CHEMICAL, POTENTIAL AND ELECTROMOTIVE: FORCE' BE' WILDER D. BANCROFT
In the paper by Gibbs on the EpuiZibrium of Heterogeneous Substances there are a few pages on the modification of the conditions of equilibrium by electromotive force, in which the electromotive force of certain cells is expressed in terms of the chemical potentials of the cation or anion at the two electrodes. The results thus obtained have received little attention, and the object of this paper is to point out the relation between the Gibbs formulas and those deduced within the past few years by Nernst, Ostwald, and others. After showing that the conditions of equilibrium, which had been found without reference to electrical considerations, will apply to an electrolytic fluid and its independently variable components, Gibbs proceeds to seek the remaining conditions of equilibrium, which relate to the possibility of electrolytic conduction.* '(For simplicity, we shall suppose that the fluid is without internal surfaces of discontinuity (and therefore homogeneous except so far as it may be slightly affected by gravity), and that it meets metallic conductors (electrodes) in different parts of its surface, being otherwise bounded by non-conductors. T h e only electrical currents which it is necessary to consider are those which enter the electrolyte at one electrode and leave it at the other. The first draft of this paper was written early in 18gg and was submitted to Professor Gibbs, who made a number of suggestions. I have withheld the paper from publication in the vain hope that Professor Gibbs would himself some day show the relation between his formula and the osmotic pressure theory of voltaic cell. The paper as published is based on the first draft and on Professor Gibbs's letter, chiefly on the latter. The original outline was mine ; but the bulk of the paper is a more or less literal transcript from Professor Gibbs's letter. Trans, Conn. Acad. 3, 5 0 2 (1878).
&emz’cal Potential an8 Electromotive Force
419
“If all the conditions of equilibrium are fulfilled in a given state of the system, except those which relate to changes involving a flux of electricity, and we imagine the state of the system to be varied by the passage from one electrode to another of the quantity of electricity ae accompanied by the quantity am, of the component specified, without any flux of the other coniponents or any variation in the total entropy, the total variation of energy in the system will be represented by the expression, (V --’)at+ ($la$,)a~,+ (Y1--Y”))a~tza,
in which V’, V”, denote the electrical potentials in pieces of the same kind of metal connected with the two electrodes, Y’, Y”, the gravitational potential at the two electrodes, and p’,, the intrinsic potentials for the substance specified. The first term represents the increment of the potential energy of electricity, the second the increment of the intrinsic energy of the ponderable matter, and the third the increment of the energy due to gravitation.’ But by (682) am, = a,ae. It is therefore necessary for eqnilibrium that (V” - V’)
+ aa(p” - p’ - Y” + Y”) =
0.
(684)
((When the effect of gravity may be neglected, and there are but two electrodes as in a galvanic or electrolytic cell, we have for any cation V” - V‘ = a a ( p t a- p l f a ) ,
(687)
V” .- V’ = a,(p“# - P I p ) ,
(688)
and for any anion where V” - V’ denotes the electromotive force of the combination. That is : When all the conditions of equilibrium are fii$lled i n a galvanic o r electrolytic cell, the electromotive force is equal to the dzference i n the values for thepotential for any ion o r apparent ion at the surfaces of the electrode mult$lied by the electYoI t is here supposed that the gravitational potential may be regarded as constant for each electrode. When this is not the case, the expression may be applied to small parts of the electrodes taken separately.
418
Wilder D.Bancmft
chemical equivalent of that ion, the greaterpotential of an anion being at the same electrode as thegreater eZectricaZpotentiaZ and the reveyse being true of any cation.” The limitation that there shall be only one solution makes the forinulas (687) and (688) apply in their present shape only to cells with concentration electrodes, in other words, to amalgam or gas cells. T o apply the formulas deduced by Gibbs, we must be able to evaluate the (intrinsic or chemical) potentials involved. If we are considering the case of a dilute solution and if we make the assumption that Avogadro’s law holds for the solute, we may write
where y, is the density of a component (in this case, massof the solute divided by the volume of the electrode), M a its molecular weight, A the constant of Avogadro’s law
t),
p v ---
and B a
quantity which depends on the solvent and the solutions, as well as the temperature, but which may be regarded as independent of ya so long as this is small, and which is practically independent of the pressure in ordinary cases. We may avoid hedging in regard to B by using the differential equation (2). We may simply say that this equation holds for changes produced by varying the quantity of ( a ) when ya is small. It is not limited to changes in which ‘I’ is constant, for the change in pu due to ‘I’ appearing in (I) (both explicitly and implicitly in B) becomes negligible when multiplied by the small quantity vu. We can then write )
f a and ylfa being the densities supposed small of the cation ( a ) in the two electrodes, which are supposed identical, except for the dissolved (a). Here a , has reference to the solution and M a to
Chemical Potentia2 and Electromotive Force
419
the electrode. I t may be more convenient to divide a, into the factors E,a,, where a, is the weight of hydrogen, which carries the unit of electricity, and E, the weight of ( a ) which carries the same quantity of electricity as the unit of weight of hydrogen. In other words, E, is Faraday's ' electrochemical equivalent ' and a, is Maxwell's ' electrochemical equivalent '. This gives
now y f a and yf',, as defined, are volume concentrations, and
E, R aHA-=Ma Fnp,' where R is the gas constant, F the number of coulombs per equivalent weight, nI the degree of polymerization of the cation in the electrode (assumed to be the same in both electrodes), and nz the valency of the cation in the solution. We may therefore write
which is the equation for concentration electrodes as deduced on the basis of the osmotic pressure theory of the voltaic cell.' This formula, as deduced, involves the assumption that the gas law holds. If this is not the case, either that molecular weight must be used for which the solute follows the gas law, the apparent molecular weight, or a correction must be introduced as was done by Cady when he took the heat of dilution into account.* Gibbs worked out the formula for the electromotive force only for the one case of concentration cells; but there were several reasons why the matter of electrolysis was not discussed V. Turin. Zeit. phys. Chem. 5,340 (1890); 7, 221 (1891); Meyer. Wied. Ann. 40, 244 (1890) ; Zeit. phys. Chem. 7, 477 (1891) ; Schaller. Zeit. Elektrochemie, 5, 259 (1898) ; Richards and Lewis. Proc. Am. Acad. 34, 87 (1898) ; Cady. Jour. Phys. Chem. 2, 551 (1898); Bancroft. Ibid. 3, 107 (1899). Jour. Phys. Chem. 2, 561 (1898).
