants. It should be recalled (‹e.g.› p. 30)
that in calculations, in general, where pressure and concentration
are not strictly proportional, the pressure is the determining
factor. A third method of expressing the solution-tension relations
consists in giving the ‹potential differences›, which exist
‹between elements› and solutions of their ‹ions, in which the ions
have unit (molar) concentration›. These potential differences
are ‹functions› of the solution-tension constants, as will be
discussed below, and the constants, in terms of concentrations or
osmotic pressures, may be easily calculated, from the potential
differences, with the aid of this function (see below, and see the
table at the end of Chapter XV).

[526] According to Wilsmore's tabulation (‹Z. phys. Chem.›,
«36», 92 (1901)), the potential difference ε_{Cu, Cu^{2+}}
of copper against a 0.5 molar solution of cupric sulphate,
in which [Cu^{2+}] = 0.11, is +0.584 volt. Inserting these
values for [Cu^{2+}] and ε_{Cu, Cu^{2+}} in the equation
ε_{Cu, Cu^{2+}} = (0.0575 / 2) log([Cu^{2+}] / K) (see below)
and solving the equation for K, we find K = 8E−22. For
[Cu^{2+}] = 0.24, ε_{Cu, Cu^{2+}} is +0.594 volt and K = 8E−22. In
regard to the convention determining the signs used (in the present
case ε_{Cu, Cu^{2+}} is ‹positive›), see the footnote below, p.
262, and in regard to the definition of zero potential, to which
the potential differences used in this book refer, see the table
and summary at the end of Chapter XV.

[527] Nernst, ‹loc. cit.›, p. 151.

[528] In other words, the greater the concentration of cupric-ion,
the greater its osmotic pressure must be, and the repelling
electric force, required to overcome the pressure of the
cupric-ion, would be correspondingly greater.

[529] ‹Cf.› Nernst, ‹Theoretical Chemistry› (1904), pp. 720–723, in
regard to the derivation and the general form of his formula.

[530] For elements that form ‹negative ions›, ‹e.g.› for chlorine,
bromine, oxygen, etc., the ‹equation reads› (see pp. 273, 275 and
the table at the end of Chapter XV):

ε_{Elem., Electrolyte} = −(0.0575 / ‹v›) log(C / K).

Note the ‹changed› sign of the expression on the right. The
difference in sign expresses the fact that, when negative ions
discharge on an electrode, they render it negative, and when they
are formed by an electrode, they leave the latter positive; for
positive ions, it will be recalled, the conditions are just the
‹reverse› (see above).

Where a ‹soluble› element (‹e.g.› chlorine) or a solution of
a metal (‹e.g.› sodium amalgam) is used as an electrode, its
concentration, in general, is not constant, as in the case of
a pure, solid metal like copper (p. 258). In such cases, the
quantity in the denominator of the ratio in the logarithm cannot
be expressed by a constant K, but is expressed by K × C_{Element},
C_{Element} being used to indicate the concentration of the element
in the experiment in question.

[531] The convention, adopted in the text, for the use of the
positive and negative signs in expressing potentials, is that
proposed by Luther (‹cf.› Le Blanc's ‹Lehrbuch der Elektrochemie›
(third edition), p. 212). The ‹sign› always ‹denotes› the character
of the ‹charge› on the ‹first component› written in the subscript
to ε. Thus, for a copper plate in contact with a solution of
cupric sulphate, when C > K, the logarithm, log(C / K), has
a ‹positive› value and ε_{Cu, CuSO_{4}} is ‹positive›, which
means that the ‹metal› will be ‹positive›, the electrolyte
negative. For instance, for [Cu^{2+}] = 1, ε_{Cu, CuSO_{4}} is
found to be +0.606 (see the table at the end of Chapter XV).
ε_{Cu, CuSO_{4}} = −ε_{CuSO_{4}, Cu′}. By this use of the signs
one is never in doubt as to their meaning. Unfortunately, widely
different definitions of the signs have been used (‹cf.› Le
Blanc, ‹Electrochemistry› (1896), pp. 209, 219, and Lehfeldt,
‹Electro-Chemistry› (1904), p. 159). Care must be taken, in using
the data of original papers, to be informed as to the definition
used.

