J. E. Crooks and R. S. Bulmer The Univers~ty,Canterbury, Kent England
Kinetics from Polarography An experiment for the teaching laboratory
The polarographic wave is essentially a kinetic phenomenon. Its height, i.e., the current which flows at a polarized mercury cathode, is usually dependent on the rate a t which reducible species can diffuseto the mercury surface, which, by Fick's Law of diffusion, is in turn linearly dependent on the concentration of the reducible species. I n some systems, however, the current is controlled by the rate at which the reducible species is formed by a chemical reaction. If the reducible species exists in equilibrium with nonreducible species in the bulk of the solution it will soon he depleted near the cathode, and the current can only be maintained by the production of more reducible species. If this process is sufficiently fast, the rate at which nonreducible species diffuse into the depleted region round the cathode will also affect the current.
rate of production of Y is small compared to the rate of diffusion of X into the reaction layer. From geometrical considerations A = 0.85 X rnX/d,'/r X 8/acmP
where m is the mass of mercury in grams flowing per second, and t is t,he drop time. The factor 0.85 includes T and the density of mercury, and the factor 3/6 converts the maximum area to the mean area averaged over the drop time. The correct procedure for the calulation of p has led t o much dispute. For simplicity, it may be assumed that there is equilibrium outside the reaction layer, and a stationary state inside. By Fick's first law, il,is proportional to the concentration gradient of Y at distance x = 0 from the electrode, i.e., ih
Simplified Theoretical Treatment
A sophisticated general treatment for the calculation of rate constants from polarographic currents is given by Brdicka, Hanus, and Koutecky (1). The treatment which follows is much simplified, but illustrates the general principles. Consider the system X
k,
eY
n-,
-
(3)
=
(4)
ZSAD~(~Y/&),~
where D y is the diffusion coefficient of Y. If a uniform concentration gradient across the reaction is assumed ib =
ZSADY[YI.-~/P
(5)
From eqns. (1) and (5) r' = DY[YI.-J~I[XIO
As chemical equilibrium exists at x'=
Cathode
(6)
p,
K = [YI/[XI.
so that
,' = (DyK/k,)'/. The calculation of k , from the polarographic current involves the concept of the "reaction layer," introduced by Wiesner (8). The reaction layer is the region of solution next to the cathode, extending .u cm from the mercury surface. .u is defined as the distance out from the cathode in which, on the average, no reducible species, Y, traveling to the cathode will have time to he converted to the irreducible species, X, before reaching the cathode. Therefore, all the species Y produced in the reaction layer will be reduced at the electrode. The mean current, averaged over the drop time, is thus, by Faraday's law of electrolysis, given by ir; = zSAd[Yl/dt
(1)
where charge zS coulomb is necessary for the reduction of one mole of Y, and A is the mean surface area of the electrode. The mean rate of production of Y is given by d[Yl/dl = h[XI
(7)
Substituting eqns. (2), (3), and (6) in eqn. (1) gives (8)
i, = 0.5lr3[X]0(mt)'l~(DyKkl)'/~
If the units of [XIa are moles ~ m - ~of , m, gm sec-', and of D, em2 sec-', then it will be in amperes. By polarographic convention, [XIa is usually expressed in mmoles 1-', m in mg sec-', and ix in pA. I n these units, equation (8) becomes I t is often more convenient to obtain an answer in terms of id, the mean current which would flow if the conversion of X to Y was fast compared with the diffusion of X, so that the current is controlled by the rate of diffusion of X. The Ilkovic equation gives
(2)
[XI, the concentration of X within the reaction layer, may be considered as constant, and equal to 1x10, the concentration of X in the bulk of the solution, if the
If the current is partly controlled by the diffusion rate, a slightly more complex calculation, given by Strehlow (I) gives Volume 45, Number 1 1 , November 1968
