THE THERMODYNAMICS O F THE ELECTROCAPILLARY CURVE. I1 THEVARIATION OF
THE
ELECTROCAPILLARY CURVEWITH COMPOSITION F. 0. KOENIG
Department of Chemistry, Stanford University, Stanford University, California Received August $1, 1933 I. IR'TRODUCTION
In the preceding paper on this subject,l the following general equation was deduced for the electrocapillary curve (I, p. 126): a
r
2)
2 = 1
'i=
3-
(PI1
k = l
1
-
4) - %I
(40)
It is the purpose of the present paper to deduce from this equation the special equations describing the variation of the electrocapillary curve with composition. The results are of value because they show how electrocapillary measurements are to be interpreted thermodynamically so as to yield the maximum amount of information about the composition of the electric double layer. The system to which equation 40 applies is made up of three parts as follows: (1)two phases a and /3 constituting a perfectly polarizable system (I, p. 112 ff.) ; (2) a piece of some metal Me and a series of contiguous phases connecting it with a (I, p. 126); (3) another piece of the same metal Me and a series of contiguous phases connecting it with p. In general the composition of each of the parts 1, 2, and 3 is to be regarded as independently variable. As regards the two phases a and @ of the perfectly polarizable system proper, it will be recalled that they may have no charged components Xi in common (I, p. 116), i.e., that P = P"
+ pa
(1)
1 This Journal 38, 111 (1934); this paper will henceforth be referred to as I, the present paper as 11. The equations in I and I1 are numbered as a single series: I contains equations 1to 43 inclusive; I1 equations 44 to 168 inclusive. The reader is referred to I for the meaning of all mathematical symbols except those used for the first time in 11. 339
340
F. 0. KOENIG
Furthermore, in practice one of the two phases is always metallic, the other nonmetallic. Therefore the electrically neutral components X k ,being insoluble in metals under ordinary conditions, are in practice always confined to the nonmetallic phase, i.e., if, as is the case throughout this paper, a denote the metallic, p the nonmetallic phase, then (I, p. 117) r' = 0 and r = r@
(5)
Thus in practice the two phases never have any components, charged or neutral, in common, and therefore the composition of each phase interior may be varied independently of the other. Restricting equation 40 in accordance with equation 5 yields
- €"d[('p" -
PI)
+ (PIT -
P@) -
%I
This paper will be confined t o the consequences of equation 40.1. Throughout this paper it is assumed that the metallic phase a and the arbitrary metal Me are in direct contact without intervening phases. Besides the variation of a and @ it is therefore necessary t o consider only the variation in the above-mentioned part of the system. This part of the system shall henceforth be designated as the reference electrode (cf. I, p. 111). For example, if the system under consideration is I f f P I1 Me I Hg 1 1 N KNOI solution I 1 N HC1 solution I AgCl I Ag 1 Me
then the reference electrode is Me 1 Ag I AgCl 1 1 N HC1 solution
The problem is thus reduced to that of the independent variations of a , of p, and of the reference electrode. The effects of these three kinds of variation on the behavior of the system are accordingly considered separately below. It is to be noted in advance that the equations derived below are perfectly exact only for the case in which the pressures Pa and PB of a and p, respectively, are each constant, and that in order to apply these equations to the electrocapillary curves determined in the Lippmann electrometer, in which P a varies slightly, it is necessary to assume that the quantities p: (equation 40.1) are independent of Pa within the limits defined by the inequality 42 (I, p. 127). The fact that equation 43 (I, p. 127), which rests upon this assumption, is experimentally verified, may be regarded as direct proof that there is no appreciable error connected with the assumption.
THERMODYNAMICS O F ELECTROCAPILLARY CURVE. I1
341
11. VARIATION OF T H E ELECTROCAPILLARY CURVE WITH T H E COMPOSITION O F T H E REFEREXCE ELECTRODE
Let the surface tension u, the temperature T, the pressures Pa,Pa, and the composition of the phases a and /3 be kept constant, and let the reference electrode be varied as regards both the number and the composition of all its phases. Then equation 40.1 reduces to d[(pa
- 9')
+ (GI - @) - %I
(44)
= 0
whence
+ (GI - as) - F = constant2 = - constant + + (PPI1-
(pa
F
- pl)
(46)
pb)
pl)
(pa
(45)
Now consider two different reference electrodes which may be represented by Mel 1 1 1 and Mez 1
I/I 2
where the three vertical lines with the subscripts 1 and 2 indicate the presence of an arbitrary number of different phases in each case. For these two reference electrodes equation 46 may be written as
F,= - constant +
(pa
- pl)l
Fz = - constant + (pe - PI),
+ (PPI1 + (VI1 -
pp),
(47.1)
pp),
(47.2)
Subtraction of equation 47.2 from equation 47.1 gives
F ,- f z
= (PI
- p9z
+
(pa
- 50%
+
(PI1
- P4l
+ (d-
$ 0 9 2
The quantity on the right side of equation 48 is evidently the combination >Iez
E.M.F.
