THE THERMODYNAMICS OF T H E ELECTROCAPILLARY CURVE. I THEGENERALEQUATIONS F. 0. KOENIG Department of Chemistry, Stanford University, Stanford University, California Received J u l y 31, 1933
I. INTRODUCTION If a difference of electric potential %, varying from 0 to about 1.5 volts is applied to the electrodes of a system such as
I
aqueous KNOl solution any constant nonpolarizable I reference electrode
H g i of any concentration
making the Hg/KN03solution end the cathode, the surface tension u a t the HglKN03 solution boundary varies in a regular way. The measurement of 0 for various values of % is best carried out in the Lippmann electrometer (1). The curve obtained by plotting u as ordinate and % as abscissa is called the electrocapillary curve of the system. In practice the system may be varied by replacing the mercury by an amalgam or by some other liquid metal such as gallium, and the potassium nitrate solution by aqueous or organic solutions of any electrolytes whose cations are sufficiently electropositive not to be deposited upon the metallic phase to any appreciable extent within the range used; the electrolyte must therefore in particular be as free as possible of the ions of those metals constituting the metallic phase. Since its discovery (2), two thermodynamic theories of the electrocapillary curve have been advanced. The first, due chiefly to Lippmann, Helmholtz and Planck (3), is that of the so-called “perfectly polarizable electrode.” The second, usually referred to as the “Gibbs theory,” and due chiefly to Gibbs, Thomson, Warburg, Gouy and Frumkin (4), consists in the direct application of Gibbs’ adsorption theorem to the polarized electrode. The two theories start from independent, apparently unrelated assumptions, but both lead to the same result, verified by experiment ( 5 ) , that the slope of the electrocapillary curve is equal to the surface density of electric charge on the surface of the mercury in contact with the solution. Together the two theories give a good account of most of the facts of elec111
112
F. 0. KOENIG
trocapillarity (6). Nevertheless they are unsatisfactory in at least two respects : 1. The theory of the “perfectly polarizable electrode” does not give an account of the variation of the electrocapillary curve with the composition of the two phases involved. 2. The “Gibbs theory,” although it does take account of the composition, rests on the assumption, among others, that the mercury surface is always in equilibrium with Hgz++ and Hg++ ions in the solution adjacent to it in the capillary. This assumption is open to question because, except for the lowest part of the ascending branch of the curve,l the solution in the capillary never contains an appreciable concentration of mercury salt; the values of CHg2++calculated from the potentials applied along by far ‘ I . This means that the greater part of the curve lie between and any Hgz++ or Hg++ ions a t all, which happen to diffuse into the neighborhood of the mercury surface in the capillary, are immediately and violently removed from the solution by electrolysis, so that actually there is anything but equilibrium with respect to these ions.2 That in spite of this assumption the “Gibbs theory” leads to a correct result, suggests that the error is partly or wholly counterbalanced by the other assumptions, difficult to verify individually, on which that theory rests. I n short, the first theory seems to be incomplete, and the second one a t least partly wrong-a situation which explains the disconcerting existence of two apparently unrelated theories leading to one correct result. The following is an attempt to give a complete theory starting from only one special assumption, which is sufficiently realized in the Lippmann electrometer. 11. THE THEORY A. THE ASSUMPTIONS
1. T h e perfectly polarizable system
Let A, figure 1, represent a system of two electrically conducting liquid phases, a and p, separated by a plane inhomogeneous boundary layer B, of finite thickness (lightly shaded portion of figure). The external surface of A is to be looked upon as bounded by a container of some nonconducting and chemically inert material. It is furthermore assumed that the interior of the system is electrically neutral as a whole. This means that any closed surface, D, in the interior, extending into the homogeneous regions 1 For this part of the curve to be sure, where C H ~ % may + + reach values as high as 10-3, the equilibrium Hg I Hg,++ does obtain, as shown by A. Frumkin and A. Obrutschewa (Z. physik. Chem. 136A, 248 (1928) and K. Bennewitz and K. Kuchler (Z. physik. Chem. 163A, 443 (1931)). 2 The author is indebted to Professor E. Lange for calling his attention to this difficulty .