In accordance with the convention as to signs, adopted in this
book, the ratio of concentrations (C / K), used in the logarithm
of Nernst's formula, is the ‹reciprocal› of the ratio usually
given. The change has been made in order that the algebraic signs
of the values obtained from the application of the formula should
be the same as those observed in the experimental arrangements, as
demanded by the convention.

[532] When two electrodes are combined to form an electric cell
or couple, the potential difference of the couple is always
the (algebraic) ‹difference› of the two individual electrode
potentials, and hence these are ‹subtracted› from each other
(algebraically). The electrode of the first term of the difference
(the minuend) is named first in the subscript of the potential
of the couple; then the sign of the difference represents the
character of the charge on that electrode, in agreement with the
convention (see footnote 2, p. 261). In illustration: two copper
electrodes may be taken, each of which, considered by itself,
carries a positive charge, because the concentrations of the
cupric-ion in the solutions bathing them are both greater than
K; when they are combined, each of the two electrodes will tend
to send a positive current, in ‹opposite› directions, into the
metal connecting them. But the potential of the electrode with the
heavier charge (the one dipping into the solution containing the
greater concentration of cupric-ion) will overcome the potential
of the other electrode, and the current will flow, through the
connecting metal, with a potential that represents the difference
between the two values. If the electrode of the more concentrated
solution is named first in the subscript of the potential of the
couple, its individual electrode-potential appears as the first
term of the difference (the minuend) and is reduced by the value
of the electrode-potential of the second electrode; as this is
numerically smaller than the value of the minuend, the difference
will be positive, showing that the electrode in the stronger
solution, named first in the subscript of the potential difference
of the couple, carries a positive charge. Further, if the second
electrode dips into a solution, in which the concentration of the
cupric-ion is smaller than K, the logarithmic expression for its
electrode-potential will be found to give a negative value; and
the (algebraic) subtraction of this negative quantity from the
electrode-potential of the first electrode will give a larger
potential difference, for the couple, than that possessed by the
first electrode alone—all of which agrees with the experimental
results, when such combinations are made.

Where negative elements are concerned, the same convention holds,
but the ‹logarithmic expression for the potential of such an
electrode carries a negative sign› (see footnote 1, p. 261), which
must be inserted, algebraically, when the expression is used as a
term in the difference under discussion.

[533] If C′ > C″, the logarithm will be positive and ε_{Cu′, Cu″}
will have a ‹positive› value, which means that the copper plate,
Cu′, ‹which is named first in the subscript to ε›, will be charged
positively, ‹when the system works›. If C′ < C″, the logarithm
will be negative, which means that the first plate, Cu′, mentioned
in the subscript, will receive a negative charge, ‹when the
system works›. The sign is therefore intended, by the convention
adopted (p. 261), to express any result for the ‹working system›,
irrespective of the charge on the individual plates before they are
combined. For instance, for C′ = 1 and C″ = 10^{−10}, both plates
are positive, ‹before› they are connected with each other, since
in each case C > K, and ε_{Cu, CuX} = (0.0575 / 2) log(C / K) = a
positive value. When the plates are combined, we find from
ε_{Cu′, Cu″} = (0.0575 / 2) log(C′ / C″) that the first plate,
dipping in the more concentrated solution of cupric-ion, is
‹positive›, which is confirmed by experiment.

[534] (1 / 10)-molar cupric sulphate, 100 c.c., containing some
sodium sulphate or nitrate, to reduce the resistance, is a
convenient concentration.

[535] The copper plate is best freed from adhering sulphide by
means of a strong cyanide solution, and re-introduced into the
solution.

[536] Küster, ‹Z. Elecktrochem.›, «4», 110 and 503 (1897).

[537] In a solution of zinc sulphate in which [Zn^{2+}] = 0.114,
the potential ε_{Zn, ZnSO_{4}}= −0.514 (the minus sign indicates
that the metal named ‹first› in the subscript has a negative
charge). Inserting the values for [Zn^{2+}] and ε_{Zn, ZnSO_{4}} in
the general equation given on p. 261, and solving for K, we find
K = 10^{17}. For [Zn^{2+}] = 0.022 and ε_{Zn, ZnSO_{4}} = −0.535,
we find K = 10^{16.8}. (‹Cf.› Wilsmore's tables, ‹loc. cit.›)

[538] Equilibrium will be established whenever the potential of the
system is equal to 0. The potential of the system may be calculated
according to the equation (see footnote 1, p. 262)

ε_{Cu, Zn} = ε_{Cu, CuSO_{4}} − ε_{Zn, ZnSO_{4}} =
(0.0575 / 2) (log(Cu^{2+} / K_{Cu}) − log(Zn^{2+} / K_{Zn})).