(48)
of the
I 1 Mel I 1 1 p 1 I I %lez a!
1
2
which, since a is metallic, has the same E.M.F. as the cell formed out of the two reference electrodes by joining them through 0. Changing from one reference electrode to another therefore displaces the electrocapillary curve of a given perfectly polarizable system horizontally by an amount equal to the E.M.F. of the cell formed by joining the two reference electrodes through /3. 111. VARIATION OF T H E ELECTROCAPILLARY CURVE WITH T H E COMPOSITION
O F T H E METALLIC P.HASE
Let the temperature T , the pressures Pa,Pa, the composition of the nonmetallic phase /3 and the composition of the reference electrode be Comparison of equation 45 with equation 39 (I, p. 126) shows that constant = (vu
-
a@).
342
F. 0. KOENIG
regarded as constant, and only the composition of the metallic phase a and the applied potential f be varied. Under these conditions the terms in T,p { , p i and (9'' - 96) vanish from equation 40.1 leaving
i-1
A . T h e case p a = 2 This case, the simplest possible, is trivial for the purpose in hand, because the interior of a! can consist only of a single metal and is therefore incapable of variations of composition. The terms in p : and (pa - 9') accordingly drop out of equation 49, leaving the Lippmann-Helmholtz equation 41.
B. T h e general case of a n y number of charged components p a In this case the interior of metals such that (I, p. 116)
a!
consists of a solution of a number q' of
pa = p a
-1
(2.1)
For this general case equation 49 is reducible t o a simple form, first, because a! being metallic all its charged components are of the same sign (positive) except one, namely the electron, and second, because a! and Me being in direct contact, it is always the electron with respect to which a and Me are in electrochemical equilibrium. Let p;+ and & be the chemical potentials (per equivalent) of the cations of the j t h metal and of the electrons, respectively, in the interior of a! and let rq+,I'g be the corresponding surface densities (in equivalents per unit area). The sum in equation 49 may then be written as
i=l
i=l
The introduction of the necessary relation a Pj+
a
Pj
a
- P0
(51)
where p q is the chemical potential per equivalent of the jthmetal, into equation 50 gives
From equation 33.1 it is evident that
2 r;+ - re "a
a
=-
F
(53)
THERMODYNAMICS O F ELECTROCAPILLARY CURVE. I1
343
The substitution of equation 53 into equation 52 and of the result into equation 49 gives
j=1
Now since a and Me are in electrochemical equilibrium as regards the electron,
where p i is the chemical potential of the electron in Me. But since the composition of the reference electrode and therefore that of Me is constant, p i is constant and hence
so that equation 54 becomes
The p; in this equation are of course subject to the Gibbs-Duhem relation at constant T and P a ,which may be written as o a
2N;dr;
=0
(58)
j=1
where N4 is the equivalent fraction of the jthmetal, defined in analogy t o the mole fraction by
n3 and nq, being the number of equivalents of the jthand the f t h metal in a
given mass of the interior of the phase CY. The simultaneous equations 57 and 58 give a complete thermodynamic account of the case of any number of metals a t constant temperature and pressure. C. Thecase p a = 3
In this case the interior of consists of a solution of two metals. Let the subscript 1 refer to the solvent metal; subscript 2 to the solute metal. Then equations 57 and 58 become, respectively, (Y
TEE JOURNAL OF PHYSICAL CHESIISTRY, VOL. XXXVIII, NO. 3
344
F. 0. KOENIG
du =
- Fy+dpy+ - Ti+dpi+ + eUdE
(60)
+ Nz“dp7 = 0
(61)
N:dp*P
The elimination of the chemical potential of the solvent metal, p y , from equations 60 and 61 gives
This equation yields for the vertical shift of the electrocapillary maximum with p i (by setting ea = 0 according to equation 41)
for the vertical shift of the electrocapillary curve
for the horizontal shift of the electrocapillary curve
For sufficiently small values of N %t’heterms in
ik7; a
N,
I?:+
in equations 63,64,
and 65 may be neglected; this yields the approximate relations
(E)P b
=
- r;+
2
=
2
so that from the shift of the curve with composition, the adsorption of the solute metal ions in a sufficiently dilute metallic solution may be calculated if the thermodynamic properties of the solution are known. It is to be noted that the equations lead to no relation giving I?:+ or I’: in terms of experimentally observable quantities, so that the knowledge of the composition of the metallic side of the double layer obtainable from thermodynamics alone, while valuable, is incomplete. Equation 67 is essentially identical with an equation given by Frumkin (2).
