WILLIAMH. SMYRL AND JOHNNEWMAN
4660
Potentials of Cells with Liquid Junctions by William H. Smyrl’ and John Newman Inorganic Materials Research Division, Lawrence Radiation Laboratory, and Department of Chemical Engineering, University of California, Berkeley, California 94780 (Received July 11, 1968)
The potential of cells with liquid junctions is affected by diffusion of ions in the junction region. From the laws of diffusion,concentration profiles and values of potentials have been calculated for several different junctions without the assumption of activity coefficients equal to 1. The results have been compared to existing experimental data, and the results of the calculations of others. The magnitude of the diffusion effect has been calculated also for cases where it is desirable that the effect be negligible, as with an electrode of the second kind.
Introduction The only cells used in electrochemical studies which are strictly without liquid junction are those used to study the thermodynamic properties of alloys. I n addition there are cells in which the effect of diffusion in the liquid junction is negligible, for example, a cell with an electrode of the second kind involving a solid salt which is only very slightly soluble. The theoretical analysis of the potential of cells with liquid junctions has been of interest to workers who wish to derive thermodynamic values from such cells by correcting for the effect of diffusion. Some of the basic problems of liquid junctions have been treated adequately (Taylor,’* Guggenheim,2b and Wagnera), and we give here an alternate discussion which emphasizes the quantitative treatment of the transport phenomena. In addition, we shall present a method, with examples, of the calculation of the effect of diffusion on cell potentials without the assumption of ideal-solution behavior and activity coefficients equal to unity. We shall attempt to give a clear definition of what is meant by liquid junction potentials and to give a clear treatment of the diffusion phenomena. The expression of cell potentials involves a consideration of electrode equilibria. However, the final result generally requires a knowledge of the concentration profiles and of the effect of diffusion. Therefore, we begin with the treatment of transport in electrolytic solutions and of the determination of the concentration profiles. Transport in Electrolytic Solutions The difference of the electrochemical potential, pi, of an ion between two points (or phases) is the work of transferring 1 gram-ion reversibly at constant temperature and volume from one point (or phase) to the other.4 If we regard c J p i as the driving force per unit volume for diffusion and migration of species i, neutral species included (where V p i is the gradient of the electrochemical potential of species i), and K,(v, v,) is the drag force exerted on species i by species j by virtue of their relative motion, then a force balance leads to the multicomponent diffusion equation The Journal of Phusical Chem$stry
The coefficient Kij is taken to be independent of the velocity difference v, - v,, but it may be a function of temperature, pressure, and composition of the solution. The velocity v, is the average or macroscopic velocity of species j . Instead of K,, one can define a transport coefficient Di, having the dimensions of a diffusion coefficient
where CT is the total concentration of the solution. This also serves the goal of accounting for much of the composition dependence of the coefficients Ku. Equation l has been discussed elsewhere (see, for example, Newmans). I n this force balance, K i j = K j t or ai,= D, by Newton’s third law of motion. This is equivalent to the assumption frequently made in treatments of irreversible thermodynamics. Compare Onsager,6 who wrote the equation in the form (3) In applications it is frequently desirable to use this equation in an inverted form. Toward this end it is to be noted that there are only n - 1 independent velocity differences and n - 1 independent gradients of electrochemical potentials in a solution with n species. Therefore, eq 1 can be expressed as ctvpi =
c Mi,@, j
vo)
(4)
(1) .School of Pharmacy, University of California, San Francisco, Calif. (2) (a) P. B. Taylor, J . Phys. Chem., 31, 1478 (1927); (b) E. A. Guggenheim, ibid., 33,842 (1929). (3) C. Wagner, Advan. Electrochem. Electrochen. Eng., 4, 1 (1966). (4) E. A. Guggenheim, “Thermodynamics,” North-Holland Publishing Co., Amsterdam, The Netherlands, 1959,p 374. (5) J. Newman, Advan. Electrochem. Electrochem. Eng., 5, 87 (1967). (6) L. Onsager, Ann. N. Y . Acad. Sci., 46, 241 (1945).