THERMODYNAMICS O F ELECTROCAPILLARY CURVE. I
113
of both a and p and cutting the boundary layer B perpendicularly everywhere, can contain no appreciable excess electric charge-a condition which is fulfilled by all conducting two-phase systems in equilibrium a t ordinary temperatures. Finally, the system is assumed to have the following special property: Somewhere within the inhomogeneous boundary layer B and parallel to it, there exists a thin layer C’ (heavily shaded portion of figure l), which i s
FIQ.1. SCHEMATIC REPRESENTATION OF PERFECTLY POLARIZABLE SYSTEM B = boundary layer between phases (Y and p; C’ = barrier impermeable t o charged particles; C = fictitious Gibbs surface; D = fictitious closed surface in interior.
impermeable in both directions to charged particles (ions, electrons) of a n y sort. The word “thin” in this statement means that the barrier to charged particles may be thin compared with the entire inhomogeneous layer B. The barrier C’ is evidently the layer in which the general properties of the medium change relatively abruptly from those of a metal to those of an aqueous or other solution: on one side of C’ the system, although still inhomogeneous, has entirely metallic properties; on the other side, although also inhomogeneous, it has entirely nonmetallic properties.
114
F. 0. KOENIG
A necessary condition for the above special property is the following: of all the species of electrically &harged particles (formed by dissociation of any of the components) present in one of the phases, say a , none may be present to any appreciable extent in the other phase p, and vice versa. This condition implies the complete absence of electrochemical equilibrium. The condition is, however, not sufficient, because its fulfillment does not preclude all electrolysis (e.g., the formation of gaseous hydrogen on a copper cathode in a solution containing H+ but no Cuf or Cu++, due to passage of electrons from the metal into the solution). Since electrolysis can always be brought about by applying a sufficient E.M.F. in the proper direction, a given system can possess the property mentioned only within a certain range of E . A discussion of the definition and properties of the “perfectly polarizable electrode” of Planck (3) would be inexpedient here. Suffice it to say that the assumption of the above property leads to all the consequences necessary for an exact treatment of the “perfectly polarizable electrode.” The property in question will therefore be designated as that of perfect polarizability, and any system possessing it as a perfectly polarizable system. The physical significance of this name lies in the impossibility of electrolysis and the attendant absence of electrochemical equilibrium. The perfectly polarizable system evidently constitutes a case intermediate between that of two conducting phases in electrochemical equilibrium as regards one or more charged species, and that of a conducting phase in contact with a dielectric. 2. Relation of the perfectly polarizable system to the electrocapillary curve
If to the electrodes of the system Hg [ aqueous K N 0 3 solution 1 1 N KC1 1 HgzClzI Hg potentials E are applied making the Hg K N 0 3 side the cathode, practically no electrolysis mill take place as long as 0 5 7.E 5 1.3 volts, because (1) the potassium nitrate solution contains no Hgz++ or Hg++, (2) K+ is too electropositive to be deposited in appreciable amount ( i )and , (3) even after the deposition potential of hydrogen gas a t atmospheric pressure ( E = about 0.ivolts in theexample quoted) isexceeded, overvoltage keeps the hydrogen from being formed to any appreciable extent. Similar considerations apply to the other systems giving rise to electrocapillary curves in the Lippmann electrometer. The perfectly polarizable system thus constitutes an idealization which is closely approached by the real systems in question. Consequently a thermodynamic theory of perfectly polarizable systems will a t the same time be a general theory of the electrocapillary curve. Any discrepancy between such a theory and the facts will be due to a deviation of the system from perfect polarizability. But of such deviation there exists a very direct measure; it is the density of the “depolarization current’’ which flows upon applying the potential E .
