T H E COKCEPTIOKS OF ELECTRICAL POTESTIBL D I F F E R E K C E BETWEEN TWO PHa4SES XSD T H E ISDIT’IDUAL ACTIVITIES O F I O S S ____ BY E. A . GUGGEh’HEIM
It has long been realised that one cannot accurately determine in a solution the activity or chemical potential of an individual ion, because of the existence of liquid junction potentials whose magnitude cannot be computed without a previous knowledge of the individual ionic activities that one wishes to measure. This vicious circle has generally been regarded as an unfortunate accident and attempts have frequently been made to overcome the dilemma by the use of cells in which the liquid junction potentials, though unknonrn, were probably small. For example the introduction of a bridge of a concentrated solution of potassium chloride will in many cases reduce the uncertainty to a few millivolts. It is however easy to convince oneself that under the most favourable circumstances the uncertainty due to the liquid junction potential is more than equivalent to the difference between the individual chemical potentials of two different ions of the same charge when present a t the same concentration in the same solution. It is in fact impossible by any such device to distinguish between the values of the activity coefficients of different ions with the same charge. In agreement with this point of view it has recently been shown by Taylor’ that “the E.M.F. of the cell with transference is thus a function of molecular free energies solely and is not a function of ionic free energies. It therefore can yield no information concerning ionic free energies” and that “with the possible exception of single electrode potentials and rates of reaction there appears to be no occasion for the use of ionic free energies as experimental quantities, but only as a mathematical device.? I t will be shown that Taylor‘s first conclusion may be considered as a corollary to a more general principle, from which it also follows that to his second conclusion there are no “possible exceptions.” The general principle referred t o may be expressed as follows. “The electric potential difference between two points in different media can never be measured and has not yet been defined in terms of physical realities; it is therefore a conception which has no physical significance.” The electrostatic ‘ P . B. Taylor: J. Phys. Chem., 31, 1478 (1927). * Taylor uses the expression “free energy” for Gibbs’ “chemical potential” P . Lewis and Randall (“Thermodynamics” (1923)) call it the “partial free energy” and use the symbol 8. We prefer to retain the nomenclature of Gibbs, as I.’ is related equally intimately to the “free energy” F, the “affinity” A, the “heat content” H and the “internal energy” E:. this may be seen from the relations I _
where T, S,P, V denote temperature, entropy, pressure, volume respectively and ni is the number of moles of the species in the system.
POTENTIAL DIFFEREKCE AND ACTIVITIES O F IONS
843
potential difference between two points is admittedly defined in electrostatics, but this is the mathematical theory of an imaginary fluid ‘electricity,’ whose equilibrium or motion is determined entirely by the electric field. ‘Electricity’of this kind does not exist; only electrons and ions have physical existence and these differ fundamentally from the hypothetical fluid ‘electricity’ in that the particles are a t all times in movement relative to one another; their equilibrium is thermodynamic, not static. The electrostatic potential $ determines the equilibrium or change towards equilibrium of the hypothetical fluid ‘electricity,’ whereas the chemical pobential pi determines those of an uncharged molecule of type i. But the equilibrium of an ion of type i and charge e i is determined neither by $ nor E$ which it is suggested should be called by pi but by the function ii; = pi the “electrochemical potential.” If is therefore j i that has a real physical significance. The conception of splitting the electrochemical potential of an ion of type i into the sum of a chemical term pi and a n electrical term e$ has no physical significance; for one can assign a n arbitrary value to for some point in each medium and this will for the ions of each type i determine pi, so as to give l i the value which determines all the physical processes involving ions of type i ; amongst such processes may be mentioned in particular diffusion, partition between two media, membrane equilibria, cells without and with liquid junctions, and even reaction rates. h word of explanation is required concerning our use of the word medium; the medium may be considered the same so long as the solvent is the same and all the solute species behave independently. For example two aqueous solutions of the same salt a t different concentrations may, or may not, be considered as the same medium according as we neglect, or take account of, specific ‘salt effects.’ Thus the liquid junction potential between two salt solutions in different solvents has no physical meaning; for two aqueous salt solutions in contact it can be computed only with an accuracy that neglects specific ‘salt effects.’ Although pi has no physical significance for a single ionic species, yet certain linear combinations of the pi have one; the necessary and sufficient condition for this is that they should be expressible in terms of the ,&. For example if we consider two ionic species i and k in the same medium and write formally
+
+
=
pi
pk =
pk
Pi
+ +
ti$ ek
$
then thesc equations have no physicrtl significance as the value of $ is quite arbitrary. If howevei we eliminate $ we obtain
w so that the combination 2 €1
E has a definite value although p, and pk have ek
not. I n general, the condition that the sum BX,pc,should have a definite
.value, where the X i are numerical coefficients, is that
X i ei
=
o for under
I
this condition Z: X i ui
S Xi
=
pi and this defines S X i p i .