POTENTIALS OF CELLSWITH LIQUID JUNCTIONS where v, is the velocity of any one of the species and where
Mi, = Kf, (i # j )
466 1 For this case of uniform composition, the species flux is related to the current density and the transference number by the expression
tjoi = zjFcj(v, - v,)
It further follows that M u = M j f . Bearing in mind that there are n - 1 independent equations of the form of eq 4, one can invert this equation to read v5
-IC
- vo
#O
LjkocXvp& ( j # 0)
(5)
where the matrix Lo is the inverse of the submatrix
Lo
(Mo)-’
=
Mo
(6)
and where the submatrix W is obtained from the matrix M by deleting the row and the column corresponding to the species o. The inverse matrix Lo is also symmetric, that is
Lf,” = Ljt”
(7)
It is to be expected that the Raj’sand the Dl;s will be less composition dependent than the Ltjo’s. Certain combinations of the Lajo’sare related to measurable transport properties and have particular significance in the treatment of cells with liquid junctions. The current density is related to the fluxes of ionic species as follows i
=
F ~ ; Q= N F&fctvf ~
= F&ici(vr
z
i
z
- v,)
(8)
Substitution of eq 5 yields (9)
I n a solution of uniform composition
(10)
V p k = Z$V@
where V @ is the gradient of the electric potential. Equation 9 becomes in this case
i
=
-F2V@
i#O
XtCi
kfO
LfkOXkCk
(11)
Comparison with Ohm’s law, also applicable in this case
i = -KV@ allows us to identify the conductivity K
=
F 2k#O
k#O
LQoXfc&hck
(12)
Although the Ltko’sdepend upon the reference velocity chosen, the conductivity, K , is invariant with respect to this choice. Next we can identify the transference numbers. Again, for a solution of uniform composition, eq 10 is valid and eq 5 becomes
vj
- vo
=
-Fv@
k#O
L/koXkck
(13)
= -tj°Kv@
(14) Comparison shows that the transference number t j o of species j with respect to species o is given by (15)
It is to be noted that the transference number has been defined as the fraction of the current carried by an ion in a solution of uniform composition. In a solution in which there are concentration gradients, the transference number is still a transport property related to the LijO’sby eq 15, but it no longer represents the fraction of current carried by an ion. A different choice of the reference species will change the Ldl’sand hence the transference numbers with respect to a common reference species. Equation 9 is applicable even in a nonuniform solution, and it can be rewritten in terms of the conductivity and the transference numbers, since Lfjo = Ljgo
As already noted, a different choice of reference species will change the transference numbers, but eq 16 still applies. However, it is apparent from eq 15 that the ratio t , o / z j is not zero even for a neutral species. While the reference velocity can be chosen arbitrarily to be that of any one of the species, charged or uncharged, it is usually taken to be the velocity of the solvent. In this case there is no problem if there are no other neutral components, since the ratio tiD/zf is always zero for the reference species. Equation 16 also has the same form if other reference velocities, such as the mass-average velocity or the molar-average velocity, are used. Again, care should be exercised since the ratio t J x i is then not zero for neutral species. Equation 16 is quite useful in the calculation of the potential of cells with liquid junctions. In the cases of interest the current density is supposed to be zero, but eq 16 also allows one to estimate the effect of the passage of small amounts of current. Equation 16 is generally useful only if the concentration profiles in the liquid junction are known. These are determined not from eq 16 but from the laws of diffusion (eq 1) and the method of forming the junction.
Determination of Concentration Profiles Several models of liquid junctions are popular, and to these we add one more. a. Fyee-Diffusion Junction. At time zero the two solutions are brought into contact to form an initially sharp boundary in a long, vertical tube. The solutions Volume 78, Number 13 December 1068
WILLIAMH. SMYRL AND JOHN NEWMAN
4662 are then allowed to diffuse into each other, and the thickness of the region of varying concentration increases with the square root of time. Even if the transport properties are concentration dependent and the activity coefficients are not unity, the potential of a cell containing such a junction should be independent of time. b. Restricted-Diffusion Junction. The concentration profiles are allowed to reach a steady state by onedimensional diffusion in the region between x = 0 and x = L , in the absence of convection. The composition x = 0 is that of one solution and at x = L is that of the other solution. The potential of a cell containing such a junction is independent of L (as well as time). The condition of no convection is usually not specified (i.e,, zero solvent velocity or zero mass-average velocity, etc.). c. Continuous-Mixture Junction. At all points in the junction, the concentrations (excluding, we suppose, that of the solvent) are assumed to be linear combinations of those of the solutions at the ends of the boundary. This assumption obviates the problem of calculating the concentration profiles by the laws of diffusion. d . Flowing Junction. I n some experiments the solutions are brought together and allowed to flow side by side for some distance. It is sometimes supposed that observed potentials should approximate those given by a free-diffusion boundary. e. Electrode of the Second Kind. To these we add the region of varying composition produced when a sparingly soluble salt is brought into contact with a solution containing a common ion. We might use a model similar to the free-diffusion junction if we imagine the salt to be introduced at the bottom of a vertical tube containing the solution. The sparingly soluble salt will then diffuse up the tube, and the concentration at the bottom will be governed by the solubility product. The concentration profiles in cases a, b, and e are governed by the laws of diffusion (eq 1). We propose to treat solutions so dilute that, in eq 1, we can neglect the interaction of the diffusing species with the other components except the solvent
or
chemical potentials in terms of one electrical variable. One way Lo do this is to use the electrochemical potential for one ionic species, p,, as a reference
- Xi--
pi = C L ~
The Journal of Physical Chenhtry
+
Xi
(19)
- pn
2,
The combination p i - (zipn/zn) is then the chemical potential of a neutral combination of ions and is independent oE the electrical state, depending only on the local composition. However, this choice is not convenient, particularly when the concentration of species n goes to zero. Another possibility is to express the electrochemical potential of species n as pn =
RT In c,
+ z,FCP +
p:
(20) The potential CP then has some of the characteristics of the commonly used electrostatic potential and, in fact, has exactly the same properties in infinitely dilute solutions where the activity coefficients of all neutral combinations become equal to 1. At higher concentrations, the quasi-electrostatic potential, CP, is of course arbitrary in the sense that it depends on the designation of the reference species n. I n contrast, the electrochemical potential of species n, or pn/xnF, behaves more like the potential of a reference electrode reversible to species n. I n a solution of uniform composition, both of these potentials behave like the commonly used electrostatic potential and, in fact, satisfy Laplace’s equation v2CP = 0 (21) Now the chemical potential of a neutral combination can be expressed in terms of a well-defined combination of activity coefficients
For the activity coefficients we shall use Guggenheim’s expression’ for dilute solutions of several electrolytes
where
I However, the activity coefficients will not be assumed to be unity. The electrochemical potential, pi, of an ionic species depends not only on the composition of the phase but also on the electrical state of the phase. For computational purposes it is convenient to express all the electro-
pn
Zn
=
1/2Cz?c* 1
(24)
is the ionic strength, ci is in moles per liter, and for aqueous solutions a! = 1.171 (l./m~l)’/~ a t 25’. The values of the coefficients fit, are tabulated by Guggenheim and are zero unless species i and j are ions of (7) See ref 4, p 357.