The potential ε_{Cu, Zn} is 0 whenever [Cu^{2+}] / K_{Cu} =
[Zn^{2+}] / K_{Zn}, ‹i.e.› when [Zn^{2+}] / [Cu^{2+}] =
K_{Zn} / K_{Cu}.

For ions of different ‹valence›, such as silver and cupric
ions, the equilibrium equation assumes a somewhat less
simple form. For Cu ↓ + 2 Ag^{+} ⇄ 2 Ag ↓ + Cu^{2+}, we have
[Ag^{+}]^2 / [Cu^{2+}] = (K_{Ag})^2 / K_{Cu}.

[539] ‹Vide› Ostwald's ‹Lehrbuch der allgemeinen Chemie›, 2d Ed.,
Vol. II, p. 874, for the historical data on this action. ‹Vide›
Küster's experiments, ‹Z. Elektrochem.›, «4», 503 (1897).

[540] See the footnote, p. 267, in regard to the form the
equilibrium ratio assumes when metals producing ions of different
‹valence› are used.

[541] The value of the constant is calculated from the data given
by Peters, ‹Z. phys. Chem.›, «26», 193 (1898).

[542] The fact that this equilibrium relation has been proved to
hold for the action Fe^{2+} ⇄ Fe^{3+} and that it must be taken
into account in all oxidation-reduction reactions involving these
‹ions›, in no wise excludes the possibility that other equilibrium
relations can also exist between ferrous and ferric compounds.
For instance, ferrous hydroxide Fe(OH)_{2} may well have a
characteristic tendency of its own to assume a further positive
charge (lose an electron) according to Fe(OH)_{2} ⇄ Fe(OH)_{2}^{+},
the potential of which action may, under given conditions, be a
‹main determining factor› in the course of an action, ‹e.g.› in
alkaline mixtures. It is not impossible, even, that we also must
consider negative ions FeO_{2}^{2−} and their tendency to be
oxidized. Evidence would ‹suggest› that ferrous hydroxide, ‹or
its negative ion› FeO_{2}^{2−}, may have, indeed, a very ‹great
tendency› to be oxidized, possibly much greater than the tendency
of Fe^{2+} to form Fe^{3+}. (‹Cf.› Manchot, ‹Z. anorg. Chem.›,
«27», 419 (1901), and McCoy and Bunzel, ‹J. Am. Chem. Soc.›, «31»,
370 (1909)). Closer investigations of these relations, from a
quantitative viewpoint, would probably determine this question and
bring exceedingly important relations to light.

[543] ‹E.g.› by the potential of the action Cl_{2} ⇄ 2 Cl^{−}.

[544] The potential of a solution of the iron salts is given by
ε = 0.058 log(10^{17} × [Fe^{3+}] / [Fe^{2+}]). In a solution of
a ferric salt, if [Fe^{2+}] = 0, the potential would obviously be
∞, which could not present a condition of equilibrium. Equilibrium
is established in such a solution, as will be shown further on
in the text, by the liberation of chlorine and the formation of
ferro-salt, according to 2 Fe^{3+} + 2 Cl^{−} ⇄ 2 Fe^{2+} + Cl_{2},
until the potential, resulting from the tendency of chlorine to
form chloride-ion, just balances the tendency of the ferric-ion
to form ferro-ion. But when a ferric chloride solution is used
as the source of supply of positive electricity, as in the
experiment described in the text, ‹both› the ferric-ion and the
chlorine tend to charge the platinum electrode with positive
electricity and to revert to a condition of equilibrium in
reference to their individual constants. The relations are much
like those between a cupric salt solution and a copper plate: if
[Cu^{2+}] > K_{Cu^{2+}}, equilibrium will be established, as we
have seen, by the positive charging of the plate in sufficient
degree to oppose the tendency of the cupric-ion to discharge (see
p. 259). But when the solution and plate are used as the source of
supply for an electric current (p. 264), both the positive charge
on the plate, and the tendency of the cupric-ion to discharge and
acquire the concentration [Cu^{2+}] = K_{Cu^{2+}}, will supply
the positive current. In calculations we ignore the positive
charge already deposited on the plate and deal only with the
concentration of Cu^{2+}. The chlorine, liberated in a solution of
ferric chloride, plays practically the same rôle as does the copper
plate in a cupric salt solution, and it can be ignored in the
discussion of the combination described in the text. In a ferrous
salt solution, in a similar manner, some ferric-ion must always be
formed by liberation of hydrogen (see p. 282), until equilibrium
is reached according to 2 Fe^{2+} + 2 H^{+} ⇄ 2 Fe^{3+} + H_{2}.
Hydrogen plays here the same rôle as chlorine does in the ferric
chloride solution.

