J. BRAUNSTEIN, A. ALVAREZ-FUNES, AND H. BRAUNSTEIN
2734
Polarographic and Potentiometric Evaluation of Association Constants in Low-Temperature Aqueous Melts
by Jerry Braunstein,' Alba Rosa Alvarez-Funes, and Helen Braunstein Department of Chemistry, Uniueraay of Maine, Orono, Maine
(Received February 21, 1966)
As part of an investigation of the effect of water on association in molten salt solutions, the association constant of cadmium with bromide was evaluated both polarographically and potentiometrically in fused calcium nitrate tetrahydrate at 50" as 3900 200 (moles of bromide/mole of nitrate)-'. Equations for the polarographic evaluation of successive association constants considering the change of ligand concentration at the cathode are modified for simple graphical analysis. The association constant of cadmium ion with chloride ion, here evaluated from polarographic measurements in a solvent consisting of 2 moles of water per mole of ammonium nitrate at 40", is shown to agree with a previously reported potentiometric evaluation.
*
Introduction As part of an investigation of the effect of water on association constants in molten salt solutions, association constants of cadmium ion with chloride ion, evaluated potentiometrically in a solvent consisting of 2 moles of water per mole of ammonium nitrate, have been reported previously.2 Melts such as NH4N032H20 are intermediate between concentrated aqueous electrolyte solutions and molten salts, and it was shown that a quasi-lattice model of molten salts is useful in interpreting association equilibria in these melts. Measurements were made in a concentration cell with silver-silver chloride or cadmium amalgam electrodes. Data at low solute concentrations and extrapolation to infinite dilution of metal ions and ligand have been demonstrated previously to be necessary to obtain consistent thermodynamic association constants in molten salt solutions,2, but since stable reproducible vaIues of the electromotive force of cells with amalgam electrodes are difficult t o obtain at Cd(N03)2 concentrations below mole ratio (moles of cadmium per mole of ammonium nitrate), it is difficult to carry out the required extrapolation to infinite dilution of the is the activslopes b In ~ / v x ( N o ~ ~ ~ / ~ R N H(YCd(N0dz ,cI. ity coefficient of Cd(N03)2 and R N H , is ~C the I mole ratio (moles NH4C1 per mole of ammonium nitrate) of ligand.) In this paper we report the results of potentiometric The Jourplol of Physic& Chemistry
and polarographic measurements of the association constant of cadmium ion with bromide ion at 50" in the solvent Ca(N03)?,4Hz0. Since polarographic measurements at low cadmium ion concentrations were found to be reproducible, the extrapolation to the thermodynamic association constants is facilitated. We extend the method of evaluation of successive association constants from polarographic measurements without a large excess of ligand to include polynuclear species and to obtain equations which lend themselves to simple graphical analysis. Polarographic measurements also are reported in the solvent NH4NO3-2H20 at 39.9" at concentrations below those studied potentiometrically, and the measurements are shown to agree with the previously reported potentiometric measurements.
Evaluation of Association Constants In order to evaluate thermodynamic association constants, it is necessary to extrapolate the data to a defined reference state, here infinite dilution of all solutes in the solvent Ca(N03)2.4H20or NH4N03. (1) To whom correspondence should be addressed at Reactor Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tenn. (2) J. M. C. Hess, J. Braunstein, and H. Braunstein, J . Inorg. LVucZ. C h m . , 26, 811 (1964). (3) (a) J. Braunstein, M. Blander, and R. M. Lindgren, J . A m . Chem. Soc., 84, 1529 (1962); (b) J. Braunstein and A. S. -Minano, Inorg. Chem., 3 , 218 (1964).
