GALVANIC
AND
S. W. GRINNELL*
Department of Chemistry, Stanford University, California Received June 27, 1999 I. INTRODUCTION
Objects Our object in this paper is to obtain in general and exact form the thermodynamic laws of galvanic cells subject to external gravitational fields. In attaining this object we shall make use of a therniodynamic 1 The contents of this paper are taken largely from the dissertation submitted by S. W. Grinnell to the Faculty of Stanford University in partial fulfillment of the requirements for the degree of Doctor of Philosophy, 1938. * Shell Research Fellow, Stanford University, 1937-38.
464
F. 0 . KOENIG AND 8. W. GRINNELL
theorem deduced recently by one of us (8) and called the “condition for quasi-reversible conduction” or “qrc. condition.” We shall accordingly use the terminology and mathematical notation given previously (8) throughout the present paper. This paper is, however, not on1y.a sequel to the earlier paper (8) but is also intended as a preparation for a later paper reporting experiments on “gravity cells” that have been in progress at Stanford University for a number of years. Relation to previous work Effects of gravity upon E.M.F. have been previously investigated both experimentally and theoretically by Des Coudres (1, 3) and by Tolman (112. A brief review of the experimental work of these authors has recently been given by MacInnes (9). The present study is an attempt to generalize and purify the theoretical deductions of Des Coudres and Tolman. Details concerning the work of these authors in relation to the present study are given in section IV below. 11. THE GENERAL CASE
Description of the cell We shall consider a cell in which both the gravitational potential Q and the mole fractions Ni may vary throughout the solution in any way whatever compatible with instantaneous fulfillment of ( i ) the conditions of hydrostatic and thermal equilibrium: grad P = - p grad
(1) grad T = 0 (2) and (ii) the conditions of electrochemical equilibrium between the electrodes (I, a’and the adjacent layers of solution @,8’ (cf. 8, equations 12, 15) :
c
VrP!
=
c
v,p;
+
‘p
Pi1-
(3)
We shall furthermore, as in reference 8, assume electrodes and solution to consist of mutually insoluble materials. Finally we shall suppose that the electrodes are relatively thin and so arranged that Q and therefore P may without appreciable error be regarded as constant throughout each electrode. This allows us to write without appreciable error = pQ’ =
cp@
=8
J’ =
(5)
-1
(6)
pp = Ij pa’ = pe’ = p
(7)
pa
=i
cp
(8)
’
465
GALVANIC CELLS SUBJECT TO FIELDS OF FORCE
Formulae for the potentials of the components in a gravitational jield
For the purposes in hand we need the following general formulae: For the potentials pal and p n t of the electrolyte and neutral components, respectively, of a subphase in a gravitational field it is true that =
Tel
f Md‘p
Pnt =
Tnt
+
pel
MntP
(9)
(10)
nhere ~ , 1 , rnt are functions of P , T , and composition independent of ‘p, and M,l , M n 1are the molar weights (first pointed out by Gibbs (4); for a simple proof see reference 7 ) . If, as will be assumed throughout, the electrolyte components in a given subphase are a t each instant in dissociativr equilibrium with the charged components, it is evidently always correct to write for the potential of any charged or neutral component i L(t
=
+ Mt‘p +
7%
Z,F$
(11)
where 7 %is a function of E’, T , and composition independent of ‘p, and IC. is a function of the electrochemical state3 of the subphase called the electric potential. R e note that for neutral components the term in IC. disappears because z , is zero. It is often convenient to expand T , in the form 7 %= T:
+ RT log N , + RT logf,
(12)
where 7 ; is a fuiAion only of P and T , N , is the mole fraction of i, and f, is a function of P , T , and composition called the activity coefficient. The potential p , may then be written as pb =
+ RT log N , + RT logf, + M,(o + z,F$
(13)
The function $ is thermodynamically indeterminate and so are, for charged components, the functions T, and f $ . From equations 11 and 13 the following theorems, needt.d later, are readily evident. If A, are a set of scalars such that
,
zixi
=0
then, although for such of the components i as are charged the pi are functions of the electrochemical state, it is nevertheless true that Xipi i
= a function of the chemical state and the Xi
(15)
a For the exact definition of “electrochemical state” and “chemical state” in the sense used in this paper, see reference 8, equations 7, 8.