POTENTIALS O I CELLS ~ WITH LIQUIDJUNCTIONS opposite charge. We shall use these expressions with concentrations instead of molalities, as used by Guggenheim. Finally, then, the electrochemical potential of an ionic species is expressed as
To determine the concentration profiles in liquid junctions involves solving the diffusion eq 18 in conjunction with eq 25 and with the material balance equation
the electroneutrality equation
Cwt z
=
0
and the condition of zero current. I n a following section we illustrate how to use the concentration profiles or the values of A@ to calculate cell potentials. Substitution of eq 18 and 25 into eq 26 yields an equation describing difEusion, migration, and convection of an ionic species but including the activity coefficients in the driving force
This equation applies to solutions so dilute that interactions except with the solvent in the multicomponent diffusion equahion can be ignored and eq 23 can be used for the activity coefficients. This problem can be solved numerically for the various models of the liquid junction. I n the case of restricted diffusion, the equations are already ordinary differential equations. For free diffusion and for an electrode of the second kind, the similarity transformareduces the problem to ordinary differtion Y = Y/& ential equations. These coupled, nonlinear, ordinary differential equations can readily be solved by the method of Newman? The equations can be linearized about a trial solution, producing a series of coupled, linear differential equations. I n finite difference form these give coupled, tridiagonal matrices which can be solved on a high-speed, digital computer. The nonlinear problem can then be solved by iteration.
4663 Numerical Results We present here calculated values of A@ for the several models for the junctions between solutions of various compositions. No detailed concentration profiles will be given, since the potentials of cells with liquid junctions can be calculated directly from the tabulated values of A@, without further reference to the concentration profiles, as indicated in the next section. The tabulation of the values of A@, rather than the potentials of complete cells, is convenient because these values relate to the junction itself, whereas more than one combination of electrodes is possible for a given junction. I n addition to A@, only thermodynamic data are needed to calculate potentials of complete cells, the entire effect of the transport phenomena being included in A@. The value of A@ depends upon the choice of the reference ion n. I n each case this is the last ion for a given junction in the tables. For infinitely dilute solutions, A@ becomes independent of this choice and, furthermore, depends only on the ratios of concentrations of the ions in the end solutions. Solutions of zero ionic strength (fi = 1) are indicated by an asterisk, but the concentrations are given nonzero values so that these ratios will be clear. These junctions also provide a basis for comparison with more concentrated solutions, to indicate the effect of the activity coefficients. Table I gives values of A@ for the continuous-mixture, restricted-diff usion, and free-diffusion junctions. Table I1 gives values of A@ for an electrode of the second kind, where AgC1, with a solubility product of 10-lO (mol/l.) 2, diffuses into hydrochloric acid solutions of various concentrations. For solutions of zero ionic strength, the values of A@ for the continuous-mixture and restricted-diff usion junctions agree with the values calculated by the methods of Henderson9 and Planck,'O respectively. I n Figures 1-6 are presented the results of more extensive calculations on the HC1KC1 junction. Cells with Liquid Junction Once the concentration profiles are known for a liquid junction region, it is then possible to calculate the effect of the nonuniform composition on the cell potential. This effect is considered in the following subsections for cells of increasingly complex liquid junctions. It will always be assumed that the electrodes are in equilibrium with the adjacent solutions and that regions of nonuniform composition lie outside the immediate vicinity of the electrodes. The procedure then involves first the treatment of electrode equilibria, in the manner of Guggenheim." (8) J. Newman, Ind. Eng. Chem., Fundamentals, 7 , 614 (1968). (9) P. Henderson, 2. Phys. Chem. (Leipzig), 59, 118 (1907); 63, 325 (1908). (10) M.Planck, Ann. Phys. Chem., 39,161 (1890); 40, 561 (1890).