POLAROGRAPHIC AND POTENTIOMETRIC EVALUATION OF ASSOCIATION CONSTANTS
2H20.3& For reliable extrapolations, data at low ligand concentrations are required; on the other hand, the frequently imposed polarographic condition of a large excess of ligand over metal ion to minimize relative changes of ligand concentration at the surface of the mercury drop (so that concentrations at the electrode surface may be approximated by bulk concentrations) has been shown to lead to appreciable errors in t h e calculated association constants. The change of ligand concentration at the surface of the dropping mercury electrode has been discussed by Tomes for the case where a single complex species is the predominant form.4 Ringbom and Eriksson5 have modified the equations of DeFord and €Iume6 to take into account the change of ligand concentration at the cathode for the case of successive equilibria. However, the equations of Ringbom and Eriksson are expressed in the concentrations of associated and unassociated species at the surface of the mercury drop, neglect polynuclear complexes, and lead to equilibrium quotients rather than thermodynamic association constants since they do not include extrapolation to a defined reference state. Solution of their equations requires successive approximations and may be difficult if many species are present. Here we extend their treatment by deriving equations for the thermodynamic association constants, also taking into account the change of ligand concentration at the surface of the mercury drop. Our equations lend themselves to a simple graphical analysis since they are expressed in the stoichiometric concentrations of the components in the bulk solution rather than the concentrations of the unassociated species at the electrode surface. We may write' for the potential of a polarographic cell in which c,? is the concentration of reducible metal ion of charge Z at the surface of the dropping mercury electrode and Oc, is the concentration of metal in the amalgam at the surface of the mercury drop
Denoting the concentration of unassociated ligand L as cL, of unassociated metal ion MZf as cg, and of the associated species ML, ML2, M2L, etc., as CI, c2, c12, etc., respectively, the current along the polarographic wave may be written
i
=
+ - c?) + kz(c2 - + 2k12(CIZ + . . . = rz [(co + + cz + + . . . > - + + + + . . .)1 ko(c0
- COO)
k1(c1
c20)
stants of the species, and is the weighted mean diffusion current constant multiplied by the capillary constant.6 Substituting the association constants (4) J. Tomes, Collection Czech. Chem. Commun., 9, 81 (1937); J. J. Lingane, Chem. Rev., 29, 1 (1941); C. G . Butler and R. C. Kaye, J . EZectroanal. Chem., 0, 463 (1964). (5) A. Ringbom and L. Eriksson, Acta Chem. Scand., 7, 1105, 1146 (1953). (6) D. D. DeFord and D. N. Hume, J . Am. Chem. Sac., 73, 5321 (1951). (7) We consider the polarographic cell to consist of the dropping mercury electrode a t which reduction of a cation PIz+to metal hl takes place, a diffusion layer, the bulk solution, and a reference half-cell in the solvent SD-Hz0 where S+ and D - are the cation and anion of the supporting electrolyte.
l
1
11
I
MDz I MDz ' Hg(M)i SD-HzO i SD-H*O I diffusion layer j bulk solution 1 1
1,
I
I
I
I1
RlDz SD-H,O AD
1
A
reference half-cell
The reference electrode A is considered to be reversible to the cation A + which is present at a low Concentration. Reduction of 1 equiv of the cation AIz corresponds to the over-all process +
Hg(M)
+A +V
M D z ( 1 )
1
tsSD(I1) +pM(Hg)
+ ~ M D z ( I I+)
+ (1 - ~ A ) A D ( I I+)
+
~ A A W I ) tsSD(1) if the transference numbers ( t M , t A , and ts) are independent of the concentrations through the diffusion region. If the concentrations of .MDz and AD are very small so that the current is carried almost entirely by the ions of the supporting electrolyte, the transference numbers t M and t A will be negligible and the process may be approximated by
.4
+ Hg(M) + SD(II) + 31 MDz(I) +21 M(Hg) + AD(I1)
+ SD(1)
The potential of such a cell may be written
if the potential is assumed to have the same dependence on the activities of the components with the polarographic current flowing as in a reversible cell. The ai are the activities of the reactant and product components; it is not necessary to resort to single-ion activities in the equations. If the transference numbers are not zero or constant, there will be an additional term in eq i
In a dilute amalgam, the activity of the mercury, aHE(M),is (nearly) equal to the activity of pure mercury. The activity of the metal A is constant (unity); the activity of SD (the supporting electrolyte) is constant, and the activity of AD in the reference half-cell is constant. The potential may therefore be written
e1
c120)
2c12
2735
(COO
c10
c20
2c120
(2)
where the superscript zero denotes concentrations at the surface of the mercury drop, the k are the products of the capillary constant by the diffusion current con-
or
The activity coefficient of the metal in the dilute amalgam is a constant (unity, if the reference state is infinite dilution in the amalgam).
Volume 70,Number 9 September 1966
J. BRAUNSTEIN, A. ALVAREZ-FUNES, AND H. BRAUNSTEIN
2736
(3)
etc., into (2)
+
i = Eco(1
KlCL
+ + + . . .) - WCOO(1 + + K~KzcLO~ + ~KIKI~C$CLO + . . .) KIK2CL2
KlCLO
2KIK12COCL
(4)
If the species do not follow Henry's law, the equations will also contain the activity coefficientsof the species.3a Since the diffusion current i d is the current when the concentrations of the reducible species a t the surface of the mercury drop are zero id
- i = ECOO(1
+
KlCLO
+ +
K1K2CLOZ
2KIK12COOCLO
+
* *
.)