466
F. 0 . KOENIG AND S. W. GRINNELL
Furthermore, although for such of the components i as are charged the 7 , and f L are thermodynamically indeterminate functions of P, T , and composition, it is nevertheless true that AL7,
= a thermodynamically determinate function of
P, T,
I
c
and composition and the A,
(16)
A, log j , = a thermodynamically determinate function of P , T ,
i
and composition and the X,
(17)
Equations 11 to 17 are the obvious extensions to the case of systems in a gravitational field of the considerations of Guggenheim ( 5 ) concerning electrochemical potentials and the electric potential of a phase interior.
Calculation of the
E.M.F.
as a junction o j chemical state and conjiguration
The E.M.F. may now be calculated by the same method as that leading to equations 106 and 109 in reference 8. The E.M.F. is always given by
By virtue of the equilibrium conditions, equations 3 and 4, this becomes
Recalling from reference 8 that the subscripts q, q' refer to electrolyte and neutral components, we note that the sums v,!&' are funcv , ~ :and '1
9
tions of the chemical states a t CY and a', respectively. The subscripts r, r', however, refer to charged and neutral components with (cf. reference 8, equations 10, 14) &VI = -1 (20) I
and therefore the sums
Y+!
v7pp!: are functions of the
and
electro-
I
chemical states a t /3 and p', respectively. We can nevertheless express the diflerence vr,pfl: (22) I' ,Vri
c
c
as a function of the chemical state a t every point of the solution, and of the configuration. For this purpose we have recourse to reference 8, equation 104
GALVANIC CELLS SUBJECT TO FIELDS OF FORCE
467
the deduction of which by means of Gauss's theorem and the definition of the specific electrolytic current vector J (8, equations 36 to 38) applies unchanged to the present case. We obtain for the difference 22 4
Kow from equations 20,21 we have
-
~rtvrt I'
zrvr
=
r
o
and therefore equations 14 and 15 show that the difference
C V+P!. - C v r d r
(26)
I'
which occurs on the right side of equation 24, is a function of the chemical state a t fl. It remains to show that the integral
is a function of chemical state and configuration. For this purpose we introduce the qrc. condition in the form of equation 73, in reference 8: namely,
and are therefore enabled to write
--.
Since: (i) the vector J a t any point is fixed by the chemical state a t all points of the solution and by the configuration (8, auxiliary assumption 5), (ii) the transport numbers t i a t any point are functions of the pressure, the temperature, and the composition a t that point (8, auxiliary assumption 4), (iii) according to reference 8, equation 87, we have 1 i
Zj
so that, by equation 21,
it is evident from equations 14 and 15 that the integral on the right of equation 29 is a function of the chemical state a t every point of the solution
468
F. 0. KOENIG AND S. W. GRINNELL
and of the configuration. From equations 19, 24, and 29 we therefore obtain for the E.M.F. as a function of chemical state and configuration
+ J, j - ( F
+ C $ grad
grad fir'
vr'
i
pi)
Zi
SV]
(32)
which corresponds to equation 106 in reference 8. As an alternative to equation 23 we can of course use
which yields for the E.M.F.
E""'=
1[
v4p;
-
v,.p::
4'
+
-
vr,p::
r'
C vrp!' ).
+ / j.(cvr grad + C !grad fir
I
ZI
pi)
SV]
(34)
corresponding to equation 109 in reference 8. In the special case that the solution is simply connccted and chemically lamellar we use instead of equations 23 and 33 the equations
C v+p{:
=
t'
Cvrd
r'
+ /, 5:v+ grad firl.6Z
vr~p?~
C vrp!'