Volume 78, Number 18 December 1968
WILLIAMH. SMYRL AND JOHN NEWMAN
46M Table I: Values of A@ for Various Junctions and Various Models a t 25"" Soln 1
Ion
H+ c1-
K+ c1-
K+ H+ c1K+
Free diffusion
s01n 2
Restrioted diffusion
Continuous mixture
-. .-.. .-..
.-. .-.. .-.. .-..
0.01 0 0.01
-33.50 34.67*
-32.65 -33.80"
-33.75 34.95*
0.1
-27.31, (-27.08); (-28.25, 18'),d (-28.3),@ -26.69*
-27.45
-27.47 (-28.10,
0.2 0.2
0.1 0.1
0.2 0.2
0.1 0.1
0 0.02 0.02 0
H+
0.1 0.1
0 0.1
K+ H+
0 0.2 0.2
0.2 0 0.2
0 0.1 0.1
f.
*
-
*
-10.31 -11.43* 1.861 (2.05)* 0.335*
-
180)d
-26.85*
-26.85*
-27.92 -26.69*
-28.04 -26.85*
-28.09 -26.85*
0.2 0 0.2
-22.58 -20.24*
-23.03 -20.74*
-22.31 -19.96*
0.05 0 0.05
-20.70 - 18.50*
-21.09 18.97*
-
-20.23 -18.02*
c1-
0 0.02 0.02
K+ H+ c1-
0 0.02 0.02
0.1 0 0.1
-18.02 -14.05*
-17.89 -14.12*
-16.84 12.90*
K+ H+ c1-
0 0.01 0.01
0.1 0 0.1
-15.91 -10.85*
- 10.30'
-14.99
-14.04 -9.09*
K+ H+ c1-
0 0.09917 0.09917
0.1 0 0.1
-27.24, (-27.98)' -26.60*
-27.38 -26.77*
-27.40 -26.76*
K+ H+ NOac1-
0 0.09917 0 0.09917
0.1 0 0.05 0.05
-27.39 -26.53*
-27.48 -26.62*
-27.55 -26.70'
K+ NOac1-
0.1 0.05 0.05
0.1 0 0.1
-0.423*
-0.157 0,423*
-
-0.157 0.423*
Na + H+ c104c1-
0.1 0 0 0.1
0 0.05 0.05 0
28.58 26 72*
29.64 27.90*
28.10 26.22*
Na+ H+ clodc1-
0.1 0
0 0.1
0
0.1
32.83 32.35*
33-50 33.11*
33.05 32.57.
0.1
0
Na + H+ c104c1-
0.2 0 0 0.2
0 0.2 0.2 0
33.29 32.35*
33.88 33.11*
33.53 32,57*
Na+ H+ c104c1-
0.05 0 0 0.05
0 0.1 0.1 0
38.77 39.96*
38.31 39.58*
39.26 40.48*
c1c1-
K+ H+ c1-
K+ H+
The Journal of P h y h a l Chemhtry
-0.157
I
-
-
POTENTIALS OF CELLSWITH LIQUIDJUNCTIONS
4665
Table I (Continued)
-
01 QP, m V -
Ion
s01n 1
Soh 2
Cut+ Ag+ NO*-
0 0.2 0.2 0
0.1 0 0 0.2
c10,-
Free diffusion
Continuous mixture
Restricted diff usion
-6.22*
-6.22*
-6.22'
Values for f~ = 1 are indicated by an asterisk. The last ion is the reference ion. Experimental values are given in parentheses. T. Shedlovsky and D. A. MacInnes, J . A m e r . Chem. Soc., 59,503 (1937). J. B. Chloupek, V. Z. Kanes, and B. A. Danesova, Collect. Czech. Chem. Commun., 5,469, 527 (1933). E. A. Guggenheim and A. Unmack, Kgl. Danske Videnskab. Selskab, Mat-Fys. Medd., 10, 1 (1931). e D. C. Grahame and J. I. Cummings, Office of Naval Research Technical Report No. 5, 1950. N. P. Finkelstein and E. T. Verdier, Trans. Faraday Soc., 53,1618 (1957).
'
Table 11: Values of A* for an Ag-AgC1 Electrode in HC1 Solutions at 25""
10-4
- %-, mV cc1-O/Cc1 ( P C I - ~ - PCI-O)/F,mV %O
-rn
4
0.0198 1.00961 -0.226
6 X 10-8
0.0737 1.0392 -0.914
2
x
[HCll (bulk), M
0.359 1.200 -4.32
x
10-6
6 X 10-6
2
0.915 1.604 -11.22
1.780 2.539 -22.16
3.21 5.499 -40.58
10-6
10-6
Chloride is the reference electrode, and j3 values are taken to be zero.
-A% (mV) -A4(mV) Figure 1. -A* (mV) for free-diffusion boundary between HC1 and KCI, calculated for two different concentrations of KC1.