THERMODYhTAMICS O F ELECTROCAPILLARY CURVE. I1
345
IV. VARIATION OF T H E ELECTROCAPILLARY CURVE WITH T H E COMPOSITION
O F T H E NONMETALLIC PHASE
Let the temperature T , the pressures Pa,Pfl the composition of the metallic phase a , and the composition of the reference electrode be kept constant, and only the composition of the nonmetallic phase p and the applied potential 3I be varied. Under these conditions the terms in T , p:, and (pa - p’) vanish from equation 40.1, leaving
k=l
i = l
Owing to the possibility of more than one kind of charged component of each sign in p as well as of neutral components, and furthermore of various types of electrochemical behavior at the junction between p and the reference electrode, this case is more complicated than that of the metallic phase a. As a result equation 69, in contrast to equation 49, is not reducible to a simpler form without added restrictions. The discussion is therefore profitably limited t o a number of special cases of physical interest. ‘
A . The case pfl
=
2
This case, the simplest possible, is trivial because the interior of p can consist only of a single electrolyte3 dissociating into two kinds of ions (e.g., a molten salt), so that variations of composition are impossible. Equation 69 reduces to the Lippmann-Helmholtz equation (41).
B. The case
pb =
2, r b = 1
In this case the interior of p contains one electrolyte dissociating into two kinds of ions and one nonelectrolyte. The best examples are electrolyte solutions in certain organic solvents, e.g., lithium chloride in acetone. Aqueous electrolyte solutions come under this case only in so far as the dissociation of the water may be neglected; as shown later (11,p. 361) this is possible without serious error in certain special cases. In the following treatment the nonelectrolyte is regarded as the solvent in the interior of /3, the electrolyte as the solute, because this is the situation met with in practice. Let p$ and p! be the chemical potentials (per equivalent) of the cation and anion of the electrolyte, and p { the chemical potential (per mole) of a I n the present discussion of the nonmetallic phase, the term “electrolyte” will be used in its special, and usual, sense, and not in the general sense of “electrolyte component,” as defined in I, p. 115.
346
F. 0. KOENIG
the nonelectrolyte. Let I'f, I!', and I't be the corresponding surface densities. Equation 69 then becomes
- e"d(# - 96) + cad%
- rfdpf - rad,!? - rfdpf
do =
(70)
The substitution of = I*@
- PB
(71.1)
re = lL@ - Pf7
(71.2)
P!
or alternatively of
where p @is the chemical potential per equivalent of electrolyte, yields du =
- rfdpf
- r$dfi@f (rf - )F!