I
THERMODYNAMICS O F ELECTROCAPILLARY CURVE. I
115
If there were no depolarization current, the system would be perfectly polarizable. I n a Lippmann electrometer containing alkali salt solution, the current density (at the small electrode) is very small, of the order of to ampere X B. T H E COMPOSITION O F T H E SYSTEM
1. Three definitions concerning composition
(la) Charged components. In a system containing electrically charged species it is possible to regard the latter as components in the thermodynamic sense. For such “charged components,” as for the electrically neutral components of a system, the chemical formulas assumed are to some extent arbitrary, the number of components alone being fixed. Let any system in equilibrium be bounded by a non-conducting chemically inert container of variable volume. Then the number of charged components is the number of electrically charged species, the total amount of which within the container may be varied independently. Thus, the system: aqueous Hi304 solution 1 vapor has three charged components, OH-, which may be taken as H+, OH-, SO4-, or H30+,OH-, SO4-, or Hf, HSOI-, etc. I n a binary alloy there are three charged components: the ions of the two metals and electrons. (lb) Electrolyte components. No matter in what proportions the charged components are present in any system as a whole, including its inhomogeneous boundary regions, all phase interiors will remain electrically neutral, because excess charges always collect in boundary layers. The interiors of the conducting phases of a system may therefore be regarded as containing a certain number of neutral components capable of dissociation, of which the charged components are dissociation products. These neutral but dissociating components will henceforth be referred to simply as “electrolyte components,” whether they are electrolytes in the ordinary sense or not. Thus the system T1 amalgam I aqueous KC1 solution, which contains two electrolytes in the ordinary sense, potassium chloride and water, contains four electrolyte components, which may be taken as thallium, mercury, potassium chloride, and water. The chemical formulas assigned to the electrolyte components are to some extent arbitrary, only the number of these components in a given system being fixed. Thus, for a system consisting of a single phase and its boundary layers, and containing p a charged components, the number of electrolyte components is in general p a - 1 . 3 a Owing to electroneutrality the amounts of only p a - 1 of the pa species can be varied independently in the phase interiors. This means that the presence of pa charged components in the system demands the presence of pa - 1 (corresponding) components for the phase interior. But whether these pa - 1 components of the interior are taken to be electrically charged species, or neutral species giving rise t o
116
F. 0. KOENIG
(2) Neutral components. Any uncharged components which do not dissociate appreciably into charged species will be referred to simply as “neutral components.” The point of these definitions lies in the fact that whereas a t constant temperature and pressure the state of a system as a whole depends upon the amounts of the neutral components and of the charged components, the state of all the phase interiors4 may be regarded as depending only upon the amounts of neutral components and of the electrolyte components, that is, the electrolyte components are convenient alternatives to the charged components in the description of phase interiors apart from boundary layers. 2. Description of the composition of the perfectly polarizable system
Since both phases a and p of the system, figure 1,are electrically conducting, they will both contain charged components. Let p a be the number of charged components in a , pa the number in p, and p the total number of charged components of the system. Then, since cy. and p may have no charged components in common, p = pa
+ pb
(1)
Let q a be the number of electrolyte components in a , q b the number in p, and q the total number of electrolyte components of both phase interiors. Then, since the perfectly polarizable system behaves, as regards its charged components, like two entirely separated single phases,
(3)
q =qa+qs=p-2
Let r‘ be the number of neutral components in a , rs the number in p, and r the total number of neutral components of the system. Then, since there is theoretically no restriction on the number or nature of the neutral components, so that the two phases may have a certain number, r a rB T , in common,
+
r Ir a
+ rB
(4)
the charged species by dissociation, or some of each, is immaterial, because only the number of the components is significant. Thus, if a one-phase system contains the charged components K+, Na+, C1-, NO 3-, the amounts of only three of these ions can be varied independently in the phase interior, and so there are only three electrolyte components, KC1, KNO NaNO 3, or KCl, NaCl, NaNO 3, etc. 4 By the state of a phase interior is meant the totality of the factors affecting any events that might be caused to take place in the phase interior.
THERMODYNAMICS OF ELECTROCAPILLARY CURVE. I
117
I n practice one of the two phases is always metallic, so that the r neutral components will practically always be restricted to one phase, the nonmetallic one, i.e., if a denotes the metallic phase, then: r = rB
(5)
The treatment will, however, proceed on the more general assumption (equation 4). The total number, J, of components of the system is given by J = pn+pB+r = qa+qfl+r
=p+r
+2 =q+r +2
and the number, K , of components of the two phase interiors is given by K = pa+pB+r
-2
=
p+r - 2
=q"+qp+r=q+r
(7.1) (7.2)
so that, for the perfectly polarizable system, K = J - 2
(8)
I n what follows, the subscript i will refer to the charged components, j will refer to the electrolyte components, and k will refer to the neutral components. In accordance with this notation, the (charged) species representing the charged components of the system will be designated by X i u , Xifl or, without allocation to phase, by Xi;the electrolyte components by Xia, Xf,or by X i ; and the neutral components by X k a , X k ' , or by xk. C. GENERAL EQUATIONS O F T H E PERFECTLY POLARIZABLE SYSTEM
1. T h e extensive properties of the boundary layer