RT --
ZF
In
i i t - i k,-(1 KlcL0 K~K~cLO' ~ K ~ K ~ ~ c .o. .) ~ c (6) L~
+
+ +
___ id
+
RT
E
-In -(1 Z F k,
+ KICL~I/,+ +
KIKZCL01/22
+ . .) *
(7)
where the subscript indicates the potential and the Concentrations at i = &/2. Subtracting EO,,,,the value of E,,, in the absence of ligand, from (7)
RT ZF
E Lo
KlcLo1/,
The coefficients of the stoichiometric concentrations in the Taylor's series are obtained by first expanding the right-hand side of eq 9 in the series In (1 x) = x
+
- '/2x2 + . . . .
In Fo
= KIcL0i/,
+
(K1K2
- 1/2K1z)cLo21/p +
+
2K1K12CLO~,,C00~/2
I
..
(10)
~~
2K1K12C0~1/,CLO1/,
AEi,, = El/, - E",,, = -- In-(1
lim (In Fo) = 0 To-0 TL-0
The first derivatives with respect to the stoichiometric concentrations are
When the current is half the diffusion current =
are evaluated from eq 9 with the material balance of ligand and metal ion at the surface of the electrode and in the bulk solution. The coefficient of the constant term in the expansion is zero since
(5)
The concentration of metal in the amalgam at the surface of the mercury drop is c,O = ilka where IC, is the product of the capillary constant by the diffusion current constant of metal in the amalgam. Substituting the expressions for cao and COO into the equation for the potential gives
E =
Equation 9, which relates FO to the concentrations of the species at the surface of the dropping mercury electrode, is an extension of eq 7 of Ringbom and Eriksson which does not neglect the possibility of formation of polynuclear species. Solution of eq 9 for the association constants requires successive approximations. However, the evaluation of the constants can be greatly simplified and the need for successive approximations eliminated by considering the Taylor's series expansion of eq 9 in powers of the stoichiometric concentrations of the metal ion and ligand in the bulk solution, which we designate To and TL. The coefficients in the expansion
+
(Reference 7 continued) If the deviation from Henry's law (Le., from the Nernst equation, which is observed to hold in these solutions for cadmium ion in the absence of halide, or halide in the absence of cadmium ion)2 which are observed when ligand is added are due t o the formation of associated species, and if the associated species as well as the unassociated metal ions and ligand follow Henry's law, we my write U M D = ~ CMCD~. C M is the concentration of "unassociated" metal ion and will be equal to the stoichiometric concentration of metal ion in the absence of ligand. Hence
+ K I K ~ C L+~ ~KIKIZCO~I/,CL~I/, ~~/, + . . .) (8)
or, using the function Foof DeFord and Hume6
The Journal of Physical Chemistry
(If the species do not follow Henry's law, the species activity coefficients should be included.) Noting that C D , the concentration of the anion of the supporting electrolyte, is virtually constant, and rewriting the concentrations of metal in the amalgam at the surface of the mercury drop and the concentration of M z + in the solution at the surface of the mercury drop as C M ( H ~ )= c80 and c X z + = coo, we have
POLAROGRAPHIC AND POTENTIOMETRIC EVALUATION OF ASSOCIATION CONSTANTS
2737
and
(2)
(-)b In FO bCo01/,
(d.TL)To
+
In FO= KITL Substitution of (10) in (11) and (12) and taking the limit as 7'0 and TI,vanish gives
TL
T-0 TL-0
T-0
TL-0
TL
(13)
and
The concentration of unassociated ligand at the surface of the electrode may be related to the concentrations of unassociated species in the bulk solution and to the stoichiometric concentrations. If the current is carried through the solution only by the supporting electrolyte, a charge balance on all the solute species leads to the relation analogous to that of Tomes4 for one species
- ie = ZJO+ (2 - 1)J1+ . . . + (22 - 1)Jri + . . . -J L = Z/CO(CO - coOi/J + (Z - l)ki(~i- ci0i/J + . . . - ~ L ( C L - cLO:/,) - ._. .
0 = ie
(15) JO, J1, etc., are fluxes of species of charge 2, 2 - 1, etc., and zk and ie are the current densities of the positive and negative solute species. Substituting eq 3 in (15), differentiating, and passing to the limit, we have
Hence
Tn
=
KI
The coefficients of the higher terms in the expansion may be found in a similar manner (retaining the appropriate higher powers of the concentrations until passing to the limit). The resulting expansion, up to quadratic terms, is
b In Fo = ( ~ ) c o o l , , ( ~ ) T o
b In Fo
T-0
Ti-6
TL
CL01/2
and
b In Fo
m il (),
(11)
+ (KIKz- '/ZKI~)TL~+ (2K1K12
- Kl2)ToT~+ ...