dP$
- e"d(pII - ps) + end%
(72)
where the two alternative equations are written as a single equation with double signs. Since by equations 33.2 and 10.2 rf -
€B
-F
=
(73.1) (73.2)
equation 72 becomes 0
do =
- rfdpf - r$dw@T ead-PFF - ead(pl' - po) + cad%
(74)
The chemical potentials p { and p @are subject to the Gibbs-Duhem relation NfdGf
+ N@d,u@= 0
(75)
in which N : , N Bare defined similarly to mole fractions by (76.1) (76.2)
where n [ and n @are the number of moles of nonelectrolyte and equivalents of electrolyte, respectively, in a given mass of the interior of 8. Since the nonelectrolyte is regarded as the solvent, i t is expedient to eliminate p i from equations 74 and 75; this gives dv =
(5
- F):
dfip T: ead PLBF
- ead(pI1 - q8) + sad%
F
For the vertical shift of the electrocapillary maximum 77 yields
p @ ,equation
(E@
(77) = 0) with
347
THERMODYNAMICS O F ELECTROCAPILLARY CURVE. I1
The double sign remaining in this equation can be eliminated, because on setting ea = 0 in equation 73.2 it appears that rfmax
=
(79)
rsmax
so that it is permissible to set
*
r P m s r = r@max
(80)
electrolyte
and equation 78 becomes
I n sufficiently dilute solutions of the electrolyte the term
N O I'trnar is negNt
ligible and equation 81 becomes
This equation may be taken as the basis for the distinction between capil-
-
lary-inactive -
-< 0) electrolytes in
0) and capillary-active
a given nonelectrolyte solvent. The further consequences of the alternative equations 77 depend upon the electrochemical behavior of the junction between the phase p (of variable composition) and the reference electrode (of constant composition). The following two cases are of physical interest: Special case 1. The potential difference ((~11- (PO) is given either by the equation B
pII
- pfi = constant + ti F
(83.1)
pI1
r! - pp = constant - -
(83.2)
or by the equation F
If ((PI' - (pa) is given by equation 83.1, then the reference electrode will be said to respond simply to the cation of p. Example: I
a
B
I
I
Me I Hg KNOI solution of variable concentration K amalgam of constant
I
I1
concentration Me
And if (PI' - (PO) is given by equation 83.2, the reference electrode will be said to respond simply to the anion of p. Example: I
I
a
B
I
1
I1
Me Hg I KC1 solution of variable concentration 1 HgCl Hg Me
384
F. 0. KOENIG
Special case 2. The junction between /3 and the reference electrode is a “liquid-liquid junction.” Example: I
I
a
if
I
Me Hg I KNOI solution of variable concentration 1N $,SO4
I1
1 Hg2S041 Hg I Me
in which the soIvent4 in the phases @ and y is the same. Of these two special cases, the first is the simpler theoretically, but special case 2 is the only one which has hitherto been investigated experimentally. Special case 1. T h e yeference electrode responds simply either to the cation or the anion of 6. If it responds simply to the cation it follows from equation 83.1 that d
(p”
- 98)
=
d
P!
F
(84.1)
and hence from equation 77 with the lower sign that (85.1)
If, on the other hand, the reference electrode responds simply to the anion of /?,it follows from equation 83.2 that d (&
- 96)
= -
d
F
(84.2)
and hence from equation 77 with the upper sign that (85.2)
Equations 85.1 and 85.2 yield the following relations for the shift of the electrocapillary curve with pb : if the reference electrode responds simply to the cation of P
anion of P
4 I n so far as the solutions used in the examples for the above cases 1 and 2 are considered to be aqueous, the electrolyte dissociation of the water is neglected here. As shown in 11, p. 361, this assumption leads to no serious error in a number of cases.
THERMODYNAMICS O F ELECTROCAPILLARY CURVE. I1
349
For sufficiently small values of N @equations 86.1 to 87.2 lead to the approximate relations
2
(?T)u =
(89.1) (89.2)
so that from the shift of the curve with composition, the adsorption of the solute cation or anion in a sufficiently dilute solution may be calculated, if the thermodynamic properties of the solution are known. Any one of the equations 88.1 to 89.2 together with equations 73.1 and 73.2 and the Lippmann-Helmholtz equation (41) evidently leads to a complete knowledge of the ionic content of the nonmetallic side of the double layer in this case. Special case 2. The junction between p and the reference electrode i s a liquid-liquid junction. This case may be represented schematically as follows:
I 11
in which Me y is the reference electrode, and the liquid-liquid junction @ y is made up in some definite and reproducible manner. The system thus represented will be referred to as system S. In order to apply equation 77 it is expedient to consider besides system S also the following cell, henceforth designated as system S’ :
1
1I1
I n this cell the electrode Me y and the liquid-liquid junction p 1 y are supposed to be identical with the reference electrode and the junction p y, respectively, in system S. The electrode Me 1 of system S’