=
S,
-
vI grad pLr.SZ
(35) (36)
where C is any path through the solution from 6 to p'. The corresponding form of the qrc. condition is that of equation 77 in reference 8: namely, :grad
(37)
p,.62 = 0
For the E.M.F. we thus obtain instead of equations 32 and 34 the simpler equations
E""' =
;[F
vqp; -
c
VP'p::
4'
+ E""' =
1[F
v,p;
-
+c
(F
& v,fp;: + 9
vr,p!,
-
7'
vr'
grad pr' v+p!:
I'
c
+
vrp!
grad p.) .Sa]
-
(38)
vVp!" ).
which correspond to cquations 132 and 133, respectively, in reference 8.
469
GALVANIC CELLS SUBJECT TO FIELDS O F FORCE
l'he special case grad cp = 0 In this case cp and P are both constant throughout the cell. We should therefore expect the general formulae obtained above to reduce to formulae for cells with liquid-liquid junctions of the type obtained in reference 8. In order to show this we write for the electrolyte and neutral components q, q', in accordance with equations 9, 10, P9
=
79
Pq'
=
79'
+
(40)
Mqcp
+ Mg5c
(41)
and for the charged and neutral components T , T ' , i, in accordance with equation 11 P, = rr M,rp z,F+ (42) Pr'
=
71'
+ + + M,,rp + z+F+ + MiP + ziF+
(43)
(44) On substitution of equations 40 to 44 into equation 34 the terms in J. pi =
Ti
cancel because of equations 25 and 31, and we obtain, since 'p is constant,
r
7
Conservation of mass in the electrode processes requires that
C v q M g9
+
v , ~ M g ~C v+Mr* 9'
I'
C vrMr = 0
(46)
I
and therefore the term in cp disappears from equation 45. Furthermore, for given electrode materials the 7," are in general functions of P and T, and the r,9' functions of F' and T . When, as is here the case, P is constant throughout, we may therefore set
Equation 45 accordingly reduces to
470
F. 0. KOENIG AND S. W. GRINNELL
in which the terms in T?, T+, 7 %evidently constitute a thermodynamically determinate function of the pressure and temperature of the system, the composition of the solution at every point, and the configuration. Equation 48 is therefore a general formula for a cell with liquid-liquid junction uncomplicated by gravitational effects. That it is equivalent to the equations obtained in reference 8 for the case in which not only grad cp but also cp itself is zero is readily verified by substituting equations 40 to 44 without the terms in cp into equations 106 and 85 of reference 8. We point out that the deduction of equation 48 constitutes an exact theoretical proof of the intuitively obvious fact that in the absence of a aravitational field the E.M.F. of any galvanic cell becomes independent of the gravitational potential. I t is of interest that in this proof the law of the conservation of mass plays a part very similar tc? that which it plays in the proof that the equilibrium constant is independent of the gravitational potential (cf. 6).