(17) Since eq 17 has the same form as the expansion of the stoichiometric activity coefficient In 1 / 7 0 derived in an earlier paper, the same simple graphical analysis may be employed to calculate the association cons t a n t ~ . Only ~ ~ the stoichiometric concentrations are used in the calculations so that successive approximations are not required. Extrapolation to zero concentration leads to the thermodynamic association constants. Experimental Section Polarographic measurements were made with a controlled-potential and derivative voltammeter made by Indiana Instrument and Chemical Corp. (ORNL Riodel Q-1988)s and a Sargent SR recorder with a 12.5mv range plug. Initial and span potentials as well as some potentials along the polarographic wave were measured with a Keithley Model 660 differential voltmeter or a Leeds and Northrup Type K potentiometer. The polarographic cell consisted of a 100-ml Berzelius beaker clamped in a water bath controlled at 39.9 or 50.0'. The cell contained about 0.5 mole of ammonium nitrate in 1 mole of water or 0.5 mole of calcium nitrate tetrahydrate (which melts at 42.7'), the dropping mercury electrode, a platinum foil counter electrode (about 1 cm2), and a silver-silver chloride or silver-silver bromide reference half-cell. In order to minimize uncertainties due to liquid junction potentials, the reference half-cell was prepared with the same solvent as was used in the cell: either NH4N03-2H20 or Ca(NO&.4Hz0. The reference half-cell was prepared by a method similar to the method of preparation of reference half-cells used in molten salt investigatiom3b An asbestos fiber was sealed through the end of a length of 6-9-mm Pyrex tubing. The tubes were tested for leakage overnight in distilled water and then filled with solvent saturated with silver chloride
b In F , T-0
TL-0
TL
(8) M. T. Kelley, H. C. Jones, and D. J. Fisher, Anal. Chem., 31,
1262 (1960).
Volume 70, Number 9 September 1966
J. BRAUNSTEIN, A. ALVAREZ-FUNES, AND H. BRAUNSTEIN
2738
or silver bromide. A silver wire previously coated with silver halide was then inserted. The silver wire was coated with halide by first flaming the end to remove oxide and then quenching it in a dilute solution of KC1 or KBr; a dilute solution of silver nitrate was then added. Groups of three to five reference halfcells prepared in this manner remained stable within several tenths of a millivolt over periods of several months when compared with each other and with similar electrodes in aqueous potassium chloride (bromide) solutions. Cadmium nitrate mas added to the cell either as the solid tetrahydrate or, with a micrometer syringe, as an aqueous solution. Ammonium chloride was added as weighed pellets of the dried salt. Potassium bromide was added as an aqueous solution by means of a micrometer syringe. The quantity of water added with the solute was insufficient to change the concentration or the half-wave potentials significantly. The solution in the cell was deoxygenated with nitrogen which was first bubbled through a large bulb of solution of the same composition as the solvent in the cell in order to prevent loss of water from the cell by evaporation. The half-wave potential was determined from plots of E us. log i / ( i d - i) or by the method of M e i t e ~ . ~ The potentiometric measurements were made in the same cell used for the polarographic measurements and using the same silver-silver bromide reference halfcells. The cadmium amalgam electrode consisted of a J-tube of 9-mm Pyrex tubing containing enough cadmium amalgam, prepared as described previously,Z to just fill the short arm of the J, which was immersed in the solution. Electrical contact was made by means of a platinum wire in the long arm of the J. Saturated nitrogen was passed through the solution which was stirred vigorously during the measurements. Solutes were added as described above.
Cd-Br-Co(N@3)2-4H20 A A POLAROORAPHIC
-00
6
IO
8
YO
00
58
'
~ c d---XI0 [ - x l o ~A.o
Figure 1. Extrapolation of the polarographic limiting slopes [ ( b 1n R / b R B r ) ] and the potentiometric limiting slopes [(a In ~ / Y C ~ ( N O ~ ) ~ / ~ ) R B ~ )to ] R infinite B~=O dilution of cadmium nitrate to obtain the thermodynamic association constant.
plotted as a function of the concentration of cadmium ion as the circles in Figure 1 and extrapolated to the association constant.lv2 Concentrations are expressed as the mole ratios
and
R B= ~
nKBr ~
2ma(NOs)?