The values of the extensive properties such as energy, entropy, or the amount of a component, may be defined according to Gibbs for any boundary layer as follows. Imagine a geometrical surface C, lying parallel to the boundary layer B (figure 1) at an arbitrary level. This level, being arbitrary, may be regarded as defined by a variable x, denoting the distance from C to any parallel fixed level, say that of a certain mark on the containing wall of the system A. If G is the value of an extensive property such as energy, entropy, etc., for the entire space (matter filled) enclosed by the surface D (figure l), then the corresponding value Gwfor the boundary layer is defined by: G
=
G"
+ GB + G"
(9)
where Ga denotes the energy, entropy, etc., which the space bounded by C and that part of D which lies in a would have i f the matter contained in it
118
'
F. 0. KOENIG
remained homogeneous up to C, and G@denotes the same for p. Any 0. thus defined evidently depends upon the position of the fictitious surface C, and is therefore a function of x. I n a perfectly polarizable system the only Owvalues of much physical interest are those for which the surface C is placed within the physical barrier C' separating the metallic from the nonmetallic media. The exact definition of this level, which will be designated as the physical dividing surface of the system, and which, for a given state of the boundary layer, is characterized by a constant value of x, x = x', is given on p. 124. From p. 124 on, all Gwvalues will definitely and finally be referred to the physical dividing surface x = d,but until then 2 will be regarded as arbitrary. It may be noted that according to the above definition
v = V" + V @
(9.1)
where T/' denotes the volume of the space within D (figure 1). As a result the volume of the boundary layer, Bo,is always zero, and thus occupies a special position among the extensive properties Gw. 2. T h e charge of the double layer
The space bounded by the fictitious surface C and that part of D which lies in a in general contains a certain net electric charge carried by charged particles accumulated near C; denote this charge by e a . If D is the area cut out of C by the closed surface D, the surface density of the charge in question is P / D ; set e a / D = e a . Quantities e@and a@ = e @ / D may be similarly defined. 6*, e@and ea, a@ are regarded as the charges and charge densities, respectively, of the two sides of the double layer, referred to the arbitrary surface C; since these quantities are so referred, they are, like the Gw,functions of x, of greatest physical interest for x = x'. 3. T h e electroneutrality of the double layer
From the initial assumption (p. 112) that the space bounded by the closed surface D is electrically neutral as a whole, it follows that if this space is divided into two parts by any surface whatever, as, for instance, C, the excess charges in the two parts are equal and opposite. Hence (10.1)
(10.2)
Le., the double layer as a whole is electrically neutral. This condition of electroneutrality can be obtained in a different form as follows. Let ni denote the number of equivalents of any charged component, Xi,contained within D. Then, by equation 9: nl = nta
+ n,b + nzw
(11)
THERMODYNAMICS OF ELBCTROCAPILLARY CURVE. I
119
Furthermore let the quantity wi be defined by Zi
wi =
where z i denotes the valence of Xi with sign included (e.g., zca++ = 2, z S o d ~= - 2); wi is evidently equal to 1 for each positively charged
+
component and to - 1 for each negatively charged component. If equation 11 is multiplied by wi and the sum taken over all p charged components, the result is
i=l
i=l
i=l
i=l
The term on the left vanishes because the space within D is electrically neutral as a whole, and so do the first two terms on the right because the phase interiors are electrically neutral; this leaves (14.1)
expressing the electroneutrality of the double layer. The introduction of the surface densities of the X i (in equivalents per unit area) defined by = n i ~ / into Q equation 14.1, gives the alternative form
2
wiri
=
o
(14.2)
i=l
4. I n a perfectly polarizable system the charge of the double layer can be varied independently of the state ofthe interiors ofthe two phases
If this proposition is true for ea, 06 referred to the physical dividing surface x = x', it is obviously true for any other value of x. Its truth for x = x' is evident from the following. Suppose there were in contact with the phase CY a series of electrodes, one for each of the Xia: permitting its addition to or removal from the system. Suppose the same for p. Now through an electrode in contact with CY cause An%aequivalents of Xia with an electric charge of wiaFAnia ( F = Faraday's equivalent) coulombs to enter the system, simultaneously introducing an equal number of equivalents Ani@of some oppositely charged component X,@into 6. Since no electrolysis can take place, the result will be that an electric charge of wfFAnia will distribute itself (1) along the CY side of C, (2) along the free surface of CY, and that an equal and opposite charge, wiFAni6, will dis-
120
F. 0. KOENIG
tribute itself similarly in fi.6 Moreover, although only one of the Xia be originally introduced into each phase, all of them will in general take part in this distribution (cf. equations 14.1 and 14.2). Consequently the concentrations of the X ja, X in the phase interiors will in general be changed by the introduction of the charges wiaFAnia, w,@FAni@. Finally, owing to shifting of the absorption equilibria of the X b by the change in charge density at the double layer and a t the free surfaces, the concentrations of the X I ,in the phase interiors will also be changed. But the concentrations of the X i and X k in the phase interiors may be restored to their original AO", values, leaving the charge of the double layer at a new value, 0" O@ f A@. Such a process can be carried out only in the absence of electrochemical equilibrium, because otherwise electrolysis will always take place. In systems with electrochemical equilibrium, the state of the boundary layer including Oa, e@is always completely determined by the state of the adjacent phase interiors. The process described is essentially the one taking place in a Lippmann electrometer when a potential is applied, except that in the electrometer it is not necessary artificially to restore the concentrations in the phase interiors to their original values, the concentration changes produced being insignificant as long as deviations from perfect polarizability remain small.
