Sept., 19G2
CATALYTIC POLAROGRAPHIC
REDCCTION O F VANADIUM(III)
1699
CATALYTIC POLAROGRAPHIC CURRENTS FOR THE REDUCTION OF VANADIUM(II1) IN THE PRESENCE OF VANADIUM(TV)l BY JOHN W. OLVER~AXD JAMES W. Ross, JR. Department of Chemistry and Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge 39, Mass. Receiced April
(9,
1968
Polarogrzphic catalytic currents have been treated for the case where the primary reduction product of the diffusing electroactive species is produced cia a second reduction path a t the electrode surface. The preciirsor species for the second path is assumed to be present a t high, constant concentration. Rate constants for the reaction V(I1) V(1V) ZV(II1) have been calculated from catalytic current,s for the polarographic reduction of vanadium( 111)in the presence of vanadium( IV) in sulfate media.
+
.-f
Introduction Equations describing polarographic catalytic currents have been developed by several workers using a linear diffusion approximation to the dropping mercury e l e ~ t r o d e . ~ - Koute~k$~,’ ~ has treated the same problem using a more rigorous expanding sphere approximation. Reaction rate constants >whichare in good agreement with those calculated by independent experimental methods may be calculated from catalytic currents using Kouteckfs solution, provided the following conditions are met. The reaction must follow the path
than 0 so that reaction 2 may be considered pseudounimolecular in R and concentration polarization may be neglected in reaction 3. h: is the unimolecular rate constant for reaction 2. K o ~ t e c k $ ~has J tabulated values of io/& as a function of IC for the case where Z is not reduced directly. io and id are the experimentally observed currents for the reduction of 0 in the presence and in the absence of 2, respectively. In the present case we consider the expression
+ ne- +R a t the electrode R + Z --+ 0 in the solution phase
where i’, is the current due to the reduction of 0 which would be obtained for our reaction scheme if the direct electroreductioii of Z did not take place. i~ is the experimentally observed total current due to the simultaneous reduction of both 0 and 2. i, is the experimentally observed current for the reduction of Z in the absence of 0. f(iJ is the enhancement of the total current due to the increased rate of regeneration of 0 when Z is reduced directly a t the electrode. All currents are assumed to be measured a t the same potential. Attempts to extend Kouteck$’s treatment to include the effect of reaction 3 were abandoned due to the complexity of the mathematics. We have chosen instead to use the simpler linear diffusion approximation to calculate f(i.). Rate constants, however, are calculated from Kouteck9’s tables using our values of i’o/id. The boundary value problem which applies to our model, assuming the diffusion coefficients D of 0 and R are equal, is
0
I n addition, the second step must be pseudo-unimolecular with respect to species R, i.e., Z must be present a t high, constant concentration in the electrode vicinity. We have recently examined several inorganic systems giving rise to catalytic currents in which the species Z is reduced irreversibly to R when 2 is present in sufficiently high concentration to ensure unimolecular regeneration of 0. In the present paper we have developed a method for correcting catalytic currents for the reduction of Z and have used the method in calculating rate constants for the oxidation of vanadium(I1) by vanadium( [V) in sulfate media. Theoretical The reaction scheme considered here is
0
+ rile- = R
a t the electrode
(1)
in the soh. phase
(2)
a t the electrode
(3)
IC
+ 2 +,0 Z + me- = R
R
We assume the concentration of Z is much larger (1) Taken in part from the Ph.D. thesis of John W. Olver, Massachusetts Institute of Technology, June, 1961. Presented in part at the September, I961 National Meeting of the American Chemical Society. This work was supported in part by the U. 9. Atomic Energy Commission under Contract AT(30-1)905. (2) American Chicle Co. Fellow, 1959-1960. National Science Foundation Summer Fellow, 1960. (3) P. Delahay and G. Stishl, J . A m . Chem. Soc., 74, 3500 (1952). (4) Z.Pospisil, Collection Czech. Chem. Commun., 18,337 (1953). (5) S. L. Miller. J . Am. Chem. Soc., 74, 4130 (1952). (6) J, Koutecltl, Collection Czech. Chem. Commun., IS, 311 (19.53). (7) J: Koutecli? and J. CiEek, ibid., 21, 1063 (1966).