where the n are numbers of moles of the components indicated by the subscripts. The polarographic function In Fo = - (2F/RT) AE,,, In (ko/K),defined by eq 9, was calculated from the cumulative change, AE,,,, of the polarographic half-wave potential with successive bromide additions and from the were diffusion currents: the slopes ( b In FO/bR~r)RB,-O evaluated graphically3* and are plotted as the triangles in Figure 1. Extrapolation to zero cadmium concentration gives the thermodynamic association constant3~ for the formation of CdBrf in Ca(N0&.4H20. I n FigResults in,NH4=O ure 2 the polarographic slopes (bIn F O / d R ~ l ) ~ C From the potentiometric measurements, the logaNO3-2H20 at 40" are plotted together with the potenrithm of the activity coefficient of Cd(i\r'O&, In from tiometric slopes (b In l/Ycd(SOa),/bRCI)RC,=O YC'd(SOa)? = 2F/RTAE, was calculated from the the previously reported potentiometric measurements. cumulative change of emf, AE, on addition of bromide In order to demonstrate the agreement of the polaroto the potentiometric cell containing a fixed concentragraphic and potentiometric results at the lowest contion of cadmium nitrate' in the solvent Ca(N03)2.4H20 centrations, the slopes at the lowest cadmium concenat 50". In the absence of bromide, the emf of the cell trations have been plotted in Figures 1 and 2 on an followed the Nernst equation in the concentration of expanded abscissa. The filled circles are potentiometric cadmium nitrate. At each cadmium concentration, results and the filled triangles are polarographic results. the limiting slope It is seen that the slopes (and their extrapolation to the association constant, with eq 16) of the polarographic and potentiometric data are consistent.
WLS
evaluated graphically and these slopes were then
The Journal of Physical Chemistry
+
(9) L. Meites, J . Am. Chem. Soe., 72, 2293 (1950).
2739
POLAROGRAPHIC AND POT~NTIOMETRIC EVALUATION OF ASSOCIATION CONSTANTS
Cd-CI-NH4N03-2H20 ;Ta40.C A A
poLAR00RAPHIC
o POTENllOYLTRlC 0
3
8
400
POTIWTIDMMRIO
e3 I
I
f
2
3
4
,
5
Red[=$:
Figure 2. Extrapolation of the polarographic limiting slopes [ ( bIn F o / M 2 c i ) ] ~ ~ and ~ d the potentiometric limiting slopes [ ( b In ~ / Y C ~ ( N O , ) ~ / ~ R C I ) ] R C , -to O infinite dilution of cadmium nitrate to obtain the thermodynamic association constant.
1
2
1
4
Figure 4. Plots of log Fo (polarographic, A ) and log l/YCd(NO,), (potentiometric, 0) vs. the concentration of bromide at fixed concentrations of cadmium nitrate.
1500 f 200. The association constant for CdC1+ in NHdN03-2HzO at 40" is 300 f 15, in agreement with the value reported previously, from emf measurements with cadmium amalgam electrodes, as 315 20.2 In Figure 3 it is seen that reversible waves were obtained with excess ligand or with excess metal ion, In Figure 4 are plotted some of the curves of In Fo (polarographic) and In l/-,a (potentiometric), The agreement between the polarographic and potentiometric results confirms that a large excess of ligand, which would vitiate the method for the evaluation of thermodynamic association constants, is not necessary in the polarographic measurements, but that extrapolation to infinite dilution of the solutes is required. The polarographic and potentiometric data are available as an AD1 document.lO
*
, 01 , -0.0
-06
1
op
-0.2
0.2
Oh
I
os
LogFigure 3. Potential of the dropping mercury electrode os. log ;/(id - i).
The association constant for the formation of CdBr+ in the solvent Ca(NO&.4H20 a t 50" is 3900 f 200 (moles of bromide/mole of nitrate)-'. Using the graphical methods described previously,2*aathe dinuclear association constant Klz is found to be 50 f 200, indicating the absence of dinuclear species. The association constant Kz for the formation of CdBrz is
Acknowledgments. This work was supported under USAEC Contract No. AT-(30-1)-2873, Report NYO2873-13. We are grateful to Dr. Douglas Inman for valuable discussions concerning the change of ligand concentration at the surface of the dropping mercury electrode. (10) The data have been deposited as Document No. 8883 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington, D. C. A copy may be secured by citing the Document number and by remitting 81.25 for photoprints, or $1.25 for 36" microfilm. Advance payment is required. Make checks or money orders payable to: Chief, Photoduplication Service, Library of Congress.
Volume 'YO, Number S Sepletnber 1988