,@
+
5 . Deduction of the fundamental equations for the boundary layer and for the surface tension
A '(fundamental equation" of the Gibbsian type for the energy of the boundary layer E o (defined according to equation 9), will express Eo as a function of the entropy of the boundary layer So,the extent of the surface 0,and the amounts of the various components adsorbed. In order to obtain this equation, consider the change in the total energy E of the region bounded by the fictitious surface D, figure 1, when the system as a whole is subjected to any infinitesimal change whatever, in which equilibrium is preserved. The energy change in question can be due only to (1) the heat absorbed or evolved by the system, (2) the mechanical work done on or by the system through changes in the volumes Val V @ and in the area a, (3) The charges ABa and AO@added to the two sides of the double layer will by equation 10.1 be equal and opposite. Hence the charges added t o the free surfaces of a and p will also be equal and opposite; they will furthermore in general be numerically much smaller than ABa, A@ because charges of opposite sign tend t o come as close together as possible. Therefore, the condition AO" 1 = wiP P Ania = wLBF AntB1 = I AO@ 1 will be very nearly fulfilled in most cases.
1
1
1
I
THERMODYNAMICS OF ELECTROCAPILLARY CURVE. I
121
the change in energy due to changes in the amounts, ni and nk of the components Xi and XI, respectively. Consequently d E = TdS
- PadVa - PsdVfl + udn +
2
q,dn2
+ 2' gkdnk
(15)
k=l
i=l
Pa and PS are the pressures of the phases a and p respectively,6 u is the surface tension a t the boundary layer B, qi the electrochemical potential' ni, and nk per equivalent of Xi, and pi the chemical potential per mole of Xk, being measured in equivalents and moles respectively. Equation 15 contains the general condition for chemical equilibrium, namely, that a particular qi or p k has the same value in all the parts of the system-Le., both in the homogeneous phase interiors and in the boundary regionsin which the corresponding component is present to an appreciable extent. Since a given Xi is appreciably present in only one of the two phases, each of the vi refers to only one phase; the same applies to the pk of any X k present in only one of the two phases. In order to obtain from equation 15 an expression for E", the following relations, all special cases of equation 9, are necessary:
+ dEa + dE" d S = dS" + dSs + dS" dni = dnia + dn,@+ dn," d E = dE"
dnk = dnka
dnka
(17) (18)
(19)
+ dnk"
(20)
and also the following equations for the energy change that each phase would undergo if it remained homogeneous up to the arbitrary surface C:
i=l 6
k=l
I n general, P" and PB are related according t o
pa
- ps
=
(r
(ir1 + 'rz)
where T I , ~2 are the principal radii of curvature of the boundary layer. It follows that in the special case of a plane boundary layer (rl = ra = m ) represented in figure 1, Pa = P @ . Equation 15 and the equations deduced from it hold independently of whether the boundary layer is curved or plane, as long as T I , r2 are large compared with the thickness of the boundary layer. The electrochemical potentials 174 play exactly the same r81e for the charged components of a system as the chemical potentials p k do for the neutral components; they were first defined by E. A. Guggenheim (J. Phys. Chem. 33, 842 (1929)). For further discussion of the electrochemical potential, see reference 8.