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725
0.81(tKkl)L/1[1 - 0.81(tKkl)Vs] (12) which reduces to (10) if &/id is small, i.e., the reaction is slow. More sophisticated calculations give other values for the numerical factor 0.81; e.g., Vesely and Brdicka (3) obtained a value of 0.573 and Weber and Koutecky (4) obtained a value of 0.886. An important diagnostic test for an observed polarographic current being chemically controlled is provided by the variation of the current with the height of the mercury column. Equation (10) can be written as %/id
=
id = 607z[X]ODx'/*(mt)'/arnm/~
mt is the mass of the mercury drop, which depends only on the conditions at the cathode, e.g., the capillary bore and the surface tension, and is independent of the column height. The rate of flow of mercury is proportional t o the mercury pressure, and therefore to the height, h, of the column, so that id .r h'/% (13) On the other hand, none of the terms on the right hand side of eqn. (8) varies with column height, so that ih is independent of h. Typical Polarographic Reaction-Rate Studies
Much work has been published on the measurement of reaction rates using polarography, in which the theoretical treatment is of various degrees of sophistication. Strehlow (6) gives a comprehensive survey of the results of such measurements. Both inorganic and organic reactions have been studied, with rate constants ranging from 10 to 10'0 1 mole-' sec-'. The rate of dissociation of an acid can he measured if the acid molecule is easily reducible a t the mercury electrode (e.g., an a-ketocarboxylic acid) and the conjugate anion is only reducible at a much more negative potential. Pyruvic acid provides the clearest example of this. I n a buffer a t pH 4 the half-wave potential of pyruvic acid is -1.08 v, whereas in a buffer a t pH 7 the half-wave potential is -1.58 v. At intermediate pH values both waves are observed, and are of equal height at pH 6.15. As the pK, of pyruvic acid is 2.4, these waves cannot be simply explained in terms of changing acid and conjugate anion concentrations with changing pH. The wave a t -1.08 v is controlled not by the bulk concentration of reducible acid, but rather by the rate a t which the anion reacts with hydrogen ion in the reaction layer. Similar waves have been studied for over thirty ketocarboxylic acids, ( 5 ) , and rate constants of the order of 101° 1 mole-' sec-' have been calculated for the combination of the conjugate anion with hydrogen ion. As the rate of production of the reducible species is comparable with the rate of diiusion of the nonreducible species into the reaction layer, a more sophisticated method of calculation than that given in the previous section is required. For such strong acids as phenylglyoxylic acid, pK, 1.2, the reaction layer treatment gives rates of the order of 10121 mole-' sec-', much greater than the diffusion controlled limit. The cause of the error can he seen if the reaction layer thickness is calculated. This turns out to he a fraction of an Angstrom, which is physically absurd. The reaction layer model is in726
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lournol of Chemical Education
valid under these conditions. However, if a reasonable minimum pogsible value for the reaction layer thickness, say 10 A, is substituted into the equations, a rate constant of the expected value of 10" 1 mole-' sec-' is then found. If the acid is not easily reducible, the reduction of hydrogen ions may be observed instead. If a reducible base, e.g., azobenzene, is present, the current due to the reduction of the protonated base gives a measure of the rate of production of hydrogen ions from the acid. Using this technique Ruetschi (6) found the rate of combination of acetate ion with hydrogen ion to be 4.4 X 101° 1 mole-' sec-I. The half-wave potential of a metal ion is, in general, considerably altered by