1
11
1
is supposed to be any electrode whatever which responds simply to either the cation or the anion of p as defined under special case 1 (11, p. 347)) and will henceforth be referred to as electrode 1. The Roman numerals I, 11, 111, I V serve merely to label, for the purposes of discussion, the four separate pieces of the same metal Me which form the terminals of S and S’. The quantity (I.“ - 1.6) occurring in equation 77 may now be transformed as follows:
# - $0B -
- $09,= (PIV - (or),, ($1 - $0pa),, + (‘p (47111 - $08),, + E’ (PI1
(90)
(91)
(92)
350
F. 0. KOENIG
where the subscripts S and S‘ indicate the systems to which the corresponding quantities in parenthesis refer, and %’ is the E.M.F. of the cell, system S’, defined by X’ ($p - p),, (93) I
The introduction of equation 92 into equation 77 gives
If now electrode 1 responds simply to the cation of fi it follows in analogy to equation 84.1 that
and hence from equation 94 with the lower sign that
Similarly if electrode 1 responds simply to the anion of p it follows in analogy to equation 84.2 that
and hence from equation 94 with the upper sign that
Equations 96.1 and 96.2 yield the following relations for the shift of the electrocapillary curve with pB :if electrode 1responds simply to the cation of p (97.1)
(98.1)
anion of @ (97.2)
(08.2)
THERMODYNAMICS O F ELECTROCAPILLARY CURVE. I1
351
N P
Inrsufficiently dilute solutions the terms in - F[ may be neglected as in N{ special case 1. The comparison of equations 97.1 to 98.2 with equations 86.1 to 87.2 shows that special case 2 leads t o exactly the same information regarding ?I%’ the double layer as special case 1, provided - is known. I n general, bPP 3%’ the only way of obtaining accurate values of - is by experiment, be-
w
cause the cell S’ contains a liquid-liquid junction (variable with p) and the exact thermodynamic equations for the E.M.F. of such cells contain so many unknown factors as t o be useless practically. The situation may be illustrated by considering a particular system, for instance, the system given as an example of special case 2 in 11, p. 348. A possible choice for the corresponding system S’is the following: P
111
Me [ K amalgam of constant concentration 1 KNOa solution of variable concentrationIV
Y
I
1
1 N H2S04 Hg2SOa1 Hg Me
The exact equations for the fi. M. %’
F.
%’ of this cell are:5
= FEl
+ %I)
f
%S
(99)
where %EI = %o
%D =
9
RT
-In F
i i , B f K + d In cK+
c&+
RT
-In ego; 2F
(100)
+
%El is the ideal value of that part of the total EXF. which is due to the two electrode potentials of the cell S’, EDis the ideal value of the diffusion 5 The exact thermodynamic treatment of cells with liquid-liquid junctions is due t o P. B. Taylor ( 5 ) and E. A. Guggenheim (3). Equations 99 to 102 were derived by the methods of these authors, so that the inclusion of the derivation here is unnecessary.
3 52
F. 0. KOENIG
potential at the liquid-liquid junction between p and y, and takes account of the deviations of the two electrode potentials and of the diffusion potential from their ideal values; c and t denote the concentration in equivalents per liter of solution and the transference number in equivalents per faraday, respectively, of the ion indicated in the subscript; f denotes the mean activity coefficient of the salt corresponding to the ions indicated in the subscript; %O is a constant. The evaluation of the integrals in equations 101 and 102 necessitates a knowledge of the values of e, t , and f both in the interiors of p and y and at every point of the intermediate diffusion layer, and is therefore at present impossible. It follows that 3%’ the calculation of accurate values of __ from such equations as 99 to 102 3PD
is also impossible at the present state of knowledge. As a rule, therefore, the complete thermodynamic interpretation of the electrocapillary data in special case 2 necessitates supplementary measurements of the E. M. F. of cells of the type S’. In at least one case of physical interest, however, the theoretical expression for
a%’ __
w
is simple enough to be of practical value. This case is
defined by the following two conditions: (1) The phase y of the reference electrode contains only one electrolyte which is the same as that in p. (2) The reference electrode responds simply to either the cation or the anion of y,Le., if the concentration of the electrolyte in y were varied, then (either: constant
- coy)B=
$1
(pp”
- ,pY)#
=
i
or:
constant
+4 F
(103.1)
-d -
(103.2)
F
The simplicity of this case arises from the circumstance that electrode 1 of system S’ may always be chosen so that the ion of to which it responds simply is the same as the ion of y to which the reference electrode responds simply; for such cells S’ it is readily shown6that d%’ =
t! dF@if the two electrodes respond simply to the cation F
--
d%’ =
tf
- dpB if
F
the two electrodes respond simply to the anion
(104.1) (104.2)