This allows the expression of the cell potential in terms of a difference in the electrochemical potential of ions in the solutions adjacent to the two electrodes. The evaluation of this difference involves the integration of eq 16 across the junction region. This equation can be rewritten in the form
Figure 2. Calculated values of -A* (mV) for freediffusion boundary between HC1 and KCI, at a constant ratio of CKCl to C H C I . The dashed line represents the (constant) ideal-solution calculation, the solid line includes activity coefficient corrections.
a. Cell with a Single Electrolyte of Varying Concentration. Cells containing a single electrolyte whose concentration varies with location in the cell constitute the simplest of the so-called cells with transference. An example is e
The sum on the right now involves only the gradients of electrochemical potentials of neutral combinations of ions and can be determined from a knowledge of the concentration profiles.
I
i I
HC1 in HzO AgCl(s) Ag(s) Pt(s) @'
(30)
(11) See ref 4, p 382.
Volume 78, Number 13 December 1988
WILLIAMH. SMYRL AND JOHN NEWMAN
4666
0.10-
CONTINUOUS M l h U R E
RESTRICTED
BOUNDARY
DIFFUSION BOUNDARY
0.05'
.
cHCI
0. 0
.
.
.
.
IO
30
20
- A $ (mV)
-A$ (m VI
Figure 3. Results for the restricted-diffusion boundary HC1KCl, given for two different concentrations of KC1.
/111
Figure 5. Values of -A* (mV) for the continuous-mixture boundary. The dashed line corresponds to the Henderson calculation; the solid line includes activity coefficient corrections.
I I
[
I
CONTINUOUS MIXTURE BOUNDARY
I I
RESTRICTED DIFFUSION BOUNDARY
/ / -=
CROl
I
C"Cl
21 0
22 5
28.0
- ~ (mv) d
'
Figure 4. Results for the restricted-diffusion boundary HCI-KC1, for a constant ratio of C K C l to C m i . The dashed line represents the ideal-solution calculation, the solid line includes activity coefficient corrections.
where the platinum leads and the silver-silver chloride electrodes have identical compositions on both sides of the cell. I n the transition region or liquid junction, the concentration of HC1 varies from the value in the e phase to that in the X phase. At both electrodes there is equilibrium among the cy, p, 6, and E phases, for example 6
(31) Combination of these relations with the definitions of the chemical potentials of the neutral silver and silver chloride, for example p c i - = pci-';
PA:
=
pAg+'
/LA,?
+
Pe-';
=
p ~ g + @ ;
PAgCl
6
=
p r a = pE-'
kAg+6
+
PC1'
(32)
- Pe-a' = P A : 6 PAgC1 + PCl-€ - PA:' + PAgC16' - POI-' If)
=
The Journal of Physical Chemistry
(33)
'
s
-
since the difference in electrochemical potential of electrons in the two leads is related to the cell potential as indicated. Since the electrodes are of identical composition, the expression for the cell potential reduces to
-F(V
- 9"')
= Pcl-c
- pc1-x
(34)
This difference in the electrochemical potentials of chloride ions can be evaluated with the aid of eq 29, which becomes in this case
(35) Equation 34 becomes
-F($"
Pe-a
*
Figure 6. Continuous-mixture boundary calculations for a constant ratio of C K C i to CHC1. The dashed line is the ideal-solution calculation, the solid line includes activity coefficient corrections.
yields an expression for the cell potential
-F(+* -
c
- $'"')
tHto
dpml
(36)
On the other hand, one could express the difference in
POTENTIALS OF CELLSWITH LIQU~D JUNCTIONS electrochemical potentials of chloride ions in terms of the quasi-electrostatic potential @ based on chloride ions as the reference ion
-F($"
- $*'I
=
-F(@' -
@')
+
RT In (cc1-'/cc1-")
(37)
As shown in eq 36, the cell potential is independent of the method of forming the junction for the case of a single electrolyte of varying concentration; that is, the integral is independent of the detailed form of the concentration profile. From measured cell potentials, eq 36 may be used to determine activity coefficients if the transference number is known, or it may be used to determine the transference number if the activity coefficient is known. Both types of determination are common practice. From the tabulated values of A@ (which tabulation requires prior knowledge of the transference number and the activity coefficient) one can calculate the cell potential from eq 37. For example, for CHC1' = 0.2 M and CHC? = 0.1 M we obtain -($" - $"') = 28.11 mV. If the silver-silver chloride electrodes were replaced by hydrogen electrodes, the expression for the potential of such a cell would become
-F($" -
$"I)
= PHC? -
PHC1'
PCl'
- PCl-'
i s /
l lriXla'
Pt(s), H2(g)HC1 in H2O transition region
HC1 in HzOAgCl(s) Ag(s) Pt(s)
saturation of AgCl and zero flux of hydrogen ions into the solid phase. From the conditions of phase equilibria at the electrodes and the definitions of the chemical potentials of neutral species, the cell potential can be written
-F($"
(39)
it is assumed that the two platinum electrodes are of identical composition and that the two solutions p and 6 differ in the concentration of AgC1, phase 6 being saturated. The transition region, in the model used here, is formed by contacting the solution /3 with the solid AgC1, and a diffusion layer develops by free diffusion into a stagnant medium. The concentrations of AgCl and HC1 in phase 6, adjacent to the solid surface, are determined by the laws of diffusion and the conditions of
-
=
$"I)
- PHCl'
'/2PH2"
+
+
-
- PCl-") (40) If the chemical potentials of hydrogen and HCI are PA:
PAgCl'
(PCl-'
expressed as PH% =
PH~B
+ RT In
PH*
and PHCl
= PHC?