L
Since the activity ai of an ion of type i is defined by pi = R T log, ai
+
pio
where pio depends on the temperature and pressure, but not on the medium, and its activity coefficient f i is defined by ai it follows that the relation P p l
=
ei =
fiCi
0 is also the necessary and sufficient
1
condition for the products II ai
xi
and II f i
1
to be physically defined although
1
the individual ai and f, are not. Thus, in particular, the 'mean activity coefficient' of a salt is defined, as is also the ratio of the activities or activity coefficients of two ionic species with the same charge. We shall now illustrate our general principle by showing that the phenomena of diffusion, partition between tJvo media, membrane equilibria, cells without and with liquid junctions, and reaction rates are completely describable in terms of the electrochemical potentials pi (or alternately such linear combinations of the pi as are expressible and so definable in terms of the jL)and that in no case does either pior+occur separat.ely from the other. Diffusion. The driving force on an uncharged molecule of type i determining it's diffusion is -grad pL,the component in the 5 direction being - & ax For an ion of type i and charge e i the corresponding chemical force is the same, but there is also an electrical force --ei grad $ with an x-component - ei a+ - Hence the total driving force on the ion determining its diffusion is
ax
-grad pi-ei grad $
=
- grad (pi
+
E;+)
= -
grad pi with an x-component
S o experiment on diffusion can distinguish between the hypothetical component parts of i i .
Partition b e t r e e n two media. If a salt AB is distributed between two phases I and I1 the equilibrium is completely defined and described by the conditions
p:
=
py
p;
=
p;
+ +
and any decomposition of Fit into the sum pi e, is arbitrary and without physical significance. S o measurements can give any inforniation about or the separate p,.
+
POTESTIAL D I F F E R E N C E AfiD ACTIYITIES O F I O S S
845
Menzbrane equilibria. If a membrane permeable to some ions, but impermeable to others and to the solvent' separates t,wo phases I and 11, then the equilibrium of any diffusible ion of type i is completely determined by the condition
p; = 3; Only if t'he medium is the same in both phases (same solvent with neglect of specific salt effects) has it any physical significance to split pi into a part p ) depending only on the concentration of i and an electrical part e i 4. Thus in the special case of an ideal solution
nhere K is the gas constant, T the absolute temperature and c', the concentration of the species z But in general for a non-ideal solution any analogous equation would hai-e no physical meaning
Cell u i t h o u t liquid e.;aniple the cell
It
junction
1s
here con\enient t o take as a concrete
Solution containing c'u-- and Zn+I1
('U
I
Zn I11
C'U
IT-
with the circuit broken in the copper; the irreversible deposition of copper on the zinc electrode is to be ign1)rctl. Let the phaws be numbered in ordcr I , 11, 111, IT. Then the two trorle-.solution junctions and the metal-metal junction may be regarded :I; iacmbranes pernieable respectively t o copper ions ( ' ~ i - - , zinc ions Zn-- and electrons El- only. The equilibria at t h e v thrw junctions are completely described by the conditions
>&-
=
E;'--
j$+
=
p;'+
p+
=
and the vexed question whether the electrical potential difference is seated at the electrodes or at the metal-metal junction has no physical meaning. Since the phases I T and I are the same medium, copper, the potential difference between then1 has a physical meaning as in e1ectroatatic.q. lye h a w in fact ,&?.
=
pc'. -. =
#-.+ e cu-&-.+ cc _ _
+I>' $1
By subtraction we obtain for 4 the I 111--. It might be suggestea that the electric potential $ be defined by the relation
FIT and then tlie
pi
$
= e1;r
could be defined by y, = (Li,
-
e,$ =
-
e,
/.i!
-
y El-
eL!-
There is no physical or logical objection t o t h k , but this u.se of the expression 'electrical potential' would be so far rciiiowd froiii that of 'electrostatic potential' that it is not to be rccorriniended. C'onsider for esaniple the description of the T-olta effect. If we define as "uncharged" a piece af metal 'Debye and Huckel: Phyaik. Z.,21, 1 8 j 11923
POTESTIAL D I F F E R E S C E A S D ACTIVITIES O F I O S S
849
in which the number of free electrons is exactly equivalent to the number of metallic ions, then the electrochemical potential ji~,of electrons vi11 not be the same in a piece of uncharged zinc and a piece of uncharged copper. If two such pieces of zinc and copper be brought into contact there mill then iq be a flow of electrons from the zinc to the copper until the difference in ,ii~,annulled. On the other hand in electrostatic theory the flow of electrons is usually said to create, not annul, an electrostatic potential difference. Hence if instead of ‘electrochemical potential pE]- of electrons’ we were to write ‘electric potential $’, the above description TT-ould be likely to sound strange. I t is therefore probably best to restrict the expression ‘electric potential’ to mean the ‘electrostatic potential’ of electrostatics, and avoid its use in the description of physical phenomena involving ions and electrons. In conclusion the author wishes to thank Professor Bjerrum for his kind criticism. Copenhagen, Decerrr ber 22. 1928.