122
F. 0. KOENIG
The substitution of equations 17 to 22 inclusive into equation 15 gives:
Equation 23 was derived for any change a t all in which equilibrium is preserved and hence is generally true. Since the preservation of equilibrium includes that of electroneutrality, the quantities nzwin equation 23 cannot all vary independently but only subject to the electroneutrality condition (equation 14.1) : (14.1) i=l
Aside from this restriction, however, the variables niware all independent, and so are Sw,Q, nkw. The number of independent variables of which Eo is a function is thus p r 1= J Is (cf. equation 6.1). The simultaneous equations 23 and 14.1 constitute a general thermodynamic solution of the problem of the perfectly polarizable system. This solution may be obtained in the form of a single equation by solving equation 14.1 for one of the ni", say nlw:
+ +
+
and substituting this in equation 23, giving
i=2
k=l
where
It may be noted that, since the variables whose differentials appear on the right side of the "fundamental equation" (equation 25) are all independent, (27.1)
8 A proof of the theorem that in the perfectly polarizable system any extensive variable G" is in general a function of J 1 independent variables will be given in a subsequent communication.
+
THERMODYNAMICS O F ELECTROCAPILLARY CURVE. I
(%)
s w , niw,
(27.2)
= a
nk"
123
(27.4)
where the subscripts nitwand nk'w mean that all the variables niw (i = 2 , . . . p ) and nkw are kept constant except the one with respect to which the differentiation is carried out. The application of the general equations, 23 and 14.1, or 25, to the problems arising in practice is facilitated by transformation into the GibbsDuhem type of equation, The independent variables in equation 25 are all extensive quantities, so that E" is a homogeneous function of the first order in these variables. Hence, by Euler's theorem
k=l
i=2
Differentiation of equation 28 and subtraction of the result from equation 25 yields the desired relation of the Gibbs-Duhem type: 0 = SWdT
+ Qdu +
P
niWdBi
r
+
nkwdpk
(29)
k=l
i s 2
The most convenient form of this equation is that obtained by solving for da: du =
- s"dT
-
r
2)
FidB, i=2
-
2
rkdPk
k=l
in which (31.1, 31.2, 31.3)
+ +
+
It may be mentioned that of the p r 1 = J 1variables whose differentials appear in equation 30, J are independent9 if the curvature of the boundary layer is regarded as varizbIe,lO so that u may in general be regarded as a function of the J variables T, Bi (i = 2, . . . p ) , pk. The 1to J in passing decrease in the number of independent variables from J from equation 25 to equation 30 is due to the fact that the variables whose differentials appear in equation 25 are extensive, whereas those whose differentials appear in equation 30 are intensive.
+
8
Proof in a subsequent communication. Cf. footnote6, p. 121.
10
124
F. 0. KOENIO
Finally, it is to be noted that the substitution of equation 26 into equation 30 shows that just as equation 25 is equivalent to the simultaneous equations 23 and 14.1, so is equation 30 equivalent to the simultaneous equations do = -sWdT
-
ridqi -
f:wiri
2
rkdpk
k=I
i=l
=
o
(14.2)
i=1
Equation 32 together with the electroneutrality condition in the form of equation 14.2 or 10.2 furnishes the best starting point for the special applications of the theory. The fundamental equation for E o having been obtained, it is of course possible to develop the theory of the perfectly polarjzable system further along Gibbsian lines, obtaining expressions for the other characteristic functions of the system. This is, however, not particularly necessary, because from the foregoing it is already evident that all the equations will be of the form of the general Gibbsian equations for any boundary layer, with electroneutrality conditions attached, the simultaneous equations obtained by keeping the electroneutrality conditions explicit (e.g. equations 23, 14.1 and equations 32, 14.2) being more symmetrical than the corresponding equations with the electroneutrality conditions implicit (e.g., equations 25 and 30). 6. Specialization of the general equations
(1) Introduction of the physical dividing surface. The general equations of the perfectly polarizable system have so far been deduced with no special assumptions as to the position of the arbitrary dividing surface C, figure 1, with respect to which the extensive properties G u and the charge of the double layer are defined. From now on all equations will be limited to the one case of physical interest, mentioned on p. 118, in which the fictitious surface C is placed within the physical barrier C’, characteristic of the perfectly polarizable system. The exact level, 5 = 2’ at which C is to be placed within C’ is defined as that in which the following equations, expressing the impermeability of the barrier to charged particles, (33.1)
(33.2) i=l
125
THERMODYNAMICS O F ELECTROCAPILLARY CURVE. I
are most nearly fulfilled. ria, I't@ are the surface densities (in equivalents per unit area) of Xia, Xi@with respect to the level in question, and ea, e @ are the corresponding charge densities of the double layer. In order to introduce the condition (equations 33.1 and 33.2) into the general equation (32), it is expedient formally to split up the electrochemical potentials (per equivalent) of the charged components X i u , X i @ as follows: qta = pSa
+
+w
qla = P!