The electric potmtial difference produced in a solution by a gravitational field In order to combat misunderstanding of a type still prevalent, we shall calculate as a function of chemical state and configuration the difference of electric potential in the solution, J” - @’. Since grad T is zero, we obtain from equation 13 grad pi =
r$)
grad P
+ RT grad log Nifi + Mi grad cp + z,F grad +b
(49)
We may set
and regard Vp as the partial molar volume of i a t infinite dilution. By virtue of equations 1 and 50 equation 49 becomes grad pi = RT grad log Nifi
+ (Mi - V p p ) grad cp + ziF grad IC. (51)
Substituting equation 51 into the qrc. condition in the form of equation 28 and taking account of equation 30 we obtain
+
~ ? . ~ i $ [Z iR T g r a d l o g N i f i
(Mi
- Vpp) gradcplGV
+ F / ?.grad
$dV = 0 (52)
According to the deduction of equation 117 in reference 8, we have
GALVANIC CELLS SCBJECT TO FIELDS OF FORCE
47 1
Equations 52 and 53 give for the electric potential difference in question
@ - @‘ = 1
1 2. v
i
2 [RT grad log
Ni f i
Zi
+ (Mi- VPp) grad q]6V
(54)
If the solution is simply connected and chemically lamellar we use equations 51 and 37 and obtain instead of equation 54 the simpler equation
When grad 9 is zero, equations 54 and 55 reduce respectively to equations 119 and 135 of reference 8 for the potential difference of liquid-liquid junctions. It is thus evident from the manner in which the f, occur in equations 54 and 55 that the potential difference in a solution subject to a gravitational field is thermodynamically indeterminate, just as is the potential difference due to a liquid-liquid juiiction. This is, moreover, true even if the composition of the solution as measured by the mole fractions is constant throughout, for in that case t h e f t still vary throughout the solution because of the gradient of pressure. When the grad N , are zero, equations 54 and 55 may be put into more compact form as follows: We define the partial molar volume V , by the equation
where the subscript “comp.” denotes constancy of the mole fractions. We note that for charged components the V ,so defined are thermodynamically indeterminate. When grad N , = 0 we have, from equations 12, 50, 56, and 1,
=
1
RT
ZF
-
grad P
T,comp.
Substitution of equation 57 into equations 54 and 55 yields for the case grad N i = 0 ,b8 -
@‘
= 1
1 Y
Its - $@‘
=
J.
i
/ 4f-
5 ( M i - V i p ) grad ppsV Zi
1 (Mi - Vip) grad p.62 F c i z i
(59)
472
F. 0. KOENIG AND 9. W. GRINNELL
These expressions are thermodynamically indeterminate, owing to the Vi. We conclude that the electric potential difference produced in a solution by a gravitational field is a concept just as meaningless experimentally as is the electric potential difference due to a liquid-liquid junction. 111. SOME SPECIAL CASES OF EXPERIMENTAL INTEREST
Two restrictions and their consequence The cases of greatest experimental interest are evidently included within the following two restrictions: (i)the solution is simply connected and chemically lamellar; (ii)the two electrodes consist of the same materials. Owing to restriction (i) we need consider only equations 38 and 39. Since restriction (ii)means that the components q and q' are identical, and likewise the components r and f ' , the equations in question both reduce to
In order to apply this equation to the special cases described below it is expedient to transform it as follows: We substitute equation 40 and obtain
where Acp is defined as Ap = 8'
-8
Since we have assumed electrodes and solution to be mutually insoluble, we may take the total number of moles, n, of any component q of an electrode to be independent of the pressure of the electrode. We therefore have 7:'
- 7;
=
7,(P', T,n,)
=
r'(2)
- 7*(P,T , n,) dP
Tons
where V , is the partial molar volume of q in the electrode and is of course thermodynamically determinate, because the p are electrolyte or neutral
GALVANIC CELLS SUBJECT TO FIELDS O F FORCE
473
components. Substitution of equation 63 into equation 61 gives the desired result.
Three special cases The following three special cases appear to be of experimental interest: Case 1: the gradients of the mole fractions in the solution are zero. Case 2: the solution is in sedimentation equilibrium. Case 3: a cell with solution originally in sedimentation equilibrium has been rapidly transported to a region in which grad (o is zero, the temperature of the cell and the volume and shape of the solution having been kept constant. Case 1 is that of the "gravity cells" that have been studied by Des Coudres and by Tolman; it is, moreover, the only case which has been experimentally realized hitherto. Opposing the experimental realization of case 2 and therefore of case 3 are serious practical difficulties, especially the prevention of convection and mixing over the long periods of time required for sedimentation under available gravitational or centrifugal fields. Theory of case 1 I
From equation 11 we have grad
pi
= grad
+ Mi grad + ziF grad @
~i
(p
(65)
Since grad T and the grad Ni are zero, we have, from equation 56, grad ri = Vi grad P From equations 65, 66, and 1, we obtain
+ ZiP grad $
(67)
gradP+z,Fgrad@
(68)
grad P and therefore, of course,
Substituting equations 67 and 68 into the integral in equation 64 and noting that the terms in @ cancel because of equations 20 and 30, we obtain
474
F. 0. KOENIG AND S. W. GRINNELL
Since at constant I' and Ni the quantities Vr, Vi, ti, and p are functions only of P , the line integral on the right becomes a definite integral between the limits P and P', so that
=
l'[c (+ vr
- V.)