An example of a system S fulfilling the above conditions is I
a
P
h/le Hg i KC1 solution of variable concentration
I
I 1N
Y
I1
KC1 I HgCl Hg 1 Me ~
6 By the methods used in deriving equations 99 to 102. See 11, footnote 5 ; also reference 4.
353
THERMODYNAMICS OF ELECTROCAPILLARY CURVE. I1
The corresponding system S' is I11
P
I
I
Y
I
IV
Me 1 Hg 1 HgCl KC1 solution of variable concentration 1 N KCl HgCl Hy Me
which obeys equation 104.2. The introduction of equations 104.1 and 104.2 respectively into equations 96.1 and 96.2 yields the equations
(105.2)
which give for the shift of the electrocapillary curve, if the reference electrode responds simply to the cation of
y
(106.1)
(107.1)
anion of y
(106.2)
(107.2)
NO In sufficiently dilute solutions the terms in - I'f may be neglected as s{ before. Finally a semiquantitative general result useful in dilute solutions is obtainable by the following approximations: (1) The deviation of the solution in p from ideality is neglected. ( 2 ) The liquid-liquid junction between p and y is neglected.
NP
(3) The term - I't is neglected. From
n:!
approximation 1it follows that dpf
d In cP
(108.1)
- RT - d In c p
( 108.2)
= 2
dp!
=
f
Z!
354
F. 0. KOENIG
where zf, z! are the valences (with sign included) of the cation and anion in p and cB is the equivalent concentration of the electrolyte. Furthermore from equation 93 it follows that d%’ = d(pIV =
- pIII)g
- d(&I -
@)at
(110)
+ d(qIv -
@)sf
(111)
which on introduction of equation 95.1 or 95.2 becomes de, =
-d B
d x ’ = d E”-
F
9+
d(plr
- q@),r
+ d(ppIV- @),!
if electrode 1responds simply t o the cation of p
if electrode 1 responds simply t o the anion of p
(112.1) (112.2)
By approximation 2 the term d(cpIv - ( P B ) ~ ! drops out; the substitution of equations 108.1 and 108.2 respectively into equations 112.1 and 112.2 then gives d%’ =
- zRT d In c6 fF
(113.1)
d%’ =
- Z!F RT d In cb
(113.2)
Introducing equation 113.1 into equation 96.1, equation 113.2 into equation 96.2, and furthermore equation 109 and approximation 3 into both equation 96.1 and 96.2 yields the alternative equations
That these equations (114) are alternative is proved by the fact that one of them is obtainable from the other by the substitution of
rf = rC F F €a
(115)
obtained from equation 73.2. Equation 114 gives for the shift of the electrocapillary curve with concentration
355
THERMODYNAMICS O F ELECTROCAPILLARY CURVE. I1
For the special case of a capillary-inactive electrolyte a t points along the electrocapillary curve sufficiently far removed from the maximum equation 118 may be integrated with the aid of a further approximation. I n such an electrolyte the adsorption r$,r!. of the two ions is due wholly to the electrostatic attraction of the charged metal surface, specific chemical forces being absent. Along the ascending branch e a > 0, so that the metal surface attracts anions and repels cations, whence .!?I > 0, ??$ < 0. At points far removed from the maximum, where E" is relptively large, I?$, because it is negative and the solution is dilute, is still relatively small numerically, so that for the lower part of the ascending branch the ratio B
€a
becomes negligible in equation 118. Setting
5
= 0 in equation 118
ta
(upper sign) gives (119.1)
which may be integrated at constant u, giving for the horizontal displacement of the ascending branch with concentration, RT c! In --j z!!F c1
--
(120.1) B
Similarly for the lower part of the descending branch
may be neglected ta
and equation 118 (lower sign) gives
( % z - % 1 = - -
=
-
RT
(119.2)
RT cc In 7 z p F CI
(120.2)
Equations 120.1 and 120.2 have already been given by Frumkin (1, 2).
C . The case p a
=
3
I n this case the interior of p consists of two electrolytes of two ions each, one of these ions being common to both electrolytes. The common ion may be either a cation or an anion. The best examples are aqueous solutions of acids (cation H+ in common) and of metallic hydroxides (anion OH- in common). Special case 1 . The common ion i s a cation. In this case equation 69 becomes du =
- rtdpf
- r,!dp,!