(34.1)
wiaFqa
, ~ F ~
(34.2)
~ ppi@ , are the chemical potentials of X i u , Xis, and 'pa, ps are the absolute electric potentials in the two phase interiors.lI Writing equation 32 in the form
2
-
2 nB
na
du = -s"dT
r,"dq,"
-
r,@dq,@ -
2
rkdpk
(35)
k=t
%=I
1 3 1
and substituting equation 34.1, equation 34.2 gives
2
-
2
Pa
PB
Pa
du = -sWdT
rzadpla
-
rl@dpL,b -F
2 PB
Wlarladpa - F
w,@r,@dq@
i-1
i=l
i = l
i=l
2
k=l
into which equations 33.1 and 33.2 may now be introduced, along with the electroneutrality condition (equation 10.2), giving 00
0a
da = -s"dT
-
2 i=l
riudpiQ-
2' i=l
r
ripdp,io -
2
rkdpk
-
-
'pp)
(37)
k=l
11 It is to be noted that, as E. A. Guggenheim (J. Phys. Chem. 33, 842 (1929); 34, 1540 (1930)) has shown, the quantities pi and 'p are in general thermodynamically indeterminate. According to Guggenheim, the difference, (om - q @ , in electrical potential between two phases is thermodynamically defined only if (i) the two phase interiors are of identical chemical composition, (ii) the two phase interiors consist of ideal solutions in the same solvent, (iii) one or both phase interiors are non-ideal solutions in the same solvent, but the deviations from ideality depend only upon the valence type of the electrolyte components (as in the theory of Debye and Huckel) and are otherwise not specific. Changes in 'pa - q @are furthermore thermodynamically defined for two phases of a n y composition if the surfaces of the phases are altered without affecting the interiors; this is possible in the perfectly polarizable system (cf. pp. 119 and 120) and also in systems where the two phases can be separated by insoluble films of various sorts (cf. A . Frumkin: 2. physik. Chem. 116, 485 (1925)).
126
F. 0. KOENJG
which is essentially the general equation (30), specialized for the physical dividing surface, x = x', characteristic of the perfectly polarizable system. (2) Introduction of the applied potential E . Imagine the phase a to be electrically connected with a piece of some arbitrary metal, Me: whether the connection is established by direct contact of a and Me, or alternatively a series of other conducting phases in contact is interposed between Q! and Me, is immaterial, as long as the absolute potential difference Ap across each phase boundary between the interior of a and the interior of Me depends only on the temperature, pressure, and composition of the two phases adjacent to it, i.e., as long as chemical effects and polarization effects are absent. Imagine the phase p to be similarly connected to another piece of the same metal Me. Let 9' denote the absolute electric potential of the Me connected with a, that of the Me connected with p. Then the electromotive force 35 between the two pieces of Me is given by
P may evidently be varied at will by connecting the two pieces of Me to the poles of a potentiometer. If, as will always be the case from now on, Q! is taken to be the metallic phase, /3 the nonmetallic phase, then the quantities defined by equation 38 are the abscissas of the electrocapillary curves of the system. P may be introduced into equation 37 by writing (a"
- a9
= (a"
- a9
+ (a11 - d ) - P
(39)
the substitution of which into equation 37 gives
+ ($011 - (,B)
- %I
(40)
This equation is immediately applicable to the problems arising in practice and is to be regarded as the general equation of the electrocapillary curve. (3) T h e Lippmann-Helmholtz equation. A t constant T , Pa, PS and composition the quantities p i a , pia, pk, pa - p', ,PI' - p b are constant; consequently the differentials of these quantities vanish from equation 40 along with that of T , leaving
(%)T,Pa,PS,
composition
E
e"
(41)
This equation, the so-called Lippmann-Helmholtz equation, is thus shown to depend upon no assumptions save that of perfect polarizability as defined in this paper; it is, therefore, true for any perfectly polarizable system irrespective of composition.