+ zi2 I
(5
- Vi)] d P (70)
Substitution of equation 70 into equation 64 gives
From equations 20, 30, 14, 16, and 56 it is evident that the linear combination of the functions VT, V i in equations 69, 70, and 71 is thermodynamically determinate. Equation 71 is the exact law for case 1. The integral in equation 71 can in general be evaluated only approximately. A convenient approximation consists in regarding V,, Vi, V,, and t,, a t constant T and Ni, as independent of P. This permits us to replace V, , Vi,V?, t,, and also p by their values a t the same temperature and composition but zero pressure, which will be indicated by the superscript *. Equation 71 then integrates to
E""'
=
-A{?
v,M,A(o
+ [Tv,V,*
where A P is defined as Ap =
P' - p
(73)
An equation more suitable than equation 72 for comparison with experiment is obtained if AP is eliminated in favor of Ap. Since the solution is regarded as incompressible, the relation between A P and Ap is (74)
AP = -p*Ap
The desired equation is therefore
-
i t* (Mi - P* v
q
(75)
475
GALVANIC CELLS SCBJECT TO FIELDS OF FORCE
We shall apply equation 75 to three special types of electrode equilibrium to be known as A, B, and C, respectively. Electrode equilibrium -4 is that of the system Me I solution of a single salt of Me in a non-electrolyte solvent The chemical equation for this equilibrium, written according to equation 9 in reference 8, is
where Me"' is the formula of the ion of the metal Me, and z+ is the valence of this ion. Equation 75 then yields, by a serics of obvious transformations, E""' = A 2 [MMe- p* VGe ( M , - p* V : ) ] (77) z+ F b+ where t? is the Hittorf number a t zcro pressurc of the solute anion, s refers to the solutc salt, and b+ is thc number of moles of Me"' per mole of salt. Electrode equilibrium H is that of thc system Ag I AgCl I solution of a single chloride in a non-electrolyte solvent The chemical equation for this equilibrium is to be written in the form C1-
=
AgCl
- Ag
+ El-
(78)
Equation 75 then yields, by a series of obvious transformations, E""' =
-9 [Ma F
- p*(VITgcl - VZJ
-
z+
(M, -
p*
V:)]
(79)
where t: and z+ are, respectively, the Hittorf number a t zero pressure and the valence of the solute cation, and s refers to the solute salt. Electrode equilibrium C' is that of the system
Pt I solution of iodine and an excess of a single iodide in a non-electrolyte solvent In this case we may describe the electrochemically active components by two alternative sets of chemical formulae: namely, (i) I-, I, El-, or (ii)I-, 1 2 , El-. The corresponding chemical equations for the electrode equilibrium are in description (i)
? 1- 2
1 ; = El-
and in descript,ion (ii)
I- - Iz = El-
476
F. 0. KOENIG AND 6. W. GRINNELL
Equation 75 yields, by a series of obvious transformations, for description
(i)
and for description (ii)
in which and z+ are, respectively, the Hittorf number a t zero pressure and the valence of the solute cation, and the subscripts s and s' refer, respectively, to the iodide and triiodide salt. If the excess of iodide over triiodide (or iodine) is sufficiently great, t:F (or t:,) may be neglected and equations 82 and 83 become respectively
Inspection of the two descriptions readily shows the transformation connecting equations 82, 84 with equations 83, 85 to be =
t;T.(i:)
$Ji)=
$3")
t:(d)
t:;i)
=
- tI,
*(ii)
*(ii)
(86) (87) (88)
t1,
Mal - M, = z+MI,
(89)
v:: - v: = z+v::
(90)
Theory of case 2 This case is characterized by the equilibrium conditions grad
fii =
0
(91)
grad
fir =
0
(92)
Equation 64 therefore reduces to
which is the exact law for case 2.