complex formation. At suitable concentrations of metal and ligand two waves may he observed, the metal being reduced at the less negative potential. The height of the metal wave sometimes gives a measure of the rate a t which the complex ion dissociates. Complexes studies include Cd(CN)42-(7) for which the association rate is 1081mole-' sec-' and Cd(EDTA)2- (8) for which the association rate is lOSl mole-' sec-'. The rates of oxidation-reduction reactions of transition metal ions have also been studied. The phenomenon here is usually somewhat different to that which has so far been considered. After the reducible ion has been reduced to the nonreducible form a t the electrode, it is then re-oxidized by the oxidizing agent in solution, which itself is not reduced at the electrode, e.g., for the reduction of hydroxylamine by ferrous ion, the scheme is Fe3+
-
-
+ HxNOH k
2Fe2+
Fe" at the cathode
+ NH, + 20H-
2Fe8+
Polarographic currents which are the consequence of such a reaction are called catalytic currents, as the electro-reducible species (e.g., Fe3+) acts as a catalyst for the electrolytic reduction of the electro-inactive oxidizing species (e.g., H2NOH). The theoretical treatment has to be modified to allow for the cyclic nature of the process. The rate constant k in the example cited has been calculated as 198 1 mole-' sec-', which is in good agreement with the value found by direct observation (9). The theory of kinetic currents was first devised to explain the waves due to the reduction of sugars such as glucose. Wiesner (10) showed that the height of the wave was controlled, not by the concentration of free, open-chain aldose, but rather by the rate a t which the cyclohexose ring opened to give the reducible tautomer. Wiesner and co-workers (11) have worked out the complete kinetics of the system a glucose
= aldme =@ glucose kl
kl,
kl'
k,
I n 0.07 M phosphate buffer, a t pH 6.9 and ionic strength 0.27
Student Experiment
The observation that the height of a polamgraphic wave may be determined,by the rate of a chemical re-
action is not only of interest to kineticists. Anyone who uses polarography should be aware that the wave height is not necessarily an unambiguous measure of concentration. An experiment demonstrating a kinetic wave and the calculation of a rate constant from it is included in the set of experiments in polarography performed by students at this University. It is difficult to find a system which gives a clear demonstration of a kinetic wave for which only a simple calculation is needed to give the rate constant. All too often the reaction in systems previously studied is so fast that the theoretical treatment is complex. The simplest reaction to investigate is the dehydration of formaldehyde hydrate.
The polarographic wave obtained from dilute aqueous solutions of formaldehyde (Ezl, = -1.7 v), may be observed with the simplest polarograph. The wave height is controlled by the rate a t which irreducible formaldehyde hydrate reacts to give reducible formaldehyde. This reaction has been much studied, both by polarography (3, 13-14), and also by a scavenger method (15). I n the scavenger method the rate of reaction is measured by the rate of formation of a carhonyl condensation product, e.g., the semicarbaaone. A review is given by Bell (16). The dehydration is subject to general acid-base catalysis, so that the ohsenred first order rate constant kt is given by kt = ka
+ kx[H+] + kox[OH-I + Zkr[Al + ZkalBI
where A and B are other acidic and basic species present. Under suitable conditions it is possible to observe $, the rate due to water catalysis. Bell and sec-'. This is Evans (15) found $ = 5.1 X probably the most reliable value, as its calculation required the fewest theoretical assumptions. Landqvist ( I d ) ,using a polarographic technique and a sophisticated method of calculation found ko = 3.4 X 10W3sec-' (result corrected for a modern value of the dehydration equilibrium constant). As the reduction of formaldehyde produces hydroxyl ions the reaction in unbuffered solutions is autocatalytic HCHO