CO* is the bulk solution concentration of 0, F the faraday, and A the electrode area. Except for the boundary condition a t n: = 0 this problem is identical with that solved by Delahay and Stiehl.3 The
1700
JOHN W.OLVERASD JALIES W.Ross, JR.
T’ol. 66
The last term in (6) represents the enhancement f(iJ evaluated a t the end of the drop life. Substitution in (4) yields
- E,Volts vs SCE. Fig. l.-Vanadium polarograms in sulfate medium: id = 1.0 X LUvanadium(II1); i, = 0.010 M vanadium(1V); i~ = 1.0 X M vanadium(II1) and 0.010 J4 vanadium(IV). All polarograms obtained in 0.4 M HzS04 and 0.15
N aHS04.
I
70
I
I
90
I80
Ratio,
I
270
Gt
Fig. 2.--kf app as a function of the dimensionless parameter C,/Co2& app., showing the approach to pseudo-unimolecular behavior.
solution is readily obtained by a Laplace transformation yielding
+
where g(kt) = { I - exp(kt/2)[Io(kt/2) I1 (Atla) I}. I,(kt/2) and 11(kt/2) are modified Bessel functions, tables of which are readily available.* The quantity
zL[;;
is identical to the gradient of concentration of 0 a t the electrode surface if 2 were not reduced. Following Delahay and Stieh13 we transpose this result to the case for the dropping mercury electrode by multiplying ( 5 ) by the factor (7/3)”2 and replacing t by the drop time r . Since
we have ( 8 ) Janke, Emde, and Ldsoh, “Tables of Higher Functions,” 6th Ed., McGraw-Hill Book Go., New Yark, N, Y., 1960,
*k‘,Loasoh. Editor,
The unimolecular rate constant k may be calculated from (7) by first assuming g ( k 7 ) is zero and determining a trial k from Kouteck9’s tables. IJsing this trial k the bracketed terms in (7) may be evaluated and k recalculated from the corrected value of io’/id. In the present work one or two cycles were sufficient to define 7c to within experimental error, Experimental Reagents and Apparatus.-Electrolyte solutions were prepared from reagent grade sodium sulfate, sodium bisulfate, and sulfuric acid t o give the desired pH values in the range 0 to 2 while maintaining the ionic strength of the electrolyte a t 0.6. A stock solution approximately 0.6 M in vanadium(1V) and 1 J4 in sulfuric acid was prepared from reagent grade vanadyl sulfate. A stock solution approximately 0.5 M in vanadium(II1) and 1 M in sulfuric acid was prepared from the vanadium(IV) solution by controlled potent,ial electrolysis until a polarogram of the solution showed no vanadium( IV) or vanadium( 11). Both vanadium stock solutions were standardized by potentiometric titration with potassium permanganate. Because vanadium(111) solutions are oxidized by air, precautions were taken to minimize air oxidation of the main V(II1) stock solution although the presence of a small amount of vanadium(Is’) in the vanadium( 111) stock solutions was not detrimental to the kinetic interpretation. All solutions were deaerated with prepurified tank nitrogen and were thermostated to 1 0 . 2 ” in a water bath before polarograms were determined. -411 polarograms were obtained using an E. H. Sargent and Co. Model X X I polarograph without damping. The Brown recorder had a full-scale deflection time of 10 see. All currents were measured a t the maximum deflection of the recorder pen. The capillary used had a 6-see. drop time and an rn of 0.92 mg./sec. over the potential range covered in this work. Experimental Method.-Typical polarograms required for a single determination of the unimolecular reaction rate are shown in Fig. 1 . The difference between curves i~ - i, and i, represents the catalytic portion of the total current due to the regeneration reaction
The formal second-order reaction rate constant kf = k / C * v c ~ v ,may be determined from thme polarograms as outlined in the Theoretical section, provided the C * V ( I ~ is sufficiently high to ensure pseudo first-order kinetics in (8). This condition is satisfied when vanadium(1V) is not reduced a t the electrode, and the following condition holds6
c*v(lv)
>> 0.665kf
C*VCIII,
Since vanadium(1V) is reduced in the preapnt case, this condition is undoubtedly not sufficiently stringent. In order to establish that reaction 8 meets the pseudofirst order restriction, apparent rate constants ha,, were determined in each electrolyte used for different ratios of C*V~IV,/C*V~III,. As expected, Icf values increased with increasing ~ * v c ~ v ) / ~ *approecimg v ~ ~ ~ ~ ) ,e limiting value btk
CATALYTIC POTAROGRAPHIC REDUCTION OF VAUADLCM(ILI)
Sept., 19P2
large ratios. When plotted as shown in Pig. 2 all the experimental values of kf&pp in the various electrolytes fell in the same curve. At values of
EFFECT OF pH
C*V(IV)
PH
greater than 30, kf Rpg is equal to 0.95 of its limitin value k f . -411 lei values reported in this work were obtainecfunder conditions where the limiting value is approached to within the experimental error of 5%. The validity of the linear diffusion approximation used in the derivation of f(&) was tested by determlning kf as a function of iB. k f is independent of potential although both i T and i, increase with increasing potential. If calculated values of Icf are independent of potential, it may be assumed that the correction .termf(iB)is a good approximation to the enhancement caused by reaction 3. The results are shown in Table I.