Wizaer D.Bancrdt
420
more fully in the paper on the Equilibrium of Heterogeneous In the first place, cases of true equilibrium (even Systems. for open circuit) are quite exceptional. Thus the simple case of unequal concentration of the electrolyte cannot be one of equilibrium since the process of diffusion cannot be stopped. Cases in which equilibrium does not subsist were formally excluded and indeed could not be satisfactorily treated without the introduction of new ideas quite foreign to those necessary for the treatment of equilibrium. Again the consideration of electrical potential in the electrolyte, and especially the consideration of difference of potential in electrolyte and electrode involves 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. Nevertheless, with some hedging in regard to the definition of the electrical potential, we may apply V" - V' =;an(/Lf,-/.Pa) ((
((
((
to points in electrolyte (') and electrode (").
This gives
or
This is the Nernst formula for the potential difference between a metal and a solution of one of its salts, Gibbs using log G where Nernst uses log P. This method of deducing this formula has perhaps one advantage over that adopted by Nernst. I t brings out the fact that the so called solution pressure P is a function of at least three variables, the nature of the metal electrode, the nature of the solvent, and the temperature. The specific effect of the solvent, if any, is contained in the term B. Since we know that the chemical potential of a dissolved substance varies with the solvent,I it is reasonably certain a priori 1
Miller.
Jour. Phys. Chem.
I,
636 (1897).
'
Chemical Potential and Electromotive Force
421
that B does not have the same value for different solvents.' This differs from the case of the cells with concentration electrodes, in which no terms for the solvent appears, and in which, experimentally, the nature of the solvent is immaterial. Ostwald' has shown that the Nernst formula applies even when an insoluble or complex salt is formed, but that, with such a system as Ag I AgCl KC1, it is easier to measure the concentration of chlorine as ion than it is to measure that of silver as ion. T h e relation between the concentration of chlorine as ion and silver as ion, which was deduced by Ostwald, can also be obtained from the Gihbs formulas by making the same assumptions. For the case of silver chloride dissociated into silver as ion and chlorine as ion we have the three equations
The three potentials are also connected by the relation M,Pl+ M2P2 = MBPLB.
If we postulate that the potential of undissociated chloride is constant when there is solid silver chloride present, we have
Substituting in equation (4)we have S - - M,B, AT making
Since this was first written the experiments of Kahlenberg have demonstrated this fact conclusively. a Lehrbuch, 2, I., 877.
and remembering that MI = Ma! we have
(‘The case of
unequal concentration, or in general cases in which the electrolyte is not homogeneous, can be treated as follows : Supposing, for convenience, that the cell is in the form of a rectangular parallelopiped with edge parallel to axis of x and cross-section of unit area. The electrolyte is supposed homogeneous in planes parallel to the ends, which are formed by the electrodes. Of course we should get equilibrium if proper forces could be applied to prevent the migration of the ions, and also of the part of the solution. which is not dissociated. What would these forces be ? For the moleculesz (12) which are not dissociated, the force per unit of mass would be 12. dX
(The problem is prac-
tically the same as that discussed on pp. 203A E. H, S.) If the unit of mass of the cation has the charge cI, the force necessary to prevent its migration would be
For an anion
(2)
the force would be
Now we may suppose that the same ion in different parts of a dilute solution will have velocities proportional to the forces which would be required to prevent its motion. We may therefore write for the velocity of the cation (I)
Since the rest of the paper, with the exception of the last paragraph, is practically a single quotation, it has seemed simpler t o omit the quotation marks. I n the succeeding pages the cation is referred to as ( I ) , the anion as ( 2 ) , and the undissociated molecule as (12).