- r!-dp!-
- e"d(;'
- ~9+ cad+
(121)
356
F. 0. ICOENIG
+
in which the subscript refers to the cation, and the subscripts 1- and 2- to the anions of the solvent and the solute respectively. The substitution into equation 121 of the two relations P
L41-=
0
L42-
=
L41
P
- L4+B
(122.1)
L4t
- P+
4
(122.2)
where p1 and p2 are the chemical potentials per equivalent of solvent and solute respectively, yields do =
- F!-
dpf
- r!-dpi
- (r! - I?!-
-
- I$-.)
d,u(.,8 Ead(ppI1- I@)
+ €ad-&: (123)
From equations 33.2 and 10.2 it follows that 4 - r2P =r,P - rl-
(124.1)
= - -E"
(124.2)
CB
F
F
and hence from equation 123 that do =
- rf-dpf
-
4 - dpz-
B
+ e*d -F - e"d(p" P+
- pa) + cod%
(125)
which, on elimination of p f by means of the Gibbs-Duhem relation: N!dp[
+ NtdPf
(126)
=0
gives du =
($ N 1
9
B
- r z - d p i +sad
'+F
-
-
ead(pl'
- pa)
+ cad%
(127)
where Nf and AT[ are the equivalent fractions (cf. equation 59) of solvent and solute respectively. For the shift of nmaX with pz equation 127 yields
in which the term
%!!xf
is negligible in sufficiently dilute solutions,
giving the approximate relation --
aL4
pmax - - r2-
The further consequences of equation 127 depend upon the nature of the junction between p and the reference electrode. The following four special cases l a , lp, l y , 1 6 are of physical interest.
357
THERMODYNAMICS O F ELECTROCAPILLARY CURVE. I1
Special case l a . T h e reference electrode responds simply to the cation of
p, Le., B
d(#
- 1.6)
=
d P+ -
(130)
F
The introduction of equation 130 into equation 127 gives du =
9
r2-
($rf--
dM{+end%
This equation yields for the shift of the electrocapillary curve with
in which the term
(131) p!
NO I'd- may be neglected in sufficiently dilute solutions Nl
3
giving the approximate relations
Special case l p . T h e reference electrode responds simply to the anion of the solvent in p, Le., B
- $,3)
II+ ,,d (
=
- d "1F
(136)
The substitution of the relation B
P$ = M l
- P lb-
(137)
into equation 127, and of equations 136 and 126 into the result yields
This equation gives for the shift of the electrocapillary curve with p{
in which the term
may be neglected in sufficiently dilute
solution giving again equations 134 and 135.
358
Special case 1 y. the solute, Le.,
F. 0. KOENIG
T h e reference electrode responds simply to the anion of 0
d(,$
- ,@) - d 'F!!-
(141)
L
The substitution of the relation P f = '2
B
- bl2-B
(142)
into equation 127, and of equation 141 into the result yields
This equation gives for the shift of the electrocapillary curve with
,u!
NP Nf
in which the term 2 I'f- may be neglected in sufficiently dilute solutions giving the approximate relations
From equations 129, 134, 146, and 41, it is evident that from the shift of the electrocapillary curve with composition, the adsorption of the solute anion in a sufficiently dilute solution may be calculated in the cases considered, if the thermodynamic proper ties of the solution are known. It is to be noted that the equations lead to no relation giving I?$ or Ff- in terms of experimentally observable quantities, so that the information obtainable, while valuable is incomplete. The situation regarding the nonmetallic side of the double layer for p a = 3 is therefore similar to that regarding the metallic side in the case pa = 3 (11, p. 344; cf. particularly equations 128, 129, 131, 132, 133, 134 and 135 with equations 63, 66, 62, 64, 65, 67, 68). Special case 16. T h e junction between p and the reference electrode i s a liquid-liquid junction. By means of supplementary E.M.F. measurements
THERMODYNAMICS OF ELECTROCAPILLARY CURVE. I1
359
of suitable cells this special case can always be reduced to one of the foregoing special cases l a , lp, 17, in exactly the same way as for pa = 2, rp = 1, special case 2 is reducible to special case 1 by means of the cells S’ (11, pp. 349-355). Special case W (of the case pa = 3 ) . T h e common ion i s a n anion. The results for this special case are immediately obtainable from those for special case 1 by replacing the word “cation” in the discussion of special case 1 by “anion” and vice versa, and, in equations 121, 128, 129, 131 to 135, 138 to 140, 143 to 147, replacing the in the subscripts by - and vice versa. D. T h e case p @ = 4
+