GALVANIC CELLS SUBJECT TO FIELO8 O F FORCE
477
Taking V , to be independent of P , we may integrate equation 93 and obtain the approximate formula
c
1 Y,(M,AQ F ,
E""' = --
+ V:AP)
(94)
where the asterisk as before indicates zero pressure. In order to eliminate AP from equation 94 we write, in accord with equation 1,
AP =
-I
pgrad p.67
= -[plAcp (95) where [ p ] is in general merely a suitable mean value of the density of the solution. It is readily shown that for the only two cases of practical interest,-namely, (i) a cylindrical cell with the electrodes a t the ends, in a uniform gravitational field, and (ii) an annularly sectorial cell with the electrodes a t the ends, in a centrifugal field,-the quantity [ p ] is exactly given by M [PI = (96) V
where M is the total mass and V the total volume of the solution in sedimentation equilibrium. From equations 94 and 95 we obtain E""' =
-2 c v p ( M p- [p]V:) p
(97)
q
For electrode equilibrium A, equation 97 gives
For electrode equilibrium B, equation 97 gives
For electrode equilibrium C we obtain without approximations frGm the exact equation 93, since components q are absent, Baal
=o
In this case equation 64
rp
(100)
Theory of case 3 and P are constant throughout, so that we obtain from ,
478
F. 0. KOENIG AND 8. W. GRINNELL
and from equation 13
+ ziF grad J. grad pr = RT grad log N,f, + z,F grad J. grad
pi =
RT grad log N i f i
(102) (103)
On substitution of equations 102, 103 into equation 101 the terms in J. cancel and we obtain
E""' =
71(c
V?
grad log N,f,
+ 5 grad log Nifi ).6f (104) i
Zi
In order to relate this equation to the sedimentation equilibrium from which the cell is derived, we mark quantities referring to this sedimentation equilibrium with the superscript e and deduce from equations 91, 92, and 51
+ (Mi- p e Vpe) grad qe + z i F grad RT grad log N:fl + ( M , - p e Ville)grad qe + z,F grad whence we obtain, since the terms in +' cancel, RT grad log N:fl
+e
=
0 (105)
= 0 (106)
Subtraction of equation 107 from equation 104 gives
E""' = KT F
IC(T
vr
grad log
9 + Nrfl
i
t . grad log ,).63 Nifi zi N&
+
t v I ( M , - p e V",) [ M , - p e Vpe] grad p e e62 (108) FJc 2, This is the exact law for case 3. For an approximate evaluation of the integrals in equation 108, it is convenient to regard V,, V , as independent not only of pressure but also of composition. According to this approximation we have
-1
T
L
v, = V ; = v;e = V;*
(109)
where Vp* is the partial molar volume a t infinite dilution and zero pressure. The approximation furthermore permits us to regard the solution as incompressible and therefore to neglect the elastic displacement within the solution on passing from sedimentation equilibrium to the state grad p = 0. This means that a t each point of the solution
N: = Ni
(110)
GALVANIC CELLS SUBJECT TO FIELDS OF FORCE
479
Finally since, in general, from equations 12, 50, 56,
vi - vp -~ T,romp.
RT
it is evident from equations 109 and 110 that we also have a t each point of the solution
fl = f i
(112)
On substitution of equations 109, 110, and 112 (along with similar equations for the components r ) into equation 108, the first integral vanishes and the second integral can be evaluated to give
in which [pel has exactly the same significance as [ p ] in equations 95, 96, 97 for case 2, and the [ t J are suitable mean values of the transport numbers. For electrode equilibrium A, equation 113 gives
For clectrodc equilibrium B, equation 113 gives
E""' = -? ! [t+]( M , - [pel Vp*) (115) z+ F For electrode equilibrium C, equation 113 gives, in terms of components I-, IT,
or alternatively, in terms of components I-,
in which [tis] and
[tI,]
Iz,
become negligible if the excess of iodide is large.
