OK THE COSCEPTIOS O F ELECTRICAL POTESTLAL D I F F E I I E S C E RETITEES TWO PHASES. I1 BY E. A . G T G G E S H E I 1 \ I
In a previous paper' the author pointed out that, all equilibria and isothermal changes towards equilibria of ions, including electrons, are completely describable in terms of certain thermodynamic functions, which he called the electrochemical potentials of the ions. The condition that two phases I and I1 should be in equilibrium as regards ions of type i, is that the electrochemical potential E of ions of this type should be the same in both phases, that is -1
Pi
=
-11
Pi
(1)
This formula replaces the more usual formula PI1
+
Z&$I
=
pi1r- z,e$*I
where pi represents the cheinical potential, zi the valency, taking regard of sign, of ions of type i, --E is the charge of the electron and $ is the electrical potential. Similarly in the description of kinetic processes, such as diffusion or chemical reaction, the electrochemical potential is sufficient. The author concluded as a fundamental principle that the decomposition of the electrochemical potential G i l into the sum of a cheniical term pi and an electrical term zie$ is quite arbitrary and without phyiical significance. In other words the chemical potentia1,or the activity of a single ion,and the electric potential difference between two points in different media are conceptions without any physical significance. -in arbitrary value can be assigned to the electric potential $ in every phase and the corresponding value? of the chemical potentials are then defined by pi =
Pi
- Z,E$
(3
For instance one iiiight, in the absence of any estcrnal electric field, set $ = o in every phase, in which case pi beconies idcntifitd with G i . This convention amounts virtually t o abolishing the use of the expression electrical potential except to describe external fields. Ah alternative convention would be at every point to set - e$ equal to the electrochemical potential of the electron at that point. This convention was discussed and reasons given for its rejection. More recently a paper by Rronsted* has appeared on the same subject. He also introduces the electrochemical potential which he however denotes by A, and obtains as the condition for the equilibrium between two phases Ai and 13 as regards ions of type I A,,,,
* J. Phys. Chem., 33, 842
1y29 8 . * Z. physik. Chem., 143, 301 1929 I
8.
= L I B )
4)
ELECTRICAL POTENTIAL DIFFERENCE BETWEEN TWO PHASES
I 541
identical with ( I ) . He nevertheless attaches considerable importance to the decomposition ofXinto the sum of a chemical termpand an electrical term $CY. The author wishes in the present paper so to modify the principle put forward in his former paper as to fall into line with the point of view of Bronsted. Let us consider, for example, a piece of copper and a piece of zinc in equilibrium as regards electrons in the absence of any external field and let us ignore the comparatively slow interdiffusion of the metallic ions and atoms, The electrons will be so distributed that their electrochemical potential is the same in both phases. This description of the equilibrium however tells us nothing about the asymmetry in the distribution of the electrons between the copper and the zinc. Even if there is complete symmetry between the two metals as regards size, shape, and relative position, the electric charge on each will generally not be the same. The actual distribution of charge will depend both on the intrinsic properties of the two metals and also on their sizes, shapes and relative positions. The question therefore suggests itself whether there is not some function of the equilibrium charge distribution, which is independent of the size, shape and relative positions of the two pieces of metal. Such a function does exist and will now be defined. Let us imagine two geometrical surfaces of the size, shape and relative positions of the two pieces of metal and consider a distribution of “electricity” within each of these surfaces such that themeanelectric charge density over any element of volume, large compared with an atom, is the same as that Tithin the corresponding element in each of the metals. The mean density oyer any volume element will incidentally be zero unless the element be in the neighborhood of one of the surfaces. From this hypothetical distribution of electricity one might calculate the electrostatic potential at any point, on the nssiinzption oj” a dielectric constant zinity throughout. It would be found that the electrostatic potential, so calculated, would be constant throughout the volume corresponding to each metal, except in the neighbourhood of the surface. Moreover the difference in the electrostatic potential thus calculated for a point Tithin the surface corresponding to the copper and that for a point within the surface corresponding to the zinc, nil1 be independent of the size, shape and relative positions of the two metals. I t therefore seems not unreasonable to define the electric potential difference between the two phases as equal to the difference between the two electrostatic potentials calculated as above. AAccordingto this conventional definition of electric potential difference, the electric potential will be the same in any phase which is insulated and uncharged, and this value may of course be chosen as zero. This agrees with the conventional definition of electric potential chosen by Bronsted’. The electric potential, as thus defined, is a potential function in the mathematical sense that it is a single valued function of position, but it is not a potential in the strict dynamic sense, for it does not measure the work done in any real process. Whereas the reversible work of transference 1
cf. Schottky: “Thermodynamik,” 1x9, 123, 156 (1929).