[545] The condition for equilibrium is [Fe^{2+}] : [Fe^{3+}] =
10^{17}, in a solution considered for itself.

[546] This ratio need not be 10^{17}, since we have two solutions
combined with each other and the total potential will be expressed
by:

ε = ε_{1} − ε_{2} = 0.058 (log(10^{17} × [Fe^{3+}]_{1} /
[Fe^{2+}]_{1}) − log(10^{17} × [Fe^{3+}]_{2} / [Fe^{2+}]_{2}))

= 0.058 log([Fe^{3+}]_{1} × [Fe^{2+}]_{2} /
([Fe^{2+}]_{1} × [Fe^{3+}]_{2})).

Equilibrium is reached when the total potential is 0. Then

[Fe^{3+}]_{1} × [Fe^{2+}]_{2} / ([Fe^{2+}]_{1} × [Fe^{3+}]_{2}) = 1

and [Fe^{2+}]_{1} / [Fe^{3+}]_{1} = [Fe^{2+}]_{2} / [Fe^{3+}]_{2}.

[547] In order to have very decided differences in the speeds of
the action in the absence and presence of fluoride, it is best to
use an old ferrous sulphate, or ferrous ammonium sulphate, solution
which contains considerable ferric salt.

[548] ‹Vide› Peters, ‹loc. cit.›, p. 236.

[549] Ostwald [‹Lehrbuch d. allgem. Chem.›, 2d Ed., II, 883
(1893)], first emphasized the fact that potential differences are a
‹measure› of oxidizing and reducing powers.

[550] The constant is calculated from the data of Küster and
Crotogino on the potential of solutions of iodine in potassium
iodide [‹Z. anorg. Chem.›, «23», 88 (1900)]. Owing to the formation
of complex ions I_{3}^{−}, for which due allowance has not been
made in the calculation, and owing to some uncertainty as to
the vague definition of the concentration of iodine used, the
estimation of the constant can only be considered a rough one.
The value given expresses the order of the equilibrium ratio
sufficiently well for our present purposes. In a recent paper, Bray
and MacKay [‹J. Am. Chem. Soc.›, «32», 914 (1910)] have determined
the constant for the formation of the complex ion according to
I_{3}^{−} ⇄ I_{2} + I^{−}, which might be used to correct the data
of Küster and Crotogino; but in view of other uncertainties and
inaccuracies, the correction has not been considered advisable.