2 RT In
CHClfHCl
then the standard cell potential, E", can be identified as a collection of thermodynamic quantities
- PAge + p A g c ?
FE" = ' / ~ P He~ - PHC;
(41)
and the cell potential becomes
-F($" -
+ '/2RT In 2RT In CHC~P~HC? + (pel-' =
$"I)
FE"
P H , ~
PCI-")
For the evaluation of the difference P C ~ - @eq 29 becomes in the absence of current
(38)
for identical partial pressures of hydrogen over the two electrodes. Equations 34 and 38 thus show the relation between the potentials of two cells with the same liquid junction but different electrodes. b. Cell with Two Electrolytes, One of Nearly Uniform Concentration. With an electrode of the second kind, as used in the prievious example, the solubility of the sparingly soluble salt will, strictly speaking, lead to diffusion of this salt from the electrode. At high concentrations of the other electrolyte, the solubility of the sparingly soluble salt is depressed and the effect on the cell potential is expected to be small. However, this effect becomes more important as the concentration of the second electrolyte is decreased. For the cell o!
4667
=
VPCl-
tH+'VPHCI
+
tAg+'VPAgCl
(42) 8 POI-,
(43)
Integration gives PCl-@
- Pc1-6
=
J
@
bPHCl
dx
~ H + O-
dX
+ bPAgC1
@
tAg+O
dx 7
(44)
The evaluation of these integrals requires a knowledge of the concentration profiles, as well as the transference numbers and thermodynamic properties as functions of the concentrations. For high concentrations of HC1, v P H C 1 and tAg+' approach zero, and the term p a - @ - gc1-S may be neglected in comparison to the other terms in eq 42. The difference in electrochemical potentials of the chloride ion can be expressed in terms of the quasielectrostatic potentials (referred to the chloride ion), differences of which are given in Table I1 PCl-@
- PCl- 6 = F(@'
- @@) + RT
In (cc1-B/cc1-6) (45)
For a bulk HCl concentration of M , one obtains (pc1-O - pcl-6)/F = 0.0198 - 0.2457 = -0.226 mV. This small error is not of much practical significance, since very few measurements have been made in this range of concentration. Thus it is seen that the effect of the solubility of the slightly soluble salt will be to cause the potential of the V o h m 78, Number 1.9 December 1868
WILLIAMH. SMYRL AND JOHN NEWMAN
4668 chloride electrode to be more negative with respect to the other electrode than would otherwise be the case. Hence the measured potential of the above cell will be lower than if silver chloride were more insoluble. Smyrl and Tobias12have discussed several nonaqueous systems where the effect is much larger in more concentrated solutions, since the effect becomes important for bulk concentrations on the order of the square root of the solubility product. The problem arises because the determination of standard cell potentials involves an extrapolation to infinite dilution. Smyrl and Tobias took the diffusion coefficients to be equal (hence A@ = O) and assumed that the concentration Of the second electrolyte uniform to the surface Of the soluble salt. C. Cells with Two EEectrolYtes, Both Of 'Oncentration. Cells of this type may still be divided into two groups according to Ivhether Or not the two trolYtes have an ion in common. A junction between CuSO4 and ZnSOc is an example where there is a common ion; a junction between NaCl and HC104 is an example where there is not. The former class will be discussed first. Consider the cell
CY
Pt(s), H,(g) NaCl in HzO,AgCl(s)Ag(s) lPt(s) (49) From the conditions of phase equilibria at the electrodes and the definitions of the chemical potentials of neutral species, the cell potential can be written
-F($'"
= t H +OVPHCl
+
t K +'VPKCl
(47)
and integration gives PCl-'
- pc1- x
=
+
- (pH+'
PC1')
- $'a') = '/ZPC(HZ"- PA: + PAgCl' (PH+' - PH+') - ( ~ c i --~ ~ c 1 - I )- PHC:
+
Pc1-S = PHC:
F(O'
The Journal of Physical Chemistry
+
(51)
The electrochemical potential differences in eq 51 can now be related through eq 29 to integrals of transference numbers multiplied by gradients of chemical potentials of neutral combinations. The integrals can be evaluated from the concentration profiles in the junction along with the concentration dependence of the t,ransport and thermodynamic properties. The cell potential is, of course, independent of the choice of the intermediate solution I. In this case it may be particularly convenient to use the quasi-electrostatic potential, here referred to the chloride ion. This allows one t o write PHto