Chemicad Potentiad and EZectromolive Fowe
423
and for the flux of the cation (I)
and for the flux of the anion (2) & = - A yk
cz
a
( dP~ - c 2 ~ ) = - x! A A dy T + ~k z -dV y z , x dx c,M, dx dx
(6)
where K I , k, are constants (migration velocities) depending on the solvent, the temperature, and the ion. Now whatever the number of ions the flux of electricity is given by the equation
+ *
4 c c141, where the upper sign is for cations and the lower for anions, and the summation for all ions. This gives k, d Y1 - dV + = ATZ 7 MI dx dx
-
clklyl.
That is,
The form of this equation shows that since $ is the current, dx
Ec,k,y,
is the resistance of an elementary slice of the cell, and )
the next term the (internal) electromotive force of that slice. Integrating from one point to another in the electrolyte
The evaluation of these integrals which denote the resistance and electromotive force for a finite part of the electrolyte depends on the distribution of the ions in the cell. For one salt with varying concentration
424
Wilder D.Bancroft
or, since,
The resistance depends on the concentration throughout the part of the cell considered, but the electromotive force depends only on the concentration at the terminal points (' and 'I), For cTMI and c2M2we may write
V
and
V ?,
where v r and v? are the
va-
aH
aH
lencies ' of the molecules.
This gives
_ kl - _ kz V" - - 17'
a,AT
4 + kz
1'
2'
log
5 for 4 ==o (circuit open). Y l
Ya
(7)
When the two ions have the same valency this is identical with Nernst's equation' for the potential difference between two concentrations of the same electrolyte, the kI and k* of the Gibbs equation being the u and v of the Nernst formula. The general formula for the potential difference between two concentrations of the same electrolyte has not hitherto been obtained, Planck's formulation being limited explicitly to cases in which the cation and anion have the same valence,2 while Loven3 considers the simplified case in which there are two different electrolytes having the same osmotic pressure. The case in which there are two electrolytes and no concentration difference is a less simple problem. We may regard 1
Zeit. phys. Chem. 4, 137 (1889). Wied. Atin. 40, 576 (1890). Zeit. phys. Chem. ao, 593 (1896).
Chemical Potential aizd Electromotive Force
425
it as relating to a tube connecting two great reservoirs filled with different electrolytes of the same concentration, i. e., cocoy; = zoc,y;'.
We use (,) for any ion, (,) for any cation, (,) for any anion. The tube is supposed to have reached a stationary state and dissociation is complete. The number of ions is immaterial, but they all must have the same valency v. Now by Equations ( 5 ) and (6), since coMo= ,.iv
or, writing N for the constant
aAT
__ V
,
The first equation makes d20coyo - constant throughout the tube, dx
and since Z o c o y ~= ' Xocoyo, Gco.yo must be constant throughout dV the tube. The second equation thus makes 2; constant, throughout the tube.
Let X = - dV -
Our original equation is
dx'
+,=-Nk
d 2 . 3= Xk0yo. dx
Now with X constant, this is easily integrated.
4 0 --E
dvo fyo,
X dx
Xk
*--N x dx yorXK,' dY0
__40
yor
XK,
40
X dx, = fN
Wilder D.Bancroft
426
=fX x+
log ( y q = & )
log H,,
N
0
Yo=F---”,(J $0 Xkll
* -XNx .
T o determine Howe have y,”-y,’=Ho(a
fX - x”
N
* XN
-0
-XI).
If we put the origin of coordinates in the middle of the tube, we X - x”, have x’ = - x”. Let. P = crN yO”-yox’ = fH,(P - P-I).
Let A, =yo” -yo’,
The condition of no electric current gives X
= ‘co
~ O ~ O ~ O Y O
*
f - X
a N coho% p - p-
I
’
Apply to both ends and add
If we set, to abridge
4’ = e l C l ~ l Y 1 ’ kzx‘= e,C&zY,’
k,” = x,clklyl” k,” = ~,c,k,y,”
where the summations are for cations or anions separal‘edy, the last equation may be written
c
Chemical PotentiaZ and Eleclromotive Fovce k,”
427
+ P-1 (k,” - k,’ - K,” + K,’) , + k,” + k,’ + k,’ = P __ I
which gives p’ = k,” ~
K,’
-+ k,’ + K,”’
k1’is the part of the conductivity of the first electrolyte which is due to the cations. If the first electrolyte contains only one cation (I) and one anion (2) and the second only one cation (3) and one anion (4) we have
or since c,yl’ = c,y,” = c,y,” = c4y;1
which is the Planck formu1a.I It is thus clear that the Gibbs conception of the electromotive force as a measure of the chemical potential leads to the results actually discovered by Nernst and Planck. While the original equation of Gibbs is absolutely accurate, this is not the case for Equations (1-8) which contain the same explicit assumptions which were made first by Nernst and afterwards by Planck. Cornell Universily. Wied. Ann. 40, 574 (1890).