This case is so complicated that a general treatment is inexpedient. There is only one special case simple enough to be worth considering here. Special case I. This case is defined by two conditions : (1) The interior of p consists of two electrolytes of two ions each, there being no common ion. (2) The ions of the solvent electrolyte are not specifically adsorbed. This special case is of importance because it includes the aqueous solutions of certain neutral salts. Under condition 1equation 69 becomes du =
P P P 8 P P - rl+drl+ - r-ldP2- r ZP + d PP 2+ rz-dPz- eud(q1’ - a@)+ P d % (148)
where the subscripts I + , 1 - refer to cation and anion of the solvent electrolyte, 2+, 2- to cation and anion of the solute electrolyte. Condition 2 means that P rl-
(149)
rf* = rf
(150)
r!+
=
so that it is permissible to set where I‘f is the surface density of the solvent molecules (in equivalents per unit area). The substitution of equation 150 and of the relation P Pl+
+
P
P
Pl- = Pl
into equation 148 gives du =
- r!dqf
P P - T [ + d p z8+ - rz-dpz-
- ced ((0’ - @)
+,
+ eade
(152)
If the subscripts 1, 2+, 2- are replaced by 0, -, respectively, equation 152 becomes identical with equation 70 for the case p @ = 2, rp = l (11, pp. 345-355). Moreover, under condition 1 above the interior of p contains only two independently variable components, so that the GibbsDuhem equation for the case in hand is similar to equation 75 for the case pa = 2, rfl = 1 in that it contains only two terms, namely NfdPf
+ Nid,u{ = 0
(153)
360
F. 0 . KOENIG
where p f is given by equation 151 and p z by B
PZt
+
B
P2-
=
B
P2
(154)
and N f , N t are the equivalent fractions of solvent and solute. It follows that all the consequences of equations 152 and 153 are identical in form with those of equations 70 and 75, Le., under the above conditions 1 and 2 the effect of the dissociation of the solvent electrolyte disappears and the case pa = 4 reduces to the case pp = 2, rB = 1. Frumkin (2) has shown how it is always possible to ascertain experimentally whether the above condition 2 is approximately fulfilled by an aqueous solution of a neutral salt. The method consists in investigating the shift in the electrocapillary curve of the salt solution produced by adding to the solution a small quantity of either that acid or that metallic hydroxide of which the salt in question is the product of neutralization. The junction between the salt solution (p) and the reference electrode must be a liquid-liquid junction. If no appreciable shift is found, equation 149 applies; if there is a shift, equation 149 does not apply. The theory of the method is best explained by means of a special case, say sodium sulfate solution. For this case equation 148 becomes do =
P B - r HP + d p HB + - FoH-dpOH-
B - I'Na+dP&a+ - rioydp&);
- cod ((,I1 - @)
f red%
(155)
The substitution of B
B
PH+ = PHz0
B - PQH-
(156)
gives d o = @&+
- r6H-l
B b B B - r HB + d P BH 2 0 - rNa+dP'Na+ - rso;dPsoa - Pd ((PI' - @) f cad%
P &OH-
(157)
Consider the change in u a t constant % produced by adding to a sodium sulfate solution of constant concentration, say 1 N , an amount of sodium hydroxide, say 0.01 equivalent per liter, so small that the values of L&O, pio;, ((PI' - 9s) are not appreciably affected, but yet large enough and pJH- appreciably. For such a change equation 157 gives to change
If no shift of the curve is observed,
dU
= 0 and it follows that
- @
dPOHr Hb + N
P roH-
THERMODYXAMICS O F ELECTROCAPILLARY CURVE. I1
361
i.e., condition 2 is at least approximately fulfilled. But if an appreciable shift is observed then
dU
-P 9POH-
# 0 and ? rH- # r6H-
(160)
Similarly it can be shown that the absence of a shift on adding a small amount of sulfuric acid to the sodium sulfate solution is also a criterion for the approximate fulfillment of condition 2. Frumkin has found that 1 N sodium sulfate 0.01 N sulfuric acid and 1 N sodium sulfate 0.01 N sodium hydroxide both have electrocapillary curves practically identical with that of 1 N sodium sulfate, and that 1 N potassium thio0.01 N thiocyanic acid and l N potassium thiocyanate cyanate 0.01 N potassium hydroxide both have electrocapillary curves practically identical with that of 1 AT potassium thiocyanate (2). It follows that the equations for the case pP = 2, r @ = 1 apply to these solutions.
+
+
+
+
Case E.
T h e concentrations in the phase /3 are constant save for that of adilute, powerfully adsorbed nonelectrolyte
A typical example of this case is the system OL
I
P
+ 0.02N HpSO4 + tert. CBH,~OHin the concentration c 8
Hg 1 N NapSO4
where O