Here, as with eq 44 and in contrast to eq 36, the integral depends on the detailed form of the concentration profiles in the junction region. As in the preceding examples, the potential can again be expressedin terms Of the quasi-electrostatic potential, referred to the chloride ion, eq 37, and the values of AO in Table I allow the cell potential to be calculated. By means of the various models, the detailed form of the concentration profiles has already been taken into account in the tabulated values of A@. Many cells of practical importance contain two electrolytes Of varying concentration with no Commonion' Such a cell is
- PA:
(50) The cell potential is again related to the thermodynamic properties of electrically neutral components, but a new term has appeared. Instead of the difference of electrochemical potential of a single ion between the two solutions, there is now a combination of electrochemical potentials of two ions, This complicated situation can be analyzed if the ionic strength does not go to Zero anywhere in the junction (as must also be the case with the junctions treated earlier). Choose some solution in the junction and denote it as I. The quantities P c l a - Pel-I and pH+@ - pH: are both well defined if the intermediate solution I has nonzero concentrations of both ions C1- and H+. The cell potential can be written, then
-F($'"
VPCl-
= '/zPH/
PAgCl'
€
The cell potential is again given by eq 34. The effect of the nonzero solubility of AgC1, discussed in subsection b, will be ignored here. In this case, however, eq 29 becomes
- $'"I
f RT In
cH+BcC1-6
+
- Os)+ RT In (fHt'fCl-')
(52)
The last term in eq 52 is well defined, although it is somewhat unusual, Here f H tpf CI-' represents the activity coefficient of hydrogen ions referred to chloride ions in a solution of vanishing &loride concentration. According to eq 23, this term would be given by InfH+BfCI-P
=
-2 a P 2 ~
1
+ PZ+
2(PHC104
+
PKC1)CHt'
(53)
where I now refers to the ionic strength in solution p. This procedure is justified by the fact that it is no (12) W. H. Smyrl and C . W. Tobias, Electrochim. Acta, 13, 1581 (1968).
POTENTIALS OF CELLSWITH LIQUIDJUNCTIONS longer necessary to select an intermediate solution in the junction. The cell potential can now be written
-F(+" - +"')
= FE"
RT In
+ '/zRT In
cH+Bcc1-6
3- F ( d
(54)
PA~C?
- PHC?
(55)
A determination of the standard cell potential by means of this cell would be affected by the uncertainties in the +@fcl a problem which is avoided values of A 3 and jH with the cell discussed in subsection b. If E" is known from measurements with that cell, any Uncertainty in the calculated value of A+ will cause a consequent uncertainty in any value of fH+Bfcl-' obtained from measurements on the present cell. This discussion reveals some of the difficulties involved in the use of such cells for thermodynamic measurements. We could check the calculated value of A@ in Table I by subtracting from the cell potential the standard cell potential and the terms in activity coefficients and concentrations. The potentials of such cells have not been measured. The cell -@,
@ I
a
i
Pt(s), Hz(g)HCI in HzOtransition region
I
Iff'
KNOS in HzO K in eHg(1) Pt(s) (56)
is very similar to that of the previous example but differs in that both electrodes involve phase equilibria of cations. Again, from the phase equilibria and the relevant thermodynamic identities, the cell potential may be written
-F($'" - $'=')
- MYf + P K +6 - PH+' = '/2PHsa - PK' + ( P K + 6 - PK?) (pH+' - pH+') + ( P K ? - P H : ) (57) = '/ZPHT
where I denotes an intermediate solution where both K+ and H + ions are present. The quantities P Y . ~PK? and p ~ + ' - p ~ + can ' be related through eq 29 to integrals of transference numbers multiplied by gradients of chemical potentials of neutral combinations. The last term in eq 57 is well defined and is given by I
PY+
- P H +I
=
PKCl
I
- $"'> = PHC?
RT In
- 3') -
where the standard cell potential has been identified
+
-F($"
PH;
RT In fH+'fCl-'
FE" = ' / z P H , ~ - PA:
4669
- PHCl I
T o demonstrate the usefulness of tabulated values of A@, let the quasi-electrostatic potential be based on the
chloride ion. Equation 57 can be rewritten
'/ZP€I:
-
+ RT In
PKC
+
PKC?
(cK+~/cH+')
cfR.bfC1-6/fH+BfCI-B)
-
+
F ( a 6 - a')
(58)
Any uncertainty in A@ and fK+%1-6 would be reflected in the uncertainty in a derived value of a standard cell potential. Thus the use of such a cell to determine standard cell potentials is justified only if the junction is well characterized and if the thermodynamic properties of one of the end solutions are well known. Other cells could be analyzed, but the analysis would involve only the principles and procedures which have been used above.