THERMODYNAMICS O F ELECTROCAPILLARY CURVE. I
127
I n the determination of an electrocapillary curve by the Lippmann electrometer, the (variable) surface tension Q is measured by observing the (variable) height of a column of mercury (1) necessary to bring the boundary layer between the mercury (or amalgam) and the solution to a fixed position in the capillary of the instrument, the pressure Pa of the solution remaining constant (cf. equation 16). Actually, therefore, Pa is somewhat variable along the electrocapillary curves determined by the Lippmann electrometer, so that equation 41 does not apply rigorously to such curves. The limits of the variation of P a are, however, small, being always such that (at constant Pa), 0
< P a - Pp < 1 atmosphere
(42)
and within these limits the variation of y p with Pamay without appreciable error be neglected. If the pia are assumed to be constant, then (oa (PI is also constant, and since the electrically neutral components are in practice always confined to the solution p, whose pressure PP remains constant, yk, y i p and ,PI' - (ob are constant as before. On the assumption that the variation of the pia with Pa is negligible, equation 40 therefore gives for the slope of the electrocapillary curves determined by the Lippmann electrometer T,P,composition
= ,a
(43)
Equation 43 is the best known equation concerning electrocapillary curves and has been verified by experiment in various ways ( 5 ) . In concluding it may be mentioned that the equation 40 leads directly to the exact equations for the variation of the electrocapillary curve with the composition of the two phases; these equations will be deduced in a subsequent communication.
111. SUMMARY 1. Attention is called to a number of inconsistencies in the existing theory of the electrocapillary curve. 2. These inconsistencies are removed by developing the theory afresh, starting from only one assumption, that of perfect polarizability, which is defined in this paper as meaning impermeability of the boundary layer between two conducting phases to electrically charged particles. This definition is known to correspond closely to the actual physical conditions in the Lippmann electrometer. 3. The general equations resulting from the initial assumption of perfect polarizability are the Gibbsian equations for any boundary layer in equilibrium, with electroneutrality conditions attached. 4. The Lippmann-Helmholtz equation is found to depend upon no
128
F. 0 . KOENIG
assumptions whatever save the initial one of perfect polarizability; it thus holds for all perfectly polarizable systems regardless of their composition. The author is indebted to Professor E. Lange, formerly of Munich, now of Erlangen, for calling his attention to some of the difficulties which his paper is an attempt to solve; and to Mr. E. A. Guggenheim, formerly of Copenhagen, at present Visiting Professor in Stanford University, for much valuable aid and advice. REFERENCES (1) For descriptions see GOUY,G.: Ann. chim. phys. [7] 29, 145 (1903); SMITH,S.W. J.: 2. physik. Chem. 32, 433 (1900); ZEHNDER,L.: Z. Instrumentenk. 30, 274 (1910); KOENIQ,F. 0.: Z. physik. Chem. 164A, 454 (1931). (2) LIPPMANN,G.: Pogg. Ann. 149,547 (1873). (3) LIPPMANN,G.: Pogg. Ann. 149, 547 (1873); Ann. chim. phys. 6, 494 (1875); 12, 265 (1877); Wied. Ann. 11,316 (1880). v. HILMHOLTZ,H.: Wiss. Abhandl. physik. tech. Reichsanstalt I, p. 925 (1879). PLANCK, M. : Ann. Physik 44,413 (1891). (4) GIBBS,J. W . : Collected Works, Vol. I, p. 336. Longmans, Green and Co., New York (1928). THOMSON, J. J.: Applications of Dynamics to Physics end Chemistry, p, 191. The Macmillan Co., London and New York (1888). WARBURQ, E.: Wied. Ann. 41, 1 (1891). GOUY,G.: Ann. phys. [9] 7, 129 (1917). FRUMKIN, A , : Z. physik. Chem. 103,55 (1923). (5) FRUMKIN, A,: Z. physik. Chem. 103, 55 (1923); Ergebnisse exakt. Naturw. 7, 235 (1928). (6) See KOENIQ,F. 0.: Z. physik. Chem. 164A, 421 (1931), for a critical discussion of the two theories and of the assumptions on which they rest. See FRUMKIN, A. : Ergebnisse exakt. Naturw. 7,235 (1928), for an account of the application of the Gibbs theory. (7) KOENIQ,F. 0.: Z. physik. Chem. 164A, 436 (1931). (8) GUQQENHEIM, E. A. : J. Phys. Chem. 34,1540 (1930). SCHOTTKY, W. : in Handbuch der Experimentalphysik, Vol. 111. Part 11, p. 147 et seq., by W. Wien and F. Harms, Leipzig (1928). WAQNER, C.: Ann. Physik [5] 3, 629 (1929); [5] 6, 370 (1930). BRBNSTED,J. : Z. physik. Chem. 143A, 301 (1929). LANQE,E., AND MISCENKO, K . P.: Z. physik. Chem. 149A, 1 (1930).