TABLE r RATECONSTAiVTS WITH POTENTIAL
VARIATION O F CALCULATED
kr, E, $7.
vs. s.c.e.
-0.68 - .70 - .72 - .74 - .76
iT.
i z,
pa.
pa.
id, pa.
3.20 3.59 4.04 4.60 5.30
0.86 1.15 1.51 2.00 2.66
1.23 1 ,24 1.24 1.25 1.25
1. mole-’ sec. -1
14.9 15.3 15.5 15.5 15.
Although current i, varies from 0.86 to 2.66 &a.over this range, the calculated values of kf are constant to within experimental error. At potentials outside the range, there is considerable scatter in calculated values of kf because of t,he lower precision when measuring currents in steeply rising portions of the i~ wave.
Results Kinetic Results.-The order of reaction 8 with respect to each of the reacting species was determined by measuring the magnitude of the catalytic contribution to the total current as a function of the concentration of one reacting species while holding the concentration of t,he other species constant. ‘The reaction was first order with respect to both vanadium(I1) and vanadiuni(1V) over 10-fold concentration ranges. Rat’econstants obtained a t various temperatures and pH values are summarized in Table 11. Discussion of Kinetic Results.-From published hydrolysis data for vanadium(II1) and vanadium(IV),g between pH values of 0 and 2, aquovanadium(11I) and VO+.2greatly predomina,te. There(9) J. Rjerrum, G . Schwareenbach. and L. G. SillBn, “Stability Constants,” Part I, ‘The Chemical Society, London, 1057.
0 0 2 6 .8 1 3 2 0
AND
TABLE I1 TEVPGRATURE ON RATE (in 1. mole-’ 8ec.-1)
1’701
CORSTAXTS
r-------Tetnperature, 18 25
OC.33
43
3 0
6 4 10 15
13 19 25
28
1s
37 45
6.G
7 7 8.9 9 5
19
fore the several-fold change in the magnitude of the reaction rate constant over that pH range cannot be explained by the formation of higher hydrolyzed species a t higher pH values. Furthermore, the difference in sulfate ion concentration between the two solutions of lowest pH is practically negligible, thereby indicating that the variation in rate constant is not a function of sulfate ion concentration. On the other hand, the variation in bisulfate ion concentration between the two solutions of lowest pH is large (0.99 M a t pH 0 and 0.54 M a t p H 0.2). Also, the Concentration of bisulfate ion becomes quite small a t a pH value of 2 where the reaction rate is nearly independent of pH. In view of the above facts, the data can be explained adequately by postulating the existence of a vanadium(1V)-bisulfate complex, the stability constant of which is only of the order of 1. The proposal that the VO+2 is being complexed is supported by the fact that the El/, for VO+2 shifts anodically with increase in HS04- or H+. The rate constant for the reaction path involving the bisulfate complex probably is small compared with that for the simple hydrolyzed species. Such a view would be reasonable if one of the hydroxy groups in the vanadium(1V) species were replaced, since usually a dihydroxy complex reacts much more rapidly than a monohydroxy complex of the same central species. A rate expression for the reaction between V(I1) and V(1V) in acidic sulfate media can be postulated as Rate
=
kf1(VO+2)(V+2)4- h2(VOHHS04+Z)(V+2)
In order to say anything more definite about the rate expression for this reaction, knowledge of the nature and stability of all possible sulfate or bisulfate complexes of the reacting species would be required.