Several related methods may be used to calculate the equilibrium
constant for [I^{−}]^2 : [I_{2}] = K from the data of Küster and
Crotogino. Perhaps the simplest method is the following: A solution
of iodine ([I] = 1 / 32 normal, and therefore [I_{2}] = 1 / 64
molar) in 1/8 molar potassium iodide, in which, the degree
of ionization being taken into account, [I^{−}] = 0.109, was
observed to show a potential ε_{I_{2}, I^{−}} = + 0.860 (the
convention as to signs, discussed on p. 261, is used here and
the potential, observed against a so-called "calomel electrode,"
is reduced to the so-called "absolute potential"; ‹cf.› Le
Blanc, ‹Lehrbuch der Elektrochemie›, p. 214). Now, ‹there must
be a certain concentration of iodide-ion›, which we will call
[C], ‹with which iodine› of the above concentration ‹would be
directly in equilibrium› and would give no potential at all
(‹cf.› pp. 261 and 258 in regard to copper). With a change in the
concentration of the iodide-ion, a potential would be produced
according to ε_{I_{2}, I^{−}} = 0.0575 log([C] / [I^{−}]).
This relation is of exactly the same nature as that developed
for the potential of copper plates, immersed in solutions of
cupric-ion of different concentrations (but see footnote 1, p.
261, concerning the ‹sign› of the new relation). In the present
case, we are dealing with univalent ions, I^{−}, in place of
bivalent ions Cu^{2+}, and the factor 0.0575 is used instead
of 0.0575 / 2 (see p. 261). If we insert the observed values,
[I^{−}] = 0.109 and ε = 0.860, of the experiment described above,
into the equation ε_{I_{2}, I^{−}} = 0.0575 log([C] / [I^{−}])
and solve the equation for [C], we find [C] = 10^{14}. ‹That
means›, 1 / 64 molar iodine ‹would be directly in equilibrium
with a concentration of iodide-ion› = 10^{14} (if this value
is inserted for [I^{−}] in the logarithmic equation, the
potential is found to be 0). For the condition of equilibrium for
I_{2} ⇄ 2 I^{−}, according to [I^{−}]^2 : [I_{2}] = K, we have then
(10^{14})^2 : (1 / 64) = K = 6.4E29. Similarly, for [I^{−}] = 0.109
and [I_{2}] = 1 / 512 the potential ε = 0.831 is observed, and the
equilibrium constant is found to be 5.1E29. When [I^{−}] = 0.109
and [I_{2}] = 1 / 128, the potential is 0.850 and the constant is
calculated to be 5.3E29. The mean value for K is 5.6E29. In these
calculations, the formation of ions I_{3}^{−}, affecting the values
for [I^{−}] and [I_{2}], has not been considered, and there is some
doubt whether the concentrations of iodine, given by Küster and
Crotogino, do not represent [I_{2}] rather than [I], as assumed in
the calculations. If the former be the case, the mean value of the
above experiments would be 2.8E29. The value, used in the text, is
considered sufficiently accurate for the purposes of this book.

[551] This relation of the equilibrium constant and the
solution-tension constants may be deduced in a manner similar to
that for the analogous equilibrium constant for the oxidation of
zinc by the cupric-ion, as given in footnote 1, on page 267. The
‹exact› value of the equilibrium constant is uncertain, since
K_{I^{−}, Iodine} has not yet been determined with a sufficient
degree of accuracy; but the value, used, gives the order of the
constant sufficiently well for our purposes, especially when it is
considered in connection with the constant given below for the same
relation, when the chloride-ion is substituted for the iodide-ion.

[552] This is the value of the constant as calculated from the
data given by Wilsmore (‹Z. phys. Chem.›, «36», 91 (1900))
for the solution-tension of chlorine under atmospheric
pressure at 18°. The calculation may be made exactly as in
the case of the similar constant for iodine (p. 273). ‹There
must be a concentration of chloride-ion›, which we will
call [C], ‹with which chlorine, of one atmosphere pressure
at 18°›, would be directly in equilibrium. The potential of
chlorine, against any other concentration of chloride-ion,
would be ε_{Cl_{2}, Cl^{−}} = 0.0575 log([C] / [Cl^{−}]). For
[Cl^{−}] = 1, ε is +1.694 (see the table at the end of Chapter
XV), and inserting these values in our equation and solving it
for [C], we find [C] = 2.88E29. That means, that chlorine, at
18° and of atmospheric pressure, would be in equilibrium with
chloride-ion of the concentration given. Since chlorine, at this
temperature and pressure, has a concentration of 1 / 23.9 moles
(at 18°, one mole is contained in 23.9 liters, instead of in
22.4 liters, at O°), we have for the condition of equilibrium:
[Cl^{−}]^2 : [Cl_{2}] = (2.88E29)^2 : (1 / 23.9) = 2E60. [Cl_{2}]
represents, thus, in the calculation of this constant, the
concentration of chlorine gas (see Chapter XV concerning gas
electrodes) and not the concentration of the dissolved chlorine;
the latter, however, is proportional to the gas concentration
(Chapter VII).

[p277]