Discussion The analysis of the cells in the previous section revealed the relation between measured cell potentials and the thermodynamic and transport properties of the materials in the cells. For cells with liquid junctions, the cell potential depends on the concentration profiles in the liquid junction and the transport and thermodynamic properties of the junction region, in addition to the standard cell potential and the composition of activity coefficients of the end solutions. Alternatively, for the junction one could specify the concentration profiles, the value of A@ which characterizes the junction, and the ion to which 3 is referred. Once single junctions have been characterized, the behavior of combinations of these junctions in other cells, e.g., cells with salt bridges, may be predicted. Calculations for several single junctions have been made, and the results are given in Table I. The only one of the junctions for which our calculations may be compared with other calculations and with experimental results is the 0.1 M HCl-O.1 M KC1 junction. MacInnes and Longsworth13have made calculations for this junction of the free-diffusion type and reported a value of 28.19 mV to compare with 27.31 mV of the present study. Spiro14 has discussed cells with liquid junctions, including salt bridges, for junctions of constant ionic strength across the junction and of the continuous-mixture type and has included activity coefficient corrections. For this HC1-KC1 junction, Spiro calculated 29.07 mV, and we calculate 27.47 mV. The experimental results are given in Table I. MacInnes and Longsworth used eq 29 and the known activity coefficients and transference numbers for this junction and an assumed concentration profile to make their calculation. It is not clear whether the difference between their results and ours is due to our assumption about activity coefficients, our assumption about the ionic diffusion coefficients, or their assumption about the concentration profile. (13) D. A. MacInnes and L. G. Longsworth, Cold Spring Harbor ~~ b~ ~ 4, .18 ~, (1936). ~ , (14) M Spiro, Electrochim. Acta, 11, 669 (1966).
smP.Q
Volume 72?Number I S
December 1968
WILLIAMH. S m R L AND JOHN NEWMAN
4670
It is known's that Doc1 varies by 17% across the 0.1 M KC1-HC1 junction, whereas it is the same in both solutions at infinite dilution. Their use of experimental data on transference numbers takes this variation of Doc1 into account, but our treatment does not. On the other hand, they have assumed a concentration profile in order to make their calculations which will certainly lead to some error. It is not possible at present to estimate the relative importance of these differences, but it is gratifying that the two treatments agree so well. Spiro calculates about the same activity coefficient correction (0.55 mV) as we do (0.62 mV), but his calculation neglecting activity coefficients (- 28.52 mV) is higher than ours by about 1.7 mV. The major differencesin the results are caused by different assumptions about the ionic diffusion coefficients. We have assumed the ionic diffusion coefficients to be constant throughout the junction and have used the values corresponding to infinite dilution in making the Henderson calculation. Spiro has used the Lewis and Sargent equation and has utilized conductivity data for the 0.1 M solutions. The restrictions in the use of the Lewis and Sargent equation lead one to conclude that this is not a valid way to correct for the variation of Doc1 mentioned earlier. The restrictions come about in the following way. The Lewis and Sargent equation is derived from the Henderson equation, which in turn is derived with an assumption that the ionic diffusion coefficients are constant across the junction. Thus one must use data consistent with this assumption. The conductivity data used by Spiro do not fulfill this requirement. Since the results reported here for the other junctions may not be compared to experimental data, we can only propose that they are at least as accurate as for the HC1-KC1 junction. The variation of the ionic diffusion coefficients across the junction may be important for the junctions containing hydrogen ion, but less important for the others. It will be necessary to make the comparison with experimental data to verify this. The comparison must be made also before calculations are made at higher concentrations-the concentration region where the junction might be used as part of salt bridges. A salt bridge is often used to separate two electrolytic solutions and sometimes the stated purpose is "to eliminate liquid junction potentials." We should now be in a position to evaluate whether this purpose is achieved, if we could define the liquid junction potential which is supposed to be eliminated. Such a salt bridge might be
1
1
HC1 (0.1 M transition KCl (0.2 M in HzO) region in HzO)
1
It seems clear that the salt bridge does not make the value of pcl- equal in the two hydrochloric acid solutions. The value of A@ (referred to the chloride ion) for this combination of junctions is 5.78 mV if the junctions are of the continuous-mixture type. This can be compared with the value A@ = 10.31 mV for a single, direct junction between 0.1 and 0.2 M HC1 solutions. If the transference numbers of KC1 were equal to 0.5 and if departures of activity coefficients from unity could be ignored, the liquid junction potential of the combination of two junctions of the salt bridge should decrease as the concentration of KC1 increases. If one insists on using salt bridges, one might consider as an alternative the series of junctions
1
1
1
HC1 (0.1 M transition KC1 (0.1 M transition region in HzO) region in H2O)
1
1
1
KC1 (0.2 M transition HC1 (0.2 M in H2O) region in HzO) for which A@ = 1.24 mV and for which the value of A@ would approach zero as all the concentrations were reduced in proportion if the transference numbers of KC1 were 0.5. Although cells with salt bridges are not useful for determining activity coefficients, they are useful for determining standard cell potentials. A cell which is particularly appropriate for such studies, but which has not been utilized extensively, is one in which the electrolyte of the salt bridge is present throughout. An example of this cell is
i
6 LiNO3, KN03 in HzO
!
transition region e (KN03 in H,O) AgN03, KN03 in HzO in which KN03 is present throughout the cell a t the same concentration. The transition region contains concentration gradients of both LiX03 and AgN03. The cell potential may be expressed as
-F($'" - $'*')
= PL?
PLiNOa6
PAP'
+
-
PAgNOa'
- PNOs-'
+
PNOs-e
We adopt the following approximations which essentially fix the range of concentrations tAg+O
=
0
1Li+O
=
0
= CLi+'