An approximate relation connecting cases 1 , 2, and 3 The following considerations apply to the cylindrical cell with electrodes at the ends in a uniform gravitational field such as that near the earth, and to the annularly sectorial cell with electrodes a t the ends in a centrifugal field. In the case of the cylindrical cell, the gravitational field is applied
480
F. 0. KOENIG AND 5. W. GRINNELL
by placing the cell parallel to the field and may always be effectively removed by placing the cell perpendicular to the field. In the case of the sectorial cell, the field is of course applied and removed by starting and stopping rotation. We shall assume the cell contents to be so nearly incompressible and the cell container to be so nearly rigid that the changes in volume and shape of the solution and the electrodes on applying or removing the field are negligible. Let the cell be initially filled with a uniform solution and a steady gravitational field applied. We then have case 1 and the E.M.F. is very nearly that given by equation 75. Now let the gravitational field act until the solution has practically reached sedimentation equilibrium. We then have case 2 and the E.M.F. is very nearly that given by equation 97. Finally let the field be removed. We then have case 3 and the E.M.F. is very nearly that given by equation 113. Under the circumstances we have the relations &case
1
PCaSe 1
=
= AP&
3
(118)
,= [PICSSS 2 = [PICese 3
(119)
&case
2
We also have, a t least to a first approximation,
Comparison of the three equations 75, 97, and 113 therefore yields the result that E&1
- EEL2 + E 6 2 3
N
0
(122)
This approximate relation holds for any type of electrode equilibrium. In particular for electrode equilibrium C, moreover, equation 122 simplifies to E521
+ EEL3
-
0
(123)
owing to the exact equation 100. IV. CONCLUDING REMARKS
On the relatiolt between “gravity cells” and cells with liquid-liquid junctions
By a “gravity cell” we mean the type of cell considered under case 1 above. Physically the relation between “gravity cells” and cells with liquid-liquid junctions consists in the fact that in both cases the solution is subject to diffusion. I n the foregoing we have arrived at a general theory of galvanic cells which includes both “gravity cells” and cells with liquid-liquid junctions as special cases. From this theory it is evident that analytically the relation between “gravity cells” and cells with liquid-
GALVANIC CELLS SUBJECT TO FIELDS O F FORCE
481
liquid junctions expresses itself in the fact that in both cases the calculation of the E.M.F. as a function of chemical state and configuration depends upon the qrc. condition. It is the qrc. condition which in both cases causes transport numbers to appear in the equations for the E.M.F. In reference 8 it was proved that the qrc. condition is invariant under a change of the convention required to fix the numerical values of the transport numbers a t each point of a given solution. It follows that, whenever the solvent is a non-electrolyte, one may set its transport number equal to zero and thus obtain exact equations involving the Hittorf numbers of the other components. The Hittorf numbers are thus sujkient but not necessary for an exact theory of “gravity cells” as well as of cells with liquid-liquid junctions. On previous theoretical work
Des Coudres published his researches on galvanic cells subject to gravitational (or centrifugal) fields in two papers appearing in 1893 (1) and 1896 (3), respectively. In his first paper Des Coudres studies the cells Cd amalgam I solution of Cd chloride or iodide I Cd amalgam the electrode equilibrium of which is of our type A. He points out the possibility of the physical situations designated by us as cases 1, 2, and 3, and derives equations for the corresponding E.M.F’S. by calculation of the work done on reversible flow of electricity through the cell. Except for slight differences of notation, these equations of Des Coudres are identical with the equations which would result from our approximate