I542
E. A. GUGGENHEIM
of an uncharged molecule is measured by its chemical potential and that of an ion or electron by its electrochemical potential, the electric potential, as above defined, measures either the work of transfer of “electricity” without matter in the actual media or alternatively the work of transfer of any charged particle in the hypothetical electrostatic field described above. But “electricity” has no existence apart from matter and the electrostatic field described above could not possibly exist, if the dielectric constant were unity throughout. Only in the special case of two points in the same medium is the electric potential difference a measure of a work of transference. The fundamental difference between the cases of two points in the same medium and of two points in different media may be made clearer by the following considerations. In vacuo and in the absence of gravity the only forces acting on a slowly moving charged particle are electrostatic and so the electrochemical potential of any ion is equal to its charge multiplied by the electrostatic potential. I n vacuo the electrostatic potential is therefore a measure of the work of transference of any charged particle. Now consider a point A in medium I and a point B in medium 11, the two media being somewhere in contact. Let h B denote any path from A to B lying entirely in the media I and 11. Let A’, B’ respectively be points very near A, B such that AX‘, BB’ are perpendicular to the tangents to the path AB at, X,B. Let A’ B’ denote any path also lying entirely in the media I and I1 and not cutting AB. Suppose now a narrow strip of the media removed including within it the whole of A’B’ but, none of AB, its surface thus cutting AA’ and BB’. Let ‘?AB) @ A , B , , @ A A , , @ B B , represent the electric potential differences between the various pairs of points. Since X‘B‘ is in vacuo, @A,B. is clearly a measure of the work of transference of a charged particle from A’ to B‘. Further the value of @AB will not be appreciably different from the value of the corresponding potential difference before the removal of the strip containing A’B’. Clearly the value of @ABdifferSfrom that of @A,Btbythe amount - @BB,. But however near A’.B‘ respectively are to A,B the difference qAAt- @ B B , will differ from zero if the two media I and I1 are different. Only if the two media are identical may we say that @AB is a measure of the work of transference of a charged particle from A’ to B‘ or virtually from A to B. In this connection a quotation out of a letter written by Gibbs‘ in 1899 is of interest. “Again, the consideration of the rlectrical potential in the electrolyte, and especially the consideration of the difference of potential in electrolyte and electrode, involve the consideration of quantities of which we have no apparent means of physical measurement, while the difference of potential in pieces of metal of the same kind attached to the electrodes is exactly one of the things which we can and do measure.” The probleni of measuring the electric potential difference, as defined above, between t v o phases thus resolves itself into a determination of the distribution of the electric charge density in the two phases. Suppose, for “‘The Collected FVorks of J. \Illlard Gibbs,” 429 (1928)
ELECTRICAL POTENTIAL DIFFERENCE BETWEEN TWO PHASES
I543
example, we wish to determine the electric potential difference between copper and zinc a t equilibrium as regards electrons in the absence of an external electric field. The simplest distribution of electric density will be obtained by assuming the zinc and copper to be spherical masses at a considerable distance apart. To each metallic sphere can be connected a long thin metallic wire of the same metal and these two wires can somewhere be joined. There will then be equilibrium throughout as regards electrons and the connecting wires can then be removed without disturbing the equilibrium. The charges remaining on each metal will then be distributed practically with spherical symmetry over its surface. The problem is thus reduced to determining the total charge on each sphere, the electric potential then being obtained by dividing the total charge on each sphere by the radius of the sphere. The experimental difficulties of carrying out such an experiment 154th sufficient accuracy to give a positive result, will not be discussed here. It seems unlikely that the problem can be reduced to a simpler form than this. Accepting then, Bronsted's conventional definition, we see that the electric pot.entia1 difference between two phases in equilibrium is a measure of the asymmetrical distribution of electric charge between the two phases. KO measurements of E.M.F., partition coefficients of electrolytes, diffusion rates, reaction rates or the like, can give any information about the electric potential difference between two phases or the chemical potential of a single ion. The electric potential difference is in fact of no significance for such phenomena.' It is a conception that is concerned purely with the distribution of the electric charge and may possibly be determinable by some experiment, such as that outlined above, involving the direct measurement of the electric charge on a phase. It need hardly be said that no experiment of this nature has yet been performed and we therefore have no knowledge of the value of the elect.ric potential between any pair of phases, nor therefore of the chemical potential, the activity or the activity coefficient of any individual ion. The author has profited by discussions of the question here treated with Professor Bronsted, Dr. Onsager and Mr. Giintelberg, to all of whom he is grateful. It must however not be presumed that they are necessarily in entire agreement with the author on all points. The author is also indebted to the R a s k - h t e d Foundation for a research grant. The Royal Agricultural College, Copenhagen, October 23, 1925.
' FT-agner in an interesting paper on the thermodynamic treatment of non-isothermal systems (Ann. Phys., 3, 629 (1929)) uses a definition of electric potential differing slightly from Bronsted's when the medium contains electrically polar molecules. The essential part of the present conclusions applies equally to Bronsted's and Wagner's definitions. Wagner's formulae for non-isothermal systems can all be rewritten in terms of electrochemical potentials and so confirm t h a t no measurements of the Soret effect or of thermoelectric cells can give any information about the electric potential difference between two phases.