+ 2Hn0 + 2e-
-
CH.OH
+ 20H-
To obtain a value for the water-catalyzed rate it is necessary to use dilute phosphate buffers, of pH 7, and extrapolate to zero buffer concentration. In 5 X M buffer, k, is 20% greater than ko. Suitable concentrations for use in a student experiment are [Formaldehyde] = 6 X 10-'M(i.e., 5 ml36% formalin in 1 1) [H~POA-]= [HPOI-] = 2.5, 5.0, and 7.5 X 10-aM
The reaction may be treated as first order because the concentration of free formaldehyde is so low, around 10-6M. I n a typical experiment a t 20°C, the values below were found [Jformaldehyde] = 6 . 2 X 10-¶M [&PO,-] ik =
=
[HP042-l = 2 . 5 X 10-SM
1.1 p 1 ; t = 2.4sec; m = 1.98 mgsee-I
It is essential that t be measured at the same applied voltage as that for which iris measured. m is measured
most simply by allowing the mercury to fall through the air into a weighed bottle for a measured time, as the rate of flow depends only on the column height, to a first approximation. D is taken to be that for methern2 sec-I at 20°C. Valenta anol, namely 1.6 X (17) found K to be 4.4 X lo-'. Substituting these values into eqns. (10) and (11) gives sec-I. This value of kl corresponds to a ko of 5.1 X The excellent agreement with the result obtained by Bell and Evans is probably the result of cancellation of errors. The much more accurate data and elaborate calculations of Landqvist give ko to be 3.4 X sec-I. The independence of inwith column height may be clearly demonstrated for formaldehyde solutions, and comparisons made with 10-3M Pb(NO& for which a plot of idagainst h"' is a good straight line through the origin. Typical results are tabulated below. Solution A is the same formaldehyde solution for which the rate constant was calculated, and solution B is 10W3M Pb(NO& in 0.1M KCI. h '
i for solution A i for soludion B
2
28.6 15.3 1.15 4.45
43.0 6.56 1.15 5.48
58.3 7.63 1.15 5.94
72.2 em 8 . 5 0 em'b 1.30a+ 6.94pA
At large column height the drop time is so short that the simple equations no longer apply. A further demonstration that the wave is chemically controlled is given by the large variation with temperature, much larger than the variation to be expected from charging solvent viscosity. If a thermostatted cell is available, the activation energy, approximately 15 kcal (16) can be measured. Literature Cited (1) BRDICKA, R., HANUS,V., AND KOUTECKY, J., "Progress in Polaroeranhv.\Vol. I." P. ANDKOLTHOFF. ",, , (Editors: ZUMAN. I. M.), Int,erscienee(a divi~ionof John Wiley & Sons, Inc.) New York, 1962, p. 145. K., Chem. listy., 41.6 (1947). (2) WIESNER, R.. Collection Czechoslou. Chem. (3) VESELY,K., A N D BRDICKA, Communs., 12,313 (1947). J., Collection Czechoslou. Chem. (4) WEBER,J., A N D KOUTECKY, Communs., 20,980 (1955). (5) STREHMW, H., "Technique of Organic Chemistry, Vol. 111, Investigation of Rates and Mechanisms of Reactions, Part 2," (Editors: FRIESS, S. L., LEWIS,E. S., AND WEISSBURGER, A,, Interscience (a division of John Wiley &Sons, Inc.), New York, 1963, p. 799. P., Z.Physik. Chem., N . F., 5,323 (1955). (6) RUETSCHI, J.. Z. El~etmchem..61,423 (1957). (7) KORYTA. (8) Z.. Collection Czechoslov. . . KORYTA.J.. AND ZABRANSKY. ~ h e m . ~ o k m u n25,3153 ., (1960). ' (9) KORYTA, J., Collection Czechoslou. Chem:Commnzms., 19, 666 (IPR4I. (10) WIESNER, K., Collection Cxechoslov. Chem. Commum., IZ,64 (1947). L. B., AND WEISNER,K., J . Am. Chem. (11) Los, J. M., SIMPSON, Soc., 78,1564 (1956). (12) BEIRER,R., AND TRUMPLER, G., Helv. Chim. Acta, 30, 706 (1947). R., Collection Czeelwslov. Chem. Communs., 20, (13) BRDICKA, 2x7 (19551 ~----,(14) LANDQ~IST, N., Acta Chem. Seand., 9,867 (1955). . Soc. (London), (15) BELL,R. P., A N D EVANS,^. G., P ~ o cROY. AZ91,297 (1966). (16) BELL,R. P., Ad". Phys. Org. Chem., 4, 1 (1966). (17) VALENTA, P., Collection Czeehoslov. Chem. Communs., 25,853 (1960).
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