equations 77, 98, and 114 on dropping the terms p*V& p*Va*, [p]V&,and [pel V:*. The absence of these terms is due to Des Coudres’s failure, in reference 1, to take into account the work due to “buoyancy forces” in the solution. In his second paper (3) Des Coudres, a t the suggestion of Nernst, corrects this error. With the theory thus improved he studies, in addition to the cadmium amalgam cells mentioned above, also the cells Hg 1 HgCl 1 solution of a single chloride 1 HgCl I Hg the electrode equilibrium of which is of our type B (with mercury instead of silver). In reference 3 Des Coudres confines his attention to case 1 (i.e., to the “gravity cell”), for which he obtains equations essentially identical with our equations 77 and 79. These equations he verifies experimentally. It is worth mentioning that Des Coudres in 1895 (2) was the first to obtain the differential equation for rate of sedimentation in a gravitational field and to solve it approximately. From his solution he was able to show that the rate is much too small to interfere with the measurement of the E.M.F. of “gravity cells” in the earth’s field. Following Des Coudres the only work of importance has been that of
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F. 0. KOENIG AND S. W. GRINNELL
Tolman, who in 1910-11 (11) published a very thorough experimental and theoretical study of “gravity cells” with electrodes I’t j I-, Iz (Le., case 1, electrode equilibrium C). Tolman was the first to deduce what is essentially equation 85 above. He did this by two methods, of which the first is a thermodynamic one similar to that of Des Coudres, and the second combines a thermodynamic calculation of the contribution of the electrodes t o the E.M.F. with a kinetic calculation of the contribution of the solution, after the manner of Kernst (10) for cells with liquid-liquid junctions. By virtue of this derivation Tolman is in fact the first to point out the fundamental similarity between “gravity cells” and cells with liquid-liquid junctions. Tolman gives a careful account of the approximations involved in equation 85. V. SUMMARY
With the help of the condition for quasi-reversible conduction (a), general equations are deduced which express the E.M.F. of a galvanic cell subject to arbitrary gradients of the gravitational potential and of the mole fractions, as a function of the chemical state a t every point and of the configuration (equations 32, 34, 38, 39). It is shown that, when the gravitational field strength is zero, these equations reduce to the general equations for cells wit,h liquid-liquid junctions (equation 48), whence the E.M.F. becomes independent of the gravitational potential. The electric potential difference in a solution subject to a gravitational field is calculated as a function of chemical state and configuration (equations 54, 55, 58, 59) and is shown t o be thermodynamically indeterminate. From the general equations the laws for certain special types of cell of experimental interest are deduced in detail (equations 71, 75, 77, 79, 82 to 85, 93 to 100, 108, 113 to 117, 122, 123). Previous t’heoretical work on galvanic cells subject to gravitational fields is reviewed. REFERENCES (1) DES COUDRES, T.: Wied. Ann. 49, 284 (1893). (2) DES COUDRES, T.: Wied. Ann. 66, 213 (1895). (3) DES COGDRES, T.: Kied Ann. 67, 232 (1896). (4) GIBES,J. W.:T h e Collected W o r k s of J . WiZZardGibbs, etc. Vol. I, pp. 144-7. Longmans, Green and Co., S e w York, London, Toronto (1928). ( 5 ) GUGGENHEIM, E. A , : J. Phys. Chem. 33, 842 (1929); 34, 1540 (1930). (6) GGGGENHEIM, E. A,: Modern Thermodynamics by the Methods of Willard Gibbs, p. 158. Methuen, London (1933). (7) KOENIG,F. 0.: J. Phys. Chem. 40, 373 (1936). (8) KOENIG,F. 0.: J. Phys. Chem. 44, 101 (1940). (9) MACINNES,D. A,: T h e Principles of Electrochemistry, pp. 17480. Reinhold Publishing Corporation, N e w York (1939). (10) NERNST,W.: Z. physik. Chem. 2, 613 (1888). (11) TOLMAN, R. C.: Proc. Am. Acad. -41%. Sci. 46, 109 (1910); J. Am. Chem. SOC. 33, 121 (1911).
