obtained for lower values of the dissociation constant k'. I t should be emphasized that the influence of concentration polarization of HA or (kr)h== cn,,'l%',' % ) k r ' ' l (12) *I-was not taken into account in the present treatBy combining (11) and (12) one obtains ment, and consequently that even larger errors than these calculated from (15) could possibly be observed. In the case of polarographic electrolyses the Equation (13) can be simplified by introducing current densities are of the order of to the value k, = 3.3 X 10ls (moles per clll.sj - I sec. - I amp. cni.?and the resulting ( A C H ,+)~% is negligible calculated in the previous section, and the approui- (see (15 ) ) . mate value D H . ~= 0.6 X 10 ciii.? sec. '. From the present treatment it is not possible to Thus after riumer ical transformations decide whether the hydronium ion concentration actually varies a t the electrode surface. But, it c m be coricluded that if the pH does not vary a t the electrode surface as predicted from (15) it IS likely Frorri (14) m e calculates J relative variatioii in that dissociation uf acid HA does not precede thr the coiicentratiori (Jf hydro~liurriion\ J ( the surface electrode process, and that inolecules of H-4 Are of the electrode directly consumed in the electrode process. Dissociation would then occur in some subsequent step of the electrode process. On the other hand, which is expressed in per cent. of concentration if undissociated molecules of HA are not directly used in the electrode process, one can predict on the CH,O +. Values of ( A C H , ~ % obtained froin (13) are by basis oi equation (15) that the pH a t the electrode no means negligible when HA is a very weak acid. surface is different from that in solution. The Consider for example the following case: m = n, difference in pH is appreciable (see above) when HA mole per liter) and the k' = 10-l2 mole per c i a J , ;.e., A-= 10P9mole per is very weak ( K = liter, C H A = lou4mole per c ~ n . under ~; these condi- current density is not too low (lo--? amp. cm. tions (ACIla(lt)%is equal to -580 ic I). per cent. of or abovej. CH~O. If ic D islfor example amp. cm-2 Acknowledgment.---It is a pleasure to ac;* fairly low current density in electrolysis-- knowledge the support of the Office of Naval Kr ( ~ C ~ ~ 3 is ~ +5.8OP+. j', Even larger errors would I)e search iri the course of this investigation P I)rl.iti.~r I H I \ I U I 74, i i 0 b , l q i i fi\lo\ KlJ[' 100 sec. it is iound that the current is independent of the head of mercury as one would expect from equation (20) (factor 7iiY1~rTI a). When k = 0 the total current i s simply the diffusion current which is proportional to 11' *. When kCz = 0 the first integral in (18) takes the form 0 X 0 and the second integral is equal to zero. By calculating the limit of the first integral for kCz = 0 , one reaches the conclusion that the average catalytic current becomes inversely proportional to the head of inerctiry when kCz approaches zero.
Verification for the Reduction of Ferric Ion in Presence of Hydrogen Peroxide Thc theory developed in the previous sections was applied t o the catalytic wave obtained in the reduction of ferric ion i n presence of hydrogen peroxide. This case of catalytic wave was thoroughly studied by Kolthoff and Parry,' and only the results pertaining to the present theory will be discussed. In this case, the catalytic process is the oxidation of ferrous ion by hydrogen peroxide. This reaction involves two ferrous ions for each molecule of hydrogen peroxide, but the rate of the oxidation process is proportional to the product of the activities of ferrous ion and hydrogen p ~ r o x i d e . ~Consequently, the present theory is a.pplirablr \r.ithout any change (scc reaction ( 2 ) ) .
Experimental Thc experimental methods described by Kolthoff and Parry were followed without change. Waves were recorded with a Sargent Polarograph model XXI, and potentials were measured with a Leeds and Northrup student potentiometer. An H cell was used for all recordings; the calomel. electrode arm of this cell was filled with 0.25 molar sulfuric acid, and the cell was connected to a saturated calomeI electrode by a bridge filled with a 1 molar potassium nitrate solution containing 4% of agar-agar. U n p otherwise indicated the temperature was 31.40 f 0.03
.
Description and Discussion of Experimental
Re-
sults
15C
Determination of Rate Constant k.-Values of the catalytic current for concentrations of hydrogen p e r ~ x i d e ' ~ ranging from 0.0145 to 0.362 mole per liter are listed iri Table I together with the corresponding rate constants determined from equation (17) and Fig. 3. The supporting rlectrolyte was 0.25 molar Ei2SO4, and the following data were used in the calculations: m = 1.52 mg. sec.-I, T = 1.02 sec., CE..+.+= 2.5 X rtiole per liter. The diffusion coefficient of ferric ion was determined from the diffusion current for ferric ion in the absence of hydrogen perositlc, arid thc valiie D y e . , + = 0.73 X I O 6 c m . 2 see:-' w;th obt.;tincrl froin t h c Ilkovic cquation.
e
,a
in the case of the Ilkovic equation.18 Nevertheless, the writing is rather heavy and it is much easier to calculate K a t various temperatures from the experimental current (see above). The energy of activation for the catalytic process is obtained from a, conventional log k vs. ( l / T )plot.
It
%0
& I25 s
a c
is)
F5
0
TABLE I
IOC
I>A'rA POR CATALYrIC CURRENTS O F liERRIC E S C E OF IIYDROCEN
10s
I N PRES-
PEROXIDE AT 31.4"
Experimental IC
( 'HdJt
7e
microamp.
0.0145 .0355 ,0724 I181
1.13
.:$ti2
500
300
700 900 Hood of Morcvy (m ml,
big 4. -\-aridtioris of catalytic current with the hcad of mercury: v t = 1
